Hinf delay-scheduled control of linear systems with time-varying delays

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time-varying delays

Corentin Briat, Olivier Sename, Jean-François Lafay

To cite this version:

Corentin Briat, Olivier Sename, Jean-François Lafay. Hinf delay-scheduled control of linear systems

with time-varying delays. IEEE Transactions on Automatic Control, Institute of Electrical and

Elec-tronics Engineers, 2009, 54 (9), pp.2255-2260. �hal-00402241�

Delay-Scheduled Control of Linear Systems With Time-Varying Delays

Corentin Briat, Olivier Sename, and Jean-François Lafay

Abstract—This paper deals with delay-scheduled control of linear systems with time-varying delays. First, a new model transformation is given which allows to provide a unified approach to stability analysis and state-feedback control synthesis for time-delay systems represented in “linear fractional transformation (LFT)” form. A new type of controller is then synthesized, where the state-feedback is scheduled by the value of the delay (delay-scheduled controllers). The results are provided in terms of linear matrix inequalities (LMIs) which are known to be efficiently solvable.

Index Terms—Linear fractional transformation (LFT), linear parameter varying (LPV) control, robust control, time-delay systems (TDS).

I. INTRODUCTION

Since several years, many papers have been devoted to the study of time-delay systems (TDS) with constant delays (see for in-stance [1]–[3] and references therein). More recently, systems with time-varying delays arising for instance in network controlled systems, have attracted more and more attention (See [5]–[7] and references therein). Indeed, lags in communication channels may destabilize such systems, or at least, deteriorate performance. Different approaches devoted to the study of time-delay systems stability have been de-veloped in the literature. Let us mention, among others, the use of Lyapunov–Krasovskii functionals (or Lyapunov–Razumikhin func-tions) [1], [4], [7], robust analysis [7], [8], well-posedness of feedback systems [9], spectral approaches [7], [8] etc.

The approaches developed in this paper are mainly based on the no-tions of robust stability, robust and LPV control of linear dynamical systems. Some new results for the stability and control of such systems are provided and suggest that the robust control approach can be used to derive linear parameter varying (LPV) controllers for LTI time-delay systems.

In the last two decades, LPV systems have been of growing interest since they allow to approximate nonlinear and LTV systems [10]–[16]. Three main approaches are usually considered in the study and control of LPV systems involving LMIs: the polytopic approach [17], the use of parameter-dependent LMIs [18] or the linear fractional transformation (LFT) approach [10]–[12].

While both first approaches have been recently extended to time-delay systems [19]–[22], the latter has been very few used in the con-text of time-delay systems and especially for bounded time-varying delays. This is due to the fact that the LFT formulation for LPV/un-certain time-delay systems is difficult to apply and may result in untractable conditions. Indeed, by applying classical robust control theorems (such as projection lemma) on bounded-real lemmas ob-tained from Lyapunov–Krasovskii functionals yields, in many cases,

Manuscript received December 02, 2008; revised April 12, 2009. First pub-lished August 25, 2009; current version pubpub-lished September 04, 2009. Recom-mended by Associate Editor G. Chesi.

C. Briat and O. Sename are with the GIPSA-Laboratory, Automatic Con-trol Department, ENSE , Domaine Universitaire-BP46, Saint Martin d’Hères 38402, France (e-mail: corentin@briat.info; olivier.sename@gipsa-lab.inpg.fr). J.-F. Lafay is with the Institut de Recherche en Communication et Cyberné-tique de Nantes, Centrale de Nantes, Nantes 44321, France (e-mail: jean-fran-cois.lafay@irccyn.ec-nantes.fr).

Digital Object Identifier 10.1109/TAC.2009.2026859

nonlinear matrix inequalities (due to the supplementary decision matrices) whose solving is known to be an NP-hard problem.

This paper proposes a new approach to study delay-dependent sta-bility and control of LTI time-delay systems, within a unified frame-work involving robust control theory for LPV systems. The contribu-tions of the paper are the following:

• First, a new model transformation for TDS with time-varying de-lays is introduced which turns a time-delay system into an uncer-tain LPV system in “LFT” form. Using this formulation, a stability test based on the scaled bounded real lemma allows to conclude on asymptotic stability of the time-delay system. The interest of such a reformulation resides in the fact that many classical robust control tools are available and can be used in order to derive suf-ficient conditions for the controller existence.

