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Robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems subject to actuator saturation
Accepted Manuscript
Robust finite-time non-fragile memory H control for discrete-time singular Markovian jumping systems subject to actuator saturation Yuechao Ma, Xiaorui Jia, Qingling Zhang PII: DOI: Reference:
S0016-0032(17)30540-9 10.1016/j.jfranklin.2017.10.019 FI 3192
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
4 August 2016 10 October 2017 17 October 2017
Please cite this article as: Yuechao Ma, Xiaorui Jia, Qingling Zhang, Robust finite-time non-fragile memory H control for discrete-time singular Markovian jumping systems subject to actuator saturation, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.10.019
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ACCEPTED MANUSCRIPT Highlights
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• A new control problem for discrete singular Markov system is investigated. • Lyapunov function method, LMI technique and convex optimization are used. • The non-fragile memory controller is designed. • Both actuator saturation and time-varying delay are considered. • The derived conditions are less conservative have wider application range.
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Robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems subject to actuator saturation
Yuechao Ma a , Xiaorui Jia a,∗ , Qingling Zhang b
of Science, Yanshan University, Qinhuangdao Hebei, 066004, P.R.China;
b College
of Science, Northeatern University, Shenyang Liaoning, 110004, P.R.China;
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a College
Abstract
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This paper considers the problem of robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems with actuator saturation. By using the multiple Lyapunov function approach and the linear matrix inequalities (LMIs) techniques, novel sufficient conditions are obtained that ensure the systems singular stochastic finite-time bounded and singular stochastic finitetime H∞ bounded are obtained. Then the non-fragile memory feedback controllers are designed and the results are extended to LMI convex optimization problems to get the optimal values of attenuation level and the estimation of saturation attraction domain. Finally, some numerical examples are provided to illustrate the viability and effectiveness of the proposed methods.
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Key words: Finite-time stability and boundedness; discrete-time singular Markovian jumping system; non-fragile memory control; actuator saturation.
? This work is partially supported by the National Natural Science Foundation of China, under Grant number 61273004, and the Natural Science Foundation of Hebei province No. F2014203085. ∗ Corresponding author. Email address: [email protected] (Xiaorui Jia).
Preprint submitted to Elsevier
28 October 2017
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Introduction
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Singular systems, which are also referred to as descriptor systems, differentialalgebraic systems, or generalized state-space systems, have been extensively studied over the past decades due to the fact that these systems have been widely applied in many scientific areas. For example, the circuits systems, chemical processes and many other aspects. During recent years, many results about singular systems have been researched. For example, stabilization and stability analysis [1,2], filtering problems [3,4], observer-design [5], control problems [6] and so forth. When systems experience random abrupt changes in their structures parameters, it is natural to model them as Markovian jumping systems or semi-Markovian jumping system. Semi-Markovian jumping system are characterized by a fixed matrix of transition probabilities and a matrix of sojourn time probability density functions. Considerable attention has been devoted to the analysis and research of this kind of systems. And in recent decades many important and significant results have been obtained. In [7], Ding etc. studied the problem of filtering for discrete-time singular time delay Markovian jumping systems. The problem of delay-dependent stochastic admissibility for discrete-time singular Markovian jump systems was investigated in [8]. Quantized control for nonlinear semi-Markovian networks system was considered in [9], siding control for semi-Markovian jumping system was researched in [10]. [11] has studied the observer-based control for singular Markovian jump system and [12] investigated the stabilization analysis for fractional uncertain Markovian system. And many other results can be found in [13-16] and the references therein.
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It should be pointed out that although traditional Lyapunov stability, which pays more attention to the asymptotic pattern of systems over an infinitetime interval, has been successfully applied in many control systems. In many practical systems, researchers prefer to consider the behavior of systems over a fixed finite time interval. Thus finite-time stability and finite-time boundedness were introduced [17,18]. And in recent years, with the development of Lyapunov function approach and linear matrix inequality techniques, many works about finite-time stability and finite-time control have been obtained. In [19], the problem of finite-time stabilization about uncertain singular Markovian jump systems was considered. Zhang, etc. investigated the finite-time unbiased filtering design problem in [20]. And many other results in [21-26] and so on. It is well known that nearly all practical systems are subject to actuator saturation which will cause instability and bad system performance. Due to this, many efforts were devoted to the saturation systems. Ma and Chen studied the problem of memory dissipative control for singular T-S fuzzy systems with actuator saturation and time-varying delay in [27]. Control problem for discrete3
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time Markovian system subject to actuator saturation was investigated in [28]. Observer design in [29], stability and stabilization in [30], output feedback control in [31] and so on. In addition, time-delay is also an unavoidable factor that causes bad system performance in various practical systems. The study of delay systems has attracted much attention these years. See examples in [32-36] and the references therein.
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On the other hand, the non-fragile controller design method is becoming an attractive topic in theory analysis and practical implementation, because it can minimize the cost of implementation, and allow for the adjustment online of control parameters. The main purpose of non-fragile control is to design a feedback controller that will be insensitive to some errors in control gains. Some results about non-fragile control can be found in [37-40]. Furthermore, a memory state feedback controller that contains input constraints can obtain less conservative conditions and have wider practical applications. And many researchers considered the memory controller designing. Zhao, etc. studied the memory feedback controller design in [41], the memory feedback control for uncertain singular T-S fuzzy system investigated in [42]. The problem of nonfragile control with memory state feedback for uncertain singular systems was researched in [43].
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Considering the above analysis, there were few results about finite-time nonfragile memory control for discrete-time singular Markovian jump systems with actuator saturation which is important and significant in engineer applications. Motivated by these, we carry out our study. In this paper, we research the robust finite-time non-fragile memory control for discrete-time singular Markovian systems with input saturation. First, based on proper Lyapunov functional and the LMI approach, sufficient conditions that ensure the system singular stochastic finite-time boundedness and finite-time boundedness are obtained. Then in order to get the optimal values of attenuation level and the largest domain of saturation attraction, the results are extended to convex optimization problems. At the end of the paper, some numerical examples are presented to demonstrate the effectiveness of the proposed methods. 4
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Problem formulation
Considering the following singular uncertain Markovian jumping systems with actuator saturation and time-varying delay: Ex (k
z (k)
¯ (rk ) sat (u (k)) + B ¯ω (rk ) ω (k) , + 1) = A¯ (rk ) x (k) + A¯d (rk ) x (k − d (k)) + B
¯ (rk ) sat (u (k)) + D ¯ ω (rk ) ω (k) , = C¯ (rk ) x (k) + C¯d (rk ) x (k − d (k)) + D
x (k) = ϕ (k) , k ∈ {−d2 , −d2 + 1, · · · , 0} ,
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(1) where x (k) ∈ R is the state vector, u (k) ∈ R is the control input, z (k) ∈ Rq is the control output vector and sat: Rp → Rp is the standard function defined as follows: n
sat (u (k)) = [sat (u1 (k)) , sat (u2 (k)) , · · · , sat (up (k))]T ,
ε
X
k∈N
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where sat (ui (k)) = sign (ui (k)) min {1, |ui (k)|} , ω (k) ∈ Rq is the disturbance input satisfies: |ω (k)|2
≤ d2 .
(2)
where 0 ≤ πij ≤ 1,
s P
j=1
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rk is discrete-time Markov stochastic process taking values in a finite space S = {1, 2, · · · , s} and the transition matrix Π = (πij )s×s , πij = Pr {rk+1 = j |rk = i }, πij = 1 for all i ∈ S.
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To simplify the notational presentation of this paper, for each possible rk = i, i ∈ S, a matrix F (rk ) will be represented by Fi ; for example, A (rk ) will be represented by Ai , by Ad (rk ), and so forth.
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d (k) is the time-varying delay satisfies 0 < d1 ≤ d (k) ≤ d2 , E ∈ Rn×n may be singular and we assume rank (E) = r ≤ n. A¯i = Ai + ∆Ai , A¯di = ¯i = Bi + ∆Bi , B ¯ωi = Bωi + ∆Bωi , C¯i = Ci + ∆Ci , C¯di = Cdi + Adi + ∆Adi , B ¯ ¯ ∆Cdi , Di = Di +∆Di , Dωi = Dωi +∆Dωi . And Ai , Adi , Bi , Bωi , Ci , Cdi , Di , Dωi are known real constant matrices with appropriate dimensions; ∆Ai , ∆Adi , ∆Bi , ∆Bωi , ∆Ci , ∆Cdi , ∆ and ∆Dωi are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form:
∆Adi ∆Bi ∆Bωi H1i = F1i (k) E1i E2i E3i E4i , ∆Ci ∆Cdi ∆Di ∆Dωi H2i
∆Ai
(3)
where H1i , H2i , E1i , E2i , E3i , E4i are known real constant matrices with appropriate dimensions and F1i (k) is unknown real and possibly time-varying matrices satisfying F1iT (k) F1i (k) ≤ I. 5
ACCEPTED MANUSCRIPT Now consider the following non-fragile memory state feedback controller: ¯ i x (k) + K ¯ di x (k − d (k)) , u (k) = K
(4)
¯ i = Ki + ∆Ki , K ¯ di = Kdi + ∆Kdi and Ki , Kdi are gain matrices to be with K determined, in addition ∆Ki , ∆Kdi are uncertain matrices described as:
∆Ki ∆Kdi = H3i F2i (k) E5i E6i ,
(5)
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where H3i , E5i , E6i are known matrices and F2i (k) is unknown matrix satisfying F2iT (k) F2i (k) ≤ I.
For a matrix Hi , Hdi , donate the j th row of Hi , Hdi as hij , hdij and define L (Hi , Hdi ) as L (Hi , Hdi ) = {x (k) ∈ Rn : |hij x (k) + hij x (k − d (k))| ≤ 1, j = 1, 2, · · · , p} .
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Let P ∈ Rn×n be a symmetric matrix and E T P E ≥ 0, α > 0 be a scalar, and donate by Ω E T P E, α the following set: n
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Ω E T P E, α = x (k) ∈ Rn : xT (k) E T P Ex (k) ≤ α .
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Let D be the set of p × p diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as Dj , j = 1, 2, · · · , 2p , and denote Dj− = I − Dj . Clearly, if Dj ∈ D, then Dj− ∈ D. Lemma 1 (Hu et al. [44]) Let Fi , Hi ∈ Rn×n . Then for any x (k) ∈ L (Hi ), n
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sat (Fi x (k)) ∈ co Dj Fi x (k) + Dj− Hi x (k) , j = 1, 2, · · · , 2p ,
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or, equivalently,
sat (Fi x (k)) =
p
2 X
αj (k) Dj Fi + Dj− Hi x (k) ,
j=1
(6)
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where co stands for the convex hull, αj (k) are some scalars which satisfy 0 ≤ αj (k) ≤ 1 and
2p P
j=1
αj (k) = 1.
Denoting the state trajectory of system (1) with initial condition x (k) = ϕ (k) ∈ Cn,d2 [−d2 , 0] by x (k, ϕ), then the domain of attraction of the origin is: ∆ Γ = ϕ (k) , k = −d2 , −d2 + 1, · · · , 0 : lim x (k, ϕ) = 0 . k→∞
6
ACCEPTED MANUSCRIPT It should be pointed out that in practical it is impossible to determinate the exact domain of attraction. An estimate Bδ ⊂ Γ of the domain of attraction is given by: (
)
Bδ = ϕ (k) ∈ Cn,d2 [−d2 , 0] : max kϕ (k)k ≤ δ . [−d2 ,0]
¯ i x (k) + K ¯ di x (k − d (k)) sat K =
2p P
j=1
αj (k)
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From the above lemma, for any x (k) ∈ L (Hi , Hdi )
i
¯ i + Dj− Hi x (k) + Dj K ¯ di + Dj− Hdi x (k − d (k)) . Dj K
(7)
Applying (6) and (7) to system (1), we can obtain the close-loop system as follows: + 1) =
z (k) = x (k)
2p P
j=1
2p P
j=1
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Ex (k
h
i
¯ ωi ω (k) , αj (k) C¯ij x (k) + C¯dij x (k − d (k)) + D
= ϕ (k) , k ∈ {−d2 , −d2 + 1, · · · , 0} ,
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where
i
¯ωi ω (k) , αj (k) A¯ij x (k) + A¯dij x (k − d (k)) + B
(8)
¯i Dj K ¯ i + Dj− Hi , A¯dij = A¯di + B ¯ i Dj K ¯ di + Dj− Hdi , A¯ij = A¯i + B
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¯ i Dj K ¯ i + Dj− Hi , C¯dij = C¯di + D ¯ di Dj K ¯ di + Dj− Hdi . C¯ij = C¯i + D
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Lemma 2 (Zhang et al. [23]) Let matrix X, Y and Z of appropriate dimensions, where X is a symmetric matrix, then X + Y F (k) Z + [Y F (k) Z]T < 0,
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holds for all time varying matrix function F (k) satisfying F T (k) F (k) ≤ I for all k ∈ Zk≥0 , if and only if there exists a positive constant ε, such that holds X + εY Y T + ε−1 Z T Z < 0 holds. Lemma 3 (Ma et al. [27]) Given matrices X, Y, Z with appropriate dimensions, and Y is symmetric. Then the following inequality holds: −Z T Y Z ≤ X T Z + Z T X + X T Y −1 X. 7
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Lemma 4 (Ding et al. [7]) For any constant positive semi-definite symmetric matrix W ∈ Rn×n , two positive integers d2 and d1 satisfying d2 ≥ d1 ≥ 1, the following inequality holds:
i=d1
xT (i)W x (i) ≥
d2 X
i=d1
T
x (i) W
d2 X
i=d1
x (i) .
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(d2 − d1 + 1)
d2 X
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Definition 1 (Zhang et al. [5]) (1) The matrix pair (E, Ai ) is said to be regular if, for each i ∈ S, the characteristic polynomial det (sE − Ai ) is not identically zero; (2) The matrix pair (E, Ai ) is said to be causal if, for each i ∈ S, deg (det (sE − Ai )) = rank (E) .
Definition 2 (Singular stochastic finite-time boundedness (SSFTB)).The discretetime singular jumping system (8) is said to be SSFTB with respect to (c1 , c2 , N, Ri ). With 0 < c1 < c2 and Ri > 0, if the singular stochastic system is regular and causal in time ∀k ∈ {1, 2, · · · , N } and satisfies n
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E xT (k1 ) E T Ri Ex (k1 ) , kEk2 xT (k1 ) Ri x (k1 ) ≤ c21 ⇒ E xT (k2 ) E T R (rk ) Ex (k2 ) ≤ c22 , (9) where k1 ∈ {−d2 , −d2 + 1, · · · , 0} , k2 ∈ {1, 2, · · · , N }.
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Definition 3 (Singular stochastic H∞ finite-time boundedness(SSH∞ FTB)).The discrete-time singular jumping system (8) is said to be SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ). With 0 < c1 < c2 and Ri > 0, if the system is SSFTB with respect to (c1 , c2 , N, Ri ), and under the zero-initial condition the control output z (k) satisfies the following constrained condition: E
(N X
T
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z (k) z (k) < γ E
k=0
3
)
(N X
k=0
T
)
ω (k) ω (k) , ∀k ∈ {1, 2, · · · , N } .
(10)
Main results
In this section, we present a set of sufficient conditions for finite-time bundedness and finite-time H∞ boundeness of the closed-loop system (8).
