R H C : M - Home | POLITesi - Politecnico di Milano Introduction and motivations 1 I Models 9 1 Plant Modeling with Tonal/Multitonal Disturbances 11 1.1 Multi-harmonic and Multivariable

POLITECNICO DI MILANODIPARTIMENTO DI ELETTRONICA, INFORMAZIONE E BIOINGEGNERIA

DOCTORAL PROGRAMME IN INFORMATION TECHNOLOGY(SYSTEMS & CONTROL AREA)

ROBUST HARMONIC CONTROL FOR

DISTURBANCE REJECTION: METHODS AND

APPLICATIONS

Doctoral Dissertation of:Roberto Mura

Supervisor:Prof. Marco LoveraTutor:Prof. Andrea CastellettiThe Chair of the Doctoral Program:Prof. Carlo Fiorini

2015 – XXVIII

Abstract

HARMONIC CONTROL techniques aimed at reducing tonal disturbances havebeen extensively studied in the last few decades, with particular attentionto a representation of the system as a linear model constructed in the fre-

quency domain, the T -matrix model. The precise knowledge of its elements isnecessary for a proper functioning of the overall control system, and classical em-ployed controllers resorting to the linear quadratic theory have not be framed todeal with model and parametric uncertainties or nonlinearities, possible causes fordegraded performance or instability of the closed-loop system. Adaptive controlvariants, coupled with a suitable offline or online identification method, have beenemployed in literature to handle this problem, but very little effort has been de-voted to the analysis of the trade-off between robustness and adaptation in theirdeployment.

In this dissertation, a discrete-time H∞ approach and a systematic methodol-ogy to the design of a robust Harmonic Control algorithm is proposed, for bothSISO and MIMO system representations. The proposed control solution allowsto account for model and parametric uncertainty in the control design problem,and provides a further benefit when dealing with the tuning problem, in particularwhen a multivariable plant is considered and different performance requirementsare associated to the disturbance attenuation on each considered output. Indeed,specifications in terms of steady attenuation levels and desired transient perfor-mance can be directly incorporated in the robust problem statement. Moreover,the control design approach has been modified to deal with the explicit accountingfor also predictable changes of the system, such as the ones induced by actuatorcharacteristics and nonlinearities. Their role and effects on closed-loop perfor-mance and stability have been considered by resorting to analysis and synthesisframeworks, as the Describing Function and Linear Parameter Varying approachesrepresent. Both are introduced as modifications of the original T -matrix model.While the first is used to analyze the cascade connection of a static nonlinearfunction with the plant matrix, the second allows to recast the problem both froma modeling and control design perspectives, combining the advantages of the ro-bust control solution with the accounting of predictable plant changes handledwithin a gain-scheduling framework.

The Thesis ends with a validation of the proposed methodologies on a clas-sical Harmonic Control application like the helicopter’s rotor vibration problem.As known, it can be formulated in terms of compensating a periodic disturbanceat rotor frequency acting at the output of an uncertain (possibly time-periodic) lin-ear system. Three case studies are proposed, including the general rotor-inducedvibration problem, its variant based on semi-active lag dampers and the structuralnoise/vibration problem.

I

Sommario

NEGLI ultimi decenni notevoli progressi sono stati raggiunti nellambito delletecniche di controllo armonico volte a ridurre disturbi tonali e multitonali,con particolare attenzione a una rappresentazione del sistema come un

modello lineare costruito nel dominio della frequenza, il modello T -matrix. Laconoscenza esatta dei suoi elementi fondamentale e necessaria per un correttofunzionamento del sistema di controllo generale, e tecniche classiche quali lateoria quadratica lineare si sono rivelate finora non adatte a trattare in manieraadeguata incertezze di modello o parametriche, piuttosto che non linearit che pos-sano essere causa di una riduzione notevole di prestazioni fino all’instabilit delsistema in anello chiuso. Varianti adattative sono state impiegate in letteraturaper gestire questo tipo di problema, ma in generale non si dato il giusto pesoall’analisi del compromesso tra robustezza e adattamento.

In questo lavoro ci si concentrati sullo sviluppo di un approccio H∞ a tempodiscreto e una metodo sistematico per la progettazione di algoritmi di controlloarmonico con propriet di robustezza, valutandone le performance a fronte di rap-presentazioni sia SISO che MIMO del sistema. La soluzione di controllo pro-posta consente di inserire a monte, in fase di design, le incertezze parametrichee di modello, e fornisce un ulteriore vantaggio in termini di tuning, in particolarequando si tratta un plant fortemente multivariabile in cui siano richiesti requisitidi prestazioni diverse sulle molteplici uscite considerate. Infatti, le specifiche intermini di livelli di attenuazione e durata del transitorio possono essere diretta-mente incorporati nella formulazione del problema robusto. Inoltre, l’approccioal progetto del controllore stato modificato per trattare in maniera esplicita leincertezze che siano in qualche modo prevedibili, come quelli indotte dalle carat-teristiche degli attuatori e dalla presenza di non linearit. Il loro ruolo e gli effettisulla stabilit e le prestazioni in anello chiuso e sono stati considerati ricorrendo ametodi di analisi e sintesi quali la funzione descrittiva e rappresentazioni LPV delsistema sotto controllo. Entrambi sono introdotti come modifiche al modello T -matrix originale. Mentre il primo viene utilizzato per analizzare il collegamentoin cascata tra lelemento di non linearit e il plant, il secondo permette di riformu-lare il problema di controllo a monte, unendo i vantaggi di una soluzione robustaaffiancata a un framework gain-scheduling.

La tesi si conclude con alcuni esempi atti a validare le metodologie proposte,concentrandosi in particolare sul problema delle vibrazioni indotte dal rotore diun elicottero.

II

Nomenclature

Variables and symbols

u(t) input of the Harmonic Control system in time-domainy(t) output of the Harmonic Control system in time-domaind(t) disturbance of the Harmonic Control system in time-domainU vector of cosine and sine input componentsY vector of cosine and sine output componentsD vector of cosine and sine disturbance componentsv(t) intermediate nonlinear output controlp dimension of the output and disturbance vectorm dimension of the input vectorω generic frequencyT steady frequency domain Harmonic Control response matrixT estimate of matrix Tr uncertainty scalar factorWm uncertainty ellipsoid matrix∆ normalized system uncertaintyθ LPV parameter vectorΘ Polytope for the LPV system formulationnθ dimension of vector θQ output weight matrix in the LQ-based control designR input weight matrix in the LQ-based control designJk LQ cost functionλ RLS forgetting factorp RLS constant speed factorAT closed-loop state matrixKM ,KM Harmonic Control matricesτ time interval between contiguous control updatesρs spectral radiusVk inverse of the RLS covariance matrixW∆,k uncertain input in the robust frameworkWY output robust shaping functionWU input robust shaping functionTs settling timeΛ(·) relative gain array matrixγ(·) Condition numberUS left unitary matrix of the SVD approachV H right unitary matrix of the SVD approachΣ SVD diagonal matrixΨ(·),Φ(·) nonlinear CharacteristicΓ(·) describing functionγ∞ BRL bound on the L2 norm from D to Z

III

C(s) Pre-compensator for the Decoupling solutionC(s) Post-compensator for the Decoupling solutionϑ Blade pitch angleψ azimuth angleN number of bladesΩ rotor frequencyµ advance ratioPi pressure in the valve chamberF (t) force provided by the damperX valve displacement from the valve seatV orifice flowz hydraulic fluid compressibilityro inner loop HC reference

Subscripts(·)c cosine component(·)s sine component(·)k discrete-time step(·)ns non-square(·)∗ relative to the diagonalized MIMO system(·)CL closed-loop(·)LQ based on the LQ (LTI) control

AcronymsSISO Single-Input Single-OutputMIMO Multi-Onput Multi-OutputLTI Linear Time-InvariantLPV Linear Parameter-VaryingLTP Linear Time-PeriodicLMS Least Mean SquaresRLS Recursive Least SquaresHTF Harmonic Transfer FunctionDF Describing FunctionBRL Bounded Real LemmaCQLF Common Quadratic Lyapunov FunctionLMI Linear Matrix InequalityRGA Relative Gain ArraySVD Singular Value DecompositionHC Harmonic ControlHHC Higher Harmonic ControlIBC Individual Blade ControlACSR Active Control of Structural ResponseS-AVLD Semi-Active Valve Lag Damper

IV

Contents

Introduction and motivations 1

I Models 9

1 Plant Modeling with Tonal/Multitonal Disturbances 111.1 Multi-harmonic and Multivariable T -matrix model . . . . . . . . 121.2 Uncertainty representation . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Nominal T -matrix model . . . . . . . . . . . . . . . . . . 161.2.2 Uncertain T -matrix . . . . . . . . . . . . . . . . . . . . . 17

1.3 Loop Nonlinearities in the Frequency-domain . . . . . . . . . . . 181.4 Overall T -matrix model . . . . . . . . . . . . . . . . . . . . . . 191.5 A T -matrix Linear Parameter Varying Framework . . . . . . . . 20

1.5.1 Affine/Polytopic T -matrix model . . . . . . . . . . . . . 21

II Methods 25

2 Harmonic Control Algorithms - An Overview 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Baseline T -matrix algorithm . . . . . . . . . . . . . . . . . . . . 292.3 Optimal LQ-based algorithm derivation . . . . . . . . . . . . . . 31

2.3.1 Convergence and Robustness Analysis . . . . . . . . . . . 322.4 Adaptive Harmonic Control Solutions . . . . . . . . . . . . . . . 35

2.4.1 Recursive Least Square (RLS) Methods . . . . . . . . . . 37

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2.4.2 Kalman Filter Identification . . . . . . . . . . . . . . . . 382.5 Continuous-time Harmonic Control . . . . . . . . . . . . . . . . 39

3 Robust Harmonic Control - SISO case 433.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 SISO T -matrix model with uncertainty . . . . . . . . . . . . . . 443.3 Discrete-time H∞ Harmonic Control Design . . . . . . . . . . . 45

3.3.1 Robust Control Tuning . . . . . . . . . . . . . . . . . . . 483.4 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Control parameters - sensitivity analysis . . . . . . . . . . 503.4.2 Numerical example - Stability . . . . . . . . . . . . . . . 523.4.3 Numerical example - Performance . . . . . . . . . . . . . 53

4 Robust Harmonic Control - MIMO case 594.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Decoupling methods: from MIMO to n-SISO . . . . . . . . . . . 61

4.2.1 Pairing rules via RGA/SVD techniques . . . . . . . . . . 614.2.2 Non-Square plants extension . . . . . . . . . . . . . . . . 644.2.3 Decoupling techniques and compensator design . . . . . . 65

4.3 Multivariable H∞ Control Design . . . . . . . . . . . . . . . . . 694.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.1 Example - Decentralized MIMO Robust Harmonic Control 734.4.2 A Numerical Experiment - Decoupled Non-Square Plant . 75

5 Nonlinear Harmonic Control 795.1 Loop Nonlinearities - Describing Function Method . . . . . . . . 80

5.1.1 A Numerical Experiment - HCLQ vs HCH∞ comparison . 825.2 An LPV Harmonic Control framework . . . . . . . . . . . . . . 87

5.2.1 LPV via LMI robust control design . . . . . . . . . . . . 885.2.2 Global LPV/H∞ solution . . . . . . . . . . . . . . . . . . 915.2.3 LPV control synthesis with a-posteriori guarantees . . . . 93

5.3 Simulation results and discussion . . . . . . . . . . . . . . . . . 965.3.1 2-vertices Affine-LPV control . . . . . . . . . . . . . . . 965.3.2 4-vertices Polytopic-LPV control . . . . . . . . . . . . . 99

III Applications 107

6 Harmonic Control for the Rotor-induced Vibration Control Problem 1096.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Active Vibration Control Methods . . . . . . . . . . . . . . . . . 111

6.2.1 HHC/IBC technologies . . . . . . . . . . . . . . . . . . . 1126.2.2 ACSR controls . . . . . . . . . . . . . . . . . . . . . . . 113

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6.3 Overview of the rotor out-of-plane blade model . . . . . . . . . . 1146.4 Helicopter Periodic Models . . . . . . . . . . . . . . . . . . . . 117

6.4.1 LTI Representation of LTP systems: Harmonic Balance . . 1196.4.2 Rotor Vibration Control - T -matrix definition . . . . . . . 119

6.5 Helicopter rotor blade simulation study . . . . . . . . . . . . . . 1206.5.1 Robust HHC-SISO . . . . . . . . . . . . . . . . . . . . . 1216.5.2 A multivariable HHC application . . . . . . . . . . . . . 124

7 Harmonic Control for the Structural Vibration Control Problem 1277.1 Experimental set-up description . . . . . . . . . . . . . . . . . . 1287.2 Control Design - Multi-Channel Harmonic Control . . . . . . . . 131

7.2.1 Black-box T -matrix model . . . . . . . . . . . . . . . . . 1327.2.2 Decoupling strategy - pairing evaluation . . . . . . . . . . 1347.2.3 Two-steps Compensator design . . . . . . . . . . . . . . . 1367.2.4 Multi-channel Robust-HC Design . . . . . . . . . . . . . 137

8 Harmonic Control with Semi-active Lag Dampers 1438.1 Introduction and Overall Control Architecture . . . . . . . . . . 1448.2 Semi-active Lag-Dampers Modeling . . . . . . . . . . . . . . . . 1458.3 Inner-loop Control - Gain Scheduling . . . . . . . . . . . . . . . 1488.4 Outer HC Controller and plant modeling . . . . . . . . . . . . . 149

8.4.1 Multi-blade coordinate transformation . . . . . . . . . . . 151

IV Concluding remarks 155

9 Conclusions and Future Perspectives 157

V Appendix 159

A From LTP to LTI - Harmonic Transfer Function 161

B Bounded Real Lemma 165B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

B.1.1 Singular value decomposition and positive matrices . . . . 165B.2 Lyapunov inequality LMI for discrete-time systems . . . . . . . . 167B.3 Bounded Real Lemma for Discrete Time systems . . . . . . . . . 169

C Global MIMO-VAF Data 173

Bibliography 183

VII

Introduction and motivations

Feedback control for disturbance rejection plays a central role for an increasingnumber of applications, and control techniques aimed at improving the overalleffectiveness represent a well-established topic of the ongoing research since theearly ’80s. Different algorithms’ families arose in the years, and their classifica-tion has been generally based on two main points (see [26]): the knowledge ofthe system plant, and the disturbance properties. For plants known without uncer-tainty and affected by broadband disturbances, shaping functions might be usedin the classical Linear Quadratic Gaussian (LQG) theory in accordance with per-formance objectives and disturbance spectrum. As generally recalled, see [52],trade-offs need to be accounted for, implying that disturbance rejection over agiven spectrum zone enforces amplification at different frequency regions. Whenthe disturbance is instead tonal or multi-tonal with known spectrum, a model ofthe exogenous signal can be embedded in the controller to produce high-gainfeedback at frequencies that comprise the disturbance spectrum. As a result, thecontroller could apply an infinite gain at the disturbance frequencies to obtain acomplete disturbance reduction. Alternatively, the feedback signal can equiva-lently be realized by updating the coefficients of the harmonic signals, providingin this way much greater robustness properties, especially when faults occur, asshown in [80] and [16]. As a limitation and an implementation issue, guaranteesof closed-loop stability, in both cases, require a good knowledge of both the gainand phase of the plant at the disturbance frequencies. If the plant and the distur-bance are poorly known and modeled, the control problem can be significantlymore challenging from the design point of view. Adaptation mechanisms are usu-ally resorted in facing the uncertain knowledge. Actually, in the active noise con-trol literature, numerous adaptive algorithms have been developed, based on LMS

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updating of finite-impulse response (FIR) filters (see [62] and [46]) and its vari-ants, like the FxLMS algorithm introduced in [39] and [14], recovered in [51],able to take into account the information of the so-called secondary path transferfunction. Modifications to these algorithms have been extensively produced toimprove convergence and stability properties, as in [99] for example. As a mat-ter of fact, just approximate stability results for the FxLMS method are till nowavailable (see [41] and [63] for details).

Another interesting approach, which is applicable when the plant is affectedby tonal or multi-tonal disturbances with known spectrum, allows the system toeffectively reach harmonic steady-state, i.e. approximate sinusoidal steady re-sponse, and uses measurements of the steady-state response amplitude and phaseto determine the required amplitude and phase of the control signal. Well knownapplications are the active control of rotor-induced vibration in helicopters, inwhich the technique is referred as harmonic or multicycle control; and the activerotor balancing control, known as convergent control (see [60]). These two con-trol problems are, of course, closely related: active disturbance rejection could beused to suppress the effect of harmonic disturbances due essentially to load im-balance. A vast class of algorithms referring to harmonic or multicycle control is

Figure 1: Feedback control for disturbance rejection. Block diagram offrequency-domain control system (Harmonic Control).

based on a linear, quasi-static, frequency-domain model of the plant response tocontrol inputs. Figure 1 outlines the general control system. Its implementationrequires knowledge of the frequency response of the transfer function between thecontrol input and the measurements at the disturbance frequency, and this infor-mation can be obtained through online or offline identification.

This dissertation focuses on the last introduced control approach, which isgenerically referred in the following as Harmonic Control (HC). In particular, asa starting point and motivation for the present Thesis, the helicopter rotor vibra-

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tion problem has been considered, in the light of the limitations that classical HCdeclinations have suffered in the years. Although this particular application, asshown few lines above and in Figure 1, the overall control problem can be treatedfor the generic harmonic disturbance rejection problem. In this respect, the thesisis structured in such a way the problem statement and the proposed control solu-tions are addressed from a methodological and general viewpoint, and only at theend they’re specialized in the light of the particular case study. It’s likewise evi-dent, however, that introducing motivations and focused discussions can be betteraddressed with reference to the real application.

Harmonic Control - the vibration control problem

Among the main problems affecting modern helicopters, vibrations generated bythe main rotor are possibly one of the most important. Various derivations ofHarmonic Control (HC) algorithms have been considered for many years as themost effective and valid approaches for the design and implementation of controllaws aimed at reducing the rotor-induced vibrations and the improvement of rotorperformance. In this context, the basic idea is to attenuate the vibratory compo-nents at the blade-passing frequency in the fuselage accelerations or in the rotorhub loads by adding suitably phased harmonic components to the rotor controls.Since the early ’60s, several studies have been carried out to determine the feasi-bility of these control laws both from the theoretical and the experimental pointsof view. As for the implementation of the designed controllers, a discrete-timeadaptive algorithm known in the rotorcraft literature as the T -matrix algorithmis typically used by defining the problem in the frequency-domain and tuningthe controller using a classical linear quadratic cost function, both in the time-invariant and adaptive variants. Surprisingly, however, very little effort has beenplaced to its analysis and in particular to the trade-off between robustness andadaptation in its deployment. Moreover, besides the robustness analysis of theT -matrix algorithm carried out in recent works [26], the problem has never beentackled in a robust control framework. As a main drawback of the classical linearquadratic control law, there’s the assumption that exact knowledge of the plantmatrix is available. Not surprisingly, an erroneous model of it can result first indegraded performance, till possible instability in the worst scenario. To deal withmodel uncertainty, a posteriori analysis could of course be carried out to prove ro-bustness qualities, but it can be argued that the uncertainty in the plant matrix canbe more adequately handled within a robust framework. Based on these consider-ations, a robust approach could be used to design control laws which incorporateall the uncertainties during the synthesis process. This represents an alternative toconventional HC control, which deals with performance degradation in presenceof model uncertainty by introducing some adaptation mechanisms. Many differ-

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ent algorithms have been developed in this sense in the last few years, mainlybased on the estimation of the plant matrix at specific time steps during the flightoperations. In this respect, the interest in investigating a robust control design ap-proach is motivated also by the possibility to relax the need for continuous updateof the plant matrix. Moreover, in many harmonic control applications nonlinear-ities are introduced in the control loop by actuator limitations. In particular, it isfrequently the case that actuator dynamics are negligible, but changes in the char-acteristic regions of the actuator, e.g. when they vary as a function of the inputmagnitude, have to be accounted for. As an example, hydraulic actuators couldpresent a non-negligible nonlinear behavior in both the ”low” vibratory load re-gion (because of intrinsic actuator issues), and the ”high” vibratory load region(because of physical saturation). The role of nonlinearities in the loop and itsanalysis and effects on closed-loop performance have not been considered in theliterature on Harmonic Control. Static nonlinearities, however, can be accountedfor in the T -matrix modeling framework by resorting to well defined mathemat-ical tools and frameworks. In addition, by considering both the predictable andunpredictable uncertainties acting on the system, other solutions could be appliedto recast the Harmonic Control problem in a different form.

In view of this, the aim of this dissertation is to propose a methodological ap-proach to the design of robust harmonic control laws, both in the SISO and MIMOsystem configurations, which can be useful to reduce the need of adaptation andguarantee, more specifically and in the synthesis process, the nominal stability ofthe closed-loop system and a prescribed robustness level to model and paramet-ric uncertainties due to, e.g., changes in the flight condition, configuration etc;and guaranteed performance for the closed-loop system, i.e., a desired level ofvibration attenuation.

Research Objectives

In the light of the preceding discussion, the present dissertation pursues as majorobjectives:

1. to understand the role of the uncertainties of the system matrix in terms ofperformance and stability;

2. to provide a systematic methodology to the design of Harmonic Controlstrategies with enhanced robustness properties;

3. to compare and generalize a single-input single-output formulation of thecontrol problem to the multivariable context;

4. to evaluate the influence of loop nonlinearities like the ones introduced byactuator characteristics;

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5. to provide a methodology to cast the control problem in the Linear ParameterVarying framework, both by the modeling and control viewpoints,

6. to validate the proposed control strategies on the rotorcraft vibration controlproblem.

Principal Contributions

As a principal and most significant contribution to the state-of-the-art in the fieldof Harmonic Control algorithms for disturbance attenuation, this study presentsa first comprehensive and detailed perspective of the robustness properties of theclosed-loop system, including in particular:

1. a detailed systematic and automatic approach to design robust harmonic con-trol laws for a SISO representation of the system, described as a linear (dis-crete) quasi-static model constructed in the frequency domain. Contextually,the control synthesis is obtained by defining a suitable optimization algo-rithm produced to automatically find control matrices coherent with classicalcontrol architectures.

2. an extension of the provided robust approach to the multivariable case, byresorting to decentralized and decoupling strategies with the aim of eliminatecross-interactions which could degrade the closed-loop performance. A two-step decoupling strategy is proposed in order to obtain a compensator ableto diagonalize the T -matrix plant and to use, similarly to the robust singlevariable case, the same instruments and tools.

3. an investigation on the application of a useful tool as the Describing Func-tion in the T -matrix framework, in order to understand the role of loop non-linearities on the closed-loop performance and stability. This in view of thefact that the Describing Function is very used as a standard mathematicaltool for analyzing limit cycles in closed-loop controllers, and it naturallywell fits in the T -matrix system representation.

4. a Linear Parameter Varying approach to the modeling and control synthesisof the system affected by both predictable and unpredictable uncertainties.In particular it is shown how the framework can provide a suitable means toaccount for variations in the plant such as the ones induced by, e.g, changingconditions or actuator nonlinearities.

5. a set of validation experiments of the proposed control strategies in the rotorvibration problem on helicopters. The Harmonic Control system is evaluatedin three different case studies with the aim to confirm the results obtainedwith the numerical tests used when presenting the proposed control methods.

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Thesis Outline

The present dissertation is structured in three main parts and nine chapters. PartI is dedicated to present in detail the so called T -matrix model and its modifica-tions in view of the methodological approaches discussed in the following. PartII is instead devoted to present and discuss the methods exploited to overcomesome of the issues and limitations of classical control laws. Part III is used toillustrate application results, with particular reference to the helicopter vibrationcontrol problem. As for each Chapter, the path is the following.

Chapter 1 introduces and extends the classical so called T -matrix model, de-rived for a multi-tone disturbance acting on a generic multivariable plant. Un-certainties are included in the system modeling, as well as loop nonlinearities,with the aim at defining an overall system matrix which can be used as basis forthe control methodologies developed in the thesis. Finally, an Linear ParameterVarying (LPV) modeling framework is introduced to account for the variation ofthe elements of matrix T and in particular to the predictable variations due, forexample, to actuator nonlinearities.

Chapter 2 presents an overview on the classical Harmonic Control laws, start-ing from the original baseline T -matrix algorithm proposed by Shaw in [109].The optimal LQ-based control approach, well established in the harmonic controlliterature, is then presented and analyzed in terms of convergence and robustness.Moreover, adaptive variants of the algorithms are introduced and defined, withparticular attention to the advantages and drawbacks introduced by their imple-mentation. Finally, the classical continuous-time implementation of HarmonicControl law is briefly discussed and commented.

Chapter 3 discusses in detail, with reference to the single-input single-outputsystem formulation, a robust control framework for the design of Harmonic Con-trol laws. The problem is introduced in the light of the already presented LQ-based algorithm, whose structure is maintained for the constructed optimizationalgorithm, used because of the non-convexity and non-smoothness of the H∞problem.

Chapter 4 illustrates a generalization of the robust approach to the multivari-able case. As a key point, a formulation adherent as much as possible to thealready solved SISO-robust control problem is pursued. In view of this, decen-tralized and decoupling control strategies are proposed, with particular focus onthe design of a compensator able to deal with non-square and full plants, in orderto compensate process cross-interactions and allow to simplify the tuning of thecontroller.

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Chapter 5 focuses on the presence in the control loop on nonlinear elements asthe ones represented by actuator characteristics. In the first part of the Chapter, theDescribing Function method is presented and used to analyze the influence of theactuator on closed-loop performance. In the second part, instead, an LPV frame-work is used to cast the problem in a form which could separate the T -matrixuncertainty contributions in predictable and unpredictable, with the aim at findinga suitable gain scheduling control strategy with robustness guarantees.

Chapter 6 is the first dedicated to the application. In particular the HarmonicControl problem is specialized in the helicopter rotor-induced vibration problem.The out-of-plane model is briefly presented, in the LTI form, to be then used tovalidate in simulation the closed-loop performance results obtained with the pro-posed robust control law, both in the SISO and MIMO cases.

Chapter 7 presents a second application, focused on a slightly different paradigmof the Harmonic Control problem. The study is related to the experimental facil-ity used in the Friendcopter project [32], a structural vibration problem where thelarge dimensions of the system make central the role of decoupling techniquesapplied on the original identified multivariable structure.

Chapter 8 introduces the framework for a third application, which investigatesthe use of so-called semi-active valve lag dampers, able to adapt the dampinglevel, and used along with Harmonic Control laws in order to reduce the fuselagevibration.

Chapter 9 presents the concluding remarks of the study and recommendationsand perspectives for future work.

Research Works

Some of the subjects, the methods and the results presented in this Dissertationcan be found in the following reviewed and under review works:• R. MURA, A. Ghalamzan. E., M. Lovera,”Robust harmonic control for helicopter vibration

attenuation”, in Proceedings of the AACC-ACC 2014.

• R. MURA, M. Lovera, ”Baseline vibration attenuation in helicopters: robust MIMO-HHC con-trol”, in Proceedings of the IFAC-WC 2014.

• R. MURA, M. Lovera, ”An LPV/Hinf framework for the active control of vibrations in heli-copters”, in Proceedings of the IEEE-MSC 2014.

• D. Duc, R. MURA, L. Piroddi, M. Lovera, G. Ghiringhelli, ”Robust harmonic control: anapplication to structural vibration reduction in helicopters”, in Proceedings of the ACNAAVWorkshop 2015.

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• R. MURA, M. Lovera, ”Rotorcraft vibration control: adaptive vs LPV method”, in Proceedingsof the IFAC-LPVs 2015.

• R. MURA, M. Lovera, ”Robust Harmonic Control for disturbance attenuation”, submitted asJournal paper

• M. Lovera, L. Piroddi, R. MURA, E. Vigoni, G. Ghiringhelli, ”Identification and robust har-monic control for rotorcraft structural vibration reduction,submitted as Journal paper

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Part I

Models

9

CHAPTER1Plant Modeling with Tonal/Multitonal

Disturbances

The Harmonic Control framework is based on the representation of the systemas a linear quasi-static model constructed in the frequency domain, applicableduring steady-state. As a classical included assumption, the exact and preciseknowledge of the plant matrix is considered available. However, it can be the casethat an erroneous model of it can result first in degraded performance and possibleinstability. Uncertainty has to be included in the system modeling, with the aimto clarify is role and the effects of poor knowledge of the system plant on theclosed-loop behaviour.

This first Chapter is devoted to present and modify the classical model usedin the Harmonic Control framework. In the first part of the Chapter the structureof the so called T -matrix model is described and extended from a basic SISOand single tone disturbance formulation to the more general multivariable andmulti-tone one. Uncertainty and the way to be accounted for are then discussed,leading to models which represent deviations from the nominal condition. Theoverall T -matrix model is then completed by introducing, by means of suitablemathematical tools, nonlinearities in the close-loop system. The second part ofthe Chapter is instead dedicated to present a different T -matrix model framework,

11

Chapter 1. Plant Modeling with Tonal/Multitonal Disturbances

able to deal with both the uncertainties and nonlinearities of the control problem,by resorting to the Linear Parameter Varying (LPV) theory.

1.1 Multi-harmonic and Multivariable T -matrix model

Referring to Figure 1.1, let u ∈ Rm be the vector of control inputs and y ∈ Rpthe vector of measured outputs. Assuming that the dynamics relating u to y couldbe considered linear time-invariant (LTI), then it can be described by a nominaltransfer function which in the following will be denoted asGyu(s) =

gyi,uj (s)

,

i = 1, . . . , p, j = 1, . . . ,m. It will be further assumed thatGyu(s) corresponds toan asymptotically stable system. Let the control input and the baseline disturbance

Figure 1.1: Multivariable and multi-harmonic T -matrix model. Mixed Time-Frequency domain representation in terms of input-output transfer matrixGyu(s).

d ∈ Rp, acting on the system output y, be multi-harmonic with frequencies ωh,h = 1, . . . ,H , so that at steady state signals d(t), u(t) and y(t) can be expressedas

d(t) = [d1(t) d2(t) · · · dp(t)]T

y(t) = [y1(t) y2(t) · · · yp(t)]T

u(t) = [u1(t) u2(t) · · ·um(t)]T

(1.1)

with

di(t) =∑Hh=1

[d

(h)ci cos(ωht) + d

(h)si sin(ωht)

], i = 1, · · · , p

uj(t) =∑Hh=1

[u

(h)cj cos(ωht) + u

(h)sj sin(ωht)

], j = 1, · · · ,m

yi(t) =∑Hh=1

[y

(h)ci cos(ωht) + y

(h)si sin(ωht)

], i = 1, · · · , p,

(1.2)

12

1.1. Multi-harmonic and Multivariable T -matrix model

where y(h)cj and y(h)

sj are given by

y(h)cj =

2

Th

∫ Th

0

y(h)j cos(ωht)dt (1.3)

y(h)sj =

2

Th

∫ Th

0

y(h)j sin(ωht)dt, (1.4)

with Th = 2πωh

, and similarly for d(h)ci , d(h)

ci and u(h)cj , u(h)

cj . Then, define the vector

Y(h)i =

[y

(h)ci

y(h)si

], i = 1, . . . , p (1.5)

and similarly for U (h)i and D(h)

j ,

U(h)i =

[u

(h)ci

u(h)si

], i = 1, . . . ,m (1.6)

D(h)i =

[d

(h)ci

d(h)si

], i = 1, . . . , p. (1.7)

As a result, the expression

Y(h)i = T

(h)i,j U

(h)j +D

(h)i (1.8)

is obtained, with

T(h)i,j =

[Re(gyi,uj (jωh)) Im(gyi,uj (jωh))

−Im(gyi,uj (jωh)) Re(gyi,uj (jωh))

]. (1.9)

The MIMO steady state response at frequency ωh can then be represented bydefining the vector

Y (h) =[Y

(h)T1 . . . Y

(h)Tp

]T, (1.10)

and similarly defining vectors D(h) and U (h), so that, by letting

T (h) =

T

(h)1,1 · · · T

(h)1,m

......

T(h)p,1 · · · T

(h)p,m

(1.11)

the relationY (h) = T (h)U (h) +D(h). (1.12)

13


Figure 1.2: Multivariable and multi-harmonic T -matrix model. Frequency-domain representation in terms of Fourier Coefficients vectors U , Y , D.

can be obtained. For the sake of simplicity, but without any loss of generality, inthe following the focus will be on the problem of rejecting a disturbance consistingof a single harmonic, i.e., H = 1, so that the index h will be dropped. When thedisturbance is a sum of sinusoids of multiple frequencies, the exploited analysiscarries through with minor modifications. Matrix T , defined in equation (1.11), isusually denoted as the T -matrix for the system (at frequency ωh) and representsthe model usually employed in the classical literature on the Harmonic Controlproblem. In the following subsections equation (1.11) will be used as a nominalmodel for the system to be controlled, while the actual design model will be ob-tained by augmenting (1.11) with suitable representations for model uncertaintyarising from different sources.

