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Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays
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Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays✩ Lifeng Ma a , Zidong Wang b,c, *, Qing-Long Han d , Yurong Liu e a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, UK d School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia e Department of Mathematics, Yangzhou University, Yangzhou 225002, China b c
article
info
Article history: Received 8 July 2008 Received in revised form 8 July 2018 Accepted 14 August 2018 Available online xxxx Keywords: Dissipative control Nonlinear Markovian jump systems Actuator failures Mixed time-delays Hamilton–Jacobi inequality
a b s t r a c t This paper addresses the dissipative control problem for nonlinear Markovian jump systems subject to actuator failures and mixed time-delays, where the mixed time-delays consist of both discrete and distributed time-delays and are mode-dependent. The purpose of the problem under investigation is to design a state feedback controller such that, in the presence of actuator failures and mixed timedelays, the closed-loop system is asymptotically stable in the mean square sense while achieving the prespecified dissipativity. By constructing a Lyapunov–Krasovskii functional and using a completing square approach, sufficient conditions are proposed for the existence of the desired controller in terms of the solvability of certain Hamilton–Jacobi inequalities. Finally, an illustrative numerical example is provided to demonstrate the effectiveness of the developed control scheme. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Markovian jump systems (MJSs) have been attracting everincreasing research interest within the systems science and control communities owing primarily to their ability in modeling random variations, see Aberkane and Dragan (2012), Bian, Jiang, and Jiang (2016), Boukas (2006), Terra, Ishihara, Jesus, and Cerri (2013), Vamvoudakis and Safaei (2017) and the references therein. On the other hand, in engineering practice, actuator failure is one of the most frequently encountered phenomena which could give rise to performance degradation or even instability (Lunze & Steffen, 2006; Ma, Wang, & Lam, 2017a; Seo & Kim, 1996). Moreover, since time-delays are ubiquitous in a variety of practical systems (e.g. chemical, biological and engineering systems), considerable attention has been paid to the analysis and synthesis issues for systems with time-delays. According to the ways they occur, timedelays can be categorized into discrete and distributed delays (Liu, Liu, Obaid, & Abbas, 2016; Scarciotti & Astolfi, 2016). So far, much effort has been devoted to an investigation on linear systems ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hiroshi Ito under the direction of Editor André L. Tits. Corresponding author at: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. E-mail addresses: [email protected] (L. Ma), [email protected] (Z. Wang), [email protected] (Q.-L. Han), [email protected] (Y. Liu).
*