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ACCEPTED MANUSCRIPT Theorem 1 For given scalars µ ≥ 1, the closed-loop system (8) for ev T 2 ery initial condition belongs to Ω E Pi E, γ µ is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist scalars γ > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and nonsingular matrices Ni such that following conditions holds:
n
0
0
0
0
A¯T dij
∗
−µd1 Q3
∗
∗
Ω33 Ω34
0
0
∗
∗
∗ Ω44
0
0
∗
∗
∗
∗ −µ
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
sup λmax P¯i
o
c21 +
i∈S
−Pˆi−1
0
∗
∗
−Z −1
∗
∗
∗
c21
+
i∈S
i∈S
n
2kEk2
c21
n
c21 + µ−N γ 2 d2 ≤ inf λmin P¯i i∈S
(11)
o
µ−N c22 . (12)
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i∈S
o
kEk2
c21
< 0,
−I
¯ 2i )} d2 µd2 −1 sup{λmax (Q
¯ 3i )} d12 (d1 +d2 −1)µd2 sup{λmax (Q
+2µd2 −1 d212 (d1 + d2 + 1) sup λmax Z¯i with
c21 +
C¯ijT C¯ T 0 0 T ¯ Dωi 0 0
0
∗
γ I
dij
0
T ¯ωi d12 B
2
kEk2
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kEk2
i∈S
d12 A¯T dij
T ¯ωi B
−N
¯ 1i )} d1 µd1 −1 sup{λmax (Q
i∈S ¯ 3i )} d2 µd2 −1 sup{λmax (Q
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Ω1 =
T ¯T A¯T ij d12 Aij − E
¯ωi NiT RT B
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T T ¯ Ω11 Ni R Adij
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T T ¯ Ω11 = −µE T Pi E + (1 + d12 ) Q3 + Q1 + Q2 + A¯T ij RNi + Ni R Aij , d12 = d2 − d1 ,
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s P Ω33 = −µd1 Q1 − µd1 +1 E T ZE, P˜i = πij Pj , j=1
Ω34 = µd1 +1 E T ZE, Ω44 = −µd2 Q2 − µd1 +1 E T ZE,
¯i Dj K ¯ i + Dj− Hi , A¯dij = A¯di + B ¯i Dj K ¯ di + Dj− Hdi , A¯ij = A¯i + B
¯ i Dj K ¯ i + Dj− Hi , C¯dij = C¯di + D ¯ di Dj K ¯ di + Dj− Hdi . C¯ij = C¯i + D And R is any constant matrix with appreciate dimensions and satisfying: T T E R = 0. And Ω E Pi E, α ∈ L (Hi , Hdi ). Moreover, the domain of at9
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s
γ 2 µ1−N − µ−N γ 2 d2 , Υ
(13)
where n
Υ = sup λmax E T Pi E i∈S
o
+ d1 µd1 −1 λmax (Q1 ) + d2 µd2 −1 λmax (Q2 ) + d2 µd2 −1 λmax (Q3 ) +
Proof: As
2p P
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+d12 (d1 + d2 + 1) µd2 λmax (Q3 ) + 2µd2 −1 d212 (d1 + d2 + 1) λmax E T ZE . αj (k) = 1, it follows that p
2 X
αj (k) Ω1 < 0.
(14)
j=1
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First, we should prove that the system is regular and causal. From (13), it is known that p
T
−µE Pi E+(1 + d12 ) Q3 +Q1 +Q2 +
2 X
p
αj (k) A¯T ij RNi +
j=1
2 X
αj (k) NiT RT A¯ij < 0.
T
−1
j=1
Ir 0
U EV =
0 0
,
U
j=1
¯ ¯ Aij1 Aij2
αj (k) A¯ij V =
A¯ij3 A¯ij4
,
R U
= R1 R2 ,
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Ni V = Ni1 Ni2 ,
2p P
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(15) For rank (E) = r ≤ n, there exist two nonsingular matrices U and V such that
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since E T R = 0, we have R1 = 0. Pre- and post-multiplying (15) by V T and V , respectively, this results in the following matrix inequality holds:
?
?
T T ¯ ? A¯T ij4 R2 Ni2 + Ni2 RAij4
< 0,
(16)
where ? will represents the matrix block we do not need. Then we have T T ¯ ¯ A¯T ij4 R2 Ni2 + Ni2 RAij4 < 0, which means Aij4 is nonsingular and we have proved the system (8) is regular and causal. Now, it is time to prove the system (8) is SSFTB with respect to (c1 , c2 , N, Ri ). Define that y (k) = x (k + 1) − x (k) , and choose the Lyapunov-Krasovskii functional as follows: 10
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V (k) =
4 X
Vi (k),
i=1
with
V1 (k) = xT (k) E T Pi Ex (k) , µk−1−s xT (s)Q1 x (s) +
s=k−d1 k−1 X
V3 (k) =
µ
k−1 X
µk−1−s xT (s)Q2 x (s) ,
s=k−d2 −d X1
k−s−1 T
x (s)Q3 x (s) +
k−1 X
µk−s−1 xT (s)Q3 x (s) ,
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k−1 X
V2 (k) =
θ=−d2 +1 s=k+θ
s=k−d(k)
V4 (k) = d12
−d 1 −1 k−1 X X
µk−s−1 y T (s)E T ZEy (s) .
θ=−d2 s=k+θ
∆V1 (k) = E {V1 (k + 1)} − V1 (k) = xT (k + 1) E T
N X
j=1
πij Pj Ex (k + 1) − µxT (k) E T Pi Ex (k) + (µ − 1) V1 (k) ,
k X
s=k+1−d1
k−1 X
k−1 X
s=k−d2 T
µk−s xT (s)Q1 x (s)
s=k−d1 k X
µk−s−1 xT (s)Q1 x (s) +
s=k−d1
µk−s xT (s)Q2 x (s) + (µ − 1)
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−
k−1 X
µk−s xT (s)Q1 x (s) −
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+ (µ − 1)
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∆V2 (k) = E {V2 (k + 1)} − V2 (k) =
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Then
µk−s xT (s)Q2 x (s)
s=k+1−d2 k−1 X k−s−1 T
µ
x (s)Q2 x (s)
s=k−d2
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= x (k) (Q1 + Q2 ) x (k) − µd1 xT (k − d1 ) Q1 x (k − d1 ) − µd2 xT (k − d2 ) Q2 x (k − d2 ) + (µ − 1) V2 (k) ,
∆V3 (k) = E {V3 (k + 1)} − V3 (k)
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=
k X
s=k+1−d(k+1)
+ (µ − 1) −
−d X1
µk−s xT (s)Q3 x (s) −
k−1 X
θ=−d2 +1 s=k+θ T
µk−s xT (s)Q3 x (s)
s=k−d(k)
µk−s−1 xT (s)Q3 x (s) +
−d X1
k X
µk−s xT (s)Q3 x (s)
θ=−d2 +1 s=k+θ+1
s=k−d(k) k−1 X
k−1 X
µk−s xT (s)Q3 x (s) + (µ − 1) d1 T
−d X1
k−1 X
µk−s−1 xT (s)Q3 x (s)
θ=−d2 +1 s=k+θ
≤ x (k) (1 + d12 Q3 ) x (k) − µ x (k − d (k)) Q3 x (k − d (k)) + (µ − 1) V3 (k) , 11
ACCEPTED MANUSCRIPT ∆V4 (k) = E {V4 (k + 1)} − V4 (k) −d 1 −1 X
k X
θ=−d2 s=k+θ+1 −d 1 −1 k−1 X X
+ (µ − 1) d12 =
d212 y T
−d 1 −1 k−1 X X
µk−s y T (s)E T ZEy (s) − d12
µk−s y T (s)E T ZEy (s)
θ=−d2 s=k+θ
µk−s−1 y T (s)E T ZEy (s)
θ=−d2 s=k+θ T
(k) E ZEy (k) − d12
k−d 1 −1 X
θ=k−d2 k−d 1 −1 X d1 +1
≤ d212 y T (k) E T ZEy (k) − d12 µ From lemma 4, we have
µk−s y T (θ)E T ZEy (θ) + (µ − 1) V4 (k)
θ=k−d2
k−d 1 −1 X
∆V4 ≤ d212 y T (k) E T ZEy (k) − µd1 +1
y T (θ)E T ZEy (θ) + (µ − 1) V4 (k) .
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= d12
k−d 1 −1 X
y T (θ)E T Z
Ey (θ)
x (k − d2 )]
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θ=k−d2 θ=k−d2 T d1 +1 2 T [x (k − d1 ) − x (k − d2 )]T E T ZE [x (k − d1 ) − = d12 y (k) E ZEy (k) − µ = d212 y T (k) E T ZEy (k) − µd1 +1 xT (k − d1 ) E T ZEx (k − d1 ) + 2µd1 +1 xT (k − d1 ) E T ZEx (k − d2 ) − µd1 +1 xT (k − d2 ) E T ZEx (k − d2 ) .
Considering that E T R = 0, then
2xT (k + 1) E T RNi x (k) = 0,
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let ξ (k) = x (k) x (k − d (k)) x (k − d1 ) x (k − d2 ) ω (k) ,
CE
PT
∆V (k) − (µ − 1) V (k) =
T T ¯ Ω11 Ni R Adij
T ξ (k)
0
0
¯ωi NiT RT B
0
0
0
∗
−µd1 Q3
∗
∗
Ω33 Ω34
0
∗
∗
∗ Ω44
0
∗
∗
∗
0
T ˆ + ΛT 1 Pi Λ1 + Λ2 ZΛ2 ,
∗
ξ (k)
(17)
AC
with " p # 2 2p 2p P P P ¯ωi ω (k) , αj (k) A¯ij x (k) αj (k) A¯dij x (k − d (k)) 0 0 αj (k) B Λ1 = Λ2 =
" j=1 2p P
j=1
j=1
αj (k) A¯ij − E x (k)
j=1
2p P
j=1
#
2p P ¯ωi ω (k) . αj (k) A¯dij x (k − d (k)) 0 0 αj (k) B j=1
According to Schur complement and Eq. (14), we have
E {V (k + 1)} − V (k) < (µ − 1) V (k) + µ−N γ 2 ω T (k) ω (k) ≤ (µ − 1) V (k) + µ−N γ 2 ω T (k) ω (k) , 12
ACCEPTED MANUSCRIPT then
n
o
E {V (k + 1)} < µE {V (k)} + µ−N γ 2 E ω T (k) ω (k) , notice that µ ≥ 1, we have
i∈S
≤ µk E {V (0)} + µk−N γ 2 d2 . 1
1
1
µk−j−1 ω T (k) ω (k)
j=0
(19)
CR IP T
E {V (k)} < µk E {V (0)} + sup {λmax (Qi )} E
k−1 X
(18)
1
1
1
− − − ¯ 1i Ri− 2 , Q2 = Ri− 2 Q ¯ 2i Ri− 2 , Q3 = Letting Pi = Ri 2 P¯i Ri 2 , Q1 = Ri 2 Q n o 1 1 1 −1 ¯ 3i Ri− 2 , Z = Ri− 2 Z¯i Ri− 2 , and noticing that E xT (k1 ) E T Ri Ex (k1 ) , kEk2 xT (k1 ) Ri x (k1 ) ≤ Ri 2 Q
1 2
1
V (0) = x (0) E Ri P¯i Ri2 Ex (0) + T
T
AN US
c21 , k1 ∈ {−d2 , · · · , 0} and the initial value of the Lyapunov-krasovskii functional can be described as:
−1 X
1
1
¯ 1i Ri2 x (s) µ−1−s xT (s) Ri2 Q
s=−d1 −1 X
+
µ
1 2
1 2
¯ 2i Ri x (s) + x (s) Ri Q
−1−s T
s=−d2 1
n
o
¯ 2i + µd2 −1 sup λmax Q
PT
i∈S
n
¯ 3i + µd2 sup λmax Q
CE
i∈S
n
o
o
−1 X
AC
1
1
n
−1 o X
i∈S
n
θ=−d2 +1 s=θ −1 1 −1 X o −d X
≤2
θ=−d2 s=θ
θ=−d2 s=θ
xT (s + 1) E T Ri ExT (s + 1) + xT (s) E T Ri Ex (s) ,
then 13
xT (s) Ri x (s)
s=−d2
(x (s + 1) − x (s))T E T Ri E (x (s + 1) − x (s)) ,
(x (s + 1) − x (s))T E T Ri E (x (s + 1) − x (s))
θ=−d2 s=θ −dP 1 −1 −1 P
−1 o X
xT (s)Ri x (s)
considering −dP 1 −1 −1 P
xT (s) Ri x (s)
s=−d1
¯ 3i xT (s) Ri x (s) + µd2 −1 sup λmax Q
s=−d2 −d −1 X1 X
1
µ−s−1 y T (s) E T Ri2 Z¯i Ri2 Ey (s)
θ=−d2 s=θ
i∈S
+ d12 µd2 −1 sup λmax Z¯i i∈S
−d −1 1 −1 X X
¯ 1i xT (s) E T Ri Ex (s) + µd1 −1 sup λmax Q
ED
≤ sup λmax P¯i n
1
¯ 3i Ri2 x (s) + d12 µ−s−1 xT (s)Ri2 Q
M
−1 X
θ=−d2 +1 s=θ
i∈S
1
¯ 3i Ri2 x (s) µ−s−1 xT (s)Ri2 Q
s=−d(0)
−d X1
+
−1 X
ACCEPTED MANUSCRIPT
n
V (0) ≤ sup λmax P¯i i∈S
d2 µ
d2 −1
kEk
i∈S
c21 +
sup λmax i∈S
+
n
n
¯ 1i d1 µd1 −1 sup λmax Q
o 2
¯ 3i Q
o
kEk
2
o
d2
i∈S 2
+
n
+ 2µd2 −1 d212 (d1 + d2 + 1) sup λmax Z¯i i∈S
o
2kEk
i∈S
kEk µd2 −1 d212
2
o
n
n
i∈S
c21 +
2kEk
o
ED
Z¯i
o
c21
!
T
n
n
≥ inf λmin P¯i
PT
i∈S
o
n
o
+ µk−N γ 2 d2 ,
T
kEk2
1
1 2
T
o
c21
c21
(21)
E {V (k)} ≥ E x (k) E Pi Ex (k) = E x (k) E Ri P¯i Ri2 Ex (k) T
c21
c21
¯ 2i d2 µd2 −1 sup λmax Q
i∈S 2
c21 +
i∈S
note that
o
¯ 3i d12 (d1 + d2 + 1) µd2 sup λmax Q
(d1 + d2 + 1) sup λmax
n
kEk2
M
k
i∈S
n
i∈S
c21 +
kEk o ¯ 3i Q
o
(20)
¯ 1i d1 µd1 −1 sup λmax Q
¯ 3i d2 µd2 −1 sup λmax Q
+ µk + 2µ
n
o
2
CR IP T
n
¯ ≤ µk sup λmax Pi
n
AN US
c21 .
Then we have
E {V (k)}
i∈S
c21 +
d12 (d1 + d2 + 1) µ sup λmax c21
n
¯ 2i d2 µd2 −1 sup λmax Q
o
E xT (k) E T Ri Ex (k) ,
(22)
it follows that n
n
+ µk + 2µ
k
i∈S
o
E xT (k) E T Ri Ex (k)
n
¯ ≤ µk sup λmax Pi
AC
i∈S
o
CE
inf λmin P¯i
n
o
2
kEk µd2 −1 d212
i∈S
c21 +
¯ 3i d2 µd2 −1 sup λmax Q i∈S
n
¯ 1i d1 µd1 −1 sup λmax Q o
kEk2
o
n
i∈S
c21 + n
kEk2
¯ 3i d12 (d1 + d2 + 1) µd2 sup λmax Q i∈S 2
c21 + n
(d1 + d2 + 1) sup λmax i∈S
14
2kEk
Z¯i
o
c21
!
¯ 2i d2 µd2 −1 sup λmax Q o
+ µk−N γ 2 d2 . (23)
c21
o
c21
ACCEPTED MANUSCRIPT Based on Eq. (12), it obtained that n
o
E xT (k) E T Ri Ex (k) < c22 , ∀k ∈ {1, 2, · · · , N } ,
(24)
from definition 2, we know the singular system is SSFTB with respect to (c1 , c2 , N, Ri ). On the other hand, one can derive from (18) that the following inequality:
holds for all i ∈ S. It follows that
n
(25)
CR IP T
E {V (k + 1)} − µV (k) + z T (k) z (k) − γ 2 µ−N ω T (k) ω (k) < 0, o
n
o
E {V (k + 1)} < µE {V (k)} − E z T (k) z (k) + µ−N γ 2 E ω T (k) ω (k) , (26) then n
o
µk−j−1 E z T (j) z (j) +γ 2 µ−N
j=0
AN US
k−1 X
E {V (k)} < µk E {V (0)}−
k−1 X
n
j=0
(27) Under the zero initial condition and noticing that V (k) ≥ 0, we can obtain that n
o
µk−j E z T (j) z (j) < γ 2 µ−N
j=0
as µ ≥ 1, it follows that E
(N X
)
T
2
z (k) z (k) < γ E
k=0
T
n
o
µk−j E ω T (j) ω (j) ,
)
ω (k) ω (k) , ∀k ∈ {1, 2, · · · , N } .
