Dissipative reliable controller design for uncertain systems and its application

Applied Mathematics and Computation 263 (2015) 107–121

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Dissipative reliable controller design for uncertain systems and its application R. Sakthivel a, M. Rathika b, Srimanta Santra b, Quanxin Zhu c,∗ a

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea Department of Mathematics, Anna University-Regional Centre, Coimbatore 641 047, India c School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China b

a r t i c l e

i n f o

Keywords: Reliable control Dissipative performance Delay fractioning approach Linear fractional uncertainties

a b s t r a c t This paper address the reliable robust strictly (Q, R, S) dissipative control problem for a class of uncertain continuous time systems with input delay and linear fractional uncertainties despite possible actuator failures in system model. In particular, by employing a novel Lyapunov functional together with delay fractioning approach, a reliable control law is designed in terms of the solution of certain linear matrix inequalities which makes the considered system strictly (Q, R, S) dissipative which can be easily solved by using the available software. At the same time, as special cases the H, passivity, mixed H and passivity control problems can obtained from the proposed dissipative control formulation. This explains the fact that the dissipative control unifies H control, passive control, and mixed H and passivity control in a single framework. Finally, a numerical example based on a mechanical system is given to demonstrate the effectiveness and applicability of the proposed design approach. More precisely, the proposed results have been compared through numerical simulation which reveals that the obtained criteria are considerably less conservative than some existing results. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In real real-world problems, time delay is unavoidable and it is well known that the time delay is one of the main sources of instability and poor performance [2,15,25,30,33–35]. For instance, in many control systems, delays may occur in the feedback loop of a plant, in the states, inputs or outputs. In the literature, dynamical linear and nonlinear systems with time-delay are mathematically modeled by differential or difference equations. Moreover, in modeling of dynamical systems, parameter uncertainties are unavoidable because of variations in system parameters, modeling errors or some ignored factors, so considerable attention has been given to both the problems of robust stability and robust control for uncertain systems with various types of time-delays [4,12,14,16,17]. Also, the time-delay systems cannot be always stabilized independent of time-delay, so delaydependent approaches are developed and some improved delay dependent stability and stabilization criteria for time-delay systems are reported in [5,6]. On the other hand, since actuator faults often occur in real process and may lead to unsatisfactory performance or even instability. A control system is said to be reliable if it retains certain properties when there exist failures [10]. It should be noted that in normal cases, a controller with fixed gain is easily implemented, and could meet the requirement in practical applications.

∗

Corresponding author. E-mail address: [email protected] (Q. Zhu).

http://dx.doi.org/10.1016/j.amc.2015.04.009 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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But when failure occurs, the traditional feedback control designs will become conservative and plant may result in unsatisfactory system performance. To improve system reliability and safety, it is necessary and important to design reliable control law so that the performance of the closed-loop system becomes well even in the presence of some actuator faults [1]. Therefore with the rapid development of LMI approach, the study on the reliable control problem has received much attention [18,27,29,32]. Alwan et al. [1] discussed the problem of designing a robust reliable H control for a class of uncertain stochastic systems with time delay and time-varying norm-bounded parametric uncertainties in the system states by using Lyapunov functions together with the Razumikhin methodology. The dissipative theory was introduced in [26] which plays an important role in system and control areas and many results have been reported so far, see [9,11,31]. It is worth mentioning that due to its simplicity in analysis and convenience in simulation, the LMI method has gained particular attention in dissipative control problems. Feng and Lam [10] studied the problem of robust reliable dissipative filtering for uncertain discrete-time singular system with interval time-varying delay and sensor failures. The problem of reliable filter design with strict dissipativity has been investigated in [24] for a class of discrete-time T-S fuzzy time-delay systems. Li and Liu [20] studied the problem of delay-dependent dissipative filtering for genetic regulatory networks with norm-bounded parameter uncertainties and time-varying delays. Shen et al. [23] designed a retarded feedback controller such that the nonlinear Markovian jump systems is stochastically stable and strictly (Q, R, S)-dissipative. Recently, the delay fractioning technique becomes an increasing interest of many researchers because it provides less conservative results while studying the dynamical behaviors of systems via Lyapunov approach when the fractioning number goes thinner [8,19,21,22]. More recently Mathiyalagan et al. [21] investigated the asymptotic stability issue for a class of stochastic Takagi–Sugeno fuzzy systems with time-varying delays by utilizing a delay-fractioning method. Many results are proposed for the stability and stabilization conditions for uncertain dynamical systems with delays by the use of the delay-fractioning or delay partitioning method [13]. Furthermore, how to guarantee the closed-loop control system performance in the case of actuator failures and external disturbances will be more tough and meaningful. However, to the best of our knowledge, the reliable strictly (Q, R, S) dissipative control problem for uncertain linear systems with actuators fault and input time varying delay is not fully investigated. Motivated by this consideration, in this paper we study the reliable strictly (Q , R, S) dissipative control problem against actuator failures for a class of uncertain linear continuous system with input time-varying delay. Further, a set of conditions are obtained for strictly (Q, R, S) dissipative performance of the closed-loop system by using the new Lyapunov–Krasovskii functional and LMI technique which are dependent on the lower and upper bounds of the time-varying delay. Then, we design a state feedback reliable dissipative control law based on the obtained LMI conditions. The controller gain is characterized in terms of the solution to a set of LMIs which can be easily solved by using available numerical software. Finally, the simulation results based on a mechanical system is given to show the effectiveness of the proposed design technique, where we can get a larger upper bound than other existing results. The main contribution of this paper are summarized as follows • A novel Lyapunov functional is introduced to the reliable strictly (Q, R, S) dissipative control systems. • Based on the delay fractioning approach, a less conservative condition is obtained for uncertain mechanical systems based on the Newton–Leibniz formula together with free weighting matrix approach. • Also, a unified delay-dependent LMI framework is developed which consisting of possible actuator failures, disturbances, time delays and LFT uncertainties. Notations: The notations used in this paper are fairly standard. The superscripts T and (−1) stand for matrix transposition and matrix inverse respectively; the notation P > 0 means that P is real, symmetric and positive definite; L2 [0, ∞) is the space of square-integrable vector functions over [0, ); Rn×n denotes the n × n dimensional Euclidean space; < z, Qz > T represents T T 0 z (t)Qz(t)dt; I and 0 represent the identity and zero matrix with compatible dimensions; diag{ · } denotes the block-diagonal matrix; we use an asterisk ∗ to represent a term that is induced by symmetry. sym(A) is defined as A + AT . Matrices which are not explicitly stated are assumed to be compatible for matrix multiplications col [.] denotes a matrix column with blocks given by the matrices in [.]. 2. Problem formulation and preliminaries In many real-world problems, input delays are frequently encountered in its dynamics and many researchers have studied problems of designing control law for the dynamical systems with input delays. In this paper, we consider the linear continuous system with input time varying delay in the following form

x˙ (t) = Ax(t) + Buf (t − d(t)) + Eω(t), z(t) = Cx(t), x(t) =

(1)

φ(t), ∀t ∈ [−d2 , 0] ,

where x(t) ∈ Rn , uf (t) ∈ Rm , z(t) ∈ Rl and ω(t) ∈ L2 [0, ∞) represents the state vector, the control input of actuator fault, the output vector and the disturbance input vector respectively; A, B, E and C are known real matrices of appropriate dimensions. Further, the unknown time-varying delay d(t) satisfies the following conditions

0 ≤ d1 ≤ d(t) ≤ d2 ,

d˙ (t) ≤ μ < 1,

(2)

where d1 , d2 and μ are positive constants and φ (t) is the initial function  t  [ − d2 , 0]. The time varying delay is represented into two ways: constant part and time varying part as d(t) = d1 + d∗ (t), 0  d∗ (t)  d2 − d1 .

