Circuits Syst Signal Process https://doi.org/10.1007/s00034-018-0831-7
Simultaneous Finite-Time Control and Fault Detection for Singular Markovian Jump Delay Systems with Average Dwell Time Constraint Mengzhuo Luo1 · Shouming Zhong2 · Jun Cheng3
Received: 10 June 2017 / Revised: 20 March 2018 / Accepted: 23 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Finite-time control and fault detection are treated simultaneously as a single problem for a class of continuous-time singular Markovian jump delay systems. To develop the objectives for both control and detection, mode-dependent fault detection filters and dynamic feedback controllers were designed for a certain set of performance indices. Based on the average dwell time approach and using some novel integral inequalities, new sufficient conditions for the presence of a fault detection/controller unit are presented as a set of linear matrix inequalities. This problem is formulated as a finite-time H∞ optimization problem, and both the stochastic finite-time stability and fault detection filter are analyzed, employing a switching control approach. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.
This work was supported in part by the National Natural Science Foundation of China under Grants 11661028, 11661030, and 11502057, the Natural Science Foundation of Guangxi under Grants 2015GXNSFBA139005, and 2014GXNSFBA118023.
B
Mengzhuo Luo
[email protected] Shouming Zhong
[email protected] Jun Cheng
[email protected]
1
College of Science, Guilin University of Technology, Guilin 541004, Guangxi, People’s Republic of China
2
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China
3
School of Science, Hubei University for Nationalities, Enshi 445000, Hubei, People’s Republic of China
Circuits Syst Signal Process
Keywords Fault detection filter · Markovian jump delay systems · Stochastic finite-time stability · Singular systems · Average dwell time
1 Introduction Singular systems, which are also referred to as generalized state-space systems, descriptor systems or implicit systems, are important systems that have received extensive attention during the past decade because of their powerful applications in many practical systems, such as economic systems, robotic systems, biological systems network control systems, and chemical systems. Different from other regular systems, singular systems are more general and complex because one needs to consider not only system stability but also regularity and the absence of impulses or causality. Because of this circumstance, singular systems better describe and analyze the behavior of some physical systems than regular systems using standard state-space models [14,38]. Moreover, the presence of time delays often causes undesirable behavior such as the degradation in stability of the dynamical system. Hence, the major issues of stability and control analysis for time-delay singular systems have been studied extensively in actual problems [9,21]. Recently, finite-time stability has received increasing attention and was proposed in practical processes such as in avoiding saturation or in exciting non-linear transient dynamics [1,2]. Different from the classical Lyapunov stability concept, finite-time stability is defined as the behavior of the dynamical systems that can be tracked over a fixed finite-time interval, that is, the system state does not exceed a certain bound during a fixed finite-time interval. The introduction of such a stability concept is very necessary and important in many practical problems. Hence, the study of finite-time stability is not only for academic reasons but also has practical import in applications. Many interesting results have been obtained for this type of stability. For example, Cheng et al. [3] investigated the problem of robust finite-time boundedness of H∞ filtering for switch systems with time-varying delay; Wen et al. [29] addressed a finite-time stabilization problem for a class of continuous-time Markovian jump delay systems with switching control approach; He and Liu [11] studied the observer-based state feedback finite-time control for non-linear jump systems with time delay and Sreten [27] analyzed the problem of finite-time stability for uncertain time-varying delay systems with non-linear perturbations and parametric uncertainties and switched time-delay systems. As is well known, the most important features of any practical systems are reliability and safety. System performance can be improved by focusing on operations. However, in general, faults are unavoidable under practical conditions such as hotspot faults, sensor faults, short circuits faults [5,15,35], and the unavoidable faults that impair reliability and safety. Therefore, it is essential for engineers to be able to detect faults quickly. Hence, over the past few decades, much attention has been paid to fault detection techniques because of the demands for higher safety and reliability in modern society. To date, various kinds of fault detection techniques have been developed, e.g., model-based approaches, knowledge-based schemes and signal-based methods. In particular, the problem of H∞ optimization-based fault detection has been an active
Circuits Syst Signal Process
research area [6,12,13,30]. Nonetheless, compared with the design problem with its two separate facets, detection and control, the main ideal behind simultaneous control and fault detection is to merge these units into a single detector/controller unit to lessen the overall complexity [7,16,22,25,31,39,40]. Shokouhi-Nejad et al. [25] dealt with this problem for linear switched systems with state delay and parameter uncertainties; Zhong and Yang [39] were concerned with this problem for continuous-time switched systems subject to dwell time constraint; Zhai et al. [40] investigated the problem for switched linear systems under a mixed H∞ /H− framework and Li et al. [16] presented this problem for switched systems with two quantized signals. Stochastic models are well known to play important roles in many branches of science and engineering applications [4,17,20,24,26,41]. The Markovian jump system is one such important hybrid that have received considerable research attention. In practice, the structure and parameters of the dynamic systems may be changed abruptly because of randomly occurring phenomena such as component failure and repairs of components, change in the interconnections of the subsystems, and abrupt environmental changes [8,23,32,34]. Note that the majority of the results concerning Markovian jump systems with constant jump rates, and such assumptions, limit their actual use because in some physical or manufactured systems, the control moves may affect the transition probabilities or jump rates. That is, the transition probabilities or jump rates could change over time in some actual models. To the best of our knowledge, the problem of simultaneous finite-time control and fault detection for the jump system with time-varying transition probabilities, especially for the singular switch system, has not been thoroughly studied. There is much room for improvement. Motivated by these observations, based on some novel integral inequalities and improved methods, we treat the problem of simultaneous finite-time control and fault detection for a class of singular Markovian systems with any deterministic switching signal and stochastic jumping process. The main challenge is overcoming complicating factors such as the presence of jumps, time-varying delays, and disturbances occurring in singular Markovian jump systems. The objective is to design a mode-dependent detector/controller unit such that the augmented system is not only stochastic singular finite-time stability but also satisfies different H∞ performance indices. A novel stochastic Lyapunov function and a set of strict linear matrix inequalities (LMIs) are used to derive sufficient conditions guaranteeing that the desired detector/controller unit is constructible. Finally, a numerical example is proposed to illustrate the effectiveness of the obtained results. A summary of the main contributions are: (1) a class of general singular systems that includes both a stochastic switch and deterministic switch, as well as their timevarying delay, are considered simultaneously. (2) Based on the average dwell time (ADT) scheme, the simultaneous finite-time control and fault detection problem for a class of singular Markovian jump system is considered, and some sufficient conditions for the stochastic finite-time stability of these systems with time-varying jump rates are obtained via new integral inequalities. (3) A mode-dependent controller/detector, which is subject to the ADT constraint, is designed by introducing extra slack matrices and eliminating the product terms between Lyapunov matrices and system matrices. (4) Based on the proposed method, the stated problem of (1) is solved by the convex
Circuits Syst Signal Process
optimization technique. Finally, the effectiveness of the proposed method is verified by a numerical example. Notation The notation used throughout is quite standard. Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrix. X ≥ Y (respectively, X > Y ) means that X and Y are symmetric matrices and that X − Y is positive semi-definitive (respectively, positive definite). L 2 [0, +∞) is a square integrable function over [0, +∞). · is the Euclidean norm in Rn . I is an identity matrix with dimensions given by the context of use. X + X T is denoted by H e (X ) for simplicity. If A is a matrix, λmax (A) (respectively, the λmin (A)) means largest (respectively, smallest) eigenvalue of A. Moreover, let , F , (Ft )t≥0 , P be a complete probability space with a filtration; then, (Ft )t≥0 satisfies the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). E {·} stands for the mathematical expectation operator with respect to the given probability measure. L 2F0 ([−d2 , 0] : Rn ) denotes the family of all F0 measurable C ([−d2 , 0] : Rn )-valued random variables ϕ = {ϕ (s) : −d2 ≤ s ≤ 0} such that sup−d2 ≤s≤0 E ϕ (s)2 < ∞. The asterisk * in a matrix is used to denote the term that is induced by symmetry. If not explicitly specified, matrices are assumed to have dimensions appropriate to the context of use. Sometimes, when no confusion arises, the arguments of a function are omitted in the analysis.
2 Problem Formulation and Preliminaries Consider a class of singular Markovian jump delay systems (SMJDSs) described by the following model: ⎧ E x˙ (t) = A (rt , st ) x (t) + Ad (rt , st ) x (t − d (t)) + B (rt ) u (t) ⎪ ⎪ ⎪ ⎪ +Bh (rt ) h (t) + B f (rt ) f (t) , ⎨ y (t) = C1 (rt ) x (t) + D (rt ) x (t − d (t)) + Dh (rt ) h (t) + D f (rt ) f (t) , ⎪ ⎪ z (t) = C (rt ) x (t) + Cd (rt ) x (t − d (t)) + C h (rt ) h (t) + C f (rt ) f (t) , ⎪ ⎪ ⎩ x (t) = φ (t) , t ∈ [−d2 , 0],
(1)
where x (t) ∈ Rn is the system state vector, y (t) ∈ Rm the measured output, z (t) ∈ Rr the regulated output, h (t) ∈ Rl the disturbance input, u (t) ∈ Rg the control input, and f (t) ∈ Rv the fault vector. The matrix E ∈ Rn×n may be singular, with rank (E) = r ≤ n assumed. φ (t) is a vector-valued initial continuous function defined on the interval [−d2 , 0], and st is a piecewise-constant switching signal taking values then in S = {1, 2, . . . , s}, s ∈ N. Suppose t0 < t1 < · · · is the switching sequence, the system switches at instants t0 < t1 < · · · , and s (t) = s (tl ) for ∀t ∈ tl , tl+1 ), and l = 0, 1, . . .. {rt , t ≥ 0} is a continuous-time discrete state Markov process with right continuous trajectory values in a finite set K = {1, 2, · · · , k}, k ∈ N with TPs Pr (rt+ = j |rt = i, st = p ) =
( p)
πi j + o () ,
( p) 1 + πii + o () ,
i = j i = j,
(2)
Circuits Syst Signal Process
where > 0,
( p) lim o () = 0, and πi j is the transition rate from
→0+
( p)
mode i at time t to mode j at time t + that satisfies πi j ( p) − kj=1, j=i πi j (∀i, j ∈ K, ∀ p ∈ S).
