Neural Computing and Applications https://doi.org/10.1007/s00521-018-3580-4

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S.I. : EMERGENCE IN HUMAN-LIKE INTELLIGENCE TOWARDS CYBER-PHYSICAL SYSTEMS

Robust guaranteed cost control for continuous-time uncertain Markov switching singular systems with mode-dependent time delays Huanli Gao1 • Xueliang Liu2 • Fuchun Liu1 Received: 1 March 2018 / Accepted: 30 May 2018 Ó The Natural Computing Applications Forum 2018

Abstract The guaranteed cost control problem for mode-dependent time-delay Markov switching singular systems with normbounded uncertain parameters is discussed. Based on delay-dependent linear matrix inequalities, sufficient conditions which ensure the nominal Markov switching singular system to be regular, non-impulsive and stochastically stable are derived by quoting a mode-dependent Lyapunov functional and applying Moon’s inequality for cross terms. The sufficient conditions involve two cases. Case 1: the time delays are known; Case 2: the time delays are unknown, but the difference of the largest and the smallest time delay is known. Then, the problem is solved to design a state-feedback control law such that the closed-loop system is stochastically stable and the corresponding cost function value is not bigger than a specified upper bound for all the admissible uncertainties. Finally, optimization algorithms are given to find the optimal performance indexes. Keywords Singular time-delay systems  Mode-dependent  Markov switching parameters  Stochastically stable

1 Introduction Singular systems, which are also referred to as generalized systems, implicit systems or descriptor systems [1], contain finite dynamic modes, non-dynamic modes and impulsive modes and have extensive applications in many actual systems, such as circuit systems, aerospace engineering systems, power systems, social economic systems, chemical systems, and so on [1–5]. Many basic concepts and

This work was supported by the National Natural Science Foundation of China (61703167), the Fundamental Research Funds for the Central Universities (2017MS058), the Guangdong Province Science and Technology Department Project (Grant No. 2016A010102010) and the Science and Technology Program of Guangzhou (Grant No. 201802020025). & Fuchun Liu [email protected] 1

College of Automation Science and Engineering, South China University of Technology, Guangzhou, China

2

School of Electronic Engineering and Intelligentization, Dongguan University of Technology, Dongguan 523808, China

control theories have been successfully extended to singular systems, such as stability, robust stabilization and LMI-based design approaches [6–11]. While for singular systems, we must consider the regularity, impulsive and stability simultaneously, the study of singular systems is more difficult and complicated than that of regular systems. When dealing with the stability and controller design problems, it is difficult to get the strict LMIs conditions; furthermore, equivalent transformation is always needed. For example, Hongquan Lu et al. [12] dealt with the delaydependent robust H1 control problem for uncertain descriptor time-delay system with Markov switching parameters and the results are not-strict LMI conditions. Based on a series of equivalent transformations, Ma et al. [6] discussed the robust H1 control problem for time-delay discrete-time singular systems by delay-dependent LMI. In references [7, 8], the authors deals with the stability problem and the robust H1 control problem for discretetime Markov switching systems, respectively, and the results are non-strict LMI conditions. In [9], authors discussed the robust stability and stabilization problems of time-delay discrete-time Markov switching singular systems in terms on state transformation and the results are also non-strict LMIs conditions.

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Once the disturbance abrupt variations affect physical systems, Markov switching systems, as a special kind of random hybrid systems, can better express the actual physical change process. Studies [13–16] and the references therein have discussed such systems extensively. Recently, many experts and scholars have devoted themselves to the stability of such systems and the design of controllers, which has produced many achievements in this field, see studies [17–22]. Obviously, the delay parameter is an important factor that cannot be avoided in the process of physical process modeling. It often appears in various engineering processes and can lead to system instability and even deterioration of system performance. Generally, the results of time delay systems include delay-dependent conditions and delay-independent ones [23–29]. Because the system stability obviously depends on the time delay parameters, especially if the delay parameter is very small, the delay-dependent results must have lower conservatism than the delay-independent conditions. Very recently, more scholars have focused on mode-dependent systems with delays and Markov switching parameters. Chen et al. [30] report the guaranteed cost control problem for uncertain Markovian jump systems with mode-dependent time delays. In [31], Wang et al. discussed the design problem of reduced-order H1 filtering for Markovian jump systems with mode-dependent time delays. Ma and Boukas [11] deal with the robust H1 filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay. Balasubramaniama et al. [32] report delay-dependent stability criterion for a class of nonlinear singular Markovian jump systems with mode-dependent interval time-varying delays. However, there is no reported research that discusses both continuous-time singular systems and mode-dependent time delays and this theory needs further improvement. This motivates the present research. In this paper, we deal with the robust guaranteed cost control problem for continuous-time uncertain Markov switching singular systems with mode-dependent delays. First, based on delay-dependent LMIs, sufficient conditions are given to ensure the nominal Markov singular system to be regular, non-impulsive and stochastically stable. Here two cases are considered. Case 1: the time delays are known; Case 2: the time delays are unknown, but the difference of the largest and the smallest time delay is known. Then, state-feedback controllers are designed to guarantee that the closed-loop system is stochastically stable and the corresponding cost function value is not bigger than a specified upper bound for all the admissible uncertainties. Finally, optimization algorithms are given to find the optimal performance indexes. Notations Rn denotes the n-dimensional Euclidean space; Rnm is the set of all n  m real matrices;

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X  0ðX [ 0Þ means that the symmetrical matrix X is positive semidefinite (positive definite); The superscript T stands for transpose of a matrix; Cn;d ¼ Cð½d; 0; Rn Þ denotes the Banach space of continuous vector functions mapping the interval ½d; 0 into Rn . And xt :¼ xðt þ hÞ; h 2 ½d; 0 denotes the function family on ½d; 0, which is generated by n-dimensional real valued continuous function xðtÞ; t 2 ½d; þ1Þ. Obviously xt 2 Cn;d . The following norms will be used: k  k refers to Euclidean vector norm or spectral matrix norm. k/kc :¼ supd  t  0 k/ðtÞk stands for the norm of a function / 2 Cn;d . L2 ½0; 1Þ stands for the space of square integrable functions on ½0; 1Þ. ðX; F ; PÞ is a probability space, X is the sample space, F is the algebra of events and P is the probability measure defined on F . E½ stands for the mathematical expectation.