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

109

The control input uf (t) of actuator fault can be described as

uf (t) = Gu(t) = GKx(t),

(3)

where K is the feedback gain to be designed and G is the actuator fault matrix defined as follows 0 ≤ G = diag{g 1 , . . . , gp } ≤ G = diag{g1 , . . . , gp } ≤ G = diag{g1 , . . . , gp } ≤ I in which the variables gi (i = 1, . . . , p) quantify the failures of the actuators. Let

us denote G0 = diag{g01 , . . . , g0p } = matrix G can be written as

G+G 2

= diag{

G = G0 +  = G0 + diag{θ1 , . . . , θp },

gp +gp g1 +g1 2 ,..., 2 },

G1 = diag{g11 , . . . , g1p } =

G−G 2

= diag{

|θi | ≤ g1i (i = 1, . . . , p).

gp −gp g1 −g1 2 ,..., 2 }.

The (4)

Moreover, uncertainty is frequently encountered in the dynamical systems, taking this into consideration, the system (1) with linear fractional transformation [LFT] uncertainty formulation can be written as [8]

x˙ (t) = (A + A(t))x(t) + (B + B(t))uf (t − d(t)) + Eω(t), z(t) = Cx(t),

(5)

where the LFT uncertainty satisfies the following condition (H1) The LFT uncertainty A(t) is assumed to be of the form A = Ma (t)[Na Nb ], where Ma , Na and Nb are known constant matrices of appropriate dimensions. The term (t) can be written in the form of linear transformation as follows (t) = [I − F(t)J]−1 F(t), where J is a known matrix satisfying I − JJT > 0 and F(t) is an unknown time varying matrix with Lebesgue measurable elements bounded by FT (t)F(t)  I. The following definition and lemmas are useful in the proof of the main results. Definition 2.1. [10] Given some scalar α > 0, matrices Q, R and S with Q and R real symmetric, system (1) is strictly (Q , R, S) dissipative if for any T > 0 under zero initial state, the following condition is satisfied:

< z, Qz >T +2 < z, Sω >T + < ω, Rω >T ≥ α < ω, ω >T .

(6)

 Without loss of generality, we assume that the matrix Q  0 and Q¯ = −Q.

Remark 2.2. It should be mentioned that the dissipative control problem under consideration here is more general than the H and passivity-based control problem. In fact, if Q = −I, S = 0, R = α I + γ 2 I the expression in (6) reduces to a H control problem; while if Q = 0, S = I and R = α I + γ I then the expression in (6) degenerates into a passivity control problem. If Q = −θ I, S = (1 − θ )I and R = θ γ 2 I, θ  [0, 1] or Q = −γ −1 θ I, S = (1 − θ )I and R = γ θ I, where θ  [0, 1] is a given scalar weight representing a trade off between H and passivity performance, then strictly (Q, R, S) dissipative conditions reduces to a mixed H and passivity performance. Lemma 2.3. [8] Given a positive definite matrix Z ∈ Rn×n , Z = ZT > 0 and scalars τ 1 < τ (t) < τ 2 , for vector function x(t) = [x1 (t) x2 (t) . . . xn (t)]T , we have

 −

t−τ1 t−τ2



−

x˙ T (s)Z x˙ (s)ds ≤ −

−τ1 −τ2



t t+θ

1 τ2 − τ1



x˙ T (s)Z x˙ (s)dsdθ ≤ −

t−τ1

t−τ2

T

x˙ (s)ds

2 τ22 − τ12



 Z

t−τ1 t−τ2



t−τ1 t−τ2

t t+θ

 x˙ (s)ds ,

x˙ (s)dsdθ



T Z

t−τ1 t−τ2



t t+θ

 x˙ (s)dsdθ .

Lemma 2.4. [21] Given the constant matrices 1 , 2 , 3 , where 1 = 1T and 0 < 2 = 2T . Then 1 + 3T 2−1 3 < 0 if and only if [

1 3T − ] < 0 or equivalently [ T 2 3 ] < 0. 3 − 2 3 1

Lemma 2.5. [29] Let x ∈ Rn and y ∈ Rn , then for any positive scalarε > 0, we have

xT y + yT x ≤  xT x +  −1 yT y. Lemma 2.6. [3] Given matrices = T , S and N of appropriate dimensions, the inequality + S(t)N + NT (t)T ST < 0 holds for F(t) such that FT (t)F(t)  I if and only if for some ε > 0,

⎡



⎣∗ ∗

S − I ∗

⎤

 NT  JT ⎦ < 0. − I

3. Robust reliable dissipative control design In this section, first we discuss the problem of strictly (Q, R, S) dissipativity performance for the considered system for the given control gain matrix and subsequently the result is extended to obtain the desired controller. In order to discuss strictly (Q,

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R, S) dissipative performance of the closed-loop form of system (5), first we consider the case in which the matrix A is fixed, i.e., A(t) = 0, B(t) = 0. Subsequently, we consider the following nominal closed-loop system

x˙ (t) = Ax(t) + BGKx(t − d(t)) + Eω(t), z(t) = Cx(t).

(7)

Now, we will develop condition which ensures that the above closed-loop system (7) is strictly (Q, R, S) dissipative. Our main aim is to determine the controller gain K such that the uncertain closed-loop form of system (5) is strictly (Q, R, S) dissipative subject to uncertainties and external disturbance when the actuator fault matrix G is known as well as unknown. Theorem 3.1. Let the scalars μ > 0, α > 0, an integer m  1, matrices Q = QT , R = RT and S are given with known actuator failure parameter matrix G. Then the nominal closed-loop system (7) is strictly (Q, R, S) dissipative for the given control gain matrix K and any time varying delay d(t) satisfying (2), if there exists matrices P > 0, Qi > 0, Rj > 0, Sk > 0, Tk > 0, i = 1, 2, 3, j = 1, 2, k = 1, 2 and for ¯ M, ¯ N¯ with appropriate dimensions such that the following inequality holds; any real matrices L,



˜1



∗



˜1

∗

where



(d2 − d1 )M¯ < 0,

(8)

−S2





(d2 − d1 )N¯ < 0,

(9)