( p)
> 0, πii
=
Remark 2.1 Based on the definition of the transition rate matrix , one knows that ( p) every element πi j of the matrix is a function of switch mode st = p; thus, this matrix is defined as ⎡ ( p) ⎤ ( p) ( p) π11 π12 · · · π1k ⎢ ( p) ( p) ( p) ⎥ ⎢ π21 π22 · · · π2k ⎥ (3) =⎢ .. .. ⎥ ⎢ .. ⎥. ⎣ . . ··· . ⎦ ( p)
πk1
( p)
πk2
···
( p)
πkk
Note that matrix is time varying and subject to the ADT constraint. Such a time varying transition rate is able to describe a wide variety of real MJDSs such as economic systems and failure-prone systems with less complexity if a single MJDS is considered. For notational conciseness, for each rt = i ∈ K and st = i ∈ S, a matrix A (rt , st ) will be represented by Ai p ; other system’s matrices will be similarly described. Assumption 2.1 For the matrix Bi ∈ Rn×g of full-column rank, there exists a matrix Bi+ such that Bi+ Bi = I . Assumption 2.2 The time-varying delay d (t) is a differentiable function, satisfying for all t ≥ 0, d1 ≤ d (t) ≤ d2 d˙ (t) ≤ d < 1, (4) where d1 , d2 , and d are known positive constants; we set d¯ ≡ d2 − d1 . The aim next is to design a detector(filter)/controller(state feedback), which is described as ⎧ ⎨ E x˙ f (t) = Aˆ i x f (t) + Bˆ i y (t) , (5) rˆ (t) = Cˆ ri x f (t) + Dˆ ri y (t) , ⎩ ˆ u (t) = K i x f (t) , where x f (t) ∈ Rn is the state of the detector/controller, rˆ (t) ∈ Rl2 is the residual signal, the matrices Aˆ i , Bˆ i , Cˆ ri , Dˆ ri , and Kˆ i are the detector/controller gains to be determined. By combining the detector/controller unit (5) and the system (1), an augmented system is established, ⎧ ⎨ E˜ x˙˜ (t) = A˜ i p x˜ (t) + A˜ di p x˜ (t − d (t)) + B˜ hi h (t) + B˜ f i (t) , rˆ (t) = C¯ ri x˜ (t) + D¯ di x˜ (t − d (t)) + D¯ hi h (t) + D¯ f i f (t) , ⎩ z (t) = Ci K x˜ (t) + Cdi K x˜ (t − d (t)) + C hi h (t) + C f i f (t) ,
(6)
where x˜ (t) =
x (t) x f (t)
E˜ =
E 0
0 E
A˜ i p =
Ai p Bˆ i C1i
Bi Kˆ i Aˆ i
A˜ di p =
Adi p Bˆ i Di
0 , 0
Circuits Syst Signal Process
Bhi Bˆ i Dhi K = I 0 B˜ hi =
B˜ f i =
Bfi Bˆ i D f i
D¯ hi = Dˆ ri Dhi
C¯ ri = Dˆ ri C1i
Cˆ ri
D¯ di = Dˆ ri Di
0 ,
D¯ f i = Dˆ ri D f i .
Remark 2.2 Recently, much focus has been given to the problem of fault detection and fault tolerance for the class of either linear or non-linear systems, and most of the existing results basically assume that the fault detection system is open-looped [5,6,15,35]. However, in many practical cases, the fault detection system has feedback control, i.e., the fault detection system is usually closed looped, and if it is designed separately from the control algorithms, faults may be hidden by the control actions and the early detection of faults is clearly more difficult, especially with low frequency faults. Therefore, compared with Refs. [5,6,15,35], the problem pursued below is much more general and closer to practical situations. Remark 2.3 The stated problem is to design a detector/controller unit in the form of (5) such that augmented system (6) is finite-time stability and the following properties should be guaranteed where there exist a disturbance and a fault under zero initial conditions: 1. For H∞ optimization purposes, the effects of the disturbance and fault on the regulated output z (t) are minimized; also, if the effects of disturbance on the residual signal rˆ (t) are minimized, then
T
0 T 0 T
0
z (t) z (t) dt ≤ T
γ12
T
0
0
z T (t) z (t) dt ≤ γ22 rˆ T (t) rˆ (t) dt ≤ γ32
T
T
h T (t) h (t) dt,
(7)
f T (t) f (t) dt,
(8)
h T (t) h (t) dt.
(9)
0
2. For H∞ model matching purposes, if the effects of the fault on signal r f (t) = rˆ (t) − rw (t) is minimized, then
T 0
r Tf (t) r f (t) dt ≤ γ42
T
f T (t) f (t) dt,
(10)
0
where rw (t) = W (t) f (t) supposed to be ⎧ ⎨ E x˙w (t) = Aw xw (t) + Bw f (t) , rw (t) = Cw xw (t) + Dw f (t) , ⎩ xw (0) = 0,
(11)
where Aw , Bw , Cw , and Dw are known constant matrices. Remark 2.4 In Ref. [25], the authors discussed the problem of finite-time control and fault detection for a linear switched systems using the performance indices (7)–(10). However, for the SMJDSs with an ADT constraint, compared with state-space systems
Circuits Syst Signal Process
or non-singular systems, singular systems are well known to preserve the structure of the physical systems more accurately by describing the dynamic part and the static part. Therefore, they provide a more general representation than standard state-space systems in the sense of modeling [37]. This is the main motivation in developing further the simultaneous finite-time control and fault detection problem for the singular fault detection systems. Some definitions specific for our analysis are now given: Definition 2.1 [42] For any switching signal and any k0 < ks < k, let Nδ (ks , k), denote the number of switching signals over the time interval [ks , k). For given N0 > 0, τa > 0, then k − ks , (12) Nδ (ks , k) ≤ N0 + τa where τa is called the ADT and N0 denotes the chatter bound. ˜ A˜ i p ) is said to be: Definition 2.2 [33] system E˜ x˙˜ (t) = A˜ i p x˜ (t) (or pair E, 1. regular if det z E˜ − A˜ i p is not identically zero for any i ∈ K, p ∈ S ; = rank E˜ for any 2. impulse free if it is regular and degree det z E˜ − Ai p i ∈ K, p ∈ S . Definition 2.3 [18] The augmented system (6) is said to be singular stochastic finite˜ T . With 0 < a1 < a2 , G˜ > 0, if the time stable (SSFTS) with respect to a1 , a2 , G, stochastic system is regular and impulse free in time t ∈ [0, T ] and satisfies: sup
−d2 ≤t≤0
E x˜ T (t) G˜ x˜ (t) , x˙˜ T (t) G˜ x˙˜ (t)
≤ a1 ⇒ E x˜ T (t) G˜ x˜ (t) < a2 , t ∈ [0, T ] .
(13)
The following lemmas play an important role in the derivation of our main results. Lemma 2.1 [36] For a differentiable function x : [α, β] → Rn , a positive definite matrix R ∈ Rn×n , a vector ξ ∈ Rk , and any matrices Ni ∈ Rn×n (i = 1, 2), the following inequality holds: −
β α
1 x˙ T (s) R x˙ (s) ds ≤ ξ T (α − β) N1 R −1 N1T + N2 R −1 N2T 3 + H e (N1 E 1 + N2 E 2 ) ξ,
where E 1 ξ = ϒ1 (α, β) = x (β) − x (α), E 2 ξ = ϒ2 (α, β) = x (β) + x (α) −
2 β −α
β α
x (s) ds.
(14)
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Lemma 2.2 For any appropriately dimensioned matrix Z = Z T > 0, Z ∈ Rn×n , M ∈ Rm×n and positive scalars d1 and α, the following inequality holds: −
t
eα(t−s) x˙˜ T (s) E˜ T Z E˜ x˙˜ (s) ds ≤ ξ T (t) εM Z −1 M T ξ (t) + 2ξ T (t) M E˜ x˜ (t) − E˜ x˜ (t − d1 ) ,
t−d1
where ε =
(15)
1 − e−αd1 .
1 α
Proof First, we can find that the following inequality is true
t
t−d1
ξ (t) E˜ x˙˜ (s)
T
α(t−s) 2 α(t−s) 2
e− e
M Z
ξ (t) α(t−s) α(t−s) Z −1 e− 2 M T e 2 Z ds ≥ 0, E˜ x˙˜ (s)
(16)
which implies inequality (15) is satisfied.
Remark 2.5 Lemmas 2.1 and 2.2 represent some novel results in handling the integral term of quadratic quantities in the estimation of the Lyapunov–Krasovskii functional (LKF) derivative. Lemma 2.1 includes more information of the state and the timevarying delay into the augmented vectors ξ (t); Lemma 2.2 contains the exponential information and does not use the approximation −eα(t−s) < −1, t − d2 ≤ s ≤ t − d1 , which provides less precision. Similar inequality techniques are given in Fortuna et al. [10] and used to obtain some satisfactory results; therefore, concerning the issues under study, Lemmas 2.1 and 2.2 help to reduce significantly the conservation of our results. Lemma 2.3 [28] For any constant matrix M > 0, any scalars a and b with a < b, and a vector function x (t) : [a, b] → Rn such that the integrals concerned are well defined, then the following inequality holds: a
b
T x (s) ds
M
b
x (s) ds ≤ (b − a)
a
b
x T (s) M x (s) ds.