• A new kind of controller, which has been referred to as delay-scheduled state-feedback controller, is developed when an approx-imate value of the delay is known or estapprox-imated. The error on the delay knowledge value is taken into account to ensure the robust-ness of the closed-loop system with respect to this uncertainty. The interest and advantage of the provided methodology rely on the fact that, for the first time, a unique LFT formulation to design dif-ferent types of controllers for difdif-ferent classes of time-delay systems is proposed. Even if the present paper is devoted to (delay-scheduled) state-feedback controllers only, the approach can be easily extended to the case of (delay-scheduled) dynamic output feedback [11]. Moreover, even if only the single-delay problem is addressed, the methodology is also valid in the multiple-delay case and for LPV TDS. Finally, this method describes a new original way to control time-delay systems and the authors stress that this approach may be of great interest for systems with large variation of the delay since a controller gain adaptation will be provided accordingly.

The paper is structured as follows, Section II gives the paper ob-jectives and preliminary results. Section III introduces the new model transformation. In Section IV the new LPV based control method for time-delay systems is exposed. Finally Section V concludes on the paper and gives future works.

Along the paper, the notation is standard andKer[A] stands for a basis of the null-space ofA.

II. PAPEROBJECTIVES ANDPRELIMINARYRESULTS Let us consider in this section LPV/uncertain linear systems repre-sented in “LFT” form _x(t) z0(t) z1(t) = A B0 B1 C0 D00 D01 C1 D10 D11 x(t) w0(t) w1(t) + Bu D0u D1u u(t) w0(t) = 2(z0(t)) (1)

wherex 2 n; u 2 m; w12 r; z12 sare respectively the state, the control input, the exogenous inputs and the controlled outputs. The signalsw0=z0 are ‘virtual’ signals describing the interconnection of the LTI system and2.

In this work,2 will contain both linear operators and multiplicative parameters. In consequence, it is chosen to have the following diagonal structure:

2 := diagi[i(t)Ip] 0

0 diag_i[Li( 1 )Io ] (2)

wherep_iando_iare respectively the number of occurrences of theith parameteri and operatorLi( 1 ). The parameters i(t) are assumed

to belong to the interval [01; 1] and the linear operators to have an H1norm (orL2-induced norm) less than 1. With such a formulation it is possible to consider both polynomial and rational dependence on parameters and operators.

The upcoming results in this section are developed for LPV systems only (without time-delay) and will be used in Section III. Let us con-sider now the general uncertain LPV system

_x(t) z(t) =  A B Bu  C D  Du x(t) w(t) u(t) w0(t) = 2(z0(t); t) w1(t) = 1(z1(t); t) (3)

withz = col(z₀; z₁; z₃); w = col(w₀; w₁; w₃) and  D = D00 D01 D02 D10 D11 D12 D20 D21 D22  B = [B0 B1 B2]  C = CT 0 C1T C2T T  Du= DT0u DT1u D2uT T (4)

wherew0=z0; w1=z1andw3=z3are respectively the LPV channel, the uncertain channel and the performance channel which has to be opti-mized.2 are scheduling parameters/operators defined in (2) and 1 the uncertainties obeying to

1 := fdiagn

i=11i; k1ik1< 1; i = 1; . . . ; nug : (5)

Definition 2.1: The aim of the paper is to design a gain-scheduled

state-feedback (GSSF) control law of the form u(t)

zc(t) = K

x(t)

wc(t) wc(t) = f(h(t))zc(t) (6)

wheref( 1 ) is a scheduling function to be defined/determined, wc=zc

the controller scheduling channel, such that the closed-loop LPV time-delay system (3) is asymptotically stable andkz3kL  kw3kL .

The following theorem gives a new sufficient condition for the ex-istence of a gain-scheduling state-feedback controller (similar to the sufficient condition for the existence of a dynamic output feedback pro-vided in [11]).

