linear parameter varying feedforward control synthesis using parameter-dependent lyapunov function

Nonlinear Dyn (2014) 78:2293–2307DOI 10.1007/s11071-014-1544-5

ORIGINAL PAPER

Linear parameter varying feedforward control synthesisusing parameter-dependent Lyapunov function

Yusuf Altun · Kayhan Gulez

Received: 27 January 2014 / Accepted: 12 June 2014 / Published online: 12 August 2014© Springer Science+Business Media Dordrecht 2014

Abstract This paper presents the dynamic feedfor-ward control synthesis for linear parameter varying(LPV) systems. It is assumed that all system matri-ces are dependent on varying parameters, which aremeasurable with sensor or observable. The parame-ters have bounded variation rates. Parameter-dependentLyapunov function is used for the feedforward controlsynthesis such that the robust stability is assured forall varying parameters at the time of the operation. Themethod is formulated in terms of linear matrix inequali-ties for LPV feedforward controller that guarantees thestability of the transfer matrix having L2-gain. Thiscompensator is designed by adding on the feedbackcontroller in two degrees of freedom control configu-ration. This controller can be used for the disturbanceattenuation or decreasing the tracking error. The numer-ical examples and simulations are given to provide theapplicability of the proposed solution.

Keywords Linear parameter varying systems ·Gain scheduling · H∞ optimal control · Feedforwardcontrol · L2-gain controller

Y. Altun (B)Computer Engineering Department, Engineering Faculty,Duzce University, Konuralp Campus, Duzce, Turkeye-mail: [email protected]

K. GulezControl and Automation Engineering Department, YildizTechnical University, Davutpasa Campus, Esenler,Istanbul, Turkeye-mail: [email protected]

1 Introduction

Recently, linear parameter varying (LPV) gain-scheduling control problem for LPV systems has beenwidely studied in academia as an alternative controlmethod for nonlinear systems. This problem has beeninvestigated by different approaches in the literature asin [1–9] for the past 30 years. It has been practicallyapplied to different control problems.

For instance, [10] considers the LPV control of aninverted pendulum system whereas [11] considers LPVcontrol of a 6-degrees of freedom (6-DOF) vehicle.LPV-H∞ control of a permanent magnet synchronousmotor system has been designed in [12]. [13] introducesthe LPV control of an F-16 aircraft via a controller statereset system. Utilizing a H-infinity gain-schedulingcontroller, [14] has considered the control of the airpath systems of a diesel engine using LPV techniques.In [15], new LPV techniques have been developed forthe control of magnetic bearing systems. Robust LPVcontrol of AMB systems has been introduced in [16].Finally, LPV control of a quadrotor system has beenstudied in [17]. Thus, the LPV control method has beenextensively used in various dynamical systems requir-ing high performance, such as vehicles, active magneticbearing systems, flight control systems, etc.

In view of the existing literature, one can recog-nize that the design of LPV controllers is generallyemployed by using L2-gain control method, whichtakes into account the parameter variations, distur-bances, sensor noises in the considered system. Hence,

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2294 Y. Altun, K. Gulez

this enables the designed controller to guarantee the sta-bility, performance, and robustness against all parame-ter variations changing with operating circumstances.However, it is well known that using L2-gain controlmethods for the synthesis of LPV controllers causeto the non-convex optimization problems. In the lit-erature, various methods have been used to formulatein terms of a convex finite-dimensional linear matrixinequality (LMI) problem such as the multi-convexityargument [18], s-procedure techniques [19], griddingmethods [20], full-block multipliers [21], and the non-smooth dissipative systems framework [22]. Besides,there are different approaches as in [23,24]. However,using gridding techniques for the design of LPV con-trollers does not guarantee the stability of the controlsystem for all varying parameters due to its samplingnature.

When the literature has been examined, it is observedthat older techniques used in LPV controller synthe-sis are mostly based on the polytopic representation ofthe system as in [3,6,25], or a single Lyapunov func-tion as in [26]. In recent years, parameter-dependentLyapunov functions (PDLF) are mostly employed toovercome the conservatism problems arisen by usingthe single Lyapunov functions [18–20,27,28], wherethese controllers are mostly obtained by interpolatingthe local controllers [29–31].

Feedforward controllers are usually used to improvethe tracking performance of the system or to attenu-ate the external disturbances which are available formeasurement. In many control applications, feedfor-ward controllers are employed together with a feed-back controller, which stabilizes the dynamic system,in two degrees of freedom (2-DOF) control structure[32–42]. There are two ways for the 2-DOF design offeedforward controllers. The first one is the simultane-ous design of the feedback and feedforward controllersas in [37]. As for the second one, which is used inthis paper, it is assumed that the feedback controller isfixed, and then feedforward controller is concentratedon it as in [38,42]. Besides, the feedback controller isgenerally designed first to provide the stability and theacceptable response of the closed-loop system. After-wards, the feedforward controller is included to the sys-tem in order to improve the closed-loop performance.For instance, [38] considers the feedforward controlof a linear time invariant (LTI) system subjected touncertainties and disturbances by using dynamic inte-gral quadratic constraints (IQCs).

In [41], a simple synthesis for the static feedfor-ward part of a 2-DOF LPV/LFT control system isachieved on the feedback controller. The feedforwardpart is designed by using the different approach from2-DOF structure in [38] since it is designed withoutconsidering the feedback controller. It proposes a sta-tic feedforward controller for LPV systems, but thepaper emphasizes the feedforward controller that is notdependent on parameters. In addition, this paper is dif-ferent problem from [41]. This is because, the feedfor-ward part is designed by considering the feedback con-troller previously designed in this paper as in [38,42].Therefore, the feedforward is constructed on the feed-back controller. In [42], the design of static feedfor-ward controller depending on scheduling parameters isalso achieved for LPV systems using different approachfrom [41]. It is similar to 2-DOF structure in [38] sinceit is designed with considering the feedback controller.Moreover, the dynamic controller is proposed in thispaper, while static feedforward controller is designedin [41] and [42]. It is well known that the dynamiccontroller is more useful according to the static one.

