Available online at www.sciencedirect.com

Journal of the Franklin Institute 355 (2018) 2221–2242 www.elsevier.com/locate/jfranklin

Static output-feedback robust gain-scheduling control with guaranteed H 2

performance

Ali Khudhair Al-Jiboory

a , b , ∗, Guoming Zhu

b

a Mechanical Engineering, University of Diyala, Baqubah 32001, Iraq b Mechanical Engineering, Michigan State University, East Lansing, Michigan 48824, USA

Received 21 November 2016; received in revised form 1 October 2017; accepted 26 December 2017 Available online 2 February 2018

Abstract

This paper develops synthesis conditions for Static Output-Feedback (SOF) Gain-Scheduling (GS) control with guaranteed upper bound of H 2 performance for continuous-time Linear Parameter Varying (LPV) systems, where measurements of scheduling parameters are affected by uncertainties or mea- surement noise. The control problem is solved through an iterative two-stage design procedure. In the first stage, parameter-dependent state-feedback controller is obtained to minimize upper-bound of the H 2 performance. Then, this controller is used as input to the second stage to synthesize Robust Gain-Scheduling (RGS) static output-feedback gain with minimal H 2 performance. In both stages, the synthesis conditions are given in terms of Parametrized Linear Matrix Inequalities (PLMIs). Robust SOF (parameter-independent) controller can be synthesized as special case of the developed synthesis conditions. Two examples have been presented to illustrate the benefit of the proposed approach. One is an academic numerical example from literature and the other one is a realistic LPV model of Electric Variable Valve Timing (EVVT) actuator for automotive engines that utilizes engine speed and vehicle battery voltage as time-varying scheduling parameters. © 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1

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. Introduction

The systematic design of Gain-Scheduling (GS) controllers for Linear Parameter-VaryingLPV) systems received significant attention in the control community literature for the past

∗ Corresponding author at: P. O. box 9, College of Engineering, University of Diyala, Baqubah, 32001 IQ, +964 73 907 5768.

E-mail address: ali_khudhair@uodiyala.edu.iq (A.K. Al-Jiboory).

ttps://doi.org/10.1016/j.jfranklin.2017.12.037 016-0032/© 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

2222 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

two decades. The effectiveness of this interesting design method is proved in many practicalengineering applications [1] . When scheduling parameters are accessible (in real-time) through

measurements or estimations for feedback control, improved performance can be achieved [2] . Furthermore, formulation of the controller design problem as a convex optimization problem

with Linear Matrix Inequalities (LMIs) constraints, is very tempting feature since optimality

is guaranteed. Application examples that proved the efficacy of LPV control include aircrafts [3] , automotive [4] robotic manipulators [5] , and miscellaneous mechatronic systems [6,7] .

In the majority of GS design work, a popular assumption is adopted in which exact mea-surements of scheduling signals are available for feedback control. Since in practical control applications perfect measurements or estimations of the time-varying parameters are hard (if possible) to obtain, exact values of scheduling parameters are non-accessible due to calibration

errors, sensors impression, etc. In other words, the mismatch between the measured and ac-tual scheduling parameters is a reality. Furthermore, as pointed out in [8] , small measurementerrors in the scheduling parameters could not only degrade system performance but could

also destabilize the overall system. Therefore, the Gain-Scheduling (GS) controllers must be insensitive to measurement errors of scheduling parameters. The solution of this interesting

control problem is the focus of this paper in the Static Output-Feedback (SOF) framework. Although there are different methods in literature [8–11] that address uncertainties in

scheduling parameters, this control problem is still a hot research topic and need to beexplored. In [8] , gain-scheduling synthesis conditions to guarantee prescribed performance bound with noisy scheduling parameters, are developed. However, these conditions assume uncertainties are multiplicative and cannot deal with bias errors that are widely common tomany measurement systems. In [9,12] , dynamic output-feedback controllers are studied to de- velop synthesis conditions for the same control problem. These developed conditions guarantee closed-loop stability and performance with parameter-independent Lyapunov matrix (which is well-known as quadratic stability approach). As pointed out in [13] , such approach is con-servative when the rates of change of scheduling parameters are known, furthermore, some practical systems cannot be stabilized with constant Lyapunov matrix. As a remedy, a good

results are reported in [11] that utilizes Parameter-Dependent Lyapunov Function (PDLF) for controller synthesis with guaranteed performance. However, it is difficult, if possible, to

obtain a controller that do not depend on the derivative of the scheduling parameters whenusing PDLF without restricting one of the Lyapunov matrices to be parameter-independent (see [14] for more details). For state-feedback controllers, synthesis conditions are developed

in [15,16] . Despite of the aforementioned research work and to the best authors’ knowledge, there is

no result reported in literature on the design of Static Output-Feedback (SOF) LPV controllerswith uncertain scheduling parameters. It is well-known that designing SOF controller is chal- lenging even for Linear Time-Invariant (LTI) system since it implies NP-hard, non-convex

optimization problem with Bilinear Matrix Inequalities (BMIs) (see [17] for a recent survey). The main contribution of the paper is the development of novel Parametrized Linear Matrix

Inequalities (PLMIs) conditions to design Robust Gain-Scheduling (RGS) SOF controller for continuous-time LPV systems subject to uncertain scheduling parameters with guaranteed H 2

performance 1 . The proposed design method consists of two stages controller design approach.

1 The “H 2 performance” is slightly abused terminology here since the root of the H 2 control theory trace back to Linear Time-Invariant (LTI) systems. However, since the current problem setup is in the LPV framework, we utilize this term to facilitate problem setting for the reader. The strict definition will be given later.

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2223

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n the first stage, parameter-dependent state-feedback control that minimizes the upper-boundf the H 2 performance is designed, then this controller is utilized as input to the conditionsf the second stage to obtain the RGS SOF control gain which minimizes the upper-bound ofhe H 2 performance again. The synthesized SOF controller utilizes the measured schedulingarameters which are assumed to be corrupted by measurement noise. The interesting featuref the developed conditions is the controller gain matrix is computed through extra slackariables and independent of Lyapunov matrix and the system open-loop matrices. As aesult, the developed approach allows all the open-loop system matrices to be affected by theime-varying parameters. As the authors know, the existing results in literature require theontrol input and/or the measurement output matrices to be parameter-independent to designGS output-feedback controller. Additionally, robust SOF controller can be synthesized in a

imilar way by restricting the synthesis variables to be parameter-independent (constants). Thencertainties in scheduling parameters are modeled as time-varying parameters independentf the scheduling parameters. The time-varying parameters and associated uncertainties areodeled in a convex domain through multi-simplex approach [18] . It is assumed that the

ncertainties are independent of the scheduling parameters. Iterative procedure (Iterative Staticutput-Feedback Design (ISOFD) algorithm) is also developed to reduce the achievable H 2

ound gradually. The paper is organized as follows, introductory concepts of LPV systems, definitions,

odeling approach are given in Section 2 . The formulation of the RGS SOF control Problem islso presented in this section. Characterization of PLMI conditions for synthesizing RGS SOFontrollers are given in Section 3 , along with the ISOFD algorithm. To illustrate the benefit ofhe developed RGS SOF synthesis conditions, two examples are presented in Section 4 . Ones an academic example from literature and the other is a realistic model of Electric Variablealve Timing (EVVT) actuator for automotive engines. Section 5 concludes the paper withome remarks and conclusions.