ED
k=0
(N X
k X
j=0
M
k X
(28)
(29)
Then from definition 3, the system is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ).
PT
In addition, according to (17) and (18), we have
CE
xT (k) E T Pi Ex (k) ≤ V (k) ≤ µk V (0) + µk−N γ 2 d2 ≤ γ 2 µ,
(30)
AC
then it follows that kϕ (k)k2 ≤
γ 2 µ1−N − µ−N γ 2 d2 ∆ = = (ϕ) , Υ
(31)
for any ϕ ∈ = (ϕ), it follows that = (ϕ) ≤ γ 2 µ. From (25), we can further get xT (k) E T Pi Ex (k) ≤ γ 2 µ for any ϕ ∈ = (ϕ), which means that all the trajectories of x (k) which start from the set = (ϕ) always stay in the domain T Ω E Pi E, γ 2 µ . Based on the above discussion, the proof is thus complete.
15
o
µk−j−1 E ω T (j) ω (j) .
ACCEPTED MANUSCRIPT Theorem 2 For given scalars µ ≥ 1, ε > 0 the closed-loop system (8) for every initial condition belongs to Ω E T Pi E, η 2 u is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and nonsingular matrices Gi such that Eq. (12) and the following conditions holds:
T T ˆ Ω2 εO1 O2
where ˆ2 = Ω
∗ −µ
d1
0
GT i Q3 Gi
0
0
R Bωi
ˆT GT i Aij Wi
0
ˆT GT i Adij Wi
T
0
∗
∗
Ω33 Ω34
0
∗
∗
∗ Ω44
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
ED
R Aˆdij Gi T
d12 GT i
T AˆT ij − E
ˆT d12 GT i Adij
ˆT GT i Cij
ˆT GT i Cdij
AN US
ˆ Ω11
M
(32)
CR IP T
< 0, Ω2 = ∗ −εI 0 ∗ ∗ −εI
0
0
0
0
0
0
T Bωi Wi
T d12 Bωi
T Dωi
−P −1
0
0
∗
−Z −1
0
∗
∗
−I
,
O2 =
PT
O1 = O11 O12 0 0 E4i 0 0 0 , T H1i R
h
0000
T H1i Wi
T d12 H1i
CE
¯ i + Dj− Hi O11 = E1i + E3i Dj K
i
T H2i
, h
¯ di + Dj− Hdi Gi , O12 = E2i + E3i Dj K
i
Gi ,
AC
ˆ 11 = −µGT E T Pi EGi + (1 + d12 ) GT Q3 Gi + GT Q1 Gi + GT Q2 Gi + GT AˆT R + Ω i i i i i ij RT Aˆij Gi , Wi =
h√
πi1 ,
√
πi2 , · · · ,
√
i
n
o
πis , P −1 = diag P1−1 , P2−1 , · · · , Ps−1 ,
¯ i + Dj− Hi , Aˆdij = Ai + Bi Dj K ¯ di + Dj− Hdi , Aˆij = Ai + Bi Dj K
¯ i + Dj− Hi , Cˆdij = Ci + Di Dj K ¯ di + Dj− Hdi , Cˆij = Ci + Di Dj K Ω33 , Ω34 , Ω44 are the same as theorem 1, R is any constant matrix with appre16
ACCEPTED MANUSCRIPT
ciate dimensions and satisfying: E T R = 0. And Ω E T Pi E, η 2 u ∈ L (Hi , Hdi ) , the saturation attraction domain is given by (13). n
T Proof: Let Gi = Ni−1 , Pre- and post-multiplying (26) by diag GT i , Gi , I, I, I, I, I, I and its transpose, we have
Ω0 2
0
¯ωi RT B
∗ −µd1 GT i Q3 Gi 0
0
0
¯T GT i Adij
¯T d12 GT i Adij 0
¯T GT i Cij T ¯T G C
CR IP T
=
RT A¯dij Gi
T T ¯T ¯T GT i Aij d12 Gi Aij − E
0
0 Ω 11
i
dij
∗
∗
Ω33 Ω34
0
0
∗
∗
∗ Ω44
0
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
−Pˆi−1
∗
∗
∗
∗
∗
∗
−Z −1
0
∗
∗
∗
∗
∗
∗
∗
−I
¯T B ωi
< 0,
0
¯T d12 B ωi
¯T D ωi
0
0
(33)
M
with
0
0
AN US
ED
T T T T T ¯T Ω011 = −µGT i E Pi EGi + (1 + d12 ) Gi Q3 Gi + Gi Q1 Gi + Gi Q2 Gi + Gi Aij R + RT A¯ij Gi . Considering(3) we can obtain that
ˆ 2 + O1T F1iT (k) O2 + O2T F1i (k) O1 < 0. Ω02 =Ω
(34)
PT
As F1iT (k) F1i (k) ≤ I, according to lemma 2, there exists a positive constant ε such that
CE
O1T F1iT (k) O2 + O2T F1i (k) O1 ≤ εO1T O1 + ε−1 O2T O2 ,
(35)
AC
therefore, the following inequality implies that (32) holds: ˆ 2 + εOT O1 + ε−1 OT O2 < 0. Ω 1 2
(36)
Using Schur complement lemma, Eq. (36) is equality to Eq. (32), then the proof of the theorem is completed.
¯ i Gi = Remark 1 If we only consider the memory controller, we can let K ¯ i , Hi Gi = Yi , K ¯ di =X ¯ di , Hdi Gi = Ydi . Then using the above theorem, the gain X 17
o
ACCEPTED MANUSCRIPT ¯i = X ¯ i G−1 ¯ ¯ −1 matrices can be obtained and K i , Kdi = Yi Gi . Theorem 3 Let Ki Gi = Xi , Kdi Gi = Xdi , Hi Gi = Yi , Hdi Gi = Ydi , for given scalars µ ≥ 1, ε, η > 0, the closed-loop system (8) for every initial conT 2 dition belongs to Ω E Pi E, η u is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist scalars θ, β1 , β2 , β3 , β4 > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and matrices Gi , Xi , Yi , Xdi , Ydi such that the following conditions holds: T ˜ ˜T T Ω2 εO1 O2 ηO3
Ω3 =
∗ −εI
0
0
∗
∗ −εI
∗
∗
∗ −ηI
∗
∗
∗
0
∗
εE3i Dj H3i 0 0
−ηI
< 0,
(37)
AN US
O4T
CR IP T
θRi−1 < Pi−1 < Ri−1 , 0 < Q1 < β1 Ri , 0 < Q2 < β2 Ri , 0 < β3 Ri−1 < Q−1 3 , 0 < Z < β4 Ri , (38)
1 θ
+
d1 µd1 −1 β1 kEk2
+
d2 µd2 −1 β2 kEk2
+
d2 µd2 −1 kEk2 β3
+
d12 (d1 +d2 −1)µd2 2kEk2 β3
˜ Ω11
˜ 12 Ω
0
RT Bωi
˜ 16 Ω
˜ 17 Ω
0
0
˜ 26 Ω
˜ 27 Ω
0
0
∗
∗
∗ −µd1 GT i Q3 Gi 0 ∗
∗
Ω33 Ω34
0
∗
∗ Ω44
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
PT
∗
AC
CE
˜2 = Ω
0
ED
where
M
+2µd2 −1 d212 (d1 + d2 + 1) β4 c21 + µ−N γ 2 d2 < µ−N c22
c21
˜ 18 Ω
˜ 28 Ω
0
0 0 0 , T T T Dωi Bωi Wi d12 Bωi −1 −P 0 0 −1 ∗ −Z 0
−I
T T T T ˜ 11 = −µGT Ω i E Pi EGi + (1 + d12 ) Gi Q3 Gi + Gi Q1 Gi + Gi Q2 Gi
h
+ Ai Gi + Bi Dj Xi + Dj− Yi
iT
h
(39)
R + RT Ai Gi + Bi Dj Xi + Dj− Yi
˜ 12 =RT Adi Gi + Bi Dj Xdi + Dj− Ydi Ω
18
h
i
,
˜ 16 = Ai Gi + Bi Dj Xi + Dj− Yi , Ω
iT
Wi ,
ACCEPTED MANUSCRIPT h
˜ 17 = d12 Ai Gi + Bi Dj Xi + Dj− Yi Ω h
˜ 26 = Adi Gi + Bi Dj Xdi + Dj− Ydi Ω h
˜ 28 = Cdi Gi + Di Dj Xdi + Dj− Ydi Ω
iT
iT
iT
h
− T ˜ − d12 GT i E , Ω18 = Ci Gi + Di Dj Xi + Dj Yi
h
˜ 27 = d12 Adi Gi + Bi Dj Xdi + Dj− Ydi Wi , Ω ,
˜1 = O ˜ 11 O ˜ 12 0 0 E4i 0 0 0 , O
i
h
, O12 = E2i Gi + E3i Dj Xdi + Dj− Ydi
O3 = E5i Gi E6i Gi 01×6 ,
T T T Dj Bi Wi H3i
T T T Dj Bi R H3i
T T T Dj Bi d12 H3i
i
CR IP T
h
O11 = E1i Gi + E3i Dj Xi + Dj− Yi
T T T Dj Di H3i
,
AN US
, 0000 and the other variables are the same as theorem3, R is any constant matrix with appreciate dimensions and satisfying: E T R = 0, and Ω E T Pi E, η 2 u ∈ L (Hi , Hdi ) , the saturation attraction domain is given by (13). And the con−1 troller gain matrices are Ki = Xi G−1 i , Kdi = Xdi Gi . O4 =
0
M
Proof:Considering Eq. (5), we can obtain that Ω2 = Ω 3 + O3 0 0
ED
T
F2iT
+ O4 εE3i Dj H3i 0
PT
< 0,
AC
CE
where
T
(k) O4 εE3i Dj H3i 0
F2i (k) O3 0 0
(40)
˜ ˜T T Ω2 εO1 O2
Ω03 = . ∗ −εI 0 ∗ ∗ −εI
According to lemma 2, the following inequality is true:
O3 0 0
T
F2iT
(k) O4 εE3i Dj H3i 0
+ O4 εE3i Dj H3i 0
≤ η O3 0 0
T
T
F2i (k) O3 0 0
O3 0 0 + η −1 O4 εE3i Dj H3i 0
19
T
O4 εE3i Dj H3i 0 .
iT
,
iT
,
ACCEPTED MANUSCRIPT Then the following inequality guarantee (40) holds:
0
Ω3 ≤ Ω 3 + η O3 0 0 +η
−1
< 0,
T
O3 0 0
O4 εE3i Dj H3i 0
T
O4 εE3i Dj H3i 0
(41)
1
1
1
1
1
CR IP T
using Schur complement lemma and the above inequality is equivalent to (37). 1 −1 −1 −1 ¯ 1i Ri− 2 , Q2 = On the other hand , noting that Pi = Ri 2 P¯i Ri 2 , Q1 = Ri 2 Q 1
− ¯ 2i Ri− 2 , Q3 = Ri− 2 Q ¯ 3i Ri− 2 , Z = Ri− 2 Z¯i Ri− 2 , Ri 2 Q
n
it follows (38) that 1 < inf λmin P¯i n
¯ 2i β1 , sup λmax Q i∈S
o
i∈S
n
o
i∈So
¯ 3i < β2 , sup λmax Q i∈S
n
, sup λmax P¯i
o
i∈S
n
< β3 , sup λmax Z¯i i∈S
n
¯ 1i < θ, sup λmax Q o
< β4 ,
o
AN US
then it easily obtained that (39) is a sufficient condition to guarantee (12). This completes the proof.
Remark 2 Noting that (37) may be difficult to solve employing the LMI toolbox of Matlab, therefore we make some transitions below:
M
Using lemma 3, we have
ED
T T T −1 −µGT i E Pi EGi ≤ µGi E + µEGi + µPi ,
PT
d1 T d1 d1 −1 −µd1 GT i Q3 Gi ≤ µ Gi + µ Gi + µ Q3 .
AC
CE
ˆ3 = In addition considering the fact −X −1 ≤ −2ρI +ρ2 X, and let Pˆi = Pi−1 , Q −1 Q3 we can replace (37) by the following inequality:
Ω4 =
˜ 0 εO ˜ 1T O2T ηO3T Ω 2
∗ −εI
0
0
∗
∗ −εI
∗
∗
∗ −ηI
∗
∗
∗
∗
∗
∗
0
O4T
M
0
0
0
0
∗
−ηI
0
∗
∗
N
20
εE3i Dj H3i 0
< 0,
(42)
<
ACCEPTED MANUSCRIPT where ˜ 02 Ω
˜0 Ω11
0
0
RT Bωi
˜ 16 Ω
˜ 17 Ω
d1 d1 ˆ ∗ µd1 GT i + µ Gi + µ Q3 0
0
0
˜ 26 Ω
˜ 27 Ω
∗
∗
Ω33 Ω34
0
0
0
∗
∗
∗ Ω44
0
0
0
∗
∗
∗
T ∗ −µ−N γ 2 I Bωi Wi
∗
∗
∗
∗
∗
−Pˆ
∗
∗
∗
∗
∗
∗
−2ρ1 I + ρ21 Z
∗
∗
∗
∗
∗
∗
∗
with h
T d12 Bωi
0
− T ˜ 11 = µGT ˆ Ω i E + µEi Gi + µPi + Ai Gi + Bi Dj Xi + Dj Yi
h
+RT Ai Gi + Bi Dj Xi + Dj− Yi
i
,
˜ 18 Ω
˜ 28 Ω
0
0 , T Dωi 0 0
CR IP T
=
˜ 12 Ω
AN US
iT
−I
R
˜ 2, and the other variables are the same as the in Ω
M =
GT i
M
N = diag {−2ρ2 I + ρ22 Q1 , −2ρ3 I + ρ23 Q2 , −2ρ4 I + ρ25 Q4 } , √ T 1 + d12 Gi
GT i
CE
PT
ED
. 08×1 08×1 08×1 And using Schur complement lemma Eq. (39) is equivalent to
υ1 c 1 I υ 2
< 0, ∗ −θ 0 ∗ ∗ β3
(43)
AC
where υ1 =
υ2 =
d1 µd1 −1 β1 kEk2
r
d2 µd2 −1 kEk2
+
+
d2 µd2 −1 β2 kEk2
+ 2µd2 −1 d212 (d1 + d2 + 1) β4 c21 +µ−N γ 2 d2 −µ−N c22 ,
d12 (d1 +d2 −1)µd2 c1 . 2kEk2
On the other hand, using the same method in [21], Ω E T Pi E, γ 2 u ∈ L (Hi , Hdi ) 21
ACCEPTED MANUSCRIPT is equivalent to
1 − γ2µ
hik
hdik
∗
−E T Pi E
0
∗
∗
−E T Pi E
≤ 0,
(44)
here hik is the k th row of Hin , and hdik isothe k th row of Hdi . Pre-and post T and its transpose, we have -multiplying Eq. (44) by diag I, GT i , Gi 1 − γ2µ
Yik
Ydik
∗
T −GT i E Pi EGi
0
∗
∗
T −GT i E Pi EGi
and a sufficient condition for (45) is 1 − γ2µ
Yik
Ydik
≤ 0,
AN US
CR IP T
∗
T ˆ GT i E + EGi + Pi
0
∗
∗
T ˆ GT i E + EGi + Pi
≤ 0.
(45)
(46)
ED
M
Remark 3 With all the ellipsoids satisfying the set invariance conditions of Theorem 1, we are interested in finding a large estimate of the domain of attraction. Based on the results obtained above, an optimization problem with LMI constraints can be formulated as follows: min κ
PT
ω1 I ∗
AC
CE
s.t.
with
ET Pˆi
ˆ Q3 I
≥ 0,
∗ ω4
≥ 0,
(47)
ω2 I − Q1 ≥ 0, ω3 I − Q2 ≥ 0, ω5 I − E T ZE ≥ 0,
κ = ω1 + d1 µd1 −1 ω2 + d2 µd2 −1 ω3 + d2 µd2 −1 ω4 + 0.5d12 (d1 + d2 + 1) µd2 ω4 + 2µd2 −1 d212 (d1 + d2 + 1) ω5 .