It is well known that there are many studies ondynamic LPV output-feedback controller design forLPV systems such as in [1–9]. There are also manystudies about LTI H-infinity feedforward control forLTI systems under uncertainty or without under uncer-tainty using different LMI techniques and approachessuch as in references [32–40]. There is a static con-troller design which is independent from schedulingparameters for LPV systems in [41], and there is a sta-tic controller design which is dependent on schedulingparameters for LPV systems in [42]. However, there isnot a solution of the dynamic feedforward controllerchanging with parameters for LPV systems since LPVfeedforward controller design for LPV systems hasnot been tackled in the literature. Therefore, dynamicparameter-dependent H-infinity (LPV) feedforwardcontroller design is discussed for LPV systems, whichis stabilized by any LTI or LPV feedback controller, inthis paper. Briefly, the focus of this paper is to solvethe design problem of the dynamic LPV feedforwardcontroller to improve LPV or LTI feedback controller.

The proposed controller structure is in the 2-DOFone where the feedback controller is previously cho-sen as a simple LTI or LPV controller to provide theinternal stability of the generalized system G withoutthe feedforward part. The G contains the feedback con-troller and plant. In other words, it is assumed that the

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LPV feedforward control synthesis 2295

feedback controller is previously designed and is inthe generalized system G. Afterwards, the LPV feed-forward controller design is proposed for the track-ing problem or disturbance rejection problem. Thecontroller assures the stability of the G system. Thismeans that the controller do not cancel the stability ofG system which is stable. The controller synthesis isobtained by using bounded real lemma (BRL) in LPVsystems and the change of variable methods. The pro-posed controller is based on LMIs, which are obtainedby using induced L2-gain control method, such as in[43]. Thus, the affine LPV system is used to composethe affine PDLF, which is mostly employed to over-come the conservatism problems arisen by using thesingle Lyapunov functions. The proposed controller isless conservative compared to the controllers, whichwere designed by utilizing parameter-independent Lya-punov functions. It is admitted that the first solution ofthe control problem in Theorem 1 does not include thesensor noises. Therefore, it is the limited solution forthe problems. Since second and third solutions in The-orem 2 and 3 enable the designer to take account of thesensor noises for the problems, they are global solutionfor the problems.

2 Problem formulation

LPV system is represented by (1), where x(t) ∈ Rn

are the states, u (t) ∈ Rnu are the control input signals,

z (t) ∈ Rnz are controlled outputs, y (t) ∈ R

ny aremeasured outputs for control, ω ∈ R

nw are the distur-bance signals, and all of the matrices are of compatibledimensions and depend on time varying parameter θ (t)and θ (t) whose sets and the admissible parameter tra-jectories are defined by (2).

x (t) = A (θ (t)) x (t) + B1 (θ (t)) ω (t)

+B2 (θ (t)) u (t) ,

z (t) = C1 (θ (t)) x (t) + D11 (θ (t)) ω (t) (1)

+D12 (θ (t)) u (t) ,

y (t) = C2 (θ (t)) x (t) + D21 (θ (t)) ω (t)

+D22 (θ (t)) u (t) ,

D �{θ ∈ R

n : θi ≤ θi ≤ θi , ∀i = 1, ..., n},

E �{θ ∈ R

n : θ i ≤ θi ≤ θi , ∀i = 1, ..., n}

,(2)

xc (t) = Ac(θ (t) , θ (t)

)xc (t) + Bc (θ (t)) ω (t) ,

u f f (t) = Cc (θ (t)) xc (t) + Dc (θ (t)) ω (t) .(3)

Fig. 1 The Feedforward LPV control scheme

As mentioned in previous section, the different con-trol synthesis of LPV output feedback is obtained forthe LPV system in (1) in the literature [1–9,18,20–28].LTI feedforward controller designs are tackled for LTIsystems by different LMI techniques and approachessuch as [38–40]. For example, the feedforward controlis designed for a LTI system subjected to uncertain-ties and disturbances in [38] where feedback controlleris fixed and feedforward controller is concentered onfeedback controller. However, dynamic LPV feedfor-ward controller design has not been discussed for LPVsystem. Thus, this paper focuses on a design of dynamicLPV feedforward controller in (3) for the LPV systemG in (1) where A is Hurwitz matrix. Figure 1 showsthe block diagram of the control system consisting ofG generalized system in (1) and KFF LPV feedforwardcontroller. In this approach, it is assumed that the feed-back controller was previously designed, and it is in G.Afterward, the feedforward controller is constructedon it as in [38]. Therefore, the controller synthesis pro-posed shows the KFF design of feedforward part in thisstructure. The optimal KFF is designed by minimizingL2 gain of transfer matrix from ω to Z . Here, Z repre-sents the outputs z and y, and ω is the disturbance orreference input.

Figure 2 shows the detailed of Fig. 1 for reference-tracking problem where Pr(θ) represents the parameter-dependent nominal plant. G generalized system includ-es the feedback controller, input–output weighting

Fig. 2 The closed loop of the reference-tracking problem

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functions, and nominal system. However, the weight-ing functions are not in the figure. It is well known thatthey are used during the controller design, and they arenot included in the closed-loop control system after thecontroller KFF is designed. The obtaining of the matri-ces of G generalized system in (1) is given in numericalexamples.

xcl (t) = Acl(θ (t) , θ (t)

)xcl (t) + Bcl (θ (t)) ω (t) ,

Z (t) = Ccl (θ (t)) xcl (t) + Dcl (θ (t)) ω (t) . (4)

Firstly, let obtain transfer matrix from ω to Z . When thecontroller in (3) is combined with the LPV plant (1), thestate space matrices in (4) of closed-loop control systemfrom ω to Z and transfer matrix in (7) are obtainedby considering Fig. 1. In (4), the subscript ‘cl’ denotesclosed loop, xcl denotes x and xc, Z denotes the outputsz and y, and ω is the disturbance input for disturbancerejection problem while ω is the reference input for thereference-tracking problem.