−S2

⎡ ⎢ ⎢ ⎢ ⎢ ˜

1 = ⎢ ⎢ ⎢ ⎢ ⎣

1 +  2 ∗ ∗ ∗ ∗ ∗ ∗

−I ∗ ∗ ∗ ∗ ∗

3

4

5

6

7

0 −S1 ∗ ∗ ∗ ∗

0 0 −S2 ∗ ∗ ∗

0 0 0 −T1 ∗ ∗

0 0 0 0 −T2 ∗

0 0 0 0 0 −S1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(10)

with





¯ P + WQT QW ¯ Q + WRT RW ¯ R + WTT T¯1 WT1 + WTT T¯2 WT2 + sym MWM ,

1 = WPT PW 1 2   Wγ = 0n,(m+6)n In ,   0P A 0n,mn BGK 0n,(m+3)n E − C T S , WP = , P¯ = P0 In 0n,(2m+5) ⎡ d2 − d1 Imn ⎤ ⎡ 0mn,(m+6)n Imn ⎢ ⎢ ⎢ 0mn,n (m+3)n Imn 0mn,(m+5)n ⎥ ⎢ 0mn, ⎥ ⎢  ⎢ ⎥ ⎢ I 0 d1 ⎢ n n, ( 2m+5 ) n ⎥ , WR = ⎢ I WQ = ⎢ m n ⎢ 0n,(m+2)n In 0n,(m+3)n ⎥ ⎢ ⎥ ⎢ ⎢ 0 ⎦ ⎣ n,(m+4)n In 0n,(2m+5)n ⎢  ⎣ 0n,(m+1)n 1 − μIn 0n,(m+4)n 0n,(m+5)n

 = −(R − α I)WγT Wγ ,

Q¯ = diag {Q1 , −Q1 , Q 2 , −Q 2 , Q 3 , −Q 3 } ,  In 0n,(2m+5)n , WT1 = 0n,(m+3)n In 0n,(m+2)n ⎡

In

0n,(2m+5)n In In

1

⎤

In

0n,(2m+5)n  1 I d2 −d1 n  1 I d2 −d1 n

R¯ = diag {R2 , −R2 , R1 , −R1 , −R1 } , ⎤ ⎡ −(T1 + T1T ) dm1 (T1 + T1T ) ⎦, T¯1 = ⎣ ∗ − dm2 (T1 + T1T )

0m,2n ⎦ , WT2 = ⎣ 0n,(m+4)n 0n,(2m+4)n 0n ⎤ ⎡ d −d 1 1 T 2 1 − (T2 + T2 ) d2 +d1 (T2 + T2T ) (T2 + T2T ) d2 +d1 ⎥ ⎢ d2 +d1 ⎢ 1 1 T T ⎥ ∗ − d2 −d − d2 −d T¯2 = ⎢ 2 (T2 + T2 ) 2 (T2 + T2 ⎥ , 2 1 2 1 ⎦ ⎣ 1 T ∗ ∗ − d2 −d 2 (T2 + T2 ) 2 1 ⎤ ⎡ −I 0 I n n n, ( 2m+4 ) n   ⎥ ⎢ −In 0n,(m+3)n ⎦ , M = M¯ N¯ L¯ , WM = ⎣0n,(m+1)n In 0n,mn

Omn,(m+6)n  d1 I m mn

−Imn

0n,(m+4)n

⎤ ⎥ 0mn,3n ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ 0n,(m+1)n ⎥ ⎥ ⎦ 0n,mn

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

111

  T T T T 0 , N¯ = N1T N2T N3T . . . N5+2m 0 , M¯ = M1T M2T M3T . . . M5+2m ⎤ ⎡   d1 T d1 d1 T ⎦ T ¯ T ⎣ A S1 0n,mn E S1 , col [C Q 0n,(m+6)n ], 3 = col (BGK ) S1 0n,(m+3)n m m m     col d2 − d1 AT S2 0n,mn d2 − d1 (BGK )T S2 0n,(m+3)n d2 − d1 ET S2 ,           1 d1 1 d1 1 d1 T T T A T1 0n,mn E T1 , col (BGK ) T1 0n,(m+3)n 2 m 2 m 2 m   ⎤ ⎡  2 2 d22 − d21 T d22 − d21 d − d d1 T 2 1 T ⎦ ⎣ ¯ A T2 0n,mn E T2 , 7 = L. col (BGK ) T2 0n,(m+3)n 2 2 2 m

T  L¯ = LT1 LT2 LT3 . . . LT5+2m 0 ,

2 =

4 =

5 =

6 =

Proof. Consider the nominal closed-loop system (7). In order to prove the required result, we define the Lyapunov functional candidate in the following form

V1 (t, x(t)) = xT (t)Px(t),  t  T V2 (t, x(t)) = γ ( s ) Q γ ( s ) ds + 1 d V3 (t, x(t)) = V4 (t, x(t)) = V5 (t, x(t)) =

  

t−

1 m

−d1 −d2 0 −d1 m

0



t+θ

 

−d1 m

t

t t+θ 0

β

 where γ T (t) = xT (t) xT (t −

t

xT (s)Q 2 x(s)ds +

t−d2



xT (s)R1 x(s)dsdθ +

x˙ T (s)S1 x˙ (s)dsdθ +



t

t+θ

0 −d1 m



−d1

 



m−1 m d1

t−



t+θ t



t t−d(t)

xT (s)Q 3 x(s)ds,

γ T (s)R2 γ (s)dsdθ , x˙ T (s)S2 x˙ (s)dsdθ ,

t+θ

−d2

x˙ T (s)T1 x˙ (s)dsdθ dβ +

d1 T m ). . . x

t



−d1



0



β

−d2

t

x˙ T (s)T2 x˙ (s)dsdθ dβ ,

t+θ

and m  1 denote the number of fractions.

Calculate the derivatives Vi (t, x(t)), i = 1, 2, 3, 4, 5 along the trajectories of the closed-loop system (7), we have

V˙ 1 (t, x(t)) = 2xT (t)P x˙ (t), V˙ 2 (t, x(t)) ≤

(11)

d1 d )Q1 γ (t − 1 ) + xT (t)[Q2 + Q3 ]x(t) − xT (t − d2 )Q2 x(t − d2 ) m m −(1 − μ)xT (t − d(t))Q3 x(t − d(t)),

γ T (t)Q1 γ (t) − γ T (t −



t−d1 −d (t) d1 T γ (t)R2 γ (t) − xT (s)R1 x(s)ds m t−d2  t xT (s)R1 x(s)ds − γ T (s)R2 γ (s)ds, d

V˙ 3 (t, x(t)) = (d2 − d1 )xT (t)R1 x(t) +  −

t−d1 t−d1 −d∗ (t)

V˙ 4 (t, x(t)) = x˙ T (t)



 V˙ 5 (t, x(t)) = x˙ T (t)

t−

(13)

1 m

 t  t−d1 d1 T ˙ ˙ S1 + (d2 − d1 )S2 x˙ (t) − ( s ) S ( s ) ds − x x x˙ T (s)S2 x˙ (s)ds, 1 d m t−d2 t− m1 1 2



d1 m

2 T1 +

(12)

∗

  0  t  −d1  t d22 − d21 T2 x˙ (t) − −d x˙ T (s)T1 x˙ (s)dsdβ − x˙ T (s)T2 x˙ (s)dsdβ . 1 2 β −d β t+ t+ 2 m

