Frequency Response Data Based LPV Controller Synthesis

1

Frequency Response Data Based LPV ControllerSynthesis Applied to a Control Moment Gyroscope

Tom Bloemers, Roland Toth and Tom Oomen

Abstract—Control of systems with operating condition-dependent dynamics, including control moment gyroscopes, oftenrequires operating condition-dependent controllers to achievehigh control performance. The aim of this paper is to develop afrequency response data-driven linear parameter-varying controldesign approach for single-input single-output systems, whichallows improved performance for a control moment gyroscope.A stability theory using closed-loop frequency response functiondata is developed, which is subsequently used in a synthesisprocedure that guarantees local stability and performance. Ex-perimental results on a control moment gyroscope demonstratethe performance improvements.

I. INTRODUCTION

Control of systems with operating condition-dependent dy-namics, including control moment gyroscopes (CMGs), oftenrequires operating condition-dependent controllers to achievehigh control performance. CMGs are attitude control devicesused, e.g., to control the attitude of spacecraft [1]. A CMG, seeFigure 1a, consists of a rotating disk, which, when spinning,generates an angular momentum. The disk is mounted ina gimbal assembly which can rotate around multiple axes.Changing the direction of the angular momentum vector,through actuation of the gimbals, generates a gyroscopictorque [2]. This torque can be used to, e.g., change the attitudeof a spacecraft. The associated dynamics are nonlinear andcharacterized by coupled behavior and challenging rotationaldynamics that change based on the operating conditions of thesystem. Locally, these behaviors manifest in terms of operat-ing condition-dependent resonant dynamics, also commonlyencountered in mechatronic systems [3]. Flexible phenomenaintroduce severe practical limitations on the achievable perfor-mance, which become even more severe in case of operatingcondition-dependent dynamics. Achieving stability and highperformance for these systems requires operating-conditiondependent controllers [4], [5].

The paradigm of linear parameter-varying (LPV) systemshas been established to provide a systematic framework toefficiently handle operating condition-dependent nonlinear dy-namics. LPV systems are characterized by a linear input-output (IO) map, whose dynamics depend on an exogenoustime-varying signal. This scheduling variable p can be usedto capture the nonlinear or operating condition-dependent

T. Bloemers (corresponding author) and R. Toth are with the ControlSystems Group, Department of Electrical Engineering, Eindhoven Universityof Technology. T. Oomen is with Control Systems Technology, Departmentof Mechanical Engineering, Eindhoven University of Technology, email:t.a.h.bloemers, r.toth, [email protected]. R. Toth isalso with the Systems and Control Laboratory, Institute for Computer Scienceand Control, Kende u. 13-17, H-1111 Budapest, Hungary.

This work has received funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme(grant agreement nr. 714663).

dynamics of a system. Typically, a priori information on thescheduling variable is known, such as the range of variation.LPV systems are supported by a well-developed model-basedcontrol and identification framework, with many successfulapplications, see [3], [6]. Model-based control techniquesrequire an accurate parameter-dependent parametric model ofthe system suitable for LPV control design. In fact, obtainingsuch a high accuracy model is a challenging task, even forlinear time-invariant (LTI) systems [7].

Frequency response function (FRF) measurements enablesystematic design of controllers directly from measurementdata and are commonly employed in the industry [7]. Afrequency response function estimate provides an accuratenonparametric description of the system that is relativelyfast and inexpensive to obtain [8]. Also the nonparametricidentification of local FRF measurements for LPV systemshas been investigated in [9], assuming that the underlyingbehavior is a smooth function of the scheduling variable.For the CMG, FRFs of the local dynamics can be accuratelycaptured at a set of operating points. FRFs enable the use ofclassical techniques such as loop-shaping, alongside graphicaltools including the Bode diagram or Nyquist plot, to designcontrollers [10]. These controllers often have a proportional-integral-derivative (PID) structure in addition to higher-orderfilters to compensate parasitic dynamics. These methods havein common that the design procedure can be difficult as theyare based on design rules, insight and experience.

Data-driven control design based on FRF measurementsprovides systematic approaches to design and synthesize LTIcontrollers. From a modeling perspective, data-driven controlsynthesis provides an alternative to control-oriented identifi-cation [11]. At first, the development of these methods havebeen along the lines of the classical control theory to tunePID controllers [12]. Later, these methods have been tailoredtowards more general control structures that focus on H∞-performance [13]. The incorporation of model uncertaintiesinto the control design enables the synthesis of stabilizingcontrollers that achieve sufficient robustness to account for thevariations in the plant [14], [15]. Robust control methods areattractive to accommodate the operating condition-dependentresonant behaviors encountered in CMGs. A major drawbackis a tradeoff between robustness and performance.

