Dynamic output feedback control for Markovian jump ... - IEEE Xplore

www.ietdl.org Published in IET Control Theory and Applications Received on 8th February 2011 Revised on 2nd December 2011 doi: 10.1049/iet-cta.2011.0089

ISSN 1751-8644

Dynamic output feedback control for Markovian jump systems with time-varying delays J.Yu1,2 J.Tan3 H. Jiang4 H. Liu2 1 School

of Information Science andTechnology,YanchengTeachers University,Yancheng 224002, People’s Republic of China 2 Intelligent Systems and Biomedical Robotics Group, University of Portsmouth, Portsmouth PO1 2DJ, UK 3 School of Computing, University of Portsmouth, Portsmouth PO1 3HE, UK 4 School of Mathematics,YanchengTeachers University,Yancheng 224002, People’s Republic of China E-mail: [email protected]

Abstract: This study investigates the problems of delay-dependent H∞ output feedback control for Markovian jump systems with time-varying delays. Firstly, the improved delay-dependent stochastic stability is obtained by constructing a new Lyapunov–Krasovskii functional. Neither free-weighting matrix approach nor model transformation is involved in the derivative of the criterion. Then, a new bounded real lemma (BRL) for Markovian jump systems is established. Based on the proposed BRL, a full-order dynamic output feedback controller is designed to ensure the stochastic stability and a prescribed H∞ performance level for the closed-loop systems. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed approaches.

1

Introduction

Time delays are often the main sources of instability and poor performance of a control system. Therefore the stability issue of delayed systems has recently drawn particular research interests in control-related communities. Generally speaking, state-of-the-art in the stability can be classified into two types: delay-independent stability and delaydependent stability. Since the delay-independent criterion allows the time delay to be arbitrarily large, it tends to be conservative in general [1]. Many analysis and synthesis results using the delay-dependent concept have been widely reported in concern of conservatism, see for example [2–7]. A free-weighting matrices method based on the linear matrix inequality (LMI) technique is proposed in [8] to study the delay-dependent stability problems time-delay systems. The method has been shown to be effective in reducing conservatism, thanks to the introduction of free-weighting matrices. On the other hand, Markovian jump systems, modelled by a set of subsystems with transition among all the modes governed by a Markov chain, are often employed to describe many practical systems whose structure and parameters are subjected to abrupt variations such as states transitions in manoeuvred target, tracking repairs of machines in manufacturing systems [9]. Since the states take continuous values and the jumping parameters take discrete values in a system simultaneously, Markovian jump systems can be regarded as a special case of hybrid systems. In the past two decades, substantial research has been devoted, in Markovian jump systems, to problems of stability, [10– 14], stabilisation, [15, 16], control, [17–19], state estimation, IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 803–812 doi: 10.1049/iet-cta.2011.0089

[20] and filtering [21, 22]. The stability analysis and stabilisation problems are investigated for a class of discretetime Markov jump linear systems with partially known transition probabilities and time-varying delays [23], the transition probabilities of the mode jumps are considered to be partially known, which relax the traditional assumption in Markovian jump systems that all of them must be completely known a priori. By exploiting a new Lyapunov–Krasovskii functional, the robust stabilisation problem for a class of Markovian jump systems with non-linear disturbances and time delays was studied in [24], the time-varying are in intervals and depend on system mode. A dynamic output feedback controller that ensures the exponential mean-square stability and a prescribe H∞ performance level for the closed-loop system is designed in [25]. In [26], the globally exponential stabilisation problem was investigated for a general class of stochastic systems with both Markovian jumping parameters and mixed time delays. In all literatures the free-weighting matrix method are used to reduce the conservatism. However, the free-weighting matrix method often needs to introduce many slack variables in obtained LMI criteria, and thus leads to a significant increase in the computational demand [27]. Moreover, as pointed out in [28, 29], in some cases free-weighting matrices may not be useful to the reduction of conservatism. Therefore to establish some new stability criteria with less conservatism and less slack variables is a very meaningful work. This motivates the present research. This paper is concerned with the output-feedback H∞ controller design problem for a class of Markovian Jump Systems with time-varying delays. By constructing a 803 © The Institution of Engineering and Technology 2012