In order to obtain the LMI constructing L2-gain con-troller, BRL for LPV systems is used in Lemma 1.

Lemma 1 [18] If P(θ) in (6) is positive definite sym-metric matrix, the stability of the system (4) having L2

gain γ is assured for all θ (t) ∈ D and θ (t) ∈ E.

The vertices of D and E in (2) are denoted in (5).

Dvex �{θ : θi = θi or θi = θi , ∀i = 1, ..., n

},

Evex �{θi : θi = θ i or θi = θi ,∀i = 1, ..., n

}.

(5)

LMI in (6) is of infinite dimensional due to parameter-dependent matrices P (θ1) , . . . , P (θn). We can reduceto the finite-dimensional LMI using the multi-convexityconditions in Lemma 2 which are fully mentioned inSect. 4.

Lemma 2 [18] θ ∈ Rn; the quadratic form which

depends on θ (t) varying with the time can be written.

The form is known as multi-convex functions as followswhere α, β and γ symbolize the coefficients of all θ

combinations.

f (θ1, . . . , θn) = α0 +∑

i

αiθi +∑

i< j

βi jθiθ j +∑

γi

θ2i ,

The stability is provided for all values of θ constrainedby the parameter box in (2), providing that the follow-ing multi-convexity condition is obtained together withf (.) function which is negative

2γi = ∂2 f

∂θ2i

(θ) ≥ 0, i = 1, ..., n.

3 The LPV feedforward control synthesis

The design problem of LPV feedforward controllerKFF in Fig. 1 is solved using PDLF. We presentparameter-dependent L2-gain controller, which isdefined by (3), for LPV systems. We present solutionfor the disturbance rejection and the reference-trackingproblems.

Three theorems are presented for the problems. InTheorem 1, the generalized system does not contain thesensor noises for the problems. This solution concernswith the design of a LPV feedforward controller forLPV systems to attenuate the disturbances affectingthe feedback and to improve reference-tracking per-formance, where the generalized system is composedwithout including sensor noises. However, the secondand third theorems enable the designer to consider thesensor noises when the generalized system is com-posed for the problems. Thus, if the designer wantsto take account of the sensor noises, he can use the sec-ond or third solution for the problem. In this paper,Theorem 1 gives main solution for the problems. Theother theorems are based on it using some procedures.

⎛

⎝AT

cl (θ) P (θ) + P (θ) Acl (θ) + P (θ) P (θ) Bcl (θ) CTcl (θ)

BTcl (θ) P (θ) −γ I DT

cl (θ)

Ccl (θ) Dcl (θ) −γ I

⎞

⎠ ≺ 0, (6)

(Acl (θ (t)) Bcl (θ (t))Ccl (θ (t)) Dcl (θ (t))

)

=

⎛

⎜⎜⎝

A (θ (t)) B2 (θ (t)) Cc (θ (t)) B1 (θ (t)) + B2 (θ (t)) Dc (θ (t))0 Ac

(θ (t) , θ (t)

)Bc (θ (t))

C1 (θ (t)) D12 (θ (t)) Cc (θ (t)) D11 (θ (t)) + D12 (θ (t)) Dc (θ (t))C2 (θ (t)) D21 (θ (t)) Cc (θ (t)) D21 (θ (t)) + D22 (θ (t)) Dc (θ (t))

⎞

⎟⎟⎠ , (7)

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⎛

⎜⎜⎜⎜⎝

AT (θ) X (θ) + X (θ) A (θ) + X (θ) K (θ) + AT (θ) L (θ)

∗ A (θ) Y (θ) − B2 (θ) M (θ) + (∗)T − Y (θ) B1 (θ) + B2 (θ) D (θ)

∗ ∗ −γ I∗ ∗ ∗∗ ∗ ∗

CT1 (θ) CT

2 (θ)

Y (θ) CT1 (θ) − MT (θ) DT

12 (θ) Y (θ) CT2 (θ) − MT (θ) DT

22 (θ)

DT11 (θ) + DT (θ) DT

12 (θ) DT21 (θ) + DT (θ) DT

22 (θ)

−γ I 0∗ −γ I

⎞

⎟⎟⎟⎟⎠

≺ 0

, (8)

(X (θ) I

I Y (θ)

) 0, (9)

Y (θ) � δ (θ) + X−1 (θ)

K (θ) � X (θ) A (θ) Y (θ) − X (θ)(

Ac(θ, θ

)

+B2 (θ) Cc (θ)) δ (θ) + X (θ) X−1 (θ) (10)

L (θ) � X (θ) (B1 (θ) + Bc (θ) + B2 (θ) D (θ))

M (θ) � Cc (θ) δ (θ) ,

Dc (θ) = D (θ)

Cc (θ) = M (θ) δ−1 (θ)

Bc (θ) = X−1 (θ) L (θ) − B1 (θ) − B2 (θ) Dc (θ)

Ac(θ, θ

) = A (θ) Y (θ) δ−1 (θ) − B2 (θ) Cc (θ)

− X−1 (θ) K (θ) δ−1 (θ)

+ X−1 (θ) X (θ) X−1 (θ) δ−1 (θ) .