Including operating condition-dependent behavior in thedata-driven control design framework is promising to over-come the tradeoff between robustness and performance. In[16], a time-domain approach is employed to identify an LPVcontroller such that the closed-loop mimics an ideal behavior.In [17]–[19], frequency-domain control synthesis approachesare investigated. Common drawbacks are their limitations to

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stable systems only, conservative stability and performanceconstraints and the controller parameterization only allows forshaping of the zeros and not the poles.

Although frequency-domain data-driven controller synthesisenables powerful and systematic design approaches in the LTIframework, methods within the LPV framework are limitedand conservative. The aim in this paper is to develop a data-driven LPV control design method that allows both for stableand unstable systems, applicable to an experimental CMGsetup. Key steps are (i) a global LPV controller parameteriza-tion, which allows tuning of both the zeros and poles based onlocal information, and (ii) developing necessary and sufficientstability and performance analysis conditions.

The main contributions of this paper areC1) A procedure to synthesize LPV controllers for (possi-

bly) unstable single-input single-output (SISO) plantsfrom frequency-domain measurement data, with localinternal stability and H∞-performance guarantees.

C2) Highlighting the advantages of using an LPV controllerthrough application to an experimental CMG setup.

This is achieved by the following sub-contributions.C3) Developing of a local LPV frequency-domain stability

analysis condition.C4) Development of a local LPV frequency-domain H∞-

performance analysis condition.Contributions C3) and C4) are generalizations to the resultspresented in [15], [20]. Specifically, when both the plant andcontroller are LTI the results in [15] are recovered, and theresults in [20] are recovered as a special case for stablesystems. Other important differences in this paper are newinsights and proofs of these sub-contributions, which establishlinks to the robust control theory and the Bezout identity. Aglobal LPV controller parameterization in combination withC3) and C4) constitutes to C1). Application of the developedprocedures on an experimental CMG constitutes to C2).

Notation: Let R denote the set of real numbers and C theset of complex numbers. Let C0 denote the imaginary axisand C+ the open right half-plane. The real part of a complexnumber z ∈ C is denoted by <z. The set of proper, stableand real-rational transfer functions is denoted by RH∞.

Remark. Although the theory in this paper is presented incontinuous-time, a discrete-time equivalent is conceptuallystraightforward. Simply replace the variables s with z, iω witheiω and evaluate the frequencies along the unit circle insteadof the imaginary axis, i.e., for the set Ω := ω | 0 ≤ ω < 2π.

II. PROBLEM FORMULATION

A. Control Moment Gyroscope

Figure 1a depicts the considered 3 degree of freedom (DOF)control moment gyroscope. It is comprised of a disk, D,which is mounted in a gimbal assembly consisting of threegimbals C,B and A, corresponding to the schematic overviewin Figure 1b. The disk D rotates with velocity q1, generatingan angular momentum proportional to q1. Angle q2 of gimbalC is controlled through input torque τ2. Gimbal B is assumedto be fixed in place such that q3 ≡ 0, as depicted in Figure1b. Angle q4 of gimbal A is controlled through a gyroscopic

A

B

CD

(a)

A

D

C

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q2

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q1

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q4

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q3

(b)

Fig. 1: Laboratory scale control moment gyroscope by Quanser (a);and schematic overview of the 3-DOF gyroscope (b).

torque, generated by changing angle q2. As the disk tilts, achange in angular momentum causes gyroscopic torque, whichis used to position gimbal A.

The equations of motions are of the form

M(q(t))q(t) + C(q(t), q(t))q(t) = τ2(t), (1)

where q> =[q1 q2 q4

]are the angular positions, τ2 is the

input torque, M and C are the inertia and Coriolis matrices.In the used configuration of the CMG, the goal is to control

the position of gimbal A by actuating gimbal C throughinput torque τ2. The driving factor in this setting is thevelocity of the disk D, which directly relates to the amount ofgyroscopic torque that can be exerted on gimbal A. In [4], it isshown that local linear approximations describe the nonlineardynamics accurately. The aggregated collection of these localapproximations is described by the following representation

x(t) = A(q1(t))x(t) +Bu(t), (2a)y(t) = q4(t), (2b)

where x> =[q4 q2 q4

]is the state, u = τ2 the input and

y = q4 the output. The A matrix depends on the velocity ofthe disk, which can range anywhere in q1 ∈ [30, 50] rad/s.

The local description of the behavior is in line with theavailability of measurement data and the considered controlsynthesis techniques in the sequel. Furthermore, the depen-dence of the system on the disk velocity makes the LPVframework a suitable choice for modeling and control.B. LPV systems

Consider a single-input single-output, continuous-time (CT)LPV system. The LPV state-space representation

Gp :

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

y(t) = C(p(t))x(t) +D(p(t))u(t),(3)

is adopted to represent the system, see also [21]. Here, x :R → X ⊆ Rnx denotes the state variable, u : R → U ⊆ Ris the input signal, y : R → Y ⊆ R is the output signal andp : R→ P ⊆ Rnp the scheduling variable.