2 Problem formulation and preliminaries Given a probability space ðX; F ; PÞ, where X is the sample space, F is the algebra of events and P is the probability measure defined on F , the singular time-delay Markov switching system considered in this paper is described by the following dynamics: _ ¼ ðAðrt Þ þ DAðrt ÞÞxðtÞ þ ðAd ðrt Þ þ DAd ðrt ÞÞ ExðtÞ xðt  dðrt ÞÞ þ ðBðrt Þ þ DBðrt ÞÞuðtÞ

ð1Þ

 0 xðtÞ ¼ /ðtÞ; rðtÞ ¼ r0 ; t 2 ½d; where xðtÞ 2 Rn is the system state, uðtÞ 2 Rm is the control input. dðiÞ [ 0 is a constant and denote the delay when the system is in the mode i. And d ¼ maxfdðiÞ; i 2 Sg. /ðtÞ 2 Cn;d is a compatible initial function. frt ; t  0g is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in a finite set S ¼ f1; . . .; sg with transition probability matrix P ¼ fpij g given by  pij h þ oðhÞ; j 6¼ i Prfrtþh ¼ jjrt ¼ ig ¼ 1 þ pii h þ oðhÞ; j ¼ i where h [ 0; limh!0 oðhÞ h ¼ 0 and pij  0, for j 6¼ i, is the transition rate from the mode i at time t to the mode j at P time t þ h and pii ¼  sj¼1;j6¼i pij : r0 2 S is the initial condition of the mode. The matrix E 2 Rnn is singular and 0\rankE ¼ r  n. Aðrt Þ; Ad ðrt Þ and Bðrt Þ are appropriate dimensional and known real constant matrices. The uncertainties matrices DAðrt Þ; DAd ðrt Þ and DBðrt Þ are assumed be norm-bounded and for every i 2 S they satisfy the following structure

Neural Computing and Applications

½ DAi ðtÞ

DAdi ðtÞ

MBi ðtÞ  ¼ Mi Fi ðtÞ½ N1i

FiT ðtÞFi ðtÞ 

I;

Ndi

N2i 

uðtÞ ¼ Kðrt ÞxðtÞ

8i 2 S ð2Þ

where for any i 2 S, Mi ; Ni ; Ndi and N2i are appropriate dimensional and known real constant matrices, while matrix functions Fi ðtÞ are unknown. In association with the system (1), we define the guaranteed cost function as follows: Z 1  J¼E xT ðtÞR1 ðrt ÞxðtÞ þ uT ðtÞR2 ðrt ÞuðtÞdtj/ðtÞ; r0 0

ð3Þ where R1 ðrt Þ and R2 ðrt Þ are given positive-definite symmetric constant matrices with appropriate dimensions. For notational simplicity, in the sequel, for each possible i 2 S, matrices Aðrt Þ, Ad ðrt Þ, Bðrt Þ, R1 ðrt Þ and R2 ðrt Þ will be respectively denoted by Ai , Adi , Bi , R1i and R2i . Throughout this paper, the following definitions are needed: Definition 2.1 [7] 1. For a given scalar d [ 0, the singular Markov switching time-delay system _ ¼ Ai xðtÞ þ Adi xðt  dÞ ExðtÞ  0 xðtÞ ¼ /ðtÞ; t 2 ½d;

ð5Þ

where Ki is a constant for every i 2 S and is designed to present an upper bound for the cost function (3) for stabilizing the uncertain singular Markov switching systems. For this purpose, the next Lemma is needed. Lemma 2.1 For any matrices D, E, F with appropriate dimensions and a scalar e [ 0, where F satisfies: F T F\I, the following inequality holds: DEF þ ET F T DT  e1 DDT þ eET E

3 Analysis of the robust performance In this section, we will firstly talk about the case of uðtÞ ¼ 0, that is _ ¼ ðAðrt Þ þ DAðrt ÞÞxðtÞ þ ðAd ðrt Þ þ DAd ðrt ÞÞxðt  dðrt ÞÞ ExðtÞ  0 xðtÞ ¼ /ðtÞ; rðtÞ ¼ r0 ; t 2 ½d;

ð6Þ Associated with the system (6), the cost function is as follows, Z 1  J¼E xT ðtÞR1 ðrt ÞxðtÞdtj/ðtÞ; r0 ð7Þ 0

ð4Þ

is said to be regular and non-impulsive for any constant  if the pairs ðE; Ai Þ and ðE; Ai þ Adi Þ d satisfying 0  d  d, are regular and non-impulsive for any i 2 S; 2. The singular Markovian jump time-delay system (4) is said to be stochastically stable, if there exists a scalar Mðr0 ; /ðÞÞ such that Z t  2  limt!1 E kxðsÞk dsjr0 ; xðsÞ ¼ /ðsÞ; s 2 ½d; 0 0

 Mðr0 ; /ðÞÞ 3. The singular Markovian jump time-delay system (4) is said to be stochastically admissible, if it is regular, nonimpulsive and stochastically stable. Associated with the given cost, the robust guaranteed cost controller is defined as follows: Definition 2.2 Consider the uncertain singular Markov switching time-delay system (1). If there exist a controller u(t) and a positive scalar J  such that for all admissible uncertainties, the corresponding closed-loop system is stochastically admissible, and the closed-loop value of the cost function (3) satisfies J  J  , then J  is said to be a guaranteed cost and the controller u(t) is said to be a robust resilient guaranteed cost controller.