Theorem 2.1: Consider the LPV system (3)–(4) of ordern. If there

exist symmetric positive definite matricesX; L3; J3and a scalar > 0 such that KT i Ni+ TiTMiTi Ki< 0; i = 1; 2 L3 I I J3  0 L3; J32 L2 (7) with N1= 0diag(L3; In ; Iw ) N2=  AX + X AT _{X C}T ? 0VJ VJ= diag(J3; In ; Iz ) T1= D K1 = I; T2 = [ BT DT] M2= diag(J3; In ; Iz ) K2= Ker BuT DuTL

andL2is the set of positive definite matrices commuting with2. Then there exists a gain-scheduled state-feedback of the form (6) which sta-bilizes (3)–(4) and ensures anH1 performance index lower than , according to Definition 2.1.

Proof: A sketch of a proof is presented in Appendix A.

Since the conditions of theorem 2.1 are stabilizability conditions which do not depend explicitly on the controller matrix, it is then com-puted separately. The computing technique is described in [11] and re-called in Appendix B for simplicity.

III. NEWMODELTRANSFORMATION FORDELAY-DEPENDENT STABILITYANALYSIS

This section introduces a new model transformation which turns a time-delay system with time-varying delays into an uncertain LPV system represented in an “LFT” form. This transformation allows to use classical robust stability analysis and control synthesis on the trans-formed system, in order to derive a delay-dependent stability test ob-tained from the scaled-bounded real lemma. Similar approaches can be found for instance in [23] where the maximal value of the delay ap-pears explicitly in the comparison model. In this paper, the comparison model is an uncertain parameter varying system which is then studied in the robust/LPV framework. This is a real novelty in the analysis and control of time-delay systems with time-varying delays.

A. Model Transformation

Let us consider the LTI time delay system

_x(t) = Ax(t) + Ahx(t 0 h(t)) + Bw3(t) + Buu(t)

z3(t) = Cx(t) + Chx(t 0 h(t)) + Dw3(t) + D3uu(t)

x() = ();  2 [0hM; 0] (8)

wherex; w3; z3; u and  are respectively the state, the exogenous

in-puts, the controlled outin-puts, the control input and the functional initial condition. The delayh(t) is assumed to belong to the set

H := fh(t) 2 C1₍ +_{; [h}

m; hM]); 0  hm< hM < +1g (9)

whereC1(J₁; J₂) is the set of differentiable functions mapping J₁to J2. Let us consider next the operator

Dh: w(t) ! t

t0h(t)(h() + hM0 hm)

01_{w()d (10)}

whereh 2 H.

Lemma 3.1: The operatorDhenjoys the following properties: 1) D_h( 1 ) is linear;

2) Dh( 1 ) has an L2-induced norm less than 1.

Proof: The proof is similar to the one given in [7].

By the mean of this model transformation and the use of the scaled bounded real lemma, it is possible to provide the first main result of the paper.

Proposition 3.1: Assumeu  0, the LPV delay free system (11) is

a comparison system for the LTI time-delay system (8) _x(t) = (A + Ah)x(t) 0 Ahw1(t) + Bw3(t) z1(t) = (t)(A + Ah)x(t) 0 (t)Ahw1(t) + (t)Bw3(t) z3(t) = (C + Ch)x(t) 0 Chw1(t) + Dw3(t) w1(t) = Dh(z1(t)) x() = ();  2 [02hM; 0] (t) = h(t) + hM0 hm (11)

where () is the new functional initial condition which coincides with () over [0hM; 0].

Proof: First note that the dynamical equation of system (8) can

be rewritten as

wherew₁ = x 0 x_h. Then defining z1(t) = (t)[(A + Ah)x(t) 0 Ahw1(t) + Bw3(t)] = (t) _x(t) with(t) = h(t) + h_M0 h_mwe have t t0h(t) 01_()z 1()d = t t0h(t) _x()d = x(t) 0 x(t 0 h(t)) = w1(t): (13)

Thus we obtain the first and second line of system (11). The third line is obtained by the same way.

Remark 3.1: Since (11) requires an initial functional condition on

[02hM; 0] then, under some particular conditions, unstable additional

dynamics may be created, making the comparison system unstable even if the original one is stable. This will result in conservatism and indi-cates that the stability of system (11) is only a sufficient condition for the stability of (8); this fact is usual when model transformations are used. However as emphasized in [7] the study of additional dynamics is not easy in the time-varying delay case and remains an open problem. The interest of the new model transformation is to turn the LTI time-delay system (8) into an uncertain linear parameter varying system (11) where the operatorD_h( 1 ) plays the role of a norm-bounded uncer-tainty, and the delayh(t) the role of a time-varying parameter through the term(t). Then LPV/robust control tools can be used in order to study such system.