With a slight abuse of notation introduce

Gp =

(A(p) B(p)C(p) D(p)

)(4)

representing the LPV system with state-space form (3). IfD−1(p) is well-defined for all p ∈ P, then the LPV system

3

G

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uG

Fig. 2: Typical 1 DOF feedback interconnection, including 4-blockshaping problems, depending on the scheduling signal.

Gp has an inverse operator

G−1p =

(A(p) +B(p)D−1(p)C(p) B(p)D−1(p)

D−1(p)C(p) D−1(p)

), (5)

such that GpG−1p = G−1

p Gp = 1 for all p ∈ P.If the scheduling signal p(t) ≡ p is constant, the scheduling-

dependent matrices in (4) become time-invariant, i.e.,

Gp =

(A(p) B(p)C(p) D(p)

)(6)

represents an LTI system for constant scheduling. For a givenp ∈ P, (6) describes the local behavior of (3). Hence, (6) isreferred to as the frozen behavior of (3). Taking the Laplacetransform of (6) with zero initial conditions results in

y(s) =(C(p)(sI −A(p))−1B(p) +D(p)

)u(s), (7)

where Gp(s) = C(p)(sI − A(p))−1B(p) + D(p) and sis the Laplace variable. The frozen behavior (6) also has acorresponding Fourier transform

Y (iω) = Gp(iω)U(iω), (8)

where i is the complex unit, ω ∈ R is the frequency andGp(iω) represents the frozen Frequency Response Function(fFRF) of (3) for every constant p(t) ≡ p ∈ P [22].C. Problem statement

The problem addressed in this paper is to design an LPVcontroller directly from fFRF measurement data obtainedfrom the considered CMG. We denote the data DN,pτ =Gp(iωk),pτNk=1, obtained at the set of operating pointsP = pτNloc

τ=1 ⊂ P. We assume the frequencies are sufficientlydense such that it suffices to check a finite number of discretepoints to draw conclusions on the underlying continuouscurve. Consider the feedback interconnection in Figure 2. Theobjective is to design a controller Kp such that the followingrequirements are satisfied.

R1) The closed-loop system in Figure 2 is internally stablein the local sense for all p(t) ≡ p ∈ P .

R2) The performance channels of the closed-loop systemare bounded in the local H∞-norm sense for all p ∈ P .

In the next section, a rational controller parameterization isintroduced that allows for a specific formulation of internalstability. This forms the basis to develop analysis conditionsfor internal stability and H∞-performance. The theory is firstformulated for p ∈ P for the sake of generality. This alsoensures R1) and R2) for p ∈ P .

III. STABILITY AND PERFORMANCE ANALYSIS

In this section, we develop local LPV stability and perfor-mance analysis conditions. These results form the basis for adata-driven synthesis procedure. First, a continuous frequencyspectrum Ω = R ∪ ∞ is considered, which will be

restricted later to a finite frequency grid ΩN = ωkNk=1

corresponding to DN,pτ .A. Stability

The selection of input-output (IO) pairs in Figure 2 corre-sponds to the problem of internal stability [23, Chapter 3]. Fora fixed p ∈ P, we define the IO map T (Gp,Kp) : (r,−d) 7→(e, u) in Figure 2 by

T (Gp,Kp) =

[Sp SpGp

KpSp Tp

], (9)

with Sp = (1 + GpKp)−1 and Tp = 1 − Sp. If Gp,Kp ∈RH∞, then T (Gp,Kp) is internally stable if all elements inthe IO map T (Gp,Kp), defined by (9), are stable. This isimplied by Sp ∈ RH∞ [23, Chapter 3]. If T (Gp,Kp) ∈RH∞ holds for all p ∈ P, then the closed-loop LPV systemis called locally internally stable. Internal stability is imporantto prevent hidden pole-zero cancellations. To assess internalstability for unstable Gp or Kp, introduce the factorization

Gp = NGpD−1Gp, NGp

, DGp ∈ RH∞. (10)

The two transfer functions NGp, DGp

are a coprime factor-ization over RH∞ if there exist two other transfer functionsXp, Yp ∈ RH∞ such that they satisfy the Bezout identity

NGpXp +DGp

Yp = 1. (11)

Consequently, XpQ,YpQ are coprime iff Q,Q−1 ∈ RH∞.Correspondingly, Kp admits the coprime factorization

Kp = NKpD−1Kp, NKp

, DKp ∈ RH∞. (12)

Using these representations, (9) can be written as

T (Gp,Kp) = D−1p

[DGp

DKpNGp

DKp

DGpNKp

NGpNKp

], (13)

with characteristic equation

Dp = DGpDKp

+NGpNKp

. (14)

The feedback system is internally stable if and only if D−1p ∈

RH∞. If we set NKp = Xp and DKp = Yp, then thecharacteristic equation (14) equals the Bezout identity (11),thus the feedback system is internally stable as D−1

p = 1 andthe rest of the terms are stable by design in (13). Similarly,the closed-loop LPV system is called locally internally stableif these conditions hold for all p ∈ P.