(14)

(15)

By applying Lemma 2.3, the integrals in Eqs. (13), (14) and (15) can be written as

 −

t−d1 −d∗ (t) t−d2



t−d1

xT (s)R1 x(s)ds ≤ −

1 d 2 − d1

1 − x (s)R1 x(s)ds ≤ − ∗ d − d1 2 t−d1 −d (t)  −

t t−

m x (s)R2 x(s)ds ≤ − d1





d1 m

t−d1 −d∗ (t)

x(s)ds

T x(s)ds

 R1

x(s)ds





t−

R1

t−d1 t−d1 −d∗ (t)

 ds ,

(16)

 x(s)ds ,

(17)



t

R2

t−d1 −d∗ (t) t−d2

T

t−d1

T

t

t−

t−d1 −d∗ (t) t−d2

T

T

d1 m



d1 m

x(s)ds ,

(18)

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R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121



0

−

−d1 m

 −



t+β

−d1 

x˙ (s)T1 x˙ (s)dsdβ ≤ α (t) T 1

(−T1 − T1T )

t

⎢ x˙ (s)T2 x˙ (s)dsdβ ≤ α2T (t) ⎢ ⎣





α1T (t) = xT (t)

t−

∗

d1 m

( (



⎤

1 ) d2 +d (T2 + T2T ) 1 ⎥ 1 T ⎥ ) − d2 −d 2 (T2 + T2 ) ⎦ α2 (t), 2 1 1 T − d2 −d ( T + T ) 2 2 2

∗

α2T (t) = xT (t)

(19)

1 T2 + T2T d2 +d1 1 T − d2 −d 2 T2 + T2 2 1

∗ 

xT (s)ds ,



) α (t), ) 1

1 (T2 + T2T ) − dd22 −d +d1



t

( (

m T1 + T1T d1 m − d2 T1 + T1T 1

∗ ⎡

t+β

−d2

where



t

t−d1 −d∗ (t) t−d2

2



xT (s)ds

(20)

1



t−d1

t−d1 −d∗ (t)

xT (s)ds .

On the other hand, by the Newton–Leibniz formula, for any arbitrary matrices L, M, N with compatible dimensions, we have



2ζ

T

d (t)L x(t) − x(t − 1 ) − m





t t−

d1 m

x˙ (s)ds = 0,

 2ζ (t)M x(t − d1 − d (t)) − x(t − d2 ) − ∗

T

 2ζ T (t)N x(t − d1 ) − x(t − d1 − d∗ (t)) − d1 T ζ (t)LS1−1 LT ζ (t) − m



t

t−

d1 m



t−d1 −d∗ (t)

t−d2



 x˙ (s)ds = 0,

(22)



t−d1 t−d1 −d∗ (t)

x˙ (s)ds = 0,

(23)

ζ T (t)LS1−1 LT ζ (t)ds = 0,

(d2 − d1 − d∗ (t))ζ T (t)MS2−1 MT ζ (t) − d∗ (t)ζ T (t)NS2−1 NT ζ (t) −



(21)

t−d1 t−d1 −d∗ (t)



t−d1 −d∗ (t) t−d2

(24)

ζ T (t)MS2−1 MT ζ (t)ds = 0,

(25)

ζ T (t)NS2−1 NT ζ (t)ds = 0,

(26)

   T T T T T where L = LT1 LT2 LT3 . . . LT5+2m , M = M1T M2T M3T . . . M5+2m , N = N1T N2T N3T . . . N5+2m . Combining (11) to (26), we obtain

V˙ (t, x(t)) ≤

¯ ζ (t) + (d2 − d1 − d∗ (t))ζ T (t)MS−1 MT ζ (t) ζ T (t) 2 +d∗ (t)ζ T (t)NS2−1 NT ζ (t) −  −  −

t−d1 −d∗ (t) t−d2 t−d1 t−d1 −d∗ (t)



t t−

d1 m

{Lζ (t) + S1 x˙ (s)}T S1−1 {Lζ (t) + S1 x˙ (s)}ds

{Mζ (t) + S2 x˙ (s)}T S2−1 {Mζ (t) + S2 x˙ (s)}ds

{Nζ (t) + S2 x˙ (s)}T S2−1 {Nζ (t) + S2 x˙ (s)}ds ≤ ζ T (t) ζ (t) < 0,

where

¯ + (d2 − d1 − d∗ (t))MS−1 MT + d∗ (t)NS−1 NT ,

= 2 2 

ζ T (t) = γ T (t) xT (t − d1 ) xT (t − d(t)) xT (t − d2 ) with

⎡ ⎢ ⎢ ⎢ ¯ =⎢ ⎢ ⎢ ⎣

¯1  ∗ ∗ ∗ ∗ ∗

¯2 

−S1 ∗ ∗ ∗ ∗

¯3 

¯4 

¯5 

0 −S2 ∗ ∗ ∗

0 0 −T1 ∗ ∗

0 0 0 −T2 ∗

¯6 

⎤

0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ −S1



t

t−

d1 m

γ T (s)ds



t−d1 −d∗ (t)

t−d2

xT (s)ds



t−d1 t−d1 −d∗ (t)

 xT (s)ds

(27)

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

113

¯ 1 = U T PU ¯ P + UQT QU ¯ Q + URT RU ¯ R + UTT T¯1 UT1 + UTT T¯2 UT2 + sym(UU ¯ M ),  P 1 2  UP =

A In

⎡

0n,mn

BGK

0n,(2m+4) Imn 0mn,n In

⎢ ⎢ ⎢ UQ = ⎢ ⎢0n,(m+2)n ⎢ ⎣ In 0n,(m+1)n

 0 P , , P¯ = P 0  ⎡ ⎤ d2 − d1 Imn ⎢ ⎢ 0mn,(m+3)n 0mn,(m+4)n ⎥ ⎢  ⎥ ⎢ ⎥ d1 ⎥ , UR = ⎢ I m n ⎢ 0n,(m+2)n ⎥ ⎢ ⎥ ⎢ 0n,(m+4)n ⎦ ⎣ 0n,(m+3)n 0n,(m+5)n