The goal of this section is to present a sufficient condition to guarantee that the system (6) is regular, impulse free and stochastically stable and the value of the cost function (7) is no bigger than some specified bound. First we assume that the time delays di ; i 2 S are known, the following Theorem can be obtained. Theorem 3.1 For each i 2 S, if there exist a positive scalar ei , and matrices Q1 [ 0; Q2 [ 0; Q3 [ 0 and matrices Fi ; Gi ; Ji and nonsingular matrices Pi such that for any i 2 S, ET Pi ¼ PTi E  0 i H

2

Hi11

6 6 H 6 ¼6 6 H 6 4 H H

ð8Þ

PTi  FiT þ ATi Gi

FiT Adi þ ATi Ji

ei FiT Mi

N1iT

ai Q3  GTi  Gi H

GTi Adi  Ji  Q1 þ JiT Adi þ ATdi Ji

ei GTi Mi ei JiT Mi

0 NdiT

H

H

 ei I

0

H

H

H

 ei I

3 7 7 7 7\0 7 7 5

ð9Þ where Hi11 ¼

s X

pij ET Pj þ lQ1 þ ai Q2 þ FiT Ai þ ATi Fi þ R1i :

j¼1

and k ¼ maxfjpii j; i 2 Sg, d ¼ minfdi ; i 2 Sg, l ¼ 1þ kðd  dÞ, ai ¼ di þ k2 ðd2  d 2 Þ:

Now the objective of this paper is to develop a controller

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Neural Computing and Applications

Then, the system (6) is regular, impulse free and stochastically stable and the cost function (7) satisfies the following bound: J  J1 ðPr0 ; Q1 ; Q2 ; Q3 Þ ¼ xT ð0ÞET Pðr0 Þxð0Þ þ þ þ

Z

dðr0 Þ Z 0 dðr0 Þ

þk

Z

d

d

Z

0

/T ðaÞQ1 /ðaÞda

dðr0 Þ

Z

0

Z

0

/T ðaÞQ2 /ðaÞdadh h 0

which implies that Ai22 and Pi22 are nonsingular for every ~ A~i Þ is regular and noni 2 S. This implies the pair ðE; impulsive and P~i is nonsingular, that is Gi is nonsingular for every i 2 S. Because it is clear that detðsE~  A~i Þ ¼ detðsE  Ai Þ, the pair ðE; Ai Þ is also regular and non-impulsive for every i 2 S. Now, Pre-multiplying and post-multiplying (11) by ½ In In In In  and ½ In In In In T , respectively, we can obtain pii ET Pi þ FiT Ai þ ATi Fi þ ATi Gi þ GTi Ai þ Pi þ PTi

_ /_ T ðaÞET Q3 E/ðaÞdadh

h

Z

 Fi  FiT  Gi  GTi þ FiT Adi þ ATdi Fi

0

½/T ðaÞQ2 /ðaÞða  hÞ

þ GTi Adi þ ATdi Gi þ ATi Ji þ JiT Ai  Ji  JiT þ JiT Adi þ AT Ji þ kðd  dÞQ1 þ ai ðQ2 þ Q3 Þ þ R1i \0

h T

_  hÞ þ /T ðaÞQ1 /ðaÞdadh þ /_ T ðaÞE Q3 E/ðaÞða Proof Now the primary task is to prove the stochastic admissibility of the system (6). For this purpose, We quote notations       E 0 0 In 0 ~ ~ ~ E¼ ; Ai ¼ ; Adi ¼ ; 0 0 Adi Ai  In     Pi 0 Pi 0 P~i ¼ ; Pi ¼ ; Fi G i F i þ Ji G i     0 lQ1 þ ai Q2 þ R1i 0 ~i ¼ J~i ¼ ;Q : Ji 0 ai Q 3 Then from the conditions (8) and (9), the following inequalities are true E~T P~ ¼ P~T E~  0 "

~ i11 þ P~T A~i þ A~T P~i pii E~T P~i þ Q i i H

ð10Þ #

P~Ti A~di þ A~Ti J~i \0  Q1 þ J~iT A~di þ A~Tdi J~i

ð11Þ

pii E~T P~i þ P~Ti A~i þ A~Ti P~i \0

ð12Þ

Since rankE~ ¼ rankE ¼ r  n, there are nonsingular  and H,  such that matrices G   Ir 0   ~  E ¼ GE H ¼ ð13Þ 0 0

Ai ,

Ai11 Ai21

Adi12 Adi22



; Pi ,



Pi11

Pi12

Pi21

Pi22

 :

From (10), it is clear that Pi12 ¼ 0, for every i 2 S. Pre  T and H, multiplying and post-multiplying (12) by H respectively, it’s easy to get ATi22 Pi22 þ PTi22 Ai22 \0

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½ In

In ½pii E~T Pi þ PTi ðA~i þ A~di ½ In 0 Þ þ ðA~i þ A~di ½ In þ kðd  dÞQ1 þ ai ðQ2 þ Q3 Þ þ R1i ½ In In T \0

0 ÞT Pi

Note that Q1 [ 0; Q2 [ 0; Q3 [ 0; R1i [ 0, the following inequality is true pii E~T Pi þ PTi ðA~i þ A~di ½ In

0 Þ þ ðA~i þ A~di ½ In

0 ÞT Pi \0:

Similar to the above proof process, the pairðE; Ai þ Adi Þ is also regular and non-impulsive for any i 2 S. So, according to Definition 2.1, the system (4) is regular and non-im pulsive for any constant time delay d satisfying 0  d  d. Next we need to prove that the system (4) is stochastically stable. And we quote a Lyapunov–Krasovskii function as Vðxt ; rt ; tÞ ¼ V1 ðxt ; rt ; tÞ þ V2 ðxt ; rt ; tÞ þ V3 ðxt ; rt ; tÞ þ V4 ðxt ; rt ; tÞ þ V5 ðxt ; rt ; tÞ

ð14Þ

V1 ðxt ; rt ; tÞ ¼ xT ðtÞET Pðrt ÞxðtÞ Z t V2 ðxt ; rt ; tÞ ¼ xT ðaÞQ1 xðaÞda V3 ðxt ; rt ; tÞ ¼ V4 ðxt ; rt ; tÞ ¼