www.ietdl.org new Lyapunov–Krasovskii functional, the delay-dependent stochastic stability criterion for the Markovian jump systems with time-varying delays is proposed in terms of LMIs. Neither the free-weighting matrix approach nor model transformation is involved in the derivative of Lyapunov functional. Based on this, a new bounded real lemma (BRL) for Markovian jump stochastic systems is established. Two numerical examples are given to illustrate the effectiveness and advantage of the proposed methods. The remainder of this paper is organized as follows. The problem to be addressed is formulated in Section 2, the controller design is presented in Section 3. The simulation examples are provided in Section 4 to show the effectiveness of the developed results and we conclude the paper in Section 5. Notation: The following notations and concept are used throughout this paper. A real symmetric matrix P > 0 (≥ 0) denotes P being a positive-definite (positive semidefinite) matrix, and A > (≥)B means A − B > (≥)0. I is used to denote an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by ∗. The superscript ‘T’ represents the transpose. 04×7 means a matrix with 4 rows and 7 columns, all the elements are 0. λi (·), λmin (·), λmax (·) denote the ith eigenvalue, the minimum eigenvalue, the maximum eigenvalue of corresponding matrix, respectively. · denotes the generalised Euclidean norm of a vector and its induced norm of a matrix. L2 [0, ∞) is the space of square-integrable vector functions over [0, ∞). (, F, {Ft }t≥0 , P) is a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all P-null sets). E is the mathematical expectation with respect to probability measure P. C([−τ , 0]; Rn ) denote the family of continuous functions from [−τ , 0] to Rn and CFb 0 ([−τ , 0]; Rn ) the family of all bounded, F0 -measurable and C([−τ , 0]; Rn )valued random variables. If ξ(t) is a continuous Rn valued stochastic process on t ∈ [−τ , ∞), we let ξt = {ξ(t + θ ) : −τ ≤ θ ≤ 0} for t ≥ 0, which is regarded as a C([−τ , 0]; Rn )-valued stochastic process.

2

Problem formulation and preliminaries

Consider the following class of Markovian jump systems with time-varying delays x˙ (t) = A(t, rt )x(t) + Ad (t, rt )x(t − τ (t)) + B1 (t, rt )u(t) y(t) = C(t, rt )x(t) + Cd (t, rt )x(t − τ (t)) + B2 (t, rt )u(t) (1)

z(t) = E(rt )x(t) + Ed (rt )x(t − τ (t)) + B3 (rt )u(t) + D3 (rt )ω(t) x(t) = ϕ(t),

A(t, rt ) = A(rt ) + A(t, rt ) Ad (t, rt ) = Ad (rt ) + Ad (t, rt ) B1 (t, rt ) = B1 (rt ) + B1 (t, rt ) C(t, rt ) = C(rt ) + C(t, rt ) Cd (t, rt ) = Cd (rt ) + Cd (t, rt ) B2 (t, rt ) = B2 (rt ) + B2 (t, rt ) A(rt ), Ad (rt ), B1 (rt ), D1 (rt ), C(rt ), Cd (rt ), B2 (rt ), D2 (rt ), E(rt ), Ed (rt ), B3 (rt ) and D3 (rt ) are known real constant matrices representing the nominal system for each rt ∈ S. In order to prevent the notations from becoming too complicated, for each possible rt = i, i ∈ S, a matrix A(rt ) will be denoted by Ai . For example, A(t, rt ) is denoted by Ai (t), A(t, rt ) is denoted by Ai (t), and so on. Ai (t), Adi (t), B1i (t), Ci (t), Cdi (t) and B2i (t) are unknown matrices representing time-varying parameter uncertainties, and assumed to be admissible. That is to say, there are of the form M1i Ai (t) Adi (t) B1i (t) F (t)[N1i N2i N3i ] = Ci (t) Cdi (t) B2i (t) M2i i (3) for any i ∈ S, and Fi (t) are the uncertain time-varying matrices satisfying FiT (t)Fi (t) ≤ I ,