The notation used in this paper is standard. A positive (negative) definite matrix A isefereed by A > 0 ( A < 0). The set of real numbers is denoted by R . The symbol � indicatesymmetric terms in matrix inequalities. The trace of the matrix A is denoted by trace ( A ).dentity matrix of size n ×n is defined as I n . Similarly, 0 n×n u refers to zero matrices of size ×n u . When the size of a matrix can be inferred, subscripts will be dropped. The transposef a matrix A is written as A

′ . The notation A + (•) ′ = A + A

′ is used to shorten formulas.sup’ is used to refer to the supremum and E { ·} is the mathematical expectation of a randomariable.

. LPV systems and preliminaries

Consider the following linear parameter varying system :

˙ x (t ) = A (ρ(t )) x(t ) + B u (ρ(t )) u(t ) + B w

(ρ(t )) w(t )

z(t ) = C z (ρ(t )) x(t ) + D zu (ρ(t )) u(t )

y(t ) = C y (ρ(t )) x(t ) ,

(1)

here ρ(t ) = [ ρ1 (t ) , ρ2 (t ) , . . . , ρm

(t ) ] ′ is a vector of independent m time-varying schedulingarameters. x(t ) ∈ R

n , u(t ) ∈ R

n u , w(t ) ∈ R

n w , z(t ) ∈ R

n z , and y(t ) ∈ R

n y is the states, theontrol input, the disturbance input, the performance output, and the measured output, respec-ively. It is assumed that x(0) = 0. The open-loop system matrices have compatible dimensions

2224 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

A (ρ(t )) ∈ R

n×n , B u (ρ(t )) ∈ R

n×n u , B w

(ρ(t )) ∈ R

n×n w , C z (ρ(t )) ∈ R

n z ×n , D zu (ρ(t )) ∈ R

n z ×n u ,

and C y (ρ(t )) ∈ R

n y ×n . The parametrization of the open-loop system matrices in Eq. (1) are assumed to be affine,

i.e., each of these open-loop matrices can be represented by the following parametrization,

A (ρ(t )) = A 0 +

m ∑

j=1

ρ j (t ) A j . (2)

The measured scheduling parameters, referred by ˜ ρ(t ) , are assumed to be affected bymeasurement noises, ˜ δ(t ) , such that

˜ ρ(t ) = ρ(t ) + δ(t ) (3)

or in the scalar form ˜ ρ j (t ) = (ρ j (t ) + δ j (t )) , j = 1 , 2, . . . , m, where δj ( t ) is the j th uncer-tainty associated with the j th scheduling parameter. These scheduling parameters, uncertain- ties, and their rates of variations are assumed to be independent of each other and boundedas,

ρ j (t ) ∈

[−ρ̄i , ρ̄ j ], δ j (t ) ∈

[−δ̄i , δ̄ j ],

˙ ρ j (t ) ∈

[−b ρ j , b ρ j

], ˙ δ j (t ) ∈

[−b δ j , b δ j

], j = 1 , 2, . . . , m,

(4)

Remark 1. It has been assumed that the scheduling parameters and their rates of variationsare bounded as defined in Eq. (4) . This assumption is not restrictive since all physical variablesin practical engineering applications and their rates are limited. However, the rate of change of measurement noise is not straight forward to obtain. If it is possible to measure thescheduling parameters prior to controller design, the rate of change of measurement noise can

be calculated based on the bandwidth of the noise spectrum. If this scenario not possible, itis recommended to set these bounds to be sufficiently large.

For the open-loop LPV system (1) , the goal is to design RGS SOF gain-scheduling con-troller,

u(t ) = K ( ̃ ρ(t )) y(t ) (5)

that robustly stabilizes the closed-loop system

˙ x (t ) = A (ρ, ˜ ρ) x(t ) + B w

(ρ) w(t )

z(t ) = C (ρ, ˜ ρ) x(t )

A (ρ, ˜ ρ) := A (ρ) + B u (ρ) K ( ̃ ρ) C y (ρ) ,

C (ρ, ˜ ρ) := C z (ρ) + D zu (ρ) K ( ̃ ρ) C y (ρ) ,

(6)

that achieves a minimum bound on closed-loop H 2 -norm. Additionally, the synthesized RGS

controller utilizes the provided (measured) scheduling parameters for feedback control. The controller matrix in Eq. (5) is assumed to have affine parametrization with respect to

the measured scheduling parameters. In other words, this matrix, K ( ̃ ρ(t )) , is parameterizedas

K ( ̃ ρ(t )) = K 0 +

q ∑

i=1

˜ ρi (t ) K i . (7)

Therefore, the goal is to obtain the controller coefficient matrices K i for i = 0, 1 , . . . , m toimplement the RGS controller by using only the measured scheduling parameters ˜ ρi (t ) .

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2225

Transformation to multi-simplex

Rate of variation modeling

SF controller synthesis

SOF controller synthesis

Inverse Transformation

Controller implementation

Fig. 1. The outline of the propose design approach.

D

�

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b

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2

vE

efinition 1. Unit-simplex [18] : a unit-simplex is defined as follow

r =

{

β(t ) ∈ R

r : r ∑

i=1

βi (t ) = 1 , βi (t ) ≥ 0, i = 1 , 2, . . . , r

}

,

here each variable β i ( t ) varies in the unit-simplex �r that has r vertices.

efinition 2. Multi-simplex [19] : a multi-simplex � is the Cartesian product of a finite numberf q simplexes that is defined as

N 1 × �N 2 × · · · × �N m =

m ∏

i=1

�N i � �.

he dimension of the multi-simplex � is defined as the index N = (N 1 , N 2 , . . . , N m

) and forimplicity of notation, R

N denotes for the space R

N 1 + N 2 + ···+ N m . Thus, any variable β( t ) inhe multi-simplex domain � can be decomposed as (β1 (t ) , β2 (t ) , . . . , βm

(t )) , and each β i ( t ),elonging to a unit-simplex �N i , can be further decomposed as (βi1 (t ) , βi2 (t ) , . . . , βiN i (t ))or i = 1 , 2, . . . , m.

The general outline of the proposed approach is shown in Fig. 1 . Initially, transformationf the scheduling parameters from the original parameter space into convex multi-simpleomain. Then, another convex domain is constructed to model the rates of changes of thecheduling parameters. These transformations are the core of the current section. It is wortho emphasize that some of the notations used in this section traced back to [18] and [20] .