with discrete and/or distributed delays, see e.g. Basin, RodriguezGonzalez, Fridman, and Acosta (2005), among which the most exploited algorithm is arguably the linear matrix inequality (LMI) framework. Nevertheless, when it comes to the nonlinear delayed systems (especially with Markovian jump parameters), the widely used LMI-based Lyapunov–Krasovskii functional method is no longer applicable. To date, the issues of stability analysis, control and filtering have not been adequately studied for nonlinear MJSs subject to both actuator failures and mixed time-delays, which constitute the first motivation of the current research. Ever since the seminal work in Willems (1972), the theory of dissipative systems has been playing a paramount role in the study of dynamical systems. In particular, the dissipative control/filtering problems have stirred an increasing research interest leading to a multitude of results reported in the literature. For linear systems, stabilization, control and filtering problems with desired dissipativity have been extensively investigated, see e.g. Feng, Lam, and Shu (2013) and Tan, Soh, and Xie (1999). It should be pointed out that, however, limited work has been done for nonlinear systems with or without Markovian jumping parameters. Those few available results include stability and dissipativity conditions established in (1) Aliyu (1999) for nonlinear Markovian jump systems in virtue of the Hamilton–Jacobi inequality (HJI) approach; and (2) Sheng, Gao, & Zhang (2014) for a class of nonlinear MJSs using an LMI-based method. When the nonlinear MJS with modedependent mixed time-delays is concerned, the corresponding dissipative control problem has not been fully examined due mainly
https://doi.org/10.1016/j.automatica.2018.09.028 0005-1098/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
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to the technical difficulties stemming from the coupling between the nonlinear dynamics and the mode-dependent time-delays. As such, in this paper, we are motivated to design a state feedback controller for a class of nonlinear MJSs to ensure the expected stability and dissipativity in the presence of actuator failures and mode-dependent mixed time-delays. 2. Problem formulation
P {rt +∆ = j|rt = i} =
πij ∆ + o(∆), if i ̸ = j, 1 + πij ∆ + o(∆), if i = j,
where ∆ > 0,∑ and πij > 0 is the transition rate from i to j if j ̸ = i while πii = − j̸=i πij . On a probability space (Ω , F , {Ft }t ≥0 , P ), we consider the following class of nonlinear delayed systems:
⎧ x˙ (t) =f (x(t), rt ) + m(x(t), rt )ϕ (u(t)) + D(rt )v (t) ⎪ ⎪ ∫ t ⎪ ⎪ ⎪ ⎪ ⎪ h(x(s))ds + g(x(t − τ )) + 1 , r t ⎨ t −τ2,rt [ ] ⎪ C (rt )x(t) ⎪ ⎪ y(t) = ⎪ ⎪ u(t) ⎪ ⎪ ⎩ x0 (t) =φ (t), t ∈ [−τ , 0]
G(t) ≜ diag{χ1 (t), χ2 (t), . . . , χµ (t)}, 0≤χ
≤ χl (t) ≤ χ
(l)
≤ +∞,
l = 1, 2, . . . , µ
(2)
where χ (l) (0 ≤ χ (l) < 1) and χ (l) (1 ≤ χ (l) < +∞) are known scalars serving as the lower and upper bounds on χl (t), respectively. (l) (i) Denoting χ0 ≜ (χ (l) + χ (l) )/2, G0 ≜ diagµ {χ0 }, ξl (t) ≜ (l) 0 )
(l) 0 ,
(χl (t) − χ /χ Ξ (t) ≜ diagµ {ξi (t)}, λl ≜ (χ − χ )/(χ + χ ) and Λ ≜ diagµ {λi }, we can reformulate ϕ (u(t)) by ϕ (u(t)) = (G0 + G0 Ξ (t))u(t) with G0 > 0,
Ξ T (t)Ξ (t) ≤ ΛT Λ ≤ I .
Definition 2 (Aliyu, 1999). A function J(v (t), y(t), rt ) : V × Y × M ↦ → R is said to be a supply rate to system (1) if J(·, ·, ·) satisfies w
{∫ E
} ⏐ ⏐ ⏐J(v (t), y(t), rt )⏐dt < ∞,
∀w ≥ s ≥ 0.
(4)
s
In this paper, we consider a supply rate associated with system (1) of the following form:
(l)
+ v T (t)Φ (i)v (t),
(l)
(l)
(l)
(3)
Definition 1 (Liu et al., 2016). The equilibrium x(t) = 0 of the unforced system (1) (i.e., u(t) = 0 and v (t) = 0) is said to be asymptotically stable in the mean square sense if the solution x(t) to (1) satisfies limt →∞ E{|x(t)|2 } = 0.