where x(t) ∈ R is the state; ω(t) ∈ R is the noise signal which is assumed to be an arbitrary signal in L2 [0, ∞); y(t) ∈ Rq is the measurement; z(t) ∈ Rm is the signal to be estimated; {rt }t≥0 is a continuous-time Markov process with right continuous trajectories and taking values in a finite set S = {1, 2, . . . , N }. Let = {πij : i, j ∈ S} be the density matrix of theMarkov chain {rt }t≥0 . Thus, πij ≥ 0 for i = j and πii = − Nj=1,j =i πij . Furthermore, the transition p

804 © The Institution of Engineering and Technology 2012

∀i ∈ S

In system (1), τ (t) denotes the time-varying delay when the mode is in rt and satisfies 0 < τ (t) ≤ τ < ∞,

τ˙ (t) ≤ μ ∀rt ∈ S

(4)

Definition 1: The Markovian jump system x˙ (t) = A(rt )x(t) + Ad (rt )x(t − τ (t)) x(t) = ϕ(t),

∀t ∈ [−τ , 0] n

where > 0, lim →0 (o( )/ ) = 0. A(t, rt ), Ad (t, rt ), C(t, rt ), Cd (t, rt ), B1 (t, rt ), and B2 (t, rt ) are matrix functions of rt , and for each rt ∈ S

Throughout the paper the following concept of stochastic stability is used.

+ D1 (rt )ω(t) + D2 (rt )ω(t)

probability from mode i at time t to mode j at time t +

can be described as i = j πij + o( ), (2) P{rt+ = j|rt = i} = 1 + πii + o( ), i = j

∀t ∈ [−τ , 0]

(5)

is said to stochastically stable, if for finite ϕ(t) ∈ Rn defined on [−τ , 0], and r0 ∈ S, the following inequality (6) is satisfied t lim E xT (s, ϕ, r0 )x(s, ϕ, r0 ) ds < ∞ (6) t→∞

0

where x(t, ϕ, r0 ) denotes the solution to system (5) at time t under the initial conditions ϕ(t) and r0 . IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 803–812 doi: 10.1049/iet-cta.2011.0089

www.ietdl.org 3

Main results 1R

The following lemmas are needed in the proofs of the main results in the paper. Lemma 1 Jensen inequality [10]: For any constant matrix M ∈ Rn×n , M = M T > 0, scalars r1 and r2 satisfying r1 < r2 , and a vector function ω : [r1 , r2 ] → Rn such that the integrations concerned are well defined, then T

r2 ω(s)ds r1

≤ (r2 − r1 )

r2 M

2R

3R

ω(s)ds r1

r2

ωT (s)M ω(s)ds r1

4R

To drive a less conservative stability criterion, we will use the following lemma.

⎡ −R ⎢ ∗ =⎣ ∗ ∗ ⎡ −3R ⎢ ∗ =⎣ ∗ ∗ ⎡ −R ⎢ ∗ =⎣ ∗ ∗ ⎡ −R ⎢ ∗ =⎣ ∗ ∗

⎤ 0 0 ⎥ R⎦ −R ⎤ 0 0 R 0 ⎥ −2R R ⎦ ∗ −R ⎤ 0 0 0 3R ⎥ −R R ⎦ ∗ −4R ⎤ 0 0 0 R ⎥ −3R 3R ⎦ ∗ −4R

R 0 −4R 3R ∗ −4R ∗ ∗ 3R −4R ∗ ∗ R −4R ∗ ∗ R −2R ∗ ∗

with Lemma 2: For any scalars W1 ≥ 0, W2 ≥ 0, τ (t) is a continuous function and satisfied τ1 < τ (t) < τ2 , then

3W1 + W2 W1 + 3W2 W2 W1 ≥ min + , τ (t) − τ1 τ2 − τ (t) τ2 − τ 1 τ2 − τ 1

+

N

πij Pj + Q0 + Qi11 +

j=1

τ Q11 2

2i = −(1 − μ)Q0

Proof: See the appendix.