B. Delay-Dependent Stability Test

The comparison model (11) is used in this section to develop a delay-dependent stability test with guaranteedL₂performances. This test is based on the application of the scaled-bounded real lemma [11] and is useful to obtain stabilization results.

Theorem 3.1 (Delay Dependent Scaled-Bounded Real Lemma): The

system (11) withh 2 H is asymptotically stable with an L2 perfor-mance index on channelw₃! z₃lower than if there exist symmetric positive definite matricesX1; X2and a positive scalar such that the LMIs ^ AT_X 1+ X1A 0X^ 1Ah X1B iA^TX2 CT ? 0X2 0 0iAThX2 0ChT ? ? 0 Iw iBTX2 DT ? ? ? 0X2 0 ? ? ? ? 0 Iz < 0 (14) hold fori = 1; 2 with ₁ = h_M; ₂= 2h_M0 h_m; ^A = A + A_hand



C = C + Ch.

Proof: The proof is given in Appendix C.

The latter theorem concludes on the stability of system (11) for a delay taking values in the separated interval0 [ [h_m; h_M]. Indeed, if the LMIs (14) are satisfied then the left-upper term is negative definite and finally the system with zero delay is asymptotically stable.

IV. DELAY-SCHEDULEDCONTROLLERSYNTHESIS

In this section, a new way to control linear time-delay systems of the form (8) is provided under the assumption that an approximate delay value is known. This approach differs radically from the common state-feedback with delayed state which consists in adding to the instan-taneous state-feedback, a supplementary term involving the delayed state (e.g.,u(t) = Kx(t) + Khx(t 0 h(t))). The implementation

of such a control law is difficult since it is generally difficult to know the exact value of the delay in real time, and the robustness analysis

with respect to delay uncertainties is also not an easy task. Moreover, since past states need to be stored in memory, then the controller with memory is more expensive from a memory space point of view.

This motivates the introduction of the delay-scheduled controller which is midway between the instantaneous state-feedback and the state-feedback with delayed state. Indeed, while the delay-scheduled state-feedback depends only on the instantaneous state, it also depends on the delayh(t) in a scheduling fashion

u(t) = K(h(t))x(t): (15)

This method does not need any dynamical model of the delay nor its exact value but only an approximative measurement. Indeed, since in the comparison model (11) the delay is viewed as a parameter, then the robust analysis with respect to delay uncertainties on the implemented delay is brought back to a perturbation on the parameters, which is simple to deal with. Hence, using this approach it is possible to ensure the robust stability of the closed-loop system in presence of delay un-certainties.

To design these controllers, the role of the model transformation de-fined in Section III is crucial since it turns a linear time-delay system into an uncertain LPV system (11). Due to the structure of this system (finite-dimensional), it is possible to use available tools in analysis and control of LPV systems (and hence theorem 2.1).

A. Controller Existence

The time-delay is assumed to belong to the setH defined in (9). In order to apply the scaled-small gain theorem, the first step is to nor-malize the set of values of the parameters such that it coincides with the closed unit ball. Note first that the system delayh(t) can be ex-pressed ash(t) = hc(t) + he(t) where hc(t) is the measured delay

that will be used in the controller andh_e(t) the bounded measurement error satisfyingjhe(t)j < e. Then denoting respectively h and hthe mean value and the radius of the delay interval[h_m; h_M] defined as h = (hm+hM)=2 and h= hM0h, the normalized delay parameter

hn(t), which takes value in [01; 1], is defined by hc(t) = hhn(t)+h.

The following proposition provides a comparison model for (8) where the above defined notations are used instead of “absolute” notations used in (11). Indeed, a new comparison model is necessary to derive controller existence conditions by taking into account the delay uncertainty.