i∈M
(5)
where Σ (i), Ω (i) and Φ (i) are real-valued constant matrices with Σ (i) and Φ (i) being symmetric. Here, we assume that Σ (i) ≤ 0 and Φ (i) ≥ 0. Moreover, for convenience of subsequent derivation, we assume that Σ (i) and Ω (i) can be decomposed as follows:
Σ1 (i) Σ (i) = Σ2T (i)
] Σ2 (i) , Σ3 (i)
[
[ ] Ω1 (i) Ω (i) = . Ω2 (i)
Definition 3. System (1) with supply rate J(v (t), y(t), rt ) defined in (5) is said to be dissipative on [s, +∞) if there exists a non-negative continuous function V : Rn × T × M ↦ → R+ such that
E{V (xw , w, rw )} − V (xs , s, is ) ≤ E (1)
where x(t) ∈ Rn is the system state, u(t) ∈ Rµ is the control input, v (t) ∈ Rν is the exogenous disturbance belonging to L2 [0, +∞) and y(t) ∈ Rp is the controlled output; f (·, ·) : Rn × R ↦ → Rn and m(·, ·) : Rn × R ↦ → Rµ are, respectively, known continuous vector-valued and matrix-valued functions with f (0, rt ) = 0 and m(0, rt ) = 0; g(·) : Rn ↦ → Rn and h(·) : Rn ↦ → Rn are known continuous vector-valued functions with g(0) = h(0) = 0; C (rt ) and D(rt ) are known real-valued matrices with compatible dimensions; τ1,rt stands for the discrete mode-dependent time-delay satisfying 0 < τ1,rt ≤ τM where τM is a known constant; τ2,rt describes the distributed mode-dependent time-delay; the parameter τ ≜ max{τij |i = 1, 2; j = 1, 2, . . . , q} and φ ∈ C ([−τ , 0], Rn ) is the initial condition. We also assume that nonlinear functions f (·), g(·) and h(·) are locally Lipschitz, which indicates that the existence of the solution to system (1) is guaranteed. In (1), the nonlinear function ϕ (·) : Rµ ↦ → Rµ with ϕ (0) = 0 is introduced to characterize the actuator failure phenomenon. ϕ (u(t)) is defined as ϕ (u(t)) ≜ G(t)u(t) where G(t) ∈ Rµ×µ is a time-varying matrix describing how the failures affect the control input. G(t) is assumed to have the following form:
(l)
–
J(y(t), v (t), i) ≜yT (t)Σ (i)y(t) + 2yT (t)Ω (i)v (t)
Let rt (t ≥ 0) be a right-continuous Markovian chain on a probability space (Ω , F , {Ft }t ≥0 , P ) taking values in a finite state space M = {1, 2, . . . , q} with generator Π = {πij } given by
{
)
w
{∫
J(v (t), y(t), rt )dt
}
s
for all w ≥ s ≥ 0, xs ∈ Rn and rs = is ∈ M. It is our objective in this paper to design a state-feedback control law u(t) = k(x(t), rt ) such that, the equilibrium x(t) = 0 of the closed-loop system with v (t) = 0 is asymptotically stable in the mean square sense, and meanwhile, the closed-loop system is dissipative on [s, +∞) with respect to supply rate (5). 3. Main results Defining xt ≜ x(t + θ ), we consider a functional V (xt , t , rt ) ∈ C 1 (Rn × T × M) associated with system (1) as follows: V (xt , t , rt ) ≜V1 (x(t), rt ) +
t
∫
g T (x(s))P1 g(x(s))ds
t −τ1,rt
+ π¯
∫
τ1
τ1
τ2,rt
∫
t
∫
g T (x(θ ))P1 g(x(θ ))dθ ds t −s
∫
t
hT (x(θ ))P2 h(x(θ ))dθ ds
+ 0
+ π¯
∫
t −s
τ2 τ2
ϑ
∫ 0
∫
t
hT (x(θ ))P2 h(x(θ ))dθ dsdϑ
(6)
t −s
where V1 (0, rt ) = 0, V1 (x(t), rt ) > 0 for x(t) ̸ = 0, and P1 and P2 are positive definite matrices with compatible dimensions. Moreover, for V (xt , t , i) (i ∈ M), we define:
∂ V (xt , t , i) Vx (i) ≜ ∂ x1
[
∂ V (xt , t , i) ∂ x2
···
] ∂ V (xt , t , i) . ∂ xn