1.2 Uncertainty representation

The multivariable plant model Gyu(s) cannot describe perfectly the real systemdue to various contributions of uncertainty. These can trivially derive from thesimplification of the control-oriented model; or because of the unknown non-modeled dynamics. Moreover, if the plant is obtained through a linearizationin the neighborhood of an operating point, or the identification experiments turnout to be erroneous or not adequate, to account for model uncertainty it can beassumed that the plant dynamics are described by a set Π of possible perturbedmodels Gp from the nominal one, G (see [111] and [45] for details). To illustratehow uncertainty translates into frequency domain, consider in Figure 1.3 the polardiagram generated by the generic perturbedGp at different frequencies ω. At eachof these frequencies, a region of complex numbers is produced, leading to uncer-tainty regions with complicated shapes and non-trivial mathematical descriptions.Such complex regions can be approximated as discs, resulting in a multiplicativeuncertainty description,

Π : Gp(s) = (1 + wO(s)∆O(s))G(s); |∆O(jω)| ≤ 1, ∀ω (1.13)

14

1.2. Uncertainty representation

Figure 1.3: Uncertainty regions illustrated in the Nyquist plot at given fre-quency. Disc-shape approximation.

where wO(s) is a scaling factor, a rational transfer function seen as a weight in-troduced to normalize the perturbation ∀ω; ∆O(s) represents any stable transferfunction with bounded unity-norm magnitude ∀ω. At each ω, ∆O(jω) gener-ates a disc-shaped region centered in the origin with unitary radius, then (1 +wO(s)∆O(s))G(s) represents a disc-shaped area centered at G(jω) with radius|G(jω)wO(jω)| as shown in Figure 1.3.

As for the multivariable plant Gyu with m inputs and p outputs,

G(s) =

gy1,u1

(s) gy1,u2(s) . . . gy1,um(s)

gy2,u1(s) gy2,u2

(s) . . . gy2,um(s)

......

. . ....

gyp,u1(s) gyp,u2(s) . . . gyp,um(s)

(1.14)

the generic output multiplicative uncertainty can be included as

Gp(s) = (I +WO(s)) ·Gyu(s) (1.15)

with WO(s) given by

WO(s) =

w

(1,1)O (s)∆(1,1)(s) . . . w

(1,m)O (s)∆(1,m)(s)

w(2,1)O (s)∆(2,1)(s) . . . w

(2,m)O (s)∆(2,m)(s)

.... . .

...

w(p,1)O (s)∆(p,1)(s) . . . w

(p,m)O (s)∆(p,m)(s)

, (1.16)

15


and|∆(i,j)

O | ≤ 1, ∀ω, i = 1, . . . , p j = 1, . . . ,m

. In the presented form, considering all the elements w(i,j)O 6= 0, ∀i, j, i.e. an un-

Figure 1.4: Multiplicative output uncertainty representation.

structured large uncertainty, would lead to non trivial control design procedures.In the light of output performance consideration, WO(s) can be included as a di-agonal matrix, with the non-null elements are relative to the output considered.Consequently, the uncertainty matrix WO(s) can be defined as

WO(s) = wO(s)∆O(s) (1.17)

with ∆O(s) given by∆O = diag[∆i] (1.18)

The obtained description can be extended to non-square plants, with the approachexplained in detail in Chapter 4.

1.2.1 Nominal T -matrix model

Assuming that the transfer function of the plant to be controlled is perfectlyknown, then so is its frequency response matrix and so a nominal design prob-lem can be formulated in terms of Gyu(jωh). Consequently the nominal sensitiv-ity matrix T (i), related to Gyu(jωh) as in equation (1.9), can be defined. Giventhat, under steady state conditions, the above defined h-th harmonic componentsof variables u and y are related by the following linear equation, the nominalT -matrix model,

Y = TU +D (1.19)

where T is a 2p × 2m constant coefficient matrix and D represents the i-th har-monic component of the disturbance affecting the system.

The assumption of exact knowledge of the T -matrix, which is implicit in mostof the literature on Harmonic Control, is of course not realistic in many appli-cations, so that models to represent deviations from this ideal situation must beworked out. In this respect, the present study accounts for deviations of the T -matrix from its nominal value caused by two distinct contributions:

16

1.2. Uncertainty representation

• the first one is the model uncertainty, supposed small and unpredictable;

• the second one, in a way predictable, is the variation due rather to a nonlinearcharacteristic Φ(·) in the loop, which can be expressed ad a function of thecontrol variable u.

1.2.2 Uncertain T -matrix

If the knowledge of the T -matrix is affected by uncertainty, a robust frameworkcan be adopted to take it into account. In particular, by translating the uncer-tainty definition based on Gyu to the T -matrix system representation, an outputmultiplicative representation of uncertainty in the T -matrix is considered, see Fig-ure 6.3. More precisely, the uncertain T is written as

T =(I2p +Wm∆

)T , ‖∆‖ < 1, (1.20)

where I2p is the identity matrix of dimension 2p, ∆ is a normalized representationof uncertainty andWm represents the matrix which describes the uncertainty ellip-soids affecting the frequency responses G(yi,uj)(jωh), i = 1, ..., p, j = 1, ...,m,at the disturbance frequency ωh. Matrix Wm can be designed with a block diago-nal structure due to the collection of uncertainties related to each of the p outputs

Wm = blkdiag(W

[1]m W

[2]m · · · W

[p]m

)(1.21)

where

W [i]m = r[i]

[α[i] β[i]

−β[i] α[i]

], i = 1, ..., p (1.22)

with r[i] a scalar scale factor and α[i], β[i] the parameters related to the considereduncertainty of the specific output. A block diagram of the uncertain feedback

Figure 1.5: Block diagram of the uncertainty definition in the T -matrix systemrepresentation.

system corresponding to the model (1.19), its uncertainty (4.29) and the controller(2.16) is represented in Chapter 3, with ∆ defined as

∆ = blkdiag(I2δ

[1] I2δ[2] · · · I2δ

[p]). (1.23)

17


1.3 Loop Nonlinearities in the Frequency-domain

In many harmonic control applications nonlinearities are introduced in the controlloop by actuator limitations. In particular, it is frequently the case that actuatordynamics are negligible, but changes in the slope of the static characteristic ofthe actuator as a function of the input magnitude have to be accounted for. As anexample, hydraulic actuators could present a non-negligible nonlinear behaviorin both the ”low” vibratory load region (because of intrinsic actuator issues), andthe ”high” vibratory load region (because of physical saturation). In quantitativeterms, this is equivalent to a model of the plant to be controlled which is given bythe cascade connection of a static nonlinear function ψ(·) with the linear dynam-ics Gyu(s), as shown in Figure 1.6. The role of nonlinearities in the loop and itsanalysis and effects on closed-loop performance have not been considered in theliterature on harmonic control. Static nonlinearities, however, can be accountedfor in the T -matrix modeling framework by resorting to the notion of DescribingFunction (DF). Indeed, focusing initially on the single-input case and consider-ing the static nonlinearity ψ(·) in Figure 1.6, if its input u is given by a singleharmonic at the generic angular frequency ω,

u(t) = uc cos(ωt) + us sin(ωt), (1.24)

the corresponding harmonic of the output of the nonlinearity can be written as

v(t) = vc cos(ωt) + vs sin(ωt). (1.25)

Equivalently, in exponential form, we have

u(t) =1

2

(U1e

jωt + U−1e−jωt) , (1.26)

v(t) =1

2

(V1e

jωt + V−1e−jωt) , (1.27)

where U−1 = −U1, V−1 = −V1 and uc = ReU1, us = −ImU1 and vc = ReV1,vs = −ImV1. Introducing now the DF for the static nonlinearity ψ(·), defined as

Ψ(U) =2

πU

∫ π

0

ψ(U sin(ωt)) sin(ωt)dt, (1.28)

we have thatV1 = Ψ(U)U1 (1.29)

which can be equivalently written as

vc − vsj = (ReΨ(U) + jImΨ(U))(uc − usj), (1.30)

18

1.4. Overall T -matrix model

which, in matrix/vector form, leads to

V =

[ReΨ(U) ImΨ(U)

−ImΨ(U) ReΨ(U)

]U = TΨU. (1.31)

Note, however, that for symmetric nonlinearities the describing function is real, soTΨ reduces to TΨ = ReΨI2 = ΨI2 and therefore the effect of the static nonlin-earity on the steady harmonic steady state representation of the system is simplya scalar gain, which coincides with the describing function itself. In the case of a

Figure 1.6: Nonlinear block diagram. Time domain representation.

multiple-input system with m control variables, the effect of static nonlinearitieson the individual actuators can be represented on the basis of the previous conclu-sion. Indeed if the m actuators are identical, so that ψj(·) = ψ(·), j = 1, . . . ,m,then TΨ = ΨI2m. If, on the contrary, the system is driven by heterogeneous ac-tuators, as may be the case if different technologies are employed simultaneouslyto achieve disturbance suppression, then

TΨ = blkdiag (Ψ1I2Ψ2I2 . . .ΨmI2) (1.32)

Remark 1. By translating this block diagram in the T -matrix form, Ψ(·) couldbe view, in the frequency domain, as a control-varying gain which contributesto the uncertainty considered already. In this sense, the Describing Functionmethod gives us the instruments to investigate this extension of the problem. Abrief overview on this theory, along with details of Ψ(·) form and particularizationis reported in Chapter 5.

1.4 Overall T -matrix model

The combination of the above mentioned uncertainty contributions leads to a over-all T -matrix of the form

T =(I +Wm∆

)TΓ(‖U‖). (1.33)

19


The relation obtained derives trivially from the previous paragraphs. With refer-ence to equation (1.8), consider the function Γ(·) such that

|yj | cos (ωit+ ψj) =m∑h=1

|uh|Γ(uh)Gyj ,uh(jωi) cos (ωit+ ηh) +

+ |dj | cos (ωit+ φj) . (1.34)

It is not hard to see that in the end the MIMO open-loop system is given by

Y = TΓ(U)U +D, (1.35)

where Γ(U) ∈ R is a scalar which enters the control loop as a time-varying gainwhich depends on the control variable norm, and which scales the matrix T sim-metrically with respect to the real and the imaginary parts of the original transfermatrix Gyu(s). In the light of this, some comments are needed:

• the inclusion in the loop of nonlinearity modifies the system in such a waythe controller copes no more with ’slow’ variations of matrix T , but alsowith ’fast’ variations due to the describing function gain;

• being Γ(·) ∈ R scalar, if one considers its contribution only in equation(1.33), there’s equivalence to consider a linear uncertainty in the complexplan, instead of the ellipsoid affecting the frequency response matrix;

• on the other hand, if one considers both the uncertainty contributions, thescaling factor due to Γ(·) affects matrix T again on a scaled ellipsoid, equiv-alent to consider higher bounds on the initialized uncertainties.

1.5 A T -matrix Linear Parameter Varying Framework

To achieve a complete solution to the problem of considering both the uncer-tainty and loop nonlinearities, the idea to resort to the Linear Parameter Vary-ing (LPV) theory has been investigated. As known from the relevant literature,see [2, 22, 23, 67], LPV models could represent an interesting compromise be-tween the global accuracy of nonlinear models, including the uncertainty, and thedesign of controllers with well known techniques available for LTI systems’ rep-resentations. The LPV framework, in this sense, provides an appealing meansto account for variations in the plant such as the ones induced by, e.g, changingconditions or loop nonlinearities. As already introduced, but this will be dis-cussed much more in detail in the following, this aspect has not been studied inthe relevant Harmonic Control literature, as it is generally assumed that such per-turbations can be taken care of, for example, by an adaptation mechanism. While

20

1.5. A T -matrix Linear Parameter Varying Framework

this is certainly a possibility, predictable changes in the plant can be surely han-dled, with relative confidence, within a gain-scheduling framework. Based onthese considerations and borrowing some ideas from the recent extensive use oflinear parameter-varying approaches in different application contexts (see amongothers the works of Bruzelius [22], Sename et al. [34] and others in [2, 91, 114]),an LPV-robust approach for reducing harmonic disturbances can been derived.

Surprisingly, no example of its application has been proposed in the T -matrixcontext. In the classical LPV framework, matrices of the dynamics depend onmeasurable exogenous time-varying parameters θ(t) ∈ Rnθ , which are usuallyconfined to a given (convex) set, and the corresponding maximum time rate-of-variation may be assumed known. A state-space representation with non-stationary parameters is given by

x(t) = A(θ(t))x(t) +B(θ(t))u(t)

y(t) = C(θ(t))x(t) +D(θ(t))u(t). (1.36)

where u is the input, y is an output, and θ = [θ1, θ2, · · · , θnθ ] is an exoge-nous parameter vector that can be time dependent. The dependence of matricesA,B,C,D from parameter θ can have various forms. The most common are theaffine and polytopic ones. These two formulations can be considered equivalent,relying on the assumption

A(θ) =

nθ∑1

ρi(θ)Ai ρ(θ) ≥ 0

nθ∑i

ρ(θ) = 1 (1.37)

and similarly for the other matrices. Hence, a polytope Θ can be defined such that[A(θ) B(θ)

C(θ) D(θ)

]∈ Co

[A1 B1

C1 D1

]· · ·[An Bn

Cn Dn

](1.38)

where each of the systems (A,B,C,D)i represent a vertex of Θ,

Θ =θ : θi ≤ θi ≤ θi, i = 1, · · · , nθ

(1.39)

1.5.1 Affine/Polytopic T -matrix model

Consider the discrete-time open-loop system

Y = TU +D. (1.40)

As discussed in the previous sections, variations of matrix T represent the mainmotivation to the synthesis of adaptive or robust controllers. It also makes sense,moreover, to consider this variation as composed by two contributions:

21


Figure 1.7: Linear Parameter Varying framework. Polytopic T -matrix modelrepresentation.

• the first one is model uncertainty, assumed small and unpredictable, dueto configuration changes (weight, centering...), so that, by choosing for ex-ample a output multiplicative representation of the uncertainty, being T thenominal T -matrix, one can define

T =(I +Wm∆

)T , ‖∆‖ < 1, (1.41)

where ∆ is a normalized representation of the uncertainty and Wm rep-resents the matrix of the uncertainty ellipsoids affecting the frequency re-sponse matrix at the disturbance frequency, both on the real and imaginaryparts.

• the second one, in a way predictable, is the variation due to varying plantcondition and actuator characteristics and nonlinearities: these could de-pend, for example, on the control norm ‖U‖, so that

T = T f (‖U‖) . (1.42)

As will be shown in the following, the above model can be represented in again-scheduling framework for control design purposes.

The combination of the two variation contributions leads to a derived matrixT (θ1, θ2) in the LPV framework defined as

T (θ1, θ2) =(I +Wm∆

) [TRe θ1 + T Im θ2

](1.43)

where matrix T has been decomposed in its real and imaginary parts as

TRe =

[Re(G(i,j)(jω)) 0

0 Re(G(i,j)(jω)

]

T Im =

[0 Im(G(i,j)(jω))

−Im(G(i,j)(jω)) 0

],

(1.44)

22

1.5. A T -matrix Linear Parameter Varying Framework

with the parameters θ1, θ2 defined as

θ1 = f1(‖U‖), θ2 = f2(‖U‖). (1.45)

with f1, f2 functions dependent on the specific control application and the actuatorcharacteristics. This yields an affine/polytopic nominal system description

T (θ1, θ2) = ρ(θ1, θ2)T θ1,θ2 + ρ(θ1, θ2)T θ1,θ2 + · · ·· · ·+ ρ(θ1, θ2)T θ1,θ2 + ρ(θ1, θ2)T θ1,θ2

θ1, θ2 ∈ Θ,∑ρ(θ1, θ2) = 1.

(1.46)

where matrices T θ1,θ2 with (θ1, θ2) ∈ θ1, θ1 × θ2, θ2 represent the verticesof the two-dimensional polytope Θ generated by scheduling parameters (θ1, θ2).More precisely, if the system can be described by means of two parameters, poly-tope Θ consists in four vertices with

ρ(θ1, θ2) = |θ1 − θ1||θ2 − θ2|/δρ(θ1, θ2) = |θ1 − θ1||θ2 − θ2|/δρ(θ1, θ2) = |θ1 − θ1||θ2 − θ2|/δρ(θ1, θ2) = |θ1 − θ1||θ2 − θ2|/δ

(1.47)

and δ = |θ1−θ1||θ2−θ2|. The obtained model represents a basis for the LPV con-trol design which will be discussed in detail in Chapter 5. Advantages and draw-backs, in particular with respect to an adaptive control solution, will be shown andcommented.

23

Part II

Methods

25

CHAPTER2Harmonic Control Algorithms - An Overview

Harmonic Control (HC) has been the subject of extensive research over the morerecent decades, especially in the rotorcraft community. In this field, studies priorto the ’80 has been collected and reviewed by Johnson [55] as part of an pre-cise and detailed summary of different algorithms and implementation techniques.These remain extremely relevant to this date, especially in the industrial panorama.A complete overview of the Harmonic Control techniques, both from a theoreti-cal and practical point of view, is completed by the interesting (and more recent)survey papers produced by Friedmann and Millott [43] and Teves et al. [115].

This Chapter is devoted to present a brief overview of the Harmonic Controlalgorithms starting from its original version, known as T -matrix algorithm, andarriving to the adaptive variants risen especially in the recent years.

2.1 Introduction

One of the first HC applications roses in terms of vibration suppression (in he-licopter applications) and was proposed by Shaw in [106]. He investigated thefrequency response function between the Fourier coefficients of the harmonic in-puts and the Fourier coefficients of the measured vibration outputs, exploiting the

27

Chapter 2. Harmonic Control Algorithms - An Overview

linear relationshipY = TU +D. (2.1)

already introduced in the previous Chapter. As a results, the relationship between

Figure 2.1: General block diagram of discrete frequency-domain control sys-tem, including the Fast Fourier Transform block and the sample and holdcircuit.

control inputs and system response translates from a general complex, non-linearproblem to a simple linear association in the frequency domain. Figure 2.1 showsa high level schematic of a typical HC system. While this is not the only possibleapproach, nor is it necessarily the best, it is important for historical and practicalreasons, and has been extensively studied both in theory and practice (e.g., in[79], [107], [108], [109]). As a focal point, an harmonic analysis, typically aFast Fourier Transform, is used to extract some of the outputs of interest. Then,a controller computes the more suitable values for harmonic controls, which arethen injected as a part of the control signals. Since the controller is designed todeal only with the harmonics of output components and controls, it is thereforeconsidered to effectively operate in the frequency-domain. In this sense, controlsystem is designed to regulate the variables that define amplitude of the harmonicoutput signal, rather than the output itself.

The steps necessary to carry out the transformation between the time and fre-quency domains are not instantaneous, but are carried out over finite time inter-vals. In this regards, the closed-loop HC algorithm is typically defined, not as acontinuous system, but rather at specific instants tk, where tk = kτ . The controlsare updated at every tk, and kept steady during the entire time interval τ , allowingsufficient time for the system to sample the output signal, execute the harmonicanalysis, and compute updated control amplitude and phase quantities; while atthe same time allowing transient dynamics to dissipate.

The underlying assumption is that harmonic controls are updated slowly enoughthat transient dynamics have a minimal impact on the closed loop dynamics. Con-sequently, the majority of investigations dedicated to open or closed-loop HC per-formance or stability analysis have, up to date, generally relied on the simplifying

28

2.2. Baseline T -matrix algorithm

assumptions, in particular from [120], of quasi-steady LTI-plant response to har-monic inputs.

2.2 Baseline T -matrix algorithm

As discussed, Harmonic Control assumes that the measurement of the plant outputand the update of the control input are not performed continuously, but rather atspecific times tk = kτ , where τ is the time interval between contiguous updatesduring which the plant output reaches a steady level. In this regards, let define thevector Uk of the harmonics of the control signals computed by the HC system attime tk. Consider also the vector Yk which contains the cosine and sine harmonicsof the output signal. The matrix T of relation (2.1) can be either estimated frommeasured data, using on-line or off-line identification algorithms, or computed onthe basis of a mathematical model of the plant. At each discrete time step k, theHC controller selects the value of the input harmonics Uk to reduce the effect ofthe disturbance D (considered constant over the time step τ ) on the output Yk,

Yk = TUk +D. (2.2)

Given that, the optimal open loop solution can be written as

Uk = −T−1D. (2.3)

Since in general disturbance D cannot be measured directly, the same result canbe obtained by using a discrete-time integral control law in closed loop, i.e., basedon the measurements of Yk,

Uk+1 = Uk − T−1Yk. (2.4)

Clearly, the introduced integral action ensures that Yk → 0 when k → ∞. Byreplacing k by k+ 1 in (2.2) and subtracting the resulting equation from (2.4), thedisturbance-free model

Yk+1 = Yk + T (Uk+1 − Uk), (2.5)

is obtained, which recalls the equivalent local response model which relates thediscrete change in the output to a step change in the control input between twotime samples.

Figure 2.2 depicts the general control scheme of the baseline T -matrix algo-rithm, where it is included the delay operator z−1 such that

Uk = Uk−1 − T−1Yk−1 = z−1(Uk − T−1Yk) (2.6)

A more detailed implementation description is shown in Figure 2.3, which re-

29


Figure 2.2: General block diagram of discrete baseline T -matrix algorithm.

cover a classical HC implementation as introduced by Shaw in [107] and [109].Based on the already introduced signal definitions, the components of the har-monic amplitude vector Y are described by the Fourier coefficients of the mea-sured vibration at the frequency ω. These Fourier coefficients are obtained byintegrating the demodulated vibration signal over one sampling period T s. Asfor the control adjustments ∆uc and ∆us, they are obtained as the product of thecomponents of Y with the inverse of the control response matrix T , and are thensampled and consequently added to the control signal amplitudes of the previ-ous time step, generating the current control amplitude vector U . The output ofthe controller is then obtained by summing the two modulated components of thecontrol vector.

Figure 2.3: A detailed block diagram for implementing Shaw’s discrete-timeHC algorithm.

30

2.3. Optimal LQ-based algorithm derivation

2.3 Optimal LQ-based algorithm derivation

The discrete-time T -matrix algorithm, used traditionally for the attenuation of theeffect of D on Y , has known modifications and improvements during the years,with the aim to minimize a defined cost-function or by resorting to adaptationmechanisms aimed at quickly recover from improper initialization or enhancedrobustness properties.

An LQ-based Harmonic Control version has been derived by minimizing ateach discrete-time step k the cost function

Jk = YkTQYk + 2Y Tk SUk + Uk

TRUk (2.7)

where weighting matrices Q = QT ≥ 0, Q ∈ R2p×2p, S ∈ R2p×2m and R =RT > 0, R ∈ R2m×2m are defined such that[

Q S

ST R

](2.8)

is positive definite. In most applications, however, the cross-weighting term in(2.8) is neglected and the cost function simplifies to

Jk = YkTQYk + Uk

TRUk, (2.9)

equivalent, by substituting Yk from (2.2) into (2.9), to

Jk = UkTV Uk + +2UTk T

TQD +DTQD, (2.10)

where V is defined asV = TTQT + R. (2.11)

The optimal control law is found by differentiating (2.10) with respect to Uk

∂Jk

∂Uk= 2V Uk + 2TTQD = 0, (2.12)

leading to the open-loop control algorithm

Uk+1 = −(TTQT + R)−1(TTQ

)D. (2.13)

which can be equivalently written as

Uk+1 = −TKUk +KYk, (2.14)

withK = −V −1(TTQ). (2.15)

As for the implementation of this discrete algorithm, the following operationsneed to be carried out:

31


1. the determination of the i-th harmonic component of y, namely the compu-tation of the modulated integrals (1.3) and (1.4) of the output at frequencyωi at time tk;

2. the update of the i-th harmonic componentUk of the control variable u usingequation (2.13);

3. the determination of the time domain value of the control variable u via amodulation of the sine and cosine components at frequency ωi.

The control law (2.13) can be then parameterised as

Uk+1 = KMUk +KNYk, (2.16)

whereKM = −KNT. (2.17)

Remark 2. In this respect it is interesting to point out that the structure of matrixT (see (1.9)-(1.11)) implies a similar structure of matrices KM and KN .

In particular, it can be shown that the structure of every submatrix Tj,h, beingof the form

Tj,h =

[aj,h bj,h

−bj,h aj,h

], (2.18)

is extended to matrix KN (its submatrices) by means of equation (2.13) and con-sequently also KM inherits the same type of structure. This means that, in theface of 2m · (2m + 2p) entries in the matrices defining the control law, only 2pfree parameters are exploited in the LQ control law. In the following Chapters,however, a different approach will be presented and a control law based on fullparameterised matrices will be discussed.

Note also that the control algorithm (2.16) still introduces a discrete-time inte-gral action, and with Q = I and R = 0 deadbeat control (i.e., the output goes tozero after one discrete time step) could be achieved if matrix T is known withoutuncertainty. This assumption is in general not satisfied, as T can only be esti-mated. Moreover, the control law (2.16) reduces to (2.4), which can be given aminimum variance interpretation, neglecting to bound the control effort and min-imizing the cost function Jk = Y Tk Yk.

2.3.1 Convergence and Robustness Analysis

The disturbance-free update model in relation (2.5) has been employed because ofthe dependence of Uk on a stationary disturbance D, whose measurement is notgenerally available. At each time tk, the relation

D = Yk − TUk (2.19)

32

2.3. Optimal LQ-based algorithm derivation

holds true. By setting k = 0 it can be written

D = Y0 − TU0, (2.20)

and hence, substituting (2.20) into (2.5) yields

Yk = Y0 + T (Uk − U0). (2.21)

The optimal control law can be expressed recursively as

Uk+1 = −V −1(TTQ)(Yk − TUk), (2.22)

and an equivalent state-space representation of the system dynamics with the op-timal control can be given by [

Yk+1

Uk+1

]= AT

[Yk

Uk

], (2.23)

where AT is defined as

AT =

[I +KT −T (I +KT )

K −KT

]. (2.24)

The reader could note that AT is actually an idempotent matrix, so that

A2T =

[(I + TK)2 + (I + TK)TK −T (I + TK)2 − T (I + TK)TK

K(I + TK)−KTK −K(I + TK)T + (KT )2

]=

=

[I +KT −T (I +KT )

K −KT

]= AT ,

(2.25)and its eigenvalues are 0 or 1. This means that matrix AT can be written as theproduct of

AT =

[I T

0 I

] [I 0

K 0

] [I −T0 I

](2.26)

and consequently, with initial conditions Y0 and U0, (2.23) is equivalent to[Yk+1

Uk+1

]= AT

[Y0

U0

]. (2.27)

Remark 3. Of course this result, where the optimal control and output are at-tained after the first update, holds true on the assumption that matrix T is knownwith precision and no uncertainty. This is not the general case, as plant condi-tions and nonlinearities could easily destabilize or deviate the system behaviorfrom the nominal path. If the sensitivity matrix T is uncertain, Harmonic Con-trol could result in degraded performance and possible instability. Moreover, theone-step convergence property decays.

33


As a consequence of relation 2.27, the steady output can be written by meansof the relation

Yk = KLQ(Y0 − TU0) (2.28)

whereKLQ = Q−1(Q−1 + TR−1TT )−1 (2.29)

is a gain which allows to estimate the distance from the complete reduction of thedisturbance D. If a unit norm D is applied on the system, KLQ represents thesteady output of the closed-loop system.

Consider an uncertainty representation as the one defined in Chapter 1, so thatthe estimated value T of matrix T involves a multiplicative error,

T = T (I + ∆T ). (2.30)

Hence, matrix AT can be defined as

AT =

[I T

0 I

] [I 0

K −KT∆T

] [I −T0 I

]. (2.31)

The stability of the HC algorithm requires that the eigenvalues ofKT∆T are suchthat

ρs(KT∆T ) < 1 (2.32)

with ρs spectral radius, which by definition could be evaluated by means of themaximum and minimum singular values of the matrix AT ,

σ(AT ) = max(√

λA∗TAT

), σ(AT ) = min

(√λA∗TAT

), (2.33)

with A∗T conjugate-transpose matrix of AT (A∗T = AT

). The stability conditioncan be written as

ρs(KT∆T ) = ρs

((TTQT + R)−1TTQT∆T

)≤σ(T )σ(Q)σ(∆T )(σ(∆T ) + 1)

σ(R)< 1

(2.34)

which results in the condition

σ(∆T ) <1

2

−1 +

√1 +

4σ(R)

σ2(T )σ(Q)

. (2.35)

The obtained relation allows to understand one of the most important drawbacksof the Harmonic Control algorithm in the presented form. There’s evidence from(2.35) of the presence of a trade-off between robustness and performance proper-ties of the closed-loop system. As shown in Figure 2.4, the higher the ratio Q/R,usually set as a performance indicator, the lower the robustness degree, meaningthat a smaller relative uncertainty could be included in the system.

34

2.4. Adaptive Harmonic Control Solutions

Figure 2.4: Trade-off Robustness/Performance with the Optimal LQ-basedHarmonic Control Algorithm. The higher the ratio Q/R, the lower therobustness degree.

2.4 Adaptive Harmonic Control Solutions

Identification of plant dynamics can be a relevant element for HC algorithms’design. As far as system identification is included in the control loop, two optionsare in general possible. Plant dynamics can be identified either off-line, and usedfor designing fixed control gain algorithms, or on-line, typically by means of anadaptive recursive estimator.

Off-line identification of the system response matrix T is commonly based onthe application of the least-squares fit method to a succession of measurements ofthe system output, in response to a known input. Input signals used as test bench-marks are represented by harmonic functions and are described by their Fourierseries. Some examples are shown in references [31], [79], [107], [108], [109],which focus is on the design of fixed-gain controllers for an harmonic controlproblem as the active reduction of structural vibration.

Adaptive T -matrix algorithms require the knowledge of the control inputs andthe measured vibration outputs at each time step in order to produce estimates ofthe transfer matrix. Existing methods reflect the stochastic nature of the actualoperating conditions.

As for the adaptive filters typically used for the on-line identification methods,two main categories are found in the literature:

• recursive versions of the least-squares (RLS) method or Kalman filters, whichoperate directly on the signals or measurements;

• gradient descent methods such as the least-mean squares (LMS) filter, whichdepend on the stochastic characteristics of the signals.

35


A theoretical analysis by Shaw [107] considers the influence of errors in the es-timate of T on controller stability, showing that either gain-scheduled feedbackgains, or online identification, was required to achieve good disturbance rejectionperformance in the face of varying plant conditions. Among others, these studiesmotivated the comprehensive review by Johnson in [55]. Here, some of the recur-sive algorithms commonly employed for on-line or adaptive system identificationare described in detail: in particular, a Kalman filter method and various recur-sive adaptations of the basic least-squares algorithm. As a result, in [55] Johnsonshowed that, although a possible erroneous estimation of matrix T is a primarycause of closed loop instability, adaptation amplifies control response time de-lays, making it slightly slower, and potentially incapable of reacting to rapidlychanging vibration amplitudes. Considering this restriction, it may be advanta-geous to implement a gain-scheduled, fixed-gain control law to account for thevarying operating conditions. This is precisely one of the premises in the work byHall and Wereley [120].

Classical recursive adaptation techniques, such as those mentioned in the pre-vious discussion, bring some unpleasant consequences which complicate the anal-ysis of HC stability by introducing nonlinear feedback control laws into the reg-ulator loop dynamics. Actually, recursive algorithms for on-line system identifi-cation create dependence of the estimated T -matrix (and therefore the feedbackgain) on the controls. This is not unexpected, since the adaptive algorithms dis-cussed require the knowledge of the controls in order to estimate T from the mea-sured response. These nonlinearities make the analytical stability evaluation ofthe coupled, closed loop higher harmonic control system (i.e. regulator) unsuit-able for analysis by means of classical control theory. Rather, their analysis isheavily dependent on simulation studies (e.g., see [31]).

The problem of system parameter identification and its effects on algorithmperformance is thoroughly investigated by Shaw in [107]. Since this study, the fo-cus on the HC algorithm has largely centered on the convergence characteristicsof the controller, in the face of model uncertainty. More recently, Chandrasekaret al. [26] consolidate and extend the existing literature into a common frame-work for algorithm performance analysis. Patt et al. in [87] followed the samemethodology by exploring numerical techniques with the purpose of improvingthe disturbance rejection robustness properties of the HC algorithm. In particularthey propose the so-called relaxation technique by introducing a weight coeffi-cient which multiplies the optimal control step change at every time step tk, lead-ing to an effective reduction of the magnitude of the feedback gain. Basically, thiscan be view in some way as the rate limits imposed in the optimal control gainsynthesis, with the advantage of making it less responsive or sensitive to errorsin the estimate of the plant dynamics by slowing down the response speed of thecontroller (see [107] and [31] for more details).

36

2.4. Adaptive Harmonic Control Solutions

2.4.1 Recursive Least Square (RLS) Methods

As introduced, in an alternative version of the Harmonic Control algorithm, theT -matrix can be identified online to cope with situations in which it is either notexactly known or is time-varying due to, e.g., varying plant conditions.

To this purpose, considering the incremental model

Yk+1 = Yk + T (Uk+1 − Uk) (2.36)

where T is the k-estimate of the sensitivity matrix, define the variables

∆Yk = Yk − Yk−1, ∆Uk = Uk − Uk−1. (2.37)

with respect to which the model can be written as

∆Yk = T∆Uk. (2.38)

Introducing now the matrices

∆Yk = [∆Y1 · · ·∆Yk]

∆Uk = [∆U1 · · ·∆Uk](2.39)

we have the relation∆Yk = T∆Uk, (2.40)

which can be used as an estimation model for the recursive update of the T -matrix.By defining matrix Vk = ∆Uk∆UT

k , assumed nonsingular, the least squaresestimate Tk is given by

Tk = ∆Yk∆UkV−1k . (2.41)

and by using relations as in [7] the following RLS identification algorithm is ob-tained

Tk = Tk−1 + εkKkεk = ∆Yk − Tk−1∆Uk

Kk = ∆UTk Vk

Vk = Vk−1 −Vk−1∆Uk∆UTk Vk−1

1 + ∆UTk Vk−1∆Uk.