For the transfer w 7→ z, with w ∈ r, d and z ∈ e, u, let

Tz,w(Gp,Kp) = NpD−1p , (15)

with Np, Dp ∈ RH∞ and Tz,w(Gp,Kp) ∈ RH∞, de-fines the corresponding SISO element of (13). For example,Tr,e(Gp,Kp) = NpD

−1p with Np = DGpDKp defines the

sensitivity Sp in (9) and (13).The following theorem presents analysis conditions to verify

internal stability of a closed-loop system locally, given theplant and controller only. As a special case, [20, Theorem1] is recovered. Here, the idea of coprime factorizations overRH∞ is used to allow for unstable plants or controllers, whilealso extending the result to the class of LPV systems.

Theorem 1. Let Gp and Kp be as defined in (10) and (12),respectively, and let Dp ∈ RH∞ be as defined in (14). Thenthe following conditions are equivalent. For all p ∈ P

4

1a) D−1p ∈ RH∞.

1b) Dp(s) 6= 0, ∀s ∈ C+ ∪ C0 ∪ ∞.1c) There exists a multiplier αp ∈ RH∞ such that

<Dp(iω)αp(iω) > 0, ∀ω ∈ Ω.

Proof. For a proof of equivalence between 1a) and 1b), see[23, Chapter 3]. Regarding the equivalence between 1a) and1c) for all p ∈ P, note the following reasoning:

(⇒) Assume 1a) and let Q = D−1p . This implies that the

Bezout identity (11) is satisfied for Xp = NKpQ and Yp =DKp

Q. Hence, 1c) is satisfied by setting αp = Q because<NGp

Xp +DGpYp = 1 for all ω ∈ Ω.

(⇐) Assume 1c) and let V = Dpαp. Note that V, V −1 ∈RH∞ because 1c) implies that Dpαp is bi-proper and has noright half-plane (RHP) zeros. Then Dp = V α−1

p satisfies theBezout identity (11), therefore D−1

p ∈ RH∞. Thus 1c) implies1a) and consequently 1b). This completes the proof.

Remark. A direct result of Theorem 1 is that α−1p ∈ RH∞.

This is easy to prove becausei) There does not exist a strictly proper αp ∈ RH∞ such

that 1c) holds. Indeed 1c) is violated at ω =∞.ii) There does not exist an αp ∈ RH∞ with α−1

p /∈ RH∞such that 1c) holds. This can be seen as α−1

p /∈ RH∞implies that there exists some RHP zero s0 such thatαp(s0) = 0. Consequently, there exists some frequency ω0

such that <Dp(iω0)αp(iω0) < 0 and 1c) is violated.

Theorem 1 gives an analysis condition that provides a localstability result for the closed-loop system if instead of aparametric model, NGp

and DGpare only given in terms of

local frequency-domain data. The next subsection presents theextension towards a performance analysis condition.B. Performance

In this subsection, analysis conditions to assess locally theH∞-performance of an LPV system, given the plant andcontroller only, are presented. This constitutes contributionC4). To derive performance analysis conditions, the main looptheorem is of importance and is presented first.

Consider the transfer function Tz,w(Gp,Kp) ∈ RH∞ ofinterest in Figure 3a, such that w 7→ z : Tz,w(Gp,Kp), andlet ∆ ∈ B∆, with

B∆ :=

∆ ∈ RH∞∣∣∣ |∆(iω)| < 1, ∀ω ∈ Ω

(16)

a fictitious uncertainty, represent the H∞-performance crite-rion. Then, the H∞-performance of the system in Figure 3ais equivalent to Figure 3b [24, Theorem 8.7]. This is statedin terms of the following theorem, where the weighting filterWT is introduced to specify the frequency-dependent designrequirements on the map w 7→ z.

Theorem 2 (Main loop theorem). Let WT ∈ RH∞ andTz,w(Gp,Kp) be defined as in (15). The following statementsare equivalent. For all p ∈ P

2a) supω∈Ω|WT (iω)Tz,w(Gp,Kp)(iω)| ≤ γ.

2b) 1− γ−1WT (iω)Tz,w(Gp,Kp)(iω)∆(iω) 6= 0,

∀ω ∈ Ω, ∀∆ ∈ B∆.

Theorem 2 is a special case of [25, Theorem 11.7].

P

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K(p)

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p

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p


(a)

P

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K

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w

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z

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y

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u

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z

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w

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wK

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zK

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(p)

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K(p)

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p


p


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(b)

Fig. 3: Generalized LPV plant (a); and performance of the SISOclosed-loop map w 7→ z (b).