0n,(m+3)n

0mn,(m+5)n Imn 0n,(2m+4)n In 0 n, ( 2m+4 )n  1 − μI n



O mn,(m+5)n d1 I m mn 0n,(2m+4)n  1 In  d2 −d1 1 I d2 −d1 n

⎤ ⎥ 0mn,2n ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0n,mn ⎥ ⎦

Q¯ = diag {Q1 , −Q1 , Q2 , −Q2 , Q3 , −Q3 } , R¯ = diag {R2 , −R2 , R1 , −R1 , −R1 } ,    −(T1 + T1T ) dm1 (T1 + T1T ) In 0n,(2m+4)n UT1 = , , T¯1 = ∗ − dm2 (T1 + T1T ) 0n,(m+3)n In 0n,(m+1)n 1 ⎤ ⎡ 0n,(2m+4)n In In 0m,n ⎦ , UT2 = ⎣ 0n,(m+4)n 0n,(2m+4)n In ⎡ ⎤ 1 1 − d2 −d1 (T2 + T2T ) d2 +d (T2 + T2T ) (T2 + T2T ) d2 +d1 1 ⎢ d2 +d1 1 1 T T ⎥ ∗ − d2 −d − d2 −d 2 (T2 + T2 ) 2 (T2 + T2 ⎥ , T¯2 = ⎢ ⎣ ⎦ 2 1 2 1 1 T ∗ ∗ − d2 −d 2 (T2 + T2 ) 2 1 ⎤ ⎡ −In 0n,(2m+3)n In −In 0n,(m+2)n ⎦ , U¯ = [M N L] , UM = ⎣0n,(m+1)n In 0n,mn In −Imn 0n,(m+3)n  T T T T T T  T T T  T T T L = L1 L2 L3 . . . L5+2m , M = M1 M2 M3 . . . M5+2m , N = N1T N2T N3T . . . N5+2m , ⎤  d d 1 1 T T A S1 0n,mn col ⎣ (BGK ) S1 0n,(m+3)n ⎦ , m m    col d2 − d1 AT S2 0n,mn d2 − d1 (BGK )T S2 0n,(m+3)n ,        1 d1 1 d1 T T A T1 0n,mn col (BGK ) T1 0n,(m+3)n , 2 m 2 m  ⎤ ⎡  d22 − d21 T d22 − d21 T ¯ 6 = d1 L. A T2 0n,mn col ⎣ (BGK ) T2 0n,(m+3)n ⎦ ,  2 2 m ⎡

¯2 =  ¯3 =  ¯4 =  ¯5 = 

By applying the idea of convex combination and time varying delay interval, the term on right hand side of (27) can be changed equivalently into the following form

¯ + (d2 − d1 )NS−1 NT < 0. ¯ + (d2 − d1 )MS−1 MT < 0,   2 2

(28)

By Lemma 2.4, the inequality (28) is equivalent to



 d 2 − d1 M < 0, ∗ S2

¯ 



 d 2 − d1 N < 0. ∗ S2

¯ 

(29)

In the view of LMIs (8) and (9), then we get V˙ (x(t), t) < 0 and hence the considered nominal closed-loop system in (7) with ω(t) = 0 is asymptotically stable. To discuss the dissipative performance of the system (7), we introduce the following performance index

JT = −z, Qz T − 2z, Sω T − ω, Rω T + αω, ω T ,  T = [−zT (t)Qz(t) − 2zT (t)Sw(t) − ωT (t)Rω(t) + αωT (t)ω(t)]dt,

(30)

0

where T > 0. It can be shown from the LMIs (8) and (9) that

V˙ (t, x(t)) − zT (t)Qz(t) − 2zT (t)Sw(t) − ωT (t)Rω(t) + αωT (t)ω(t) ˜ ψ(t) < 0, ≤ ψ T (t)

(31)

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R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

where ψ T (t) = [ζ T (t) ωT (t)] and

˜ 1 + (d2 − d1 − d∗ (t))MS ˜ = ¯ −1 M¯ T + d∗ (t)NS ¯ −1 N¯ T .

2 2

(32)

Again by applying convex combination and time varying interval, the right hand of (32) can be written as

˜ 1 + (d2 − d1 )NS ˜ 1 + (d2 − d1 )MS ¯ −1 M¯ T < 0, ¯ −1 N¯ T < 0.

2 2

(33)

˜ 1 is defined as in Eq. (10). Considering Eqs. (30) and (31), we have where



JT ≤

  −z(t)T Qz(t) − 2zT (t)Sw(t) − ωT (t)Rω(t) + αωT (t)ω(t) + V˙ (t, x(t)) dt < 0

T 0

for any non-zero ω(t) ∈ L2 [0, ∞). This implies that



0

T

[−zT (t)Qz(t) − 2zT (t)Sw(t) − ωT (t)Rω(t)] > α



T 0

ωT (t)ω(t),

and the condition (6) is satisfied. Based on Definition (2.1), we can conclude that the closed-loop system (7) is strictly (Q, R, S) dissipative. Now, let us concentrate on the design of (Q, R, S) reliable control law which guarantees the dissipativeness of the system (7) with known actuator failure G. In the following theorem, we present the LMI based sufficient conditions to derive (Q, R, S) dissipative reliable controller design for the closed-loop system (7).  Theorem 3.2. For given matrices Q, S, R with Q = QT , R = RT , the scalars μ > 0, α > 0, an integer m  1, there exists a feedback controller in the form of (3) such that the closed-loop system (7) strictly (Q, R, S) dissipative with known actuator failure matrix G, if ˆ there exists matrices X > 0, Qˆ i > 0, Rˆj > 0, Vk > 0, i = 1, 2, 3, j = 1, 2, k = 1, 2, 3, 4 and any appropriate dimensioned matrices Lˆ, M and Lˆ such that the following LMIs hold;



∗

 ˆ d2 − d1 M < 0, S2

(34)



∗

 d2 − d1 Nˆ < 0, S2

(35)

⎡

ˆ1+ ˆ ˆ2

⎢ ⎢ ⎢ ⎢ ⎢ where = ⎢ ⎢ ⎢ ⎢ ⎣

∗ ∗ ∗ ∗ ∗ ∗

−I ∗ ∗ ∗ ∗ ∗

ˆ3

ˆ4

ˆ5

ˆ6

0 −V1 ∗ ∗ ∗ ∗

0 0 −V2 ∗ ∗ ∗

0 0 0 −V3 ∗ ∗

0 0 0 0 −V4 ∗

⎤

ˆ7

0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ with ⎥ 0 ⎥ ⎥ 0 ⎦ V1 − 2X





 M , ˆ 1 = W T XW ¯ P + WQT QW ¯ Q + WRT RW ¯ R + WVT V¯ 3 WV3 + WVT V¯ 4 WV4 + sym MW

P 3 4 Vˆi = Vi − 2X, i = 1, 2, 3, 4,  A 0n,mn BGK 0n,(m+3)n WP = In 0n,(2m+5)n ⎡

In

WM = ⎣0n,(m+1)n 0n,mn

−In In In

⎡ d2 − d1 Imn ⎢ ⎢ 0mn,(m+3)n ⎢  ⎢ d1 I WR = ⎢ m n ⎢ ⎢ ⎢ 0n,(m+4)n ⎣ 0n,(m+5)n  WV3 =

In

0n,(m+3)n

0n,(2m+4)n −In −Imn

O mn,(m+6)n d1 I m mn 0n,(2m+5)n  1 In  d2 −d1 1 I d2 −d1 n

0n,(2m+5)n In

E − CT S

⎡

⎤ 0n,(m+3)n ⎦ , 0n,(m+4)n ⎤ ⎥ 0mn,3n ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0n,(m+1)n ⎥ ⎦ 0n,mn