Proposition 4.1: The system (16) is a comparison system for (8) (a

control inputu has been added) _x z =  A Bw Bu  Cz Dw Du x w u w0(t) = hn(t)z0(t) w1(t) = Dh(z1(t)) w2(t) = he_(t) e z2(t) (16)

wherez = col(z0; z1; z2; z3); w = col(w0; w1; w2; w3)

 A = A + Ah Bw= [ 0Ah 0 B3]  CT z = [A + Ah (A + Ah) A + Ah C3+ C3h]T  DT u = [Bu Bu Bu D3u]T Bu = Bu Dw= 0 0Ah 0 B3 hI 0Ah eI B3 0 0Ah 0 B3 0 0C3h 0 D33

with = h_M0 h_m+ h and where w₃(t) and z₃(t) are, respectively, the exogenous inputs and controlled outputs.

Proof: The proof consists in computing the generalized system

by substituting input/output signals w0=z0; w1=z1; w2=z2 by their explicit expression, in order to show that the computed gener-alized system finally coincides with system (8). First, note that h(t) = hc(t) + he(t); hc(t) = hhn(t) + h and let (t) := _x(t) = z0(t) = z2(t). We have w0(t) = hn(t)z0(t) = hn(t)(t) and w2(t) = (he(t))=(e)z2(t) = (he(t))=(e)(t). Then according to (16) z1(t) = (t) + hw0(t) + ew2(t) = ( + hhn(t) + he(t))(t) = (hM0 hm+ h(t))(t) Finally as(t) = _x(t) we get w₁(t) = D_h(z₁(t)) = x(t)0x(t0h(t)) which implies, from the state equation of Proposition 4.1

_x(t) = Ax(t) + Ahx(t 0 h(t))

+ B3w3(t) + Buu(t)

z3(t) = C3x(t) + C3hx(t 0 h(t))

+ D33w3(t) + D3uu(t) (17) Since (17) has the same form as (8), this shows that if system (16) is asymptotically stable then (17) and therefore (8) are hence asymptoti-cally stable.

Using Proposition 4.1 and Theorem 2.1 the second main result of the paper, which provides the solution to the delay-scheduled state feed-back design problem, can be given:

Lemma 4.1: Consider system (8). If there exist symmetric positive

definite matricesZ; L3; J3 > 0 and a positive scalar such that the

LMIs KT i Ni+ TiTMiTi Ki< 0 for i = 1; 2 L_I3 _JI 3  0 (18) hold with N1= 0diag(L3; I1; I2; Iw ) M1= diag(L3; I1; I2; 01Iz ) UJ= diag(J3; I1; I2; Iz ) M2= diag(J3; I1; I2; 01Iw ) N2= Z A T _{+ AZ Z C}T ? 0UJ ; T1= D; K1= I T2= [ BT DT]; K2= [ BTu DTu]

then there exists a stabilizing delay scheduled state-feedback according to definition 2.1 with anH₁performance index on channelw₃ ! z₃ lower than .

Proof: The proof is a straightforward application of theorem 2.1

with system (16).

B. Controller Computation

The controller can be constructed using the methodology of Appendix B whereX = Z; A = A+Ah; B = [0 0Ah 0 B3]; Bu=

Bu; DT3u= [Bu Bu Bu D3u]T  C = A + Ah (A + Ah) A + Ah C3+ C3h  D = 0 0Ah 0 B3 hI 0Ah eI B3 0 0Ah 0 B3 0 0C3h 0 D33 (19) where = hM0 hm+ h; h = (hm+ hM)=2 and h = hM 0 h.

Using the following lemma, the controller can be finally constructed:

Lemma 4.2 (Controller Construction): The controller construction

is obtained by applying Algorithm 1.1 whereY corresponds to the matrix (24) in which the matrices defined above are substituted and Z = X = P01_. C. Example Consider system _x(t) = 01 0_{1 1} x(t) + 1 01_{0 01} x(t 0 h(t)) + 1₀ w(t) + 1₀ u(t) z(t) = 0 1_{0 0} x(t) + 0₁ u(t) (20) with(h_m; h_M) = (0:1; 0:3). When the delay value is exactly known, we find anH1 closed-loop performance lower than 3 = 4:9062. Since a finite has been found, this means that the closed loop system can be stabilized by a delay-scheduled state-feedback controller. A suitable controller is given by the expression

u(t) zc(t) = K x(t) wc(t) wc(t) = hn(t)zc(t) K = K_K11 K12 21 K22 K11= [030:1953 0 14:9936] K12= [00:3196 1:1417] K21= 05:2534 02:7139_{0:3303 0:1755} K22= 00:0330_{0:0617 00:0289}0:1899 : V. CONCLUSION

A new model transformation allowing to turn an LTI time-delay system with time-varying delays into an uncertain LPV system in “LFT” form has been developed. From this reformulation, a new delay-dependent stability test, based on the application of the scaled-bounded real lemma, has been derived. The interest of such formulation resides in the similarities with the bounded-real lemma for finite dimensional systems which can be used with many robust control tools.