(2.42)

In order to emphasize the more recent data and discard the older ones, a vary-ing forgetting factor λk can be introduced, with p constant speed factor,

λk = pλk−1 + (1− p). (2.43)

As a result the evolution of matrix Vk is modified as follows,

Vk =1

λk

[Vk−1 −

Vk−1∆Uk∆UTk Vk−1

λk + ∆UTk Vk−1∆Uk

]. (2.44)

37


A sub-optimal approach to determine an estimate Tk is to replace matrix Vk

in(2.44) by Vk, starting from an arbitrary matrix V0, positive definite. The up-dated estimate Tk is used at each control update step to compute the control actionUk+1, given by

Uk+1 = Kk(Yk − TkUk) (2.45)

withKk = −(TTk QTk + R)−1(TkQ) (2.46)

2.4.2 Kalman Filter Identification

As known (see [54] and [24] among others), an alternative way for the on-lineidentification of time-varying parameters is given by resorting to Kalman filters.The output equation is again

Yk = TkUk +Dk (2.47)

where the measurement noise has zero mean, varianceE[DkDj ] = Rk and Gaus-sian probability distribution. The variation of the parameters can be modeled as arandom process,

Tk+1 = Tk + ufk (2.48)

with ufk random variable with zero mean, varianceE[ufk] = QKFk and Gaussianprobability distribution. This equation clearly implies that Tk is varying such thatan approximate change order can be estimated in one time-step. On the contrary,no information is available with regards of the dynamics governing the variationof Tk itself. The minimum error-variance estimate of Tk is then obtained by usinga Kalman filter structure,

Tk = Tk−1 +Kek(Yk − Tk−1Uk) (2.49)

with, being Mk the variance of the error in the estimate of Tk before the measure-ment, and Vk the variance after the measurement,

Mk = Vk−1 +QKFk−1

Vk =(M−1k + Uk

UTkRKFk

)−1

Kek = Vk

UkRKFk

.

(2.50)

Remark 4. Note that Vk depends on the control input Uk, but not on the outputmeasurement Yk. Moreover no matrix inversion is required, being Tk related toa single measured variable. If no process dynamics (QKFk = 0) are included,Kalman filter is equivalent to the generalized least-squares algorithm.

38

2.5. Continuous-time Harmonic Control

2.5 Continuous-time Harmonic Control

In their works in [120] and [47], Hall and Wereley proposed to substitute the in-tegration over one period, with the summation and the delay shown in Figure 2.3,with a continuous integration along with a sample and hold circuit. Moreover, byusing the continuous signals, a continuous-time implementation of the HarmonicControl was obtained as shown in Figure 2.5. By observing the structure of T

Figure 2.5: Continuous-time Harmonic Control implementation.

matrices, the inverse control response matrix can be assumed to have the similarform

T =

[a b

−b a

]. (2.51)

By defining U(s) as the Laplace transform of u(t), and Y (s) as the Laplace trans-form of y(t), the transfer function of the continuous-time HC depicted in Fig-ure 2.6 can be written as

H(s) =U(s)

Y (s)= 2k

as+ bω

s2 + ω2(2.52)

which coincides with the classical form for continuous time notch filters used forthe attenuation of single frequency disturbances. The static gain k is a design pa-rameter chosen to satisfy the stability margins and the bandwidth of the controller.In particular, for a direct comparisons between the discrete-time algorithm and thecontinuous-time algorithm, the gain should be selected as k = 1

T s . Note that witha complex conjugate pair of poles at s = ±jω, the controller H(s) is in the formof the classical controller for rejecting a sinusoidal disturbance with a frequency

39


time[s]0 5 10 15 20 25 30 35 40

outp

ut

-1

-0.5

0

0.5

1

Continuous-time HC

Figure 2.6: Example of disturbance rejection with a continuous-time HC im-plementation.

exactly equal to ω. The closed-loop pole positions can thus be shown to be ats ∼ − 1

T s ± jω where T s represents an approximation of the settling time of theclosed-loop system. As in the discrete-time case, a smaller T s corresponds to afaster response to changes in the harmonic disturbance, but the dynamics of thesystem may be destabilized.

The continuous-time HHC offers an interpretation for the relaxed version ofthe discrete-time HC algorithm described by Patt et al. in [87], obtained by ap-plying a relaxation factor α < 1 to the control update, resulting in the modifiedrelation

Uk+1 = Uk − αT−1Yk. (2.53)

Simulation results in [87] showed that the relaxed version increases the conver-gence time, but as a major advantage it could be beneficial in situations wherethe estimated T results to be uncertain, a condition which could effectively occurin maneuvering flight. The continuous-time interpretation of HC can be used tomotivate the obtained results, as it can be shown that the relaxation factor α < 1essentially decreases the static gain, resulting in a stabilization of the close-loopsystem while slowing down the response, consistent with the simulation resultsobtained by the authors.

With regard to continuous-time implementations of the Harmonic Control al-gorithm, Shin et al. in [110] examined the reduction problem of vibration at differ-ent harmonic frequencies, by using multiple modes of actuation. The used multi-mode controller has been obtained as a summation of individual continuous-timecontrollers, in the form

H(s) =∑h

2

T sh

ahs+ bhωhs2 + ω2

h

(2.54)

where the subscript h is referring to different combinations of actuation modes

40

2.5. Continuous-time Harmonic Control

with the harmonic frequency to be attenuated.

41

CHAPTER3Robust Harmonic Control - SISO case

Harmonic Control algorithms implementation requires knowledge of the frequencyresponse relating the control input to the output measurements at the disturbancefrequency. An offline identification can be performed to this aim, but unfortu-nately, when the information is uncertain or when the plant is subject to change,instability can occur. As shown in 1, adaptive schemes have been employed inliterature to handle this problem. Surprisingly, however, very little effort has beendevoted to the analysis of the T -matrix algorithm and in particular to the trade-offbetween robustness and adaptation in its deployment. Hence, the idea to apply anH∞ approach to the design of a robust T -matrix algorithm, with the aim of devel-oping a systematic approach to the design of harmonic control laws with robustproperties, has been investigated.

This Chapter is devoted to introduce and discuss a formal robust HarmonicControl approach in the single input-single output case, exploiting the advantagesof integrating the model uncertainty directly in the control design process. Inthe first part of the Chapter the problem and the robust synthesis procedure isdescribed in detail, in the second part a numerical study would demonstrate theadvantages of the proposed approach in comparison with the referenced optimalcontrol law presented in 1.

43

Chapter 3. Robust Harmonic Control - SISO case

3.1 Problem Statement

Several studies have been carried out to determine the feasibility of HarmonicControl both from the theoretical and the experimental point of view, as shown in1 with regard to the survey papers of Friedmann et al. [43] and Kessler [57, 58],where the used actuation technologies, the considered performance criteria andthe achieved performance are reviewed. As for the control implementation, adiscrete-time adaptive algorithm, known especially in the rotorcraft literature asthe T -matrix algorithm (from Shaw in [108], where this approach was originallyproposed), is typically used by defining the problem in the frequency-domain andtuning the controller using an LQ-like cost function.

Besides some robustness analysis of the T -matrix algorithm, like the interest-ing one carried out by Chandrasekar et al. in [26], the problem has never beentackled in a robust control framework. In view of this, the aim of this Chapteris to propose an approach to the design of robust HC control laws which can beuseful to reduce the need of adaptation and guarantee, more specifically:

• nominal stability of the closed-loop system;

• robustness to model uncertainty due to, e.g., changes in the plant conditions,plant configuration etc;

• guaranteed performance for the closed-loop system, i.e., a guaranteed levelof disturbance rejection.

Besides allowing to account for model uncertainty in the control design problem,the H∞ formulation of the HC problem provides an additional benefit when deal-ing with the tuning problem. Indeed, requirement specifications in terms of steadystate attenuation levels and desired transient performance (especially in the multi-variable case), contrary to an LQ-like algorithm which can turn out to be difficultto tune, can be specified directly in an H∞ problem statement.

3.2 SISO T -matrix model with uncertainty

For the sake of simplicity, in this Chapter the focus will be on the SISO and singletone formulation of the control problem. Extensions to the MIMO control systemwill be discussed in Chapter 4. As introduced in I, let u be the (scalar) controlinput and y the (scalar) measured output, in presence of a disturbance d. Definethen

yc =2

Th

∫ Th

0

y cos(ωt)dt (3.1)

ys =2

Th

∫ Th

0

y sin(ωt)dt (3.2)

44

3.3. Discrete-time H∞ Harmonic Control Design

Y =

[yc

ys

](3.3)

and similarly for u. Assume now that under steady state conditions the abovedefined ω harmonics of u and y are related by the linear equation

Y = TU +D (3.4)

where the T -matrix is now a 2 × 2 constant coefficient matrix. Recall that if thedynamics of the controlled system can be assumed to be linear time-invariant, thenmatrix T is clearly related to the frequency response Gyu(jω) associated with thesystem by

T =

[Re(Gyu(jω)) Im(Gyu(jω))

−Im(Gyu(jω)) Re(Gyu(jω)

]. (3.5)

In the formulation of the robust H∞ control synthesis, a multiplicative represen-tation of uncertainty in the T -matrix is considered

T = T(I2 +Wm∆

), |∆| < 1, (3.6)

where I2 is the 2 × 2 identity matrix, ∆ is a normalized representation of theuncertainty and Wm represents the uncertainty ellipsoid affecting the frequencyresponse Gyu(jω) = gyi,ui at the disturbance frequency. Note that matrix Wmhas the same structure as the T -matrix.

3.3 Discrete-time H∞ Harmonic Control Design

The main drawback of the LQ-like control law as introduced in (2.13) in the pre-vious Chapter is the assumption that exact knowledge of the T -matrix is available.Not surprisingly, an erroneous model of it can result first in degraded performance,till possible instability in the worst scenario. To deal with model uncertainty,a posteriori analysis could be carried out to prove robustness qualities. In thisregard, the derivation, from [26], described in 1 provides upper bounds on themaximum singular value of the additive uncertainty for which robust stability isguaranteed by using the described LQ-like control law. While such an analysisis of course informative, especially for the underlying expression of the perfor-mance/robustness trade-off, its main limitation lies in the difficulty in relatingback bounds on the uncertainty on matrix T to the actual plant model uncertainty.

Based on these considerations, a robust framework could be used to designcontrol laws which incorporate all the uncertainties during the synthesis process.This represents an alternative to conventional HC control, which deals with perfor-mance degradation in presence of model uncertainty by introducing some adap-tation mechanism. As already mentioned, many different algorithms have beendeveloped in this sense in the last few years, mainly based on the estimation of T

45


at specific time steps during the flight operations. In this respect, the interest ininvestigating a robust control design approach is motivated also by the possibilityto relax the need for continuous update of the T -matrix.

A block diagram of the uncertain feedback system corresponding to the model(1.19), its uncertainty (4.29) and the controller (2.16) is represented in Figure 3.1,where ∆ is defined as

∆ = blkdiag(I2δ

(1) I2δ(2) · · · I2δ

(p)). (3.7)

With reference to Figure 3.1, variables W∆,k and Z∆,k can be defined as

Figure 3.1: Block diagram of the uncertain feedback system.

Z∆,k = WmTUk

W∆,k = ∆Z∆,k,(3.8)

leading to the uncertain closed-loop system

Uk+1 = KMUk +KNUk

Yk = TUk +Dk +W∆,k

Z∆,k = WmTUk

W∆,k = ∆Z∆,k.

(3.9)

Letting now Zk = [Yk Z∆,k]T and Wk = [D W∆,k]T , the uncertain system(4.33) can be represented in input-output form as

Zk = GZW (z)Wk (3.10)

where GZW (z) is defined in (4.35),[Yk

Z∆,k

]=

[T (zI − (KM +KN T ))−1KN + I2p ∗WmT (zI − (KM +KN T ))−1KN ∗

] [Dk

W∆,k

](3.11)

46


Figure 3.2: Augmented plant model.

The H∞ synthesis problem is completed by defining the weighting functions WZ

and WU , used respectively on the output Zk and the control variable Uk accord-ing to the block diagram for the augmented plant model in Figure 3.2. In partic-ular the control design should be focused on the closed-loop steady-state perfor-mance, therefore, the shape of the frequency response ofWY (z) can be intuitivelydesigned on this basis (see, e.g., Figure 3.4 where the frequency responses ofcontinuous-time templates suitable to generate WY (z) are depicted). First ordercontinuous-time counterparts WY (s) and WU (s) are designed as

1

WY (s)=

(s+ ωbεYsMY

+ ωb

)kw1

WU (s)=

εUs+ ωUb

s+ωUbMU

kw

(3.12)

where parameters εY , εU and MY , MU are suitably chosen with respect to perfor-mance targets. Order kw could be increased when tighter objectives are to be met,while the role of parameters ωb, ωUb is instead blurred because of the discretisa-tion procedure. A brief sensitivity analysis would clarify the influence of each ofthese parameters on closed-loop performance in the following of the Chapter. Theinteresting advantage of the proposed approach is that requirement specifications,in terms of steady state attenuation levels and desired transient performance, canbe immediately ”encoded” in the problem statement through weighting functions.In addition, the properties of the optimal solution can provide information aboutthe actual distance between the desired and the achievable performance level.

Proposition 3.3.1. Given that, the robust synthesis problem can be formulated as

Find KM , KN

s.t.∥∥∥∥GZW (z)WZ(z)

GUD(z)WU (z)

∥∥∥∥∞≤ γ,

(3.13)

47


100 101 102 103 104

Mag

nitu

de (

dB)

-30

-25

-20

-15

-10

-5

0

5

10

15Bode Diagram

Frequency (rad/s)

Figure 3.3: Frequency response of possible 1/WY (z) weighting functions.

where GUD(z) is the control sensitivity function, obtained by re-opening the closed-loop from the disturbance D to the control variable U in the nominal case,

GUD(z) = (zI − (KM +KN T ))−1KN . (3.14)

Figure 3.4: Possible HC implementation in terms of control matricesKM ,KN .

3.3.1 Robust Control Tuning

Problem (5.37) is a structured H∞ problem, which is known to be both non-convex, so that the convergence of the algorithm may depend on the initial con-

48


troller and global optimality of the computed solution cannot be guaranteed, andnon-smooth, i.e., gradient-based descent algorithms could fail. In view of this, be-side the possibility to test sub-gradient methods to overcome the non-smoothnessissue, a randomized method has been used to solve the optimization problem: thekey point is that a control law is optimal if in its neighborhood a better control lawcannot be found (or, equivalently, can be found with null probability, see [124]).

To this purpose, assume that an initial stabilizing controller, based on the clas-sical LQ-based T -matrix algorithm, is available. To compute a new controllerK(i+1) the basic idea is to test randomly sampled controllers in a neighborhoodof K(i) and select the best one in terms of minimization of the cost function in(5.37). When it is no longer possible to find better controllers, the algorithm stopsand the optimal controller is obtained. The stopping criterion is set on the param-

Figure 3.5: Representation of an example iteration of the tuning algorithmfor two decision variables K1 and K2. The circle is indicating the boundwhere the search is applied.

eter 0 < pmax < 1, defined, in order to check the convergence, as the probabilityof not finding better controllers in the current iteration. The number Nc of MonteCarlo samples and the initial step size λ are also defined. Given a stabilizinginitial controller K(i) the optimal controller is found by means of Algorithm 1(see [124] for details). Note that the term ∆K(i)/‖∆K(i)‖ can be regarded asan approximation of the steepest descent direction, and the candidate controllerKj is randomly generated by taking into account the knowledge of the estimatedsteepest direction. Actually the controller is a random matrix, and to check con-vergence the procedure has been iterated as many times as the standard deviationin the cost function becomes below±1%, thus guaranteeing the minimality of thecost with a very small tolerance.

Remark 5. It has already been mentioned that the gains of the LQ-based con-troller are not fully parameterized 2 × 2 matrices, but inherit the structure of the

49


Algorithm 1 Tuning algorithm

1: generate Nc controllers by means of a MonteCarlo generation: for j = 1, ..., NcK(j) ← K(i) + λ‖K(i)‖

(ηj + ∆K(i)

‖∆K(i)‖

)2: compute the H∞ norm, cj ← J(K(j))3: select the candidate controller, j ← minj J(K(j))4: compute the an approximation of the rejection ratio p(i+1)

p(i+1) ← P (J(K(j)) > J(K(i))5: if ρ(GZW ) < 1 . %check nominal and robust stability%

set K(i+1) ← K j and i← i+ 1else decrease step λ and goto 1.

6: if p(i+1) > p(i)

decrease step λelse increase step λ

7: if p(i+1) ≥ pmaxreturn K

else goto 1.

T matrix itself. This structure of the control law can be either retained in the de-sign of the robust controller, or, on the contrary, can be relaxed leading to a fullyparameterized control law. Both possibilities will be considered in the followingSections.

3.4 Numerical Study

In this section some numerical examples are used to illustrate the main propertiesof the robust SISO control solution and its advantages over the classical invariantLQ-based approach. More precisely, each example underlines a specific advan-tage of the proposed approach, among them the stability robustness of the closed-loop system, the performance, the synthesis procedure. A Monte Carlo study hasbeen further carried out for the feedback system, with experiments performed onbasis of 500/1000 random values of the normalized uncertainty ∆, so that eachinitialization of the matrix T accounts for the uncertainty due to changing plantconditions, configurations and other similar causes.

3.4.1 Control parameters - sensitivity analysis

In order to select the weighting functions WY (z), WU (z), a sensitivity analysishas been carried out. Apart from parameter εY , which defines performance targets(rejection level) and should be free to set in view of the considerations of previoussection, the influence of other parameters, i.e., ωb or Ms, in view of goodnessof results and algorithm efficiency is not trivial to understand, especially becauseof the discretisation procedure aimed at finding the ’best’ discrete-time control

50

3.4. Numerical Study

outp

ut n

orm

0.05

0.1

0.15in

put n

orm

2.8

3

3.2

3.4

Monte Carlo Simulation #2 4 6 8 10 12

Set

tling

tim

e

0

5

10

15

outp

ut n

orm

0

0.2

0.4

0.6

inpu

t nor

m

2.5

3

3.5

Monte Carlo Simulation #1 2 3 4 5 6 7 8

Set

tling

tim

e

0

5

10

15

20

Figure 3.6: Robust HC sensitivity analysis with reference to ωb andMY . Out-put, control and settling time evaluation.

law. Given that, a sensitivity analysis has been produced, starting from the sim-ulation results based on a Monte Carlo study, whose details will be extensivelyreported in the following. Figures 3.6 and 3.7 summarize the results. In eachof them three quantities are shown and compared: the steady output norm ‖Yk‖,the steady input norm ‖Uk‖ and the effectively obtained settling time Ts, com-puted on basis of output variations with suitable defined tolerance. The goal ofthis sensitivity analysis is to understand how the discrete-time robust synthesis issensible to changes in the defined parameter of the weighting functions WY (z),WU (z). In particular four experiments have been produced. The first one, whoseresults are shown in Figure 3.6, focuses on the obtained results with reference toan increased bandwidth ωb of the output weight. The second one, instead, wouldevaluate how performance change when under and overshooting shapes, by meansof parameter MY , are introduced for the output weight. Results are shown againin Figure 3.6. The last two experiment follow the same reasoning and evaluate theclosed-loop results with regards to control parameters ωUb and MU , whose resultsare illustrated in Figure 3.7. Four sets are used for the sensitivity analysis,

Sωb = ωb : ωb ∈ [0.1, 500] rad/s SMY= MY : MY ∈ [0.2, 2]

SωUb = ωUb : ωUb ∈ [0.1, 200] rad/s SMU= MU : MU ∈ [0.2, 2].

(3.15)As a conclusion, reasonable bounds on weighting parameters are obtained (see

Table 3.1). Some of the drawn results are trivial, others not. In particular, from thefigures it evident how an excessive bandwidth of the output shaping function WY

can degrade the steady output, nor is possible to evaluate an appreciable settlingtime sensitivity. On the other hand, by imposing a restrictive bound MY on thetransient output a degraded steady performance is obtained. In the same way, en-

51


outp

ut n

orm

0

0.2

0.4

inpu

t nor

m

1.5

2

2.5

3

3.5

Monte Carlo Simulation #2 4 6 8 10 12

Set

tling

tim

e

0

5

10

outp

ut n

orm

0

0.2

0.4

inpu

t nor

m

1

2

3

Monte Carlo Simulation #2 4 6 8 10 12 14

Set

tling

tim

e

0

2

4

6

8

Figure 3.7: Robust HC sensitivity analysis with reference to ωUb and MU .

Output, control and settling time evaluation.

forcing a minimum control effort during the transient results in poor performanceat steady state too.

Table 3.1: Robust HC sensitivity analysis. Obtained bounds with desiredsteady performance εY = 0.02.

ωmaxb 50MminY 0.8

ωU,minb 20MminU 1

3.4.2 Numerical example - Stability

Consider a nominal T -matrix T given by

T =

[1.05 −2.75

2.75 1.05

](3.16)

and the relative uncertainty

Wm = 0.48

[0.95 −1.85

1.85 0.95

]. (3.17)

For this system both an LQ and an H∞ controller have been designed. In par-ticular, the former is based on the weights Q = I and R = 0.1I , while the latter

52


is based on the weighting function WY (z) the parameters of which are chosen asεY = 0.05, ωb = 20 rad/s, MY = 1 (no weight on the control action has beenincluded, WU (z) = 1). For the two feedback systems (subject to a unit normdisturbance) a Monte Carlo study was carried out, by randomly choosing 1000values for the normalized uncertainty ∆. The results are depicted in Figure 3.8,where the steady state magnitude of the output Yk 1 is shown for both cases. Ascan be seen from the figure, and recalling the unit-norm disturbance affecting thesystem, theH∞ controller can guarantee closed-loop stability and provides a con-sistent level of performance, while the performance of the LQ-based controller isvery sensitive to uncertainty and even instability occurs in some instances of theMote Carlo study.

Monte Carlo # Occurrence

0 100 200 300 400 500 600 700 800 900 1000

Out

put N

orm

0

0.5

1

1.5

2

2.5

3LQ controllerH

∞

controller

Figure 3.8: Numerical Example - stability. Monte Carlo study: steady statemagnitude of the output Yk, with reference to a unity-norm disturbance D.LQ-based VS H∞ control law comparison.

As shown in Figure 3.9, moreover, it is evident how the instability occurrencesof the Monte Carlo simulation increase when the uncertainty included enlarges.On the contrary, the robust algorithm is able to prevent this situation by encodingthe uncertainty radius in the synthesis procedure.

3.4.3 Numerical example - Performance

Consider a nominal T -matrix T given by

T =

[0.208 0.024

−0.024 0.208

], (3.18)

1computed numerically over a finite interval, so very large values denote diverging output

53


LQ H∞

Mon

te C

arlo

# O

ccur

ranc

e

0

200

400

600

800

1000

1200 UnstableStable

LQ H∞

0

200

400

600

800

1000

1200UnstableStable

LQ H∞

0

200

400

600

800

1000

1200UnstableStable

Figure 3.9: Numerical Example - stability. Monte Carlo study: distribution ofunstable and stable steady output Yk (computed numerically) with refer-ence to a unity-norm disturbance D. LQ-based VS H∞ control law com-parison, including an uncertainty of a) 50%, b) 30% and c) 10%.

to which the relative uncertainty

Wm = 1.5 · 10−3

[6.93 0.58

−0.58 6.93

](3.19)

has been associated.For this numerical example, a 95% attenuation of the (unit norm) disturbance

D is set to be requested as nominal performance. The tuning procedure in thenominal case is the following. Since it is easier to tune the LQ controller bychanging the values of the weights Q and R than tuning the H∞ controller, twodifferent LQ control laws are used. The first one is used to initialize Algorithm 1,to find an H∞ controller with the desired level of attenuation (the one to be com-pared between the two algorithms); the second one reflects the performance ob-tained with the already tuned nominal H∞ controller. From a practical point ofview this means that the second LQ tuning is slightly different from the first oneto get the same nominal attenuation level.

As in the previous example, a Monte Carlo simulation is performed to comparethe performance of the LQ and H∞ solutions. In order to compare the resultsthe considered metrics are the steady output norm, ‖Yk‖, its standard deviationσ(‖Yk‖) and the settling time Ts, computed on basis of output variations withtolerance ≤ 1 · 10−4.

The simulations have been carried out as follows. To check whether the min-imization algorithm converges to the best controller, the procedure has been im-plemented and run several times ending with a standard deviation σJ in the costfunction of about 1%. Note that σJ is a function of the number of points Ncconsidered in each step of the tuning algorithm. We can expect indeed that byincreasing Nc the cost variance σ2

J becomes lower and lower. In Figure 3.10 an

54


example of the resulting cost function is shown, when a relative uncertainty ofapprox 10% is included in the tuning algorithm. As stated above, the norm and

# iterations0 10 20 30 40 50 60 70 80

Cos

t fun

ctio

n J

3

4

5

6

7

8

9

10

11

12Control synthesis - J (∞-norm) minimization

Figure 3.10: Numerical Example - performance: validation of the solutionconvergence in the H∞ minimization process. Each curve represents adifferent instance of the cost function minimization process.

the settling time to converge to the steady-state solution are compared, and theresults are shown, respectively, in Figures 6.6, considering in this case a moderatelevel of uncertainty (10%) to focus on the closed-loop performance of the controlsystems. As can be seen from Figure 6.6, the two controllers can provide a com-parable level of attenuation of the external disturbance; on the other hand, somedifference in the behaviour of the feedback systems can be noticed in Figure 6.6bottom, from which it is apparent that the cost of increased robustness is a slightlyslower response in dynamic terms: on average the settling time of the robust con-trol system is longer. Moreover, the results obtained with reference to differentuncertainty radii are reported in Table 3.2.

As mentioned in previous sections, the comparison can be extended to a struc-tured version of the H∞ controller. To this purpose, Algorithm 1 has been mod-ified on the basis of the structure constraint, meaning that the cost function isoptimized, concerning the SISO case, with respect to 4 parameters only (insteadof 8 parameters as in the fully parameterised formulation). From here on and inview of these considerations, we refer to HCHs∞ and HCHu∞ for the structured andunstructured robust controllers, respectively. Table 3.2 show fairly well how anHs∞ controller loses in some way the advantage of a robust synthesis under the

performance aspect, while at the same time a further dynamic slowdown is evi-dent. On this basis, the importance of a fully parameterised formulation becomescentral in the robust synthesis.

55


0 100 200 300 400 500

Out

put n

orm

0

0.05

0.1

||Y|| - HCLQ

||Y|| - HCH∞

0 100 200 300 400 500

Inpu

t nor

m

2

4

6

||U|| - HCLQ||U|| - HCH

∞

Monte Carlo # Occurrence0 100 200 300 400 500

Set

tling

tim

e

0

5

10

Figure 3.11: Numerical Example - performance. Monte Carlo study: steadyoutput norm in the case of the LQ-based (invariant) controller andH∞ HCcontroller.

Table 3.2: Numerical Example - performance. Monte Carlo study: perfor-mance of the LQ and of the H∞ (both structured and unstructured) con-trollers.

Uncertainty 10 % 30 % 50 %avg [‖Y ‖]− HCLQ 0.054(24) 0.087(23) 0.141(23)avg [‖Y ‖]− HCHs∞ 0.052(21) 0.084(20) 0.137(16)avg [‖Y ‖]− HCHu∞ 0.047(23) 0.070(22) 0.129(18)max ‖Y ‖ − HCLQ 0.065 0.094 0.171max ‖Y ‖ − HCHs∞ 0.067 0.092 0.158max ‖Y ‖ − HCHu∞ 0.054 0.066 0.131Ts − HCLQ 3.46 3.57 3.84Ts − HCHs∞ 12.67 12.91 14.26Ts − HCHu∞ 4.21 4.83 5.68

56


Table 3.2 confirm that for a given level of steady state performance, the costof robustness is a slower dynamic response (the larger the uncertainty, the slowerthe response) and that better results can be obtained by resorting to a fully param-eterised H∞ controller.

57

CHAPTER4Robust Harmonic Control - MIMO case

A robust variant of standard Harmonic Control algorithm has been presented anddiscussed in the previous Chapter. As already introduced there in, besides al-lowing to account for model uncertainty in the control design problem, the H∞formulation of the Harmonic Control problem provides an additional benefit whendealing with the tuning problem. More specifically, output measurements could betypically taken on a large number of locations of the generic structure, hence thecontrol problem results to be strongly multivariable, with different performancerequirements associated to the disturbance rejection on the individual output.

From this point of view, the tuning of an LQ-like, possibly adaptive, algorithm,can turn out to be extremely challenging. Requirement specifications in terms ofsteady state attenuation levels and desired transient performance, on the otherhand, can be immediately ”encoded” in an H∞ problem statement and the prop-erties of the optimal solution can provide information about the actual distancebetween the desired and the achievable performance level.

This chapter is dedicated to present a MIMO-H∞ approach to the design ofa robust harmonic control law with large plant systems. The key point is that aformulation adherent as much as possible to the already solved SISO-robust con-trol problem is pursued. The first part of the Chapter focuses on methods usuallyemployed to describe a generic multivariable system as a number of independent

59

Chapter 4. Robust Harmonic Control - MIMO case

single channels. Then, robust control design procedure is discussed in the light ofthe decoupled multivariable system. Finally results are presented and discussed.

4.1 Problem Statement

When the considered plant is strongly multivariable, some cross-talking phenom-ena could arise and should be taken into account in the control design. A generalissue is represented by the influence of a single control input on different outputswith prescribed performance requirements. In this case can be difficult, even notpossible, to design a controller for each input-output channel independently.

As illustrated in Figure 4.1, decentralized control strategies can surely be adoptedfor systems whose transfer matrix is diagonal or with a diagonal dominance. Inthis context the analysis and synthesis considerations proposed in the previousChapter hold without difficulties. Indeed, the signal generated by a single actua-tor has a significant effect only for a specific output, so that it is possible to designindependent controllers for each input-output channel. The MIMO plant is actu-ally described by an n-SISO system. Such a condition is clearly desirable sinceit simplifies the multichannel control design process. When this does not hold,

Figure 4.1: Decentralized control scheme. Plant Transfer matrix G is con-sidered diagonal or already presents a diagonal dominance. The MIMOsystem can be considered as an n-SISO one.

then a centralized control (see Figure 4.2) should be adopted, meaning that theprocedure to generate the control signals takes into account all the output signalsand its interactions. Such control laws are not easy to design and their complexitygrows exponentially with the number of channels. Fortunately some techniques

60

4.2. Decoupling methods: from MIMO to n-SISO

have been developed to simplify a multichannel system so that many independentdecentralized controllers can be employed to control the given plant. Methodstypically used to simplify the system will be discussed in the following, with theaim at transforming a MIMO system to an equivalent n-SISO on which a decen-tralized Harmonic Control approach can be applied. This procedure is commonlyreferred as decoupling control and it allows to have an equivalent diagonal system.

Figure 4.2: Decoupling control strategy. The multivariable system is decou-pled to find an equivalent diagonal n-SISO representation. Decentralizedcontrol can be then applied.

4.2 Decoupling methods: from MIMO to n-SISO

To avoid loop interactions, MIMO systems can be decoupled into separate (in-dependent) loops. Decoupling may be done using several different techniques,including: restructuring the pairing of variables; minimizing interactions by de-tuning conflicting control loops; using linear combinations of manipulated and/orcontrolled variables. Well known methods for simplifying MIMO control schemesinclude the Relative Gain Array (RGA) method, the Niederlinski Index (NI) andthe Singular Value Decomposition (SVD).

4.2.1 Pairing rules via RGA/SVD techniques

Relative Gain Array method was first proposed by Bristol in [20], not only asa valuable tool for screening selection of manipulative-controlled variables pair-ings, but also to predict the behavior of controlled responses. Basically, RGA isused to determine if a decentralized control strategy can be adopted for the givenmultivariable plant (see [90], [20] and [27]). It can be viewed as a normalizedform of the gain matrix that describes the impact of each control variable on the

61


output, relative to each control variable’s impact on other variables. The idea is tomeasure the process interaction of open-loop and closed-loop control systems forall possible input-output variable pairings. Hence, a ratio of the open-loop ’gain’to the closed-loop ’gain’ is determined and the results are displayed in a matrix.

Definition 4.2.1. Considering the generic non-singular (square) matrix A ∈ C,RGA is defined by

RGA(A) = Λ(A)∆= A× (A−1)T (4.1)

in which × denotes the Hadamard product, i.e. an element-by-element multipli-cation.

Consider the (square) MIMO plant with transfer matrix Gyu(s) with elementsgij(s) being particular transfer functions from the input uj to output yi. Basically,if no interaction between the loops are present, the gain between input uj andoutput yi should remain the same when the other loops are closed, so that therelative gain results in gij/gij = 1. On the other hand, gij differs as comparedwith gij if cross-interactions are present in the system. In this case, the ratio

λij∆= gij/gij (4.2)

could be used as an interaction index, and the RGA matrix Λ(G) can be computed.As an example, let compute Λ(G) for the 2 × 2 transfer matrix

G =

[g11 g12

g21 g22

]. (4.3)

Result is given by

Λ(G) =

[λ11 λ12

λ21 λ22

]∆=

[λ11 1− λ11

1− λ11 λ11

](4.4)

where

λ11 =open-loop gain (u2 = 0)

closed-loop gain (y2 = 0)=

1

1−g12 g21

g11 g22

.