,

V¯3 =

Imn 0mn,n In

⎢ ⎢ ⎢ WQ = ⎢ ⎢0n,(m+2)n ⎢ ⎣ In 0n,(m+1)n

0mn,(m+6)n Imn 0n,(2m+5)n In 0 n, ( 2m+5 )n  1 − μI n

⎤ 0mn,(m+5)n ⎥ ⎥ ⎥ ⎥, 0n,(m+3)n ⎥ ⎥ ⎦ 0n,(m+4)n

  R¯ = diag Rˆ2 , −Rˆ2 , Rˆ1 , −Rˆ1 , −Rˆ1 ,



0n,(m+2)n

,

(Vˆ3 + Vˆ3T ) ∗

m d1 m d21



(Vˆ3 + Vˆ3T ) , (Vˆ3 + Vˆ3T )

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

⎡

⎤

⎡d

115

⎤

1 1 2 −d1 (Vˆ4 + Vˆ4T ) − d2 +d (Vˆ4 + Vˆ4T ) − d2 +d (Vˆ4 + Vˆ4T ) 1 1 ⎢ d2 +d1 ⎥ 1 1 T ˆ ˆ ⎢ ∗ (V4 + V4 ) (Vˆ4 + Vˆ4T ) ⎥ 0m,2n ⎦ , V¯4 = ⎣ WV4 = ⎣ 0n,(m+4)n d22 −d21 d22 −d21 ⎦, 1 T 0n,(2m+4)n 0n ˆ ˆ ∗ ∗ ( + V ) V 4 4 d22 −d21    0 X , Q¯ = diag Qˆ 1 , −Qˆ 1 , Qˆ 2 , −Qˆ 2 , Qˆ 3 , −Qˆ 3 , X¯ = X 0      = M, ˆ = −(R − α I)WγT Wγ , Wγ = 0n,(m+6)n In , M ˆ N, ˆ Lˆ ,     T T T T ˆ = M ˆT ˆT M ˆT M ˆT ...M , Nˆ = Nˆ 1T Nˆ 2T Nˆ 3T . . . Nˆ 5+2m Lˆ = LˆT1 LˆT2 LˆT3 . . . LˆT5+2m 0 , M 0 , 1 2 3 5+2m 0   ˆ 2 = col XC T Q¯ 0n,(m+7)n ,

⎤ ⎡   d d d 1 1 1 ˆ 3 = col ⎣ ET ⎦ ,

(AX )T 0n,mn (BGY )T 0n,(m+3)n m m m     ˆ 4 = col d2 − d1 (AX )T 0n,mn d2 − d1 (BGY )T 0n,(m+3)n d2 − d1 ET ,

          1 d1 1 d1 1 d1 ˆ 5 = col

(AX )T 0n,mn (BGY )T 0n,(m+3)n ET , 2 m 2 m 2 m   ⎤ ⎡  d22 − d21 d22 − d21 d22 − d21 T T T ˆ 6 = col ⎣ ˆ 7 = d1 Lˆ. E ⎦,

(AX ) 0n,mn (BGY ) 0n,(m+3)n 2 2 2 m

0n,(2m+5)n In In

In

Moreover, the gain matrix of the feedback reliable controller (3) can be obtained by K = YX−1 . Proof. The proof of this theorem follows immediately from Theorem 3.1. In order to obtain the feedback controller   gain matrices, ¯ = {X, X, ...X } ∈ R(5+2m)×(5+2m). Pre and post multiplying (8) and (9) by diag W, ¯ I, I, V1 , V2 , V3 , V4 , X, X , where X = P−1 , take W ˆ k = XMk X, Nˆ k = XNk X, k = 1, V1 = S1−1 , V2 = S2−1 , V3 = T1−1 and V4 = T2−1 . Let Qˆ i = XQi X, i = 1, 2, 3, Rˆj = XRj X, j=1,2, Lˆk = XLk X, M 2, 3, . . . , 5 + 2m and K = YX−1 , using the inequalities −XVi−1 X ≤ Vi − 2X, i = 1, 2, 3, 4, we obtain the LMIs (34) and (35). Then based on Theorem 3.1 and Definition (2.1), it is concluded that the considered nominal closed-loop system (7) is strictly (Q, R, S) dissipative performance. The proof is completed.  Next, we extend the result in Theorem 3.2 to obtain the robust reliable strictly (Q, R, S) dissipative control design for the uncertain system (5) with known actuator failure matrix G, where the uncertainties are formulated in LFT form. Theorem 3.3. Consider the uncertain system (5) with the assumption (H1). For the given matrices Q, S, R with Q, R are symmetric, the constants μ > 0, α > 0 and an integer m  1, there exists a reliable controller in the form of Eq. (3) such that the resulting closed-loop form of system (5) is strictly (Q, R, S) dissipative with known actuator failure matrix G, if there exists matrices X > 0, Qˆ i > 0, Rˆj > 0, Vk > 0, ˆ Lˆ with appropriate dimensions and positive scalar β 1 > 0, β 2 > 0 such that i = 1, 2, 3, j = 1, 2, k = 1, 2, 3, 4, any real matrices Lˆ, M, the following LMIs hold;

⎡

˜i

⎢∗ ⎢ i = ⎢ ⎢∗ ⎣∗ ∗

T β1 M1a −β1 I

∗ ∗ ∗

T N1a β1 JT −β1 I ∗ ∗

T β2 M1a

0 0

−β2 I ∗

⎤ T N1b 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, T⎦ β2 J −β2 I

i = 1, 2,

(36)

˜ i is defined as in Theorem 3.2 with where

 ⎤T    2 2  d − d d d 1 1 1 2 1 MT d2 − d1 MaT MaT MaT 0n,3n ⎦ , = ⎣MaT 0n,(5+2m)n m a 2 m 2     = X T NaT 0n,(12+2m)n , N1b = 0 0n,mn X T NbT 0n,(10+2m)n . ⎡

T M1a

N1a



Moreover, the constant gain of the desired robust reliable feedback controller can be constructed as K = YX−1 . Proof. In order to get strictly (Q, R, S) dissipativity of the closed-loop form of system (5) via the robust reliable control law (3), we consider same LKF functional candidate as in Theorem 3.1. By replacing the matrix A by A + A(t) and B by B + B(t) in (7) and proceeding the proof in the same way as above Theorem 3.2 and using Lemma 2.6, it is easy to get the LMIs (36). Thus, we conclude that the uncertain system (5) is strictly (Q, R, S) dissipative. Next, we will present a theorem to compute the robust reliable dissipative controller gain based on the results obtained in the previous theorems. In particular, in the following theorem, when the actuator failure matrix G is unknown but satisfying the

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R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

constraints (4), we determine the gain matrix of the reliable robust state feedback controller such that the closed-loop form of system (5) is strictly (Q, R, S) dissipative.  Theorem 3.4. Consider the uncertain system (5) with the assumption (H1). For the given matrices Q, S, R with Q, R are symmetric, the constants μ > 0, α > 0, an integer m  1, there exists a robust reliable control law in the form of Eq. (3) such that the resulting closed-loop form of system (5) is strictly (Q, R, S) dissipative with unknown actuator failure matrix G, if there exists matrices ˆ Lˆ with appropriate dimensions and scalars X > 0, Qˆ i > 0, Rˆj > 0, Vk > 0, i = 1, 2, 3, j = 1, 2, k = 1, 2, 3, 4, any real matrices Lˆ, M, β 1 > 0, β 2 > 0 ε > 0 such that the following LMIs hold;