(11)

Theorem 1 Consider the LPV plant in (1) and its para-meter trajectories constrained by (2). The closed-loopsystem has L2-gain γ , and there exists the LPV feed-forward controller in (3) if there exist the symmetricpositive definite matrices X (θ) and Y (θ) defined by(12) compatible with the parameter-dependent matri-ces K , L , M defined by (10) such that the infinite LMIsin (8) and (9) hold for ∀ (

θ, θ) ∈ D × E.

X (θ) = X0 +n∑

i=1

θi Xi ≥ 0 and

Y (θ) = Y0 +n∑

i=1

θi Yi ≥ 0.

(12)

In such a case, the parameter-dependent state spacematrices of the LPV feedforward controller are obtainedas in (11).

Remark 1 it is clear in (7) that the feedforward con-troller in (3) does not affect the stability of the closed-loop system also containing generalized system G aslong as the stability of the controller is guaranteed forall varying parameters. The feedforward controller isdesigned for the system G in the solution after thefeedback controller provides the stability of the sys-tem G. Therefore, the feedforward controller which isstable does not cancel the internal stability of the sys-tem G previously stabilized by the feedback controller.Also, it is known that the feedforward controllers do notstabilize the unstable system. Consequently, the feed-forward controller is designed to improve the track-ing performance or to attenuate the measurable exter-nal disturbances with guaranteeing the stability of theclosed-loop system for all varying parameters after itis assumed that the G generalized system is stabilizedby the feedback controller.

There exist the infinite-dimensional LMIs due tousing PDLF. Therefore, the reduction of the infinite-dimensional LMIs is necessary to obtain the finiteLMIs. This reduction is performed by the multi-convexity conditions in the Sect. 4.

If LMIs in (8) and (9) are solved by minimizing γ

in the multi-convexity conditions, we can get the con-troller matrices defined by (11) for the optimal controldesign. The definitions in (10) are obtained by usingthe method of change of variables, and so the matricesof the controller depended on parameters are computedby using them.

Proof For the control synthesis, the affine parameter-dependent Lyapunov function is selected as in (14); itstime-derivative is denoted as in (15).

123


⎛

⎝I δ + X−1 00 −δ 00 0 I

⎞

⎠

T ⎛

⎝AT



cl (θ)


⎞

⎠

⎛

⎝I δ + X−1 00 −δ 00 0 I

⎞

⎠ , (13)

V (x) = xT

(

P0 +n∑

i=1

θi Pi

)

x 0, (14)

d

dt(V (x, θ)) = xT

(P (θ) A (θ) + AT (θ) P (θ) + P (θ)

)x ≤ 0. (15)

In accordance with BRL for LPV systems in Lemma 1,let define a positive definite symmetric Lyapunovmatrix:

P (θ) = PT (θ) =(

X (θ) P12 (θ)

PT12 (θ) P22 (θ)

) 0.

Because of nonlinearity in LMI, the matrix is pre- andpost-multiplied as follows where P12 is the invertibleand square matrix.

P (θ) =(

I 00 P−1

12 (θ) X (θ)

)T (X (θ) P12 (θ)

PT12 (θ) P22 (θ)

)

×(

I 00 P−1

12 (θ) X (θ)

)

=(

X (θ) X (θ)

X (θ) X (θ) P−T12 (θ) P22 (θ) P−1

12 (θ) X (θ)

) 0.

To increase positivity, X matrix is subtracted from thematrix as follows. It is clear that this step enables to thepositiveness of eigenvalues.

δ−1 (θ) = X (θ) P−T12 (θ) P22 (θ) P−1

12 (θ) X (θ)

−X (θ) .

For simplicity, let define Y (θ) = δ (θ) + X−1 (θ).X (θ) and Y (θ) are positive symmetric matrices asin (12). Hence, the parameter-dependent Lyapunovmatrix is obtained as in (16). We also get the time-derivative of the Lyapunov matrix as in (17). It is clearthat it is used for imposing on Lemma 1, which providesthe stability, and constructing LMI for L2 controllerdesign.

P (θ) =(

X (θ) X (θ)

X (θ) δ−1 (θ) + X (θ)

) 0, (16)

P (θ) =(

X (θ) X (θ)

X (θ) δ−1 (θ) + X (θ)

). (17)

In order to construct the controller, LMI which isinfinite dimensional is obtained by using Lemma 1

and congruence transformation. Firstly, the obtainedAcl , Bcl , Ccl , Dcl closed-loop transfer matrices fromω to Z for proposed controller structure in Fig. 1 arereplaced in (6) according to Lemma 1. Nevertheless, theinequality is not in the form of LMI since it contains themultiplications of the unknown matrices. A lineariza-tion is necessary for the multiplications of the unknownmatrices. Therefore, appointed Lyapunov function in(16) and its time-derivative in (17) are replaced in (13).Afterward, the congruence transformation is used forgetting LMI, and thus the LMI (6) is pre- and post-multiplied by the transpose of ϕ and ϕ in (18), respec-tively, as in (13). The derivative of the matrix inverse isalso used in (19). Consequently, we get the inequality(8) in the form of LMIs under definitions in (10). (10)is obtained by the change of variables method.