Matrix Λ can be read as in the following:

1. λ11 = 0→ no interaction between input u1 and output y1. Hence, pairingshould be chosen along the anti-diagonal, i.e. (u1 − y2),(u2 − y1);

2. λ11 = 1→ similar to above case, but pairing is chosen along the diagonal,i.e. (u1 − y1),(u2 − y2);

62


3. 0 < λ11 < 1→ gain (i.e. gij) increases when the loops are closed. Hence,there are interactions in the system, and the worst case occurs when λ11 =0.5;

4. λ11 > 1 → gain decreases when the loops are closed. Interactions arepresent, and they get worse as larger λ11 is;

5. λ11 < 0→ sign changes when the loops are closed, leading to an extremelycritic control problem. The more negative λ is, the worse the interactionsare.

Even for high-dimension plants ( [4], [27], the general pairing rule is to considerthe RGA elements closer to unity, avoiding the negative pairings. In view of thestated considerations, Λ(·) can be used as a measure of diagonal dominance, andto characterize the chosen pairings with a simple measurement, in the form of acost function

JΛ = ||Λ(G)− I||∑ (4.5)

where the sum matrix norm is defined as

||A||∑ =∑

i,j|aij | (4.6)

if the n × n matrix A has elements aij , i, j = 1, . . . n. Clearly, JΛ can measurehow dominant the diagonal is. A drawback and a possible disadvantage of usingthe cost JΛ is its necessity to be recomputed for each alternative pairing. On thecontrary, Λ can be computed at once. Hence, an iterative RGA method can bealternatively used as proposed in [122],

Λ∞ , limk→∞

Λk(G). (4.7)

It could be proven that Λ∞ always converges to the identity matrix if G is a gen-eralized diagonally dominant matrix.

Somehow parent of the RGA ( [4]), SVD method is a common matrix de-composition algorithm, especially used in control design to determine is a sys-tem could be decoupled. It allows to express a (rectangular) transfer matrixG(s) ∈ Cp×m, at each frequency s = jω, as a product of three matrices,

G(jω) = USΣV H (4.8)

where V H is the Hermitian of the generically complex matrix V ∈ Cm×m, US ∈Cp×p is an orthogonal matrix whose columns are the eigenvectors of GGH ; Σ ∈Rp×m is a diagonal matrix composed of non-negative σi, square roots of theeigenvalues of GHG,

σi =√λi(GHG) =

√λi(GGH). (4.9)

63


Columns of matrices US and V (ui and vj) are the associated eigenvectors, repre-sentative of the input and output system directions, so that

GHGvj = σ2j vj , GGHui = σ2

i ui Gvi = σiui. (4.10)

The key point is that SVD can give a detailed description of how a system matrix(G or T ) acts on a vector (e.g. the control input U ) at a particular frequency ω.Minimum and maximum system amplifications are described by singular valuesσ and σ, respectively,

σ = min‖u‖‖y‖, σ = max

‖u‖‖y‖. (4.11)

Given that, by denoting vectors v (v) as the largest (smallest) amplitude inputdirection, u (u) as output direction corresponding with the most (least) effectiveinputs, the relation

Gv = σu, Gv = σu (4.12)

is obtained. From that, a synthetic controllability measure could be defined bymeans of the condition number γ,

γ(G)∆=σ(G)

σ(G). (4.13)

Remark 6. Ill-conditioned systems present large condition number (typically γ >10), meaning that particular input combinations could have higher influence onthe output than others. With respect to a robust control framework, it can been as-sumed that a large condition number indicates also high sensitivity to uncertainty(see [111] for details). This is not true in general but the reverse holds: smallvalues of γ could be beneficial when dealing with system uncertainty. In general,if the γ is large, then this may indicate control problems, because it can imply thethe plant has large RGA elements. Moreover, the greater the γ value, the harderit is for the system to be decoupled.

4.2.2 Non-Square plants extension

The SVD/RGA methods can be used and extended for non-square plants too.Considering an m × p matrix Gns, with inputs u and outputs y, the RGA canbe generalized by use of the pseudo-inverse,

Λ = Gns × (G†ns)T . (4.14)

Basically, for the case of many candidate inputs, one may consider to avoid thoserelative to RGA columns whose sum is smaller than 1 (see [25]). Analogously,when many candidate measured outputs are present, one may consider to disregard

64


those outputs corresponding to RGA rows whose sum is, again, smaller than 1.The reason is reported in the following for completeness (see [111] and [90] formore details).

Consider a unit column vector of lengthm, qj = [0 · · · 0 1 0 · · · 0]T . The jthinput is given by uj = qTj u. Analogously, by defining qi, the ith output is givenby yi = qTi y. If Gns either has full column rank or full row rank then

m∑j=1

λij = ‖qTi Ul‖ (4.15)

p∑i=1

λij = ‖qTj Vl‖, (4.16)

where Ul and Vl are matrices containing the first l output and l input singularvectors for Gns respectively (l=rank(Gns)). Note that input and output singularvectors can be obtained by performing an SVD of Gns,

Gns = USΣV H = UlΣlVHl

with Σl consisting in the l nonzero singular values, Ul consisting in the l firstcolumn of US , and Vl the l first columns of V . As the conclusion in [111], thecolumns in Vl represent the input directions that can effect the outputs, and simi-larly, the columns in Ul represent the output directions which can be effected bythe inputs.

Given that, qTi Ul yields the projection of an unit output yi onto the effectiveoutput space of Gns; analogously , qTj Vl can be seen as the projection of an unitinput on the effective input space of Gns spanned by the columns of Vl. Notethat input/output selection for possible control schemes can grow rapidly withrespect to a large group of candidates. Fortunately, the above results describe quiteefficient tools aimed at pairing the more suitable variables. As an example, withthe goal of eliminating some input candidate, the RGA for “full” plants shouldbe computed firstly. Then, according to (4.16), RGA columns whose sum is lowrepresent the inputs with a low impact on the overall system, so that they can bedismissed. Similarly, the selection of output candidates can be implemented basedon the RGA, and finally, by means of (4.15), outputs corresponding to low rowsums can be removed.

4.2.3 Decoupling techniques and compensator design

As shown, the simplest approach to extend the robust SISO-HC control design toa MIMO plant configuration is to use diagonal controllers. This is related to theassumption that the interaction between non-chosen pairing inputs and outputs of

65


the plant can be neglected when compared with the pairing ones. In this context,RGA and SVD methods can be used to evaluate the most suitable input-outputchannels. Of course are frequent the situations when these approaches cannotbe adopted, because the above mentioned assumption is no longer true, and off-paring channels cannot be neglected. In these cases a decoupler can be introducedin the loop as introduced in Figure 4.2. It basically compensates for process inter-actions and thus virtually reduce control loop interactions so that each output canbe controlled only by one input. Typically, decoupling controllers are designedusing a simple process model such as a steady state model or a transfer functionmodel. A very simple decoupling strategy is depicted in Figure 4.3, referring to a

Figure 4.3: Harmonic Control Decoupling strategy with 2×2 plants.

2× 2 system G,

G =

[G1,1 G1,2

G2,1 G2,2

]. (4.17)

∆d is the decoupler, whose form is given by

∆d =

[∆1,1 ∆1,2

∆2,1 ∆2,2

]=

[1 ∆1,2

∆2,1 1

]. (4.18)

For large multivariable systems a natural generalization of this decoupling strategyis given by a two-step procedure: the first step aims at designing a compensatorto deal with the interactions in the plant (can reduce the interactions in the plantbefore designing controller); the second one focuses on the design of a diagonalcontroller, considering the plant as, effectively, an n-SISO (decoupled) system.

Consider a compensator C(s), able to counteract the interactions in the squareplant G(s),

Gs(s) = G(s)C(s). (4.19)

66


The result is a shaped plantGs(s), easier to control than the original one. Based onthe shaped plant and a suitableC(s), a diagonal controllerKd(s) can be designed,leading to the overall controller

K(s) = C(s)Kd(s). (4.20)

Generally, a dynamic decoupling results when the obtained plant is diagonal at aselected frequency ω. As an example, a compensator of the form C(s) = G−1(s)(disregarding the possible problems involved in realizing G−1), would result in ashaped plant Gd(s) = I , with I identity matrix. Consequently, a possible con-troller can be given by

Kd(s) =k

sI → K(s) =

k

sG−1(s), (4.21)

known as inverse-based controller. Moreover, in some cases it is desired to keepthe diagonal elements in the shaped plant unchanged by selectingC(s) = G−1(s)Gdiag.In other situations, the control designer would obtain a diagonal plant a steadystate only, by means of a static compensator.

As shown in [111], although the decoupling idea is interesting and appeal-ing for a lot of applications, some difficulties and limitations are to be taken intoaccount. Among them, the designer should consider the nontrivial problem ofimposing a form for Gs. As an example, a compensator C(s) cannot cancel anyright-half-plane zeros of the process G(s). Moreover, as it may be expect, decou-pling strategies could be extremely sensitive to modeling errors or uncertainties.Possible solutions to overcome such limitations are represented, for example, bythe internal control (IMC) approach or the partial decoupling of upper or lowertriangular transfer function matrices. This could be achieved equivalently as inthe following:

1. by splitting the compensator design such that pairing information is in-cluded;

2. by introducing a post-compensator C(s) as in Figure 4.4.

Regarding the case 1., C(s) can be designed with the aim at minimizing the cost

minC(s)

JΛ(s) = ‖Λ(Gs(s))− I‖∞, (4.22)

where Gs(s) = G(s)C(s), which should be lower or upper triangular. A secondcompensator C(s) is then designed based on the shaped plant Gs(s),

C(s) = Gs(s)−1Gdiags (s) (4.23)

67


Figure 4.4: Pre/Post compensator decoupling strategy for a general MIMOcontroller design.

where Gdiags is a diagonal matrix taken from the shaped transfer function matrixGs(s), which already includes all the pairing information. Hence, the overallcompensator is given by

C∗(s) = C(s)C(s), (4.24)

and the final decoupled process,

G∗(s) = G(s)C∗(s). (4.25)

Clearly, being the compensator designed by minimizing a non-convex cost func-tion JΛ, it is not possible to ensure that the global optimum can be achieved. Theoverall algorithm to compute the compensator is summarized in 2. Note, simi-larly to 1, that the term ∆C(i)/‖∆C(i)‖ can be regarded as an approximation ofthe steepest descent direction, and the candidate pre-compensator Cj is randomlygenerated by taking into account the knowledge of the estimated steepest direc-tion. This compensator is actually a random matrix, and to check convergencethe procedure has been iterated as many times as the standard deviation in thecost function becomes below a suitable threshold, allowing to obtain guaranteesof minimality.

As for point 2., a slight different way to implement a decoupler is illustratedin Figure 4.4. The shaped plant is given here by C2 G C1, the controller is thenK = C1 Ks C2. An SVD interpretation can be given by defining C1 = V andC2 = UHS , where V and US are obtained from a singular value decomposition ofG. Recalling the relations UHS U = I , V HV = I , the system simplifies to the oneshown in Figure 4.5. In this way, measurements and control inputs are re-mappedas virtual signals of a diagonal system.

68

4.3. Multivariable H∞ Control Design

Algorithm 2 Two-step Compensator Design

1: generate Nc compensators by means of a MonteCarlo generation: for j = 1, ..., Nc

C(j) ← C(i) + υ∗‖C(i)‖

(ηj +

∆C(i)

‖∆C(i)‖

)2: compute the H∞ norm, cj ← J

(j)Λ = ‖Λ(G(s)C(i))− I‖∞

3: select the candidate compensator, j ← minj J(j)Λ

4: compute the an approximation of the rejection ratio p(i+1)

p(i+1) ← P (J(j)Λ > J(Λ

(i))

5: if p(i+1) > p(i)

decrease step υ∗

else increase step υ∗

6: if p(i+1) ≥ pmaxcompute the shaped plant Gs(s) = G(s)C(s)compute a second compensator C(s) = Gs(s)

−1Gdiags (s)return C∗ = CC

else goto 1.

Figure 4.5: Pre/Post compensator decoupling strategy for MIMO controllerdesign. SVD simplification.

4.3 Multivariable H∞ Control Design

In the light of the methods resumed in the previous Section, a multivariable ex-tension of the robust controller discussed in Chapter 3 is proposed. As alreadyintroduced, a formulation adherent as much as possible to the already solvedSISO-robust control problem is pursued. This is actually possible precisely bymeans of the discussed decoupling strategies. Indeed, even though in principle ageneral multivariable Harmonic Control approach can be applied to the originalMIMO model, including all its cross-interactions, this direct approach would leadto a controller which would be both very complex and very hard to design andtune.

69


Consider the global (non-square) plant transfer matrix Gyu(jω),

Gyu(jω) =

G1,1 G1,2 · · · G1,p

G2,1 G2,2 · · · G2,p

......

...Gm,1 Gm,2 · · · Gm,p

, (4.26)

and apply 2 with the aim at finding a diagonal system representation in terms oftransfer matrix,

G∗(jω) = C(jω)G(jω) =

G∗1,1 0 · · · 0

0 G∗2,2...

......

0 0 · · · G∗p,p

. (4.27)

The reader would recall the focus of Harmonic Control in pursuing closed-loopsteady-state performance requirements. Hence, the equivalent diagonal (square)system formulation is based on the output virtual scaling of the decoupler, so that anumber of virtual inputsm ≤ pwould be considered. Therefore, the multivariableT -matrix plant assumes a block-diagonal structure,

T ∗ = blkdiag[T ∗1,1, T

∗2,2, · · · , T ∗p,p

]. (4.28)

As for the multivariable robust H∞ control synthesis, similarly to the SISOformulation, an output multiplicative representation of the uncertainty in the T -matrix can be chosen (see Figure 6.3)

T ∗ =(I2p +W ∗m∆

)T∗, ‖∆‖ < 1, (4.29)

where T∗

is the nominal (diagonal) model, I2p is the identity matrix of dimension2p, ∆ is a normalized representation of the uncertainty and W ∗m represents thematrix of the uncertainty ellipsoids affecting the frequency responses G∗(i,j)(jω),i = 1...p, j = 1...p, at the disturbance frequency. Note that in this case matrixW ∗m has a block diagonal structure due to the collection of uncertainties related toeach of the diagonal system matrices T

∗i,i, i = 1...p,

W ∗m = blkdiag[W

(1)m W

(2)m · · · W

(p)m

](4.30)

where

W (i)m = r(i)

[α(i) β(i)

−β(i) α(i)

], i = 1, ..., p (4.31)

70

4.3. Multivariable H∞ Control Design

with ri a scalar scale factor and α, β the parameters relating to the considereduncertainty of the specific output. Note that the uncertainty inclusion is actuallybased on the diagonal system representation. With reference to Figure 4.6, vari-

Figure 4.6: Block diagram of the uncertain multivariable feedback system.

ables W ∗∆,k and Z∗∆,k can be defined for each SISO channel in the equivalentn-SISO system reformulation, as

Z∗∆,k = blkdiag[W

(i)m T

∗1,1U1,k · · · W (p)

m T∗p,pUp,k

]W ∗∆,k = ∆Z∆,k,

(4.32)

leading to the augmented multivariable closed-loop system

U∗k+1 = K∗MU∗k +K∗NU

∗k

Y ∗k = T∗U∗k +Dk +W ∗∆,k

Z∗∆,k = W ∗mT∗U∗k

W ∗∆,k = ∆Z∗∆,k.

(4.33)

Letting now Z∗k = [Y ∗k Z∗∆,k]T and W ∗k = [D W ∗∆,k]T , the global uncertainsystem can be represented in input-output form as

Z∗k = G∗ZW (z)W∗k (4.34)

where G∗ZW (z) is defined in (4.35),[Y ∗kZ∗∆,k

]=

[T∗(zI2p − (K∗M +K∗NT

∗))−1K∗N + I2p ∗

W ∗mT∗(zI2p − (K∗M +K∗NT

∗))−1K∗N ∗

][Dk

W ∗∆,k

].

(4.35)

Proposition 4.3.1. Given that, the H∞ multi-channel synthesis problem can be

71


formulated asFind K∗M , K

∗N

s.t.∥∥∥∥G∗ZW (z)W ∗Y (z)

G∗UD(z)W ∗U (z)

∥∥∥∥∞≤ γ,

(4.36)

where, as shown for the SISO control design, G∗UD(z) is the control sensitivityfunction obtained by re-opening the closed-loop from the disturbance D to the(virtual) control variable U∗ in the nominal case,

G∗UD(z) = (zI2p − (K∗M +K∗NT∗))−1K∗N . (4.37)

As for the definition of the weighting functions W (i)Y , i = 1...p and W (j)

U , j =1...p (note that the input U∗ has virtually dimension p) used, respectively, on theoutput Y ∗k and the virtual control variable U∗k according to the block diagram forthe augmented MIMO plant model in Figure 4.7. The shape of functions W ∗Y (z),

Figure 4.7: Decoupled Robust Harmonic Control strategy. Weighting func-tions.

W ∗U (z) can be intuitively designed based on the output performance requirements,starting from the frequency responses of continuous-time templates suitable togenerate them in the discrete-time domain. In particular, different performancerequirements associated to disturbance reduction levels in different locations onthe structure can be accounted for. From this point of view, the tuning of an LQ-like, possibly adaptive, algorithm, can turn out to be extremely time consuming,even if tuned on a diagonal system reformulation. Requirement specifications in

72

4.4. Numerical Simulations

terms of steady state attenuation levels and desired transient performance, canbe easily requested in the problem statement through weighting functions of therobust approach. More precisely, different options are available:

• one could design the same weighing function for all the considered outputs,so as to define a uniform performance bound for the entire system;

• one can define as many functions as the number of output considered, mean-ing that it could be possible to give higher penalties to specific outputs whichcan present, for example, more critical characteristics (e.g., pilot and co-pilotseats on an aircraft).

Remark 7. The presented multivariable robust Harmonic Control synthesis hasbeen applied on the decoupled (diagonal) system, but, as already introduced, inprinciple it could be applied directly to the original multivariable system, includ-ing all its cross-interactions. As it can be expected, this alternative approachwould lead to a controller which would be more complex to tune. As an example,the definition of both the uncertainty matrices and the shaping functions can beconsidered a little blurred with respect to a full MIMO system, and more straight-forward if a diagonal representation is obtained.

4.4 Numerical Simulations

In this section, a numerical analysis is provided by means of two simulation ex-periments. The first one is focused on the evaluation of the proposed robust mul-tivariable control on an 3× 3 MIMO square model. Decentralized method are ap-plied and the results are compared with an LQ-based solution based on the samepairing information. The second one wants to demonstrate the capabilities of theproposed approach on a non-square plant which imply to resort to a decouplingstrategy for the control synthesis.

4.4.1 Example - Decentralized MIMO Robust Harmonic Control

Consider the square plant G, evaluated at a specific frequency ω, relating controlinputs U and measurements Y ,

G(jω) =

+0.018− 0.013i +0.250− 0.916i −0.359 + 0.743i

−0.535 + 0.241i +0.075− 0.017i −0.087 + 0.022i

+0.041 + 0.037i −0.044− 0.011i +0.414− 0.217i

.When the SVD is performed by means of the relation

Y = GU = USΣV HU (4.38)

73


matrix Σ obtained,Σ =

[diag1.309, 0.590, 0.319

](4.39)

leading to a condition number γ,

γ =σmax

σmin= 4.09 (4.40)

which could be assumed as good controllability index. The basic idea is that thelargest element of the first column of matrix US is taken and it is paired with thecontrol variable corresponding to the largest element of the first row of matrix V .By computing their modules,

|US | =

0.9573 0.1207 0.2625

0.1066 0.9822 0.1547

0.2686 0.1439 0.9525

(4.41)

|V | =

0.0380 0.9936 0.1062

0.7069 0.0859 0.70200.7063 0.0730 0.7042

, (4.42)

as an illustrative example, the largest element of the first column of |US| is thefirst, which corresponds to the first measured output. By inspecting matrix |V |,two elements can be chosen for a paring selection. The ambiguity produced bySVD is evident. The corresponding RGA-matrix is given by

ΛT =

0.003 1.002 0.006

1.0503 0.0167 0.089

0.078 0.019 1.056

(4.43)

which doesn’t confirm the ambiguity found before, but actually allows to con-clude that the system can be considered as diagonal dominant and a decentralizedcontrol can be applied. In the light of the inferred pairing selection, an equivalentdiagonal T -matrix model can be obtained as

T =

0 0 +0.250 −0.916 0 0

0 0 +0.916 +0.250 0 0

−0.535 +0.241 0 0 0 0

−0.241 −0.535 0 0 0 0

0 0 0 0 +0.414 −0.217

0 0 0 0 +0.217 +0.414

(4.44)

Both the controllers HCLQ and HCH∞ can be applied on the decentralized andoverall T -matrix model. As an example, three different reduction levels are im-posed, targeting, on average, ‖Y1‖ = 0.05, ‖Y2‖ = 0.1 and ‖Y3‖ = 0.2. Arelative uncertainty matrix has been included considering a 10% as maximum

74


Figure 4.8: Decentralized Harmonic Control - Optimal and Robust solutions.Obtained output norm with HCLQ and HCH∞ , designed for the original Tmodel (a) and the diagonal one T ∗ (b).

variation of the T -matrix. Note that in principle, both the optimal and robust de-sign approaches can be applied as such to the original multivariable model, withthe drawbacks explained in the previous Sections.

As shown in Figure 4.8, by using the pairing information obtained from theRGA approach, the obtained output norms with both the implemented controllersslightly deviate from the desired reference. On the contrary, when the two controlstrategies are designed on the original full plant T , the tuning procedure appears tobe effectively more difficult and performance little poorer. In particular, the HCLQcontroller results extremely sensitive to the cross-interaction, although a diagonaldominance of the matrix is evident. The HCLQ controller, instead, presents smalldifferences between the two scenarios, and in general results in a smaller devi-ation, with regard to the LQ-based solution, from the desired reference. Thisconfirms, partially, the advantage expressed in the previous Sections, that is thepossibility to encode directly in the synthesis process the steady reduction levelsand desired transient performance.

4.4.2 A Numerical Experiment - Decoupled Non-Square Plant

When cross-interactions cannot be neglected and in particular when the consid-ered plant is non-square, difficulties in selecting the most suitable input-outputpairings can arise. As described, a way to eliminate the system interactions is todesign a decoupler with the aim to transform the model into a diagonally dominant

75


one. As an example, consider a non-square plant,

GT =

+0.025− 0.019i −0.076 + 0.055i +0.005 + 0.004i

−0.038 + 0.006i −0.091 + 0.084i −0.005 + 0.018i

−0.469 + 0.969i −0.114 + 0.029i +0.148− 0.144i

+0.316− 1.157i +0.095− 0.022i −0.051− 0.014i

+0.041− 0.006i +0.003 + 0.007i +0.092− 0.165i

−0.034− 0.020i −0.008− 0.001i +0.044− 0.136i

(4.45)

A first compensator C can be obtained by means of Algorithm 2, resulting in

C =

6.7898 −0.4615 0.5043

−2.8140 −89.8036 −1.6815

2.2787 −3.5017 1.2513

6.4411 2.6583 0.7618

−8.3535 5.2643 15.5458

8.9681 −10.8680 10.8254

(4.46)

Figure 4.9 illustrates the convergence of some of the C elements. As a result, the

# iteration Algorithm 2

0 10 20 30 40 50-10

-5

0

5

10

15

20

1st element2nd element3rd element4th element5th element6th element

Figure 4.9: Convergence of the 3rd column elements of compensator C.

shaped transfer matrix Gs takes the form,

Gs =

+0.591− 5.519i +6.499− 6.779i −0.000 + 0.000i

−0.000 + 0.000i +8.995− 7.665i −0.000 + 0.000i

−0.309− 0.274i −0.204− 0.504i +2.061− 4.249i

(4.47)

76


which is actually not triangular as expected. Nonetheless, being the correspon-dent RGA matrix close to the identity matrix, a second compensator C can becomputed by means of equation (4.23),

C =

+1.000 + 0.000i −1.340− 1.035i +0.000 + 0.000i

+0.000− 0.000i +1.000 + 0.000i +0.000 + 0.000i

−0.024 + 0.084i +0.042− 0.003i +1.000 + 0.000i

(4.48)

A diagonal system is hence obtained,

G∗ =

+0.591 - 5.517i +0.000− 0.000i −0.000 + 0.000i

−0.000 + 0.000i +8.995 - 7.665i −0.000− 0.000i

+0.000− 0.000i +0.000 + 0.000i +2.061 - 4.245i

(4.49)

and consequently, a diagonal T -matrix model is derived. An application of thedecoupled Robust Harmonic Control design is in detail described and discussedin Chapter 7.

77

CHAPTER5Nonlinear Harmonic Control

As introduced in Chapter 1, in many harmonic control applications nonlinearitiesare included in the control loop by actuator limitations. Actuator dynamics areactually often negligible, and are not considered in the control synthesis, but itcould be the case, however, that the static characteristic of the actuator can vary,e.g. with respect to the input magnitude. As an example, hydraulic actuators couldpresent a non-negligible nonlinear behavior in both the ”low” vibratory load re-gion (because of intrinsic actuator issues), and the ”high” vibratory load region(because of physical saturation). Surprisingly, the analysis of the effects of loopnonlinearities on closed-loop performance have not been considered in the har-monic control literature. In this Chapter, two formal methods and frameworks areintroduced and used as instruments for this type of analysis.

In the first part of the Chapter, the static nonlinearities are accounted for in theT -matrix modeling framework by resorting to the notion of Describing Function.The Harmonic Control law derived from the robust framework discussed in theprevious Chapters is analyzed in the light of the introduced static nonlinearity inclosed-loop. Comments are then addressed, in particular with respect to a com-parison between the robust control law and the optimal LQ-based HC describedin Chapter 3.

In the second part of the Chapter, both the uncertainty and loop nonlinearities

79

Chapter 5. Nonlinear Harmonic Control

are accounted for by resorting to a Linear Parameter Varying (LPV) framework.It will be shown how it can provide a suitable means to account for variationsin the plant such as the ones induced by, e.g, changing conditions or actuatorsaturations. After an initial summary on the main results of LPV control theoryin discrete-time systems, the problem of designing an LPV/H∞ controller for theHarmonic Control problem is discussed. Finally, simulation results are shown andcommented.

5.1 Loop Nonlinearities - Describing Function Method

The Describing Function (DF) method refers to an approximate procedure for ana-lyzing particular classes of nonlinear control problems (e.g., an amplifier with sat-uration, or an element with deadband effects). It is based on a quasi-linearizationprocedure, in the sense of an approximation of the non-linear system by an LTItransfer function that depends on the amplitude of the input waveform. When us-

𝑁𝐿 𝐿(𝑗𝜔)Δ𝑢∗ Δ𝑦

-

Figure 5.1: Closed-loop system with loop nonlinearity. Lur’e form.

ing the describing function method the nonlinearities of the problem are collectedin a block NL as in Figure 5.1. While the input-output behavior of the nonlinearblock is described by its response to sinusoids of different amplitudes and frequen-cies, i.e., A sin(ωt), in the system response the first harmonic only (with the sameinput frequency) is considered. In the light of that, the response can be describedby the gain and phase shift between the input sinusoid and the first harmonic ofthe output. It can be computed analytically or by numerical integration as

Γ(A,ω) = a+ jb (5.1)

where

a =ω

Aπ

T∫0

F (t) sin(ωt)dt

b =ω

Aπ

T∫0

F (t) cos(ωt)dt. (5.2)

80

5.1. Loop Nonlinearities - Describing Function Method

Figure 5.2: Derivation of some classical describing functions.

81


The Describing Function is one of the most widely-applicable methods for de-signing nonlinear systems, and is very used as a standard mathematical tool foranalyzing limit cycles in closed-loop controllers. The technique is of course ap-proximate since it only considers the fundamental frequency. This is justifiedby means of a low-pass assumption, meaning that the linear block in connectionwith the nonlinear one should have a low-pass behavior. In this way, any sus-tained oscillation at the output of the nonlinearity can be filtered by the linearblock and hence it can be argued that the input of the nonlinearity is almost si-nusoidal. Within the limits of this approximation, the method gives conditionsfor finding self-sustained oscillation, by means of the harmonic balance relation,which would verify if there exist a frequency ω and an amplitude A such that

G(jω)Γ(A,ω) = −1 (5.3)

where Γ(A,ω) is the complex number giving the gain and phase-shift of the non-linearity as a function of frequency and amplitude. To graphically describe theoccurrence of a possible limit cycle the Nyquist curve of G(jω) is plotted to-gether with −1/Γ(A,ω). Any intersection between these curves indicated a can-didate limit cycle. A simple but useful application example is the saturation. Theassociated describing function is the following

Γ(A) =

1 A ≤ A1

2

π

(arcsin

A1

A+A1

A

√1−

A21

A2

), A > A1

(5.4)

As in most cases, the Describing Function could not be a function of the frequencyand this simplifies the verification of the harmonic balance equation. Moreover,starting from (5.4), is quite simple to derive the describing functions of similarnonlinearities as shown in Figure 5.2.

5.1.1 A Numerical Experiment - HCLQ vs HCH∞ comparison

A number of describing functions can be computed by particularization of thefunction

Γ(α) =2

π

(arcsinα+ α

√1− α2

)(5.5)

in which the parameter α = U/u defines a peculiar point of the nonlinear char-acteristics. With reference to the chosen characteristic Φ(·), function Γ(·) can bedefined as

Γ(·) =

k1, u ≤ U1

k1 + (k1 − k2)Γ(u, U1), U1 ≤ u ≤ U2

k2Γ(u, U2) + (k1 − k2)Γ(u, U1), u ≥ U2.

(5.6)

82


𝑢𝑖𝑛

𝑢𝑜𝑢𝑡

𝑈1 𝑈2

−𝑈1−𝑈2

𝑀

−𝑀

𝑘1

𝑘2

Figure 5.3: Considered Actuator characteristic Φ(·).

Consider again the discrete-time open-loop T -matrix system

Figure 5.4: Describing function shape based on (5.6).

Y (i) = T (i)U (i) +D(i). (5.7)

When a characteristic of type shown in Figure 5.3 is included and consideredin the control system, unobvious behaviors could arise and affect output perfor-mance. This point can turn into a good chance to evaluate another advantage ofthe robust control framework. With reference to Chapter 1 and Figures 1.6 and5.3, a Monte Carlo simulation procedure as the ones performed in the previousChapters has been produced, before and after the inclusion of function Γ(·) in the

83


0 20 40 60 80 100

Γ(|

|u||)

- L

Q

0

0.5

1

1.5

Time [step]0 20 40 60 80 100

Γ(|

|u||)

- H

∞

0

0.5

1

1.5

Figure 5.5: Describing Function Method. Output of block Γ(‖uk‖) for 10simulations. LQ control results in strong chattering behaviors and limitcycles which are responsible for a consistent degradation of performance.

time [s]0 2 4 6 8 10 12 14 16 18 20

Fou

rier

Coe

ffici

ents

yc(t

) y s(t

)

-0.05

0

0.05

yc(t)LQ y

s(t)LQ y

c(t)H

∞ ys(t)H

∞

time [s]10 10.5 11 11.5 12

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06ZOOM

Figure 5.6: Describing Function Method - Closed-loop simulation with nosaturation included. Fourier coefficients of the output signal. Both the LQ-based and Robust harmonic Control solution doesn’t present any chatteringbehavior.

84


time [s]0 2 4 6 8 10 12 14 16 18 20

Out

put s

igna

l y(t

)

-0.05

0

0.05

HCLQ

HCH∞

time [s]10 10.5 11 11.5 12

Out

put s

igna

l y(t

)

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08ZOOM

Figure 5.7: Describing Function Method - Closed-loop simulation with no sat-uration included. Output signal. Both the LQ-based and Robust HarmonicControl solution doesn’t present any chattering behavior.

control loop. Results are summarized in Table 5.1, while Figure 5.5 shows, withrespect of time, the output of block NL.

The continuous-time nominal model used to compare results of HCLQ andHCH∞ solutions is based on rotor-blade vibration problem (find details in Chap-ter 6), and is the following

G(s) =− 0.00581s5 − 0.298s4 − 0.321s3 + 1.31s2 + 1.46s+ 1.727

0.045s5 + 1.082s4 + 1.95s3 + 2.99s2 + 1.95s+ 1.09(5.8)

while the considered parameters for the describing function Γ(·), with referenceto Fig 5.7 are

U1 = 1; U2 = 5; k1 = 0.5; k2 = 1.5 (5.9)

As is evident, the LQ-based control law shows a non-neglibile tendency to resultin strong chattering behaviors and limit cycles which are responsible for a consis-tent degradation of performance. A similar trend is slightly apparent in the H∞solution too, but the small oscillations found in this case dissolve quickly till thesteady value. As a confirm of the previous results a Simulink implementation ofthe control system has been performed, and results are shown in Figure 5.8 andFigure 5.9.

As a conclusion, the robust approach in the Harmonic Control context, asso-ciated with a model able to account for loop nonlinearities as the one described

85


Table 5.1: Describing Function Example - Simulation results: performance ofthe LQ and of the H∞ when saturation is included in the control loop.

No SAT SAT includedE[‖Y ‖]− HCLQ 0.0399(23) 0.547(32)E[‖Y ‖]− HCH∞ 0.0311(21) 0.096(13)max ‖Y ‖ − HCLQ 0.0448 0.938max ‖Y ‖ − HCH∞ 0.0379 0.251

time [s]0 2 4 6 8 10 12 14 16 18 20

Fou

rier

Coe

ffici

ents

yc(t

) y s(t

)

-0.5

0

0.5

yc(t)LQ y

s(t)LQ y

c(t)H

∞ ys(t)H

∞

time [s]10 10.5 11 11.5 12

Fou

rier

Coe

ffici

ents

yc(t

) y s(t

)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8ZOOM

Figure 5.8: Describing Function Method - Closed-loop simulation with satu-ration included. Fourier coefficients. The chattering and degraded perfor-mance is evident with the LQ-based solution.