.1. Transformation to multi-simplex domain

The goal of this transformation to convert open-loop system matrices and controllerariables from the original parameter space into convex multi-simplex domain. Based on

q. (3) and the bounds of scheduling parameters and their uncertainties defined in Eq. (4) ,

2226 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

two unit-simplexes should be defined for each measured scheduling parameter. The first unit simplex to model ρ( t ) while the second one is to model the uncertainty δ( t ). Since eachvarying parameter (scheduling or uncertainty) bounded from above and from below, each unit simples has two vertices. Following the method presented in [20] , the time-varying parameterscan be modeled as follows:

1. Scheduling parameters (ρ j (t ) ⇒ β̌ j (t )) ,

β̌ j1 (t ) =

ρ j (t ) + ρ̄ j

2 ̄ρ j ⇒ ρi ( t ) = 2 ̄ρ j β̌ j1 (t ) − ρ̄ j , (8)

then,

β̌ j2 (t ) = 1 − β̌ j1 (t ) = 1 − ρ j (t ) + ρ̄ j

2 ̄ρ j =

ρ̄ j − ρ j (t )

2 ̄ρ j ,

where,

β̌ j (t ) = ( ̌β j1 (t ) , β̌ j2 (t )) ∈ �2 , ∀ j = 1 , 2, . . . , m,

β̌(t ) = ( ̌β1 (t ) , β̌2 (t ) , . . . , β̌m

(t )) .

2. Uncertainties (δ j (t ) ⇒

ˆ β j (t )) ,

ˆ β j1 (t ) =

δ j (t ) + δ̄ j

2 ̄δ j ⇒ δ j ( t ) = 2 ̄δ j ˆ β j1 (t ) − δ̄ j , (9)

then,

ˆ β j2 (t ) = 1 − ˆ β j1 (t ) = 1 − δ j (t ) + δ̄ j

2 ̄δ j =

δ̄ j − δ j (t )

2 ̄δ j ,

where,

ˆ β j (t ) = ( ̂ β j1 (t ) , ˆ β j2 (t )) ∈ �2 , ∀ j = 1 , 2, . . . , m,

ˆ β(t ) = ( ̂ β1 (t ) , ˆ β2 (t ) , . . . , ˆ βm

(t )) .

Thus, using this change of variables, the original affine parameter-dependent system (1) as well as the gain-scheduling controller (5) can be converted from ρ( t ) and ˜ ρ(t ) into new multi-simplex (convex) variables β̌(t ) and

ˆ β(t ) . Therefore, the multi-simplex variables ˜ β(t ) can bedefined as,

˜ β(t ) = ( ̌β j (t ) , ˆ β j (t )) , j = 1 , 2, . . . , m, ˜ β(t ) ∈ �, where � = �2 × �2 × · · · × �2 ︸ ︷︷ ︸ 2m unit-simplexes

.

(10)

For instance, consider the case with one scheduling parameter, β̌1 (t ) = ( ̌β11 (t ) , β̌12 (t )) andˆ β1 (t ) = ( ̂ β11 (t ) , ˆ β12 (t )) , then the homogeneous terms in the multi-simplex variables can bewritten in terms of the new convex variables as ˜ β(t ) = ( ̌β11 (t ) , β̌12 (t ) , ˆ β11 (t ) , ˆ β12 (t )) .

For instance, let G ( ̃ ρ(t )) represents any matrix involved in the controller construction. Thismatrix affinely dependent on the measured scheduling parameter as

G ( ̃ ρ(t )) = G 0 + ˜ ρ1 (t ) G 1 = G 0 + (ρ1 (t ) + δ1 (t )) G 1 . (11)

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2227

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sing the relations (8) and (9) ; and performing homogenization procedure [18] to obtain

2

( ̃ β(t )) = β̌11 ̂ β11 G 1 , 1 + β̌11 ̂ β12 G 1 , 2 + β̌21 ̂ β11 G 2, 1 + β̌21 ̂ β21 G 2, 2 = G ( ̃ ρ(t )) , (12)

here G 1 , 1 , G 1 , 2 , G 2, 1 and G 2, 2 are coefficients of the parameter-dependent matrix G ( ̃ α) at theertices of the multi-simplex domain and can be generated for any number of schedulingarameter, m ≥1, as

r j ,s j = G 0 +

m ∑

j=1

{ (−1) r j +1 ρ̄ j + (−1) s j +1 δ̄ j } G j , (13)

ith r j = 1 , 2, s j = 1 , 2, and j = 1 , 2, . . . , m. It is worth mentioning this transformation from˜ into ˜ α is exact.

emark 2. All controller synthesis variables should be processed by these steps in order torite the controller gain (5) in terms of the multi-simplex parameters as K ( ̃ β(t )) . However,

he system matrices in Eq. (1) are dependent only on the actual scheduling parameters ρ( t ).hey are independent of the uncertainty δ( t ). However, the same procedure presented can besed to convert the open-loop system matrices from ρ( t )-space into β( t )-space by constrain-ng δ̄i = 0 in Eq. (13) . Thus, when the parameter-dependent matrix depending on the cleancheduling variables, we denote the multi-simplex variables as β( t ) instead of ˜ β(t ) to dis-inguish it from other parameter-dependent matrices that depend on the corrupted schedulingarameters. Thus, the open-loop system matrices can be written in terms of the multi-simplexariables as

(β(t )) , B w

(β(t )) , B u (β(t )) , C z (β(t )) , D zu (β(t )) , and C y (β(t )) .

.2. Rate of variation modeling

Since the rates of change of the time-varying parameters are bounded in Eq. (4) , thesesates of change should be modeled in a convex domain η( t ). As each varying parameterselong to unit-simplex such that ˙ β j1 (t ) +

˙ β j2 (t ) = 1 , j = 1 , 2, . . . , m; the following relationhould always satisfied

˙ β j1 (t ) +

˙ β j2 (t ) = 0 j = 1 , 2, . . . , m.

Following the lines presented in [21,22] , the derivatives of the multi-simplex varying pa-ameters can assume values bounded by convex polytope j

j =

{

φ ∈ R

2 : φ =

2 ∑

k=1

η jk H

(k) j ,

2 ∑

k=1

H j (k, j) = 0, η j ∈ �2

}

, i = 1 , 2, j = 1 , 2, . . . , 2m.

(14)

tilizing the bounds on the rates of changes b ρ j and b δ j , the columns of the matrix H j cane reconstructed. Consequently,

˙ (t ) ∈ = 1 × 2 × · · · × 2m

=

2m ∏

j=1

j . (15)

2 For ease of notations, time dependency will be dropped sometimes.

2228 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

t

The mapping of the derivatives of the scheduling parameters into the derivatives of themulti-simplex variables is exact and given by

−b ρi

2 ̄ρi ≤ ˙ βi1 (t ) ≤ b ρi

2 ̄ρi ,

with

˙ βi2 (t ) = − ˙ βi1 (t ) .