ϕ � blockdiag

{(I δ + X−1

0 −δ

), I, I, I

}, (18)

δ−1 = −δ−1δδ−1. (19)

�Remark 2 This solution is limited in point of takingaccount of the sensor noises for the reference-trackingand the disturbance attenuation problems since thedimensions of B1 and B2 input matrices are equal inthe solution. It is clear that the dimensions of thesematrices in Bc expression must be equal when we com-pute the controller matrices in (11). This case does notenable the designer to consider the sensor noises for theproblems because the generalized LPV system does notcontain the input of sensor noise. In order to overcomethe difficulty, we need to eliminate B2 matrix in Bc

expression. Therefore, we extend the LPV feedforwardproblem by augmenting the input matrices of the gen-eralized system G(θ(t)) for sensor noise input, and sothe extended LPV system in (20) is illustrated in Fig. 3where ωn is the sensor noise input. The transfer matrix

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Fig. 3 The extended feedforward LPV control scheme

of control system from ω and ωn to Z is also obtainedin (21). Hence, the feedforward controller, which cantake account of the sensor noises, can be gotten by thenext theorem. As a result, Theorem 1 can be used forthe reference-tracking and the disturbance attenuationproblem without sensor noises.

x (t) = A (θ (t)) x (t) + B1 (θ (t)) ω (t)

+ B2 (θ (t)) ωn (t) + B3 (θ (t)) u (t)

z (t) = C1 (θ (t)) x (t) + D11 (θ (t)) ω (t) (20)

+ D12 (θ (t)) ωn (t) + D13 (θ (t)) u (t)

y (t) = C2 (θ (t)) x (t) + D21 (θ (t)) ω (t)

+D22 (θ (t)) ωn (t) + D23 (θ (t)) u (t) ,

(Acl (θ (t)) Bcl (θ (t))Ccl (θ (t)) Dcl (θ (t))

)=

⎛

⎜⎜⎝

A (θ (t)) B3 (θ (t)) Cc (θ (t)) B1 (θ (t)) + B3 (θ (t)) Dc (θ (t)) B2 (θ (t))0 Ac

(θ (t) , θ (t)

)Bc (θ (t)) 0

C1 (θ (t)) D13 (θ (t)) Cc (θ (t)) D11 (θ (t)) + D13 (θ (t)) Dc (θ (t)) D12 (θ (t))C2 (θ (t)) D23 (θ (t)) Cc (θ (t)) D21 (θ (t)) + D23 (θ (t)) Dc (θ (t)) D22 (θ (t))

⎞

⎟⎟⎠ , (21)

⎛

⎜⎜⎜⎜⎜⎜⎝

AT (θ) X (θ) + (∗)T + X (θ) K (θ) + AT (θ) L (θ)

∗ A (θ) Y (θ) − B3 (θ) M (θ) + (∗)T − Y (θ) B1 (θ) + B3 (θ) D (θ)

∗ ∗ −γ I∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

X (θ) B2 (θ) CT1 (θ) CT

2 (θ)

B2 (θ) Y (θ) CT1 (θ) − MT (θ) DT

13 (θ) Y (θ) CT2 (θ) − MT (θ) DT

23 (θ)

0 DT11 (θ) + DT (θ) DT

13 (θ) DT21 (θ) + DT (θ) DT

23 (θ)

−γ I DT12 (θ) DT

22 (θ)

∗ −γ I 0∗ ∗ −γ I

⎞

⎟⎟⎟⎟⎟⎟⎠

≺ 0

, (22)

(X (θ) I

I Y (θ)

) 0 (23)

Y (θ) � δ (θ) + X−1 (θ)

K (θ) � X (θ) A (θ) Y (θ) − X (θ) (Ac (θ)

+B3 (θ) Cc (θ)) δ (θ) + X (θ) X−1 (θ) (24)

L (θ) � X (θ) (B1 (θ) + Bc (θ) + B3 (θ) D (θ))

M (θ) � Cc (θ) δ (θ) ,

Dc (θ) = D (θ)

Cc (θ) = M (θ) δ−1 (θ)

Bc (θ) = X−1 (θ) L (θ) − B1 (θ) − B3 (θ) Dc (θ)

Ac(θ, θ

) = A (θ) Y (θ) δ−1 (θ) − B3 (θ) Cc (θ)

− X−1 (θ) K (θ) δ−1 (θ)

+ X−1 (θ) X (θ) X−1 (θ) δ−1 (θ) .

(25)

Theorem 2 Consider the LPV plant in (20) and itsparameter trajectories constrained by (2). If there existthe symmetric positive definite matrices X (θ) andY (θ) defined by (12) compatible with the parameter-dependent matrices K , L , M defined by (24), theclosed-loop system has L2-gain γ and there exists theLPV feedforward controller (3) such that the infiniteLMIs in (22) and (23) hold for ∀ (

θ, θ) ∈ D × E .

In such case, the parameter-dependent state spacematrices Ac, Bc, Cc, Dc of the LPV feedforward con-troller are computed as in (25).

123


Similar to Theorem 1, the reduction of the infinite-dimensional LMIs is necessary to obtain the finiteLMIs, which is fulfilled in the next section.

We get the different LMIs for the solution in Theo-rem 2 by using the next Lemma.

Lemma 3 (Projection Lemma) [44] Let Q be a sym-metric matrix which has three rows/three columns andis in the form of LMI as follows:⎛

⎝Q11 Q12 + H T Q13

Q21 + H Q22 Q23

Q31 Q32 Q33

⎞

⎠ ≺ 0,

There exists a solution H of this LMI if and only if(

Q11 Q13

Q31 Q33

)≺ 0 and

(Q22 Q23

Q32 Q33

)≺ 0.