86

5.2. An LPV Harmonic Control framework

time [s]10 10.5 11 11.5 12

Out

put s

igna

l y(t

)

-2

-1

0

1

2ZOOM

time [s]0 2 4 6 8 10 12 14 16 18 20

Out

put s

igna

l y(t

)

-2

-1

0

1

2

HCLQ

HCH

∞

Figure 5.9: Describing Function Method - Closed-loop simulation with satu-ration included. Output signal. The chattering and degraded performanceis evident with the LQ-based solution.

above, present an important, often not considered, advantage. Performance degra-dation and control effort are minimized, allowing to avoid, for example, fatigueissues of mechanical control components. While this is already a valuable benefit,in the following part of the Chapter it will be shown as the nonlinear character-istic of the control actuator can be dealt in a different framework, able to, in away, transform an analysis instrument as the describing function, into a formalsynthesis procedure.

5.2 An LPV Harmonic Control framework

The aim of this second part of the Chapter is to evaluate the advantages of anLPV control approach to the design of harmonic control laws. The reader couldrefer to classical references like the works of Shamma in [102], [104], [105], andothers [84], [85], [15] for early analytical LPV works and [66] [98] for reviewpapers. As a starting motivation, in the Harmonic Control literature no distinctionswere made between predictable and unpredictable changes in the system matrix,as it is generally assumed that most plant perturbations can be taken care of, forexample, by some adaptation mechanism. While this can be certainly possible, itcan be claimed that predictable changes in the T -matrix can be more adequatelyhandled within a gain-scheduling framework. In view of this, Chapter 1 already

87


introduced a possible LPV modeling for the frequency domain system.In classical gain-scheduling approaches (see [64], [97], [103]), the controller

design is generally based on the local descriptions of the overall nonlinear sys-tem, so that these laws are guaranteed to satisfy specifications where the localmodels are valid. The obtained local controllers, however, could then be mapped,scheduled, or both together to obtain a non-local controller for the nonlinear sys-tem, with the advantage that the local properties of the nonlinear system can beapproximated by linear system properties. Hence, the local controller synthesiscan be performed using the well established linear system theory. In brief, LPVcontrol methods are able to lead to a controller for a generic nonlinear plant bypatching together a collection of linear controllers, which can be blended (e.g.,via switching or interpolation) according to the available online measurements.Often, moreover, it is desirable that time-variations of the plant do not impact onthe performance and in particular on the stability of the closed-loop system. Inthis regards, an approach to design robust controllers for LPV systems, as in [50],[89], [5], [53], has been developed, with the aim to handle time-varying uncertainparameters in the form of an optimization problem which would, e.g., minimizethe L2-induced norm of the closed-loop system (see the works in [85], [86] andreferences therein).

5.2.1 LPV via LMI robust control design

Thanks to some recent works on robust linear control ( [3], [29], [30], [101]) andits connection with the LMI theory and tools (see [69], [36], , [6], [18], [38]),robust control design methods have been extensively studied and evolved, leadingto a well established theory used for many control applications. As referred in[95] and [100], this approach is able to account for nonlinearities of a systemby considering them as parameter uncertainties, on which basis it can be built arobust controller.

Unfortunately, robust approaches can lead to a very conservative behavior ifsystem nonlinearities are large or when high-performance are desired. An answerto this problem is represented by the LPV theory, which allows both to model thenonlinearities and to make the controller performance varying with some intro-duced parameters. Since a decade ago, hence, linear parameter varying theory isincreasingly being used to extend classical robust control techniques to a largerclass of systems, keeping at the same time the instruments of linear systems.Among the studies [37] , both theoretical and practical, developed in the morerecent years we recall for the works in [95] and [100].

Consider the T -matrix system as introduced in Chapter 1, found as a combi-nation of two variation contributions,

T (θ1, θ2) =(I +Wm∆

) [TRe θ1 + T Im θ2

], (5.10)

88


and the equivalent state-space representation of the system dynamics obtained inChapter 3, [

Yk+1

Uk+1

]= AT

[Yk

Uk

]. (5.11)

An LPV state-space description of the system can be obtained byXk+1

Zk

Yk

=

AT (θ1, θ2) BD(θ1, θ2) BU (θ1, θ2)

CZ(θ1, θ2) DZ,D(θ1, θ2) DZ,U (θ1, θ2)

CY (θ1, θ2) DY,D(θ1, θ2) DY,U (θ1, θ2)

Xk

Dk

Uk

(5.12)

with Zk representing the robust performance output, including the uncertainty.The aim of the LPV robust synthesis is to minimize the H∞ norm of the system,while ensuring at the same time internal stability (see [2, 19]). In other words,the H∞ control problem for the LVP system (5.12) consists in finding an LPVcontroller K(θ1, θ2) such that the closed-loop system results to be quadraticallystable and that, for a given positive real γ∞, the L2-induced norm of the operatormapping the disturbance D into the performance output Z is bounded by for allpossible trajectories of parameters (θ1, θ2).

Following [95], the robust LPV method considered requires the following as-sumptions:

1. DY,U (θ1, θ2) = 0;

2. BU , CY , DZ,U and DY,D are parameter independent.

Remark 8. In general it could be the case thatBU is not satisfying the introducedassumption. To avoid this, a strictly proper filter can be added on the controlinput, as shown in [44].

Proposition 5.2.1. Given the polytopic structure of the system (5.12), an LPVcontroller can be designed in the form[

AK(θ1, θ2) BK(θ1, θ2)

CK(θ1, θ2) DK(θ1, θ2)

]=

nθ∑i=1

ρi(θ1, θ2)

[AK,i BK,i

CK,i DK,i

]. (5.13)

with coefficients ρi(θ) defined as

ρi(θ) =

2nθ∏j=1

|θj − Cc(Θi)j |

2nθ∏j=1

|θj − θj |, (5.14)

Cc(Θi) being the complementary of θi, i.e., the ith vertex of the polytope Θ.

89


Definition 5.2.1. ACL is called convex hull of A(θ1)CL · · ·A

(θnθ )

CL if for eachACL ∈ ACL there exist coefficients ρi(θ) such that (5.13) and (5.14) hold true.

From the theory of linear dissipative systems, it is possible to find such a con-troller by means of the (see again [2, 19]) Bounded Real Lemma for discrete-timesystems (DT-BRL). The associated LMI-based problem, as shown in [100], ingeneral, results in an infinite set of LMIs to solve, but it is simplified when thepolytopic approach is used, i.e., the parameter dependence enters in a linear wayin the system definition. In view of this, the LPV/H∞ problem consists in findinga common quadratic Lyapunov function (CQLF) P = PT > 0 and a minimal γ∞that solve the LMI problem at each vertex of the polytope Θ.

As a result, the closed-loop LPV system can be written as

XCL,k+1 = ACL(θ1, θ2)XCL,k +BCL(θ1, θ2)Dk

YCL,k = CCL(θ1, θ2)XCL,k +DCL(θ1, θ2)Dk(5.15)

where the closed-loop matrices are (being each of them function of (θ1, θ2))

ACL =

[AT +BUDKCU BUCK

BKCU AK

]BCL =

[BD +BUDKDY,D

BKDY,D

]

CCL =

[CD +DZ,UDKCU

DZ,UCK

]DCL =

[DZ,D +DZ,UDKDY,D

].

(5.16)

Two fundamental Lemma follow.

Lemma 5.2.1 (Quadratic stability). If ACL is the convex hull of A(θ1)CL · · ·A

(θn)CL

then ACL in quadratically stable iff there exists a matrix P = PT > 0 such thatA

(i)TCL PA

(i)CL − P < 0.

Lemma 5.2.2 (DT-Bounded Real Lemma - BRL). Given the discrete-time closed-loop LPV system (5.15) and letting ACL the convex hull of A(θ1)

CL · · ·A(θnθ )

CL , ifthere exists a common quadratic Lyapunov function (CQLF) P = PT > 0 and aconstant γ∞ > 0 such that[

ATCLPACL − P + CTCLCCL ATCLPBCL + CTCLDCL

BTCLPACL +DTCLCCL −γ2

∞I +BTCLPBCL +DTCLDCL

]< 0

(5.17)for all vertices of the polytope Θ, then the LPV system is quadratically stable andthe H∞ norm of the LPV system is smaller than γ∞.

90


5.2.2 Global LPV/H∞ solution

For a polytopic set of parameters, the solution of the previous defined LMI prob-lem at each vertex of the polytope Θ will give a controller. The convex combina-tion of these controllers results in the global controller for the LPV system. Giventhat, an important result is here summarized.

Proposition 5.2.2. Following [3], under the Assumptions 1 and 2, there exists again-scheduled controller

XK,k+1 = AK(θ1, θ2)XK,k +BK(θ1, θ2)YK,k

Vk = CK(θ1, θ2)XK,k +DK(θ1, θ2)YK,k(5.18)

which ensures, over all parameter trajectories, that

• the close-loop system is internally quadratic stable;

• the L2-induced norm of the operator mapping D into Z is bounded by γ∞,

if and only if there exist a scalar γ∞ and two symmetric matrices (P ,SP ) whichsatisfy the 2nθ+1 LMIs (computed offline):[

NP 0

0 I

]TM1

[NP 0

0 I

]< 0, i = 1 · · ·nθ (5.19)

[NS 0

0 I

]TM2

[NS 0

0 I

]< 0, i = 1 · · ·nθ (5.20)[

P I

I SP

]≥ 0 (5.21)

with

M1 =

APAT − P APCTZ BD

CZPAT −γ∞I + CZPC

TZ DZ,D

BTD DTZ,D −γ∞I

M2 =

ATSPA− SP ATSPBD CTZBTDSPA −γ∞I +BTDSPBD DT

Z,D

CZ DZ,D −γ∞I

(5.22)

where matrices A, BD, CZ , DZ,D are evaluated at the ith vertex of the polytopeΘ. NP and NS denote the bases of null spaces of (BTU , D

TZ,U ) and (CY , DY,D)

respectively.

91


Once matrices P ,SP and γ∞ are obtained, the controller is reconstructed ateach vertex of Θ, as shown in [44]. In this way, the LPV robust controllerK(θ1, θ2) can be obtained as a convex combination of the vertices controllersK(i), as shown in equation (5.13).

Remark 9. Note that, by reformulating the original T -matrix system by introduc-ing the free-update model, equations (5.19) - (5.20) can be relaxed and replacedas shown in the Appendix B. Actually, the (bilinear) BMI generally obtained forthe solution of the LPV-robust control problem, is slightly simplified when staticstate-feedback controllers are included, instead of a generic dynamic one. In thiscase, following [100], the Real-Bounded-Lemma for discrete-time system can bedirectly applied by means of a change of variables.

In view of the above consideration, by looking at the inequality[ATCLPACL − P + CTCLCCL ATCLPBCL + CTCLDCL

BTCLPACL +DTCLCCL −γ2

∞I +BTCLPBCL +DTCLDCL

]< 0

(5.23)with, being K the static state-feedback controller,

ACL = AT +BUK (5.24)

an equivalent formulation of the inequality of (5.23) is given (see Appendix B) by−P ATCLP 0 CTCLPACL −P PBCL DT

CL

0 BTCLP −γ2∞I 0

CCL 0 DCL −I

< 0. (5.25)

Unfortunately, being (5.25) not yet an LMI in P ,K, it can be transformed intoone by using the adjoint variables SP = P−1 and WS = KP−1 = KSP . By preand post multiplying by the symmetric positive definite matrix

SP 0 0 0

0 SP 0 0

0 0 I 0

0 0 0 I

(5.26)

the following LMI is obtained,−SP SPA

TCL 0 SPC

TCL

ACLSP −SP BCL 0

0 BTCL −γ2∞I DT

CL

CCLSP 0 DCL −I

. (5.27)

92


Once matrices SP ,WS , and γ∞ are obtained, again, the controller is reconstructedat each vertex of Θ. In this way, similarly to what stated previously, the LPVrobust controller K(θ1, θ2) is obtained as a convex combination of the verticescontrollers K(i), as shown in equation (5.13).

5.2.3 LPV control synthesis with a-posteriori guarantees

The above described synthesis procedure can be used also for analysis purposes.Actually the control solution derived from (5.23) could present strongly conser-vative behavior if the uncertainty considered is too large, not compatible with thegoal pursued in this context, e.g., the complete reduction of the harmonic distur-bance. The synthesis procedure has been thus modified, by splitting the problemin two synthesis/analysis steps. In order to simplify the control synthesis whileremaining at the same time adherent to the LPV control theory, the followingstrategy can be used:

1. the original problem is split into the respective LTI problems at the verticesof the polytope Θ;

2. Each LTI control problem can be solved in the robust framework, obtainingan LTI/H∞ controller at each vertex of the polytope Θ;

3. The LPV/H∞ controller can be computed by using relation (5.13), coher-ently with system definition;

4. Quadratic stability and LPV robust performance are then a-posteriori veri-fied on the resulting closed-loop LPV system (by translating the LMI into aSDP problem solved with SeDuMi using the Yalmip interface, see [69]).

In order to exploit the above strategy, system (5.7) can be firstly manipulatedwith the aim to confine the dependence on the parameters (θ1, θ2) in the dynamicmatrix AT (θ1, θ2) only. This is possible by introducing a first order dynamics onthe control variable Uk and augmenting the state vector as Uk+1

Xf,k+1

vk

=

[Af Bf

Cf 0

] Uk

Xf,k

ηk

(5.28)

with (being α, β, γ suitable scalar constants and ηk the new manipulable variable)

Af =

[αI βI

γI 0

], Bf =

[0

I

], Cf =

[0 I

]. (5.29)

The system is then modified by introducing ”fast” first order dynamics on the

93


Figure 5.10: Generalized P −K −∆ plant structure.

control variable Uk,

Uk+1 = Aε1 Uk + vk (5.30)

and by including a strictly proper filter

Xf,k+1 = Aε2 Xf,k + ηk

vk = Aε3 Xf,k(5.31)

where ηk ∈ R2×2 and matrices Aε1 , Aε2 , Aε3 are simply defined (being I the2× 2 identity matrix), as:

Aε1 = ε1I, Aε2 = ε2I, Aε3 = ε3I

ε1, ε2 1, ε3 ' 1.(5.32)

Therefore an equivalent formulation of the system (5.7), in state-space form, is

Yk+1 = TAε1Uk + TAε3Xf,k +Dk

Uk+1 = Aε1Uk +Aε3Xf,k

Xf,k+1 = Aε2Xf,k + ηk

Zk = Yk

(5.33)

where ηk represents the new manipulable variable in the control system.System (5.33) can thus be written in a compact form as

Xk+1 = ATfXk +Bηηk +BDDk

Zk = CTfXk +Dηηk +DDDk(5.34)

94


where

ATf = ATf (θ1, θ2) =

0 T (θ1, θ2)Aε1 T (θ1, θ2)Aε30 Aε1 Aε30 0 Aε2

Bη =

0

0

I

BD =

I00

CTf = [I 0 0] , Dη = DD = 0.

(5.35)

Note, again, that reformulating the system (5.7) into form (5.33) allows to confinethe dependence on the parameters θ1,θ2 into the dynamic matrix ATf only.

Based on the representation of the uncertainty as in (1.41), the generalizedP −K −∆ plant structure associated with the i-vertex system is found as

P −K −∆ =

A Bη BD∆ BD

CZ∆ DZ∆η DZ∆w∆ DZ∆D

CZ DZη DZw∆ DZD

=

=

0 T (i)Aε1 T (i)Aε3 0 I I I

0 Aε1 Aε3 0 0 0 0

0 0 Aε2 I 0 0 0

0 T (i)WmAε1 0 0 0 0 0

0 0 T (i)Wm 0 0 0 0

I 0 0 0 0 0 0

,

(5.36)

with self-explanatory entries with reference to Figure 5.10. The mixed sensitiv-ity problem is then solved by designing discrete shaping functions WZ(z) andWη(z), respectively on the performance output and the control variable.

Each robust synthesis problem can be formulated as

Find K(T (i))

s.t.∥∥∥∥GZD WZ

GηD Wη

∥∥∥∥∞≤ γ∞,

(5.37)

where GηD is the control sensitivity function from the disturbanceD to the controlvariable η in the nominal case T (i) at each vertex. Once obtained the robustcontrollers at each vertex of polytope Θ, the LPV controller K(θ1, θ2) is found as

95


a combination of four controllers,

K(θ1, θ2) = ρ(θ1, θ2)K(θ1, θ2) + ρ(θ1, θ2)K(θ1, θ2) + · · ·+ρ(θ1, θ2)K(θ1, θ2) + ρ(θ1, θ2)K(θ1, θ2)

θ1, θ2 ∈ Θ,∑ρ(θ1, θ2) = 1.

(5.38)

The feasibility of the LPV problem described by plant (5.36) and control (5.38),is verified by using the Lemma 5.23, and bounds on the closed-loop H∞ normare obtained. Note that the feasibility requirement assures the stability and perfor-mance of the closed-loop system, but no guarantees are obtained on the optimalityof the LPV control law based on the independent robust synthesis at each vertex(θ1, θ2).

5.3 Simulation results and discussion

In this section two numerical examples are used to show the main properties of theLPV control solution and its advantages over a classical LTI/LQ-based approachand its adaptive variant. In particular some numerical scenarios show how a stan-dard LTI control law could get into instability if the nonlinearities of the actuatorare not taken into account. On the contrary, other simulated scenarios prove howthe performance degradation is avoided, or at least strongly limited, when a LPVcontroller is used in the control system.

5.3.1 2-vertices Affine-LPV control

A first numerical example is derived from an affine T -matrix model, when thevertices considered for the LPV control synthesis are two instead of four, and thereal/imaginary part of the transfer function G are varying simultaneously with asame rate. Referring to (5.38) the obtained controller will be in the form,

K(θ) = K(θ) + ρ(θ)(K(θ)−K(θ)). (5.39)

The reference on the disturbance attenuation is set to 95%, and shaping functionWZ and Wη are defined accordingly, as well as matrices Q and R of the LTI/LQbased controller.

Consider a nominal matrix T , and the extreme variations (the vertices of thepolytope Θ) T 0 and T 1 as

T =

[1.05 −2.75

2.75 1.05

]

T 0 =

[0.42 −1.10

1.10 0.42

]T 1 =

[1.68 −4.40

4.40 1.68

].

(5.40)

96

5.3. Simulation results and discussion

Consider the uncertainty due to change plant conditions given by

Wm =

[0.0293 −0.0318

0.0318 0.0293

]. (5.41)

The application of the Bounded-Real Lemma verifies the stability properties of theclosed-loop system. In this scenario the inclusion of a LTI control solution leadsto an unfeasible problem when inequalities from (5.2.1) and (5.2.2) are tested. Onthe contrary, by including in the loop the LPV control law (5.38), a feasible prob-lem is obtained and a finite bound γ∞ on the closed-loop H∞ norm is achieved.

Clearly, the expected behavior of the closed-loop system is easily explained.The inclusion in the loop of the main characteristics of the actuator has a directinfluence on the matrix T . While a LTI control solution cannot account for thisvariation, the LPV control law is able to update its intrinsic knowledge of thematrix on the basis of the evolution of the parameter ρ, which in turn is a function,as shown, of the input norm. A Monte Carlo study has been further carried out

100 150 200 250 300 350

dist

urba

nce

norm

0

1

2

3

4

5

time0 50

ρ(th

eta)

0

0.2

0.4

0.6

0.8

1

Figure 5.11: Example realization of disturbance variation - 2-vertices Affine-LPV control.

for the feedback system, with experiments performed as follows:

• 500 random values of the normalized uncertainty ∆ have been generated, sothat each initialization of the matrix T accounts for the uncertainty due tochanging conditions, configurations and other similar causes;

• a random realization of the variation of baseline disturbance D is consid-ered in each of the Monte Carlo simulations, on a horizon which allows to

97


0 100 200 300 400 500

×104

-5

0

5

10

LPV control

HCLQ control

Monte Carlo Occurrence #

0 100 200 300 400 500

Out

put n

orm

0.02

0.04

0.06

0.08

0.1ZOOM

Figure 5.12: Simulation experiment - Monte Carlo study: steady output norm‖Y ‖.

reach the steady state. The implementation of disturbance variation is ofpiecewise-linear type

Dk+j = Dk + α Drand, 1 ≤ j ≤ Nh (5.42)

where α is a constant and Drand = U(−1, 1), while Nh represents the timeinterval where the disturbance D is not varying.

The results are depicted in Figures 5.11 and 5.12, where the steady-state mag-nitude of the output/disturbance ratio ‖Y ‖/‖D‖ (computed numerically over afinite interval, so very large values denote diverging output) is shown. As can beseen, instability does occur in the LTI/LQ control case. This is due to the lack ofadaptation by the controller on the basis of the actuator’s information. Of coursethis conclusion was already known since the BRL resulted in an unfeasible prob-lem, so that no guarantees, neither on the closed-loop performance, neither on itsstability properties, were achieved.

A second nominal matrix T can be then considered. This is based on numericalvalues extracted from a simplified single rotor blade model of the Agusta A109helicopter, the details of which are given in the following Chapter. Matrix T andits polytopic bounds are defined as

T =

[0.232 −0.050

0.050 0.232

]

T 0 =

[0.113 −0.025

0.025 0.113

]T 1 =

[0.487 −0.106

0.106 0.487

],

(5.43)

98


while the uncertainty Wm is given by

Wm =

[0.214 −0.428

0.428 0.241

]. (5.44)

In contrast with the previous scenario, the application of both (5.23) and (5.2.1)leads to a feasible problem when both the LTI/LQ and the LPV/H∞ controllersare included in the control loop. The main difference is instead focused on thedegradation of closed-loop performance. Figure 5.13 illustrates this behaviorquite clearly: even though in this case the LTI controller does not lead to sta-bility issues, it exhibits a far less satisfactory degree of performance robustnesswith respect to the LPV one, due to the lack of adaptation with respect to the vari-ation of the disturbance norm and direction. The LPV controller, instead, shows ahigher robustness degree when the disturbance is varying. This can be appreciatedmore clearly by looking at Figure 5.14, where a single Monte Carlo realization isshown and the evolution of disturbance and controlled output is compared whendisturbance D is varying.

0 100 200 300 400 5000

0.1

0.2

0.3LPV controlHC

LQ control

Monte Carlo Occurrence #0 100 200 300 400 500

Out

put n

orm

0.02

0.04

0.06

0.08

0.1ZOOM

Figure 5.13: Simulation experiment (performance) - Monte Carlo study:steady output norm ‖Y ‖.

5.3.2 4-vertices Polytopic-LPV control

In this section a four vertex polytopic T -matrix model is considered. Similarly tothe previous numerical examples, the robust control law at each vertex (θ1, θ2) isdesigned on a target attenuation of 95% and the considered uncertainty Wm is setas in Table 5.3.2. Weighting functionsWZ(z) andWη(z) are accordingly defined,as well as matrices Q and R of the LQ-based (invariant and adaptive) controllers.

99


outp

ut n

orm

50 100 150 200 250 300 350 400 450 500 550 6000

10

20

30

40

Uncontrolled OutputLTI controllerLPV controller

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

outp

utno

rmno

rmal

ized

time

LTI controllerLPV controller

Figure 5.14: Disturbance variation during one Monte Carlo simulation andrespective controlled output norm ‖Y ‖. LTI/LPV comparison.

The scenarios simulated in the following consider two different dimensions ofpolytope Θ (Θsmall and Θlarge), and experiments are performed by means of aMonte Carlo study. In particular Nr random values of the normalized uncertainty∆ have been generated, so that each initialization of the matrix T accounts for theuncertainty due to changing conditions, configurations and other similar causes.Moreover, a random realization of the varying baseline disturbance D is consid-ered in each of the Monte Carlo simulations, on a horizon which allows to reachthe steady state condition. More precisely, two different values for the horizonNh(Nh,s,Nh,l) are considered for the implementation of disturbance variation, whichis of a piecewise-linear type

Dk+j = Dk + δ Drand,k, (5.45)1 ≤ j ≤ Nh,

with δ constant and Drand = U(−1, 1), while Nh represents the length of thetime interval over which the disturbance D is not varying.

The nominal system matrix T is defined as

T =

[−0.286 0.089

−0.089 −0.286

], (5.46)

and the associated relative uncertainty matrix as

Wm =

[0.33 0.53

−0.53 0.33

]. (5.47)

100


As for the initializing parameters used for the experiments, they’re summarizedin Table 5.3.2. Results are depicted in Figures 5.15 - 5.20 in terms of steady-state

Table 5.2: Initializing parameters

Nominal T -matrix[T

(1,1)

N,N T(2,1)

N,N

]= [−0.2855 0.0892]

Uncertainty[W

(1,1)m W

(1,2)m

]= [0.33 0.53]

Weighting functions Wz(z) =z − 0.7304

z − 0.9765Wη(z) = 1

Polytope bounds Θsmall : θ ∈ [0.85 1.15]× [0.85 1.15]Θlarge : θ ∈ [0.45 1.65]× [0.45 1.65]

Dist. variation rate Nh,s = U(10, 30)Nh,l = U(40, 100)

RLS Initialization λ0 = 0.98, p = 0.99Vk = αI2, α = 0.5

T kN,N = βI2, β = 0.2

LQ weights Q = 3.8 · 102I2, R = I2Monte Carlo Simul. Nr = 500

I/O ratio ro/i computed (numerically over a finite interval) as ‖Yk‖/‖Dk‖. Foreach Monte Carlo simulation the following indicators are registered:

• IAVG, computed as the average input/output ratio on the simulated scenario;

• IMAX , computed as the maximum value of ro/i on the simulated scenario;

• ITR, computed as the average settling time when disturbance is varying.

In the following figures the resulting indicators are boxed to compactly show thecomparison between the three applied control strategies.

First scenario - Narrow Polytope Θsmall

Figure 5.15 summarizes the results in a scenario characterized by a slow variationof disturbance and system parameters θ1, θ2 in Θsmall. As in Table 5.3.2, thehorizon Nh is set as Nh,l, and variation of baseline disturbance is as in (5.45). Asshown in the figure, no difference is evident in the average input/output ratio withall the tested control laws, both the adaptive and invariant control laws show a sim-ilar behavior, very close to the LPV reference. Same considerations holds for theconvergence time to the steady output when disturbance is varying. The similaraverage results could be in some way expected because the variability of matrixT is comparable with the uncertainty ∆ included in the loop. Slightly differentconclusions can be drawn by looking at Figure 5.16, where the rate of variation

101


LPV RLS/LQ LQ

0.04

0.05

0.06

0.07

0.08

IAVG

LPV RLS/LQ LQ

0.04

0.05

0.06

0.07

0.08

0.09

IMAX

LPV RLS/LQ LQ

2.5

3

3.5

4

4.5

ITR

Figure 5.15: First scenario - Monte Carlo statistics for the IAV G, IMAX , ITR

indicators.

of the disturbance Dk and system parameters θ1, θ2 in Θsmall is increased to in-terfere with the RLS identification which might not be able to converge alwaysto the actual value of matrix T . In this case, results in Figure 5.16 show how theadaptation mechanism is degrading the performance when compared with the in-variant controller, in particular in terms of the second indicator IMAX . Moreover,as in the previous scenario, the relatively restricted variation of matrix T (withthe included uncertainty) seems to confirm the goodness of a classical invariantsolution, and at the same time the reduced advantage of the LPV solution.

LPV RLS/LQ LQ

0.05

0.06

0.07

0.08

IAVG

LPV RLS/LQ LQ0.04

0.06

0.08

0.1

0.12

0.14

IMAX

LPV RLS/LQ LQ

3

3.2

3.4

3.6

3.8

ITR

Figure 5.16: First scenario - Nh,s. Monte Carlo statistics for the IAV G,IMAX , ITR indicators.

Second scenario - Large Polytope Θlarge

The more evident advantage of a parameter-varying solution is shown when thepredictable variation of matrix T is significant, e.g.much larger than the consid-ered uncertainty ∆. In this scenario, characterized by an evolution of parametersθ1, θ2 in Θlarge, with Nh = Nh,l, the Monte Carlo simulation shows that in atleast one occurrence the invariant LQ-based controller results in instability, mean-ing that the input/output ratio becomes even higher than unity at steady-state, sothat the vibration is enhanced instead of being reduced. The role of the identifi-cation is clear in this case, and results obtained in the literature, e.g.in [87], are

102


here confirmed, as the adaptation mechanism helps to avoid the instability issuesof the invariant solution. Figure 5.17 illustrate the results. As shown, the twostable solutions are comparable in terms of average steady attenuation, but betterresults are achieved with the LPV solution, which is not depending on the onlineidentification of system parameters.

LPV RLS/LQ

0.04

0.05

0.06

0.07

0.08

0.09

0.1

IAVG

LPV RLS/LQ

0.05

0.1

0.15

IMAX

LPV RLS/LQ

1.5

2

2.5

3

3.5

4

4.5

ITR

Figure 5.17: Second scenario - Nh,l. Monte Carlo statistics for the IAV G,IMAX , ITR indicators.

When a high variation rate (Nh = Nh,s) of parameters θ1, θ2 is again con-sidered, and consequently of matrix T , e.g.a scenario in which a rotorcraft is ap-proaching a complex manoeuvre or a critical change of flight condition, differentresults are obtained between the adaptive and the gain-scheduling controls. Bothcontrol strategies encounter a little performance degradation. With reference toFigure 5.18, although the similar average performance achieved by both the con-trollers, the RLS/LQ solution presents numerous outliers and in general a shift inthe average of the maximum input/output ratio. This means that the differencebetween the actual matrix T and the identified T k could be a serious cause forperformance degradation. Again, this behavior can be explained considering theneeded convergence time of RLS, the influence of which is getting higher whenthe interval Nh is being reduced. In view of this, although the goal for reducingperiodic vibrations at steady-state, the LPV solution seems to be an interestingoption for reducing the efficiency loss of the control action during transients, es-pecially when the variation of matrix T is sudden and significant. An exampleof what happens in the mentioned scenario is shown in Figure 5.19, where theLPV solution seems to be a little more insensitive to disturbance variations. Wecan conclude that the gain-scheduling approach is able to avoid some of the prob-lems of both the invariant (instability) and adaptive (performance degradation) HCvariants.

Inclusion of measurement noise

So far we have considered a disturbance variation as in (5.45). As said, this ismotivated by the steady scenario where Harmonic Control is forced to operate,

103


LPV RLS/LQ0.04

0.06

0.08

0.1

IAVG

LPV RLS/LQ

0.05

0.1

0.15

0.2

0.25

IMAX

LPV RLS/LQ

2.8

3

3.2

3.4

3.6

ITR

Figure 5.18: Second scenario - Nh,s. Monte Carlo statistics for the IAV G,IMAX , ITR indicators.

time100 200 300 400 500 600 700 800

outp

ut/d

istu

rban

ce r

atio

0

0.1

0.2

0.3

0.4

0.5LPVRLS/LQ|| Dk || / 20

Figure 5.19: Second scenario - simulation of parameter varying (LPV) andadaptive (RLS/LQ) control laws.

104


meaning that performance is evaluated on the steady disturbance reduction witha stationary baseline disturbance Dk. Fluctuations of this disturbance from cyclek − 1 to cycle k could also be expected, as generally Harmonic Control imple-mentations consider a sliding-window average of the measured output to updatethe control action. In order to understand the role of fluctuations on the controlsystem performance, the baseline disturbanceDk can be described by introducinga white-noise term ηN ∼ N (0, σ2), so that

Dk+j = Dk + δ Drand(k) + ηN (k + j), (5.48)1 ≤ j ≤ Nh,

with σ set to 5% of Dk. This means that in each interval Nh the disturbanceDk, previously considered constant, is affected by a measurement noise whichcan degrade both the identification procedure and the final performance. Resultsare depicted in Figures 5.21,5.22 and 5.20. As shown, the conclusions previously

time100 200 300 400 500 600 700 800

outp

ut /

dist

urba

nce

ratio

0

0.1

0.2

0.3

0.4

0.5LPVRLS/LQ||Dk || / 20

Figure 5.20: Simulation of parameter varying (LPV) and adaptive (RLS/LQ)control laws, including noise on the baseline disturbance.

drawn for the noise free case hold also for the current scenario, but performancedegradation is evident in both the proposed solutions, with slow or fast variationof disturbance D and parameters (θ1, θ2), cases that present much more similarresults than the previous described scenarios. As a conclusion, are evident theadvantages of using an LPV approach to design Harmonic Control laws. Fromthe comparison between the three different control strategies, the role of the adap-tation mechanism has been in some way clarified. The adaptive and LPV ap-proaches allow to account for the variability in the system response, and thus ofthe plant matrix T , but while the first is dependent from an identification proce-dure, the second splits the variability into a predictable and unpredictable changes.

105


LPV RLS/LQ

0.06

0.08

0.1

0.12

IAVG

LPV RLS/LQ

0.05

0.1

0.15

0.2

IMAX

LPV RLS/LQ3

3.5

4

4.5

ITR

Figure 5.21: Monte Carlo statistics for the IAV G, IMAX , ITR indicators forhigh rate disturbance variation, including noise.

LPV RLS/LQ

0.06

0.08

0.1

0.12

0.14

IAVG

LPV RLS/LQ

0.1

0.15

0.2

IMAX

LPV RLS/LQ

3.2

3.4

3.6

3.8

4

ITR

Figure 5.22: Monte Carlo statistics for the IAV G, IMAX , ITR indicators forlow rate disturbance variation, including noise.