The stability test is extended to address a new control synthesis problem for time-delay systems. In this original approach, the con-troller gains are scheduled by the current value of the delay and this structure has motivated the name of “delay-scheduled controller” in reference to gain-scheduled controllers arisings in the control of LPV systems. A certain interest of the approach is the simple robustness analysis with respect to uncertainty on the delay knowledge which is actually a difficult problem when the delay is considered as an operator. In the provided approach, the uncertainty is uniquely char-acterized by a parameter variation and can be very easily handled due to the similarities with robustness analysis with respect to uncertain system matrices. Even if the results are presented for LTI systems with single time-delay, it can be easily generalized to the LPV case with multiple delays.

Further works will be devoted to the reduction of conservatism of the approach by finding better model transformations and tighter bound on the norm of the operators.

APPENDIX

A. Proof of Lemma 2.1

Consider the LPV system (3). The augmented system (see [11]) is given by (21)–(23), shown at the bottom of the page.

The signalsw_candz_crepresent the controller parameter-scheduled channel. LetK be the state-feedback matrix, as defined in definition 2.1. Then computing the closed-loop system and substituting its ex-pression into the scaled-bounded real lemma

AT clP + P Acl P Bcl CclT ? 0Lw DclT ? ? 0L01 z < 0 (24) withLw = diag(L; I1; Iw ) and Lz = diag(L; I1; 01Iz ) where

L scales the controller and system parameter-scheduled channels. Iso-lating the expression containing the state-feedback matrix from (24), this inequality may be rewritten into

9 P; Lw; L01z + PTUTKV + VTKTUP < 0 (25) where P = diag(P; I; I); U = [I j 0 j 0] V = BuT 0 DTu0 DTu1 Du2T 0 0 0 0 I 0 0 0 0 0 0 T

and the matrix9(P; L_w; L01_z ) is given by the substitution of the ma-trices of system (23) into (24). Note that this inequality is nonlinear due to the product between the matrix variablesP; K and the presence of both matricesL and L01. However, as shown in [11], it turns out that it is possible to find an equivalent LMI formulation by the mean of the projection lemma and then the following equivalent underlying problem is obtained

KT

U8KU < 0 KVT9KV < 0 (26)

where KU = Ker(U); KV = Ker(V );

8 = diag(P01_{; I; I)9diag(P}01_{; I; I) and 9 is given}

by the substitution of the matrices of system (23) into (24). Permuting lines and columns of matrix inequalities (26) yields

KU0T80KU0 < 0 KV0T90KV0 < 0

whereK_U0 = diag(NU; I) and KV0 = diag(NV; I). Due to the

iden-tity block in the matricesK_U0 andK_V0 , it is possible to simplify the expression using Schur’s complement. To this aim, let us decompose the previous matrices as90 = [9ij]i;j=1;2 80 = [8ij]i;j=1;2 and therefore we get NT U 0 0 I 811 812 ? 822 NU 0 0 I < 0 NT V 0 0 I 911 912 ? 922 NV 0 0 I < 0 (27) which are in turn equivalent to

NUT 80110 80128012280T12 NU < 0

NT

U 9110 0 90129012290T12 NV < 0 (28) which are exactly the LMIs of Theorem 2.1 and the proof of the con-troller existence is completed. The form ofK is given in Appendix B.

B. State-Feedback Computation

The state-feedback matrixK may be computed using the following algorithm:

Algorithm 1.1: Controller computation

1) Compute whole matricesL = [L1 L2 LT

2 L3] and J = L 01 ₌

[_JJ1T J2

2 J3] from the values of L3 andJ3 using a singular value

decomposition on the relationsLT₂J₂ = I 0 L₃J₃and thenL andJ are solutions of the equations:

0 I 4T 2 43 7 = 7T 2 73 0 I (29) where(4; 7) 2 f(L; J); (J; L)g.

2) The computation of the controller matrixK consists in solving the LMI (as in [11]).