Remark. By Theorem 2, nominal performance can be seen asa special case of robust stability, where a fictitious uncertaintyis connected to the performance channel, see Figure 3b.

In the data-driven setting, the absence of a parametric modelof Tz,w(Gp,Kp) makes it difficult to turn 2b) into a convexconstraint as it is generally done in LPV synthesis approachesfor gain-scheduling [6]. Hence, in that case 2b) is needed tobe evaluated for an infinite set of realizations of the fictitiousuncertainty ∆, for example, as in [26]. The contribution inthis paper is to utilize Theorem 1 together with Theorem2 to derive a single condition to analyze both stability andperformance without the need to sample ∆.

Theorem 3. Let WT ∈ RH∞ and Tz,w(Gp,Kp) be definedas in (15). Requirements R1) and R2) are satisfied if and onlyif there exists a multiplier αp ∈ RH∞ such that

<(Dp(iω)− γ−1|WT (iω)Np(iω)|)αp(iω) > 0,

∀ω ∈ Ω, ∀p ∈ P.(17)

Proof. Requirement R2) can be equivalently stated usingTheorem 2, Condition 2b), i.e.,

1− γ−1WT (iω)Tz,w(Gp,Kp)(iω)∆(iω) 6= 0,

∀ω ∈ Ω, ∀p ∈ P, ∀∆ ∈ B∆.(18)

As Dp ∈ RH∞, Dp(iω) 6= 0, ∀ω ∈ Ω and by multiplying(18) with it, the resulting non-singularity condition is:

Dp(iω)− γ−1WT (iω)Np(iω)∆(iω) 6= 0,

∀ω ∈ Ω, ∀p ∈ P, ∀∆ ∈ B∆.(19)

Based on a homotopy argument, (19) corresponds to Condition1b) in Theorem 1, which through 1c) is equivalent with

<(Dp(iω)− γ−1WT (iω)Np(iω)∆(iω))αp(iω)>0,

∀ω ∈ Ω, ∀p ∈ P, ∆ ∈ B∆.(20)

When ∆ = 0 ∈ B∆, (20) reduces to <Dp(iω)αp(iω) > 0,which is the same as Condition 1c) in Theorem 1, hence (20)implies requirement R1).

Let 1 ≥ ε > 0 and consider (20) on

Bε∆ :=

∆ ∈ RH∞∣∣∣ |∆(iω)| ≤ 1− ε, ∀ω ∈ Ω

, (21)

which is the scaled closed uncertainty ball contained in B∆.Since any ∆ ∈ Bε∆ represents a rotation and contraction

5

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Dp

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1|WT Np|Fig. 4: Illustration of stability and H∞-performance. The transferfunction αp represents, for each frequency, a line passing through theorigin. If this line does not intersect with the disks Dp−γ−1|WTNp|,then the disks exclude the origin and (18) must hold.

in the complex plane, it is necessary and sufficient to check(20) on the boundary only, i.e., for ∆ ∈ ∂Bε∆, with|∆(iω)| = 1− ε, ∀ω ∈ Ω. Note that, in (20), WT (iω)Np(iω)only represents complex scaling of this ball which is centeredat Dp(iω). Hence, (20) restricted on Bε∆ is equivalent with

<(Dp(iω)− γ−1(1− ε)|WT (iω)Np(iω)|)αp(iω) > 0,

∀ω ∈ Ω, ∀p ∈ P. (22)

This means that if (22) holds, then violation of (20) can onlyhappen in B∆ \Bε∆. As (22) is continuous in ε, by takingthe limit ε → 0, B∆ \Bε∆ → ∅ and we obtain that (17) isequivalent with (20).

Theorem 3 states that the performance condition 2a) issatisfied if and only if for each frequency ω ∈ Ω andscheduling value p ∈ P the disks with radius γ−1|WTNp|,centered at Dp, do not include the origin. This holds if thereexists αp ∈ RH∞, representing for each frequency a linepassing through the origin, that does not intersect with thedisks, see Figure 4. The analysis condition is especially usefulas it provides a local stability and performance result given acontroller and the data DN,pτ .

If the fFRFs are subject to model uncertainty, robust stabilityand performance have to be taken into account [15].C. Synthesis

It turns out that it is possible to give an equivalent formu-lation of Theorem 3 which enables controller synthesis.

Theorem 4. Given Gp = NGpD−1Gp

, with NGp, DGp

∈RH∞ coprime, as defined in (10), and a weighting filterWT ∈ RH∞, the following statements are equivalent.

4a) There exists a proper rational controller Kp thatachieves internal stability and performance as definedin requirements R1) and R2), respectively.

4b) There exists a controller Kp = NKpD−1Kp

, withNKp

, DKp ∈ RH∞, as defined in (12), such that

<Dp(iω) > γ−1|WT (iω)Np(iω)|,∀ω ∈ Ω, ∀p ∈ P.