Problem 1. Given a scalar ν > 0. Suppose that the scheduling parameters ρ( t ) are providedas ˜ ρ(t ) with uncertainties δ( t ). Find RGS static output-feedback controller in the form of

Eq. (5) for any pair ( ̃ β, ˙ ˜ β) ∈ � × that stabilizes the closed-loop system (6) and satisfies

sup

( ̃ β, ̇ ˜ β) ∈ �×

E

{∫ T

0 z (t ) ′ z (t ) dt

}< ν2 , (16)

for the disturbance input w(t ) given by

w(t ) = w 0 δ(t ) ,

where w 0 is a random variable satisfying

E

{w 0 w

′ 0

} = I n w ,

and E { ·} denotes the mathematical expectation. The δ( t ) is the Dirac’s delta function.

The outline of the solution approach of Problem 1 is based on two-stages design approach,which is a strategy that developed in [23] (robust H ∞

control for uncertain Linear Time-Invariant (LTI) systems), [24] (robust SOF H ∞

control for uncertain discrete-time systems), and [25,26] . In this paper, parameter-dependent state-feedback controller is designed in the first stage that minimizes the H 2 performance. Then, this controller is used as input matrixto the second stage to synthesize RGS SOF controller in the form of Eq. (5) that guaranteesEq. (16) .

Lemma 1. [27] Let u(t ) = 0 in Eq. (1) , for a given scalar ν > 0, if there exist continuously dif-ferentiable parameter-dependent matrix 0 < P (β) = P (β) ′ ∈ R

n×n and parameter-dependent symmetric matrix W (β) = W (β) ′ ∈ R

n z ×n z such that the following PLMIs are satisfied

3 ⎡

⎣

A (β) ′ P (β) + P (β) A (β) − ˙ P (β) �

C z (β) −I n z

⎤

⎦ < 0, (17)

⎡

⎣

W (β) �

P (β) B w

(β) P (β)

⎤

⎦ > 0, (18)

race (W (β)) < ν2 , (19)

then, for any (β, ˙ β) ∈ � × , the open-loop system defined in Eq. (1) is asymptotically stableand Eq. (16) is satisfied. The matrices A ( β), B ( β), and C ( β) are the open-loop system matriceswith ρ( t ) replaced by β( t ) using Eq. (8) .

3 These inequalities are dual version of these inequalities presented in Ref. [27] .

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2229

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[

t

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emma 2. [28] (Projection Lemma) Given a symmetric matrix � ∈ R

n×n and two matrices , V of column dimensions n , there exists an unstructured matrix Z satisfying

+ V Z U + (V Z U ) ′ < 0 (20)

f and only if the following projection inequalities with respect to Z are satisfied

v �N

′ v < 0, (21a)

′ u �N u < 0 (21b)

here N u and N v any matrices whose columns form a bases of the null spaces of U and , respectively, such that N v V = 0 and U N u = 0.

. PLMIs synthesis conditions

This section presents PLMIs for RGS controller synthesis.

.1. State-feedback control

Theorem 3 provides conditions to be used for designing parameter-dependent state-feedbackontroller satisfying the prescribed bound on H 2 performance, and then, the designed con-roller is used as the known matrix in conditions of Theorem 4 to synthesize RGS SOFontroller. It is worth mentioning that feasible solution of this theorem is necessary for theext step to obtain the SOF controller.

heorem 3. Given a positive scalar ν and a small scalar ε> 0 . If there exist a continuouslyifferentiable parameter-dependent matrix 0 < P (β) = P (β) ′ ∈ R

n×n , parameter-dependent

atrices W (β) = W (β) ′ ∈ R

n z ×n z , Z( ̃ β) ∈ R

n u ×n , F ( ̃ β) ∈ R

n×n for any ( ̃ β(t ) , ˙ ˜ β(t )) ∈ � ×

uch that the following PLMIs satisfied

A (β) F ( ̃ β) + B u (β) Z( ̃ β) + (•) ′ − ˙ P (β) � �

P (β) − F ( ̃ β) + ε(A (β) F ( ̃ β) + B u (β) Z( ̃ β)) ′ −ε(F ( ̃ β) + F ( ̃ β) ′ ) �

B w

(β) ′ 0 n w ×n −I n w

⎤

⎦ < 0 2n+ n w , (22)

F ( ̃ β) + F ( ̃ β) ′ − P (β) �

C z (β) F ( ̃ β) + D zu (β) Z( ̃ β) W (β)

]> 0 n+ n z , (23)

race (W (β)) < ν2 . (24)

he parameter-dependent controller

( ̃ β) = Z( ̃ β) F ( ̃ β) −1 , (25)

tabilizes the closed-loop system with guaranteed H 2 bound ν defined in Eq. (16) .

roof. Decoupling the state matrix from Lyapunov matrix P ( β) can be done through pro-ection Lemma via introducing extra slack variable V ( ̃ β) . Rewriting inequality (17) using

2230 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

I

W

Eq. (20) with the following definitions,

�(β) =

⎡

⎣

− ˙ P (β) P (β) 0

P (β) 0 0

0 0 I

⎤

⎦ , V ( ̃ β) =

⎡

⎣

F ( ̃ β) ′ 0

Y ( ̃ β) ′ 0

0 I

⎤

⎦ , U (β) =

[A (β) ′ −I 0

B w

(β) ′ 0 −I

],

Z =

such that N

′ u �(β) N u < 0, with N

′ u = [ I A (β) B w

(β) ] . Substituting these relations intoEq. (20) , ⎡

⎣

− ˙ P (β) P (β) 0

P (β) 0 0

0 0 I

⎤

⎦ +

⎡

⎣

F ( ̃ β) ′ 0

Y ( ̃ β) ′ 0

0 I

⎤

⎦

[A (β) ′ −I 0

B w

(β) ′ 0 −I

]

+

⎡

⎣

A (β) B w

(β)

−I 0

0 −I

⎤

⎦

[F ( ̃ β) Y ( ̃ β) 0

0 0 I

]< 0. (26)

In order to maintain convexity of the conditions, it is necessary to impose structural constraintson V ( ̃ β) such that Y ( ̃ β) = ε F ( ̃ β) , where ε is positive scalar used as tuning parameter forperformance improvement [29] . Substituting for A ( β) by the closed-loop matrix A (β, ˜ β) :=A (β) + B u (β) K ( ̃ β) with the change of variable Z( ̃ β) = K ( ̃ β) F ( ̃ β) yields ⎡

⎣

− ˙ P (β) P (β) 0

P (β) 0 0

0 0 I

⎤

⎦ +

⎡

⎣

F ( ̃ β) ′ A (β, ˜ β) ′ −F ( ̃ β) ′ 0

εF ( ̃ β) ′ A (β, ˜ β) ′ −εF ( ̃ β) ′ 0

B w

(β) ′ 0 −I

⎤

⎦

+

⎡

⎣

A (β, ˜ β) F ( ̃ β) εA (β, ˜ β) F ( ̃ β) B w

(β)

−F ( ̃ β) −εF ( ̃ β) 0

0 0 −I

⎤

⎦ < 0,

that leads to Eq. (22) . Multiplying Eq. (23) by T (β, ˜ β) from left and by T (β, ˜ β) ′ fromright with T (β, ˜ β) := [ C (β, ˜ β) − I ] and C (β, ˜ β) := C z (β) + D zu (β) K ( ̃ β) to obtain

(β) > C (β, ˜ β) P (β) C (β, ˜ β) ′ .