(17)
a
3 Design of the Detector/Controller Unit Next, the LMI conditions are presented such that the SSFTS and performance properties (7)–(10) are satisfied simultaneously for augmented system (6) with an ADT constraint. First, we design the detection/controller unit (Theorem 3.1), for which performance index (7) is considered. Theorem 3.1 For any i ∈ K, p ∈ S , α ≥ 0, μ ≥ 1, augmented system (6) with h (t) = 0 is SSFTS with respect to a given (a1 , a2 , G, T ), and performance index (7) is satisfied if there exist positive definite matrices Pi p , Q lp , l = 1, 2, 3, W1 , W¯ 1 , W21 , ˜ and W22 , symmetric matrices H1 , H2 , any real matrices Tk , k = 1, 2, 3, 4, 5, G, S p , R,
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positive scalars η, b1 , γ¯1 such that ⎡
T 0 11 Wi p A˜ di p −T51 E˜ + E˜ T T53 ⎢ ∗ 22 23 24 ⎢ ⎢ ∗ ∗ 34 33 1 = ⎢ ⎢ ∗ ∗ ∗ 44 ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎡ T T ˜ ˜ ˜ 11 Wi p Adi p −T51 E + E T53 0 ⎢ ⎢ ∗ 22 23 24 ⎢ ⎢ ∗ ∗ 34 33 2 = ⎢ ⎢ ⎢ ∗ ∗ ∗ 44 ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Pi p < μPiq Q lp < μQ lq (l = 1, 2, 3) ,
ηa1 eαT − a2 b1 < 0,
W21 H1
H1 W22
>0
W21 H2
⎡
W21 H1 ⎢ H1 W22 H2 >0 ⎢ ⎣ W22 GT
⎤ 16 26 ⎥ ⎥ 36 ⎥ ⎥ < 0, (18) 46 ⎥ ⎥ 56 ⎦ 66 ⎤ 16 ⎥ 0 26 ⎥ ⎥ 0 36 ⎥ ⎥ < 0, (19) ⎥ 0 46 ⎥ ⎥ −γ¯12 I 56 ⎦ ∗ 66 (20)
Wi p B˜ hi 0 0 0 −γ¯12 I ∗ Wi p B˜ hi
⎤
G W21 H2
(21)
⎥ ⎥ > 0, H2 ⎦ W22 (22)
and the ADT satisfying τa > τa∗ =
T ln μ , ln (a2 b1 ) − ln (ηa1 ) − αT
(23)
where 3 ˜ Q lp + d¯ 2 E˜ T W21 E, 11 = H e Wi p A˜ i p + H e T51 E˜ − α E˜ T Pi p E˜ + P¯ + 1 √ √ 16 = d1 A˜ iTp T51 , d¯ A˜ iTp d¯ A˜ iTp 0 K T CiT 22 = − (1 − d) e−αd2 Q 1 p + H e T12 E˜ + T22 E˜ − T32 E˜ + T42 E˜ + d¯ E˜ T (H1 − H2 ) E˜ + 2 E˜ T (W22 − G 2 ) E˜ , T T ˜ + T42 E˜ + E˜ T T43 − E˜ T (W22 − G 2 ) E, 23 = T32 E˜ − E˜ T T33 T T − E˜ T (W22 − G 2 )T E˜ + T24 24 = (−T12 + T22 ) E˜ + E˜ T T14
26 33
T − d¯ E˜ T H E˜ −2T E˜ + E˜ T T T + d¯ E˜ T H E˜ , − 2T22 E˜ + E˜ T T25 1 42 2 46 √ √ √ √ T T ¯ ˜T ¯ 12 ¯ 22 K T C T = d1 A˜ di d¯ A˜ di dT dT p d Adi p di p 0 , = −eαd1 Q 3 p + H e T33 E˜ + T43 E˜ − T53 E˜ + E˜ T W22 + d¯ H2 E˜ ,
Circuits Syst Signal Process
T − d¯ E˜ T H E˜ , 34 = − E˜ T G 2T E˜ 0 −2T43 E˜ + E˜ T T46 2 0 · · · 0 T53 , 36 = ! "# $ ⎡
6
11 44
⎢ ∗ 44 = ⎢ ⎣ ∗
−2T24 E˜ + E˜ TT25 + d¯ E˜ T H1 E˜ −2H e T25 E˜ − d¯ 2 E˜ T W21 E˜
⎤
0 0
−2H e T46 E˜
∗
⎥ ⎥ ⎦,
αd2 ˜ ˜ ˜ = −e Q + H e T E − T E + E˜ T W22 − d¯ H1 E, 11 2 p 24 14 44 ⎤ ⎡ √ √ ¯ 14 ¯ 24 0 0 dT dT 0 0 ⎥ ⎢ √ ⎥ 46 = ⎢ ¯ 25 0 0⎦, 0 0 0 dT ⎣ 0 0 0 √ √ T T T T , 56 = d1 B˜ hi d¯ B˜ hi d¯ B˜ hi 0 0 0 C hi −1 66 = diag −W1−1 −W22 −W1 −3W1 −I −W¯ 1−1 −ε −1 W¯ 1 , √ √ √ √ T ¯ A˜ T ¯ 32 ¯ 42 K T C T ˜T 26 = , d 0 d¯ A˜ di d dT dT A 1 p di p di di p √ √ ¯ 33 ¯ 43 0 0 T53 , 36 = 0 0 dT dT ⎡ 11 ⎤ ¯ E˜ T H1 E˜ 44 −2T24 E˜ + E˜ T T 0 25 + d ⎢ ⎥ ⎥, 0 ∗ −2H e T25 E˜ 44 = ⎢ ⎣ ⎦ 2 T ˜ ¯ ˜ ˜ ∗ ∗ −2H e T46 E − d E W21 E ⎤ ⎡ 0 0 0 ⎥ ⎢0 0 0 46 = ⎣ ⎦, √ ¯ 46 0 0 0 0 0 0 dT E 0 R 0 E T R = 0 R ∈ Rn×r , E˜ = R˜ = Wi p = E˜ T Pi p + S p R˜ T 0 E 0 R k 0 G1 ( p) T T T1T = 0 T12 P¯ = πi j E˜ T P j p E˜ G= 0 T14 0 0 0 , 0 G2 j=1
T2T = 0
T T22
0
T T24
T4T = 0
T T25
0
T T T42 T T 0 0 T46 43 λmin E˜ T P˜i p E˜ b1 = min i∈K, p∈S max λmax E˜ T Pi p E˜
η=
i∈K , p∈S
λmin G˜
0
T3T = 0
0
T T32
T T5T = T51
T T33
T T53
0
Pi p = G˜ 2 E˜ T P˜i p E˜ G˜ 2 , 1
0
1
0
0
0 ,
0
0
0
0 ,
Circuits Syst Signal Process
d2 eαd2 λmax Q 1 p +d2 eαd2 λmax Q 2 p +d1 eαd1 λmax Q 3 p p∈S
p∈S
p∈S
, λmin G˜ ˜ T W1 E˜ ˜ T W¯ 1 E˜ λ λ E E max max 1 1 , + eαd1 d12 + eαd2 d22 − d12 2 2 ˜ λmin G λmin G˜ λmax E˜ T W21 E˜ 1 ¯ αd2 2 d2 − d12 + de 2 λmin G˜ T W E˜ ˜ λ E max 22 1 ¯ αd2 2 d2 − d12 + de . 2 λmin G˜ +
Proof First, we show that system (6) is regular and impulse free. From Eq. (18), 11 < 0, and hence there exist two non-singular matrices, M˜ and N˜ , such that ⎡ (11) A˜ i p 0 I 2r M˜ E˜ N˜ = M˜ A˜ i p N˜ = ⎣ (21) 0 0 A˜ i p 0 S p1 N˜ S p = R˜ = M˜ T ˜ T , S p2 U
(12) A˜ i p (22) A˜ i p
⎤
⎡
⎦
Pi p = ⎣
(11)
Pi p
(21)
Pi p
(12)
Pi p
(22)
Pi p
⎤ ⎦,
where U˜ is any real non-singular matrix. Now, and post-multiplying pre-multiplying 11 < 0 by N˜ T and N˜ , we then find H e S p2 U˜ A˜ i22p < 0 is true, implying A˜ i22p is non-singular matrix. Then, for i ∈ K, p ∈ S , pair E˜ A˜ i p is regular and impulse free. Based on Definition 2.2, augmented system (6) is clearly regular and impulse free for any time-varying d (t) satisfying Eq. (4). Now, augmented system (6) with h (t) ≡ 0 is shown to be SSFTS. First, choosing a stochastic Lyapunov function candidate for system (6) for any i ∈ K, p ∈ S : V1 (x˜t , rt , st ) = x˜ T (t) E˜ T Pi p E˜ x˜ (t) , t α(t−s) T V2 (x˜t , rt , st ) = e x˜ (s) Q 1 p x˜ (s) ds + t−d(t) t
eα(t−s) x˜ T (s) Q 2 p x˜ (s) ds
t−d2
eα(t−s) x˜ T (s) Q 3 p x˜ (s) ds,
+
(25)
t−d1
V3 (x˜t , rt , st ) =
(24) t
0 −d1
+ V4 (x˜t , rt , st ) = d¯
t
eα(t−s) x˙˜ T (s) E˜ T W¯ 1 E˜ x˙˜ (s) dsdθ
t+θ −d1 t
−d2 −d1
−d2
eα(t−s) x˙˜ T (s) E˜ T W1 E˜ x˙˜ (s) dsdθ ,
t+θ t t+θ
eα(t−s)
E˜ x˜ (s) E˜ x˙˜ (s)
T
W21 0 0 W22
(26) E˜ x˜ (s) dsdθ. (27) E˜ x˙˜ (s)
Circuits Syst Signal Process
Then, defining the following weak infinitesimal operator L : 1 E {V ((xt+ , r (t + ) , t + ) |rt = i, st = p ) →0 (28) −V (xt , rt , st )} .