A~i H; A~di H;  T P~i H  Adi , G  Pi , G  Then, we let: Ai , G   Ai12 Adi11 ; Adi , Ai22 Adi21

that is

where

Obviously, it is clear



di

Z

tdðrt Þ Z t 0

dðrt Þ Z 0

V5 ðxt ; rt ; tÞ ¼k

tþh Z t

dðrt Þ tþh d Z t

Z

d

xT ðaÞQ2 xðaÞdadh _ x_T ðaÞET Q3 ExðaÞdadh ½xT ðaÞQ2 xðaÞða  t  hÞ

tþh

_ þ x_T ðaÞET Q3 ExðaÞða  t  hÞ þ xT ðaÞQ1 xðaÞdadh Let L½ be the weak infinitesimal operator of the stochastic process fxt ; rt g, then for each i 2 S, we have

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_ LV1 ðxt ; i; tÞ ¼ x_T ðtÞET Pi xðtÞ þ xT ðtÞET Pi xðtÞ s X _ þ ðAi þ DAi ÞxðtÞ pij ET Pj xðtÞ þ ½ExðtÞ þ xT ðtÞ

LVðxt ; i; tÞ  gT ðtÞ!i gðtÞ

j¼1

_ þ Ji xðt  di Þ þ ðAdi þ DAdi Þxðt  di ÞT ½Fi xðtÞ þ Gi ExðtÞ _ þ ½Fi xðtÞ þ Gi ExðtÞ _ þ ðAi þ DAi ÞxðtÞ þ Ji xðt  di ÞT ½ExðtÞ þ ðAdi þ DAdi Þxðt  di Þ ð15Þ For V2 ðxt ; rt ; tÞ we get

JiT Mi

LV2 ðxt ; i; tÞ ¼ xT ðtÞQ1 xðtÞ  xT ðt  di ÞQ1 xðt  di Þ Z t s X pij xT ðaÞQ1 xðaÞda þ tdj

j¼1 T

 x ðtÞQ1 xðtÞ  xT ðt  di ÞQ1 xðt  di Þ Z t s X pij xT ðaÞQ1 xðaÞda þ td

j¼1;j6¼i

Z

 j pii j

ð16Þ

t td

 x ðtÞQ1 xðtÞ  xT ðt  di ÞQ1 xðt  di Þ Z td xT ðaÞQ1 xðaÞda þk Similarly LV3 ðxt ; i; tÞ  di x ðtÞQ2 xðtÞ 

T

Z

d

Z

d

Z

t

xT ðaÞQ2 xðaÞda tdi T

x ðaÞQ2 xðaÞdadh

_  LV4 ðxt ; i; tÞ  di x_ ðtÞE Q3 ExðtÞ þk

d d

Z

Z

t

_ x_T ðaÞET Q3 ExðaÞda

By Schur’s complement, condition (9) implies !i \0; i 2 S. Now, let c0 ¼ minfkmin ð!i Þ; i 2 Sg, from (20) it is clear that for every i 2 S, LVðxt ; i; tÞ   c0 k  xðtÞ k2 : Then, using Dynkin’s formula, for any t  d; Z t    c0 E k xðsÞ k2 ds: EVðxt ; i; tÞ  EVðxd; rd; dÞ d

As mentioned before, the pairðE; Ai Þ is regular and impulse free for every i 2 S. This leads to that there exist nonsin N such that ðE; Ai Þ for every i 2 S is gular matrices M; r.s.e. to the following Weierstrass standard form  N;  Ai ¼ MA   i N; E ¼ ME       Ir 0 Ai1 Adi11 Adi12 0 E ¼ ; Ai ¼ ; Adi ¼ : 0 0 0 Inr Adi21 Adi22 So, system (4) for every i 2 S, can be equivalently transformed into y_1 ðtÞ ¼ Ai1 y1 ðtÞ þ Adi11 y1 ðt  di Þ þ Adi12 y2 ðt  di Þ;

_ x_T ðaÞET Q3 ExðaÞdadh

tþh

y2 ðtÞ ¼ Adi21 y1 ðt  di Þ þ Adi22 y2 ðt  di Þ;  0; wðtÞ ¼ N 1 /ðtÞ; t 2 ½d; ð22Þ

Moreover, LV5 ðxt ; i; tÞ ¼



k 2 ðd  d 2 ÞxT ðtÞQ2 xðtÞ 2

d

Z

d



y1 ðtÞ ¼ N 1 xðtÞ; y1 ðtÞ 2 Rr ; y2 ðtÞ 2Rnr : y2 ðtÞ  Then for any t 2 ½0; d, Z t k y1 ðtÞ k  k eAi1 t y1 ð0Þk þ k eAi1 ðtsÞ ½Adi11 y1 ðs  di Þ where yðtÞ ¼

k _ þ kðd  dÞxT ðtÞQ1 xðtÞ þ ðd2  d2 Þx_T ðtÞET Q3 ExðtÞ 2 Z td Z d Z t k xT ðaÞQ1 xðaÞda  k xT ðaÞQ2 xðaÞdadh k

ð21Þ

tdi t

ð18Þ

td Z d

NdiT

tþh

T

Z

ð17Þ

t

NdiT

d

td

T

JiT Mi

So it is true Z t  k xðsÞ k2 ds  c1 E 0 EVðxd; rd; dÞ:

xT ðaÞQ1 xðaÞda

T

þk

ð20Þ  where gT ðtÞ ¼ xT ðtÞ ðExðtÞÞ _ T xT ðt  di Þ ; P !i11 ¼ sj¼1 pij ET Pj þ lQ1 þ ai Q2 þ FiT Ai þ ATi Fi ; 2 3 FiT Adi þ ATi Ji !i11 PTi  FiT þ ATi Gi 6 7 !i ¼4 H ai Q3  GTi  Gi GTi Adi  Ji 5 T T H H  Q1 þ Ji Adi þ Adi Ji 2 T 32 T 3T 2 T 32 T 3T N1i N1i F i Mi F i Mi 6 T 76 T 7 76 7 1 6 þ ei 4 Gi Mi 54 Gi Mi 5 þei 4 0 54 0 5 

tþh

0

þ Adi12 y2 ðs  di Þds k  k1 k w kd

t

_ x_T ðaÞET Q3 ExðaÞdadh

tþh

ð19Þ Then from (15)–(19) and Lemma 2.1, it can be obtained that, for every i 2 S,

 di11 k þ kAdi12 kÞ maxt2½0;d k where k1 ¼ maxi2S f½1 þ dðkA eAi1 t kg  0: Similarly, we can get k y2 ðtÞ k  k2 k w kd