1i = Pi Ai +

ATi Pi

Lemma 3 [30]: Let X ∈ Rn , Y ∈ Rn , then there exists a positive scalar 1 such that

τ 3i = Qi22 − Qi11 + Q22 2 √ √ 2τ 2τ χ0i = RAdi RAi 2 2

T 0

0

Proof: See the appendix.

X T Y + XY T ≤ 1 X T X + 1−1 Y T Y Lemma 4 [10]: Let J = J T , M and N be real matrices of appropriate dimensions with F satisfying F T F ≤ I , then J + MFN + N T F T M T < 0 if and only if there exists a positive scalar 2 such that J + 2−1 MM T + 2 N T N < 0 Firstly, we present a new delay-dependent stability condition for the Markovian jump system (5) in the following theorem.

Remark 1: The idea of delay partitioning is used to construct the Lyapunov–Krasovskii functional candidate (15). Generally speaking, the conservatism of the results will be further reduced as the partitioning is getting thinner. In order to avoid the complexity of computation, we only partition the delay into two parts in this paper. Moreover, there are no slack matrices in the criterion. Secondly, we investigate the bound real lemma for the following Markovian jump systems with time-varying delays x˙ (t) = A(rt )x(t) + Ad (rt )x(t − τ (t)) + D1 (rt )ω(t)

Theorem 1: Given a scalar τ > 0. Then, for any delay τ (t) satisfying (4), the Markovian jump system in (5) is stochastically stable if there exist matrices Pi > 0, Q0 > Q11 Q12 Qi12 Q > 0, and > 0, such that 0, R > 0, i11 ∗ Q22 ∗ Qi22 the following LMIs holds for k = 1, 2, 3, 4, and i = 1, 2, . . . , N i + kR χ0i 0 and γ > 0. Then, for any delay τ (t) satisfying (4), the Markovian jump system in (9) is stochastically stable and satisfies z(t)E2 < γ ω2

(7) (8)

if for any non-zero ω(t) ∈ L2 [0, ∞] under the condition x(t) = 0 for all t ≤ 0, if there exist matrices Pi > 0, Q0 > Qi11 Qi12 Q11 Q12 0, R > 0, > 0, and > 0, such that ∗ Qi22 ∗ Q22 (8) and the following LMIs holds for k = 1, 2, 3, 4, and i = 1, 2, . . . , N ⎡

i + kR ⎢ ∗ ⎣ ∗ ∗

χ1i −γ 2 I ∗ ∗

χ2i T Dc3i −I ∗

⎤ χ0i τ D1iT R⎥ 0, γ > 0 and θR > 0, there exist a Markovian jump linear output feedback controller in the form (11) such that, for any delay τ (t) satisfying (4) and all admissible uncertainties, the closed-loop system is stochastically stable and z(t)E2 < γ ω2 for any nonzero ω(t) ∈ L2 [0, ∞] under the condition x(t) = 0 for all t ≤ 0, if there exist matrices Xi > 0, Si > 0, R > 0, Q0 > Q11 Q12 Qi11 Qi12 > 0, > 0, and scalars i1 > 0, ∗ Qi22 ∗ Q22 0, i2 > 0, such that (8) and the following LMIs holds for k = 1, 2, 3, 4 and i = 1, 2, . . . , N ⎤ ⎡ i χ4i χ5i χ6i χ7i 0 0 0 ⎥ k ⎢ ∗ − i2 I R ⎥ ⎢ −1 ∗ − i2 I 0 0 ⎥+ ⎢∗ ∗ ⎣∗ ∗ ∗ − i1 I 0 ⎦ −1 ∗ ∗ ∗ ∗ − i1 I