L V (xt , rt , st ) = lim
Hence, we deduce that for each i ∈ K, p ∈ S , L V1 (x˜t , rt , st ) = x˜ T (t) H e Wi p A˜ i p − α E˜ T Pi p E˜ x˜ (t) + 2 x˜ T (t) Wi p A˜ di p x˜ (t − d (t)) + 2 x˜ T (t) Wi p B˜ hi h (t) + x˜ T (t) P¯ x˜ (t) + αV1 (x˜t , rt , st ) , L V2 (x˜t , rt , st ) ≤ αV2 (xt , rt , st ) + x˜ T (t) Q 1 p + Q 2 p + Q 3 p x˜ (t) − eαd1 x˜ T (t − d1 ) Q 3 p x˜ (t − d1 ) − (1 − d) e−αd2 x˜ T (t − d (t)) Q 1 p x˜ (t − d (t)) − eαd2 x˜ T (t − d2 ) Q 2 p x˜ (t − d2 ) , L V3 (xt , rt , st ) ≤ αV3 (xt , rt , st ) + x˙˜ T (t) d¯ E˜ T W1 E˜ + d1 E˜ T W¯ 1 E˜ x˙˜ (t) % t−d2 T % t−d(t) x˙˜ (s) E˜ T W1 E˜ x˙˜ (s) ds − t−d1 x˙˜ T (s) E˜ T W1 E˜ x˙˜ (s) ds − t−d(t) %t eα(t−s) x˜˙ T (s) E˜ T W¯ 1 E˜ x˜˙ (s) ds, − t−d1
Now, from Lemmas 2.1 and 2.2, the following inequalities are true t−d2 x˜˙ T (s) E˜ T W1 E˜ x˙˜ (s) ds − x˙˜ T (s) E˜ T W1 E˜ x˙˜ (s) ds t−d(t) t−d1 1 −1 −1 T T T ≤ ξ (t) (d2 − d (t)) T1 W1 T1 + T2 W1 T2 3 1 ξ (t) + (d (t) − d1 ) T3 W1−1 T3T + T4 W1−1 T4T 3 + ξ T (t) H e T1 E˜ e2T − e4T ξ (t) + ξ T (t) H e T2 E˜ e2T + e4T − 2e5T ξ (t) + ξ T (t) H e T3 E˜ e3T − e2T ξ (t) + ξ T (t) H e T4 E˜ e3T + e2T − 2e6T ξ (t) t − eα(t−s) x˙˜ T (s) E˜ T W¯ 1 E˜ x˙˜ (s) ds ≤ ξ T (t) T5 W¯ 1−1 T5T ξ (t) t−d1 + 2ξ T (t) T5 E˜ x˜ (t) − E˜ x˜ (t − d1 ) ,
−
t−d(t)
where
eiT = 0n×(i−1)n In 0n×(7−i)n , ξ T (t) = x˜ T (t) x˜ T (t − d (t)) x˜ T (t − d1 ) % t−d1 T 1 T (t) , x ˜ ds h (s) d(t)−d t−d(t)
x˜ T (t − d2 )
% t−d(t) T 1 d2 −d(t) t−d2 x˜ (s) ds
1
L V4 (xt , rt , st ) ≤ αV4 (xt , rt , st ) + d¯ 2 x˜ T (t) E˜ T W21 E˜ x˜ (t) + d¯ 2 x˙˜ T (t) E˜ T W22 E˜ x˙˜ (t) ,
Circuits Syst Signal Process
T t−d(t) W21 0 E˜ x˜ (s) E˜ x˜ (s) ds, 0 W22 E˜ x˙˜ (s) E˜ x˙˜ (s) t−d2 T t−d1 W21 0 E˜ x˜ (s) E˜ x˜ (s) ds. −d¯ ˜˙ 0 W22 E˜ x˙˜ (s) t−d(t) E x˜ (s)
−d¯
To enlarge the accessible region covered by the criteria, one has 0 = d¯ x˜ T (t) E˜ T H1 E˜ x˜ (t − d (t)) − x˜ T (t − d2 ) E˜ T H1 E˜ x˜ (t − d2 ) & t−d(t) T T −2 x˜ (s) E˜ H1 E˜ x˙˜ (s) ds ,
(29)
t−d2
0 = d¯ x˜ T (t − d1 ) E˜ T H2 E˜ x˜ (t − d1 ) − x˜ T (t − d (t)) E˜ T H2 E˜ x˜ (t − d (t)) t−d1 (30) x˜ T (s) E˜ T H2 E˜ x˙˜ (s) ds . −2 t−d(t)
Then, −d¯
t−d(t)
t−d2
−2d¯ −d¯
−2d¯
E˜ x˜ (s) E˜ x˙˜ (s)
t−d(t) t−d2
t−d1
t−d(t) t−d1
T
W21 0 0 W22
E˜ x˜ (s) ds E˜ x˙˜ (s)
x˜ T (s) E˜ T H1 E˜ x˙˜ (s) ds E˜ x˜ (s) E˜ x˙˜ (s)
T
W21 0 0 W22
E˜ x˜ (s) ds E˜ x˙˜ (s)
x˜ T (s) E˜ T H2 E˜ x˙˜ (s) ds
t−d(t)
T E˜ x˜ (s) W21 H1 E˜ x˜ (s) ds H1 W22 E˜ x˙˜ (s) E˜ x˙˜ (s) t−d2 T t−d1 E˜ x˜ (s) W21 H2 E˜ x˜ (s) ¯ ds −d ˜˙ H2 W22 E˜ x˙˜ (s) t−d(t) E x˜ (s)
= −d¯
t−d(t)
d¯ ξ T (t) (d2 − d (t)) e5 E˜ T e2 E˜ T − e4 E˜ T ≤− d2 − d (t) ˜ T W21 H1 (d2 − d (t)) Ee 5 ξ (t) × ˜ T ˜ T − Ee H1 W22 Ee 2 4 ¯ d ξ T (t) (d (t) − d1 ) e6 E˜ T e3 E˜ T − e2 E˜ T − d (t) − d1 ˜ T W21 H2 (d (t) − d1 ) Ee 6 ξ (t) × ˜ T ˜ T − Ee H2 W22 Ee 3 2 1 W21 H1 T F2 = ξ (t) −d¯ F1 + (d2 − d (t)) F1T + F2T ξ (t) H1 W22 d2 − d (t)
Circuits Syst Signal Process
1 W21 H2 + ξ T (t) −d¯ F3 + F4 (d (t) − d1 ) F3T + F4T ξ (t) H2 W22 d (t)−d1 H W W21 H1 21 1 F1T − H e d¯ F1 F2T = ξ T (t) −d¯ (d2 − d (t)) F1 H1 W22 H1 W22 d¯ W21 H1 F2T ξ (t) F2 − H1 W22 d2 − d (t) W21 H2 W21 H2 F3T − H e d¯ F3 F4T + ξ T (t) −d¯ (d (t) − d1 ) F3 H2 W22 H2 W22 d¯ W21 H2 F4T ξ (t) , F4 (31) − H2 W22 d (t) − d1 where F1 = e5 E˜ T 0 F2 = 0 e2 E˜ T − e4 E˜ T F3 = e6 E˜ T 0 F4 = 0 e3 E˜ T − e2 E˜ T . Based on the reciprocally convex approach to system stability, the following inequality holds: −
d¯ W21 F2 H1 d2 − d (t) ⎡
≤ − F2
d¯ H1 W21 H2 F2T − F4T F4 W22 H2 W22 d (t) − d1 ⎤ W21 H1 G ⎥ FT ⎢ H1 W22 2 ⎥ ⎢ F4 ⎣ . W21 H2 ⎦ F4T GT H2 W22
(32)
Hence, when h (t) = 0, L (xt , rt , st ) − αV (xt , rt , st ) ≤ ξ (t) (t) + εT5 W¯ 1−1 T5T 1 −1 T −1 T + d2 − d (t) T1 W1 T1 + T2 W1 T2 3 1 ξ (t) + (d (t) − d1 ) T3 W1−1 T3T + T4 W1−1 T4T 3 ¯ 1 + d1 W¯ 1 + d¯ 2 W22 E˜ x˙˜ (t) , + x˙˜ T (t) E˜ T dW T
where ⎡
11 ⎢ ∗ (t) = ⎢ ⎣ ∗ ∗
Wi p A˜ di p 22 ∗ ∗
T −T51 E˜ + E˜ T T53 23 33 ∗
⎤ 0 24 ⎥ ⎥, 34 ⎦ 44
(33)
Circuits Syst Signal Process ⎡ 11 ⎤ ˜ ˜T ˜ ¯ ˜T 0 44 −2T 24 E + E T25 + d E H1 E ⎢ ⎥ ¯ 2 − d (t)) E˜ T W21 E˜ ⎥. ∗ −2H e T25 E˜ − d(d 0 44 = ⎢ ⎣ ⎦ T ∗ ∗ −2H e T46 E˜ − d¯ (d (t) − d1 ) E˜ W21 E˜
Then, by the Schur complement Lemma, Eqs. (18) and (19) guarantee
L V (xt , rt , st ) − αV (xt , rt , st ) < 0.