123

Neural Computing and Applications  H i

where k2 ¼ maxi2S fk Adi21 k þ k Adi22 kg  0: It is clear that sup k y1 ðsÞ k

0  s  d

2

 k12

kw

k2d;

sup k y2 ðsÞ k

0  s  d

2

ð23Þ

 k22 k w k2d :

2

 i11 H 6 6 H 6 ¼6 6 H 6 4 H H

PTi  FiT þ ATi Gi  GTi  Gi H

FiT Adi þ ATi Ji GTi Adi  Ji  Q1 þ JiT Adi þ ATdi Ji

ei FiT Mi ei GTi Mi ei JiT Mi

N1iT 0 NdiT

H H

H H

 ei I H

0  ei I

where

sup k xðsÞ k2  k3 k / k2d :

0  s  d

According to (14) and system (22), a scalar r [ 0 can be easily found and satisfy Vðxd ; i; dÞ  r k / k2d Then together with (21), a scalar q [ 0 can also be easily found and satisfy the following Z t Z d E k xðsÞ k2 ds  E k xðsÞ k2 ds 0 0 ð24Þ Z t 2 2 þE k xðsÞ k ds  q k / kd d

Now in terms of Definition 2.1, system (4) is stochastically  stable for any constant d satisfying 0  d  d: Now let’s try to deal with the problem of guaranteed cost performance index. From (20) and by using the Dynkin’s formula, Z T Z T JT ¼ E xT ðtÞR1 ðrðtÞÞxðtÞdt ¼ E fxT ðtÞR1 ðrðtÞÞxðtÞ 0 0 Z T LVðxðtÞ; rðtÞ; tÞdt þ LVðxðtÞ; rðtÞ; tÞgdt  E 0 Z T fxTðtÞR1 ðrðtÞÞxðtÞ þ LVðxðtÞ; rðtÞ; tÞgdt ¼E 0

 EVðxðTÞ; rðTÞ; TÞ þ Vðxð0Þ; r0 ; 0Þ Z T i gðtÞdt þ Vðxð0Þ; r0 ; 0Þ gTðtÞ! E 0

 i ¼ !i þ diagfR1i ; 0; 0g. Then by Schur’s comwhere ! i \0. So, the following is correct  plement, H\0 implies ! Z 1 xT ðtÞR1 ðrðtÞÞxðtÞdt  Vðxð0Þ; r0 ; 0Þ J¼E 0

¼ J1 ðPr0 ; Q1 ; Q2 ; Q3 Þ h

Secondly, we assume that the time delays di are unknown, we can get the following Theorem. Theorem 3.2 For each i 2 S, if there exist positive scalars ei , and matrices Q1 [ 0 and matrices Fi ; Gi ; Ji and nonsingular matrices Pi such that for every i 2 S, the condition (8) satisfied and the following inequality is true

123

7 7 7 7\0 7 7 5

ð25Þ

So, it’s easy to find a scalar k3 [ 0 such that

This completes the proof.

3

 i11 ¼ H

s X

pij ET Pj þ lQ1 þ FiT Ai þ ATi Fi þ R1i :

j¼1

Then, the system (6) is regular, impulse free and stochastically stable and the cost function (7) satisfies the following bound: J  J2 ðPr0 ; Q1 Þ ¼ xT ð0ÞET Pðr0 Þxð0Þ Z d Z Z 0 /T ðaÞQ1 /ðaÞda þ k þ d

dðr0 Þ

0

/T ðaÞQ1 /ðaÞadh h

Proof As to this case, we quote the Lyapunov–Krasovskii functional candidates as follows: Z t Vðxt ; rt ; tÞ ¼ xT ðtÞETPðrt ÞxðtÞ þ xT ðaÞQ1 xðaÞda þk

Z

d d

Z

tdðrt Þ t

xT ðaÞQ1 xðaÞdadh

tþh

Then the process of proof is similar to Theorem 3.1 therefore it is omitted here. h Remark 3.1 As to the singular Markovian jump system with mode-dependent time delay, Theorem 3.1 presents a sufficient condition to guarantee that the system is regular, impulse free and stochastically stable and the cost function has the upper bound. It’s worth noting that the bound directly depend on the system time delays. Remark 3.2 The appearance of the items ai Q2 and ai Q3 leads to that the condition (25) is affine linear in di ; i 2 S, which implies that if the conditions of Theorem 3.2 are satisfied for every di ; i 2 S, they are also satisfied for any di 2 ½0; di ; i 2 S. So, for every mode of the system time delays di ; i 2 S, once the every upper bound di is known, the Theorem 3.2 is feasible. Remark 3.3 Theorem 3.2 gives a sufficient condition to guarantee that the system (4) is regular, impulse free and stochastically stable and the cost function also has the upper bound. But here the bound depends on the supremum and infimum of the system time delays.  i 2 S in the conditions Remark 3.4 If we let every di ¼ d; of Theorems 3.1 and 3.2, this can evolve into the case that the system time delay only has one mode. For many

Neural Computing and Applications

relevant results about delay-dependent stability and controller synthesis, see reference [33–37].