Using Schur complement, Eq. (18) can be recovered. The PLMI (24) ensures that ν is theguaranteed cost (upper bound) of the H 2 norm of the closed-loop system. �

3.2. Static output-feedback control

The next theorem utilizes the state-feedback controller K ( ̃ β) obtained from Theorem 3 tosynthesize the RGS SOF controller. Scalar η is used as an extra degree of freedom to reducenumerical problems associated with the solver.

Theorem 4. Given K ( ̃ β) , positive scalar ν > 0, and sufficiently small positive scalar η. Ifthere exist a continuously differentiable parameter-dependent matrix 0 < P (β) = P (β) ′ ∈R

n×n , parameter-dependent matrices W (β) = W (β) ′ ∈ R

n w ×n w , V (β) ∈ R

n×n , F (β) ∈ R

n×n ,

Q(β) ∈ R

n z ×n z , R( ̃ β) ∈ R

n u ×n u , and L( ̃ β) ∈ R

n u ×n y for any ( ̃ β(t ) , ˙ ˜ β(t )) ∈ � × satisfying

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2231

t

⎡⎢⎢⎣

⎡⎣

t

w

t

K

a

P

b

V

�

N

a

he following PLMIs

F (β) A (β, ˜ β) + (•) ′ − ˙ P (β) � � �

P(β) − F (β) ′ + ηV (β) A (β, ˜ β) −η(V (β) + V (β) ′ ) � �

B u (β) ′ F (β) ′ + L( ̃ β) C y (β) − R( ̃ β) K( ̃ β) ηB u (β) ′ V (β) ′ −R( ̃ β) − R( ̃ β) ′ �

Q(β) ′ C (β, ˜ β) 0 n z ×n Q(β) ′ D zu (β) I n z − Q(β) − Q(β) ′

⎤

⎥ ⎥ ⎦

< 0

(27)

W (β) �

P (β) B w

(β) P (β)

⎤

⎦ > 0, (28)

race (W (β)) < ν2 , (29)

ith

A (β, ˜ β) := A (β) + B u (β) K ( ̃ β)

C (β, ˜ β) := C z (β) + D zu (β) K ( ̃ β) , (30)

hen, the RGS SOF controller

( ̃ β) = R( ̃ β) −1 L( ̃ β) , (31)

symptotically stabilizes the closed-loop system (6) and satisfies Eq. (16) .

roof. If inequality (27) holds, inequality (20) in Lemma 2 is satisfied with matrices definedelow.

:=

⎡

⎢ ⎢ ⎣

0

0

I 0

⎤

⎥ ⎥ ⎦

, Z := R( ̃ β) , U :=

[ X (β, ˜ β) 0 − I 0

] , (32)

:=

⎡

⎢ ⎢ ⎣

A (β, ˜ β) ′ F (β) ′ + F (β) A (β, ˜ β) − ˙ P (β) � � �

P (β) − F (β) ′ + ηV (β) A (β, ˜ β) −η(V (β) + V (β) ′ ) � �

B u (β) ′ F (β) ′ ηB u (β) ′ V (β) ′ 0 n u ×n w �

Q(β) ′ C (β, ˜ β) 0 n z ×n Q(β) ′ D zu (β) −Q(β) ′ Q(β)

⎤

⎥ ⎥ ⎦

,

(33)

with the following null spaces of V and U

v =

[I 0 0 0

0 I 0 0

], N u =

⎡

⎢ ⎢ ⎣

I 0 0

0 I 0

X (β, ˜ β) 0 0

0 0 I

⎤

⎥ ⎥ ⎦

(34)

nd

A (β, ˜ β) := A (β) + B u (β) K ( ̃ β)

2232 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

C (β, ˜ β) := C z (β) + D zu (β) K ( ̃ β) (35)

X (β, ˜ β) := R( ̃ β) −1 L( ̃ β) C y (β) − K ( ̃ β) .

Note that −Q(β) ′ Q(β) < I n z − Q(β) − Q(β) ′ since (I n z − Q(β)) ′ (I n z − Q(β)) > 0 [23] . Thus,

U N u =

[ X (β, ˜ β) 0 − I 0

] ⎡

⎢ ⎢ ⎣

I 0 0

0 I 0

X (β, ˜ β) 0 0

0 0 I

⎤

⎥ ⎥ ⎦

= 0, (36)

N v V =

[I 0 0 0

0 I 0 0

]⎡

⎢ ⎢ ⎣

0

0

I 0

⎤

⎥ ⎥ ⎦

= 0, (37)

V Z U =

⎡

⎢ ⎢ ⎣

0 0 0 0

0 0 0 0

L( ̃ β) C y (β) − R( ̃ β) K ( ̃ β) 0 −R( ̃ β) 0

0 0 0 0

⎤

⎥ ⎥ ⎦

(38)

Consider now Eq. (21a) ,

N v �N

′ v =

[

A (β, ˜ β) ′ F (β) ′ + F (β) A (β, ˜ β) − ˙ P (β) �

P (β) − F (β) ′ + εV (β) A (β, ˜ β) −ε(V (β) + V (β) ′ )

]

< 0. (39)

Multiplying Eq. (39) by R(β, ˜ β) from left and by R(β, ˜ β) ′ from right with R(β, ˜ β) :=[ I n A (β, ˜ β)

′ ] to obtain

A (β, ˜ β) ′ P (β) + P (β) A (β, ˜ β) − ˙ P (β) < 0, (40)

in addition to P ( β) > 0, represents Lyapunov stability condition for A (β, ˜ β) . Thus, Eq.(21a) verified. On the other hand, Eq. (21b) is