(23)

Proof. (⇒) Assume Kp = NKpD−1Kp

satisfies 4a). Then, byTheorem 3, there exists an αp ∈ RH∞ such that (17) holds.Choosing NKp = NKpαp, DKp = DKpαp results in Kp =

NKpD−1Kp

= NKpD−1Kp

and consequently 4b) holds.(⇐) Assume 4b) holds. Because Dp ∈ RH∞ and Dp(iω)

is positive for all ω ∈ Ω, NKp , DKp form Bezout factorsfor NGp

, DGp. Thus by Theorem 1, D−1

p ∈ RH∞ and

Kp internally stabilizes Gp and R1) holds. By Theorem 3,requirement R2) holds. This completes the proof.

Theorem 4 presents a local H∞-optimal controller synthesiscondition given only data DN,pτ . This is further developed inSection IV, where an optimization problem is formulated andthe controller parameterization is discussed.

Remark. Theorem 4 shows that the multiplier αp can be ab-sorbed into the controller as γ−1|WT (iω)Np(iω)αp(iω)| ⇒<γ−1|WT (iω)Np(iω)|αp(iω). Note that the absorbed mul-tiplier changes the considered Np and Dp, but αp cancels outwhen Kp = NKp

D−1Kp

is computed. The price to be paid forthis absorption is the increased order of NKp

and DKp.

Remark. [15, Theorem 1] is recovered in the special casewhen the plant and controller are LTI .

IV. CONTROLLER SYNTHESIS

In this section, we build upon the stability and performanceanalysis and synthesis conditions derived in Section III bydeveloping a procedure to synthesize LPV controllers. Thisforms Contribution C1). First, an optimization problem is setup in Section IV-A that characterizes the synthesis problembased on Theorem 4. This is followed by a discussion on thecontroller parameterization in Section IV-B and implementa-tion aspects in Section IV-C.A. Controller synthesis

Given the data DN,pτ and a controller parameterizationKp = NKp

D−1Kp

, given in the Section IV-B, an optimizationproblem is formulated satisfying requirements R1) and R2).

minθ,γ

γ

s.t. γ<Dp(iω, θ) > |WT (iω)Np(iω, θ)|∀ω ∈ Ω, p ∈ P

(24)

where θ are the controller parameters.The optimization problem (24) is in general non-convex.

However, through a linear parameterization of the controller,(24) becomes a quasi-convex optimization problem in thecontroller parameters θ and the performance indicator γ. Tosolve the quasi-convex program, a bisection algorithm overγ is utilized. This results in an iterative approach, where forevery fixed value of γ, a second-order cone program is solved.

To provide stability and performance guarantees, the con-straints in (24) need to be satisfied for all ω ∈ Ω, which is aninfinite set, leading to a semi-infinite program. One solution isto solve (24) for a finite set of frequencies ΩN = ωkNk=1 ⊂Ω. The frequency set can be chosen randomly, according tothe scenario approach [27]. This allows for the computationof confidence bounds on the constraints. In the data-drivensetting this choice is spared from the user as the data is onlyavailable at a pre-specified set of frequency points. Eitherof these methods result in a quasi-convex second-order coneprogram and can be solved as described above.B. Controller parameterization

An orthonormal basis function (OBF)-based representation[21] is a natural choice to parameterize the controller factors

NKp(s) =

∑nNi=0 wi(p)φi(s), (25a)

6

W (p)

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...

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p

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y

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u

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y0

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yn

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Fig. 5: Input-output graph of the Wiener LPV OBF structure.

DKp(s) =∑nDi=0 vi(p)ϕi(s), (25b)

Here, φinNi=0 and ϕinDi=0 with φ0 = ϕ0 = 1 and nD ≥ nNare the sequence of basis functions, with coefficient functions

wi(p) =∑m`=1 w

ìψ`(p), vi(p) =

∑m`=1 v

ìψ`(p). (26)

Here, the coefficient functions are formed through a chosenfunctional dependence, e.g., affine, polynomial or rational,characterized by the basis functions ψ`m`=1. See [21, Chapter9.2] for an overview of OBF based LPV model structures.The OBF controller parameterization enables tuning of boththe poles and zeros of the controller, in contrast to previousdata-driven frequency-domain LPV tuning methods [17]–[19].Additional controller requirements are discussed in [28].

Local aspects of (25a)-(25b) can be preserved by consider-ing a time-domain Wiener LPV OBF realization

yNKp (t) =∑nNi=0 wi(p(t))yφ(t), (27a)

yDKp (t) =∑nDi=0 vi(p(t))yφ(t), (27b)

with yφ = Φu. The parameterization of NKp and DKp canbe viewed as a bank of OBFs, whose output is weighted withparameter-dependent coefficient functions, see Figure 5.