(34)
Next, we show that augmented (6) is SSFTS with h (t) = 0. First, letting St − = lim Stk +s s→0−
k
from Eq. (20), one has
xtk = xt − k
xtk = p
xt − = q, k
(35)
V xtk , stk < μV xt − , st − .
(36)
E {V (xt , st )} < E eα(t−tk ) V xtk , stk ,
(37)
− E {V (xt , st )} < μE eα t−tk V xt − , st − .
(38)
k
k
Hence, for t ∈ tk , tk+1 )
and based on Eq. (36)
k
k
Following the same argument for the proof of Eq. (38), one obtains − − . E V xt − , st − < μE eα tk −tk−1 V xt − , st − k
k
k−1
k−1
(39)
Then, combining Eqs. (38) and (39), − . E {V (xt , st )} < μ2 E eα t−tk−1 V xt − , st − k−1
k−1
(40)
Hence, ∀t ∈ [t0 , T ] and t0 = 0, ( ' T E {V (xt , st )} < μ Ns (0,t) E eαt V (x0 , s0 ) < μ τa eαT E {V (x0 , s0 )} .
(41)
Alternatively, E {V (xt , st )} ≥ x˜ T (t) E˜ T Pi p E˜ x˜ (t) ≥ b1 E x˜ T (t) G˜ x˜ (t) ,
E {V (x0 , s0 )} ≤ ηa1 .
(42) (43)
Circuits Syst Signal Process
Conditions (41)–(43) imply that μ τTa eαT ηa 1 E x˜ T (t) G˜ x˜ (t) ≤ . b1
(44)
From Eqs. (21) and (23), we find T
μ τa ≤
a2 b1 . eαT ηa1
(45)
Substituting Eqs. (45) into (44), one obtains E x˜ T (t) G˜ x˜ (t) ≤
a2 b1 αT e ηa1 eαT ηa1
b1
= a2 .
(46)
Then, according to Definition 2.3, augmented system (6) is SSFTS. The proof is complete.
Next, system (6) with performance index (7) is shown to be guaranteed if there exist a disturbance under zero initial conditions. Step 1 Choose the same LKF as in Theorem 3.1, after some mathematical manipulation and Schur complement, one has
L V (xt , rt , st ) − αV (xt , rt , st ) + (s) < 0,
(47)
where (s) = z T (s) z (s) − γ¯12 h T (s) h (s) . Step 2 Suppose there is a scalar α > 0, for t ∈ tk , tk+1 ) such that t α(t−tk ) E {V (xt , st )} ≤ E e V xtk , stk − eα(t−s) (s) ds. tk
From Eq. (20), one has − t α t−tk − − {V − E V xt , st eα(t−s) (s) ds (xt , st )} ≤ μE e k k tk − < μ2 E eα t−tk−1 V xt − , st − k−1 k−1 tk t −μ eα(t−s) (s) ds − eα(t−s) (s) ds tk−1
tk
( ' < μ Ns (0,t) E eαt V (x0 , s0 ) − μ Ns (0,t) t2 eα(t−s) (s) ds − μ Ns (t1 ,t) −··· −
t1 t
tk
eα(t−s) (s) ds
0
t1
eα(t−s) (s) ds
(48)
Circuits Syst Signal Process
( ' = μ Ns (0,t) E eαt V (x0 , s0 ) −
t
μ Ns (v,t) eα(t−v) (s) ds
0
≤ μ Ns (0,T ) eαT E {V (x0 , s0 )} −
t
μ Ns (v,t) eα(t−v) (s) ds.
(49)
0
Step 3 Under the zero initial condition, Eq. (48) implies
t
μ Ns (v,t) eα(t−v) (s) ds ≤ 0,
(50)
0
i.e.,
t z T (s) z (s) ds ≤ μ Ns (v,t) eα(t−v) z T (v) z (v) dv 0 0 t t T ≤ γ¯12 μ Ns (v,t) eα(t−v) h T (v) h (v) dv ≤ γ¯12 μ τa eαT h T (s) h (s) ds 0 0 a2 b1 2 t T ≤ γ¯ h (s) h (s) ds, ηa1 1 0 t
2 b1 2 where γ12 = aηa γ¯ . According to the description given in Eq. (7), if Theorem 3.1 holds, 1 1 then augmented system (6) is SSFTS and satisfies the H∞ optimization performance index (7). The proof is complete.
In this section, we direct our attention to designing a detector/controller unit formed in accordance with Eq. (5) and based on the results of Theorem 3.1, which guarantee system (6) is SSFTS with H∞ performance index (7).
Theorem 3.2 For any i ∈ K, p ∈ S , α ≥ 0, μ ≥ 1, any real scalars ε1 and ε2 , augmented system (6) is SSFTS with respect to a given (a1 , a2 , G, T ) and performance index (7) is satisfied if there exist positive definite matrices Pi p , Q lp , l = 1, 2, 3, W1 , W¯ 1 , W21 , W22 , symmetric matrices H1 , H2 , any real matrices Tk , k = 1, 2, 3, 4, 5, G, ˜ W , N Y¯ A , Y¯ B , Y¯ K , and positive scalars b1 , γ¯1 such that Eqs. (20)–(23) are S p , R, satisfied and ⎡
11 ⎢ ∗ ⎢ 1 = ⎢ ⎢ ∗ ⎣ ∗ ∗ ⎡ 11 ⎢ ∗ ⎢ ⎢ 2 = ⎢ ∗ ⎢ ⎣ ∗ ∗ where
12 22 ∗ ∗ ∗
13 23 33 ∗ ∗
14 24 34 44 ∗
12
13
14
22
23
24
∗
33
34
∗ ∗
∗ ∗
44 ∗
⎤ 15 25 ⎥ ⎥ 35 ⎥ ⎥ < 0, 45 ⎦ 55 ⎤ 15 25 ⎥ ⎥ ⎥ < 0, 35 ⎥ ⎥ 45 ⎦ 55
(51)
(52)
Circuits Syst Signal Process
11 =
(11)
11 ∗
(12) 11 (22) , 11
(11) 11 = H e X¯ i Ai p + Y¯ B C1i +
k
( p)
(11)
3
(11)
πi j E T P j p E − α E T Pi p E + v=1
j=1
Q (11) vp
(11) (11) + H e T51 E + d¯ 2 E T W21 E, (12) 11 = Y¯ K + Y¯ A + AiTp X¯ iT + C1i Y¯ BT ,
(22) 11 = H e Y¯ K + Y¯ K +
12
13 14 15
(22) T51 E
k
( p)
(22)
(22)
3
πi j E T P j p E − α E T Pi p E + v=1
j=1
¯2
(22) W21 E,
+ He +d E ⎡ (11) (11) E T Pi p + S P R T − X¯ i T Y¯ T −Yi + ε2 AiTp X¯ iT + C1i ⎢ +ε A T X¯ T + C T Y¯ T B ⎢ 1 ip i 1i B =⎢ (22) (22) T T ⎣ E Pi p + S P R − X¯ i + ε1 Y¯ KT + Y¯ AT −Yi + ε2 Y¯ KT + Y¯ AT ⎡ T (11) T T (11) E + E 0 −T 51 ⎢ 51 =⎣ (22) (22) T 0 −T51 E + E T T51 0 0 0 0 X¯ i Bhi + Y¯ B Dhi , = 0 0 0 0 X¯ i Bhi + Y¯ B Dhi (11) 0 0 0 0 CiT T51 0 = (22) , 0 0 0 0 0 0 T51 T
Q (22) vp
⎤ ¯ ¯ X i Adi p + Y B Di 0 ⎥ ⎥ ⎥, ⎦ ¯ ¯ X i Adi p + Y B Di 0 ⎤ 0 0
0⎥ ⎦, 0
⎤ (11) H e −ε1 X¯ i + d¯ W1 T ¯ ¯ ¯ −Yi − ε2 X i ε1 X i Adi p + Y B Di 0 ⎥ ⎢ +d¯ 2 W (11) + d W¯ (11) 1 1 ⎢ ⎥ 22 ⎢ ⎥ (22) H e (−Yi ) + d¯ W1 ⎢ ⎥ =⎢ ε2 X¯ i Adi p + Y¯ B Di 0 ⎥, ∗ (22) (22) 2 ¯ W ¯ ⎢ ⎥ + d + d W 1 1 22 ⎢ ⎥ (33) ⎣ ∗ ∗ 22 0 ⎦ (44) ∗ ∗ 0 22 ⎡
22
(33)
(11)
(44)