The positive scalars ω1 , ω2 , ω3 , ω4 , ω 5 are introduced to bound the different items in (13) for getting a larger estimate of the domain of attraction for the system. Therefore, a maximal estimate of the attraction domain can be obtained by: 22
ACCEPTED MANUSCRIPT
δmax =
s
γ 2 µ1−N − γ 2 µ−N d2 . κmin
Remark 4 The feasibility of conditions stated in theorem 4 can be turned into the following optimization problems with a fixed parameter µ
min c22 ˆ 3 > 0, Gi , Xi , Xdi , Yi , Ydi Pˆi , Q1 , Q2 , Q
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(48)
s.t. (38) (42) (43) (45) . min γ 2 + c22
ˆ 3 > 0, Gi , Xi , Xdi , Yi , Ydi Pˆi , Q1 , Q2 , Q
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s.t. (38) (42) (43) (45) .
(49)
Numerical examples
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4
M
Remark 5 In order to get less conservative and better results, we constructed multiple and novelty Lyapunov function, which made the algorithm seem to be complex. Thus we turned the results to LMIs and LMI optimization problems so that the results can be obtained by the LMI toolbox of Matlab directly.
PT
Example 1 When Kdi , ∆Ki , ∆Kdi = 0, it turns out to be a normal state feedback control problem and and the results are also suitable. Considering the same systems as shown in [23]: Mode 1
A1 =
1.5 0 1 0 0 0 , Ad1 = , B1 = , Bω1 = 0 1 0 0.5 1 1
CE
AC
0.5
C1 =
0
0 0
,
0 0
H11 =
0.1 0
E31 =
0
0.1
,
,
0
0.5
H21 =
0 0
Cd1 =
E41
0 0
00
0.5 ,
,
D1 =
,
E11 =
0
0.1
0
0.1 0
,
0 0
1
Dω1 =
,
0
00
,
0.2
E21 =
0
0.1 0 0 = , H31 = 0 0 , E51 = E61 = 0 0 0
23
0.1 0
0 0
,
.
ACCEPTED MANUSCRIPT Mode 2 1
A2 =
C2 =
1
01
,
0
Ad2 =
1
0 0.5
,
0 1 B2 = , Bω2 = 1 0
0 0
,
0.5 0 1 0.1 0 0.5 , Dω2 = , D2 = , Cd2 = 0 0 0 0 0 0
H12 =
0 0
0.1 0
0.1
,
0
H22 =
0
00
,
0.1 0
0
E12 =
0.1
0 0
,
0 0
0
E22 =
,
0.1
0 0
,
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0 0
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. , H32 = 0 0 , E52 = E62 = , E42 = 00 0 0 0 In addition, the transition rate matrix is given by
E32 =
Pr =
0.7 0.3
0.4 0.6
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1 0 0 0 Let E = , R = and given the initial values for R1 = R2 = 01 00 I2 , c1 = 1, N = 5, d = 1, d1 = d2 = 1, µ = 2.1094, we can obtain that the controller gains are:
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K1 = −0.2389 −0.7260 , K2 = −0.9918 −0.7720 .
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And using Eq. (48) we can obtain the optimal value c2 = 19.0707 while in [18] c2 = 33.7066, which shows that the proposed methods are better than that in [23].
Example 2 Consider a two-mode time-delay singular Markovian jump system and the parameters are as follows: Mode 1 24
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40 35
γ
30 25 20 15 60 50
3.5 3
40 20
c2
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2.5
30
2 1.5
µ
Fig. 1. The local optimal bound of c2 and γ −3
12
x 10
10
u(k)
6
4
2
0
−2
0
5
10
15
20
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8
25 time(k)
30
35
40
45
50
40
45
50
40
45
50
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Fig. 2. The trajectory of u (k)
ED
2.5
1.5
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Jumping modes
2
1
0
5
CE
0.5
10
15
20
25 time(k)
30
35
Fig. 3. The operation modes
−3
x 10
14 12 10
8 x1
AC
16
6 4 2 0 −2
0
5
10
15
20
25 time(k)
30
35
Fig. 4. The trajectory of x1 (k)
25
ACCEPTED MANUSCRIPT −4
1
x 10
0
−1
x2
−2
−3
−4
−5
−6
−7
0
5
10
15
20
25 time(k)
30
35
40
45
50
−0.4 1.2 0.02 0 0.5 0.1 0.2 , , Bω1 = , B1 = , Ad1 = 0 0.01 0 0 −0.1 0 1.2
−0.7
C1 =
−0.2
0.09
1
H11 =
0.01 0
E31 =
0.1
E61 =
0.3 0
0 0
0.4
,
,
,
E41
ED
,
1.3 0 3 D1 = , Dω1 = , 0.2 0.1 2
−0.1
E11 =
0.2
0.5 0
,
0.1
E21 =
1
1 −0.7
0.09 0.8
0.4
0
,
0.5 0.1 1.3 0 1 0.03 0.05 , Ad2 = , B2 = , Bω2 = , 0.01 2.8 0 0.1 0 0 0.01 ,
0.16
Cd2 =
1.7
1.3 −0.1
,
0.1 0 1 D2 = , Dω2 = , 0.2 1.1 2
0.01 0 0.1 0.2 0.01 0.5 0.1 0.8 = , H22 = , E12 = , E22 = , 0 0 0.1 0.3 1 1 1.6 0
AC
H12
0.2
0.1 0.03
.
CE
C2 =
0.1
PT
0.25 −0.1
,
0.1 0.4 0.01 0.3 = , H31 = 0.01 0.01 , E51 = , 1 0 1 0
Mode 2 A2 =
0.6 0.3
Cd1 =
H21 =
0.1 0.4
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A1 =
M
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Fig. 5. The trajectory of x2 (k)
0.04
E32 =
1
,
E42
0.31 0.4
E62 =
0.1 0.01
0.1 0.4 0.1 0.48 = , H32 = 0 0.01 , E52 = , 1 0 0.1 0.02
.
26
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x 10
xT(k)ETRiEx(k)
2
1
0
0
5
10
15
20
25 time(k)
30
35
40
45
50
In addition, the transition rate matrix is given by
0.2
Pr =
1
0.8
0.6 0.4
0
.
0
0
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Fig. 6. The trajectory of xT (k) E T Ri Ex (k)
1
0
, and 01 23 00 N = 2, ε = 0.8, η = 0.5, d = 1, ρ1 = 0.1, ρ2 = 0.1, ρ3 = 0.1, c1 = 0.01, d1 = 1, d2 = 2. Using theorem 4, one can know that the optimal value of c22 + γ 2 relies on the value of µ. Using the LMI toolbox in Matlab and solving the LMIs in Theorem 4, remark 2 and the LMI optimization problems in remark 4, and we can find feasible solution when 1.53 ≤ µ ≤ 6.94. Fig.1 shows the optimal value with different value of µ. When µ = 1.8, we obtain the optimal value γ = 19.2759, c2 = 30.6733, the controller gains are as follows and the trajectory of the controller is shown in Fig.2.
R=
,
R1 = R2 =
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,
ED
M
Given the initial values for E =
K1 = 0.6963 −4.7419 , K2 = −0.4799 −3.8709 ,
PT
Kd1 = −0.1971 −0.1230 , Kd2 = −1.2617 −0.1330 .
AC
CE
The jumping signal is as shown in Fig.3. With corresponding Matlab program, we obtain the state response of the system that shown in Fig.4 and Fig.5. From Fig.6, it is easy to find that the jumping system is finite-time bounded. And the example shows that our proposed methods are feasible. Remark 6 It should be pointed out that, the parameter µ was introduced in the Lyapunov function to get less conservative results. And the optimal values c2 and γ depend on the parameter µ, the optimal values of c2 and γ with different value of µ was shown in Fig.1. It is also easily to find from the figure that when µ = 1.8, γ ≈ 19, c2 ≈ 31, which is consistent with the results calculated by the LMI toolbox. Remark 7 Fig. 1 shows the rk of the jump rate; Fig.3 and Fig. 4 shows 27
ACCEPTED MANUSCRIPT provide the state response of the system. From the figures provided, the controllers we design guarantee the resulted closed-loop systems finite-time stable and finite-time bounded. That is also illustrate that the proposed methods are correct and feasible. For given c1 , c2 and γ, using remark 3 we can obtain the largest estimate of the domain of the attraction, the following example can illustrate this.
−0.4 0
A1 =
0
0.5
,
0.23 0.01
Ad1 =
0
0
,
0 0.01
Bω1 =
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Example 3 Consider the following two-mode time-delay singular Markovian jump system :
0 0
,
0.3 0
Dω1 =
0.2 0.1
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−0.1 0.1 0 0.01 0.2 0.03 0 A2 = , , Bω2 = , B2 = , Ad2 = 0 0 0 0 0.1 0.01 1.5 the transition rate matrix is
,
0.1 0.9 . 0.5 0.5
M
Pr =
PT
ED
And the other matrix variables are defined similarly as example 2. We choose µ = 1.8, N = 2, ε = 0.8, η = 0.7, d = 1, ρ1 = 0.2, ρ2 = 0.3, ρ4 = 0.1, γ = 3, c1 = 0.001, c2 = 12, d1 = 1, d2 = 2. Solving the LMI optimization problem in remark 3 by the LMI toolbox in Matlab, we obtain the value of δmax = 0.3319. And the controller gain matrices are:
CE
K1 = 0.6388 1.0017 , K2 = −0.0467 1.7517 ,
AC
Kd1 = −0.3955 −0.0565 , Kd2 = −0.7021 −0.0813 .
Example 4 In order to illustrate utilization of the proposed methods, A simple economic system based on Samuelson’s multiplier-accelerator model was analyzed in the example and the detail model description can refer to [45]. A three-mode Markovian system was considered in this example. Mode 1 stands for the marginal propensity to save is in mid-range (Norm), mode 2 stands for the marginal propensity to save is in low range (Boom) and mode 3 stands for the marginal propensity to save is in high range (Slump) and the transition 28
ACCEPTED MANUSCRIPT rates is
Pr =
0.2 0.2
0.5 0.3
0.4
. 0.4
0.2 0.2 0.6
And the corresponding coefficient matrices of each mode are shown as follows: Mode 1 0.6
A1 =
0.2
0 1.2
−1.3
C1 =
0
0.2 1.2
0.1
Ad1 =
,
0.2
0 −0.1
Cd1 =
1
,
0
0.25 0.5
0.4 ,
B1 =
0
0.1
Bω1 =
0
0.1 0
,
0.5 0 1 D1 = , Dω1 = , 0.8 0.2 1
,
1.1
E31 =
0.8
,
1.1 0.5
1.3 1.4 = . 0.2 0.8
C2 =
0
0.1 1.5
,
,
0.8
Ad2 =
PT
−0.4
A2 =
0.01
H31 =
ED
Mode 2
0.5 1.2
E41 =
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0.01 0 0.6 0.6 −0.6 0.2 0.2 0 = , , E21 = , E11 = , H21 = 0.6 0.1 0.2 0 0.1 0.05 0 0
E61
,
0.01
,
0.1 0.9
E51 =
0.3 0
,
M
H11
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0
0 0.1
,
0 1 B2 = , Bω2 = 0 0
0 0
,
1.5 −0.5 1.3 1.2 1 0.3 0 , Cd2 = , D2 = , Dω2 = , 0.7 0.8 0.3 −1.1 1 0.1 0.5
CE
H12 =
0.01 0 0.1 0.2 0.3 0.5 1.2 0.6 , H22 = , E12 = , E22 = , 0 0 0.1 0.3 0.5 0.5 0.3 0
E32 =
1.9
AC
0.6
E62
,
E42 =
0.3 0.44 = . 0.5 0.2
0.6 0.4
1.3 0.2
,
0
H32 =
Mode 3 29
0.01
,
0.5 0.8
E52 =
0.61 0.2
,
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−3
4
x 10
2 0 −2
u(k)
−4 −6 −8 −10 −12 −14
0
5
10
15
20
25 time(k)
30
35
40
45
50
4
3.5
Jumping modes
3
2.5
2
1.5
1
0
5
10
15
20
25 time(k)
30
35
40
45
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0.5
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Fig. 7. The trajectory of u (k)
50
Fig. 8. The operation modes
C3 =
−0.2 0.3 1 0 0.02 0.8 0.1 , Ad3 = , B3 = , Bω3 = , 0.01 2.2 0 0.1 0 0 0
0.5 −0.6 0.9 1.8
0.01 0
0 0
0.5
E33 =
,
1.5
,
0.31 0.4
E63 =
Cd3 =
0.1 0.1
In addition, let
0.25 1.3 1 0.3 0 , D3 = , Dω3 = , 1.1 −0.2 0.2 0.2 0.1
0.2 0.1
H23 =
E43
CE
,
PT
H13 =
AC
M
A3 =
ED
0.1 0.3
,
0.2 0.5
E13 =
0.9 1.3
,
0.2 0.7
E23 =
1.3 0.1
0.2 1.4 0 0.2 0.5 = , H33 = , E53 = , 1.2 0 0.01 0.1 0.2
,
.
1 0
E=
00
,
and µ = 2, N = 2, ε = 0.8, η = 0.5, d = 1, ρ1 = ρ2 = ρ3 = 0.1, c1 = 0.001, d1 = 0, d2 = 2. Using the LMI toolbox in Matlab and solving the LMIs in Theorem 4, remark 2 and the LMI optimization problems in remark 4, we 30
ACCEPTED MANUSCRIPT
−3
16
x 10
14 12 10 8 x1
6 4 2 0 −2 −4
0
5
10
15
20
25 time(k)
30
35
40
45
50
−7
4
x 10
3
2
x2
1
0
−1
0
5
10
15
20
25 time(k)
30
35
40
45
50
45
50
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−2
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Fig. 9. The trajectory of x1 (k)
Fig. 10. The trajectory of x2 (k) −4
x 10
M
xT(k)ETRiEx(k)
2
1
0
5
10
15
ED
0
20
25 time(k)
30
35
40
Fig. 11. The trajectory of xT (k) E T Ri Ex (k)
CE
PT
can obtain the optimal values γ = 6.6958, c2 = 7.1445, the controller gains are as follows and the trajectory of the controller is shown in Fig.7.
K1 = −1.1660 8.7236 , K2 = 0.3806 3.9298 , K3 = 0.2080 5.9020 ,
AC
Kd1 = −0.4478 3.5029 , Kd2 = −0.6998 0.6338 , Kd3 = −0.7997 0.1025 .
Choose the jumping signal as shown in Fig.8. With corresponding Matlab program, we obtain the state response of the system that shown in Fig. 9 and Fig. 10. Otherwise, from Fig. 11, it is easy to find the system is finite-time bounded. In this example, by controlling the system, the states of the system stay in a 31
ACCEPTED MANUSCRIPT stable state. That is to say, the consumption and saving of people will not have great influences on a certain area.
5
Conclusion
Acknowledgments
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6
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The problem of robust finite-time robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems with actuator saturation has been discussed in this paper. Sufficient conditions have been derived, which guarantee that the systems are singular stochastic finite-time boundedness and singular stochastic finite-time H∞ boundedness. Then the non-fragile memory feedback controllers have been designed and can be obtained by solving LMIs. Moreover, the results are extended to optimization LMIs so that the optimal values of the attenuation level γ , boundeness c2 and the largest estimation of the domain of attraction can be obtained directly. Finally, some numerical examples are presented to show the effectiveness and the feasibility of the proposed methods. It should be pointed out that, as analyzed in the before, semi-Markovian jumping system has much broader applications and this system will be our future works focus on.
CE
PT
ED
This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei Province No. F2014203085. Moreover, the authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.
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ACCEPTED MANUSCRIPT [44] T. S. Hu, Z. L. Lin, B. M. Chen, Analysis and design for discrete-time linear systems subject to actuator saturation, Systems and Control Letters. 45 (2002) 97-112.
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[45] W. P. Blair, D. D. Sworder, Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria. International Journal of Control. 21 (1975) 833-844.