4 Guaranteed cost controller design The system (1) being controlled by (5) evolves into the following structure _ ¼ ðAðrt Þ þ DAðrt Þ þ ðBðrt Þ þ DBðrt ÞÞKðrt ÞÞxðtÞ ExðtÞ þ ðAd ðrt Þ þ DAd ðrt ÞÞxðt  dðrt ÞÞ  0 xðtÞ ¼ /ðtÞ; rðtÞ ¼ r0 ; t 2 ½d; ð26Þ As mentioned above, the goal is to develop the controller (5) such that the closed-loop system (26) is regular, nonimpulsive and stochastically stable and the guaranteed cost function J¼E

Z

1

xT ðtÞ½R1 ðrt Þ þ K T ðrt ÞR2 ðrt ÞKðrt ÞxðtÞdtj/ðtÞ; r0



0

is no bigger than some specified constant bound. Case 1 The time delay is known. Theorem 4.1 For each i 2 S, if there exist a positive scalar ei , matrices of appropriate dimensions  1 [ 0; Q 2 [ 0; Q 3 [ 0, Ui ; Vi ; Wi and nonsingular matriQ ces Xi such that the following conditions are satisfied XiT ET ¼ EXi  0

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

Vi 1 Q

where Xi11 ¼ pii XiT ET þ Ui þ UiT ; Xi13 ¼ Wi  UiT þ XiT ATi þ STi BTi 2      3 pffiffiffiffiffiffi Xi11 pffiffiffiffiffiffiffiffiffiffiffiffi Xi11 pffiffiffiffiffiffiffiffiffiffiffiffi Xi11 p    p p i1 iði1Þ iðiþ1Þ 6 0 0 0 7 6 7 Yi ¼ 6   7; 4 5 pffiffiffiffiffiffi Xi11  pis 0 Zi ¼ diagfX111 ;    ; Xði1Þ11 ; Xðiþ1Þ11 ;    ; Xs11 g:

Then the uncertain singular time-delay Markovian jump system (1) with the controller uðtÞ ¼ Sðrt ÞXðrt Þ1 xðtÞ is regular, impulse free and stochastically stable and the cost function (3) satisfies 1 ; Q  1 ; Q  1 Þ J  J1 ðXr1 ;Q 1 2 3 0 Proof From the proof process of Theorem 3.1, we know that the sufficient condition for the existence of the state feedback guaranteed controller is that for each i 2 S, there exist positive scalars ei , and matrices Q1 [ 0; Q2 [ 0; Q3 [ 0 and matrices Fi ; Gi ; Ji and nonsingular matrices Pi such that for every i 2 S, the condition (8) satisfied and the following inequality is true

ð27Þ

XiT N1i T þ STi N2i T 1 N T Q

ai UiT ai ViT

0

 ei I H

H

H H

H H H

Xi11

lXiT

ai XiT

XiT

STi

ai WiT

0 0

0 0

0 0

0 0

0 0

0  ei I

0 0

0 0

0 0

0 0

0 0

0 0

H

H

3  ai Q

0

0

0

0

0

H H

H H

H H

H H

 Zi H

0 1  lQ

0 0

0 0

0 0

H H

H H

H H

H H

H H

H H

H H

2  ai Q H

0  R1i

H

H

H

H

H

H

H

H

H

0 0  R2i

0

H

 1 AT  ViT þ Q di  Wi  WiT

0 ei M i

H H

H H

H H

H

H

H H

H H

3

Yi

Xi13

di

7 7 7 7 7 7 7 7 7 7 7 7\0 7 7 7 7 7 7 7 7 7 5 ð28Þ

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Neural Computing and Applications

2

Hki11 6 6 H 6  ki ¼ 6 H H 6 6 4 H

PTi  FiT þ ðAi þ Bi Ki ÞT Gi ai Q3  GTi  Gi

FiT Adi þ ðAi þ Bi Ki ÞT Ji GTi Adi  Ji

ei FiT Mi ei GTi Mi

H H

 Q1 þ JiT Adi þ ATdi Ji H

ei JiT Mi  ei I

H

H

H

H

Hki11 ¼

where

Ps

j¼1

Bi Ki Þþ ðAi þ Bi Ki ÞT Fi þ R1i þ KiT R2i Ki : Now, we introduce the following notations: 2

In 6 60 6 I ¼6 60 6 40 0

0

0

0

In

In

0

0 0 2

0 0

3

2 Pi 0 0 7 6 0 07 6 0 In 0 7 ~ 6 ~i ¼ 6 Fi Ji Gi ; P 0 07 7 6 7 6 In 0 5 40 0 0 0 In 0 0 0 3 0 In 0 0 7 0 0 0 07 7 Adi  In ei Mi 0 7 7; 7 0 0 0 05 Ndi 0 0 0 0

0

2

pij ET Pj þ lQ1 þ ai Q2 þ FiT ðAi þ

0 0 0 In 0

0

~1 P~ i

3

7 07 7 07 7; 7 05 In

Pi 6 60 6 ¼6 6 Fi 6 40 0

3 ðN1i þ N2i Ki ÞT 7 7 0 7 T 7\0 Ndi 7 7 5 0  ei I

0 In Ji

0 0 Gi

0 0 0

0 0

0 0

In 0

31 2 0 Xi 6 7 07 60 6 7 7 07 ¼6 6 Ui 6 7 05 40 In 0

0 In Vi

0 0 Wi

0 0 0

0 0

0 0

In 0

3 0 7 07 7 07 7 7 05 In

Then, respectively, pre-multiplying and post-multiplying (8) by XiT and Xi yield (27). Now let’s to deal with the following inequality

0 6 0 6 6 ~ki ¼ 6 Aki A~ 6 6 4 0 Nk1i s X ~ ~ i ¼ diagf Q pij ET Pj þ lQ1 þ ai Q2 þ R1i ; Q1 ; ai Q3 ; ei I; ei Ig

~ ~ ~1 ~ T ~  ~ ~T ~ ~ ~ ~T ~1 ~1 ~T ~ ~T ~ P~ i  I  Hki  I  Pi ¼ Aki Pi þ Pi Aki þ Pi Qi Pi \0:

That is

j¼1

2

Ps

T 6 Xi ½

j¼1

pij ET Pj þ lQ1

Xi þ Ui þ 6 þai Q2 þ R1i þ KiT R2i Ki 6 6 H 6 6 6 H 6 6 þ ½ Ui Vi Wi 0 0 T ðai Q3 Þ  ½ Ui 6 6 4 H

3 UiT

H

Vi

Vi

Wi 

þ

XiT ATki

 Q1  ViT þ ATdi H  Wi  WiT Wi 0 0 \0 H H

where Aki ¼ Ai þ Bi Ki ; Nk1i ¼ N1i þ N2i Ki . Then it is clear that ~~ \0:  ki  I T ¼ P~~T A~~ki þ A~~T P~~i þ Q I H i ki i From the above analysis of Theorem 3.1, Pi ; Gi are both nonsingular for any i 2 S. Through simple mathematical calculations, we can define that