(22)
22 = − (1 − d) e−αd2 Q 1 p (11) (11) (11) (11) (11) + H e T12 E + T22 E − T32 E + T42 E − E T G 2 E (11) (11) (11) E + 2E T W22 E, + d¯ E T H1 − H2 22 = − (1 − d) e−αd2 Q 1 p (22) (22) (22) (22) (22) + H e T12 E + T22 E − T32 E + T42 E − E T G 2 E (22) (22) (22) E + 2E T W22 E, + d¯ E T H1 − H2
Circuits Syst Signal Process
⎡ 33
(11)
33
⎢ ⎢ ⎢ =⎢ 0 ⎢ ⎣ ∗ ∗
⎤
(11) T −E T G 2 E
0 (22) 33
−E T
0
(33) 33
∗ ∗
0
(22) T G2
0 (44) 33
∗
⎥ ⎥ E⎥ ⎥, ⎥ ⎦
(11) (11) (11) (11) (11) 33 = −eαd1 Q 3 p + H e T33 E + T43 E − T53 E (11) (11) E, + E T d¯ H2 + W22 (22) (22) (22) (22) (22) 33 = −eαd1 Q 3 p + H e T33 E + T43 E − T53 E (22) (22) E, + E T d¯ H2 + W22 (33) (11) (11) (11) (11) (11) 33 = −eαd2 Q 2 p + H e T24 E − T14 E + E T −d¯ H1 + W22 E, (44) (22) (22) (22) (22) (22) 33 = −eαd2 Q 2 p + H e T24 E − T14 E + E T −d¯ H1 + W22 E, (44) (22) (22) (22) (22) (22) E, 33 = −eαd2 Q 2 p + H e T24 E − T14 E + E T −d¯ H1 + W22 ⎡
23
⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(11) + T32
+E T
(11) T42
0 0
(11) T T43
0 0 E −
(11) T T33
(11) +E T G (11) E − W22 2
0
0 (42)
23
(11) T22 −
(11) T12
0 0
0 0 E
(11) T + T24 (11) +E T G (11) E − W22 2
+E T
(11) T T14
T
0
(22) (22) (22) T (22) − T33 E + ET (42) T43 23 = T32 + T42 (22) + E T G (22) E, − W22 2 (22) (22) (22) T (22) T T (44) T E + E = T − T + T 23 22 12 14 24 (11) (11) T E, + E G 2 − W22 ⎡ 0 0 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 0 0 ⎢ ⎢ ⎢ (11) (11) −2T42 E ⎢ −2T22 E T ⎢ (11) (11) T ⎢ T 24 = ⎢ +E T 0 0 +E T T46 25 ⎢ (11) T ⎢ −d¯ E T H (11) E ¯ −d E H2 E ⎢ 1 ⎢ (22) (22) −2T22 −2T42 E E ⎢ T T ⎢ (22) ⎢ T T T T (22) 0 0 +E +E ⎣ 46 25 (22) (22) −d¯ E T H1 E −d¯ E T H2 E
0 (44)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦
23
ε1 X¯ i Bhi +Y¯ B Dhi ε2 X¯ i Bhi +Y¯ B Dhi 0
0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Circuits Syst Signal Process
⎡
25
35
0 ⎢ √ 0 (11) =⎢ ⎣ dT ¯ 12 0 ⎡ 0 ⎢ ⎢√ 0 =⎢ ¯ (11) ⎣ dT 14 0
⎡
45
34
0 ⎢0 ⎢ =⎢ ⎢0 ⎣0 0 ⎡
0 0 0 0 0
√
0 0 0 √ (22) ¯ dT 12
0 √ 0 (11) ¯ dT 22 0
0 0 √ 0 (22) ¯ dT 22
0 0 T Cdi 0
0 0 0 √ (22) ¯ dT 14
0 √ 0 (11) ¯ dT 24 0
0 0 √ 0 (22) ¯ dT 24
0 0 0 0
¯ (11) dT 25 0 0 0 0
√ 0 (22) ¯ dT 25 0 0 0
0 0 0 0 T C hi
⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ =⎢ ⎢ −2T (11) E ⎢ 24 T ⎢ ⎢ +E T T (11) 0 ⎢ 25 ⎢ (11) ⎢ +d¯ E T H E 1 ⎢ (22) ⎢ −2T24 E ⎢ ⎢ (22) T ⎢ 0 +E T T25 ⎣ (22) +d¯ E T H1 E
0 0 0 0
⎤ 0 0⎥ ⎥, 0⎦ 0
(11) T53 0 0 0
0
⎤
(22) ⎥ T53 ⎥ ⎥, 0 ⎦ 0
⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎦
0 0 0 0 0
0
(11) −2T43 E (11) T +E T T46 (11) −d¯ E T H2 E
0
0
0
⎤ ⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ (22) ⎥ −2T43 E T ⎥ ⎥ (22) 0⎥ +E T T46 ⎥ ⎥ (22) T ⎥ ¯ −d E H2 E ⎥, ⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 0⎥ ⎦ 0
(11) E H e −2T25 0 0 0 ⎢ ⎢ −d¯ 2 E T W (11) E 21 ⎢ ⎢ (22) H e −2T25 E ⎢ ⎢ 0 0 0 (22) ⎢ 2 T =⎢ −d¯ E W21 E ⎢ (11) ⎢ 0 ∗ ∗ H e −2T46 E ⎢ ⎢ (22) ⎣ ∗ ∗ 0 H e −2T46 E ⎡
44
∗
∗
∗
∗
⎤ 0 0 0 0 −γ¯12 I
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Circuits Syst Signal Process
⎡
55
⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
25
(11)
−W1 0 ∗ ∗ ∗ ∗ ∗
⎤ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥, 0 0 0 ⎥ ⎥ −I 0 0 ⎥ ⎥ (11) −1 0 ∗ −ε W¯ 1 ⎦ (22) −1 ∗ ∗ −ε W¯ 1 ⎤ 0 0 0 0 0 0 0 0⎥ ⎥, T 0 C 0 0⎦ di √ (22) ¯ dT 0 0 0
0 0 0 0 0 −W1(22) ∗ −3W1(11) 0 (22) ∗ 0 −3W1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 ⎢ 0 √ ⎢ = ⎣ ¯ (11) dT 32
0
0 0 0 √ (22) ¯ dT 32
0 √ 0 (11) ¯ dT 42
0
42
⎤ ⎡ √ (11) √ (11) (11) ¯ ¯ dT dT 0 0 0 T51 0 33 43 √ √ ⎢ (22) ⎥ ¯ (22) ¯ (22) 0 ⎢ dT dT 0 0 0 T51 ⎥ 33 43 35 = ⎢ ⎥, ⎣ 0 0 0 0 0 0 0 ⎦ 0 0 0 0 0 0 0 ⎤ ⎡ 0 0 0 0 0 0 0 ⎢0 0 0 0 0 0 0⎥ ⎥ ⎢ √ ⎥ ⎢ ¯ (11) dT 0 0 0 0⎥, 45 = ⎢ 0 0 46 √ (22) ⎥ ⎢ ¯ ⎣0 0 0 dT 0 0 0⎦ 46 T 0 0 0 0 C hi 0 0 ⎡ (11) H e −2T25 E 0 0 0 ⎢ ⎢ (22) E 0 0 0 H e −2T ⎢ 25 ⎢ ⎢ (11) H e −2T46 E ⎢ ⎢ ∗ ∗ 0 44 = ⎢ (11) −d¯ 2 E T W21 E ⎢ ⎢ (22) ⎢ H e −2T46 E ⎢ ∗ ∗ 0 ⎢ (22) ⎣ −d¯ 2 E T W21 E ∗ ∗ ∗ ∗
X¯ Y ε X¯ Y W T = ¯i i N T = 1 ¯i i ε2 X i Yi X i Yi (11) (11) 0 0 Pi j Ti j Ti j = Pi j = (22) (22) , 0 Pi j 0 Ti j (11) ¯ (11) 0 0 W1 W 1 W1 = W¯ 1 = (11) (11) 0 W1 0 W¯ 1 W22 =
(11)
W22 0
0
(11)
W22
SP =
(11)
Sp 0
W21 =
0
(22)
Sp
Q lp =
(11)
W21 0
(11)
Q lp 0
⎤ 0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0 0 0 −γ¯12 I
0
,
(11)
W21
0 (11)
Q lp
.
Circuits Syst Signal Process
Also, the detector/controller unit gain matrices are given by Aˆ i = Yi−1 Y¯ A
Bˆ i = Yi−1 Y¯ B
Kˆ i = Bi+ X¯ i−1 Y¯ K ,
(53)
Proof Consider the matrix i p =
I 010n×n
A˜ iTp b2
0n×10n , I10n×10n
(54)
T where b2 = A˜ di p 0n×4n B˜ hi 0n×4n . Pre-multiplying and post-multiplying (51) and (52) by the matrix i p and its transposition, respectively, we find that for ∀i ∈ K, p ∈ S such that (18) and (19) hold. That is, (51) and (52) are equivalent to (18) and (19). The proof is complete.