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Robust finite-time non-fragile memory H control for discrete-time singular Markovian jumping systems subject to actuator saturation Yuechao Ma, Xiaorui Jia, Qingling Zhang PII: DOI: Reference:
S0016-0032(17)30540-9 10.1016/j.jfranklin.2017.10.019 FI 3192
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
4 August 2016 10 October 2017 17 October 2017
Please cite this article as: Yuechao Ma, Xiaorui Jia, Qingling Zhang, Robust finite-time non-fragile memory H control for discrete-time singular Markovian jumping systems subject to actuator saturation, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.10.019
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ACCEPTED MANUSCRIPT Highlights
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• A new control problem for discrete singular Markov system is investigated. • Lyapunov function method, LMI technique and convex optimization are used. • The non-fragile memory controller is designed. • Both actuator saturation and time-varying delay are considered. • The derived conditions are less conservative have wider application range.
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Robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems subject to actuator saturation
Yuechao Ma a , Xiaorui Jia a,∗ , Qingling Zhang b
of Science, Yanshan University, Qinhuangdao Hebei, 066004, P.R.China;
b College
of Science, Northeatern University, Shenyang Liaoning, 110004, P.R.China;
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a College
Abstract
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This paper considers the problem of robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems with actuator saturation. By using the multiple Lyapunov function approach and the linear matrix inequalities (LMIs) techniques, novel sufficient conditions are obtained that ensure the systems singular stochastic finite-time bounded and singular stochastic finitetime H∞ bounded are obtained. Then the non-fragile memory feedback controllers are designed and the results are extended to LMI convex optimization problems to get the optimal values of attenuation level and the estimation of saturation attraction domain. Finally, some numerical examples are provided to illustrate the viability and effectiveness of the proposed methods.
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Key words: Finite-time stability and boundedness; discrete-time singular Markovian jumping system; non-fragile memory control; actuator saturation.
? This work is partially supported by the National Natural Science Foundation of China, under Grant number 61273004, and the Natural Science Foundation of Hebei province No. F2014203085. ∗ Corresponding author. Email address: [email protected] (Xiaorui Jia).
Preprint submitted to Elsevier
28 October 2017
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Introduction
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Singular systems, which are also referred to as descriptor systems, differentialalgebraic systems, or generalized state-space systems, have been extensively studied over the past decades due to the fact that these systems have been widely applied in many scientific areas. For example, the circuits systems, chemical processes and many other aspects. During recent years, many results about singular systems have been researched. For example, stabilization and stability analysis [1,2], filtering problems [3,4], observer-design [5], control problems [6] and so forth. When systems experience random abrupt changes in their structures parameters, it is natural to model them as Markovian jumping systems or semi-Markovian jumping system. Semi-Markovian jumping system are characterized by a fixed matrix of transition probabilities and a matrix of sojourn time probability density functions. Considerable attention has been devoted to the analysis and research of this kind of systems. And in recent decades many important and significant results have been obtained. In [7], Ding etc. studied the problem of filtering for discrete-time singular time delay Markovian jumping systems. The problem of delay-dependent stochastic admissibility for discrete-time singular Markovian jump systems was investigated in [8]. Quantized control for nonlinear semi-Markovian networks system was considered in [9], siding control for semi-Markovian jumping system was researched in [10]. [11] has studied the observer-based control for singular Markovian jump system and [12] investigated the stabilization analysis for fractional uncertain Markovian system. And many other results can be found in [13-16] and the references therein.
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It should be pointed out that although traditional Lyapunov stability, which pays more attention to the asymptotic pattern of systems over an infinitetime interval, has been successfully applied in many control systems. In many practical systems, researchers prefer to consider the behavior of systems over a fixed finite time interval. Thus finite-time stability and finite-time boundedness were introduced [17,18]. And in recent years, with the development of Lyapunov function approach and linear matrix inequality techniques, many works about finite-time stability and finite-time control have been obtained. In [19], the problem of finite-time stabilization about uncertain singular Markovian jump systems was considered. Zhang, etc. investigated the finite-time unbiased filtering design problem in [20]. And many other results in [21-26] and so on. It is well known that nearly all practical systems are subject to actuator saturation which will cause instability and bad system performance. Due to this, many efforts were devoted to the saturation systems. Ma and Chen studied the problem of memory dissipative control for singular T-S fuzzy systems with actuator saturation and time-varying delay in [27]. Control problem for discrete3
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time Markovian system subject to actuator saturation was investigated in [28]. Observer design in [29], stability and stabilization in [30], output feedback control in [31] and so on. In addition, time-delay is also an unavoidable factor that causes bad system performance in various practical systems. The study of delay systems has attracted much attention these years. See examples in [32-36] and the references therein.
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On the other hand, the non-fragile controller design method is becoming an attractive topic in theory analysis and practical implementation, because it can minimize the cost of implementation, and allow for the adjustment online of control parameters. The main purpose of non-fragile control is to design a feedback controller that will be insensitive to some errors in control gains. Some results about non-fragile control can be found in [37-40]. Furthermore, a memory state feedback controller that contains input constraints can obtain less conservative conditions and have wider practical applications. And many researchers considered the memory controller designing. Zhao, etc. studied the memory feedback controller design in [41], the memory feedback control for uncertain singular T-S fuzzy system investigated in [42]. The problem of nonfragile control with memory state feedback for uncertain singular systems was researched in [43].
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Considering the above analysis, there were few results about finite-time nonfragile memory control for discrete-time singular Markovian jump systems with actuator saturation which is important and significant in engineer applications. Motivated by these, we carry out our study. In this paper, we research the robust finite-time non-fragile memory control for discrete-time singular Markovian systems with input saturation. First, based on proper Lyapunov functional and the LMI approach, sufficient conditions that ensure the system singular stochastic finite-time boundedness and finite-time boundedness are obtained. Then in order to get the optimal values of attenuation level and the largest domain of saturation attraction, the results are extended to convex optimization problems. At the end of the paper, some numerical examples are presented to demonstrate the effectiveness of the proposed methods. 4
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Problem formulation
Considering the following singular uncertain Markovian jumping systems with actuator saturation and time-varying delay: Ex (k
z (k)
¯ (rk ) sat (u (k)) + B ¯ω (rk ) ω (k) , + 1) = A¯ (rk ) x (k) + A¯d (rk ) x (k − d (k)) + B
¯ (rk ) sat (u (k)) + D ¯ ω (rk ) ω (k) , = C¯ (rk ) x (k) + C¯d (rk ) x (k − d (k)) + D
x (k) = ϕ (k) , k ∈ {−d2 , −d2 + 1, · · · , 0} ,
p
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(1) where x (k) ∈ R is the state vector, u (k) ∈ R is the control input, z (k) ∈ Rq is the control output vector and sat: Rp → Rp is the standard function defined as follows: n
sat (u (k)) = [sat (u1 (k)) , sat (u2 (k)) , · · · , sat (up (k))]T ,
ε
X
k∈N
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where sat (ui (k)) = sign (ui (k)) min {1, |ui (k)|} , ω (k) ∈ Rq is the disturbance input satisfies: |ω (k)|2
≤ d2 .
(2)
where 0 ≤ πij ≤ 1,
s P
j=1
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rk is discrete-time Markov stochastic process taking values in a finite space S = {1, 2, · · · , s} and the transition matrix Π = (πij )s×s , πij = Pr {rk+1 = j |rk = i }, πij = 1 for all i ∈ S.
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To simplify the notational presentation of this paper, for each possible rk = i, i ∈ S, a matrix F (rk ) will be represented by Fi ; for example, A (rk ) will be represented by Ai , by Ad (rk ), and so forth.
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d (k) is the time-varying delay satisfies 0 < d1 ≤ d (k) ≤ d2 , E ∈ Rn×n may be singular and we assume rank (E) = r ≤ n. A¯i = Ai + ∆Ai , A¯di = ¯i = Bi + ∆Bi , B ¯ωi = Bωi + ∆Bωi , C¯i = Ci + ∆Ci , C¯di = Cdi + Adi + ∆Adi , B ¯ ¯ ∆Cdi , Di = Di +∆Di , Dωi = Dωi +∆Dωi . And Ai , Adi , Bi , Bωi , Ci , Cdi , Di , Dωi are known real constant matrices with appropriate dimensions; ∆Ai , ∆Adi , ∆Bi , ∆Bωi , ∆Ci , ∆Cdi , ∆ and ∆Dωi are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form:
∆Adi ∆Bi ∆Bωi H1i = F1i (k) E1i E2i E3i E4i , ∆Ci ∆Cdi ∆Di ∆Dωi H2i
∆Ai
(3)
where H1i , H2i , E1i , E2i , E3i , E4i are known real constant matrices with appropriate dimensions and F1i (k) is unknown real and possibly time-varying matrices satisfying F1iT (k) F1i (k) ≤ I. 5
ACCEPTED MANUSCRIPT Now consider the following non-fragile memory state feedback controller: ¯ i x (k) + K ¯ di x (k − d (k)) , u (k) = K
(4)
¯ i = Ki + ∆Ki , K ¯ di = Kdi + ∆Kdi and Ki , Kdi are gain matrices to be with K determined, in addition ∆Ki , ∆Kdi are uncertain matrices described as:
∆Ki ∆Kdi = H3i F2i (k) E5i E6i ,
(5)
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where H3i , E5i , E6i are known matrices and F2i (k) is unknown matrix satisfying F2iT (k) F2i (k) ≤ I.
For a matrix Hi , Hdi , donate the j th row of Hi , Hdi as hij , hdij and define L (Hi , Hdi ) as L (Hi , Hdi ) = {x (k) ∈ Rn : |hij x (k) + hij x (k − d (k))| ≤ 1, j = 1, 2, · · · , p} .
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Let P ∈ Rn×n be a symmetric matrix and E T P E ≥ 0, α > 0 be a scalar, and donate by Ω E T P E, α the following set: n
o
Ω E T P E, α = x (k) ∈ Rn : xT (k) E T P Ex (k) ≤ α .
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Let D be the set of p × p diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as Dj , j = 1, 2, · · · , 2p , and denote Dj− = I − Dj . Clearly, if Dj ∈ D, then Dj− ∈ D. Lemma 1 (Hu et al. [44]) Let Fi , Hi ∈ Rn×n . Then for any x (k) ∈ L (Hi ), n
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sat (Fi x (k)) ∈ co Dj Fi x (k) + Dj− Hi x (k) , j = 1, 2, · · · , 2p ,
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or, equivalently,
sat (Fi x (k)) =
p
2 X
αj (k) Dj Fi + Dj− Hi x (k) ,
j=1
(6)
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where co stands for the convex hull, αj (k) are some scalars which satisfy 0 ≤ αj (k) ≤ 1 and
2p P
j=1
αj (k) = 1.
Denoting the state trajectory of system (1) with initial condition x (k) = ϕ (k) ∈ Cn,d2 [−d2 , 0] by x (k, ϕ), then the domain of attraction of the origin is: ∆ Γ = ϕ (k) , k = −d2 , −d2 + 1, · · · , 0 : lim x (k, ϕ) = 0 . k→∞
6
ACCEPTED MANUSCRIPT It should be pointed out that in practical it is impossible to determinate the exact domain of attraction. An estimate Bδ ⊂ Γ of the domain of attraction is given by: (
)
Bδ = ϕ (k) ∈ Cn,d2 [−d2 , 0] : max kϕ (k)k ≤ δ . [−d2 ,0]
¯ i x (k) + K ¯ di x (k − d (k)) sat K =
2p P
j=1
αj (k)
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From the above lemma, for any x (k) ∈ L (Hi , Hdi )
i
¯ i + Dj− Hi x (k) + Dj K ¯ di + Dj− Hdi x (k − d (k)) . Dj K
(7)
Applying (6) and (7) to system (1), we can obtain the close-loop system as follows: + 1) =
z (k) = x (k)
2p P
j=1
2p P
j=1
h
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Ex (k
h
i
¯ ωi ω (k) , αj (k) C¯ij x (k) + C¯dij x (k − d (k)) + D
= ϕ (k) , k ∈ {−d2 , −d2 + 1, · · · , 0} ,
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where
i
¯ωi ω (k) , αj (k) A¯ij x (k) + A¯dij x (k − d (k)) + B
(8)
¯i Dj K ¯ i + Dj− Hi , A¯dij = A¯di + B ¯ i Dj K ¯ di + Dj− Hdi , A¯ij = A¯i + B
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¯ i Dj K ¯ i + Dj− Hi , C¯dij = C¯di + D ¯ di Dj K ¯ di + Dj− Hdi . C¯ij = C¯i + D
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Lemma 2 (Zhang et al. [23]) Let matrix X, Y and Z of appropriate dimensions, where X is a symmetric matrix, then X + Y F (k) Z + [Y F (k) Z]T < 0,
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holds for all time varying matrix function F (k) satisfying F T (k) F (k) ≤ I for all k ∈ Zk≥0 , if and only if there exists a positive constant ε, such that holds X + εY Y T + ε−1 Z T Z < 0 holds. Lemma 3 (Ma et al. [27]) Given matrices X, Y, Z with appropriate dimensions, and Y is symmetric. Then the following inequality holds: −Z T Y Z ≤ X T Z + Z T X + X T Y −1 X. 7
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Lemma 4 (Ding et al. [7]) For any constant positive semi-definite symmetric matrix W ∈ Rn×n , two positive integers d2 and d1 satisfying d2 ≥ d1 ≥ 1, the following inequality holds:
i=d1
xT (i)W x (i) ≥
d2 X
i=d1
T
x (i) W
d2 X
i=d1
x (i) .
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(d2 − d1 + 1)
d2 X
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Definition 1 (Zhang et al. [5]) (1) The matrix pair (E, Ai ) is said to be regular if, for each i ∈ S, the characteristic polynomial det (sE − Ai ) is not identically zero; (2) The matrix pair (E, Ai ) is said to be causal if, for each i ∈ S, deg (det (sE − Ai )) = rank (E) .
Definition 2 (Singular stochastic finite-time boundedness (SSFTB)).The discretetime singular jumping system (8) is said to be SSFTB with respect to (c1 , c2 , N, Ri ). With 0 < c1 < c2 and Ri > 0, if the singular stochastic system is regular and causal in time ∀k ∈ {1, 2, · · · , N } and satisfies n
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E xT (k1 ) E T Ri Ex (k1 ) , kEk2 xT (k1 ) Ri x (k1 ) ≤ c21 ⇒ E xT (k2 ) E T R (rk ) Ex (k2 ) ≤ c22 , (9) where k1 ∈ {−d2 , −d2 + 1, · · · , 0} , k2 ∈ {1, 2, · · · , N }.
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Definition 3 (Singular stochastic H∞ finite-time boundedness(SSH∞ FTB)).The discrete-time singular jumping system (8) is said to be SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ). With 0 < c1 < c2 and Ri > 0, if the system is SSFTB with respect to (c1 , c2 , N, Ri ), and under the zero-initial condition the control output z (k) satisfies the following constrained condition: E
(N X
T
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z (k) z (k) < γ E
k=0
3
)
(N X
k=0
T
)
ω (k) ω (k) , ∀k ∈ {1, 2, · · · , N } .
(10)
Main results
In this section, we present a set of sufficient conditions for finite-time bundedness and finite-time H∞ boundeness of the closed-loop system (8).