Instability issues are avoided with both the gain-scheduling and adaptive variants,but a slight advantage of the first strategy has been proved especially with sharpand significant variations of the plant matrix T . In this sense, based on the similarresults of the two stable control laws, the LPV approach could surely be placedas a valid alternative to the existing adaptive HC variant already used in manyapplications.

106

Part III

Applications

107

CHAPTER6Harmonic Control for the Rotor-induced

Vibration Control Problem

The main rotor of a helicopter generates the thrust and the moments necessary tofly the aircraft. However, it is also responsible for a number of side effects, mainlymechanical vibrations which are transmitted to the fuselage (and passengers). Theeffect on the fuselage of the generated vibratory loads is essentially periodic, beingthe rotor angular rate essentially constant, and the resulting angular frequencycan be proven to be equal to NΩ (classically denoted as N /rev), where N is thenumber of blades and Ω is the rotor angular frequency, also denoted as 1/rev,meaning ”per revolution”. To deal with this problem, in the years research hasfocused on improving the characteristics of the helicopter through the addition ofsuitably scaled and phased harmonics to the collective and cyclic controls of themain rotor. In view of this, the problem can effectively be formulated as the oneof compensating a periodic disturbance of a known frequency NΩ acting at theoutput of an uncertain (possibly time-periodic) linear system. It is not a surprisethat in the Harmonic Control context the primary source of literature comes fromthe helicopter research community.

This Chapter is devoted to present an application for the Robust HarmonicControl approach proposed in Part II. In particular the helicopter vibration prob-

109

Chapter 6. Harmonic Control for the Rotor-induced Vibration ControlProblem

lem is discussed with reference to a well known out-of-plane blade model, pro-posed originally by Johnson in [54], which retains the main characteristics of afull blade dynamics. In the first part of the Chapter a brief overview and back-ground on the general vibration control problem in helicopters’ applications ispresented, highlighting common points and differences between the different ap-proaches developed in the years. Then, the role of uncertainties is clarified withthe aim at introducing the robust framework to deal with situations when changesof the flight condition or configuration would cause unknown plant modificationswhich are typically handled in the literature by estimate methods. In the last partof the Chapter results are discussed in the light of a comparison between the clas-sical HC optimal control solution and the proposed robust framework.

6.1 Background

Aside from its established use in military application, helicopters’ role in the civilsociety has gradually spread since the ’60s. Thanks to their peculiarities and han-dling qualities, nowadays they are employed in several extreme conditions suchas rescue operations or emergency air medical assistance. Urban communities,however, have often expressed great dislike of noisy aircraft, so that police andpassenger helicopters can be considered still unpopular. The high noise and vi-bration levels produced are the most important factors affecting the passengersand crew comfort. As an example, in the helicopter cabin, the sound pressurelevels reach on average from 20 to 30 dB more than fixed wing aircrafts do (referto [96] and [78] for details). Moreover, noisy tonal components are typically inthe frequency range between 500 Hz and 4500 Hz, to which the human ear is mostsensitive.

Vibration loads are also a primary cause for the degradation of the structuralintegrity. In this respect, they’re responsible for non negligible increase in thefatigue of the mechanical components and a reduction of the effectiveness of theon-board equipment. Vibrations originate chiefly from the main rotor, the tail ro-tor, the engines and other machinery [113], and arise especially in forward flight.The main rotor, which generates the aerodynamic forces necessary to fly the air-craft, is rigidly mounted on the roof structure of the helicopter via connectingmembers, struts, through which vibrations are transmitted to the fuselage. Thefrequencies of the disturbance derived from the main rotor are integer multiplesof the rotor frequency. As a contextual known source of vibration, rotor bladesin rotation represent a major responsible because of the particular aerodynamicloads they are subjected to. In this respect, the intensity of the vibratory loads de-pends on the speed of the helicopter. Low speed transition flight (e.g., landing ortaking off manoeuvres) and high speed flight are the most critical. At low speed,moreover, the severe vibration level is essentially due to abrupt changes in the

110

6.2. Active Vibration Control Methods

blades load, a condition which happens when a rotor blade passes in the neighbor-hood of the vortices created by a previous blade. This is generally referred to asBlade-Vortex-Interaction (BVI), whose induced vibration is drastically reducedfor moderate and high flight speed conditions, when the vibrations are causedmainly by the main rotor.

As for the control perspective, a number of techniques have been developed toreduce noise and vibration inside helicopter cabins. They can be broadly dividedinto ”passive” and ”active” vibration control techniques. The main difference be-tween the two categories is in the employment of active devices such as actuatorsand loudspeakers.

Among the passive control category, a first approach, usually performed dur-ing the vehicle design phase, involves designing blades and fuselages in sucha way that their natural vibration frequencies lie away from the rotor harmonicfrequencies. This tuning helps avoiding blade and fuselage resonances with therotor harmonics. As shown by Nguyen in [83], however, this method requiresextensive modeling and testing of the blade and fuselage structural characteristicsthroughout the aircraft development phase. Rotor isolation is another commonlytested vibration reduction technique. Isolating materials, such as pads of rubberor springs are placed between the vibrating system and its supporting structure toreduce the force transmitted from the vibrating system to the supporting structure.Alternative passive techniques make use of mechanical components as tuned-massabsorbers and tuned-mass dampers. The first can be attached to a structure in or-der to reduce its dynamic response. As a control parameter, the frequency of theabsorber is tuned with the aim at translating the exited frequency resonance phasefrom the structural motion. A tuned-damper works in a similar manner but in-cludes an additional mechanical damper. Absorbers and dampers are tuned to thefrequencies to be removed and are placed at specific locations in the fuselage. De-spite its simplicity, the reduction obtained with such methods is limited (both inthe amount of energy absorption and frequency operation range) and involves anadditional weight. Moreover, due to their fixed design, tuned-mass absorbers anddampers are not able to adapt to operating condition changes (see [83]). Althoughtuned mass absorbers and isolators are passive devices, both can be implementedas semi-active and active systems as well. Moreover, passive vibration controllersmay also be heavy, reducing the available payload of the helicopter, and cannotremain in tune for all flight conditions [92].

6.2 Active Vibration Control Methods

Active vibration reduction techniques were introduced to overcome the limitationsimposed by passive control solutions and to increase the reduction of both noiseand vibrations. Typically, the acoustic noise signals or vibrations are measured by

111


Figure 6.1: HC architectures in the rotor frame. Higher Harmonic Control andIndividual Blade control implementation.

the sensors mounted on the fuselage. Then they are processed by a controller andused to generate a signal to drive the actuators, able to produce a vibration fieldas close as possible to the one to be removed but opposite in phase, producing adestructive interference as result. A classification of the existing active methodscan be carried out according to the actuators location:

1. Rotor Location - Individual Blade Control (IBC) and Higher Harmonic Con-trol System (HHC);

2. Fuselage Location - Active Control of Structural Response (ACSR).

Both the HHC and IBC methods aim at attenuating the vibrations by means ofactuators placed on the rotor. In this way, the vibrations are attenuated beforethey propagate to the fuselage. On the contrary, in the ACSR method, only thevibrations transmitted to the cabin are considered. Nonetheless, all these systemsrely on the Harmonic Control concept, presented extensively in both Part I andPart II. Actually, from a control theoretic perspective, both the approach groupsare similar in that the controller all implement the Harmonic Control algorithm(recall references [26] and [87]).

6.2.1 HHC/IBC technologies

Higher Harmonic Control systems, whose scheme is shown in Figure 6.1, aimsat reducing the vibrations that affect the main rotor before they propagate to thecabin ( [56] [26]). Transducers are mounted at key locations in the fuselage tomeasure the residual vibrations while the actuators are typically mounted in the

112

6.2. Active Vibration Control Methods

non-rotating swash-plate. The actuators excite the blades in order to create bladeoscillations which, when properly phased, reduce the vibrations induced by therotor during flight conditions. Typical actuators for this system are hydraulic orelectromagnetic actuators.

Based on the Harmonic Control approach, as introduced in the previous Chap-ters HHC is based on the principle of inter-harmonic coupling, due to the peri-odicity of rotor dynamics in forward flight (see [54]). Its name derives from theaddition of suitably phased and modulated higher harmonic components, such as,e.g.,N /rev, (N±1)/rev and (N±2)/rev, to the rotor controls, in order to achieve areduction of theN /rev vibratory components. Moreover, by recognizing that rotornoise and vibration are fundamentally precipitated by the same phenomena (e.g.,the unsteady blade aerodynamic loads and motion in forward flight), researchon rotor noise reduction techniques has exploited the basic principles normallystudied in the context of vibration control (see [123]). Exhaustive and completereviews and literature collections can be found in Malpica ( [75]) and Fan ( [40])works.

As for the Individual Blade Control technique, it is in many aspects very simi-lar to HHC, except for using different locations of the employed actuators ( [26]).While the HHC technique uses actuators placed in the non-rotating frame, theIBC approach employs actuators located in the rotating frame, allowing to controleach blade independently. A scheme of the IBC is shown again in Figure 6.1. Theactuators can be placed between the hub and the blades or directly on the blades,according to different implementations of the system. As reviewed by Kesslerin [57], often IBC refers to the actuation of each rotor blade independently witha broadband actuator. Alternatively ( [48], [94] ), it might be implemented withtrailing-edge flaps, individually actuated pitch-links, or active blade twist.

In addition to the application for vibration reduction, IBC with HHC algorithmmay also be used to reduce noise as shown in [112], [21] and to improve the over-all performance of a helicopter (see [61], [68], [49]). Practically, however, forthe simultaneous noise/vibration reduction problem, experience has shown it isgenerally not possible to suppress rotor low-frequency noise and vibratory loadssimultaneously at the involved frequencies, due to significant phasing discrepan-cies between independent optimal control requirements for each problem. See therelated works in [123], [28], [82] and [17] for details.

6.2.2 ACSR controls

Contrary to the HHC and IBC techniques, whose aim is to obtain a global re-duction of the vibration disturbance, the ACSR approach, introduced by Westlandhelicopters, is designed to reject the vibration in specific locations on the struc-ture. Currently, ACSR is one of the most successful helicopter vibration reduc-tion methods. Sensors are placed at key locations in the fuselage, where minimal

113


Figure 6.2: Overview of possible HC architectures for baseline vibration at-tenuation.

vibration is desired (e.g. the pilot and passenger seats). Recently, the ACSR tech-nology has been incorporated in modern production series such as the AugustaAW-101. A scheme of this technique is shown in Figure 6.2. Performance analy-sis of ACSR using a coupled rotor and flexible fuselage model was carried out byCribbs, Friedmann, and Chiu [33], which showed ACSR has low power demand.Moreover, since no modification to the rotor is required, ACSR also may havepotentially fewer airworthiness issues than HHC [88].

6.3 Overview of the rotor out-of-plane blade model

A brief overview of the governing equations of the out-of-plane bending dynamicsmodel of an helicopter rotor blade, which has been used in this study, is summa-rized in the following as derived by Johnson in [54]. Although very simple, thismodel retains the main characteristics of the full blade dynamics. More precisely,linear out-of plane bending can be described by the (partial differential) bendingequation; to reduce the model to a finite-dimensional representation, a modal ex-pansion for the blade deflection is introduced and truncated to a finite number ofmodes. Then, it is possible to show that the partial differential equation is equiv-alent to a set of ordinary differential equations forced by the aerodynamic loadacting on the blade. For the purpose of this study, the pitch angle of the blade isconsidered as input signal, while the vertical shear force at the root of the bladeis the selected output. For a more detailed discussion, the reader is referred to theabove reference or to [12] and [70].

114

6.3. Overview of the rotor out-of-plane blade model

Linear out-of-plane bending can be described by the equation

M(r, t) = E(r)I(r)d2z(r, t)

dr2, (6.1)

whereM(r, t) is the total bending moment (inertial, centrifugal and aerodynamic)acting at radial station r ∈ (0, R) of the blade at time t, z(r, t) is the out-of planeblade deflection and E(r)I(r) defines the local bending stiffness. Let now m(r)be the mass per unit length of the blade and introduce a modal expansion for z

z(r, t) =∞∑i=1

ηi(r)qi(t), (6.2)

where ∫ R

0

ηi(r)ηj(r)m(r)dr = 0, ∀i 6= j,

andηi(R) = R.

Then, it is possible to show that the partial differential equation (6.1) is equivalentto the set of ordinary differential equations

Iqk(qk(t) + ν2kqk(t)) =

∫ R

0

ηk(r)FZ(r, t)dr, k = 1, 2, . . .

where νk is the natural frequency associated with the kth bending mode, FZ(r, t)is the aerodynamic load acting at radial station r and

Iqk =

∫ R

0

m(r)η2k(r)dr.

The pitch angle of the blade (ϑ) and the vertical shear force at the root of the bladeare respectively considered as input and output signals, defined as

SZ(t) =

∫ R

0

FZ(r, t)dr −∫ R

0

m(r)z(r, t)dr =

=

∫ R

0

FZ(r, t)dr −∑k

qk(t)

∫ R

0

m(r)ηk(r)dr.

Remark 10. Note that the described procedure is applicable when a finite numberof modes is considered, so that (6.2) reduces to

z(r, t) =

Kq∑i=1

ηi(r)qi(t). (6.3)

In most applications, Kq ranges from 1 to 4. For Kq = 1 the case of a rigid beamis considered.

115


Rotor angular frequency Ω 40 rad/sRotor radius R 5.5 mMass per unit length m 5 kg/mStiffness EI 1.8 · 103 Nm2

Lift-curve slope a 5.7 rad−1

Lock number γ 7.84Blade chord c 0.3 m

Table 6.1: Mechanical an aerodynamical characteristics of the considered rotorblade.

Given that, the dynamics of one blade of a helicopter rotor can be describedby a time-varying linear state-space system of the form

x(t) = A(t)x(t) +B(t)u(t)

y(t) = C(t)x(t) +D(t)u(t), (6.4)

where x ∈ Rn is the state, y ∈ R is the output and u ∈ R is the input, defined asthe pith angle ϑ(t). The dimension n of the state vector equals twice the number ofout-of-plane bending modes q that are taken into account in the modal expansionfor the blade dynamics (n = 2q). In this work, a rotor blade with the characteris-tics given in Table 6.1 has been considered. In particular, the obtained equationsdescribe a system which turn out to be time-periodic with period T/N , where Nis the number of blades and T is the period of one rotor revolution. The azimuthangle ψ = Ωt can be then defined and used as an independent variable. Moreover,the equations vary according to the value of the advance ratio µ, defined as

µ =Va

ΩRb. (6.5)

where Va is the true airspeed of the aircraft and Rb the rotor radius, so that model(6.4) can be written as

x(t) = A(t, µ)x(t) +B(t, µ)u(t)

y(t) = C(t, µ)x(t) +D(t, µ)u(t). (6.6)

As derived in [54] and [12], in hover or in vertical flight (Va = 0), the overallmodel is time-invariant; otherwise, in forward flight Va 6= 0, it turn out to be time-periodic. In this respect, as better shown in the following, it is possible to extendand connect the analysis of the time-invariant and the time-periodic systems bymeans of the lifting technique presented in [121].

In view of this, the general rotor blade model can be put in a slightly differentform, allowing a direct conversion into a harmonic state space model. System

116

6.4. Helicopter Periodic Models

matrices can actually be expanded as Fourier series, so that the state matrix A(ψ)can be given as

A(ψ) = A0 +

Nq∑q=0

[Aqc cos(qNψ) +Aqs sin(qNψ)] (6.7)

with constant matrices A0, Aqc and Aqs, but with only A0 used for constant co-efficient approximations. The same result holds for matrices B, C and D. Inparticular, the control matrix B(ψ) is obtained assuming for the pitch control ofeach blade the form

ϑi(ψ) = ϑ0 +

Nq∑q=0

[ϑqc cos

(ψ +

2πi

N

)+ ϑqs sin

(ψ +

2πi

N

)]. (6.8)

Referring to [70], it can be proven that the time-periodic matrices have only twoharmonics and are given by

A(ψ) = A0 +A1c cos(ψ) +A1s sin(ψ)

B(ψ) = B0 +B1s sin(ψ) +B2c cos(2ψ)

C(ψ) = C0 + C1c cos(ψ) + C1s sin(ψ) + C2c cos(2ψ)

D(ψ) = D0 +D1s sin(ψ) +D2c cos(2ψ)

(6.9)

Note that the response of the main rotor to perturbations on the control inputsϑ is strongly dependent on the rotor advance ratio µ, as introduced above in theSection.

6.4 Helicopter Periodic Models

Helicopter rotor systems in forward flight are quite famous examples of natu-rally linear time-periodic (LTP) ones. First theories on LTP systems dates back toMathieu and Floquet reasearches, respectively on the natural modes of vibrationof lakes with elliptic boundaries, and the so called Floquet Theory on the stabilityand analysis properties of this type of systems. Moreover, many engineering prob-lems (see among others [65], [8] [117], [93], [35]) showed periodic behavior andrepresentation of the system. Satellite altitude control [59], electrical and thermaldiffusion systems with elliptical boundary conditions, or diffraction of acousticand electromagnetic waves around ellipsoidal objects [93] are just few examples.

A very basic LTP system is represented by Mathieu’s equation in (6.10), of-ten used as a representative model for understanding the general behavior of thistype of systems. In the Mathieu’s equation ωr and α are the so called pumpingfrequency and pumping amplitude, respectively,

mx+ 2mζωnx+mω2n [γ − 2α cos(ωrt)]x = F. (6.10)

117


If α = 0, the equation reduces to be time-invariant. The main difference betweenthe LTI and LTP systems is that the first, driven by a complex exponential input,produces a steady-state output with the same frequency content ω and a differentamplitude and phase; the second produces instead a steady output which includesnot only the forcing frequency, but also harmonics of frequency ω ±Nωr at am-plitudes yn and and phases φn respectively. Hence, the response of an LTP system

Figure 6.3: General LTP schematic representation.

to impulsive inputs, in the time domain, is observed to be dependent on the timethat the input is applied. As a result, the impulse response function of an LTPsystem is time-dependent, and a single impulse response function cannot provideenough information on the behavior of the system [35].

Both Floquet and Hill observed the mentioned issue in providing a simpleinput/output operator for LTP systems, and suggested to use complex Exponen-tially Modulated Periodic (EMP) signals as test signals for LTP systems [120].In particular, an EMP signal is defined as the complex Fourier series expansionof a periodic signal with the fundamental frequency ωr, modulated by a complexexponential signal,

u(t) =

∞∑k=−∞

uke(s+jkωr)t. (6.11)

Different methods have been proposed to provide input/output operator maps forLTP systems. Methods that are all based on the system state-space model andthe time-dependent impulse response function (see [13], [10], [11], [9], [119]).The most common referenced operator map in this context is represented by theHarmonic Transfer Function (HTF).

118

6.4. Helicopter Periodic Models

6.4.1 LTI Representation of LTP systems: Harmonic Balance

As known, the response of an LTI system to a sinusoidal excitation is sinusoidal,with the same frequency and different phase and amplitude. In the LTP case,response to sinusoidal excitations can be calculated as well. However, the period-icity of the state-space matrices induces important differences. The steady stateresponse of LTP systems to a complex exponential signal u(t) = ejωrt comprisesan infinite number of harmonics of the pumping frequency ωr, resulting in one-to-many mappings, which are difficult to analyze without a proper formalism, as theHTF. It was originally introduced by Wereley and Hall, particularly for helicopterapplications [119]. Roughly speaking, it can be shown that an LTP system mapsan EMP signal to an EMP signal as a one-to-one map, as well as an LTI transferfunction maps a complex exponential signal to a complex exponential one.

The mapping between the EMP input/output spaces with the HTF is givenin (6.12). In this equation, G0(s) is the transfer function between the input atfrequency ω and the output at frequency ω, while G1(s) represents the transferfunctions between the inputs at ω ± ωr to the output at ω.

...

Y(s− jωr)

Y(s)

Y(s+ jωr)

...

=

. . ....

......

· · · G0(s− jωr) G−1(s− jωr) G−2(s− jωr) · · ·· · · G1(s) G0(s) G−1(s) · · ·· · · G2(s+ jωr) G1(s+ jωr) G0(s+ jωr) · · ·

......

.... . .

...

U(s− jωr)

U(s)

U(s+ jωr)

...

(6.12)

For a system of order n, with input u ∈ Rm and outputs y ∈ Rp, the transfermatrix results in G0(s) ∈ Cm×p and the resultant HTF G ∈ Cm(2Nh+1)×p(2Nh+1),where Nh is the number of harmonics included.

6.4.2 Rotor Vibration Control - T -matrix definition

As derived in [75], [71], [72], the T -matrix model can be defined and related tothe elements of the HTF for the helicopter model. According to the definition ofthe control input vector u, the main rotor is considered to be subject to a steady-state harmonic control input whenever the control vector is constant. This impliesthat to define the T -matrix for the helicopter, it should be necessary to study theresponse of the periodic helicopter models to an EMP input with s = 0, meaningthat just the computation of the input/output operator G(0) is needed. Consideringthe LTP system, and a constant input u(t) = u0, the vector U is defined by

UT =[· · · 0 0 uT0 0 0 · · ·

](6.13)

and the steady-state response Y of the LTP system by

Y = G(0)U . (6.14)

119


This leads to the expression [y−N

yN

]=

[G−N,0

GN,0

]u0. (6.15)

which, by converting the N/rev harmonics of the output from exponential totrigonometric form, yields the T -matrix relation[

yNc

yNs

]=

[Re(GN,0) Im(GN,0)

−Im(GN,0) Re(GN,0)

] [uNc

uNs

]. (6.16)

6.5 Helicopter rotor blade simulation study

The design of active control strategies (see Figure 6.4) for the reduction of rotor-induced vibrations is considered as one of the more demanding control problemsin rotorcraft applications. As said, major issues derived from vibrations are mostlythe discomfort to the passengers and crew, and undesirable effects, such as degra-dation of the structural integrity, increase in the fatigue of the components andreduction of the effectiveness of the on-board instrumentation. In the recent lit-erature, various HC architectures have been considered and used as the best ap-proaches for the design and the implementation of this type of controls. Basically,the aim is to attenuate the vibratory components at the blade-passing frequency, inthe rotor hub (before fuselage propagation) loads or in the fuselage accelerations(after propagation) by adding suitably phased harmonic components to the rotorcontrols ( [43, 72, 73], [58], [57], [42]). To this end, very good knowledge of thefrequency response relating the control inputs to the output measurements at thedisturbance frequency is necessary for a proper functioning of the control system.As for the implementation of vibration control laws, most of the literature derivedfrom the so-called T -matrix algorithm both in its original Linear Time Invariant(LTI) and adaptive (recursive) variants, which are based on an LQ-like frequency-domain controller, coupled with a suitable offline or online identification of theelements of the T -matrix when this is varying or is uncertain. Obviously uncer-tainty in the elements of the T -matrix, due to varying flight condition and chang-ing configuration, has to be considered. As shown, offline or online estimationis used in the literature to this purpose, and least-squares error methods like theRLS algorithm seem the natural choice for this task, given the linear nature of themodel. From the experimental results available in the literature, however, the needfor adaptive control does not appear to be a strong requirement for the successfuldeployment of the control system (particularly so when control of the structuralresponse, rather than rotor control, is considered, see the discussion of this pointby Kessler in [57, 58]). To the contrary, vibration control systems based on theT -matrix approach seem to exhibit a fair degree of robustness. In this respect,

120

6.5. Helicopter rotor blade simulation study

Figure 6.4: Overview of possible Higher Harmonic Control declinations.

it was surprising that the problem of robustness of active rotor control systemshas received very little attention. Given that, in the following a simulation studyis performed with the inclusion of an uncertainty due to a varying, or unknown,advance ratio µ.

The aim is to illustrate the main properties of the Robust Harmonic Controlscheme proposed in Part II, with a particular focus on the Vibration Control prob-lem presented in this Chapter. The T -matrix considered, as already introduced,is based on numerical values extracted from the described simplified single rotorblade model of the Agusta-A109 helicopter (the parameters of which are given inTable 6.1). As shown, although very simple, the model describes the out-of-planebending dynamics of a helicopter rotor blade, as derived in [54] and retains themain characteristics of the full blade dynamics. An analysis based on both theSISO and MIMO descriptions of the system is proposed.

6.5.1 Robust HHC-SISO

As shown in previous Sections, the resulting out-of-plane model can be written as

x(t) = A(t, µ)x(t) +B(t, µ)u(t)

y(t) = C(t, µ)x(t) +D(t, µ)u(t),(6.17)

where system matrices A,B,C,D are (periodic, with period T = 2π/Ω) func-tions of time and of the dimensionless advance ratio parameter µ representingthe forward speed of the helicopter. The model considered here is the lineartime-invariant system obtained by averaging the time periodicity, and choosingµ = µ = 0.27. In order to get the T -matrix, one can evaluate the frequency

121


response of the model, leading to the constant matrix

µ = 0.27 =⇒ T =

[0.264 0.033

−0.033 0.264

], (6.18)

A 95% attenuation of the (unit norm) disturbanceD is set to be requested as nom-inal performance. The tuning procedure in the nominal case follows what statedin Chapter 3. Then a Monte Carlo simulation has been performed to comparethe performance of the (invariant) LQ-based and robust H∞ solutions. Reflect-ing the illustrated results of Chapters 3 and 4, the same metrics have considered,such as steady-state output norm, ‖Yk‖ including its variance σ2(‖Yk‖). More-over, the settling time Ts is computed on basis of output variations with a tol-erance ≤ 1 · 10−4. An uncertainty matrix Wm has been included in the rotor-

Mag

nitu

de (

dB)

-20

-15

-10

-5

0

5

10

10-2 10-1 100 101 102

Pha

se (

deg)

135

180

225

270

315

360

Bode Diagram

Frequency (rad/s)

Figure 6.5: Transfer function of the LTI out-of-plane rotor blade model as afunction of µ.

blade T -matrix model by definition of the range of the advance ratio µ, so thatbounds µmin ≤ µ ≤ µmax could be defined. In this study, bounds are chosen asµmin = 0.25, µmax = 0.29, and the considered LTI transfer function is repre-sented in Figure 6.5 considering a model (6.7) with modes q = 2, leading to anassociated relative uncertainty

Wm = 2 · 10−3

[8.55 0.75

−0.75 8.55

]. (6.19)

As stated above, the norm and the settling time to converge to the steady-state

122


0 100 200 300 400 500

Out

put n

orm

0

0.05

0.1

||Y|| - HCLQ||Y|| - HCH

∞


0 100 200 300 400 500Set

tling

tim

e

0

5

10

Figure 6.6: Example 2, Monte Carlo study: settling time and H2 norm of thesteady-state output - LQ controller and H∞ controller.

solution are compared, and the results are shown, respectively, in Figure 6.6, con-sidering a moderate level of uncertainty (10%) to focus on the closed-loop per-formance of the control systems. As a result, Figure 6.6 shows how both thecontrollers can provide a comparable attenuation level. The obtained results arequite similar the the numerical example provided in Chapter 3. In the same way,the same conclusions can be addressed by means of Figure 6.6, from which it isevident that the cost of increased robustness is a slightly slower response in dy-namic terms: on average the settling time of the robust control system is longer.Moreover, the results obtained with reference to different uncertainty radii are re-ported in Table 6.2. The comparison has been extended to a structured version of

Table 6.2: Example 1, Monte Carlo study: performance of the LQ and of theH∞ (both structured and unstructured) controllers.

Uncertainty 10 % 30 % 50 %avg ‖Y ‖]HCLQ 0.0586(28) 0.0884(22) 0.149(23)avg [‖Y ‖](Hs

∞) 0.0534(23) 0.0851(19) 0.148(12)avg [‖Y ‖](Hu

∞) 0.0482(27) 0.0696(24) 0.123(17)max ‖Y ‖(LQ) 0.0652 0.0958 0.176max ‖Y ‖(Hs

∞) 0.0682 0.0932 0.165max ‖Y ‖(Hu

∞) 0.0531 0.0661 0.128Ts(LQ) 3.32 3.48 3.64Ts(H

s∞) 14.28 13.66 16.92

Ts(Hu∞) 4.16 4.97 5.89

the robust controller. Evidently, the expected conclusion can be confirmed, as afully parameterised formulation of the robust synthesis turns out to be extremely

123


important in terms of both the obtained performance and the dynamics. Table 6.2confirm that for a given level of steady state performance, the cost of robustnessis a slower dynamic response (the larger the uncertainty, the slower the response)and that better results can be obtained by resorting to a fully parameterised H∞controller.

6.5.2 A multivariable HHC application

In this second experiment, the considered T -matrix is characterised by 2 inputsand 3 outputs and is therefore composed by six submatrices representing differentinput/output channels. The nominal matrix T , obtained by evaluating the modelfor the given nominal flight condition, is the following,

T =

0.348 0.038 0.221 0.024

−0.038 0.348 −0.024 0.221

0.232 0.025 0.339 0.037

−0.025 0.232 −0.037 0.339

0.174 0.019 0.332 0.036

−0.019 0.174 −0.036 0.332

,

and the relative uncertainties associated with the three system outputs Y1, Y2, Y3,defined consistently with the definition in equation (4.30), are given by

W 1m =

[0.103 0.028

−0.028 0.103

], W 2

m =

[0.068 0.019

−0.019 0.068

], W 3

m =

[0.051 0.014

−0.014 0.051

].

On the basis of the above uncertain model, two control scenarios are considered.The first one aims at attenuating vibrations on the whole set of outputs to morethan 95%, while the second one separates the required performance between out-puts Y1 and Y2 (again 95% attenuation required) and Y3 (only 90% attenuationrequired), in order to mostly penalize specific outputs because considered morecritical in terms of vibration. Thus, two weighting functions have been intro-duced,

W(low)Y (z) =

0.99z − 0.968

z − 0.779W

(high)Y (z) =

0.95z − 0.949

z − 0.887, (6.20)

for which continuous-time equivalent frequency responses are represented in Fig-ure 3.4. As for WU (z), no weight on the control action has been included in theproblem. Finally, in order to compare the resulting performance, a second LQ-based tuning is obtained guaranteeing a similar level of nominal attenuation, inboth the considered scenarios. With reference to Figure 6.7, Figure 6.8 and Ta-ble 6.3, LQ weight matrices Q and R are defined, for the first and second scenario

124


respectively, as

Q = blkdiag(5.23 · I2, 2.28 · I2, 3.16 · I2) R = 0.1 · I4Q = blkdiag(4.22 · I2, 2.31 · I2, 5.49 · I2) R = 0.1 · I4

. (6.21)

For the feedback system (subject to a unit norm disturbance) a Monte Carlo studywas carried out by randomly choosing 500 values for the normalized uncertainty∆. In the following, results of this Monte Carlo procedure in terms of steady-stateattenuation are depicted. Figure 6.7 and Figure 6.8 show the steady-state attenu-

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

||Y1|| - HHC

LQ

||Y1|| - HHC

H∞

0 50 100 150 200 250 300 350 400 450 500

Out

put N

orm

0

0.05

0.1

0.15

||Y2|| - HHC

LQ

||Y2|| - HHC

H∞


0 50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

||Y3|| - HHC

LQ

||Y3|| - HHC

H∞

Figure 6.7: Closed-loop performance comparison in the first scenario.

ation level for each output. The same figures show also the difference between thetwo control laws in terms of settling time Ts to reach the steady-state (calculatedin terms of discrete time steps), and clear evidence is posed upon the conservativeproperty of the H∞ regulation in this sense. In both scenarios a comparable levelof attenuation is obtained with both controllers, but it is also apparent that theLQ-like controller fails to find a correct trade-off to get similar levels of attenua-tion for each of the measured outputs. The proposed robust control design showsanother substantial advantage over a classical LQ-based synthesis: consideringa general ”large” MIMO system, defining weighting functions Wy that resumespecific control objectives seems to be much more convenient than iterating dif-ferent Q and R matrices until control requirements are being satisfied. Table 6.3confirms these considerations, and moreover shows how the cost of increased ro-bustness is a slightly slower response in dynamic terms. Similarly to the previousSection, the comparison has been extended also to a structured version of the H∞controller. To this purpose, the synthesis procedure has been modified on the ba-sis of the structure constraint, meaning that the cost function is optimized with

125


0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

||Y1|| - HHC

LQ

||Y1|| - HHC

H∞

0 50 100 150 200 250 300 350 400 450 500

Out

put N

orm

0

0.05

0.1

0.15

||Y2|| - HHC

LQ

||Y2|| - HHC

H∞

Monte Carlo # Occurrence0 50 100 150 200 250 300 350 400 450 500

0

0.05

0.1

0.15

||Y3|| - HHC

LQ

||Y3|| - HHC

H∞

Figure 6.8: Closed-loop performance comparison in the second scenario.

respect to 20 independent parameters instead of 40, as in the fully parameterizedformulation. Table 6.3 summarizes the results obtained in the first scenario.

Based on these results, the H∞ approach could be beneficial in the sense ofreducing the need for adaptation in the operation of the HHC system, which wouldmake the system itself easier to implement and operate, while allowing a morepredictable closed-loop behavior both in terms of stability and performance.

Table 6.3: Simulation results. HHCLQ vs HHCH∞ .