123

UiT

H H

0

T XiT Nk1i

0 ei M i

NdiT 0

 ei I H

0  ei I

7 7 7 7 7 7 7 7 7 7 7 5

ð29Þ

Furthermore, pre-multiplying and post-multiplying (29) by T diagfIn ; Q1 and diagfIn ; Q1 1 ; In ; In ; In g 1 ; In ; In ; In g , 1    respectively, and set Vi Q1 1 ¼ Vi ,Q1 ¼ Q1 ; Q2 ¼ 1  1  1  1 Q2 ; Q3 ¼ Q3 ; R1i ¼ R1i ; R2i ¼ R2i ,Si ¼ Ki Xi , we can get

Neural Computing and Applications

2

Ps

6 XiT ½ 6 6 6 6 6 6 6 6 4

j¼1

pij ET Pj þ lQ1

þai Q2 þ R1i þ KiT R2i Ki

þ ½ Ui

Vi

Wi

3 Xi þ Ui þ

UiT

Vi

Wi 

UiT

1 Q H

 1 AT  ViT þ Q di  Wi  WiT

0 ei M i

H H

H H

H H

 ei I H

0

T

0  ðai Q3 Þ  ½ Ui

Vi

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

Vi 1 Q

Ps

0 0 ei M i

7 7 7 1 N T 7 Q di 7 7 0 7 7 7 0 5  ei I

XiT N1iT þ STi N2iT 1 N T Q

Yi

lXiT

XiT

STi

0 0

0 0

0 0

H

 1 AT  ViT þ Q di  Wi  WiT

0

0 0

H H

H H

H H

 ei I H

0  ei I

0 0

0 0

0 0

0 0

H

H

H

H

H

 Zi

0

0

0

0  R1i H

0 0  R2i

H H

ð30Þ

Ui ; Vi ; Wi and nonsingular matrices Xi satisfying (27) and the following

j¼1;j6¼i

Xi13

T XiT Nk1i

0 0 \0

Wi

pij XiT ET Xj1 Xi ,   Ir 0 . without loss of generality, we assume that E ¼ 0 0

Xi11

0

H H

Noting that there exists the term

2

þ

XiT ATki

di

H H

H H

H H

H H

H H

H H

1  lQ H

H

H

H

H

H

H

H

 Xi11 0 and Xi21 Xi22 Xi11 [ 0: Through simple mathematical calculations, the following equation is established   s s X X Xi11 1 T T 1 pij Xi E Xj Xi ¼ pij Xj11 ½ Xi11 0 : 0 j¼1;j6¼i j¼1;j6¼i

3 7 7 7 7 7 7 7 7 7\0 7 7 7 7 7 7 7 5

ð31Þ



And from (27), it is clear that Xi ¼

Then by Schur’s complement, condition (28) can be easily obtained. The proof is completed. h Case 2 The time delay is unknown, but the difference of the upper and lower bounds of the time delay is known. Now according to Theorem 3.2 and similar to the proof process of Theorem 4.1, the following conclusion can be obtained.

Then the uncertain singular time-delay Markovian jump system (1) with the controller uðtÞ ¼ Sðrt ÞXðrt Þ1 xðtÞ is regular, impulse free and stochastically stable and the cost 1 Þ: function (3) satisfies J  J2 ðXr1 ;Q 1 0 Remark 4.1 According to the different cases of the system time delay, Theorems 4.1 and Theorem 4.2 provide sufficient conditions for the existence of the guaranteed cost controllers of the system (1). Performance upper bound depends on the different choices of the controllers. So, there exists a controller, which makes the performance index to be minimized. Following two theorems talk about the design problems of the optimal guaranteed cost controllers.

Theorem 4.2 For each i 2 S, if there exist a positive 1 [ 0, scalar ei , matrices of appropriate dimensions Q

123

Neural Computing and Applications

In order to get the optimal guaranteed cost controller, we are in the position to dealing the nonlinear terms in the cost function.  1 ; Q  1 ; Q 1 Þ ¼ xT ð0ÞET X 1 xð0Þ J1 ðXr1 ;Q 1 2 3 r0 0 Z 0 1 /ðaÞda þ /T ðaÞQ 1 dðr0 Þ 0

þ þ

Z

Z

Z

dðr0 Þ

Z

0

dðr0 Þ

þk

Z

0

d T

þ /_ T ðaÞE

N1

1 /ðaÞdadh / ðaÞQ 2

dðr0 Þ

1 /ðaÞða  hÞ ½/T ðaÞQ 2

h 1 _  Q3 E/ðaÞða

ð32Þ

/ðaÞ/T ðaÞda ¼ N1 N1T , then the following con-

¼

d

dðr0 Þ

4

d

Z

d Z

0

_  1 E/ðaÞða 1 /ðaÞða  hÞ þ /_ T ðaÞET Q ½/T ðaÞQ 2 3

d

1 /ðaÞÞda Traceð/T ðaÞQ 1 1 Þda Traceð/ðaÞ/T ðaÞQ 1

1 /ðaÞda þ / ðaÞQ 1 1 N3 Þ þ kðd  dÞ  d ¼ kðd  dÞ  d  TraceðN T Q

d

T

3

2

 1 N4 Þ þ kðd  dÞ  TraceðN T Q 1  TraceðN4T Q 3 3 1 N3 Þ ~ 4 Þ þ kðd  dÞ  d  TraceðM ~5Þ \kðd  dÞ  d  TraceðM ~6Þ þ kðd  dÞ  TraceðM

# Z 0 Z 0 N1T  1 /ðaÞdadh\dðr0 Þ /T ðaÞQ \0; 2 1 Q dðr0 Þ h  1 /ðaÞda ¼ dðr0 ÞTraceðN T Q  1 /T ðaÞQ 2 1 2 N1 Þ

~ 2 Þ; \dðr0 ÞTraceðM

ð33Þ if and only if

d

_ 1 E/ðaÞða 1 /ðaÞdadh  hÞ þ /T ðaÞQ þ /_ T ðaÞET Q 3 1 R0 T T  Letting and 0\a  h\d. d /ðaÞ/ ðaÞda ¼ N3 N3 and R0 T _ _T T  E /ðaÞ/ E ðaÞda ¼ N4 N , and similarly, we can get