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ACCEPTED MANUSCRIPT Theorem 1 For given scalars µ ≥ 1, the closed-loop system (8) for ev T 2 ery initial condition belongs to Ω E Pi E, γ µ is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist scalars γ > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and nonsingular matrices Ni such that following conditions holds:
n
0
0
0
0
A¯T dij
∗
−µd1 Q3
∗
∗
Ω33 Ω34
0
0
∗
∗
∗ Ω44
0
0
∗
∗
∗
∗ −µ
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
sup λmax P¯i
o
c21 +
i∈S
−Pˆi−1
0
∗
∗
−Z −1
∗
∗
∗
c21
+
i∈S
i∈S
n
2kEk2
c21
n
c21 + µ−N γ 2 d2 ≤ inf λmin P¯i i∈S
(11)
o
µ−N c22 . (12)
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i∈S
o
kEk2
c21
< 0,
−I
¯ 2i )} d2 µd2 −1 sup{λmax (Q
¯ 3i )} d12 (d1 +d2 −1)µd2 sup{λmax (Q
+2µd2 −1 d212 (d1 + d2 + 1) sup λmax Z¯i with
c21 +
C¯ijT C¯ T 0 0 T ¯ Dωi 0 0
0
∗
γ I
dij
0
T ¯ωi d12 B
2
kEk2
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kEk2
i∈S
d12 A¯T dij
T ¯ωi B
−N
¯ 1i )} d1 µd1 −1 sup{λmax (Q
i∈S ¯ 3i )} d2 µd2 −1 sup{λmax (Q
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Ω1 =
T ¯T A¯T ij d12 Aij − E
¯ωi NiT RT B
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T T ¯ Ω11 Ni R Adij
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T T ¯ Ω11 = −µE T Pi E + (1 + d12 ) Q3 + Q1 + Q2 + A¯T ij RNi + Ni R Aij , d12 = d2 − d1 ,
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s P Ω33 = −µd1 Q1 − µd1 +1 E T ZE, P˜i = πij Pj , j=1
Ω34 = µd1 +1 E T ZE, Ω44 = −µd2 Q2 − µd1 +1 E T ZE,
¯i Dj K ¯ i + Dj− Hi , A¯dij = A¯di + B ¯i Dj K ¯ di + Dj− Hdi , A¯ij = A¯i + B
¯ i Dj K ¯ i + Dj− Hi , C¯dij = C¯di + D ¯ di Dj K ¯ di + Dj− Hdi . C¯ij = C¯i + D And R is any constant matrix with appreciate dimensions and satisfying: T T E R = 0. And Ω E Pi E, α ∈ L (Hi , Hdi ). Moreover, the domain of at9
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s
γ 2 µ1−N − µ−N γ 2 d2 , Υ
(13)
where n
Υ = sup λmax E T Pi E i∈S
o
+ d1 µd1 −1 λmax (Q1 ) + d2 µd2 −1 λmax (Q2 ) + d2 µd2 −1 λmax (Q3 ) +
Proof: As
2p P
j=1
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+d12 (d1 + d2 + 1) µd2 λmax (Q3 ) + 2µd2 −1 d212 (d1 + d2 + 1) λmax E T ZE . αj (k) = 1, it follows that p
2 X
αj (k) Ω1 < 0.
(14)
j=1
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First, we should prove that the system is regular and causal. From (13), it is known that p
T
−µE Pi E+(1 + d12 ) Q3 +Q1 +Q2 +
2 X
p
αj (k) A¯T ij RNi +
j=1
2 X
αj (k) NiT RT A¯ij < 0.
T
−1
j=1
Ir 0
U EV =
0 0
,
U
j=1
¯ ¯ Aij1 Aij2
αj (k) A¯ij V =
A¯ij3 A¯ij4
,
R U
= R1 R2 ,
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Ni V = Ni1 Ni2 ,
2p P
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(15) For rank (E) = r ≤ n, there exist two nonsingular matrices U and V such that
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since E T R = 0, we have R1 = 0. Pre- and post-multiplying (15) by V T and V , respectively, this results in the following matrix inequality holds:
?
?
T T ¯ ? A¯T ij4 R2 Ni2 + Ni2 RAij4
< 0,
(16)
where ? will represents the matrix block we do not need. Then we have T T ¯ ¯ A¯T ij4 R2 Ni2 + Ni2 RAij4 < 0, which means Aij4 is nonsingular and we have proved the system (8) is regular and causal. Now, it is time to prove the system (8) is SSFTB with respect to (c1 , c2 , N, Ri ). Define that y (k) = x (k + 1) − x (k) , and choose the Lyapunov-Krasovskii functional as follows: 10
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V (k) =
4 X
Vi (k),
i=1
with
V1 (k) = xT (k) E T Pi Ex (k) , µk−1−s xT (s)Q1 x (s) +
s=k−d1 k−1 X
V3 (k) =
µ
k−1 X
µk−1−s xT (s)Q2 x (s) ,
s=k−d2 −d X1
k−s−1 T
x (s)Q3 x (s) +
k−1 X
µk−s−1 xT (s)Q3 x (s) ,
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k−1 X
V2 (k) =
θ=−d2 +1 s=k+θ
s=k−d(k)
V4 (k) = d12
−d 1 −1 k−1 X X
µk−s−1 y T (s)E T ZEy (s) .
θ=−d2 s=k+θ
∆V1 (k) = E {V1 (k + 1)} − V1 (k) = xT (k + 1) E T
N X
j=1
πij Pj Ex (k + 1) − µxT (k) E T Pi Ex (k) + (µ − 1) V1 (k) ,
k X
s=k+1−d1
k−1 X
k−1 X
s=k−d2 T
µk−s xT (s)Q1 x (s)
s=k−d1 k X
µk−s−1 xT (s)Q1 x (s) +
s=k−d1
µk−s xT (s)Q2 x (s) + (µ − 1)
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−
k−1 X
µk−s xT (s)Q1 x (s) −
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+ (µ − 1)
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∆V2 (k) = E {V2 (k + 1)} − V2 (k) =
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Then
µk−s xT (s)Q2 x (s)
s=k+1−d2 k−1 X k−s−1 T
µ
x (s)Q2 x (s)
s=k−d2
CE
= x (k) (Q1 + Q2 ) x (k) − µd1 xT (k − d1 ) Q1 x (k − d1 ) − µd2 xT (k − d2 ) Q2 x (k − d2 ) + (µ − 1) V2 (k) ,
∆V3 (k) = E {V3 (k + 1)} − V3 (k)
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=
k X
s=k+1−d(k+1)
+ (µ − 1) −
−d X1
µk−s xT (s)Q3 x (s) −
k−1 X
θ=−d2 +1 s=k+θ T
µk−s xT (s)Q3 x (s)
s=k−d(k)
µk−s−1 xT (s)Q3 x (s) +
−d X1
k X
µk−s xT (s)Q3 x (s)
θ=−d2 +1 s=k+θ+1
s=k−d(k) k−1 X
k−1 X
µk−s xT (s)Q3 x (s) + (µ − 1) d1 T
−d X1
k−1 X
µk−s−1 xT (s)Q3 x (s)
θ=−d2 +1 s=k+θ
≤ x (k) (1 + d12 Q3 ) x (k) − µ x (k − d (k)) Q3 x (k − d (k)) + (µ − 1) V3 (k) , 11
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k X
θ=−d2 s=k+θ+1 −d 1 −1 k−1 X X
+ (µ − 1) d12 =
d212 y T
−d 1 −1 k−1 X X
µk−s y T (s)E T ZEy (s) − d12
µk−s y T (s)E T ZEy (s)
θ=−d2 s=k+θ
µk−s−1 y T (s)E T ZEy (s)
θ=−d2 s=k+θ T
(k) E ZEy (k) − d12
k−d 1 −1 X
θ=k−d2 k−d 1 −1 X d1 +1
≤ d212 y T (k) E T ZEy (k) − d12 µ From lemma 4, we have
µk−s y T (θ)E T ZEy (θ) + (µ − 1) V4 (k)
θ=k−d2
k−d 1 −1 X
∆V4 ≤ d212 y T (k) E T ZEy (k) − µd1 +1
y T (θ)E T ZEy (θ) + (µ − 1) V4 (k) .
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= d12
k−d 1 −1 X
y T (θ)E T Z
Ey (θ)
x (k − d2 )]
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θ=k−d2 θ=k−d2 T d1 +1 2 T [x (k − d1 ) − x (k − d2 )]T E T ZE [x (k − d1 ) − = d12 y (k) E ZEy (k) − µ = d212 y T (k) E T ZEy (k) − µd1 +1 xT (k − d1 ) E T ZEx (k − d1 ) + 2µd1 +1 xT (k − d1 ) E T ZEx (k − d2 ) − µd1 +1 xT (k − d2 ) E T ZEx (k − d2 ) .
Considering that E T R = 0, then
2xT (k + 1) E T RNi x (k) = 0,
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let ξ (k) = x (k) x (k − d (k)) x (k − d1 ) x (k − d2 ) ω (k) ,
CE
PT
∆V (k) − (µ − 1) V (k) =
T T ¯ Ω11 Ni R Adij
T ξ (k)
0
0
¯ωi NiT RT B
0
0
0
∗
−µd1 Q3
∗
∗
Ω33 Ω34
0
∗
∗
∗ Ω44
0
∗
∗
∗
0
T ˆ + ΛT 1 Pi Λ1 + Λ2 ZΛ2 ,
∗
ξ (k)
(17)
AC
with " p # 2 2p 2p P P P ¯ωi ω (k) , αj (k) A¯ij x (k) αj (k) A¯dij x (k − d (k)) 0 0 αj (k) B Λ1 = Λ2 =
" j=1 2p P
j=1
j=1
αj (k) A¯ij − E x (k)
j=1
2p P
j=1
#
2p P ¯ωi ω (k) . αj (k) A¯dij x (k − d (k)) 0 0 αj (k) B j=1
According to Schur complement and Eq. (14), we have
E {V (k + 1)} − V (k) < (µ − 1) V (k) + µ−N γ 2 ω T (k) ω (k) ≤ (µ − 1) V (k) + µ−N γ 2 ω T (k) ω (k) , 12
ACCEPTED MANUSCRIPT then
n
o
E {V (k + 1)} < µE {V (k)} + µ−N γ 2 E ω T (k) ω (k) , notice that µ ≥ 1, we have
i∈S
≤ µk E {V (0)} + µk−N γ 2 d2 . 1
1
1
µk−j−1 ω T (k) ω (k)
j=0
(19)
CR IP T
E {V (k)} < µk E {V (0)} + sup {λmax (Qi )} E
k−1 X
(18)
1
1
1
− − − ¯ 1i Ri− 2 , Q2 = Ri− 2 Q ¯ 2i Ri− 2 , Q3 = Letting Pi = Ri 2 P¯i Ri 2 , Q1 = Ri 2 Q n o 1 1 1 −1 ¯ 3i Ri− 2 , Z = Ri− 2 Z¯i Ri− 2 , and noticing that E xT (k1 ) E T Ri Ex (k1 ) , kEk2 xT (k1 ) Ri x (k1 ) ≤ Ri 2 Q
1 2
1
V (0) = x (0) E Ri P¯i Ri2 Ex (0) + T
T
AN US
c21 , k1 ∈ {−d2 , · · · , 0} and the initial value of the Lyapunov-krasovskii functional can be described as:
−1 X
1
1
¯ 1i Ri2 x (s) µ−1−s xT (s) Ri2 Q
s=−d1 −1 X
+
µ
1 2
1 2
¯ 2i Ri x (s) + x (s) Ri Q
−1−s T
s=−d2 1
n
o
¯ 2i + µd2 −1 sup λmax Q
PT
i∈S
n
¯ 3i + µd2 sup λmax Q
CE
i∈S
n
o
o
−1 X
AC
1
1
n
−1 o X
i∈S
n
θ=−d2 +1 s=θ −1 1 −1 X o −d X
≤2
θ=−d2 s=θ
θ=−d2 s=θ
xT (s + 1) E T Ri ExT (s + 1) + xT (s) E T Ri Ex (s) ,
then 13
xT (s) Ri x (s)
s=−d2
(x (s + 1) − x (s))T E T Ri E (x (s + 1) − x (s)) ,
(x (s + 1) − x (s))T E T Ri E (x (s + 1) − x (s))
θ=−d2 s=θ −dP 1 −1 −1 P
−1 o X
xT (s)Ri x (s)
considering −dP 1 −1 −1 P
xT (s) Ri x (s)
s=−d1
¯ 3i xT (s) Ri x (s) + µd2 −1 sup λmax Q
s=−d2 −d −1 X1 X
1
µ−s−1 y T (s) E T Ri2 Z¯i Ri2 Ey (s)
θ=−d2 s=θ
i∈S
+ d12 µd2 −1 sup λmax Z¯i i∈S
−d −1 1 −1 X X
¯ 1i xT (s) E T Ri Ex (s) + µd1 −1 sup λmax Q
ED
≤ sup λmax P¯i n
1
¯ 3i Ri2 x (s) + d12 µ−s−1 xT (s)Ri2 Q
M
−1 X
θ=−d2 +1 s=θ
i∈S
1
¯ 3i Ri2 x (s) µ−s−1 xT (s)Ri2 Q
s=−d(0)
−d X1
+
−1 X
ACCEPTED MANUSCRIPT
n
V (0) ≤ sup λmax P¯i i∈S
d2 µ
d2 −1
kEk
i∈S
c21 +
sup λmax i∈S
+
n
n
¯ 1i d1 µd1 −1 sup λmax Q
o 2
¯ 3i Q
o
kEk
2
o
d2
i∈S 2
+
n
+ 2µd2 −1 d212 (d1 + d2 + 1) sup λmax Z¯i i∈S
o
2kEk
i∈S
kEk µd2 −1 d212
2
o
n
n
i∈S
c21 +
2kEk
o
ED
Z¯i
o
c21
!
T
n
n
≥ inf λmin P¯i
PT
i∈S
o
n
o
+ µk−N γ 2 d2 ,
T
kEk2
1
1 2
T
o
c21
c21
(21)
E {V (k)} ≥ E x (k) E Pi Ex (k) = E x (k) E Ri P¯i Ri2 Ex (k) T
c21
c21
¯ 2i d2 µd2 −1 sup λmax Q
i∈S 2
c21 +
i∈S
note that
o
¯ 3i d12 (d1 + d2 + 1) µd2 sup λmax Q
(d1 + d2 + 1) sup λmax
n
kEk2
M
k
i∈S
n
i∈S
c21 +
kEk o ¯ 3i Q
o
(20)
¯ 1i d1 µd1 −1 sup λmax Q
¯ 3i d2 µd2 −1 sup λmax Q
+ µk + 2µ
n
o
2
CR IP T
n
¯ ≤ µk sup λmax Pi
n
AN US
c21 .
Then we have
E {V (k)}
i∈S
c21 +
d12 (d1 + d2 + 1) µ sup λmax c21
n
¯ 2i d2 µd2 −1 sup λmax Q
o
E xT (k) E T Ri Ex (k) ,
(22)
it follows that n
n
+ µk + 2µ
k
i∈S
o
E xT (k) E T Ri Ex (k)
n
¯ ≤ µk sup λmax Pi
AC
i∈S
o
CE
inf λmin P¯i
n
o
2
kEk µd2 −1 d212
i∈S
c21 +
¯ 3i d2 µd2 −1 sup λmax Q i∈S
n
¯ 1i d1 µd1 −1 sup λmax Q o
kEk2
o
n
i∈S
c21 + n
kEk2
¯ 3i d12 (d1 + d2 + 1) µd2 sup λmax Q i∈S 2
c21 + n
(d1 + d2 + 1) sup λmax i∈S
14
2kEk
Z¯i
o
c21
!
¯ 2i d2 µd2 −1 sup λmax Q o
+ µk−N γ 2 d2 . (23)
c21
o
c21
ACCEPTED MANUSCRIPT Based on Eq. (12), it obtained that n
o
E xT (k) E T Ri Ex (k) < c22 , ∀k ∈ {1, 2, · · · , N } ,
(24)
from definition 2, we know the singular system is SSFTB with respect to (c1 , c2 , N, Ri ). On the other hand, one can derive from (18) that the following inequality:
holds for all i ∈ S. It follows that
n
(25)
CR IP T
E {V (k + 1)} − µV (k) + z T (k) z (k) − γ 2 µ−N ω T (k) ω (k) < 0, o
n
o
E {V (k + 1)} < µE {V (k)} − E z T (k) z (k) + µ−N γ 2 E ω T (k) ω (k) , (26) then n
o
µk−j−1 E z T (j) z (j) +γ 2 µ−N
j=0
AN US
k−1 X
E {V (k)} < µk E {V (0)}−
k−1 X
n
j=0
(27) Under the zero initial condition and noticing that V (k) ≥ 0, we can obtain that n
o
µk−j E z T (j) z (j) < γ 2 µ−N
j=0
as µ ≥ 1, it follows that E
(N X
)
T
2
z (k) z (k) < γ E
k=0
T
n
o
µk−j E ω T (j) ω (j) ,
)
ω (k) ω (k) , ∀k ∈ {1, 2, · · · , N } .