Scenario 1 Scenario 2HHCLQ HHCuH∞ HHCsH∞ HHCLQ HHCuH∞ HHCsH∞

avg ‖Y1‖ 0.088 0.043 0.075 0.052 0.034 0.048max ‖Y1‖ 0.149 0.099 0.0151 0.105 0.095 0.110avg ‖Y2‖ 0.041 0.037 0.044 0.058 0.049 0.056max ‖Y2‖ 0.068 0.082 0.087 0.125 0.113 0.115avg ‖Y3‖ 0.017 0.033 0.023 0.096 0.106 0.102max ‖Y3‖ 0.054 0.069 0.075 0.135 0.152 0.147Ts 3.30 7.98 9.31 2.48 7.61 10.03

126

CHAPTER7Harmonic Control for the Structural

Vibration Control Problem

In this chapter, an application of the MIMO-Robust Harmonic Control is shownstarting from the experimental facility used in the Friendcopter project (find moredetails in [32]). The actual physical system is used as an illustrative and interest-ing validation example of the control strategies proposed in II. In particular, thelarge dimensions of the system make central the role of decoupling techniques ap-plied on the original identified multivariable structure. Indications on input-outputcross-interactions are first evaluated and then compensated in order to design asuitable multi-channel robust controller.

In the first part of the chapter, the experimental set-up is described. Detailsare given in terms of actuator and sensor location on the structure, the identifieddisturbance spectrum and the acquisition system. As a result, a large multivariabletransfer matrix is obtained and used for control design, which is in detail discussedin the central part of the Chapter. Concluding remarks and comments are finallyaddressed.

127

Chapter 7. Harmonic Control for the Structural Vibration Control Problem

7.1 Experimental set-up description

The employed set-up, illustrated in Figure 7.1, is an actual fuselage of an AgustaA109 MKII helicopter, lacking of some devices like blades, tail rotors, pumps,internal materials etc., but equipped with the most important elements of the mainrotor and structural links. To better simulate on-flight conditions, an aerodynamicbrake system is mounted (see Figure 7.2). Motion of the main gearbox is producedby means of two electric motors which drive it in place of the helicopter turbines.Although the lack of some of the noise-generating devices (e.g. pumps and tail

Figure 7.1: Friendcopter project: employed fuselage mock-up. Agusta A109MKII fuselage and mounted aerodynamic brake system. The setup is lack-ing of some of the noise-generating devices (e.g. pumps and tail rotors).

Figure 7.2: Friendcopter project: mounted aerodynamic brake. The system isintroduced to simulate on-flight conditions.

128

7.1. Experimental set-up description

rotors), the current configuration allows to reproduce the disturbance tones com-monly generated by the main rotor gearbox [32]. As for the connection gearbox-fuselage, a set of rigid struts and an aluminum anti-torque plate, mounted abovethe cabin roof, has been applied. In this way, all the loads generated by the rotorare transmitted to the anti-torque plate instead of the struts. A typical shape of theanti-torque plate is shown in Figure 7.3. The generated vibrations are transmittedto the fuselage mainly through the anti-torque plate and the rear struts, elementswhich shall be given priority in mounting of control actuators, in view of design-ing an active control system. For the considered application, suitable actuators

Figure 7.3: Anti-torque plate shape example. The connection gearbox-fuselage is based on a set of rigid struts and an aluminum anti-torque platemounted above the cabin roof.

are represented by piezoelectric patches. Indeed, they’re not bulky and do not sig-nificantly affect the mechanical structure. Their location were decided on a laserscanner analysis basis (see [32] for details). A total of 40 actuators were placedand split on the two rear struts (8 on the left one and 8 on the right one) and onthe anti-torque plate (12 on the left side and 12 on the right side). No actuatorwas instead placed on the front struts because no significant vibration field wasdetected in the part of the roof corresponding to their structural attachments tothe cabin. Figure 7.4 resumes the overall leftside actuators location. As far assensors are concerned, accelerometers have been used. More specifically, a totalof 18 sensors have been placed on the cabin roof (9 per side), in correspondenceto the points of structural attachment of the anti-torque plate and the rear struts, aswell as in other locations characterized by high vibration levels (see Figure 7.5).A model of the structure has been identified, based on datasets collected on theavailable facility, balancing the historical measured data and the experiments cho-sen for the vibratory test bench. With the aim to replicate the vibratory loadsduring flight conditions, the aerodynamic brake system has been tuned to mini-mize the error between the vibration spectra collected on the experimental facilityand the corresponding spectra measured in flight conditions. Figure 7.6 illustratedthe results, showing a fair qualitative system description.

129


Figure 7.4: Actuators location on one side of the anti-torque plate and on therear struts.

Figure 7.5: Sensors location on the left side of the cabin. Accelerometers A1-A2 are located on the anti-torque plate structural attachment. Accelerome-ter A9 is located on the rear strut structural attachment.

The spectra reported in Figure 7.6 characterize the disturbances which the con-trol system is called to counteract. Indeed, besides the large low frequency (< 200Hz) rotor harmonics, there are significant peaks in the speech frequency range,

130

7.2. Control Design - Multi-Channel Harmonic Control

that are especially annoying for the human ear (in the case of manned vehicles).In this specific case the higher frequency tones namely are at 1599 Hz, 1785 Hz,2400 Hz, and 4250 Hz ( [32]). The attenuation of these high frequency tones isgenerally the main objective of active vibration control in helicopters.

Figure 7.6: Measured cabin noise spectra: test bench (solid line) and flightdata (dashed line). Speech-band peaks at frequencies 1599 Hz, 1785 Hz,2400 Hz, and 4250 Hz.

7.2 Control Design - Multi-Channel Harmonic Control

With the aim at reducing the acoustic noise and vibrations in the cabin, at leastfor a specific frequency and reduced volume of the helicopter fuselage, a concretegoal is to reduce vibrations in the volume where the passengers are supposedto sit. This can be achieved through active control techniques based on HarmonicControl concept. The particularity of the experimental setup is the large number ofinput and output variables that makes both the identification and control synthesisnon trivial.

Even though in principle a general MIMO-HC approach can be applied assuch to the original MIMO model, as extensively described in Chapter 4 this di-rect approach would lead to a controller which would be both very complex andvery hard to tune. Therefore, it would be useful to exploit the fact, mentionedin the previous Section, that the piezoelectric patches have been mounted on thestructure in such a way that they can be effectively operated in combinations tomaximize their effect of the vibratory response. Consequently, an automatic de-sign approach rather than physical insight can be used to work out the optimal

131


actuators combinations and reduce the number of actual control inputs to achievea square plant, ideally with a triangular structure.

To this purpose, a pre-compensator is introduced, according to the block dia-gram in Figure 9. The construction of the pre-compensator relies on an approachbased on the RGA and SVD approaches, in view of their well discussed nature asan interaction measure for MIMO plants (see Chapter 4).

7.2.1 Black-box T -matrix model

The acquisition system employed in the project consists of two PCI’s (PCI-6251,PCI-MIO-16E-1) that are used to acquire the accelerometric signals and to gen-erate the excitation signals for the actuators. The generated signals are low-passfiltered (with a cut-off frequency of 5 kHz) and amplified by a factor of 20. Theaccelerometers are connected to 478A16 signal conditioners that provide the volt-age supply needed by the sensors.

The software used to set the devices and to perform the experiments is NI-LabVIEW SignalExpress. The sampling frequency for all signals was set to 10kHz and the cut-off frequency of the anti-aliasing and reconstruction filters wasset to 5 kHz (KEMO 8 BenchMaster 21M filters have been employed). All sub-sequent, offline, data processing has been carried out using MATLAB. The datacollected for model identification and validation were gathered by exciting oneactuator at a time with a realization of uniform white noise (exploiting the wholevoltage range of the device) and recording the output signals of all the sensors, fora duration of 10 s.

As for the excitation signals used for identification purpose, a classical sinu-soidal chirp signal of type

ex(t) = sin

(φ0 + 2π

(f0t+

k

2t2))

(7.1)

with k being the frequency increasing rate, represent not a suitable solution, bothfor large frequency range shown in Figure 7.6 and the limitations of the acquisitionsystem (considering the number of input-output system channels). A uniformwhite noise signal has been indeed preferred.

Subspace model identification (see [74] for details) algorithms have been ap-plied to the problem of identifying a model of the structural response focusingon a set of 8 actuators and 4 sensors. In view of the linearity assumption of theresponse over the relevant frequency range and in order to simplify the identifica-tion procedure, a single dataset corresponding to the combination of the individualexperiments has been considered. Of the available 100000 samples, the first 5000have been used for identification, while the remaining 95000 have been employedto validate the identified model.

132


Model order has been selected by cross validation, in terms of the Variance-Accounted-For (VAF) metric. More precisely, given the multivariable nature ofthe system, the following definition for a vector VAF has been adopted. Lettingyl, l = 1, . . . , 4 the individual components of the 4-dimensional output vector, thescalar VAF associated with each of the measured output has been computed as

V AFl = 100

(1− V ar[yl(k)− ysim,l(k)]

V ar[yl(k)]

), (7.2)

where yl(k) is the value at time k of the l-th component of the measured outputand ysim,l(k) is the value at time k of the l-th component of the output obtainedby the identified model. Letting now

V V AF =[V AF1 . . . V AFl

]T(7.3)

the vector VAF is defined as

V AF = ‖V V AF‖/√l. (7.4)

A plot of the computed vector VAF as a function of model order is depicted inFigure 7.7; as can be seen, high order models are needed to achieve a sufficientlylarge value of the vector VAF index. In particular, orders up to 120 can be con-sidered for some input-output pairs. From the identified model, T -matrices can

n10 20 30 40 50 60 70 80

||VA

F||

50

60

70

80

90

100

Figure 7.7: Variance accounted for as a function of model order.

be extracted at each frequency of interest by exploiting the explicit connectionbetween a frequency response sample and the corresponding T -matrix.

The non-square multivariable 4×8 identified model Ggl is obtained with the

133


Actuator LocationAS07 left rear strutAS08 left rear strutTS05 left-side anti-torque plateTS06 left-side anti-torque plateAD07 right rear strutAD08 right rear strutTD03 right-side anti-torque plateTD04 right-side anti-torque plate

Table 7.1: Actuator location.

aim of attenuating the effect of the periodic disturbance acting at frequency 1599 Hz.

GTgl =

+0.021 + 0.015i −0.063− 0.046i +0.004− 0.004i −0.034− 0.025i

−0.032− 0.005i −0.076− 0.069i −0.004− 0.015i −0.036− 0.034i

−0.390− 0.808i −0.095− 0.024i +0.124 + 0.119i +0.006− 0.039i

+0.263 + 0.965i +0.079 + 0.018i −0.042 + 0.012i −0.027 + 0.016i

+0.034 + 0.005i +0.002− 0.006i +0.076 + 0.138i +0.054 + 0.045i

−0.029 + 0.017i −0.007 + 0.001i +0.037 + 0.113i +0.047 + 0.060i

+0.145− 0.100i +0.046− 0.003i +0.379 + 0.181i −0.005− 0.021i

+0.308− 0.067i +0.053 + 0.006i +0.364 + 0.135i +0.036 + 0.013i

(7.5)

7.2.2 Decoupling strategy - pairing evaluation

Following the procedure described in Chapter 4, the SVD approach is firstly ap-plied. For both the rear struts and the anti-torque plate two pairs of actuators havebeen chosen, one on the left side and one on the right side. In particular, theywere chosen in a symmetric way, in order to maximize their combined effect ofthe vibratory response and to determine the type of deformation the structure issubject to. As for the sensors, the choice was taken in such a way that each ac-celerometer would be affected significantly by a single pair of actuators. As anexample, accelerometer A2 is chosen in correspondence of the actuation couple(TS05 - TS06). As a result, a reduction of the cross-coupling effects is obtained.Results of the performed SVD decomposition are the following,

Y = GglU = USΣV HU → Σ =[diag1.411, 0.535, 0.155, 0.099.

](7.6)

A first controllability information about the system is given by its computed con-dition number,

γ =σmaxσmin

= 14.32, (7.7)

134


Accelerometer ActuatorsA2 TS05, TS06A9 AS07, AS08A10 TD03, TD04A18 AD07, AD08

Table 7.2: Pairing selection of input-output channels with both the SVD andRGA techniques.

which implies that some problems could arise in terms of cross-talking effectsand paring choice. Actually, if the main component of a chosen control variableis multiplied by σ, then its effect on governing the system output could be con-sidered negligible. In view of this, SVD can effectively give information aboutpossible input-output pairings. From a practical point of view, the measured out-put corresponding to the largest element of the first column of matrix US is takenand paired with the input variable corresponding to the largest element of the firstrow of matrix V T . Same reasoning for the other channels. Being V and UScomplex matrices, their module is computed,

|US | =

0.986 0.125 0.087 0.064

0.087 0.127 0.778 0.609

0.135 0.981 0.129 0.052

0.039 0.076 0.609 0.789

(7.8)

|V | =

0.014 0.017 0.573 0.147 0.067 0.131 0.692 0.387

0.026 0.041 0.701 0.251 0.247 0.264 0.431 0.356

0.648 0.267 0.202 0.018 0.419 0.467 0.063 0.263

0.711 0.148 0.191 0.058 0.424 0.439 0.064 0.238

0.034 0.256 0.159 0.574 0.575 0.467 0.074 0.157

0.013 0.208 0.191 0.659 0.451 0.492 0.169 0.111

0.144 0.652 0.157 0.386 0.089 0.118 0.334 0.497

0.227 0.609 0.134 0.026 0.193 0.166 0.426 0.558

. (7.9)

As an illustrative example, the largest element of the first column of |US| is thefirst, which corresponds to the first chosen sensor A2. As expected it well couplesto the left actuators of the anti-torque plate. Indeed, by inspecting matrix |V |, itsfirst column largest elements are the fourth (corresponding to actuator TS06) andthe third, which has a non negligible value too, corresponding to the actuator TS05.Table C.2 resumes the optimal pairings selected. An interesting considerationcan be drawn from Table C.2. The actual pairing selection reflects the chosensubsystem configuration. The SVD analysis, actually, confirms the assumptionthat cross-channels coupling effects can be minimized by means of a suitable

135


choice of input output variables. Of course this is not true in general, but forthe particular system configuration (i.e. the actuators location in the cabin), input-output influence can be easily inferred. Let compare the obtained results by meansof the RGA approach. As shown in Chapter 4, the RGA matrix Λ can be easilycomputed, and for the present experiment its module is given by

|ΛT | =

0.010 0.296 0.002 0.046

0.016 0.526 0.007 0.038

0.345 0.093 0.098 0.036

0.598 0.088 0.011 0.023

0.013 0.017 0.063 0.3670.014 0.021 0.039 0.4650.022 0.085 0.489 0.075

0.043 0.036 0.373 0.027

. (7.10)

From the above matrix, it is not hard to notice that the conclusions are essentiallythe same as before. Indeed, the largest element in the first column (sensor A2) of|ΛT | is the fourth, which corresponds to actuator TS06. Again, it cannot be ne-glected the third element too, very closed to the largest one, and corresponding toactuator TS05. The slight advantage of computing the RGA matrix is representedby the possibility to iterate its evaluation, as proven in Chapter 4. The RGAiterative algorithm is able to efficiently find, with relatively simplicity, the mostsuitable input-output association even when ambiguity is present. As final remark,note that the experiment has been constructed such that the actuators of interestcould effectively operated simultaneously to maximize their effect to reduce noiseand vibrations. This situation is confirmed by the SVD and RGA approaches, butproblems could arise in terms of control tuning. In particular, a possible issue,from a control tuning perspective, is represented by the mutual influence of theinput-output couples located on the rear strut. Hence, the idea is to effectivelycompensate the identified global plant Ggl by means of a compensator, able toconvert the multivariable nature of the plant in a multi-channel n-SISO system.

7.2.3 Two-steps Compensator design

As proposed in Chapter 4, the goal of introducing a compensator to eliminate thecross-channels interactions is pursued by means of a two-steps design procedure.A first compensator C is obtained by applying Algorithm 2, while Figure 7.8 il-lustrates the relative fast convergence of its elements. The achieved shaped modelGs,

Gs =

+0.328− 0.517i +0.002 + 0.000i −0.232− 0.151i +0.888 + 1.636i

+0.016 + 0.048i +0.136− 0.213i −0.076− 0.174i +0.038− 0.109i

+0.001− 0.001i −0.001 + 0.000i −2.768 + 2.016i −1.525− 2.278i

−0.001 + 0.000i −0.000− 0.000i −0.005 + 0.001i +2.829− 2.196i

(7.11)

136


Algorithm 2 - iteration #0 50 100 150 200

-20

-15

-10

-5

0

5

1st element2nd element3rd element4th element


-4

-2

0

2

4

6

8


-1

0

1

2

3

4


-2

-1

0

1

2

3

Figure 7.8: Compensator elements convergence with Algorithm 2.

is effectively close to a lower triangular structure, as in the intentions of the pro-posed approach. A second compensator C can be then computed, by means ofrelation C = G−1

s diag(Gs) to obtain a diagonal dominant system G∗. The over-all compensator is found as the product of C · C, leading to the expected diagonalsystem,

G∗ =

+0.328− 0.517i +0.000 + 0.000i −0.001− 0.000i −0.003− 0.002i

−0.000 + 0.000i +0.136− 0.213i −0.000 + 0.000i +0.001− 0.001i

+0.000− 0.000i −0.000 + 0.000i −2.768 + 2.016i +0.000− 0.002i

−0.000 + 0.000i −0.000− 0.000i −0.000 + 0.000i +2.829− 2.195i

. (7.12)

7.2.4 Multi-channel Robust-HC Design

The two-step design of a compensator as the one proposed allows to consider themultivariable, even highly-coupled, plant as a number of n-SISO independent sys-tems. Note that no assumptions or approximations are included in the equivalentdecoupled system description. The presence in the loop of the compensator sim-plifies the control tuning process. The overall control design procedure, based on

137


a diagonal plant, targets the virtual inputs that are then used to govern the real ac-tuators. The Robust Harmonic Control solution presented in Chapters 3,4 is used

Figure 7.9: Decoupling control strategy. From MIMO to n-SISO modelingand control. The scheme shows the four independent (virtual)input-outputchannels.

with the aim of attenuating the effect of the periodic disturbance acting at fre-quency 1599 Hz. The diagonal T -matrix corresponding to the frequency responseG∗ at the frequency of interest has been constructed,

T ∗ = blkdiag [T1 T2 T3 T4] (7.13)

withT1 =

[0.328 −0.517

0.517 0.328

]T2 =

[0.136 0.213

−0.213 0.136

]

T3 =

[−2.768 2.016

−2.016 2.768

]T4 =

[2.829 2.195

−2.195 2.829

].

(7.14)

Four feedback loops are thus controlled as in Figure 7.9, where the input signalsU∗ = [U∗1 U

∗2 U∗3 U∗4 ] are defined as the virtual inputs of the diagonalized system,

and Y are the (real) outputs given by accelerometers measures A2, A9, A10, A18.For the sake of comparison two different approaches to the design of the indi-

vidual control loops have been considered and compared: on one hand the clas-sical LQ-like HC algorithm has been applied; on the other hand, the robust HCsolution has been considered. The two controllers have been tuned to achieve thesame level of closed-loop steady state attenuation of the disturbance. However,while the LQ approach does not take model uncertainty into account, in the de-sign of the robust HC controller information on model uncertainty deriving fromthe identification problem has been retained. To compare the performance levelsachieved by the two design approaches a MonteCarlo study has been carried out,by randomly perturbing 500 times the T -matrix in the simulation model based onits uncertainty representation. Results are presented in Figure 7.10, with referenceto a relative uncertainty of 10 %, with a differentiated disturbance reduction target

138


for the four considered outputs. In particular the desired output norms have beendefined as ‖Y1‖ = 0.05, ‖Y2‖ = 0.1, ‖Y3‖ = 0.1 and ‖Y4‖ = 0.01. As expected,the average steady state performance level provided by the robust controller isslightly superior to the one from the LQ solution for all the considered outputs.As for the output variability, the robust solution presents a higher standard devi-ation, while the considerations relative to the settling time remain as extensivelydiscussed in the previous Sections and Chapters. To complete the addressed

0 50 100 150 200 250 300 350 400 450 500

Out

put N

orm

0

0.05||Y

1|| - HC

LQ

||Y1|| - HC

H∞

0 50 100 150 200 250 300 350 400 450 5000.09

0.1

0.11||Y

2|| - HC

LQ

||Y2|| - HC

H∞

0 50 100 150 200 250 300 350 400 450 5000

0.01

0.02||Y

3|| - HC

LQ

||Y3|| - HC

H∞

0 50 100 150 200 250 300 350 400 450 5000

0.01

0.02||Y

4|| - HC

LQ

||Y4|| - HC

H∞

Monte Carlo Occurrence #

0 50 100 150 200 250 300 350 400 450 500Set

tling

tim

e

0

5

Figure 7.10: Closed-loop performance results of the MonteCarlo study. Topfour panels: closed-loop attenuation of the disturbance for each of the fouroutput channels Bottom panel: settling time.

control comparison, three more scenarios have been tested. Results are summa-rized in Figures 7.11, 7.12 and 7.13, with reference to the desired output normas in Table 7.3. Black bars indicate the average output norm, while the grey oneis giving information about the standard deviation on the Monte Carlo study forthe specific considered variable. Although the evident limited variance obtained

139


LQ H∞

Out

put N

orm

0

0.05

0.1

0.15

LQ H∞

0

0.05

0.1

0.15

0.2

LQ H∞

0

0.05

0.1

0.15

LQ H∞

0

0.05

0.1

0.15

LQ H∞

Inpu

t Nor

m

0

0.5

1

1.5

2

LQ H∞

0

1

2

3

4

5

LQ H∞

0

0.1

0.2

0.3

0.4

LQ H∞

0

0.1

0.2

0.3

0.4

Figure 7.11: Closed loop performance comparison between HCLQ and HCH∞

approaches with target attenuation levels as in Scenario 1.

Table 7.3: Desired performance and included relative multiplicative uncer-tainty.

Desired Performance‖Y1‖ ‖Y2‖ ‖Y3‖ ‖Y4‖ Uncertainty

Scenario 2 0.1 0.1 0.1 0.1 10%Scenario 3 0.01 0.05 0.1 0.1 10%Scenario 4 0.01 0.05 0.05 0.05 30%

with the LQ-based harmonic controller, with a robust (decoupled) framework theperformance obtained for the whole set of outputs is outperformed. Maximal at-tenuation levels on the entire Monte Carlo Simulation are limited, at the cost of aslightly increased control effort. Moreover, the instability issue occurred on oneof the outputs in Scenario 3 is avoided.

140


LQ H∞

Out

put N

orm

0

0.005

0.01

0.015

LQ H∞

0

0.02

0.04

0.06

0.08

LQ H∞

0

0.05

0.1

0.15

LQ H∞

0

0.05

0.1

0.15

LQ H∞

Inpu

t Nor

m

0

0.5

1

1.5

2

LQ H∞

0

1

2

3

4

5

LQ H∞

0

0.1

0.2

0.3

0.4

LQ H∞

0

0.1

0.2

0.3

0.4


approaches with target attenuation levels as in Scenario 2.

LQ H∞

Out

put N

orm

0

0.005

0.01

0.015

LQ H∞

0

0.02

0.04

0.06

0.08

LQ H∞

0

0.02

0.04

0.06

0.08

LQ H∞

0

1

2

3

4

LQ H∞

Inpu

t Nor

m

0

0.5

1

1.5

2

LQ H∞

0

1

2

3

4

5

LQ H∞

0

0.1

0.2

0.3

0.4

LQ H∞

0

1

2

3


approaches with target attenuation levels as in Scenario 3. The output norm‖Y4‖ result to be unstable in the first case.

141

CHAPTER8Harmonic Control with Semi-active Lag

Dampers

Hydraulic dampers are generally placed between the rotor hub and the rotor blades.Their primary role is to generate a suitable damping level to avoid and minimizeundesired effects like ground resonance and instability issues during landing orhigh-speed manouvres. Classical dampers, however, are not able to produce anadaptive damping level, resulting in well known difficulties in dealing with un-desirable loads during cruise flight conditions or with the fatigue of mechanicalcomponents. A recent research line focused on vibration control problems in-vestigates the use of so-called semi-active valve lag dampers (S-AVLDs), which,contrary to the classical ones, are able to adapt the damping level by manipulat-ing the flow of hydraulic fluid between the damper chambers. Moreover, damperloads versus piston velocity profiles differ significantly when the controllable ori-fice of the damper is operated in a fixed manner to when it is operated by meansof the inner-loop force controller.

The Chapter is devoted to present an IBC-based vibration system, with theaim to investigate the changes in the damping associated with the HC vibrationcontroller by using S-AVLDs together with inner controllers to improve over forcetracking capabilities. In particular, the overall control system is designed for the

143

Chapter 8. Harmonic Control with Semi-active Lag Dampers

four-blade Puma IAR 330 helicopter.

8.1 Introduction and Overall Control Architecture

From a technology point of view, S-AVLDs (illustrated in Figure 8.1) can gen-erate higher damping levels than magnetorheological fluid elastomeric dampers,designed for similar applications. Moreover, they do not increase the complexityof the blade’s structural design, except for the active devices mounted along theblades. Anusonti-Inthra et al (see [1]) showed how a stiffness variation of rootelements could effectively reduce hub vibrations, and thus vibration propagationacross the helicopter fuselage. Previous studies, in particular for the five-blade

Figure 8.1: Lag Damper.

EH-101 helicopter (see [81]), have shown that semiactive lag dampers can be usedalong with Harmonic Control laws. In this regard, a cascade control problem isobtained, composed by a local control of the device responsible for the actuation,and a global control of the overall system.

A scheme of the general architecture of the vibration control strategy usingsemi-active valve lag dampers is illustrated in Figure 8.2. The architecture con-sists of two cascade layers. The outer loop is concerned with the design of a vibra-tion controller aiming at reducing vibrations based on a model of the main rotorvibratory system. This vibration controller, as shown in the previous Chapters,uses the information collected through vibration sensors, like accelerometers, anddetermines the damper force references r needed to be delivered by the S-AVLDactuator. The inner feedback loop is added in order to improve the tracking per-formance of the damper forces. Although there exists alternative control config-urations which do not incorporate an inner SAVLD control loop, such strategiesare expected to provide a lower performance as they do not compensate againstuncertainties, nonlinearities and external disturbances found in the S-AVLD sys-tem.

144

8.2. Semi-active Lag-Dampers Modeling

Figure 8.2: Overall architecture for the vibration helicopter control with usingS-AVLDs.

8.2 Semi-active Lag-Dampers Modeling

A classical passive damper is essentially similar to that of an automobile shockabsorber. As illustrated in Figure 8.3, the S-AVLD is comprised typically of twochambers and a pistonrod, a set of relief valves and bypass valves. The working

Figure 8.3: Schemes of the passive and semi-active lag dampers.

chambers are filled with hydraulic fluid and the damping effect is expressed bya force which opposes the piston velocity. The damper force is approximatelyproportional to the pressure difference of the fluid in both chambers. The resultingmodel equations are essentially based on the work of Wallaschek in [118]. Reliefvalves are introduced as a safety mechanism and are activated when the pressureof the fluid on any of the chamber reaches a critical value. In principle, once oneof the chambers reaches or exceeds a permissible level of pressure, the respectiverelief valve becomes open allowing the hydraulic fluid to flow towards the otherchamber. This has the ultimate effect of restoring the pressure on this chamber

145


back to normal working conditions.Referring to Figure 8.3, an equation for the force provided by the damper is

defined as

F (t) = my(t) +A(P1(t)− P2(t)) + Fcsign(y(t)) (8.1)

where Pi is the pressure in chamber i and A the cross-sectional area of the piston.The force on the piston caused by the motion y(t) applied to the damper is ob-tained as the sum of the inertia of the body my(t), the force on the piston due tothe two chambers pressure differenceA(P1(t)P2(t)) and the Coulomb friction Fc,due to the piston rubbing against the sides, assumed constant and often negligible.

The equation for the relief valve motion is given by

mvX + δX + k(X +Xc) = A(P1 − P2) (8.2)

whereX is the valve displacement from the valve seat andXc the spring pre-load.An extension to the basic model would be to include the effect of the compress-ibility z of the hydraulic fluid as

z(t) = P1(t)− P2(t) =1 + ξ

ξβV(Ay(t)− V (t)−Qv(t)) (8.3)

where V is the average volume of a chamber and ξ is a constant which takes intoaccount the different sizes of chambers. V is the orifice flow and Qv the flow atthe output of the relief valve. It can be assumed that sign(V ) = sign(z) sincethere cannot be flow in the opposite direction to the pressure gradient for a uniformpipe. From this assumption,

sign(V )z = sign(V )z sign(z) = |z| (8.4)

where the expression for V can be written as

V = sign(z)

(−D1 +

√D2

1 + 4D2|z(t)|2D2

)(8.5)

withD1 =

128lη

πd4, D2 =

cρ

2A20

(8.6)

which describe the physical characteristic of the bypass orifice, including the vis-cosity η, density ρ and the geometry l,d and A0 as length, diameter and area ofthe orifice. As for the flow Qv, it can be defined as

Qv = CpoγX

1 + γXXπdv sin(α)

√2z

ρ(8.7)

146

8.2. Semi-active Lag-Dampers Modeling

where Cpo and γ are experimentally determined, dv is the diameter of the valveand α is the half angle of the valve.

The overall system can be written as a set of three differential equations. Notethat a symmetric assumption has been made for the damper system, for com-pression and rebound. In this way, the valve properties such as the stiffness k,precompression Xc, and dimensions are considered the same for both valves.

Remark 11. Experimentally, the damper can be considered as composed by asingle valve since a simultaneous opening of both valves cannot be observed withthe low forcing frequencies.

Let define Fcrit as the force at which the relief valve opens. Two cases describethe system,

• Valve open (Avz > Fcrit)

y(t) =F (t)−Az

m

X(t) =1

mv

(Avz(t)− δX(t)− k(X(t) +Xc)

)z(t) =

1 + ξ

ξβV

(Ay(t)− sign(z)

(−D1+

√D2

1+4D2|z(t)|2D2

+R(X, t)√|z|))

(8.8)with

R(X, t) = CpoγX

1 + γXXπdv sin(α)

√2

ρ(8.9)

• Valve closed (Avz ≤ Fcrit)

y(t) =F (t)−Az

mX(t) = 0(∧X(t) = 0, X(t) = 0)

z(t) =1 + ξ

ξβV

(Ay(t)− sign(z)

(−D1+

√D2

1+4D2|z(t)|2D2

)) (8.10)

An ulterior assumption can be made by considering the contribution to the overallforce due to the inertia of the piston. Actually the maximum acceleration achievedduring a cycle results in a negligible inertia term if compared with the force gen-erated by the damper. Consequently, equation (8.1) becomes

F (t) = Az(t) (8.11)

Starting from the obtained model for a generic passive damper, the model of theS-AVLD can be written in a similar manner (see [116]). The bypass valve is

147


the component that provides the ”active” characteristics of the damper. Withoutit, the damper would behave as a passive lag hydraulic damper. The dampingcharacteristics of the S-AVLD are modified by means of a variable orifice areaon the bypass valve which either augment or reduce the flow of the fluid betweenthe chambers. Roughly speaking, it allows the damping to be either increased orreduced, but the system cannot exhibit ”negative” damping,

z(t) =1 + ξ

ξβV(Ay(t)− V (t, A0(u))−Qv(t)) (8.12)

8.3 Inner-loop Control - Gain Scheduling

A simplified model can be obtained with the aim to design the inner loop con-troller. In particular, the compressibility effect can be considered constant andgiven by a constant coefficient β,

β = B0

(1

V1+

1

V2

)(8.13)

with B0 a constant associated with the fluid within the damper and V1 and V2 thevolumes of the two working chambers of the damper. As a result the system isgiven by,

• Valve open

F = Az

z(t) = B0

(1V1

+ 1V2

)(Ad(t)− V (A0(u))−Qv

) (8.14)

• Valve closed

F = Az

z(t) = B0

(1V1

+ 1V2

)(Ad(t)− V (A0(u))

) (8.15)

with

V = sign(z)

(−D1+

√D2

1+4D2|z(t)|2D2

)Qv = sign(z)R(X)

√|z|

(8.16)

An ulterior simplification can be used by considering the maximum pressure dif-ference, responsible for the opening of the relief valves. By considering Qv = 0,it is assumed that they are always inactive. As a result,

F = Az

z(t) = β(Ad(t)− ζ

√|z|sign(z)u

) (8.17)

148

8.4. Outer HC Controller and plant modeling

where the approximation

ζ√|z|sign(z) ≈ sign(z)

(−D1 +

√D2

1 + 4D2|z(t)|2D2

)(8.18)

has been introduced. By substituting z in the second equation,

z(t) = β

(Ad(t)− ζ

√∣∣∣∣ 1

AF

∣∣∣∣sign( 1

AF

)u

)(8.19)

which can be written as

z(t) = β(Ad(t)− pu

)(8.20)

by defining

p = ζ

√∣∣∣∣ 1

AF

∣∣∣∣sign( 1

AF

). (8.21)

A linear proportional law can be chosen as,

u = −κ(r − F ) (8.22)

leading to the system

z(t) = βAd(t)− βp(−κr)− βp(κF ), (8.23)

and by means of the relation F = Az

z(t) = −βAκp(t)z + βAd(t) + βκp(t)r (8.24)

which represent an LPV system with parameter p.Figure 8.4 illustrates the normalized evolution of the parameter p and the out-

put force F with respect to a sinusoidal reference input. In Figures 8.5, 8.4, in-stead, reference tracking and the control variable are shown for both the completeand simplified damper models.