Remark 3.1 In this paper, we selected the variable diagonal matrices and such processing may affect the conservatism of the system. Nevertheless, this choice greatly reduces the complexity in computations and the cost of system implementation. Remark 3.2 Also, Bi is assumed to be of full-column rank; its left inverse matrix Bi+ exists satisfying Bi+ Bi = I . Now, it can be shown that the equation X¯ i Bi Kˆ i = Y¯ K is solvable for known matrix Bi . rank matrix; hence, there exist a singular value decompoFirst, Bi is afull-column ϒi sition Bi = Ui Vi H , where Ui ∈ Rn×n and Vi ∈ Rg×g are unitary matrices, and 0 ϒi ∈ Rg×g is a diagonal matrix with positive diagonal elements. In Theorem 3.2, it is assumed that X¯ i = ϑ I is a scalar matrix and the matrix Y¯ K has the form Ui . Hence, Kˆ i = Bi+ X¯ i−1 Y¯ K = ϑ −1 Vi ϒi−1 . Then, one has that X¯ i Bi Kˆ i = 0 ϒi ϑUi ViT ϑ −1 Vi ϒi−1 = Ui = Y¯ K . Hence, the equation X¯ i Bi Kˆ i = Y¯ K is 0 0 solvable if Bi is of full-column rank. Theorem 3.3 For any i ∈ K, p ∈ S , α ≥ 0, μ ≥ 1, any real scalars ε1 and ε2 , augmented system (6) is SSFTS with respect to a given (a1 , a2 , G, T ); moreover, performance index (8) is satisfied if there exist positive definite matrices Pi p , Q lp , l = 1, 2, 3, W1 , W¯ 1 , W21 , W22 , symmetric matrices H1 , H2 , any real matrices Tk , k = 1, 2, 3, 4, 5, ˜ W , N Y¯ A , Y¯ B , Y¯ K , and positive scalars b1 , γ¯2 such that Eqs. (20)–(23) are G, S p , R, satisfied and ⎡
11
⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
12
13
3.3 14
22 ∗ ∗ ∗
23 33 ∗ ∗
3.3 24 34 3.3 44 ∗
15
⎤
⎥ 25 ⎥ ⎥ <0 35 ⎥ ⎥ ⎦ 45 55
⎡
11
⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗
12
13
3.3 14
22
23
3.3 24
∗
33
34
∗ ∗
∗ ∗
44 ∗
3.3
15
⎤
⎥ 25 ⎥ ⎥ 35 ⎥ ⎥ < 0, ⎥ 45 ⎦ 55 (55)
Circuits Syst Signal Process
where ) ) 3.3 14 = 14 Bhi → B f i : B f i replaces Bhi in the 14 , ) ) 3.3 24 = 24 Bhi → B f i , ) 2 ) 2.3 2 ) 3.3 44 = 44 ) γ¯12 → γ¯22 , 44 = 44 γ¯1 → γ¯2 Also, the matrices Aˆ i , Bˆ i and Kˆ i can be obtained from Eq. (53). Proof Following the proof given in Theorems 3.1 and 3.2, Eq. (8) is found to be true; hence, the details are omitted here.
Theorem 3.4 For any i ∈ K, p ∈ S , α ≥ 0, μ ≥ 1, any real scalars ε1 and ε2 , augmented system (6) is SSFTS with respect to a given (a1 , a2 , G, T ), and performance index (9) is satisfied if there exist positive definite matrices Pi p , Q lp , l = 1, 2, 3, W1 , W¯ 1 , W21 , W22 symmetric matrices H1 , H2 , any real matrices Tk , k = 1, 2, 3, 4, 5, G, S p , ˜ W , N , Y¯ A , Y¯ B , Y¯C , Y¯ D , Y¯ K , Y¯ A , and positive scalars b1 , γ¯3 such that Eqs. (20)–(23) R, are satisfied and ⎡
11
⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
12 22 ∗ ∗ ∗
13 23 33 ∗ ∗
⎡
⎤
14
3.4 15
24 34 3.4 44 ∗
⎥ 3.4 25 ⎥ 35 ⎥ ⎥ ⎦ 3.4 45 55
⎥ <0
11
⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗
12
13
14
22
23
24
∗
33
34
∗ ∗
∗ ∗
44 ∗
3.4
3.4 15
⎤
⎥ 3.4 ⎥ 25 ⎥ ⎥ < 0, 35 ⎥ ⎥ 3.4 ⎥ 45 ⎦ 55 (56)
where 3.4 15 = 3.4
25 = 3.4
44 = 3.4 45 =
)* T T & ) T T T ) C1i Y¯ D ) C Di Y¯ D ) CiT 3.4 di ) 25 = 25 ) 15 ) → → , T 0 0 0 ) ¯ YC ) T T T ) C ) ¯ di → Di Y D 3.4 25 )) = 44 ) γ¯12 → γ¯32 44 0 0 ) 2 44 ) γ¯1 → γ¯32 , ) ) 3.4 ) T ) T T ¯T T ¯T 45 ) C hi → Dhi 45 = 45 ) C hi → Dhi YD YD .
Also, the gain matrices of the detector/controller unit are given by: Aˆ i = Yi−1 Y¯ A
Bˆ i = Yi−1 Y¯ B
Kˆ i = Bi+ X¯ i−1 Y¯ K
Cˆ ri = Y¯C
Dˆ ri = Y¯ D . (57)
Proof Following the proof given in Theorems 3.1 and 3.2, one finds that Eq. (9) holds; hence, the details are omitted here. Theorem 3.5 For any i ∈ K, p ∈ S , α ≥ 0, μ ≥ 1, any real scalars ε1 and ε2 , augmented system (6) is SSFTS with respect to a given (a1 , a2 , G, T ), and performance
Circuits Syst Signal Process
index (10) is satisfied if there exist positive definite matrices Pi p , Q lp , l = 1, 2, 3, W1 , W¯ 1 , W21 , W22 , Pw , symmetric matrices H1 , H2 , any real matrices Tk , k = 1, 2, 3, 4, 5, ˜ W , N , Y¯ A , Y¯ B , Y¯C , Y¯ D , Y¯ K , Y¯ A , and positive scalars b1 , γ¯4 such that G, S p , Sw R, Eqs. (20)–(23) are satisfied and ⎡
11
⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎣ ∗
12
13
3.5 14
22
23
3.5 24
∗
33
3.5 34
∗
∗
3.5 44
∗
∗
∗
3.5 15
⎤
⎥ ⎥ 3.5 25 ⎥ ⎥ ⎥ 3.5 35 ⎥ < 0 ⎥ ⎥ 3.5 45 ⎦ 55
⎡
11
⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗
12
13
3.5 14
22
23
3.5 24
∗
33
3.5 34
∗ ∗
∗ ∗
44 ∗
3.5
3.5 15
⎤
3.5 ⎥ 25 ⎥ ⎥ 3.5 ⎥ < 0, 35 ⎥ ⎥ 3.5 ⎥ 45 ⎦ 55
(58) where 3.5 14 = 0 ⎡
0
0
0
W T B˜ f i
3.5 15 = 0
0
T51 , ⎤ N T B˜ f i ⎥ ⎦, 0
T C¯ ri
0 0 0 0 ⎢ E˜ T T T − 2T22 E˜ ˜ T T T − 2T42 E˜ 3.5 = E 25 ⎣ 24 46 0 0 d¯ E˜ T H2 E˜ −d¯ E˜ T H1 E˜ 3.5 0 0 0 0 0 0 0 0 3.5 √ √ √ √ 25 = 25 = ¯ 12 ¯ 22 D¯ T 0 ¯ 32 ¯ 42 D¯ T 0 , dT dT dT dT di di ⎡ ⎤ T − 2T E˜ E˜ T T46 43 0 0 0 0⎥ ⎢ T H E˜ ⎢ ⎥ ¯ ˜ 3.5 − d E 2 34 = ⎢ ⎥, ⎣ E˜ T T T − 2T E˜ ⎦ 24 25 0 0 0 0 d¯ E˜ T H1 E˜ √ √ 3.5 ¯ 33 ¯ 43 0 T53 0 T53 dT dT √0 √0 , = = 3.5 35 35 ¯ ¯ dT14 dT24 0 0 0 0 0 0 ⎡ ⎤ H e −2T25 E˜ − d¯ 2 E˜ T W21 E˜ 0 0 0 0 ⎢ ⎥ ⎢ ⎥ 0 0 0 ∗ H e −2T46 E˜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 3.5 H e E Pw Aw 44 = ⎢ T ST E T P B ⎥ , ∗ ∗ Aw ⎢ w w⎥ w T ⎢ ⎥ −α E Pw E ⎢ ⎥ ⎣ ∗ ∗ ∗ H e (Sw ) −Sw Bw ⎦ 2 ∗ ∗ ∗ ∗ −γ¯4 I ⎤ ⎡ H e −2T25 E˜ 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 ∗ H e −2T46 E˜ − d¯ 2 E˜ T W21 E˜ ⎥ ⎢ ⎥ ⎢ 3.5 ⎥ ⎢ TP A H e E , 44 = ⎢ w w T T T ∗ ∗ −Aw Sw E Pw Bw ⎥ ⎥ ⎢ T ⎥ ⎢ −α E Pw E ⎥ ⎢ ⎣ ∗ ∗ ∗ H e (Sw ) −Sw Bw ⎦ ∗ ∗ ∗ ∗ −γ¯42 I √ ⎡ ⎡ ⎤ ⎤ ¯ 25 0 √0 0 0 0 dT 0 0 ⎢0 ⎢0 ⎥ ¯ 46 dT 0 0 0⎥ 0 0 ⎢ ⎢ ⎥ ⎥ 3.5 ⎢ ⎢ ⎥ ⎥ 3.5 T T 0 −Cw 0 −Cw 0⎥ 0⎥. 45 = ⎢ 0 45 = ⎢ 0 ⎢ ⎢ ⎥ ⎥ 0 0 0⎦ 0 0 0⎦ ⎣0 ⎣0 T T T T ¯ ¯ 0 0 D f i − Dw 0 0 0 D f i − Dw 0
Circuits Syst Signal Process
Also, the matrices Aˆ i , Bˆ i , Kˆ i , Cˆ ri and Dˆ ri can be obtained from Eq. (57). Proof Now, we choose a stochastic Lyapunov function candidate for system (6) for any i ∈ K, p ∈ S : V¯ (xt , rt , st ) =
E xw (t) E x˙w (t)
T
0 Sw
Pw 0
I 0
0 0
E xw (t) + V (xt , rt , st ) , (59) E x˙w (t)
then, L V¯ (xt , rt , st ) = L V (xt , rt , st ) + xwT (t) H e E T Pw Aw − α E T Pw E xw (t) + 2xwT (t) E T Pw Bw f (t) T T Sw E x˙w (t) + x˙wT (t) E T (H e (Sw )) E x˙w (t) − 2xwT (t) Aw
− 2 x˙wT (t) E T Sw Bw f (t) + αxwT (t) E T Pw E xw (t) .