N

′ u �N u =

⎡

⎣

I 0 X (β, ˜ β) ′

0

0 I 0 0

0 0 0 I

⎤

⎦

×

⎡

⎢ ⎢ ⎢ ⎢ ⎣

�11 � � �

P (β) − F (β) ′ + ηV (β) A (β, ˜ β) �22 � �

B u (β) ′ F (β) ′ ηB u (β) ′ V (β) ′ �33 �

Q(β) ′ C (β, ˜ β) 0 n z ×n Q(β) ′ D zu (β) �44

⎤

⎥ ⎥ ⎥ ⎥ ⎦

×

⎡

⎢ ⎢ ⎢ ⎢ ⎣

I 0 0

0 I 0

X (β, ˜ β) 0 0

0 0 I

⎤

⎥ ⎥ ⎥ ⎥ ⎦

. (41)

with

�11 = A (β, ˜ β) ′ F (β) ′ + F (β) A (β, ˜ β) − ˙ P (β) ,

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2233

�

�

�

N

w

t

t

T

t

r

22 = −η(V (β) + V (β) ′ ) ,

33 = 0 n u ×n w ,

44 = −Q(β) ′ Q(β) . (42)

Therefore,

′ u �N u =

⎡

⎣

�11 + X (β, ˜ β) ′ B u (β) ′ F (β) ′ �̄12 F (β) B u (β) �̄14

P (β) − F (β) ′ + ηV (β) A (β, ˜ β) �22 ηV (β) B u (β) 0 n×n z

Q(β) ′ C (β, ˜ β) 0 n z ×n Q(β) ′ D zu (β) �44

⎤

⎦

×

⎡

⎢ ⎢ ⎢ ⎢ ⎣

I 0 0

0 I 0

X (β, ˜ β) 0 0

0 0 I

⎤

⎥ ⎥ ⎥ ⎥ ⎦

. (43)

ith

�̄12 = P (β) − F (β) + ηA (β, ˜ β) ′ V (β) ′ + ηX (β, ˜ β) ′ B u (β) ′ V (β) ′ ,

�̄14 = C (β, ˜ β) ′ Q(β) + X (β, ˜ β) ′ D zu (β) ′ Q( ̃ β) , (44)

hat leads to

N

′ u �N u =

⎡

⎣

F (β)[ A (β, ˜ β) + B u (β) X (β, ˜ β)] + (•) ′ − ˙ P (β) � �

P (β) − F (β) ′ + ηV (β)[ A (β, ˜ β) + B u (β) X (β, ˜ β)] −η(V (β) + V (β) ′ ) �

Q(β) ′ [ C (β, ˜ β) + D zu (β) X (β, ˜ β)] 0 n z ×n −Q(β) ′ Q(β)

⎤

⎦ .

(45)

Considering Eq. (35) with

A (β, ˜ β) := A (β, ˜ β) + B u (β) X (β, ˜ β) = A (β) + B u (β) K ( ̃ β) C y (β) ,

C (β, ˜ β) := C (β, ˜ β) + D zu (β) X (β, ˜ β) = C z (β) + D zu (β) K ( ̃ β) C y (β) ,

K ( ̃ β) := R( ̃ β) −1 L( ̃ β) ,

(46)

hat yields

N

′ u �N u =

⎡

⎣

F (β) A (β, ˜ β) + (•) ′ − ˙ P (β) P (β) − F (β) + ηA (β, ˜ β) ′ V (β) ′ C (β, ˜ β) ′ Q( ̃ β)

P (β) − F (β) ′ + ηV (β) A (β, ˜ β) −η(V (β) + V (β) ′ ) 0 n×n z

Q(β) ′ C (β, ˜ β) 0 n z ×n −Q(β) ′ Q(β)

⎤

⎦ .

(47)

Multiplying Eq. (47) by T 2 from left and by T

′ 2 from right, with

2 =

[I A (β, ˜ β) ′ 0

0 0 (Q(β) −1 ) ′

](48)

hat leads to Eq. (17) with the matrices A (β, ˜ β) and C (β, ˜ β) replacing A ( β) and C z ( β),espectively, i.e.,

T 2 (47) T

′ 2 =

[A (β, ˜ β) ′ P (β) + P (β) A (β, ˜ β) − ˙ P (β) C ( β, ˜ β) ′

C ( β, ˜ β) −I n z

]< 0. (49)

2234 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

On the other hand, taking the Schur complement of Eq. (28) to obtain W (β) >

B w

(β) ′ P (β) B(β) . Thus, Eq. (29) implies that ν is an upper bound to the H 2 -norm. �

Remark 3. When a feasible solution for Theorem 4 exists, the vertex matrices of the con-troller matrix in Eq. (31) at the the multi-simplex vertices are used to calculate the controllercoefficients in Eq. (7) (that are utilized for controller implementation). Generally, the inverse transformation

4

G 0 =

1

2

2m

2 ∑

i 1 =1

2 ∑

i 2 =1

· · ·2 ∑

i m =1

2 ∑

k j =1

2 ∑

k 2 =1

· · ·2 ∑

k m =1

G i 1 ,i 2 , ... ,i m ,k 1 ,k 2 , ... ,k m

G j =

1

2

2m ρ̄ j

2 ∑

i 1 =1

2 ∑

i 2 =1

· · ·2 ∑

i m =1

2 ∑

k j =1

2 ∑

k 2 =1

· · ·2 ∑

k m =1

(−1) i j + j G i 1 ,i 2 , ... ,i m ,k 1 ,k 2 , ... ,k m ,

(50)

is used to calculate the synthesis coefficients G j for j = 0, 1 , . . . , m in Eq. (13) .

Remark 4. Since the SOF controller is synthesized via slack variables R( ̃ β) and L( ̃ β) , theconditions of Theorem 4 allows designing RGS output-feedback controller when all open-loop

matrices are affected by varying parameters. This feature is difficult, if possible, to achieve with the methods in [8,9,11] that require the control and/or measurement matrices ( B u and/orC y ) to be parameter-independent (constant matrices).

Remark 5. The conditions of Theorem 4 associated with some conservativeness due to over- bounding the (4, 4) block in Eq. (33) . To mitigate this and following some ideas in literature[17,23] , iterative procedure (ISOFD algorithm) has been developed to reduce design conserva- tiveness. Initially, Theorem 3 is used to obtain parameter-dependent state-feedback controller K ( ̃ β) . This controller is used in Theorem 4 to synthesize RGS SOF controller K ( ̃ β) . Then us-ing this RGS SOF controller, calculate new state-feedback controller as K ( ̃ β) = K ( ̃ β) C y (β)

and use it again in Theorem 4 . Each iteration of the Algorithm 1 assures a feasible solution

Algorithm 1: Iterative Static Output-Feedback Design (ISOFD) Algorithm. Initialization:

• Set i = 0, K 0 ( ̃ β) = 0. • Using Theorem 3, compute initial state-feedback controller K 0 ( ̃

β) . • Set i max and T olerance .

repeat

• Set i = i + 1 . • Given K i−1 ( ̃

β) , solve the conditions of Theorem 4 to obtain SOF controller K i ( ̃ β)

with a minimal achievable bound νi . • Given K i ( ̃ β) , calculate new state-feedback controller such that K i ( ̃ β) = K i ( ̃ β) C y (β) .

until i � i max OR | νi − νi−1 | < T olarance ;