If these LMIs hold, a solution of H is given by

H = QT32 Q−1

33 QT13 − Q21,

⎛

⎜⎜⎜⎜⎝

A (θ) Y (θ) − B3 (θ) M (θ) + (∗)T − Y (θ) B1 (θ) + B3 (θ) D (θ) B2 (θ)

∗ −γ I 0∗ ∗ −γ I∗ ∗ ∗∗ ∗ ∗

Y (θ) CT1 (θ) − MT (θ) DT

13 (θ) Y (θ) CT2 (θ) − MT (θ) DT

23 (θ)


13 (θ) DT21 (θ) + DT (θ) DT

23 (θ)

DT12 (θ) DT

22 (θ)

−γ I 0∗ −γ I

⎞

⎟⎟⎟⎟⎠

≺ 0

, (26)

⎛

⎜⎜⎜⎜⎝

AT (θ) X (θ) + X (θ) A (θ) + X (θ) L (θ) X (θ) B2 (θ) CT1 (θ)

∗ −γ I 0 DT11 (θ) + DT (θ) DT

13 (θ)

∗ ∗ −γ I DT12 (θ)

∗ ∗ ∗ −γ I∗ ∗ ∗ ∗

CT2 (θ)


23 (θ)

DT22 (θ)

0−γ I

⎞

⎟⎟⎟⎟⎠

≺ 0. (27)

Theorem 3 Consider the LPV plant in (20) and itsparameter trajectories constrained by (2). The closed-loop system has L2-gain γ , and there exists the LPVfeedforward controller (3) if there exist the symmetricpositive definite matrices X (θ) and Y (θ) defined by(12) compatible with the parameter-dependent matri-ces K , L , M defined by (24) such that LMIs in (23),(26), and (27) hold for ∀ (

θ, θ) ∈ D × E.

The parameter-dependent LPV feedforward con-troller is computed as in (25), but there exists only a

difference. This difference is that Ac is computed as in(28).

Ac(θ, θ

) = A (θ) Y (θ) δ−1 (θ) − B3 (θ) Cc (θ)

− X−1 (θ) (θ) φ−1 (θ) ζ T (θ) δ−1 (θ)

+ X−1 (θ) AT (θ) δ−1 (θ)

+ X−1 (θ) X (θ) X−1 (θ) δ−1 (θ) . (28)

Proof Let define (29) in which , μ, , �, ζ, φ aredescribed as in (32) and (33) by using the LMIs in pre-vious Theorem. If Lemma 3 is applied to the inequal-ities, (30) is equivalent to (29). Hence, we can get theLMIs in (26) and (27) for the controller design prob-lem. It is clear that the controller matrices Bc, Cc, Dc

are similar to the controller matrices given in previousTheorem, but the matrix Ac is as in (28). This is becauseK (θ) in (31) is computed similar to H in Lemma 3.

⎛

⎝ (θ) μ (θ) + K (θ) (θ)

∗ �(θ) ζ (θ)

∗ ∗ φ (θ)

⎞

⎠ ≺ 0, (29)

( (θ) (θ)

∗ φ (θ)

)≺ 0 and

(�(θ) ζ (θ)

∗ φ (θ)

)≺ 0, (30)

123


K (θ) = (θ) φ (θ)−T ζ T (θ) − AT (θ) , (31)

(θ) =(

AT (θ) X (θ) + X (θ) A (θ) + X (θ))

μ (θ) = AT (θ) (32)

(θ) = (L (θ) X (θ) B2 (θ) CT

1 (θ) CT2 (θ)

).

As mentioned aforetime, the reduction of the infinite-dimensional LMIs to the finite LMIs is necessity. It isperformed by the multi-convexity conditions in nextsection in which it is performed for this Theorem.Similarly, the procedures can be also performed forTheorem 1 and 2.

�(θ) = (A (θ) Y (θ) − B3 (θ) M (θ) + (∗)T − Y (θ)

)

ζ (θ) =(

B1 (θ) + B3 (θ) D (θ) B2 (θ) Y (θ) CT1 (θ) − MT (θ) DT

13 (θ) Y (θ) CT2 (θ) − MT (θ) DT

23 (θ))

φ (θ) =

⎛

⎜⎜⎝

−γ I 0 DT11 (θ) + DT (θ) DT

13 (θ) DT21 (θ) + DT (θ) DT

23 (θ)

∗ −γ I DT12 (θ) DT

22 (θ)

∗ ∗ −γ I 0∗ ∗ ∗ −γ I

⎞

⎟⎟⎠

. (33)

4 The reduction of the infinite-dimensional LMIs

In this section, the infinite LMIs obtained in Theo-rem 3 are reduced to the finite LMIs. It is clear thatthe LMIs obtained in (26) and (27) are infinite dimen-sional due to PDLF. These inequalities depend on para-meters θi for i = 1, . . ., n. It appears quite difficult tosolve such inequalities. This difficulty concerns withthe finite LMIs in parameter space. Yet, we are able toreduce the infinite-dimensional LMIs by using Remark3.6 in [18] and the multi-convexity condition givenby Lemma 2. Thus, the inequalities can be writtenin the finite-dimensional LMIs by this approach. Inthis approach, it is common that the PDLF varying thescheduled parameters is restricted.

According to Remark 3.6 in [18], it is tried to providean LMI condition for any parameter-dependent matrix:N (θ) ≺ 0 for all θ ∈ Ras long as S which is a positivesemi-definite symmetric matrix.

N (θ) +n∑

i=1

θ2i Si ≺ 0, (34)

implies N (θ) ≺ 0, and the multi-convexity require-ment according to Lemma 2 is

∂2 N (θ)

∂θ2i

+ 2Si ≥ 0, for i = 1, . . . , n. (35)

The below reduction is performed by using the con-ditions in (34) and (35) including the multi-convexityin Lemma 2.

The closed loop of the LPV feedforward controlsystem is affinely rewritten as in (40). The selectedPDLF and its time- derivative expressed by (36) affinelydepend on the parameters θi . If the multi-convexityconditions are imposed on (6) similar to (34) and(35), we can get the inequalities (37) and (41) for allθ ∈ Dvex × Evex where the symmetric matrix si isdescribed in (38).