(60)
Next, consider the matrix ¯ ip =
I 012n×n
A˜ iTp b3
0n×12n , 012n×12n
(61)
T where b3 = A˜ di p 0n×6n B˜ Tfi 0n×4n . ¯ i p and its transposition, Pre-multiplying and post-multiplying Eq. (58) by the matrix respectively, and similar to the proof given in Theorems 3.1 and 3.2, then Eq. (10) is true; hence, the details are omitted here.
Remark 3.3 From the proof of Theorems 3.2 and 3.5, it can be seen that matrices (54) and (61) play a very important role in our discussion. As is well known, if we want to obtain the optima gain matrices for the detector/controller unit, then how the non-linear terms are divided is crucial. This is why matrices (54) and (61) were introduced as they make our proof easier. Remark 3.4 Note that the likelihood to obtain a complete knowledge of the transition probabilities is difficult and the cost is probably high. Hence, to describe more accurately the practical problem, we need to further study more general Markovian jump systems with incomplete transition descriptions. Based on this consideration, the idea and method presented so far can be extended to cope with singular Markovian jump systems with partly unknown transition rates. Theorem 3.6 Under any switch signal, augmented system (6) with dwell time constraint (23) is SSFTS and also satisfies (7)–(10) if the LMI conditions (18), (19), (51), (52), (55), (56), and (58) hold. Moreover, matrices Aˆ i , Bˆ i , Kˆ i , Cˆ ri , and Dˆ ri can be obtained from Eqs. (53) and (57). Further, based on Theorems 3.1–3.5, the simultaneous finite-time control and fault detection problem is converted into the following optimization: for given positive constant weighs i , i = 1, 2, 3, 4 min 1 γ¯12 + 2 γ¯22 + 3 γ¯32 + 4 γ¯42 St. (18) − (23) , (51) − (52) , (55) − (56) , (58) .
(62)
Circuits Syst Signal Process
The next step is to evaluate the residual signal and compare it with some threshold values to detect the fault in the system. For this purpose, the residual evaluation function based on the root-mean-square energy of the residual signal is used, + Jrˆ (t) =
1 T
T
rˆ T (s) rˆ (s) ds,
(63)
0
and the threshold Jth is obtain by Jth = sup
f (t)=0 h(t)∈L 2
Jrˆ (t) ;
(64)
therefore, the controller/detector unit is obtained such that performance index (9) holds 2 √ . Finally, the and the constant threshold value Jth can be defined as Jth = γ3 h(t) T occurrence of fault can be detected by the following logic rule: Jrˆ (t) > Jth ⇒ alar m, Jrˆ (t) ≤ Jth ⇒ no faults.
(65)
4 Examples An illustrative example is now given to demonstrate the effectiveness of the proposed method. First, we consider the transition rate matrix with two vertices and singular matrix E: −0.6 0.6 −1.3 1.3 1 0 (1) = (2) = E= , 0.8 −0.8 1.5 −1.5 0 0 Second, introducing the following parameters into augmented system (6),
A11
=
A21
=
Ad11
=
Ad21
=
B1 = C11 = C1 =
−0.005 −0.721 1.08 −1.05 1.8 −0.24 A22 = , 0.22 −0.22 0.5 0.32 −0.2 −0.15 −0.13 0.09 Ad12 = 0.3 0.52 0.18 0.135 0.08 1.15 1.8 0.04 Ad22 = , 0.022 −0.92 0.15 −0.3 0.1 1 Bh1 = B f 1 = B2 = Bh2 = B f 2 = 0.1 0 C12 = 0.2 0.3 D1 = D2 = 0.1 0 , C2 = 1.2 −0.3 Cd1 = Cd2 = 0.1 0.5 1.88 3
−0.05 −0.72
A12 =
Dh1 = D f 1 = C h1 = C f 1 = 0.2,
1.79 1
Circuits Syst Signal Process
Dh2 = D f 2 = C h2 = C f 2 = 0.03
R=
0 . 1
The matrices of the system for model (11) are chosen as
0.1 Aw = 0
0 0.2
1 Bw = 0
Cw = 2
0
Dw = 0.1.
Setting the slack scalar ε1 = 0.5, ε2 = 0.8 and the time-varying delay d (t) = 0.1 + 0.1 sin (10t), then, one easily finds that d1 = 0.1, d2 = 0.2. Letting a1 = 1, a2 = 5, T = 30, α = 0.2, μ = 1.5, and 1 = 2 = 3 = 4 = 1, then using Theorem 3.6 yields γ¯1 = 0.6502, γ¯2 = 0.6710, γ¯3 = 0.7193, and γ¯4 = 1.1591. Moreover, the controller and detector units are obtained as follows:
Aˆ 1 Bˆ 1 Cˆ r 1 Kˆ 1
Aˆ 2 Bˆ 2 Cˆ r 2 Kˆ 2
⎤ 0.5612 −2.7609 6.3354 ⎦, −4.0942 Dˆ r 1 = ⎣ −1.0002 −0.3863 2.3090 0.0922 0.1779 3.0523 5.5174 ⎡ ⎤ 2.3099 −0.8701 4.5217 ⎦. −2.0017 Dˆ r 2 = ⎣ −5.7788 −0.0927 1.0351 0.2213 0.0769 1.3378 5.4968
⎡
For the purpose of simulation, we set , f (t) =
1, 0,
10 ≤ t ≤ 20 otherwise.
T The initial state and initial modes are taken as x˜0 = 1.5 0 , r0 = 1, s0 = 2, respectively. The simulation time is taken as 30 time units, and each unit length is taken as T=5 s. The path of the switching modes is chosen according to the ADT τa > 2.7481 constraint, and the path of the jumping modes is from time step 0 to time step 30, both paths are shown in Figs. 1 and 2, respectively. Further, the residual evaluation function, shown in Figs. 3 and 4, indicates a fault is detected. From Fig. 3, one finds that if we enlarge the parameters D f 1 and D f 2 , the effect of the fault on the output y (t) is larger, and hence the residual signal rˆ (t) becomes larger,and the detection time of fault is % 1 30 T reduced. Based on our results, then Jth = sup E 30 0 rˆ (t) rˆ (t) dt = 0.4781, h(t) =0 f (t)=0
% 10.05 T 1 r ˆ dt = 0.6207 > Jth , i.e., the fault and Fig. 4 shows us that 10.05 r ˆ (t) (t) 0 signal is detected after 0.05s. The regulated output z (t) (Fig. 5) indicates that system (6) provides a good robust performance. Finally, the phase plane plots of the closedloop system (Figs. 6, 7) show that the states of system (6) are stabilized in a finite-time interval with our proposed control strategy.
Remark 4.1 Now, if we select similar parameter settings as given in Ref. [19], and the fault signal f (t) is simulated as a square of unit amplitude occurring from 8 to 12 s, then, by a simple calculation, one finds that the threshold Jth =
Circuits Syst Signal Process 3 2.5 2 1.5 1 0.5 0
0
5
10
15
20
25
30
Time
Fig. 1 System switch signal st
2
1
5
10
15
20
25
30
times[s]
Fig. 2 System jumping mode rt 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0
5
10
15
times[s]
Fig. 3 Residual signal rˆ (t)
20
25
30
Circuits Syst Signal Process 2
fault case fault-free case
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
5
10
15
20
times[s]
25
30
Fig. 4 Residual evaluation function 0.4 0.3 0.2 0.1 0 −0.1 −0.2
0
2
4
6
8
10
times[s]
12
14
16
18
20
Fig. 5 Regulated output z(t) 0.5
1
X2
X1
0.5
0
0 −0.5
−0.5 −1
0
5
10
15
20
25
30
−1
0
5
10
k/sec
20
25
2
0.5
Xf1
30
Xf2
1
0
0
−0.5 −1
15
k/sec
−1 −2 0
5
10
15
20
25
30
k/sec
Fig. 6 System state responses with input control
0
5
10
15
k/sec
20
25
30
Circuits Syst Signal Process 5 4
a2
3
a1
2 1 0 −1 −2 −3 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Fig. 7 Phase plane plot of closed-loop system
sup E
h(t) =0 f (t)=0
% 1 30 T 30 0 rˆ
% 8.75 T 1 r ˆ dt = 0.4655 > Jth , r ˆ (t) rˆ (t) dt = 0.3416, 8.75 (t) (t) 0
i.e., the fault signal is detected 0.75 s later, but in the Ref. [19], the fault signal was detected 0.9 s later. Obviously, the present system has better sensitivity to fault signals than that given in Ref. [19]. Therefore, our results give less conservatism than Ref. [19].
5 Conclusions The problem of simultaneous finite-time control and fault detection for a class of singular Markovian jump systems with an ADT constraint was investigated. A mode-dependent detector/controller unit was designed that guarantee that the closed-loop system is finitetime stable and satisfies four H∞ performance indices. By using some novel integral inequalities and optimization techniques, the results were derived in terms of the LMIs. Finally, a numerical example was provided to illustrate the effectiveness of the proposed method. For further study in this direction, this fault detection unit and the novel techniques may be applied to deal with many more challenging issues, e.g., event-based control and fault detection for a class of singular jump systems under asynchronous switching. Acknowledgements This paper is supported by the China Scholarship Council (201608455012). All authors thank the editors and the anonymous referees for their constructive comments and valuable suggestions, which were helpful in improving the quality of this paper. Compliance with ethical standards Conflict of interest All the authors declare that there is no conflict of interest regarding the publication of this paper.
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