4 Simple algebraic manipulations of Eq. (13) leads to the relationship (50) .

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2235

w

p

m

R

s

d

R

i

r

o

t

s

r

c

o

i

o

4

a

d

a

a

s

T

u

W

4

w

v

b

d

L

ith at least νi ≤ νi−1 . Although this convergence not mathematically demonstrated in thisaper, it has been demonstrated in all conducted tests since ν is bounded from below andonotonically non-increasing.

emark 6. As a special case of Theorem 4 , robust SOF controller can be synthesized. Morepecifically, by constraining the synthesis variables R( ̃ β) and L( ̃ β) to be parameter indepen-ent (i.e., constant matrices), robust SOF controller can be obtained.

emark 7. As shown above, the synthesis conditions of Theorems 3 and 4 are characterizedn terms of PLMIs (for a fixed ε and η) that depends (continuously) on time-varying pa-ameters inside multi-simplex domain. Therefore, this controller design problem is a convexptimization problem with infinite dimensional constraints. At this end, it is necessary to relaxhe infinite dimensional optimization problem into finite set of LMIs to be solved. Fortunately,olvability of PLMIs is manageable in the presence of recent theoretical and computationalelaxation tools such as matrix Sum-Of-Squares [30] , Slack variable approach [31] , and matrixoefficient check methods [32,33] . In this paper, a specialized parser ROLMIP [34] (devel-ped as a tool to perform such PLMIs manipulation and LMI relaxations) has been used tomplement the conditions of Theorems 3 and 4 to obtain the optimal solution of the convexptimization problem. This parser works jointly with YALMIP [35] and SeDuMi [36] .

. Illustrative examples

In this section, two examples are studied to demonstrate the efficacy of the proposedpproach. First, an academic example is given to illustrate the achieved performance withifferent bounds of measurement noise using the ISOFD algorithm. Second, the developedpproach is applied to a realistic LPV model for an Electric Variable Valve Timing (EVVT)ctuator of automotive engine. The purpose of this example is to validate the developedynthesis approach through practical LPV model from engineering application point of view.he synthesis conditions are implemented in MATLAB environment (R2016a). The computersed for control design is equipped with Intel Core i 7 (2.4 GHz) processor, 8 GB RAM withindows 10.

.1. Academic example

Consider the following LPV system,

⎡⎣ A(ρ) Bu(ρ) Bw(ρ)

Cz(ρ) Dzu(ρ)

Cy(ρ) Dyw(ρ)

⎤⎦ =

⎡⎢⎢⎢⎢⎣

25.9 − 60ρ(t ) 1 3 −0.0320 − 40ρ(t ) 34 − 64ρ(t ) 2 −0.47

1 1 00 0 11 0 0

⎤⎥⎥⎥⎥⎦,

hich is a slightly modified version of the example presented in [16,37] . ρ( t ) is the time-arying parameter which is bounded by 0 ≤ρ( t ) ≤1, | ̇ ρ(t )|≤ 1 , with measurement uncertaintyound | δ( t ) | ≤ ζ , and | ̇ δ(t )|≤ 10 × ζ .

The scalar parameters ε and η in Theorems 3 and 4 are tuning parameters and extraegrees of freedom to reduce conservativeness and avoid numerical problems associated withMI solvers. Although line search can be performed on these parameters, such search requires

2236 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

Table 1 H 2 performance with different bounds of measurement noise. ε = 0. 001 , η = 0. 001 .

ζ Robust

0 0.001 0.01 0.1 0.2 0.5 1 K robust

1st stage 3.049 3.052 3.098 3.867 4.548 4.647 4.647 4.647 2nd stage 3.052 3.139 3.359 3.376 3.932 4.328 4.328 4.328

Iterations 1 3.100 3.106 3.255 3.122 3.870 4.216 4.223 4.212 2 3.096 3.104 3.240 3.118 3.821 4.127 4.132 4.115 3 3.072 3.102 3.235 3.115 3.781 4.049 4.055 4.021 4 3.002 3.099 3.230 3.113 3.744 3.982 3.987 3.953

Table 2 Number of decision variables and LMI rows for The- orems 3 and 4 .

# LMI variables # LMI rows

1st stage 28 146 2nd stage 42 274

huge computational expenses. In order to avoid such computational demands and following

similar ideas in literature [38] , ε and η are constrained to a discrete values within small set,

ε, η ∈ { 0. 001 , 0. 01 , 0. 1 , 1 } .

The designer can do few trials among these values to find the value that achieve the bestperformance.

In this example, ε = η = 0. 001 have been found to achieve best performance bound. Forthe ISOFD algorithm, the tolerance and maximum number of iterations have been set to 10

−4

and 150, respectively. Table 1 shows the guaranteed upper bound ν for the first stage and the first four iterations

of the second stage for a wide range of measurement noise bound ( ζ ). In the first stage, theconditions of Theorem 3 is used to design parameter-dependent state-feedback controller, then, this controller is fed to the conditions of Theorem 4 to synthesize RGS SOF controller. It isobvious that at each single iteration ( i ), the iterative procedure ensures νi ≤ νi−1 . Referringto Remark 6 , robust static output-feedback (parameter-independent) controller ( K robust ) is designed. The achievable performance for K robust is given in the last column of Table 1 . Notethat K robust achieve very competitive results with RGS SOF controllers for ζ ≥0.5 . This isa natural observation since when scheduling parameter is corrupted with a large noise, thereis no point to implement LPV controller as long as the achieved performance of the robustcontroller K robust is very close to the performance of the RGS controller.

For ε = η = 0. 001 and ζ = 0. 5 , Fig. 2 illustrates the ISOFD algorithm convergence andshows the benefit of using the iterative procedure to reduce the design conservativeness. It isworth mentioning that the execution time of a single iteration of the ISOFD algorithm is 2.4seconds for this example. Table 2 shows the number of LMI variables and number of LMIrows associated with Theorems 3 and 4 for this example.

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2237

10 20 30 40 50 60 70 80 90 100 110 120 130 140 1503

3.2

3.4

3.6

3.8

4

4.2

4.4

Iteration

ν

Fig. 2. Algorithm convergence.

4

E

T

s

w

o

m

a

w

b

fl

E

fi

L

.2. EVVT actuator

To validate the result of the developed approach on practical engineering applications,lectric Variable Valve Timing (EVVT) cam phaser system is investigated in this subsection.he EVVT actuator consists of two main components: an electric motor and a planetary gearet (see Fig. 3 ). The planetary gear set consists of an outer ring gear, a planet gear carrierith planet gears attached, and a sun gear. The ring gear, which is running at the half speedf the crankshaft, is driven by the crankshaft through the engine timing belt. Details of theodeling work of the EVVT can be found in [39] . The planetary gear carrier is driven by

n electric motor and four planet gears engaging both ring and sun gears at the same time,here the sun gear is connected to the camshaft. The speed of the camshaft is determinedy the ring gear speed together with the EVVT motor speed, which provides the engine withexible valve opening timing. Therefore, the cam-phase can be adjusted by controlling theVVT motor speed with respect to engine speed [4] .