⎤

⎡

˜ i  BˆT G1 Yˆ T  i = ⎣ ∗ − I 0 ⎦ < 0, ∗ ∗ − I where

⎡



Bˆ = ⎣B 0n,(5+2m)n T

i = 1, 2,

d1 T  B d 2 − d1 B T m



(37)

1 2



d1 m



 B

T

⎤ d22 − d21 T B 0n,7n ⎦, 2

Yˆ = [0n,(m+1)n Y 0n,(15+m)n ] and the other parameters are defined as in Theorem 3.3. Moreover, the gain matrix of the feedback controller (3) is obtained by K = YX−1 . Proof. If the actuator failure matrix G is unknown, then from (4) the matrices (36) in Theorem 3.3 can be calculated according to the values in G. Further i , i = 1, 2 can be obtained as

˜ i + BˆT Yˆ + Yˆ T B, ˆ i = 1, 2, i =  ˜ i , i = 1, 2 is obtained by replacing G by G0 in WP , it follows from Lemma 2.5 and (4) that where 

˜ i + 1 BˆT Bˆ +  −1 Yˆ T G2 Y, ˆ i = 1, 2. i ≤  1 1

(38)

Then by using Lemma 2.4, it is easy to see that (38) is equivalent to LMIs (37). Hence, the system (5) is strictly (Q, R, S) dissipative via the reliable robust controller (3) when G is unknown. The proof is completed.  Next, we present a corollary to show that our result is general enough to cover some existing works. In the case of constant delay, i.e., d(t) = d, the system (1) can be further rewritten as

x˙ (t) = Ax(t) + Buf (t − d) + Eω(t), z(t) = Cx(t), x(t) =

(39)

φ(t), ∀t ∈ [−d, 0] .

Corollary 3.5. Consider the system 39. For the given matrices Q, S, R with Q, R are symmetric, the scalars μ > 0, α > 0, an integer m  1, there exists a feedback controller in the form of Eq. (3) such that the resulting close-loop form of system 39 with known ˆ with actuator fault is strictly (Q, R, S) dissipative, if there exists matrices X > 0, Qˆ > 0, Rˆ > 0, V1 > 0, V2 > 0 and the real matrices M, appropriate dimension such that the following LMIs hold

⎡

θ1 +  θ2

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∗ ∗ ∗ ∗

−I ∗ ∗ ∗

θ3

θ4

0 −V1 ∗ ∗

0 0 −V2 ∗



d ˆ M m

⎤

⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎦ V1 − 2X

where

¯ X + WQT QW ¯ Q + WRT RW ¯ R + WVT VW ¯ V + sym(MW ˆ M ), θ1 = WXT XW with

0 , Q¯ = diag{Q, −Q }, R¯ = diag{R, −R}, Wγ = [0n,(m+2) In ] X   A 0n,(m−1)n BGK 0n,n E Inm 0mn,(2+m)n WX = , WQ = , In,n 0n,n 0n,n 0n,n 0n,n 0mn,n Imn 0mn,mn ⎤ ⎡   d 0n,(m+3)n In 0n,(m+2)n m ⎦ , WV =  , WR = ⎣ m 0n,(m+1)n In 0n 0n,(m+2)n In 0n,n

 = −(R − α I)WγT Wγ , X =

d



X 0

(40)

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

 WM = I n

θ2



 0n,(m+1)n ,

−In

V¯3 =

(Vˆ2 + Vˆ2T ) ∗ ⎡

  = col XC T Q¯ 0n,(m+3)n , 

θ4 = col

1 d (AX )T 2m

m d1 m d21

117



(Vˆ2 + Vˆ2T )   ˆ = M1+2m 0 , , M (Vˆ2 + Vˆ2T ) 



d θ3 = col ⎣ (AX )T 0n,(m−1)n m    1 d 1 d T 0n,(m−1)n (BGY )T 0 E , 2m 2m

d (BGY )T 0 m

⎤ d T⎦ E , m

Vˆi = Vi − 2X, i = 1, 2.

Moreover, the gain matrix of the feedback controller (3) is given by K = YX−1 . Proof. Choose a Lyapunov functional candidate as follows

V1 (t, x(t)) = xT (t)Px(t), V2 (t, x(t)) = V4 (t, x(t)) =



 d −m

t



t d t− m

γ T (s)Q γ (s)ds, V3 (t, x(t)) =

x˙ T (s)Sx˙ (s)ds, V5 (t, x(t)) =

t+θ



0 −d m



0



β

t t+θ



0 −d m



t

t+θ

γ T (s)Rγ (s)dsdθ ,

x˙ T (s)T x˙ (s)dsdθ dβ ,

where γ T (t) = [xT (t) xT (t − m1 ).....xT (t − m−1 m d1 )] and m  1 are number of fractions. By following the similar steps as in Theorem 3.2 with some modifications, we can obtain the desired result. The proof is completed.  d

4. Numerical simulation Consider the mechanical system as in [7,28] which is described by the following seconder order differential equation Mx¨ (t) + C x˙ (t) + Kx(t) = Buf (t − d(t)),

(41)

where x(t) ∈ Rn is the displacement; M ∈ Rn×n , C ∈ Rn×n and K ∈ Rn×n are the mass, damping and stiffness matrices, respectively; M assumed to be nonsingular; B ∈ Rn×m is the weight matrix of reliable control input uf (t). By introducing the disturbance signal ω(t) which belongs to L2 [0, ∞) and considering an output signal z(t) to the above system, and defining a new state vector  T , we can obtain the following uncertain mechanical control system q(t) = xT (t) x˙ T (t)

 Eq(t) = (A + A(t))q(t) + (B + B(t))uf (t − d(t)) + Bω ω(t), z(t) = Cq(t),

where

 E=

I 0 , 0 M

 A=

0 −K

(42)

I , −C

⎡ ⎡ ⎤ 1.1 0 0 2 −1 ⎣ ⎦ 0 1.8 0 , K = ⎣−1 2 M= 0 0 1.6 0 −1

⎡ ⎤ ⎤ 0 1.2 −1.6 0 ⎣ ⎦ −1 , C = −0.6 1.2 −0.6⎦, 1 0 −0.6 0.6

(43)

and the input and output matrices are

  T  T 0.1 0.1 0.5 0 0 0 B = 0 0 0 1 0 0 , Bω = 0 0 0 0 0 0.1 , C = . 0 0 0 0.1 0.1 0.5

If we take the values m = 1, d1 = 0.1, μ = 0.2, Ma = Na = Ma1 , and Nb = 0 where Ma1 = [ −0 −I ] and  = [ 0.1 J = 0.1, Q = −[ 0.2

0.2 ], 0.3

(44) −0.01 0.005 0 0.01 0.01 −0.01 ], 0 −0.005 0.01

−0.25 R = 4, S = [ −0.25 ]. Moreover, the sensor fault matrix G is assumed to satisfy 0.1  G  0.2 and by solving

the LMIs in Theorem 3.4 with the use of Matlab LMI tool box, we can get a maximum upper bound d2 = 0.7 and the corresponding state feedback dissipative controller gain matrix can be given as

  K = −0.1420 2.5580 −1.6728 −3.1670 −8.0796 −6.0604 .