Equations (27a)-(27b) reveal that requirements (i)-(iv) aresatisfied. Requirement (v) is satisfied, w.l.o.g. by vìì=0 =1, 0, . . . , 0. Because the set of bases is complete w.r.t. H2,hence any solution including the optimal solution of (24) canbe found via parameterizations (25a)–(25b) [15].

Remark. Note that (27a) and (27b) depend on time-varyingp and characterize the global behavior of the factors NKpand DKp . The concept in this paper is to tune the parameter-dependent coefficient functions based on their local behavior,i.e., (25a)-(25b) for constant p, in-line with the data DN,pτ .

Algorithm 1 presents the selection of optimal OBFs, basedon the Kolmogorov n-Width theory. Given a desired numberof poles, an optimal set of OBFs is selected based on FuzzyKolmogorov c-Max (FKcM) clustering of the poles, such thatthe decay rate of the OBFs is minimized [21, Chapter 8].

Algorithm 1: Basis function selection1 Choose arbitrary bases φinNi=0 and ϕinDi=0, solve (24) forθ = w`

im`=1 ∪ vìm`=1 and compute NKp and DKp .2 Given NKp and DKp , compute the corresponding local pole

and zero variations of the controller. Choose new bases φiand ϕi based on FKcM clustering of the poles and zeros.

3 Solve (24) and compute NKp and DKp .4 Stop if a desired performance and order of the bases has

been achieved, otherwise go to step 2.

C. Controller implementationThe OBF parameterizations admit a linear fractional rep-

resentation (LFR). In this representation, the dependency on

wD

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wN

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zD

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y

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u

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N

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D(p)

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N (p)

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D1

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p

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p


(a)

u

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p


K(p)

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K

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y

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zK

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wK

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(b)

Fig. 6: Controller realization through (a) the series connection of theLFRs NKp = Fu(N ,∆N (p)) and DKp = Fu(D−1,∆D(p)) and(b) Kp = Fu(K,∆K), with ∆K = diag(∆N ,∆D).

the scheduling variable p is extracted by formulating (27a)and (27b) in terms of LTI systems, denoted N and D, suchthat NKp = Fu(N ,∆N (p)) and D−1

Kp= Fu(D−1,∆D(p)),

respectively, where Fu is the upper linear fractional trans-formation [25], see Figure 6a. The inverse D−1 is obtainedthrough partial inversion of the IO map, see, e.g., [25, Chapter10]. The controller is formed through the series connectionof the LFRs N and D−1, resulting in the LFR K such thatKp = Fu(K,diag(∆N ,∆D)), see Figure 6b.

V. CONTROL DESIGN FOR THE CMGIn this section, a controller is designed and implemented on

the CMG. Although the theory in this paper is presented incontinuous-time, with this example we show that a discrete-time application is possible. The system identification andcontroller design are performed at a sampling rate of 200 Hz.A. Frequency-domain measurements

As described in Section II, the dynamics of the CMG aredependent on the velocity of the disk. It is therefore naturalto consider the velocity q1(t) = p(t) as a scheduling variable.The disk velocity operates in the range P = [30, 50] rad/s.To identify the local behavior at different disk velocities,an equidistant grid P = 30, 40, 50 is chosen. As thegyroscope is inherently an unstable system, the measurementsare performed in closed-loop using a stabilizing LTI controller.

The coprime factors NGp(iω) and DGp

(iω) can be cal-culated from the estimates of the process sensitivity SpGp

and sensitivity Sp, respectively [15]. This is achieved byestimating the fFRF of the mappings d 7→ y and d 7→ uG,respectively, in Figure 2. During a closed-loop experiment,the system is excited by a white-noise disturbance signal d.The position of gimbal A is measured with an optical encoder.Data records with a length of 240000 samples are collectedfor each operating point p ∈ P .

The obtained fFRFs are estimated using the empirical trans-fer function estimate, using a Hanning window, and contain1000 frequency points per operating point. Figure 7 shows theestimated fFRFs Gp. The figure highlights that the system issubject to a relatively high noise level, which has a significanteffect at higher frequencies. The scheduling dependency isalso clear to see, which manifests in terms of a shift in theresonance frequencies and the low-frequency gain.B. Data-driven controller synthesis

The goal is to control the position q4 of gimbal A by actu-ating gimbal C through torque τ2. To highlight the parameterdependence, the objective is to track a reference signal subject

7

Fig. 7: Bode plot of the estimated fFRFs of Gp for the threegrid points p ∈ P = 30, 40, 50 in blue, orange and yellow,respectively. A parameter-varying shift in resonance frequencies andlow-frequency gain is observed.

to variations in the disk velocity. To specify this objectivein terms of control design, consider the full 4-block shapingproblem in Figure 2. Based on the fFRFs in Figure 7, the firstresonance occurs at 1.7 Hz. The shaping filters are designedsuch that a bandwidth of 0.75 Hz is achieved. The sensitivity isshaped to provide a lower bound on the bandwidth and to limitthe overshoot by providing an upper bound of 6 dB for higherfrequencies. Integral action is desired to achieve 0 steady-stateerror. To suppress the effects of measurement noise whilealso limiting high-frequent control actions, a high-frequentroll-off is enforced into the controller by shaping the controland complementary sensitivities. Shaping the complementarysensitivity also provides an upper bound on the achievedbandwidth. The process sensitivity is restricted to lie below0 dB to limit the amplification of disturbances.