Y + ~UT_{KV + V}T_KT_{U < 0~} Y =  AT_X01_{+ X}01_{A X}01_{[ 0 B ] [ 0 C}T_] ? 0diag(L; In ; Iw ) 0_{0 D}0T ? ? 0diag(L01_{; I} n ; Iz ) (21) 51= (A + Ah)TX1+ X1(A + Ah) 0X1Ah X1B (hM0 hm)(A + Ah) (C + Ch)T ? 0X2 0 0(hM 0 hm)ATh 0ChT ? ? 0 Iw (hM0 hm)BT DT ? ? ? 0X01 2 0 ? ? ? ? 0 Iz (22) _x zc z0 z1 z3 x wc = A 0 B0 B1 B2 Bu 0 0 0 0 0 0 0 I C0 0 D00 D01 D02 Du0 0 C1 0 D10 D11 D12 Du1 0 C2 0 D20 D21 D22 Du2 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 x wc w0 w1 w3 u zc (23)

whereY is given in (21) and ~ U = 0 X01_B u 0 0 0 I  Du 0 T V = I 0 0 0 0 I 0 0 C. Proof of Theorem 3.1

First substitute matrices governing system (11) into the scaled-bounded real lemma [11] and performing a congruence transformation with respect to diag(I; I; I; X2; I) leads to the semi-infinite LMI

51+ h(t)52 < 0 with 51defined in (22) and

52= 0 0 0 (A + Ah) 0 ? 0 0 0AT h 0 ? ? 0 BT ₀ ? ? ? 0 0 ? ? ? ? 0 (30)

Since this LMI is affine inh(t) then it is necessary and sufficient to check the feasibility of the LMI at the vertices only (i.e., ath(t) 2 fhmin; hmaxg) and we get LMIs (14).

REFERENCES

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Optimal Preview Control of Markovian Jump Linear Systems

Kenneth D. Running and Nuno C. Martins, Member, IEEE

Abstract—In this technical note, we investigate the design of controllers, for discrete-time Markovian jump linear systems, that achieve optimal ref-erence tracking in the presence of preview (refref-erence look-ahead). For a quadratic cost and given a reference sequence, we obtain the optimal so-lution for the full information case. The optimal control policy consists of the additive contribution of two terms: a feedforward term and a feedback term. We show that the feedback term is identical to the standard optimal linear quadratic regulator for Markovian jump linear systems. We provide explicit formulas for computing the feedforward term, including an anal-ysis of convergence.

Index Terms—Markovian jump linear systems.

I. INTRODUCTION

This technical note deals with the problem of designing control sys-tems that achieve optimal reference tracking in discrete-time. More specifically, we consider the servomechanism problem, i.e., given a ref-erence, the objective is to design feedback and feedforward strategies so that the state of the plant tracks the reference optimally, according to a quadratic cost. Here, we consider a plant that is linear but varies in time according to a Markovian process that takes values in a finite alphabet, such systems are called Markovian jump linear systems.

Basic notation: Throughout the technical note, we adopt the

fol-lowing notation: 1) Boldface letters, such asx, indicate (possibly vector valued) real random variables, while the default font is used to represent particular realizations of a random variable or deterministic quantities. 2) IfG is a matrix then [G]i;j is the entry located at theith row and jth column. 3) if G is a matrix then G0_{indicates its transpose. Further}

notation will be introduced as needed.

Manuscript received April 19, 2008; revised November 17, 2008. First pub-lished August 18, 2009; current version pubpub-lished September 04, 2009. This work was supported by the Air Force Office of Scientific Research (AFOSR) Grant FA95500810120 and National Science Foundation (NSF) CMMI Grants 0727659 and 0901210. Recommended by Associate Editor I. Paschalidis.

K. D. Running is with the Aerospace Engineering (AE) Department, Univer-sity of Maryland, College Park, MD 20742 USA.

N. C. Martins is with the Electrical and Computer Engineering (ECE) De-partment and the Institute for System Research (ISR), University of Maryland, College Park, MD 20742 USA (e-mail: nmartins@isr.umd.edu).

Digital Object Identifier 10.1109/TAC.2009.2026860 0018-9286/$26.00 © 2009 IEEE