ED
k=0
(N X
k X
j=0
M
k X
(28)
(29)
Then from definition 3, the system is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ).
PT
In addition, according to (17) and (18), we have
CE
xT (k) E T Pi Ex (k) ≤ V (k) ≤ µk V (0) + µk−N γ 2 d2 ≤ γ 2 µ,
(30)
AC
then it follows that kϕ (k)k2 ≤
γ 2 µ1−N − µ−N γ 2 d2 ∆ = = (ϕ) , Υ
(31)
for any ϕ ∈ = (ϕ), it follows that = (ϕ) ≤ γ 2 µ. From (25), we can further get xT (k) E T Pi Ex (k) ≤ γ 2 µ for any ϕ ∈ = (ϕ), which means that all the trajectories of x (k) which start from the set = (ϕ) always stay in the domain T Ω E Pi E, γ 2 µ . Based on the above discussion, the proof is thus complete.
15
o
µk−j−1 E ω T (j) ω (j) .
ACCEPTED MANUSCRIPT Theorem 2 For given scalars µ ≥ 1, ε > 0 the closed-loop system (8) for every initial condition belongs to Ω E T Pi E, η 2 u is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and nonsingular matrices Gi such that Eq. (12) and the following conditions holds:
T T ˆ Ω2 εO1 O2
where ˆ2 = Ω
∗ −µ
d1
0
GT i Q3 Gi
0
0
R Bωi
ˆT GT i Aij Wi
0
ˆT GT i Adij Wi
T
0
∗
∗
Ω33 Ω34
0
∗
∗
∗ Ω44
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
ED
R Aˆdij Gi T
d12 GT i
T AˆT ij − E
ˆT d12 GT i Adij
ˆT GT i Cij
ˆT GT i Cdij
AN US
ˆ Ω11
M
(32)
CR IP T
< 0, Ω2 = ∗ −εI 0 ∗ ∗ −εI
0
0
0
0
0
0
T Bωi Wi
T d12 Bωi
T Dωi
−P −1
0
0
∗
−Z −1
0
∗
∗
−I
,
O2 =
PT
O1 = O11 O12 0 0 E4i 0 0 0 , T H1i R
h
0000
T H1i Wi
T d12 H1i
CE
¯ i + Dj− Hi O11 = E1i + E3i Dj K
i
T H2i
, h
¯ di + Dj− Hdi Gi , O12 = E2i + E3i Dj K
i
Gi ,
AC
ˆ 11 = −µGT E T Pi EGi + (1 + d12 ) GT Q3 Gi + GT Q1 Gi + GT Q2 Gi + GT AˆT R + Ω i i i i i ij RT Aˆij Gi , Wi =
h√
πi1 ,
√
πi2 , · · · ,
√
i
n
o
πis , P −1 = diag P1−1 , P2−1 , · · · , Ps−1 ,
¯ i + Dj− Hi , Aˆdij = Ai + Bi Dj K ¯ di + Dj− Hdi , Aˆij = Ai + Bi Dj K
¯ i + Dj− Hi , Cˆdij = Ci + Di Dj K ¯ di + Dj− Hdi , Cˆij = Ci + Di Dj K Ω33 , Ω34 , Ω44 are the same as theorem 1, R is any constant matrix with appre16
ACCEPTED MANUSCRIPT
ciate dimensions and satisfying: E T R = 0. And Ω E T Pi E, η 2 u ∈ L (Hi , Hdi ) , the saturation attraction domain is given by (13). n
T Proof: Let Gi = Ni−1 , Pre- and post-multiplying (26) by diag GT i , Gi , I, I, I, I, I, I and its transpose, we have
Ω0 2
0
¯ωi RT B
∗ −µd1 GT i Q3 Gi 0
0
0
¯T GT i Adij
¯T d12 GT i Adij 0
¯T GT i Cij T ¯T G C
CR IP T
=
RT A¯dij Gi
T T ¯T ¯T GT i Aij d12 Gi Aij − E
0
0 Ω 11
i
dij
∗
∗
Ω33 Ω34
0
0
∗
∗
∗ Ω44
0
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
−Pˆi−1
∗
∗
∗
∗
∗
∗
−Z −1
0
∗
∗
∗
∗
∗
∗
∗
−I
¯T B ωi
< 0,
0
¯T d12 B ωi
¯T D ωi
0
0
(33)
M
with
0
0
AN US
ED
T T T T T ¯T Ω011 = −µGT i E Pi EGi + (1 + d12 ) Gi Q3 Gi + Gi Q1 Gi + Gi Q2 Gi + Gi Aij R + RT A¯ij Gi . Considering(3) we can obtain that
ˆ 2 + O1T F1iT (k) O2 + O2T F1i (k) O1 < 0. Ω02 =Ω
(34)
PT
As F1iT (k) F1i (k) ≤ I, according to lemma 2, there exists a positive constant ε such that
CE
O1T F1iT (k) O2 + O2T F1i (k) O1 ≤ εO1T O1 + ε−1 O2T O2 ,
(35)
AC
therefore, the following inequality implies that (32) holds: ˆ 2 + εOT O1 + ε−1 OT O2 < 0. Ω 1 2
(36)
Using Schur complement lemma, Eq. (36) is equality to Eq. (32), then the proof of the theorem is completed.
¯ i Gi = Remark 1 If we only consider the memory controller, we can let K ¯ i , Hi Gi = Yi , K ¯ di =X ¯ di , Hdi Gi = Ydi . Then using the above theorem, the gain X 17
o
ACCEPTED MANUSCRIPT ¯i = X ¯ i G−1 ¯ ¯ −1 matrices can be obtained and K i , Kdi = Yi Gi . Theorem 3 Let Ki Gi = Xi , Kdi Gi = Xdi , Hi Gi = Yi , Hdi Gi = Ydi , for given scalars µ ≥ 1, ε, η > 0, the closed-loop system (8) for every initial conT 2 dition belongs to Ω E Pi E, η u is SSH∞ FTB with respect to (c1 , c2 , N, γ, Ri ), if there exist scalars θ, β1 , β2 , β3 , β4 > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z and matrices Gi , Xi , Yi , Xdi , Ydi such that the following conditions holds: T ˜ ˜T T Ω2 εO1 O2 ηO3
Ω3 =
∗ −εI
0
0
∗
∗ −εI
∗
∗
∗ −ηI
∗
∗
∗
0
∗
εE3i Dj H3i 0 0
−ηI
< 0,
(37)
AN US
O4T
CR IP T
θRi−1 < Pi−1 < Ri−1 , 0 < Q1 < β1 Ri , 0 < Q2 < β2 Ri , 0 < β3 Ri−1 < Q−1 3 , 0 < Z < β4 Ri , (38)
1 θ
+
d1 µd1 −1 β1 kEk2
+
d2 µd2 −1 β2 kEk2
+
d2 µd2 −1 kEk2 β3
+
d12 (d1 +d2 −1)µd2 2kEk2 β3
˜ Ω11
˜ 12 Ω
0
RT Bωi
˜ 16 Ω
˜ 17 Ω
0
0
˜ 26 Ω
˜ 27 Ω
0
0
∗
∗
∗ −µd1 GT i Q3 Gi 0 ∗
∗
Ω33 Ω34
0
∗
∗ Ω44
0
∗
∗
∗
∗ −µ−N γ 2 I
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
PT
∗
AC
CE
˜2 = Ω
0
ED
where
M
+2µd2 −1 d212 (d1 + d2 + 1) β4 c21 + µ−N γ 2 d2 < µ−N c22
c21
˜ 18 Ω
˜ 28 Ω
0
0 0 0 , T T T Dωi Bωi Wi d12 Bωi −1 −P 0 0 −1 ∗ −Z 0
−I
T T T T ˜ 11 = −µGT Ω i E Pi EGi + (1 + d12 ) Gi Q3 Gi + Gi Q1 Gi + Gi Q2 Gi
h
+ Ai Gi + Bi Dj Xi + Dj− Yi
iT
h
(39)
R + RT Ai Gi + Bi Dj Xi + Dj− Yi
˜ 12 =RT Adi Gi + Bi Dj Xdi + Dj− Ydi Ω
18
h
i
,
˜ 16 = Ai Gi + Bi Dj Xi + Dj− Yi , Ω
iT
Wi ,
ACCEPTED MANUSCRIPT h
˜ 17 = d12 Ai Gi + Bi Dj Xi + Dj− Yi Ω h
˜ 26 = Adi Gi + Bi Dj Xdi + Dj− Ydi Ω h
˜ 28 = Cdi Gi + Di Dj Xdi + Dj− Ydi Ω
iT
iT
iT
h
− T ˜ − d12 GT i E , Ω18 = Ci Gi + Di Dj Xi + Dj Yi
h
˜ 27 = d12 Adi Gi + Bi Dj Xdi + Dj− Ydi Wi , Ω ,
˜1 = O ˜ 11 O ˜ 12 0 0 E4i 0 0 0 , O
i
h
, O12 = E2i Gi + E3i Dj Xdi + Dj− Ydi
O3 = E5i Gi E6i Gi 01×6 ,
T T T Dj Bi Wi H3i
T T T Dj Bi R H3i
T T T Dj Bi d12 H3i
i
CR IP T
h
O11 = E1i Gi + E3i Dj Xi + Dj− Yi
T T T Dj Di H3i
,
AN US
, 0000 and the other variables are the same as theorem3, R is any constant matrix with appreciate dimensions and satisfying: E T R = 0, and Ω E T Pi E, η 2 u ∈ L (Hi , Hdi ) , the saturation attraction domain is given by (13). And the con−1 troller gain matrices are Ki = Xi G−1 i , Kdi = Xdi Gi . O4 =
0
M
Proof:Considering Eq. (5), we can obtain that Ω2 = Ω 3 + O3 0 0
ED
T
F2iT
+ O4 εE3i Dj H3i 0
PT
< 0,
AC
CE
where
T
(k) O4 εE3i Dj H3i 0
F2i (k) O3 0 0
(40)
˜ ˜T T Ω2 εO1 O2
Ω03 = . ∗ −εI 0 ∗ ∗ −εI
According to lemma 2, the following inequality is true:
O3 0 0
T
F2iT
(k) O4 εE3i Dj H3i 0
+ O4 εE3i Dj H3i 0
≤ η O3 0 0
T
T
F2i (k) O3 0 0
O3 0 0 + η −1 O4 εE3i Dj H3i 0
19
T
O4 εE3i Dj H3i 0 .
iT
,
iT
,
ACCEPTED MANUSCRIPT Then the following inequality guarantee (40) holds:
0
Ω3 ≤ Ω 3 + η O3 0 0 +η
−1
< 0,
T
O3 0 0
O4 εE3i Dj H3i 0
T
O4 εE3i Dj H3i 0
(41)
1
1
1
1
1
CR IP T
using Schur complement lemma and the above inequality is equivalent to (37). 1 −1 −1 −1 ¯ 1i Ri− 2 , Q2 = On the other hand , noting that Pi = Ri 2 P¯i Ri 2 , Q1 = Ri 2 Q 1
− ¯ 2i Ri− 2 , Q3 = Ri− 2 Q ¯ 3i Ri− 2 , Z = Ri− 2 Z¯i Ri− 2 , Ri 2 Q
n
it follows (38) that 1 < inf λmin P¯i n
¯ 2i β1 , sup λmax Q i∈S
o
i∈S
n
o
i∈So
¯ 3i < β2 , sup λmax Q i∈S
n
, sup λmax P¯i
o
i∈S
n
< β3 , sup λmax Z¯i i∈S
n
¯ 1i < θ, sup λmax Q o
< β4 ,
o
AN US
then it easily obtained that (39) is a sufficient condition to guarantee (12). This completes the proof.
Remark 2 Noting that (37) may be difficult to solve employing the LMI toolbox of Matlab, therefore we make some transitions below:
M
Using lemma 3, we have
ED
T T T −1 −µGT i E Pi EGi ≤ µGi E + µEGi + µPi ,
PT
d1 T d1 d1 −1 −µd1 GT i Q3 Gi ≤ µ Gi + µ Gi + µ Q3 .
AC
CE
ˆ3 = In addition considering the fact −X −1 ≤ −2ρI +ρ2 X, and let Pˆi = Pi−1 , Q −1 Q3 we can replace (37) by the following inequality:
Ω4 =
˜ 0 εO ˜ 1T O2T ηO3T Ω 2
∗ −εI
0
0
∗
∗ −εI
∗
∗
∗ −ηI
∗
∗
∗
∗
∗
∗
0
O4T
M
0
0
0
0
∗
−ηI
0
∗
∗
N
20
εE3i Dj H3i 0
< 0,
(42)
<
ACCEPTED MANUSCRIPT where ˜ 02 Ω
˜0 Ω11
0
0
RT Bωi
˜ 16 Ω
˜ 17 Ω
d1 d1 ˆ ∗ µd1 GT i + µ Gi + µ Q3 0
0
0
˜ 26 Ω
˜ 27 Ω
∗
∗
Ω33 Ω34
0
0
0
∗
∗
∗ Ω44
0
0
0
∗
∗
∗
T ∗ −µ−N γ 2 I Bωi Wi
∗
∗
∗
∗
∗
−Pˆ
∗
∗
∗
∗
∗
∗
−2ρ1 I + ρ21 Z
∗
∗
∗
∗
∗
∗
∗
with h
T d12 Bωi
0
− T ˜ 11 = µGT ˆ Ω i E + µEi Gi + µPi + Ai Gi + Bi Dj Xi + Dj Yi
h
+RT Ai Gi + Bi Dj Xi + Dj− Yi
i
,
˜ 18 Ω
˜ 28 Ω
0
0 , T Dωi 0 0
CR IP T
=
˜ 12 Ω
AN US
iT
−I
R
˜ 2, and the other variables are the same as the in Ω
M =
GT i
M
N = diag {−2ρ2 I + ρ22 Q1 , −2ρ3 I + ρ23 Q2 , −2ρ4 I + ρ25 Q4 } , √ T 1 + d12 Gi
GT i
CE
PT
ED
. 08×1 08×1 08×1 And using Schur complement lemma Eq. (39) is equivalent to
υ1 c 1 I υ 2
< 0, ∗ −θ 0 ∗ ∗ β3
(43)
AC
where υ1 =
υ2 =
d1 µd1 −1 β1 kEk2
r
d2 µd2 −1 kEk2
+
+
d2 µd2 −1 β2 kEk2
+ 2µd2 −1 d212 (d1 + d2 + 1) β4 c21 +µ−N γ 2 d2 −µ−N c22 ,
d12 (d1 +d2 −1)µd2 c1 . 2kEk2
On the other hand, using the same method in [21], Ω E T Pi E, γ 2 u ∈ L (Hi , Hdi ) 21
ACCEPTED MANUSCRIPT is equivalent to
1 − γ2µ
hik
hdik
∗
−E T Pi E
0
∗
∗
−E T Pi E
≤ 0,
(44)
here hik is the k th row of Hin , and hdik isothe k th row of Hdi . Pre-and post T and its transpose, we have -multiplying Eq. (44) by diag I, GT i , Gi 1 − γ2µ
Yik
Ydik
∗
T −GT i E Pi EGi
0
∗
∗
T −GT i E Pi EGi
and a sufficient condition for (45) is 1 − γ2µ
Yik
Ydik
≤ 0,
AN US
CR IP T
∗
T ˆ GT i E + EGi + Pi
0
∗
∗
T ˆ GT i E + EGi + Pi
≤ 0.
(45)
(46)
ED
M
Remark 3 With all the ellipsoids satisfying the set invariance conditions of Theorem 1, we are interested in finding a large estimate of the domain of attraction. Based on the results obtained above, an optimization problem with LMI constraints can be formulated as follows: min κ
PT
ω1 I ∗
AC
CE
s.t.
with
ET Pˆi
ˆ Q3 I
≥ 0,
∗ ω4
≥ 0,
(47)
ω2 I − Q1 ≥ 0, ω3 I − Q2 ≥ 0, ω5 I − E T ZE ≥ 0,
κ = ω1 + d1 µd1 −1 ω2 + d2 µd2 −1 ω3 + d2 µd2 −1 ω4 + 0.5d12 (d1 + d2 + 1) µd2 ω4 + 2µd2 −1 d212 (d1 + d2 + 1) ω5 .