8.4 Outer HC Controller and plant modeling

Harmonic Control can reduce the vibratory loads by exciting the swashplate atfrequencies higher than fundamental frequency, which is the main rotor angu-lar speed Ω. The main idea is to smooth the rotor aerodynamics. A rotor with anumber ofN equally spaced identical blades acts as a filter when all the blade rootloads are summed in the non-rotating reference frame. Hence only the loads at fre-quencies which are at integer multiples of the fundamental frequency multiplied

149


time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nor

mal

ized

ref

. [-]

-1

-0.5

0

0.5

1

Parameter p(t)Output force F(t)

Figure 8.4: Evolution of the normalized variable p and F .

time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Orif

ice

Are

a A

[-]

0

0.2

0.4

0.6

0.8

1

1.2Control modelA-SVLD model

Figure 8.5: Tracking performance of the gain-scheduling controller.

time [s]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Nor

mal

ized

ref

. [-]

-1

-0.5

0

0.5

1ref. r(t)Output F(t) - S-AVLD modelOutput F(t) - control model

Figure 8.6: Gain scheduling control variable.

150


by the blade number are transmitted to the fuselage. Therefore the loads on thefuselage at N /rev frequency, the most critical one, can be reduced by moving theswashplate at the same frequency. This in turn induces blade pitch variations thatcan smooth out aerodynamic loads. The vibration reduction analysis, includingelastic fuselage, servo actuators and sensors for vibration calculation, is performedin this study by using a simulation tool called MASST (Modern Aeroservoelas-tic State Space Tools) which has been developed at Politecnico di Milano for theaeroservoelastic and aeromechanical analysis of aircraft and rotorcraft [77], [76].MASST constitutes of modular models of complex linearised aeroservoelasticsystems. Models are not directly formulated in MASST but are rather composedof subcomponents collected from well-known, reliable and possibly state-of-artsources and blended together in a mathematical environment. The problem isthus formulated in state-space form. The equations of motion of the system arecast as first order time differential equations. As a consequence, generic states-pace approaches can be used to analyze aeroelastic systems. MASST has beendesigned to be modular and to incorporate heterogeneous subcomponents fromdifferent sources to model, as the deformable aircraft structural dynamics, theairframe unsteady aerodynamics, the rotor aeroelasticity, the servoactuators dy-namics and sensors/filters dynamics. When these elements are combined, theyprovide a powerful and flexible closed loop aeroservoelastic modeling capability.Each component is modeled in its most natural and appropriate modeling envi-ronment and then cast into state-space form.

8.4.1 Multi-blade coordinate transformation

The dynamics of the rotor blades in rotation are generally expressed in the rotat-ing frame. The rotor, however, responds as a whole to excitations such as aero-dynamic gusts, control inputs, and tower-nacelle motion, all of which occur in anon-rotating frame. Multi-blade coordinate transformation (MBC) helps to inte-grate the dynamics of individual blades and express it in a fixed (non-rotating)frame. Consider a rotor with N blades that are spaced equally around the rotorazimuth. In such a case, the azimuth location of bth blade is given by

ψb = ψ + (b− 1)2π

N(8.25)

with ψ azimuth of the first (reference) blade. Let qb be a particular rotating degreeof freedom for the bth blade. The MBC is a linear transformation that relates the

151


rotating degrees of freedom to new degrees of freedom defined as

q0 =1

N

∑Nb=1 qb

qnc =2

N

∑Nb=1 qb cos(nψb)

qns =2

N

∑Nb=1 qb sin(nψb)

qN/2 =1

N

∑Nb=1 qb(−1)b

(8.26)

Being in the considered study qb a lag degree of freedom, then q0 is the rotorcollective lag, q1c is the horizontal displacement of the rotor center-of-mass in therotor plane, and q1s is the vertical displacement of the rotor center-of-mass in therotor plane. The rotor modes corresponding to qnc and qns (n > 1) and qN/2 arecalled reaction-less modes because they do not cause any transference of momentsor forces from the rotor to the hub (fixed frame). The value of n goes from 1 to(N -1)/2 if N is odd, and from 1 to (N -2)/2 if N is even.

The inverse transformation, yielding the blade coordinate given the rotor coor-dinates, is given by

qb = q0 +N∑b=1

(qnc cos(nψb) + qns sin(nψb)) + qN/2(−1)b (8.27)

With reference to the blade motion as in Figure 8.7, under the assumption of

Figure 8.7: Fundamental blade motion.

isotropic rotor (the structural characteristics of the lag dampers are the same foreach blade), it can be solved, in the MASST model, the coefficient periodicityissue, by translating the d.o.f. ξi from the rotating to the non-rotating frame by

152


means of the MBC approach,

ξ0 =1

N

∑Ni=1 ξi

ξC =2

N

∑Ni=1 ξi cos(nψi)

ξS =2

N

∑Ni=1 ξi sin(nψi)

ξN/2 =1

N

∑Ni=1 ξi(−1)i

(8.28)

In particular, by using the inverse transformation, it can be possible to obtain therelation of the forces in the rotor frame and include the inner loop in the outerHarmonic Controller.

153

Part IV

Concluding remarks

155

CHAPTER9Conclusions and Future Perspectives

The subject of the Thesis is the analysis and design of Harmonic Control laws fordisturbance attenuation, specifically in plant affected by tonal disturbances withknown spectrum. In this context, the system is supposed to reach the harmonicsteady-state, i.e. approximate sinusoidal steady response, by using measurementsof the steady response amplitude and phase to determine the required amplitudeand phase of the control signal. In this regards, the considered model frameworkis based on the classical representation of the system as a linear model constructedin the frequency domain, the T -matrix model.

Although the Thesis addresses the problem from a general and methodolog-ical perspective, a particular application, i.e. the helicopter vibration problem,represented a starting point for the present research work, as it can be formulatedin terms of compensating a periodic disturbance at rotor frequency acting at theoutput of an uncertain (possibly time-periodic) linear system. In particular, theprecise knowledge of the T -matrix elements has been considered as a prerequi-site for the good functioning of the overall control system. Unfortunately, classi-cal employed controllers cannot deal with model and parametric uncertainties ornonlinearities which could cause degraded performance or instability issues forthe closed-loop system.

The aim of the present Dissertation has been to propose a systematic methodol-

157

Chapter 9. Conclusions and Future Perspectives

ogy to design Harmonic control laws in the robust framework, able to adequatelyaddress the uncertainties in the T -matrix model, both for the SISO and MIMOsystem representations. The proposed solution can reduce, for example, the needof adaptation and guarantee, more specifically and directly in the synthesis pro-cess, the nominal stability of the closed-loop system, a prescribed robustness levelto model and parametric uncertainties and a desired level of disturbance attenua-tion. As for the multivariable generalization of the provided robust approach, theidea has been to resort to decentralized and decoupling strategies with the aim ofeliminate cross-interactions which could degrade, again, the closed-loop perfor-mance. A two-step decoupling strategy has been proposed in order to obtain aglobal compensator able to diagonalize the T -matrix plant and to use, similarly tothe robust single variable case, the same instruments and tools.

In addition, the employment of the Describing Function method, in the T -matrix framework, has been useful to understand the role of loop nonlinearities onthe closed-loop performance and stability, and the advantages of the robust controlsolution with respect to other classical Harmonic Control methods. Then, the ideato resort to a Linear Parameter Varying approach to both the modeling and controlsynthesis of the system affected by predictable and unpredictable uncertainties hasbeen exploited. As a conclusion, it has been shown as the LPV approach couldsurely be placed as a valid alternative to, for example, the existing adaptive-LQHC variant already used in many applications. In particular it has been shown howthe robust gain-scheduling framework can provide a suitable means to account forvariations in the plant such as the ones induced by changing conditions or actuatornonlinearities.

Harmonic Control has been then specialized in the Vibration control problemin helicopters. Part III of the Thesis is dedicated to this problem, with the aimto validate the results obtained with the previous numerical studies. More specif-ically, three slightly different paradigms of the vibration control problem havebeen included. In this respect some future research directions can be glimpsed.The design of an active control system for reducing vibrations in helicopters isalways a challenging task which requires years of studies and experiments. Oncedefined the control framework, implementation and on-ground, wind tunnel andflight test experiments should be performed. Actually, during the flight, many un-known problems could arise. As an example, slight variations of disturbance tonescould be considered. Then, non-stationary aerodynamic conditions might be ariseduring the flight, decreasing the achievable attenuation. Finally, a generalizationof the proposed methods could be developed with respect to the briefly introducedLTP models.

158

Part V

Appendix

159

APPENDIXAFrom LTP to LTI - Harmonic Transfer

Function

Consider a continuous-time LTP n-dimensional system

x(t) = A(t)x(t) +B(t)u(t)

y(t) = C(t)x(t) +D(t)u(t)(A.1)

Each matrix could be expanded in a complex Fourier series

A(t) =

∞∑m=−∞

Amejmψ (A.2)

and similarly for B,C,D. The system can be then analyzed in the frequencydomain by means of the HTF operator. By considering the class of EMP signals,the complex signal u(t) is said to be EMP of period T and modulation s if

u(t) =

∞∑k=−∞

ukeskt = est

∞∑k=−∞

ukejkψ, t ≥ 0 (A.3)

with sk = s+jkΩ. As said, this class of signals is a generalization of the classof T -periodic signals, that is, of signals with period T . Actually, an EMP signal

161

Appendix A. From LTP to LTI - Harmonic Transfer Function

with s = 0 is just an ordinary time-periodic signal. In much the same way as atime-invariant system subject to a (complex) exponential input has an exponentialsteady-state response, a periodic system subject to an EMP input has an EMPsteady-state response. In such a response, all signals of interest can be expandedas EMP signals. By deriving Fourier expansions for A(t), B(t), C(t) and D(t)it is possible to prove that the EMP steady-state response of the system can beexpressed as the infinite dimensional matrix equation with constant elements,

sX = (A−N )X + BU , Y = CX +DU (A.4)

where X , U and Y are doubly infinite vectors formed with the harmonics ofx,u and y respectively, as:

X T =[· · ·xT−2 x

T−1 x

T0 xT1 xT2 · · ·

](A.5)

and similarly for U and Y , with N defined as N = blkdiag[jkΩIn]. A, B, Cand D are doubly infinite Toeplitz matrices formed with the harmonics of A, B,C and D respectively, as

A =

. . ....

......

· · · A0 A−1 A−2 · · ·· · · A1 A0 A−1 · · ·· · · A2 A1 A0 · · ·

......

.... . .

(A.6)

where the submatrices An are the coefficients of the Fourier expansion of ma-trix A(t). To relate these coefficients to those of the Fourier-series expansion intrigonometric form, Eq. (1), recall that the Fourier series expansion of a scalarfunction can be rewritten in complex exponential form,

a(t) = a0 +∞∑k=1

(akc cos(kωt) + aks sin(kωt)) =∞∑

k=−∞

akejkωt (A.7)

with, being k = 1, 2, . . . ,

ak =1

2(akc − jaks) a−k =

1

2(akc + jaks). (A.8)

Hence,

Ak =1

2(Akc − jAks) A−k =

1

2(Akc + jAks). (A.9)

The HTF operator G(s) is thus defined as

G(s) = C[sI − (A−N )]−1B +D (A.10)

162

which relates the input harmonics and the output harmonics. This can be seenas the periodic extension of the constant coefficient expression for the transferfunction

G(s) = C(sI −A)−1B +D (A.11)

In particular, if s = 0 (considering the steady-state response of the system to aperiodic input of basic frequency N/rev), the input-output operator becomes

G(s) = C[(N −A)]−1B +D (A.12)

It was shown that the output of an LTP system can be consider as the sum of theoutputs of infinite LTI systems translated in frequency of a factor ejk0t. In thecase of stable LTI systems it is of special interest consider what happen when aninput as the complex exponential with s = 0, namely a constant signal u(t) = U ,is used. The output is a constant signal,

Y = G(0)U (A.13)

with G(0) static gain of the system. Analogously, of special interest is analyzethe response of an LTP system to an EMP signal with s0 = 0, namely an ordinaryperiodic signal with the same period of the LTP system. As said, the output willbe an EMP signal with s0 = 0, and

Y = G(0)U (A.14)

In this case U and Y are just constant vectors of the Fourier Coefficients of u(t)and y(t) respectively. Therefore G(0) can be viewed as the steady matrix gain thatgive the Fourier coefficients of the output when the input is a T -periodic signal ofknown Fourier coefficients. G(0) is called also Harmonic Transfer Matrix, and itis for LTP systems what the static gain is for LTI.

163

APPENDIXBBounded Real Lemma

B.1 Preliminaries

B.1.1 Singular value decomposition and positive matrices

Consider the singular values of two general matrices A, B. The relation,

σ(A+B) ≤ σ(A) + σ(B)

σ(A ·B) ≤ σ(A) · σ(B)(B.1)

called triangle inequality and Schwarz inequality respectively.

Lemma B.1.1. For two symmetric positive definite matrices A = AT , B = BT ,

σ(A) < σ(B)⇒ A < B. (B.2)

Consider two positive-definite and symmetric matrices A,B, the followingproperties hold true:

• Sum of positive matrices: A > 0, B > 0⇒ A+B > 0

• Block diagonal matrices: A > 0, B > 0⇔[A 0

0 B

]> 0

165

Appendix B. Bounded Real Lemma

• Convexity property: a,B ≥ 0, λ, µ > 0⇒ λA+ µB ≥ 0

Definition B.1.1. A congruence transformation is a transformation of the formA → PTAP , where A and P are square matrices and P is non singular. Inparticular a congruence transformation preserves definiteness.

Fundamental Matrix Inequalities

A fundamental matrix inequality used in the Robust Control literature is the fol-lowing, being ∆ the unstructured uncertainty, and A, B of suitable dimensions.

Lemma B.1.2. if ∆T∆ ≤ Q∆, then, for any α > 0,

XT∆Y + Y T∆TX ≤ αXTX +1

αY T∆T∆Y ≤ αXTX +

1

αY TQ∆Y (B.3)

Corollary B.1.2.1. if ∆ is norm bounded so that

‖∆‖ = σ(∆) ≤ γ ⇔ ∆T∆ ≤ γ2I (B.4)

the inequality (B.3) is equivalent to

XT∆Y + Y T∆TX ≤ αXTX +γ2

αY TY, ‖∆‖ ≤ γ (B.5)

Lemma B.1.3. Given the matrices A = AT , M , N of suitable dimensions, and∆, ∆T∆ ≤ R∆, the inequality

A+M∆N +NT∆MT < 0 (B.6)

holds if and only if there exist a constant ε > 0 such that

A+ εMTM +1

εNTR∆N < 0 (B.7)

Lemma B.1.4 (Matrix Inversion Lemma). Given four matrices A, B, C, D ofsuitable dimensions the following equality holds true,

(A+BCD)−1 = A−1 −A−1B(DA−1B + C−1) +DA−1. (B.8)

Schur Complement and Applications

Lemma B.1.5 (Schur Lemma). Given the matrices X , Y , Z of suitable dimen-sions, the following statements are equivalent:

[X Y

Y T Z

]> 0⇔

Z > 0 ∧X − Y Z−1Y T > 0

X > 0 ∧ Z − Y X−1Y T > 0[Z Y T

Y X

]> 0

(B.9)

166

B.2. Lyapunov inequality LMI for discrete-time systems

Lemma B.1.6. For any scalar ε > 0 A B DT

BT C ET

D E −εI

> 0⇔[A B

BT C

]+

1

ε

[DT

ET

] [D E

]> 0 (B.10)

Corollary B.1.6.1. A B DT

BT C ET

D E −W

> 0⇔[A B

BT C

]+

[DT

ET

]W−1

[D E

]> 0 (B.11)

B.2 Lyapunov inequality LMI for discrete-time systems

Consider a discrete-time LTI system,

xk+1 = Axk (B.12)

Lemma B.2.1. System (B.12) is exponentially stable iff there exist a matrix P =PT > 0 such that APA− P < 0.

Equivalently, by manipulating the former inequality, it can be obtained

−P +ATPA < 0, −P < 0

m−P −AT (−P−1)A < 0, −P < 0

, (B.13)

and using the Schur Lemma the Lyapunov inequality can be expressed as[−P−1 A

AT P

]< 0⇔

[−P AT

A −P−1

]< 0 (B.14)

which can be written in the positive-definite form,[P AT

A P−1

]> 0. (B.15)

(B.14) and (B.15) are not LMIs. An alternative expression can be derived bypre and post multiplying (B.13) by S = P−1, recalling that if P = PT > 0 thenS = P−1 always exists and is also symmetric and positive definite. Consequently,

− P +ATPA < 0⇔P−1(−P +ATPA)P−1 < 0

−P−1 − (AP−1)T (−P−1)−1(AP−1) < 0

−S − (AS)T (−S−1)(AS) < 0

, (B.16)

167


which can be expressed, by means of the Schur Lemma, as a feasibility LMI onmatrix variable S, [ −S AS

SAT −S

]< 0 (B.17)

.As shown, the problem of stability verification for the DiscreteTime LTI sys-

tem (B.12) is transformed into a strict LMI feasibility test for the LMI (B.17).This LMI is an equivalent test for the (Quadratic) Stability of the considered dis-cretetime LTI system. The solution S, if exist, define the Quadratic LyapunovFunction V (xk) = xTk Pxk, which certifies the exponential (asymptotic) stability.

LMIs for discrete-time LTI systems without uncertainty

Consider the discrete-time system

xk+1 = Axk +Buk (B.18)

with a static state feedbackuk = Kxk. (B.19)

The closed-loop system is stable iff there exist a matrix P = PT > 0 such that

(A+BK)TP (A+BK)− P < 0 (B.20)

which is not an LMI in P,K, but can be transformed into one by using the vari-able change S = P−1 > 0 and W = KP−1 = KS, leading to the Lyapunovinequality transformations

S((AT +KTBT )S−1(A+BK)− S−1)S < 0⇔(SAT + SKTBT )S−1(A+BK)S − SS−1S < 0⇔

(SAT +WTBT )S−1(AS +BW )− S < 0⇔−S − (SAT +WTBT )(−S−1)(AS +BW ) < 0

(B.21)

which can be expressed as an LMI problem in terms of matrix variables S,W ,again by means of the Schur Lemma. Consequently,[ −S AS +BW

SAT +WTBT −S

]< 0 (B.22)

and the algorithm for discretetime state feedback design via LMI is just to solvethe LMI feasibility problem (B.22) for S, W and then get the static state feedbackgain as K = WS−1.

168

B.3. Bounded Real Lemma for Discrete Time systems

B.3 Bounded Real Lemma for Discrete Time systems

Consider the uncertain system

xk+1 = A(∆)xk (B.23)

with A = A + ∆A = A + M∆AN . A represents the nominal system ma-trix, while matrices M ,N are known constant matrices capturing the uncertaintystructure.

Remark 12. It is intuitively expected that the nominal matrix A has to be (Hur-witz) stable.

Note that system can be written as the feedback interconnection of the systems

xk+1 = Axk +Mwk, zk = Nxk, wk = ∆zk (B.24)

Can also generalize (B.23), (B.24) by including a feed-through term in theoutput equation,

zk = Nxk +Dwk. (B.25)

The problem consists in investigating the conditions for robust quadratic sta-bility of the norm bounded uncertain system

xk+1 = Axk +Bwk

zk = Cxk +Dwk

wk = ∆zk

(B.26)

with A Schur stable and x0 = 0, σ(∆) ≤ γ = 1µ , z the controlled output and

w the disturbance.

Theorem B.3.1. Given system (B.24) and assuming A Shur stable and x0 = 0,then

• System’s Energy gain ‖E‖∞ < µ, t.t

supwk

‖Gwk‖‖wk‖

< µ. (B.27)

• there exists a solution P = PT > 0 of the LMI

169


ATPA− P + CTC+

+(ATPB + CTD)[µ2I − (BTPB +DTD)]−1(BTPA+DTC) < 0

m[ATPA− P + CTC ATPB + CTD

BTPA+DTC BTPB +DTD − µ2I

]< 0

m ATPA− P ATPB + CTD CT

BTPA+DTC BTPB − µ2I DT

C D −I

< 0

(B.28)

An alternative formulation can be obtained by applying the Schur Lemma.

[ATPA− P + CTC ATPB + CTD

BTPA+DTC BTPB +DTD − µ2I

]< 0

m[ −P 0

0− µ2I

]+

[ATPA+ CTC ATPB + CTD

BTPA+DTC BTPB +DTD

]< 0

m[ −P 0

0− µ2I

]+

[ATP CT

BTP DT

] [P−1 0

0 I

] [PA PB

C D

]< 0

m−P 0 ATP CT

0 −µ2I BTP DT

PA PB −P 0

C D 0 −I

. (B.29)

A congruent transformation can be performed through the permutation matrix

I 0 0 0

0 0 I 0

0 I 0 0

0 0 0 I

(B.30)

170

B.3. Bounded Real Lemma for Discrete Time systems

leading to the DT-BRL form−P ATP 0 CT

PA −P PB DT

0 BTP −µ2I 0

C 0 D −I

(B.31)

Note that the lower left block reflects the well-posedeness requirement I >µ2DTD.

171

APPENDIXCGlobal MIMO-VAF Data

173

Appendix C. Global MIMO-VAF Data

A1 A2 A3 A4 A5 A6 A7 A8 A9

AS01 35 23 47 24 50 51 16 66 24AS02 54 16 73 4 -3 1 13 -7 89AS03 51 41 52 31 50 47 48 22 49AS04 63 78 10 19 68 85 71 70 27AS05 49 26 69 12 -9 5 23 18 84AS06 75 20 4 52 19 33 30 62 17AS07 38 87 18 5 64 84 71 59 30AS08 31 54 29 15 73 72 68 23 21TS01 40 63 61 68 95 85 69 63 41TS02 43 10 57 92 83 78 86 85 21TS03 25 77 5 82 79 50 86 90 43TS04 18 83 89 58 77 31 54 -7 3TS05 32 53 39 90 72 59 90 87 20TS06 42 77 58 78 91 75 92 88 46TS07 35 56 81 87 91 22 53 75 48TS08 39 64 80 59 86 16 57 76 35TS09 74 21 57 54 63 50 29 27 47TS10 61 26 33 48 0 30 20 73 81TS11 62 45 82 62 68 45 60 39 34TS12 45 59 72 89 89 52 60 90 47AD01 53 41 46 47 31 50 15 63 77AD02 68 77 56 72 39 77 53 24 46AD03 17 61 7 72 58 38 44 40 74AD04 48 56 39 54 57 47 45 64 81AD05 77 80 35 52 59 79 62 51 33AD06 62 51 57 35 36 72 37 44 72AD07 69 56 42 35 65 50 59 66 72AD08 31 58 16 57 78 44 40 47 49TD01 65 75 46 40 45 45 54 55 35TD02 47 57 50 23 50 9 19 29 58TD03 1 44 76 40 85 71 60 38 60TD04 5 47 23 56 70 48 51 27 43TD05 41 48 52 51 81 46 68 46 59TD06 44 43 75 44 83 40 37 47 70TD07 44 50 64 39 67 40 60 41 60TD08 69 35 79 33 71 36 37 46 61TD09 61 73 68 5 9 7 82 28 70TD10 61 48 76 14 50 49 48 82 82TD11 10 4 50 53 34 5 74 29 75TD12 40 20 61 53 64 -13 73 6 69

Table C.1: Simulation results of the global model estimated with the filteredsignals in terms of the vaf criterion - 1.

174

A10 A11 A12 A13 A14 A15 A16 A17 A18

AS01 8 16 22 20 9 64 36 59 75AS02 41 66 27 41 3 26 66 29 34AS03 10 76 38 61 48 42 47 14 23AS04 39 48 53 51 46 33 44 52 61AS05 54 61 56 44 8 26 71 8 37AS06 45 32 71 17 62 81 51 74 79AS07 55 36 57 51 52 30 48 48 49AS08 46 66 55 59 50 16 33 17 -3TS01 39 47 1 65 30 33 12 61 60TS02 39 58 66 64 53 21 50 33 67TS03 39 -25 -18 15 -2 53 74 77 53TS04 14 68 69 43 46 36 80 62 68TS05 28 55 38 35 41 60 58 14 84TS06 56 78 -34 17 -0 47 59 80 52TS07 35 -5 45 24 31 50 62 64 37TS08 25 27 23 23 31 64 31 46 51TS09 75 16 8 50 37 44 54 9 81TS10 -16 13 55 68 46 76 66 89 71TS11 56 14 63 60 64 68 69 66 67TS12 59 67 36 62 31 30 68 86 55AD01 42 55 17 94 29 29 45 96 12AD02 56 -3 90 2 19 24 46 44 89AD03 24 31 68 20 52 17 20 17 61AD04 44 68 27 87 18 30 46 73 42AD05 58 -25 75 -28 7 31 81 58 66AD06 15 18 7 89 37 5 76 84 -4AD07 60 80 65 34 0 40 62 60 41AD08 31 51 57 41 62 21 23 -9 58TD01 45 68 27 60 58 36 10 39 72TD02 43 64 52 69 63 48 14 60 63TD03 13 81 29 74 9 78 77 85 48TD04 14 75 14 93 73 87 45 34 16TD05 26 66 62 84 29 90 75 65 31TD06 14 80 25 70 25 88 72 77 65TD07 11 74 62 71 -5 22 90 81 21TD08 13 84 53 26 8 45 93 75 20TD09 66 20 63 93 62 39 53 35 58TD10 17 81 57 9 79 -0 -13 77 71TD11 59 27 81 26 93 55 72 36 21TD12 11 63 92 72 23 80 76 67 62

Table C.2: Simulation results of the global model estimated with the filteredsignals in terms of the vaf criterion - 2.

175

List of Figures

1 Feedback control for disturbance rejection. Harmonic Control. . . . . . . . 2

1.1 Multivariable and multi-harmonic T -matrix model . . . . . . . . . . . . . 121.2 Multivariable and multi-harmonic T -matrix model. Frequency domain. . . 141.3 Uncertainty regions illustrated in the Nyquist plot at given frequency. . . . . 151.4 Multiplicative output uncertainty representation. . . . . . . . . . . . . . . . 161.5 Block diagram of the uncertainty representation. . . . . . . . . . . . . . . . 171.6 Nonlinear block diagram. Time domain representation. . . . . . . . . . . . 191.7 Polytopic T -matrix model representation. . . . . . . . . . . . . . . . . . . 22

2.1 General block diagram of frequency-domain control system. . . . . . . . . 282.2 General block diagram of discrete baseline T -matrix algorithm. . . . . . . . 302.3 Block diagram for implementing Shaw’s discrete-time HC algorithm. . . . 302.4 Trade-off Robustness/Performance - LQ-based Harmonic Control. . . . . . 352.5 Continuous-time Harmonic Control implementation. . . . . . . . . . . . . 392.6 Example of disturbance rejection with a continuous-time HC implementation. 40

3.1 Block diagram of the uncertain feedback system. . . . . . . . . . . . . . . 463.2 Augmented plant model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Frequency response of possible 1/WY (z) weighting functions. . . . . . . . 483.4 Possible HC implementation in terms of control matrices KM ,KN . . . . . . 483.5 Representation of an example iteration of the tuning algorithm. . . . . . . . 493.6 Robust HC sensitivity analysis wrt ωb and MY . . . . . . . . . . . . . . . . 513.7 Robust HC sensitivity analysis wrt ωUb and MU . . . . . . . . . . . . . . . . 523.8 Numerical Example - stability. . . . . . . . . . . . . . . . . . . . . . . . . 533.9 Numerical Example - stability. Varying uncertainty radius. . . . . . . . . . 543.10 Numerical Example - performance: convergence of the H∞ minimization. . 55

177

List of Figures

3.11 Numerical Example - performance. Monte Carlo study: steady output normin the case of the LQ-based (invariant) controller and H∞ HC controller. . 56

4.1 Decentralized control scheme. . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Decoupling control strategy for the harmonic control. . . . . . . . . . . . . 614.3 Harmonic Control Decoupling strategy with 2×2 plants. . . . . . . . . . . 664.4 Pre/Post compensator decoupling strategy for MIMO controller design. . . 684.5 Pre/Post compensator decoupling strategy for MIMO controller design. SVD

simplification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 Block diagram of the uncertain multivariable feedback system. . . . . . . . 714.7 Decoupled Robust Harmonic Control strategy. Weighting functions. . . . . 724.8 Decentralized Harmonic Control - Optimal and Robust solutions. . . . . . . 754.9 Convergence of some of the elements of compensator C. . . . . . . . . . . 76

5.1 Closed-loop system with loop nonlinearity. Lur’e form. . . . . . . . . . . . 805.2 Derivation of some classical describing functions. . . . . . . . . . . . . . . 815.3 Considered Actuator characteristic Φ(·). . . . . . . . . . . . . . . . . . . . 835.4 Describing function based on (5.6). . . . . . . . . . . . . . . . . . . . . . . 835.5 DF closed-loop Gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.6 DF Method - Closed-loop simulation with no saturation - Frequency domain. 845.7 DF Method - Closed-loop simulation with no saturation - Time domain. . . 855.8 DF Method - Closed-loop simulation with saturation included - Frequency

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.9 DF Method - Closed-loop simulation with saturation included - Time domain. 875.10 Generalized P −K −∆ plant structure. . . . . . . . . . . . . . . . . . . . 945.11 Example realization of disturbance variation - 2-vertices Affine-LPV control. 975.12 Simulation experiment - Monte Carlo study: steady output norm ‖Y ‖. . . . 985.13 Simulation experiment (performance) - Monte Carlo study: steady output

norm ‖Y ‖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.14 Disturbance variation during one Monte Carlo simulation and respective

controlled output norm ‖Y ‖. LTI/LPV comparison. . . . . . . . . . . . . . 1005.15 First scenario - Nh,l. Monte Carlo statistics for the IAVG, IMAX , ITR

indicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.16 First scenario - Nh,s. Monte Carlo statistics for the IAVG, IMAX , ITR

indicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.17 Second scenario - Nh,l. Monte Carlo statistics for the IAVG, IMAX , ITR

indicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.18 Second scenario - Nh,s. Monte Carlo statistics for the IAVG, IMAX , ITR

indicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.19 Second scenario - simulation of parameter varying (LPV) and adaptive (RL-

S/LQ) control laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.20 Simulation of parameter varying (LPV) and adaptive (RLS/LQ) control laws,

including noise on the baseline disturbance. . . . . . . . . . . . . . . . . . 1055.21 Noise inclusion on baseline disturbance - high rate variation. . . . . . . . . 1065.22 Noise inclusion on baseline disturbance - low rate variation. . . . . . . . . 106

178

List of Figures

6.1 Higher Harmonic Control and Individual Blade control implementation. . . 1126.2 Overview of possible HC architectures for baseline vibration attenuation. . 1146.3 General LTP schematic representation. . . . . . . . . . . . . . . . . . . . . 1186.4 Higher Harmonic Control declinations. . . . . . . . . . . . . . . . . . . . . 1216.5 Transfer function of the LTI out-of-plane rotor blade model as a function of µ. 1226.6 Example 2, Monte Carlo study: settling time and H2 norm of the steady-

state output - LQ controller and H∞ controller. . . . . . . . . . . . . . . . 1236.7 Closed-loop performance comparison in the first scenario. . . . . . . . . . 1256.8 Closed-loop performance comparison in the second scenario. . . . . . . . . 126

7.1 Friendcopter project: employed fuselage mock-up. . . . . . . . . . . . . . 1287.2 Friendcopter project: mounted aerodynamic brake. . . . . . . . . . . . . . 1287.3 Anti-torque plate shape example. . . . . . . . . . . . . . . . . . . . . . . . 1297.4 Actuators location on one side - details. . . . . . . . . . . . . . . . . . . . 1307.5 Sensors location on the left side of the cabin. . . . . . . . . . . . . . . . . 1307.6 Measured cabin noise spectra: test bench (solid line) and flight data (dashed

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.7 Variance accounted for as a function of model order. . . . . . . . . . . . . 1337.8 Compensator elements convergence with Algorithm 2. . . . . . . . . . . . 1377.9 Decoupling control strategy. . . . . . . . . . . . . . . . . . . . . . . . . . 1387.10 Closed-loop performance results of the MonteCarlo study. . . . . . . . . . 1397.11 Closed-loop performance results of the Monte Carlo study - Scenario 1. . . 1407.12 Closed-loop performance results of the Monte Carlo study - Scenario 2. . . 1417.13 Closed-loop performance results of the Monte Carlo study - Scenario 3. . . 141

8.1 Lag Damper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.2 Overall architecture for the vibration helicopter control with using S-AVLDs. 1458.3 Schemes of the passive and semi-active lag dampers. . . . . . . . . . . . . 1458.4 Evolution of the normalized variable p and F . . . . . . . . . . . . . . . . . 1508.5 Tracking performance of the gain-scheduling controller. . . . . . . . . . . . 1508.6 Gain scheduling control variable. . . . . . . . . . . . . . . . . . . . . . . . 1508.7 Fundamental blade motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 152

179

List of Tables

3.1 Robust HC sensitivity analysis for a desired steady performance εY = 0.02. 523.2 Numerical Example - performance. Monte Carlo study: performance of the

LQ and of the H∞ (both structured and unstructured) controllers. . . . . . . 56

5.1 Describing Function Method - Optimal VS Robust Performance . . . . . . 865.2 Initializing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1 Mechanical an aerodynamical characteristics of the considered rotor blade. 1166.2 Example 1, Monte Carlo study: performance of the LQ and of theH∞ (both

structured and unstructured) controllers. . . . . . . . . . . . . . . . . . . . 1236.3 Simulation results. HHCLQ vs HHCH∞ . . . . . . . . . . . . . . . . . . . . 126

7.1 Actuator location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2 Pairing selection of input-output channels. . . . . . . . . . . . . . . . . . . 1357.3 Desired performance and included relative multiplicative uncertainty. . . . 140

C.1 Estimated global model vaf criterion - 1. . . . . . . . . . . . . . . . . . . . 174C.2 Estimated global model vaf criterion - 2. . . . . . . . . . . . . . . . . . . . 175

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