�

P (θ) = P0 +n∑

i=1

θi Pi and

d

dt(P (θ)) = θ1 P1+, . . . ,+θn Pn, (36)

(Pi Acli + AT

cli Pi Pi Bcli

BTcli Pi 0

)+ si ≥ 0 (37)

for i = 1, . . . , n,

si ≥ 0 for i = 1, . . . , n. (38)

Now then, the infinite-dimensional LMIs obtained in(26) and (27) can be reduced to the finite LMIs usingthe argument. Supposing that D13, D23, C1, and C2 areparameter-independent in order to simplify the expres-sions, we can get the finite-dimensional LMIs definedin (42)–(45) for all θ ∈ Dvex × Evex by the argument.(42) and (43) are obtained similar to (34), while (44)and (45) are obtained by using the multi-convexity con-dition similar to (35). Si and Ti are positive definitematrices defined as in (39). These steps can be also per-formed for the LMIs in Theorem 1 and 2. In numericalexamples, Theorem 1 is performed for one parametervarying system (θi = θ1 for i = 1).

Si = STi ≥ 0 and Ti = T T

i ≥ 0 ∀i=1, 2, 3, (39)

123


(Acl (θ) Bcl (θ)

Ccl (θ) Dcl (θ)

)=

(Acl0 Bcl0Ccl0 Dcl0

)+ θ1

(Acl1 Bcl1Ccl1 Dcl1

)+, · · · , +θn

(Acln BclnCcln Dcln

), (40)

⎛

⎝AT



cl (θ)


⎞

⎠ +n∑

i=1

θ2i si ≺ 0, (41)

⎛

⎜⎜⎜⎜⎝

AT (θ) X (θ) + X (θ) A (θ) + X (θ) L (θ) X (θ) B2 (θ) CT1 CT

2∗ −γ I 0 DT11 (θ) + DT (θ) DT

13 DT21 (θ) + DT (θ) DT

23∗ ∗ −γ I DT12 (θ) DT

22 (θ)

∗ ∗ ∗ −γ I 0∗ ∗ ∗ ∗ −γ I

⎞

⎟⎟⎟⎟⎠

+ ∑ni=1 θ2

i

⎛

⎜⎜⎜⎜⎝

S1 0 S2 0 0∗ 0 0 0 0∗ ∗ S3 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ ∗ 0

⎞

⎟⎟⎟⎟⎠

≺ 0

, (42)

⎛

⎜⎜⎜⎜⎝

A (θ) Y (θ) − B3 (θ) M (θ) + (∗)T − Y (θ) B1 (θ) + B3 (θ) D (θ) B2 (θ) Y (θ) CT1 − MT (θ) DT

13∗ −γ I 0 DT

11 (θ) + DT (θ) DT13

∗ ∗ −γ I DT12 (θ)

∗ ∗ ∗ −γ I∗ ∗ ∗ ∗

Y (θ) CT2 − MT (θ) DT

23DT

21 (θ) + DT (θ) DT23

DT22 (θ)

0−γ I

⎞

⎟⎟⎟⎟⎠

+n∑

i=1θ2

i

⎛

⎜⎜⎜⎜⎝

T1 T2 0 0 0∗ T3 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ ∗ 0

⎞

⎟⎟⎟⎟⎠

≺ 0

, (43)

(∂2

∂θ2i

{AT (θ) X (θ) + X (θ) A (θ) + X (θ)

}∂2

∂θ2i

{X (θ) B2 (θ)}∗ 0

)

+(

S1 S2

∗ S3

)≥ 0, (44)

(∂2

∂θ2i

{A (θ) Y (θ) − B3 (θ) M (θ) + (∗)T − Y (θ)

}∂2

∂θ2i

{B1 (θ) + B3 (θ) D (θ)}∗ 0

)

+(

T1 T2

∗ T3

)≥ 0. (45)

5 Numerical examples

In this section, we carry out the simulations for pro-posed feedforward controller design in Theorem (1).It is combined with feedback PI controller for thedynamic system considered in (46) where x p repre-sents the states x1 and x2.

x p =( −10 1

−0.02 −2θ

)x p +

(02

)u, yp = (

1 0)

x p ,

(46)

θ (t) ∈ [1 5

] ∀t ∈ [ 0 ∞ )and θ (t) = yp (t) . (47)

We aim to obtain the controller for the system (1).All numerical tests are performed with YALMIP parser[45] and SEDUMI solver [46] for obtaining the solu-tions of LMIs. In addition to the reduction of Theo-rem 3, the reduction of the infinite LMIs in Theorem 1is performed in this example. (47) defines the lower andupper bounds of the varying parameter for the systemin (46). We can combine PI feedback controller withfeedforward controller for the system, which is illus-trated in Fig. 4. z1 and z2 symbolize z in (1), KFF(θ)

symbolizes feedforward controller, and Pr(θ) symbol-izes the dynamic system in (46).

123


Fig. 4 The closed loop of the reference-tracking problem

Firstly, we obtain the matrices of the generalizedsystem in (1) using control structure in Fig. 5. Thestate space parameters of PI controller in Fig. 4 aregiven in (48), where bPI = 14.142, cPI = 14.142, anddPI = 180. The weighting functions whose cutoff fre-quencies are 10 and 0.19 rad/sn are in (49). Their statespace forms are also as in (50). The input function isused to keep the low frequency range just as the outputfunction is used to keep the closed-loop bandwidth ata demanded value and to shape the supplementary sen-sitivity function. All of the parameter-dependent func-tions are affinely formed for the example system. Forinstance, A(θ) is formed by (51).

xP I = bP I y (48)

yF B = cP I xP I + dP I y,

Ws = 1

s + 10, WT = 1

s + 0, 19, (49)

xT = AT xT + BT u f f

z1 = CT xT + DT u f f

xs = As xs + Bs yz2 = Cs xs + Ds y

, (50)

A (θ) = A0 + θ1 A1, (51)