Using the approach presented in this paper, an LPV andLTI controller are synthesized, for which the results are givenin Figures 8 and 9. Both controllers are parameterized bydiscrete-time Laguerre bases of orders nK = nD = 5 withpole z = 0.7. The LPV controller has affine scheduling-dependance, and the LTI controller is scheduling-independent.The achieved performance levels are γLPV = 1.2097 andγLTI = 3.1792. The LTI controller does not meet the per-formance criteria for all operating points and, therefore has tosacrifice performance in order to achieve robust performance.The LPV controller achieves good performance for the con-sidered operating space by compensating for the parameter-dependent low-frequency gain and resonance behavior.

C. Results

First, the tracking performance is evaluated locally, whenthe scheduling variable operates at constant velocities P =30, 40, 50 rad/s. Figure 10 shows the measured step re-sponses using the designed LPV and LTI controllers. The maindifferences are observed for p = 30 and p = 50 rad/s. Atp = 30 rad/s, the step response shows a significant oscillationwhen using the LTI controller. This oscillation correspondsto the resonance frequency at 1.7 Hz in Figure 8 and it issignificantly larger compared to the LPV case. For p = 50rad/s, a slightly higher bandwidth is achieved when using theLPV controller, which corresponds to a faster rise and settling

Fig. 8: Magnitude plots of the fFRFs of the 4-block (9). The LPVand LTI designs are shown in blue and orange, respectively. Theweighting filters are shown in black. The LTI controller design doesnot meet the performance specifications.

Fig. 9: Magnitude and phase plots of the LPV controller for frozenscheduling-values P = 30, 40, 50 in blue orange and yellow,respectively. The LPV controller compensates the parameter-varyinglow-frequency gain and resonance behavior observed in Figure 7.

time. Finally, the responses when using the LPV controller arevery consistent, with only a small variation in settling time.

Next, the performance is evaluated for a time-varyingscheduling variable. A Square wave reference signals, filteredwith a 3rd order low-pass filter with a cut-off frequency of0.7 Hz, are used to challenge the system. The amplitude of thereference is 15. The scheduling variable, i.e. the disk velocity,tracks a similar, but faster square wave trajectory in the rangeP = [30, 50] rad/s. Implementation of the controller is doneaccording to the LFR representation described in Section IV-C,where the controller is scheduled at each sampling interval.

Figure 11 shows the reference signal, tracking performance,scheduling variation and control effort for the designed LPVand LTI controllers. The results indicate that the LPV con-troller performs significantly better than the LTI controller. Areduction in overshoot and settling time are observed. Morespecifically, we obtain a 39% and 33% decrease between the `2and `∞ norms of the error signals, respectively. These resultsexperimentally validate the capabilities of the proposed controlmethodology, including the benefit of using an LPV controllerover an LTI controller for the CMG. However, it is imperativeto note that stability and performance guarantees are providedonly locally. Hence, stability and performance of the nonlinear

8

Fig. 10: Measured local step responses of the CMG for constantscheduling variables P = 30, 40, 50 rad/s, from top to bottom,respectively. The angle of gimbal A is shown in blue and orangewhen using the LPV and LTI controller, respectively. The LTI designloses performance for p ∈ 30, 50 rad/s, whereas the LPV designdisplays consistent results for the considered operating points.

Fig. 11: Experimental results of the CMG. The top figure showsthe reference (black) and the angle of gimbal A when using theLPV and LTI controllers in blue and orange, respectively. The otherfigures shows the error, scheduling and input signals. The LPV designsignificantly improves the performance by decreasing the overshoot.

system can only be guaranteed for sufficiently slow variationsof the scheduling variable.

VI. CONCLUSION

The LPV controller synthesis approach in this paper en-ables the design of operating condition-dependent controllersdirectly from frequency-domain data. Experimental demon-strations on a control moment gyroscope show that significantincrease in performance can be achieved via the proposedapproach for operating condition-dependent systems. In com-parison to existing methods in the literature, this approachenables the design of rational LPV controllers, for which localstability and performance analysis certificates are provided.

Future research aims at global stability and performanceguarantees.

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Documents

Frequency Response Data Based LPV Controller Synthesis