A series of system identification experimental tests were conducted ( Fig. 4 ) at a range ofxed values of engine speed ( N ) and battery voltage ( V ) at Energy and Automotive Researchab of Michigan State University (see [40] for more details). It was found that the identified

2238 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

Timing beltBold : Role in planetary systemItalic : Role in VVT system

Ring GearVVT Pulley

CarrierVVT Motor

Sun GearCamshaft

Planetary Gears

Fig. 3. Electric planetary gear EVVT system.

Fig. 4. Engine test bench used in the EVVT modeling.

Table 3 Range of the time-varying parameters.

ρ1 ( t ) ∈ [0.2529 0.6472] ρ2 ( t ) ∈ [6.975 14.540]

model of the EVVT actuator is of the following form,

G (N (t ) , V (t ) , s) =

ρ1 (N (t ) , V (t ))

s(s + ρ2 (N (t ) , V (t ))) , (51)

where ρ1 ( N ( t ), V ( t )) and ρ2 ( N ( t ), V ( t )) are time-varying coefficients as functions of enginespeed and battery voltage. For notational simplicity, ρ1 ( t ) and ρ2 ( t ) are used to refer to ρ1 ( N ( t ),V ( t )) and ρ2 ( N ( t ), V ( t )), respectively. Note that ρ1 ( t ) is associated with the DC gain of thetransfer function (51) , and ρ2 ( t ) is the location of the open-loop pole of the 2nd order system.In other words, the DC gain and pole location of the transfer function (51) are time-varyingcoefficients and functions of engine speed and battery voltage. It is worth emphasizing that thevalues of ρ1 ( t ) and ρ2 ( t ) were obtained experimentally over specified fixed values of batteryvoltages and engine speeds that cover the entire range of engine operating conditions. The ranges of the varying coefficients ρ1 ( t ) and ρ2 ( t ) are given in Table 3 . In order to performcontroller design using the conditions developed earlier in Section 3 , the EVVT plant model

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2239

(

w

a

t

d

b

w

r

b

a

t

ta

t

o

sE

p

K

F

t

o

g

51) needs to be realized in state-space form as follows: [ ˙ x 1 ˙ x 2

]=

[0 1

0 −ρ2 (t )

][x 1 x 2

]+

[0

ρ1 (t )

]u,

y =

[1 0

][x 1 x 2

],

here x 1 represents the cam-phase angle and u is the EVVT motor speed command. Toccommodate for performance output, control energy, and disturbance input, we define

C z (ρ) =

[1 0

0 0

], D zu (ρ) =

[0

1

], B w

(ρ) =

[0. 1

0

],

hen, the state-space realization of the overall LPV system corresponding to the matricesefined in Eq. (1) are

⎡⎣ A(ρ) Bu(ρ) Bw(ρ)

Cz(ρ) Dzu(ρ)

Cy(ρ) Dyw(ρ)

⎤⎦ =

⎡⎢⎢⎢⎢⎣

0 1 0 0.10 −ρ2(t ) ρ1(t ) 01 0 00 0 11 0 0

⎤⎥⎥⎥⎥⎦.

In order to design the controller, the parameters ranges which are given in Table 3 haveeen considered and the following scheduling signals have been defined,

ρ1 (t ) = 0. 197 sin (2t ) + 0. 450,

ρ2 (t ) = 3 . 783 sin (2t ) + 10. 756 , (52)

ith measurement noise bounds | δ1 ( t ) | ≤0.08 and | δ2 ( t ) | ≤0.8 associated with ρ1 ( t ) and ρ2 ( t ),espectively. From Eq. (52) , the rates of change of the scheduling parameters are boundedy | ̇ ρ1 (t )|≤ 0. 3942 and | ̇ ρ2 (t )|≤ 7 . 564. The rates of change of the measurement noises aressumed to be bounded by

˙ δi (t ) ≤ 100× |δi (t )| with i = 1 , 2. First, the open-loop system matrices are converted from ρ( t )-space into β( t )-space using

he procedure presented in Section 2 . Then, the conditions of Theorems 3 and 4 are usedo synthesize RGS SOF controller with ε = 0. 001 and η = 0. 01 . The achieved H 2 bound ν

ssociated with the first stage and the second stage are 0.759 and 0.767, respectively. It is clearhat the guaranteed bound using the RGS SOF controller is very close to the performancef the state-feedback controller. After the SOF gain (31) has been obtained in the multi-implex domain, the inverse transformation (50) is used to calculate controller coefficients in

q. (7) which are used to implement the controller in real-time using the measured schedulingarameters. These coefficients are obtained as

0 = −0. 7126 , K 1 = −0. 5879 , K 2 = −0. 6736 . (53)

With this controller, the time-domain simulations of the EVVT actuator are shown inig. 5 . A disturbance input (torque) generated as w(t ) = 30 × exp(−2. 3 t ) sin (t ) to disturb

he cam phase angle of the EVVT. The simulation results demonstrate not only robustnessf the synthesized controller against uncertainties in scheduling parameters but also a fairlyood ability to attenuate disturbance torque on the cam-phase angle (performance output).

2240 A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242

0 2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

θ 1(t

)

Measured scheduling parameterActual scheduling parameter

0 2 4 6 8 10 12 146

8

10

12

14

16

θ 2(t

)

0 2 4 6 8 10 12 140

2

4Performance outputDisturbance input

0 2 4 6 8 10 12 14−4

−3

−2

−1

0

Time (sec.)

u(t

)

Fig. 5. Time-domain simulation of the EVVT actuator.

5. Conclusion

Parameterized Linear Matrix Inequalities (PLMIs) conditions to design Robust Gain- Scheduling (RGS) Static Output-Feedback (SOF) controller with inexactly measured schedul- ing parameters are presented in this paper. Multi-simplex approach are utilized to model the time-varying parameters and associated uncertainties. The foundation of the developed

conditions is based on a two-stage design approach by designing parameter-dependent state- feedback H 2 controller in the first stage, and the resulting controller is then used for syn-thesizing RGS SOF controller in the second stage. Iterative procedure is developed to reduce

A.K. Al-Jiboory, G. Zhu / Journal of the Franklin Institute 355 (2018) 2221–2242 2241

t

d

s

s

c

s

a

R

[

[

[

[

[

[

[

[

[

he bound on H 2 performance of the closed-loop system. The developed conditions can han-le the general case where the varying parameters affecting all open-loop system matricesince the controller is synthesized using extra slack variables independent of Lyapunov andystem matrices. Robust (parameter-independent) SOF controller can be handled as a specialase of developed conditions. Two examples are presented, one is academic example and theecond one is a realistic Electric Variable Valve Timing (EVVT) actuator. Numerical resultsnd simulations of these examples demonstrate the validity of the developed approach.

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