⎛

⎜⎜⎜⎜⎝

AT (θ) X (θ) + X (θ) A (θ) + X (θ) K (θ) + AT (θ) L (θ)

∗ A (θ) Y (θ) − B2 M (θ) + (∗)T − Y (θ) B1 + B2 D (θ)

∗ ∗ −γ I∗ ∗ ∗∗ ∗ ∗

CT1 CT

2Y (θ) CT

1 − MT (θ) DT12 Y (θ) CT

2 − MT (θ) DT22


12 DT21 + DT (θ) DT

22−γ I 0

∗ −γ I

⎞

⎟⎟⎟⎟⎠

+ θ2

⎛

⎜⎜⎜⎜⎝

S1 S2 0 0 0∗ S3 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ 0 0∗ ∗ ∗ ∗ 0

⎞

⎟⎟⎟⎟⎠

≺ 0

, (52)

(AT

1 X1 + X1 A1 0∗ A1Y1 + Y1 AT

1

)+

(S1 S2

∗ S3

)≥ 0,

(53)

Fig. 5 The generalized system structure of the reference-tracking problem

(52) and (53) finite LMIs are obtained from infiniteLMIs (8) and (9) in Theorem 1 as (42), (43), (44), and(45) are obtained for Theorem 3. All of the parameter-dependent matrices in (52) and (53) are formed as in(51). In order to construct the controller, the LMIs (9),(52), and (53) are solved by minimizing γ under therestricted parameter in (47) for the variables X0, X1,Y0, Y1, M0, M1, L0, L1, K0, K1, D0,D1,S1, S2, S3. Thestate space matrices of the parameter-dependent con-troller are obtained by using (11). Consequently, whenwe simulate the system in Fig. 4 for step response, wecan compare PI with LPV feedforward controller com-bined with PI controller. As shown in the Fig. 6, theproposed LPV feedforward controller combined withPI controller is better performance than alone PI con-troller in point of the settling time and overshoot.

Another example is given for the disturbance rejec-tion problem in Fig. 7 which is commonly for theprocess control. In this example, the feedforward con-

troller is imposed on state feedback LPV controllerwhich is symbolized as KLPV in Fig. 7. See [47] toconstruct state feedback LPV controller.

123


Step response

Time (seconds)

Am

plitu

de

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

X: 0.1831Y: 1.359

X: 0.1841Y: 1.058

PI

PI with LPV feedforward

Fig. 6 Step response of closed-loop control system

Fig. 7 The closed-loop system of disturbance rejection problem

Consider Van der Pol equations in (54), where ini-tial conditions are 0.5. (55) defines the lower and upperbounds of the varying parameter for the system. Wecan combine feedback controller with feedforward con-troller for the problem, which is illustrated in Fig. 7. Anormalization coefficient (NC) is between controllerand external disturbance for disturbance rejection. Inthis example, it is 1100. This is determined accordingto the system behavior.

Fig. 8 The generalized system structure of the disturbance rejec-tion problem

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Fig. 9 The applied disturbances to the control system

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t (sn)

thet

a

Fig. 10 The change of theta

Firstly, we obtain the matrices of the system in (1)using control structure in Fig. 8 where the weightingfunctions are defined in (56).

In order to construct the controller, the LMIs of theform (9), (52), and (53) including the multi-convexityconditions are solved by minimizing γ under therestricted parameter in (55). The parameter-dependentstate space matrices of the controller Ac, Bc, Cc, Dc

can be obtained by (11).

x p =(

0 −11 −0.3θ

)x p +

(01

)u, yp = x p

θ = 1 − x21

, (54)

θ (t) ∈ [0 1

] ∀t ∈ [ 0 ∞ ), (55)

Ws = 0, 1

s + 0, 5, WT = 0, 05s

s + 0, 5. (56)

123


The disturbances applied to the control system areas in Fig. 9. The initial conditions are 1 for the sim-ulation. As a result of the simulating of the controlsystem, we can compare LPV state feedback controllerwith LPV feedforward controller combined with it. Fig-ure 10 shows the change of parameter θ during the sim-ulation. Figure 11 shows the response of the alone LPVstate feedback controller against the disturbances. Fig-ure 12 shows the response of the feedforward controllercombined with it. Therefore, the effects of the exter-nal disturbances are efficiently decreased as shown inFig. 12. Consequently, it is shown that the LPV feedfor-ward controller is better performance than alone LPVstate feedback controller against the disturbances.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t (sn)

Sta

te v

aria

bles

Fig. 11 The response of only LPV state feedback controller

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t (sn)

Sta

te v

aria

bles

Fig. 12 The response of LPV feedforward controller combinedwith LPV state feedback controller

6 Conclusion

The results show that a design of LPV feedforwardcontroller has been accomplished for the disturbancerejection and reference-tracking problem in LPV sys-tems.

Theorem 1 is used to design the controller withoutincluding the sensor noises for reference-tracking ordisturbance attenuation problems, whereas Theorems 2and 3 are used to design the controller with includ-ing the sensor noises for the problems. Theorem 1was applied to the tracking problem for the numeri-cal implementation, and thus it has been shown thatproposed LPV feedforward controller is of much bet-ter performance than alone feedback PI controller in2-DOF control system. Furthermore, when we appliedto the disturbance rejection problem, it has been shownthat proposed LPV feedforward controller combinedwith LPV state feedback controller is of better per-formance than alone LPV state feedback controlleragainst disturbances. Thus, the effects of the exter-nal disturbances are efficiently decreased. As a result,the applicability of proposed solution was successfullyprovided with numerical and simulation examples.

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http://www.dcsc.tudelft.nl/~cscherer/lmi/notes05.pdf

http://www.dcsc.tudelft.nl/~cscherer/lmi/notes05.pdf

linear parameter varying feedforward control synthesis using parameter-dependent lyapunov function

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