Also, by solving the LMIs in Theorem 3.4 based on Remark 2.2, the calculated minimum allowable γ for different values of d2 is provided in Table 1. Table 1 Calculated minimum γ for various values of d2 .

H case Passivity case Mixed H and passivity case

d2

0.2

0.4

0.6

γ min γ min γ min

0.1870 0.0625 0.1051

0.2702 0.1399 0.1715

0.5832 0.3540 0.3349

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R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

0.2

0.3

x (t)

x (t)

x (t)

x (t)

2

2

0.1

x3(t)

0.1

Velocity

Displacement

1

0.15

1

0.2

0

x (t) 3

0.05 0

−0.05

−0.1

−0.1 −0.2

−0.15 0

10

20

30

40

−0.2

50

0

10

20

30

40

50

Time T

Time T Fig. 1. Dissipativity performance of the uncertain system.

0.2

0.3 x (t)

0.15

x (t)

0.1

x (t)

0.05

1

0.2

3

Velocity

displacement

2

0.1 0

x1(t) x2(t) x3(t)

0

−0.05 −0.1 −0.1 −0.2

−0.15 −0.2 0

10

20 30 Time (seconds)

40

50

0

10

20 30 Time (seconds)

40

50

Fig. 2. H performance of the uncertain system.

Moreover, when d2 = 0.6 we can obtain the H controller gain matrix with minimum performance index γ min = 0.5832 and passivity controller gain matrix with minimum performance γ min = 0.3540 respectively as follows

  K = 0.4981 1.9774 −1.7893 −3.8878 −10.9866 −9.9986 ,   K = 0.5464 1.5881 −1.1332 −3.3554 −8.1795 −5.8508 .

Finally, when d2 = 0.6, the mixed H and passivity controller gain matrix can be calculated as

  K = 0.5596 1.5945 −1.1295 −3.3536 −8.1720 −5.8393

(45)

with minimum performance γ min = 0.3349. Figs. 1–4 represents the simulation result for trajectories of displacement and velocity of uncertain mechanical system for the above control gains with the disturbance input ω(t) = 0.01 sin t and for the initial   condition 0.1 −0.2 0.2 . Fig. 5 denotes the controller and output trajectories of the system. Further, in order to compare the obtained result with the existing works, we consider the system (42) with constant delay in the following form

 Eq(t) = Aq(t) + Buf (t − d) + Bω ω(t), = Cq(t), z(t)

(46)

By solving the LMI in Corollary 3.5 with the same values which are given in Eqs. (43) and (44) and the sensor fault G = 0.5 one can easily obtain a feasible solution. The maximum upper bound d for various values of H performance γ is provided in Table 2 and the minimum H performance for various values of the upper bound d is given in Table 3. It is observed from the Tables 2 and 3 that our results are much less conservative than in [7,28], the calculated upper bound d increases as the fractioning number m increases and disturbance performance index γ decreases as the fractioning number m increases. The state responses of the displacement and velocity of the system (42) without control are given in Fig. 6. It is concluded from the simulation results that the state trajectories of the system without control do not converge to equilibrium point but

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121

0.3

0.2

x1(t)

0.1

x2(t)

0.15

x (t)

x3(t)

0.1

x2(t)

0

1

x3(t)

0.05

Velocity

Displacement

0.2

119

0

−0.05

−0.1

−0.1

−0.2 −0.15

0

10

20

30

40

−0.2

50

Time T

0

10

20 30 Time T

40

50

Fig. 3. Passivity performance of the uncertain system.

0.3

0.2

x1(t)

x (t) 1

0.2

x (t)

0.15

x (t)

0.1

x (t)

2

2

3

3

0.1

0.05 Velocity

Displacement

x (t)

0

0

−0.05

−0.1

−0.1

−0.2 −0.15

0

10

20

30

40

50

−0.2

Time T

0

10

20

Time T

30

40

50

Fig. 4. Mixed H and passivity performance of the uncertain system.

2 0.1 1

Output Responses

Controller Responses

1.5

0.5 0 −0.5

0

−0.05

−1 −1.5 −2

0.05

−0.1 0

10

20

30

40

50

0

10

Time T Fig. 5. Control and output responses with dissipativity.

20

Time T

30

40

50

120

R. Sakthivel et al. / Applied Mathematics and Computation 263 (2015) 107–121 Table 2 Calculated maximum upper bound d for various values of γ .

[7] [28] m=1 m=2

γ

0.2

0.5

0.8

1.0

d d d d

0.072 0.202 0.288 1.013

0.140 0.331 0.314 1.014

0.158 0.336 0.321 1.015

0.164 0.338 0.323 1.015

Table 3 Calculated minimum γ for various values of d.

[7] [28] m=1 m=2

d

0.05

0.1

0.15

γ γ γ γ

0.147 0.082 0.0305 0.0015

0.278 0.119 0.0344 0.0029

0.624 0.156 0.0428 0.0043

1

0.25 x1(t)

0.8

x (t) 1

x (t)

0.2

x3(t)

0.15

x (t) 2

2

0.6

3

0.1

0.2 Velocity

Displacement

0.4

x (t)

0

0.05 0

−0.2 −0.05 −0.4 −0.1

−0.6

−0.15

−0.8 −1 0

10

20 Time T 30

40

50

−0.2 0

10

20 Time T 30

40

50

Fig. 6. Displacement and velocity responses of the unforced uncertain system.

state trajectories converges to equilibrium point quickly with the proposed controller which demonstrates the applicability of our controller design method. 5. Conclusion In this paper, robust reliable strictly (Q, R, S) dissipative control problem for a class of uncertain continuous time systems with input delay is studied. By implementing the delay fractioning technique and Lyapunov approach, a new set of conditions are obtained which ensures that the resulting closed-loop system is strictly (Q, R, S) dissipative for all possible actuator failures. In particular, the reliable strictly (Q, R, S) dissipative control law is designed in terms of the solution of certain linear matrix inequalities. More precisely, the solvability of the addressed problem has been expressed as the feasibility of a set of LMIs. Moreover as particular cases, the H, passivity, and mixed H and passivity control problems have been obtained from the proposed results. Finally, we have provided numerical study with an example to validate the effectiveness of the proposed design techniques. Acknowledgments This work was jointly supported by the National Natural Science Foundation of China (61374080), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. References [1] M.S. Alwan, X. Liu, W.C. Xie, On design of robust reliable H control and input-to-state stabilization of uncertain stochastic systems with state delay, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1047–1056. [2] P. Balasubramaniam, R. Krishnasamy, Robust exponential stabilization results for impulsive neutral time-delay systems with sector-bounded nonlinearity, Circuits Syst. Signal Process. 33 (2014) 2741–2759.

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