The positive scalars ω1 , ω2 , ω3 , ω4 , ω 5 are introduced to bound the different items in (13) for getting a larger estimate of the domain of attraction for the system. Therefore, a maximal estimate of the attraction domain can be obtained by: 22
ACCEPTED MANUSCRIPT
δmax =
s
γ 2 µ1−N − γ 2 µ−N d2 . κmin
Remark 4 The feasibility of conditions stated in theorem 4 can be turned into the following optimization problems with a fixed parameter µ
min c22 ˆ 3 > 0, Gi , Xi , Xdi , Yi , Ydi Pˆi , Q1 , Q2 , Q
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(48)
s.t. (38) (42) (43) (45) . min γ 2 + c22
ˆ 3 > 0, Gi , Xi , Xdi , Yi , Ydi Pˆi , Q1 , Q2 , Q
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s.t. (38) (42) (43) (45) .
(49)
Numerical examples
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4
M
Remark 5 In order to get less conservative and better results, we constructed multiple and novelty Lyapunov function, which made the algorithm seem to be complex. Thus we turned the results to LMIs and LMI optimization problems so that the results can be obtained by the LMI toolbox of Matlab directly.
PT
Example 1 When Kdi , ∆Ki , ∆Kdi = 0, it turns out to be a normal state feedback control problem and and the results are also suitable. Considering the same systems as shown in [23]: Mode 1
A1 =
1.5 0 1 0 0 0 , Ad1 = , B1 = , Bω1 = 0 1 0 0.5 1 1
CE
AC
0.5
C1 =
0
0 0
,
0 0
H11 =
0.1 0
E31 =
0
0.1
,
,
0
0.5
H21 =
0 0
Cd1 =
E41
0 0
00
0.5 ,
,
D1 =
,
E11 =
0
0.1
0
0.1 0
,
0 0
1
Dω1 =
,
0
00
,
0.2
E21 =
0
0.1 0 0 = , H31 = 0 0 , E51 = E61 = 0 0 0
23
0.1 0
0 0
,
.
ACCEPTED MANUSCRIPT Mode 2 1
A2 =
C2 =
1
01
,
0
Ad2 =
1
0 0.5
,
0 1 B2 = , Bω2 = 1 0
0 0
,
0.5 0 1 0.1 0 0.5 , Dω2 = , D2 = , Cd2 = 0 0 0 0 0 0
H12 =
0 0
0.1 0
0.1
,
0
H22 =
0
00
,
0.1 0
0
E12 =
0.1
0 0
,
0 0
0
E22 =
,
0.1
0 0
,
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0 0
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. , H32 = 0 0 , E52 = E62 = , E42 = 00 0 0 0 In addition, the transition rate matrix is given by
E32 =
Pr =
0.7 0.3
0.4 0.6
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1 0 0 0 Let E = , R = and given the initial values for R1 = R2 = 01 00 I2 , c1 = 1, N = 5, d = 1, d1 = d2 = 1, µ = 2.1094, we can obtain that the controller gains are:
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K1 = −0.2389 −0.7260 , K2 = −0.9918 −0.7720 .
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And using Eq. (48) we can obtain the optimal value c2 = 19.0707 while in [18] c2 = 33.7066, which shows that the proposed methods are better than that in [23].
Example 2 Consider a two-mode time-delay singular Markovian jump system and the parameters are as follows: Mode 1 24
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40 35
γ
30 25 20 15 60 50
3.5 3
40 20
c2
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2.5
30
2 1.5
µ
Fig. 1. The local optimal bound of c2 and γ −3
12
x 10
10
u(k)
6
4
2
0
−2
0
5
10
15
20
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8
25 time(k)
30
35
40
45
50
40
45
50
40
45
50
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Fig. 2. The trajectory of u (k)
ED
2.5
1.5
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Jumping modes
2
1
0
5
CE
0.5
10
15
20
25 time(k)
30
35
Fig. 3. The operation modes
−3
x 10
14 12 10
8 x1
AC
16
6 4 2 0 −2
0
5
10
15
20
25 time(k)
30
35
Fig. 4. The trajectory of x1 (k)
25
ACCEPTED MANUSCRIPT −4
1
x 10
0
−1
x2
−2
−3
−4
−5
−6
−7
0
5
10
15
20
25 time(k)
30
35
40
45
50
−0.4 1.2 0.02 0 0.5 0.1 0.2 , , Bω1 = , B1 = , Ad1 = 0 0.01 0 0 −0.1 0 1.2
−0.7
C1 =
−0.2
0.09
1
H11 =
0.01 0
E31 =
0.1
E61 =
0.3 0
0 0
0.4
,
,
,
E41
ED
,
1.3 0 3 D1 = , Dω1 = , 0.2 0.1 2
−0.1
E11 =
0.2
0.5 0
,
0.1
E21 =
1
1 −0.7
0.09 0.8
0.4
0
,
0.5 0.1 1.3 0 1 0.03 0.05 , Ad2 = , B2 = , Bω2 = , 0.01 2.8 0 0.1 0 0 0.01 ,
0.16
Cd2 =
1.7
1.3 −0.1
,
0.1 0 1 D2 = , Dω2 = , 0.2 1.1 2
0.01 0 0.1 0.2 0.01 0.5 0.1 0.8 = , H22 = , E12 = , E22 = , 0 0 0.1 0.3 1 1 1.6 0
AC
H12
0.2
0.1 0.03
.
CE
C2 =
0.1
PT
0.25 −0.1
,
0.1 0.4 0.01 0.3 = , H31 = 0.01 0.01 , E51 = , 1 0 1 0
Mode 2 A2 =
0.6 0.3
Cd1 =
H21 =
0.1 0.4
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A1 =
M
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Fig. 5. The trajectory of x2 (k)
0.04
E32 =
1
,
E42
0.31 0.4
E62 =
0.1 0.01
0.1 0.4 0.1 0.48 = , H32 = 0 0.01 , E52 = , 1 0 0.1 0.02
.
26
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x 10
xT(k)ETRiEx(k)
2
1
0
0
5
10
15
20
25 time(k)
30
35
40
45
50
In addition, the transition rate matrix is given by
0.2
Pr =
1
0.8
0.6 0.4
0
.
0
0
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Fig. 6. The trajectory of xT (k) E T Ri Ex (k)
1
0
, and 01 23 00 N = 2, ε = 0.8, η = 0.5, d = 1, ρ1 = 0.1, ρ2 = 0.1, ρ3 = 0.1, c1 = 0.01, d1 = 1, d2 = 2. Using theorem 4, one can know that the optimal value of c22 + γ 2 relies on the value of µ. Using the LMI toolbox in Matlab and solving the LMIs in Theorem 4, remark 2 and the LMI optimization problems in remark 4, and we can find feasible solution when 1.53 ≤ µ ≤ 6.94. Fig.1 shows the optimal value with different value of µ. When µ = 1.8, we obtain the optimal value γ = 19.2759, c2 = 30.6733, the controller gains are as follows and the trajectory of the controller is shown in Fig.2.
R=
,
R1 = R2 =
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,
ED
M
Given the initial values for E =
K1 = 0.6963 −4.7419 , K2 = −0.4799 −3.8709 ,
PT
Kd1 = −0.1971 −0.1230 , Kd2 = −1.2617 −0.1330 .
AC
CE
The jumping signal is as shown in Fig.3. With corresponding Matlab program, we obtain the state response of the system that shown in Fig.4 and Fig.5. From Fig.6, it is easy to find that the jumping system is finite-time bounded. And the example shows that our proposed methods are feasible. Remark 6 It should be pointed out that, the parameter µ was introduced in the Lyapunov function to get less conservative results. And the optimal values c2 and γ depend on the parameter µ, the optimal values of c2 and γ with different value of µ was shown in Fig.1. It is also easily to find from the figure that when µ = 1.8, γ ≈ 19, c2 ≈ 31, which is consistent with the results calculated by the LMI toolbox. Remark 7 Fig. 1 shows the rk of the jump rate; Fig.3 and Fig. 4 shows 27
ACCEPTED MANUSCRIPT provide the state response of the system. From the figures provided, the controllers we design guarantee the resulted closed-loop systems finite-time stable and finite-time bounded. That is also illustrate that the proposed methods are correct and feasible. For given c1 , c2 and γ, using remark 3 we can obtain the largest estimate of the domain of the attraction, the following example can illustrate this.
−0.4 0
A1 =
0
0.5
,
0.23 0.01
Ad1 =
0
0
,
0 0.01
Bω1 =
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Example 3 Consider the following two-mode time-delay singular Markovian jump system :
0 0
,
0.3 0
Dω1 =
0.2 0.1
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−0.1 0.1 0 0.01 0.2 0.03 0 A2 = , , Bω2 = , B2 = , Ad2 = 0 0 0 0 0.1 0.01 1.5 the transition rate matrix is
,
0.1 0.9 . 0.5 0.5
M
Pr =
PT
ED
And the other matrix variables are defined similarly as example 2. We choose µ = 1.8, N = 2, ε = 0.8, η = 0.7, d = 1, ρ1 = 0.2, ρ2 = 0.3, ρ4 = 0.1, γ = 3, c1 = 0.001, c2 = 12, d1 = 1, d2 = 2. Solving the LMI optimization problem in remark 3 by the LMI toolbox in Matlab, we obtain the value of δmax = 0.3319. And the controller gain matrices are:
CE
K1 = 0.6388 1.0017 , K2 = −0.0467 1.7517 ,
AC
Kd1 = −0.3955 −0.0565 , Kd2 = −0.7021 −0.0813 .
Example 4 In order to illustrate utilization of the proposed methods, A simple economic system based on Samuelson’s multiplier-accelerator model was analyzed in the example and the detail model description can refer to [45]. A three-mode Markovian system was considered in this example. Mode 1 stands for the marginal propensity to save is in mid-range (Norm), mode 2 stands for the marginal propensity to save is in low range (Boom) and mode 3 stands for the marginal propensity to save is in high range (Slump) and the transition 28
ACCEPTED MANUSCRIPT rates is
Pr =
0.2 0.2
0.5 0.3
0.4
. 0.4
0.2 0.2 0.6
And the corresponding coefficient matrices of each mode are shown as follows: Mode 1 0.6
A1 =
0.2
0 1.2
−1.3
C1 =
0
0.2 1.2
0.1
Ad1 =
,
0.2
0 −0.1
Cd1 =
1
,
0
0.25 0.5
0.4 ,
B1 =
0
0.1
Bω1 =
0
0.1 0
,
0.5 0 1 D1 = , Dω1 = , 0.8 0.2 1
,
1.1
E31 =
0.8
,
1.1 0.5
1.3 1.4 = . 0.2 0.8
C2 =
0
0.1 1.5
,
,
0.8
Ad2 =
PT
−0.4
A2 =
0.01
H31 =
ED
Mode 2
0.5 1.2
E41 =
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0.01 0 0.6 0.6 −0.6 0.2 0.2 0 = , , E21 = , E11 = , H21 = 0.6 0.1 0.2 0 0.1 0.05 0 0
E61
,
0.01
,
0.1 0.9
E51 =
0.3 0
,
M
H11
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0
0 0.1
,
0 1 B2 = , Bω2 = 0 0
0 0
,
1.5 −0.5 1.3 1.2 1 0.3 0 , Cd2 = , D2 = , Dω2 = , 0.7 0.8 0.3 −1.1 1 0.1 0.5
CE
H12 =
0.01 0 0.1 0.2 0.3 0.5 1.2 0.6 , H22 = , E12 = , E22 = , 0 0 0.1 0.3 0.5 0.5 0.3 0
E32 =
1.9
AC
0.6
E62
,
E42 =
0.3 0.44 = . 0.5 0.2
0.6 0.4
1.3 0.2
,
0
H32 =
Mode 3 29
0.01
,
0.5 0.8
E52 =
0.61 0.2
,
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−3
4
x 10
2 0 −2
u(k)
−4 −6 −8 −10 −12 −14
0
5
10
15
20
25 time(k)
30
35
40
45
50
4
3.5
Jumping modes
3
2.5
2
1.5
1
0
5
10
15
20
25 time(k)
30
35
40
45
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0.5
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Fig. 7. The trajectory of u (k)
50
Fig. 8. The operation modes
C3 =
−0.2 0.3 1 0 0.02 0.8 0.1 , Ad3 = , B3 = , Bω3 = , 0.01 2.2 0 0.1 0 0 0
0.5 −0.6 0.9 1.8
0.01 0
0 0
0.5
E33 =
,
1.5
,
0.31 0.4
E63 =
Cd3 =
0.1 0.1
In addition, let
0.25 1.3 1 0.3 0 , D3 = , Dω3 = , 1.1 −0.2 0.2 0.2 0.1
0.2 0.1
H23 =
E43
CE
,
PT
H13 =
AC
M
A3 =
ED
0.1 0.3
,
0.2 0.5
E13 =
0.9 1.3
,
0.2 0.7
E23 =
1.3 0.1
0.2 1.4 0 0.2 0.5 = , H33 = , E53 = , 1.2 0 0.01 0.1 0.2
,
.
1 0
E=
00
,
and µ = 2, N = 2, ε = 0.8, η = 0.5, d = 1, ρ1 = ρ2 = ρ3 = 0.1, c1 = 0.001, d1 = 0, d2 = 2. Using the LMI toolbox in Matlab and solving the LMIs in Theorem 4, remark 2 and the LMI optimization problems in remark 4, we 30
ACCEPTED MANUSCRIPT
−3
16
x 10
14 12 10 8 x1
6 4 2 0 −2 −4
0
5
10
15
20
25 time(k)
30
35
40
45
50
−7
4
x 10
3
2
x2
1
0
−1
0
5
10
15
20
25 time(k)
30
35
40
45
50
45
50
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−2
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Fig. 9. The trajectory of x1 (k)
Fig. 10. The trajectory of x2 (k) −4
x 10
M
xT(k)ETRiEx(k)
2
1
0
5
10
15
ED
0
20
25 time(k)
30
35
40
Fig. 11. The trajectory of xT (k) E T Ri Ex (k)
CE
PT
can obtain the optimal values γ = 6.6958, c2 = 7.1445, the controller gains are as follows and the trajectory of the controller is shown in Fig.7.
K1 = −1.1660 8.7236 , K2 = 0.3806 3.9298 , K3 = 0.2080 5.9020 ,
AC
Kd1 = −0.4478 3.5029 , Kd2 = −0.6998 0.6338 , Kd3 = −0.7997 0.1025 .
Choose the jumping signal as shown in Fig.8. With corresponding Matlab program, we obtain the state response of the system that shown in Fig. 9 and Fig. 10. Otherwise, from Fig. 11, it is easy to find the system is finite-time bounded. In this example, by controlling the system, the states of the system stay in a 31
ACCEPTED MANUSCRIPT stable state. That is to say, the consumption and saving of people will not have great influences on a certain area.
5
Conclusion
Acknowledgments
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6
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The problem of robust finite-time robust finite-time non-fragile memory H∞ control for discrete-time singular Markovian jumping systems with actuator saturation has been discussed in this paper. Sufficient conditions have been derived, which guarantee that the systems are singular stochastic finite-time boundedness and singular stochastic finite-time H∞ boundedness. Then the non-fragile memory feedback controllers have been designed and can be obtained by solving LMIs. Moreover, the results are extended to optimization LMIs so that the optimal values of the attenuation level γ , boundeness c2 and the largest estimation of the domain of attraction can be obtained directly. Finally, some numerical examples are presented to show the effectiveness and the feasibility of the proposed methods. It should be pointed out that, as analyzed in the before, semi-Markovian jumping system has much broader applications and this system will be our future works focus on.
CE
PT
ED
This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei Province No. F2014203085. Moreover, the authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.
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ACCEPTED MANUSCRIPT [44] T. S. Hu, Z. L. Lin, B. M. Chen, Analysis and design for discrete-time linear systems subject to actuator saturation, Systems and Control Letters. 45 (2002) 97-112.
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