j=1

i,12 i,16 i,17 i,22 i,33

j=1,j =i

τ + Q0 + Qi11 + Q11 2 Adi Adi Si D1i , i,15 = = 0 Xi Adi + i Cdi Xi D1i + i D2i T S E + BT = i i T i 3i Ei √ T 2 Ai Si + B1i i Ai = τ i X i Ai + i C i 2 √ 2 T cSi EdiT , τ i,12 = −(1 − μ)Q0 , i,26 = = i,27 EdiT 2 τ Si I 2 = Qi22 − Qi11 + Q22 , i,77 = θR R − 2θR I Xi 2

i = Xi Ai Si + (Si−1 − Xi )BKi Ci Si + Xi B1i CKi Si

08n×8n 0, so ( t−(τ/2) yT (s)ds)T R t−(τ/2) yT (s)ds > 0, t t ( t−τ (t) yT (s)ds)T R t−τ (t) yT (s)ds > 0. According to Lemma 2 1 (τ/2) − τ (t) 1 + τ (t)

t−τ (t) yT (s)ds

T t−τ (t) R yT (s)ds

t−(τ/2)

t

T t y (s)ds R

t−(τ/2)

τ y (s)R2 y(s)ds = − 2

t−τ (t)

T

t−τ

−

τ 2

yT (s)Ry(s)ds t−τ

t−(τ/2) yT (s)Ry(s)ds t−τ (t)

≤ max{3W3 + W4 , W3 + 3W4 } (29)

yT (s)ds

t−τ (t)

t−τ (t) t−τ (t) 2 ≥ min 3 yT (s)ds)T R yT (s)ds τ t−(τ/2) t−(τ/2) T t t t−τ (t) T T y (s)ds R y (s)ds yT (s)ds)T R + t−τ (t)

t−τ (t)

t

t−τ (t) ×

τ − 2

t−(τ/2)

T

t−τ (t)

and

y (s)ds + 3 T

t−(τ/2)

T −R R x(t − τ (t)) x(t − τ (t)) W3 = ∗ −R x(t − τ ) x(t − τ ) T τ τ x t− x t− −R R 2 2 W4 = ∗ −R x(t − τ (t)) x(t − τ (t))

T t T y (s)ds R

t−τ (t)

t−(τ/2)

with

T

y (s)ds

t−τ (t)

Then So −

τ 2

t yT (s)Ry(s)ds ≤ max{3W1 + W2 , W1 + 3W2 } t−(τ/2)

where T x(t − τ (t)) x(t − τ (t)) −R R τ τ W1 = ∗ −R x t − x t− 2 2 T −R R x(t) x(t) W2 = ∗ −R x(t − τ (t)) x(t − τ (t)) Then τ T x t− τ2 T −R R 2 LV3 (xt , rt ) = x˙ (t)R˙x(t) + ∗ −R 2 x(t − τ ) τ x t− 2 × + max{3W1 + W2 , W1 + 3W2 } x(t − τ ) (28) Similarly, when τ (t) ∈ [(τ/2), τ ], we have τ − 2

t x˙ T (s)R˙x(s)ds t−(τ/2)

≤

x(t)

τ x t− 2

T

−R R ∗ −R

⎡ i,11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ i = ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

x(t)