1 /ðaÞdadh  hÞ þ /T ðaÞQ 1 Z 0 _  T ðaÞQ 1 E/ðaÞ 1 xðaÞ þ d/_ T ðaÞET Q dx \kðd  dÞ 2 3

if and only if ~1 M N1 Z 0

h

_ 1 E/ðaÞða 1 /ðaÞdadh  hÞ þ /T ðaÞQ þ /_ T ðaÞET Q 3 1 Z d Z 0 1 /ðaÞða  hÞ ½/T ðaÞQ \k 2

d

dðr0 Þ 1 N1 Þ\TraceðM ~ 1 Þ; ¼ TraceðN1T Q 1

"

ð35Þ

It is clear that Z d Z 0  1 /ðaÞða  hÞ k ½/T ðaÞQ 2

k

0

dðr0 Þ Z 0

_ 1 E/ðaÞda /_ T ðaÞET Q 3

if and only if " # ~3 M N2T \0; 3 N2 Q

d

dðr0 Þ

¼

0

dðr0 Þ

clusions are correct. Z 0  1 /ðaÞda /T ðaÞQ 1 Z

Z

_  1 E/ðaÞdadh /_ T ðaÞET Q 3

1 N2 Þ\dðr0 ÞTraceðM ~ 3 Þ; ¼ dðr0 ÞTraceðN2T Q 3

xT ð0ÞET Xr1 xð0Þ ¼ ½T2 xð0ÞT T2 Xr1 T1 ½T2 xð0Þ\a 0 0

R0

0 h

\dðr0 Þ

_ 1 E/ðaÞdadh / ðaÞET Q 3

if and only if " # a xð0ÞT T2T \0: T2 xð0Þ  T2 Xr0 T1

Z

dðr0 Þ

ð34Þ

_ /_ T ET ðaÞda ¼ N2 N T , and similarly, we E/ðaÞ 2

dðr0 Þ

can get Z 0

1 /ðaÞdadh  hÞ þ /T ðaÞQ 1   Ir 0 and denote Without loss of generality, we let E ¼ 0 0   I T1 ¼ r , T2 ¼ ½ Ir 0 : Then from the Schur’s comple0 ment, the following conclusion is correct.

Let

R0

Let

_T

h d Z 0

# N1T \0; 2 Q

~2 M

T

h 0

"

the last inequality is established, if and only if " # " # ~4 ~5 M N3T N4T M \0; \0; 2 3 N3 Q N4 Q " # ~6 N3T M \0; 1 N3 Q

ð36Þ

ð37Þ

Now, from the above analysis, we can get the guaranteed cost controllers which can minimize the guaranteed cost.

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Neural Computing and Applications

Theorem 4.3 Considering the uncertain singular timedelay Markovian jump systems (1) and the cost function (3), if the following optimal problem has a solution  1 [ 0; Q 2 [ 0; Q 3 [ 0, ~ l [ 0; l ¼ f1; 2; 3; 4; 5; 6g, M Q  Ui ; Vi , Wi ,Xi ,i 2 S, scalars a [ 0 and ei [ 0, i 2 S. min

~ l ;l¼1;2;3;4;5;6 a;M

~ 1 þ dðr0 Þ  ðM ~2 þ M ~ 3Þ J  ¼ a þ TraceðM ~4 þ M ~ 5 Þ þ kðd  dÞ  M ~ 6Þ þ kðd  dÞ  d  ðM subject to ð27Þ; ð31Þ; ð32Þ; ð33Þ; ð34Þ; ð35Þ; ð36Þ and ð37Þ:

Then the system (1) with the controller uðtÞ ¼ Sðrt Þ Xðrt Þ1 xðtÞ is regular, impulse free and stochastically stable and the cost function (3) has an optimal performance index J  . Theorem 4.4 Considering the uncertain singular timedelay Markovian jump systems (1) and the cost function (3), if the following optimal problem has a solution 1 [ 0; Q  2 [ 0; Q 3 [ 0, ~ 1 [ 0; M ~ 6 [ 0, M Q Ui ; Vi , Wi ,Xi ,i 2 S, scalars a [ 0 and ei [ 0, i 2 S. ~ 1 þ kðd  dÞ  M ~ 6Þ min J  ¼ a þ TraceðM

~1 ;M ~6 a;M

subject to ð27Þ; ð31Þ; ð32Þ; ð33Þ and ð37Þ: Then the system (1) with the controller uðtÞ ¼ Sðrt ÞXðrt Þ1 xðtÞ is regular, impulse free and stochastically stable and the cost function (3) has an optimal performance index J  . Remark 4.2 Combined with the famous Schur’s Complement Lemma and Theorems 4.1, 4.2, the proofs of Theorems 4.3 and 4.4 can be easily obtained. So they are omitted here. Remark 4.3 It is worthy of noting that the optimization problems in Theorems 4.3 and 4.4 are both convex. Therefore, once the optimization problem has a solution, it is globally optimal.

5 Conclusion In this paper we discuss the robust guaranteed cost control problem for mode-dependent time-delay continuous-time Markov switching singular systems with norm-bounded uncertain parameters. Firstly, by quoting a mode-dependent Lyapunov functional and applying Moon’s inequality for cross terms, we give the two cases sufficient conditions which ensure the nominal Markov switching singular system to be regular, non-impulsive and stochastically stable. The two cases are as following, Case 1: the time delays are known; Case 2: the time delays are unknown, but the difference of the largest and the smallest time delay is known.

Then, the problem is solved to design a state-feedback control law such that the closed-loop system is stochastically stable and the corresponding cost function value is not bigger than a specified upper bound for all the admissible uncertainties. Finally, optimization algorithms are given to find the optimal performance indexes.

Compliance with ethical standards Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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