τ x t− 2 i,12 i,22 ∗ ∗ ∗ ∗ ∗

Qi,12 + (τ/2)Q12 0 i,33 ∗ ∗ ∗ ∗


T x(t) τ2 T −R R τ LV3 (xt , rt ) = x˙ (t)R˙x(t) + ∗ −R x t− 2 2 x(t) τ × + max{3W3 + W4 , W3 + 3W4 } x t− 2 (30) T x(t) x(t) τ Q11 Q12 τ τ LV4 (xt , rt ) = ∗ Q22 x t − 2 x t− 2 2 T t x(s) Q11 Q12 τ − ∗ Q22 t−τ/2 x s − 2 x(s) τ × ds (31) x s− 2 According to (8) and (16)–(31), we have ⎛ ⎡ T⎤ ⎡ ⎤⎞ Ai Ai ⎜ ⎢AT ⎥ τ 2 ⎢A ⎥⎟ di ⎥⎟ ⎜ ⎢ ⎥ ⎢ LV (xt , rt ) ≤ ζ T (t) ⎜i + kR + ⎢ di ⎥ R ⎢ ⎥⎟ ζ (t) ⎝ ⎣ 0 ⎦ 2 ⎣ 0 ⎦⎠ 0 0 with k = 1, 2, 3, 4, ζ T (t) = [xT (t) xT (t − τ (t)) xT (t − (τ/2)) xT (t − τ )]. From (7), using the Schur complement lemma, we can conclude that LV (xt , rt ) < 0. Using the same method in [18], we have that the Markovian jump system in (5) is stochastically stable. This completes the proof. Remark 3: In most literature, the terms −(τ/2/(τ/2 − τ (t))), −(τ/2/τ (t)) (when τ (t) ∈ [0, (τ/2)]) and −(τ/2/ 0 0 −Qi,12 −Qi,12 ∗ ∗ ∗

i,15 0 0 0 −γ 2 I ∗ ∗

i,16 i,26 0 0 T Dc3i −I ∗

⎤ i,17 i,27 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎦ i,77

IET Control Theory Appl., 2012, Vol. 6, Iss. 6, pp. 803–812 doi: 10.1049/iet-cta.2011.0089

www.ietdl.org (τ (t) − τ/2)), −(τ/2/(τ − τ (t))) (when τ (t) ∈ [(τ/2), τ ]) are all directly increased to 1 in equations (28) and (30), it can lead to some conservatism. So, the Lemma 2 is used to reduce this kind of conservatism. 8.3

S then i I

χ1i − i2 I ∗

⎤ k χ2i 0 ⎦ + R ∗ − i2−1 I

08n×6n 06n×6n

Ai Si + B1i i Ai −1 = Pi Aci i Xi Ai + i Ci 1i ⎛⎡ ⎤ πii Si πii I N N ⎜⎢ ⎥

−T 1i ⎝⎣ π I π X − πij Sj−1 ⎦ ii ij j

(32)

+

N

πij

j=1,j =i

⎤ k χ2i R ⎦ 0 + ∗ −1 − i2 I

=

08n×6n 0, Xi

−T 1i

Proof: According to Schur complement lemma, (13) can be rewritten as i ⎣∗ ∗

Si I

− 2θR i,77 = I > 0, so, Pi > 0. It is easy to obtain Xi

since

Proof ofTheorem 3

⎡

θR2 R

Then, performing congruence transformations to (26) −1 −1 −1 −1 by diag{ −1 1i , 1i , 1i , 1i , I , I , 1i }, respectively, the following inequality holds (see (27)) τ Q12 2

0 3i ∗ ∗ ∗ ∗

0

Pi Dc1i

EciT

0 −Qi12 −Qi22 ∗ ∗ ∗

0 0 0 −γ 2 I ∗ ∗

T Ecdi 0 0 T Dc3i −I ∗


τ ATci Pi

⎤

⎥ τ ATcdi Pi ⎥ ⎥ k ⎥ 0 ⎥ + R 0 ⎥ ∗ ⎥ 0 ⎥ ⎦ 0 2 θR R − 2θR Pi

04n×4n 0, from (θR R − Pi )R−1 (θR R − Pi ) > 0

After replacing the term θR2 R − 2θR Pi with −Pi R−1 Pi in (27), and performing congruence transformations by diag{I , I , I , I , I , I , R−1 Pi }, then (10) are satisfied. Therefore all the conditions in Theorem 2 are satisfied. The proof is completed.

then −Pi R−1 Pi < θR2 R − 2θR Pi