Before giving the main results, we first denote
τ i ≜ max{τi,j |i = 1, 2; j ∈ M}, τ i ≜ min{τi,j |i = 1, 2; j ∈ M}, π¯ ≜ max{|πii |}, β Υ˜ (i) ≜ − Σ3 (i) + α −1 I + ΛT Λ, 4
η˜ (i) ≜ − 2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 , ϖ ˜ (i) ≜ − 2xT (t)C T (i)Ω1 (i) + Vx (i)D(i), q ∑ H1 (i) ≜Vx (i)f (i) + πij V1 (x(t), j) j=1
Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
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+
Vx (i)P1−1 VxT (i)
+
τ2,i Vx (i)P2−1 VxT (i)
2
2
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i) ˜ −1 (i)ϖ η˜ (i)Υ˜ −1 (i)η˜ T (i) ϖ ˜ (i)Φ ˜ T (i) − − . 4
Theorem 1. Consider the nonlinear time-delay system (1). For the supply rate (5), if there exist positive scalars α > 0, β > 0 and a series of functionals V (xt , t , i) (i ∈ M) solving the following set of HJIs: H1 (i) ≤ 0,
˜ (i) ≜ −Φ (i) + α Φ
Ω2T (i)Ω2 (i)
(7)
< 0,
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT ( ) β + uT (t) −Σ3 (i) + α −1 I + ΛT Λ u(t) 4 ( ) + −2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 u(t) ) ( + v T (t) −Φ (i) + α Ω2T (i)Ω2 (i) v (t) ( ) + −2xT (t)C T (i)Ω1 (i) + Vx (i)D(i) v (t). LV (xt , t , i) − J(y(t), v (t), i)
≤Vx (i)f (i) + +
(8) u(t) = k(x(t), i) = − Υ˜ −1 (i)η˜ T (i), 2 the equilibrium at the origin is asymptotically stable in the mean square sense, and meanwhile, the closed-loop system is dissipative with respect to the supply rate (5) over [s, +∞). Proof. First, let us investigate the dissipativity of system (1) with the feedback controller (8). By utilizing the completing square method shown in Ma, Wang, Liu, and Alsaadi (2017b), along the trajectory of system (1), LV (xt , t , i) (i ∈ M) satisfies LV (xt , t , i) ≤Vx (i)f (i) + Vx (i)m(i)ϕ (u(t)) q ∑ + Vx (i)D(i)v (t) + πij V1 (x(t), j)
+
τ2,i Vx (i)P2
VxT (i)
.
4 4 Next, for a given β > 0, it follows from (3) that
(9)
(10)
≤ − x (t)C (i)Σ1 (i)C (i)x(t) − 2x (t)C (i)Σ2 (i)u(t) T
T
Taking (9)–(11) into consideration, we acquire LV (xt , t , i) − J(y(t), v (t), i)
πij V1 (x(t), j)
(14)
}
J(y(t), v (t), rt )dt ,
(15)
which indicates that the closed-loop system is dissipative with respect to the supply rate (5) over [s, +∞). In the following, we shall proceed to examine the asymptotic stability of the equilibrium at the origin in the mean square sense. ˜ (i) < 0, we have First, letting v (t) = 0 and noticing α > 0 and Φ q ∑
πij V1 (x(t), j)
j=1
(11)
+
Vx (i)P1−1 VxT (i)
+
τ2,i Vx (i)P2−1 VxT (i)
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t)) 2
− xT (t)C T (i)Σ1 (i)C (i)x(t)
j=1
4
w
T
+ v T (t)(−Φ (i) + α Ω2T (i)Ω2 (i))v (t).
+
T
With given xs ∈ R and rs = is ∈ M, integrating both sides of (14) from s to w (w ≥ s ≥ 0) and taking mathematical expectation lead to
T
+ u (t)(−Σ3 (i) + α I)u(t) − 2x (t)C (i)Ω1 (i)v (t)
Vx (i)P1−1 VxT (i)
(13) 1 Φ −1 (i) 2
LV (xt , t , i) − J(y(t), 0, i)
− (yT (t)Σ (i)y(t) + 2yT (t)Ω (i)v (t) + v T (t)Φ (i)v (t)) −1
4
s
Likewise, for a given α > 0, it is known from (5) that
+
4
˜ (i)v˜ (i) + u˜ T (i)Υ˜ (i)u˜ (i) + v˜ T (i)Φ ˜ (i)v˜ (i) =H1 (i) + u˜ T (i)Υ˜ (i)u˜ (i) + v˜ T (i)Φ
E{V (xw , w, rw )} − V (xs , s, is )
4
q ∑
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i) ˜ −1 (i)ϖ η˜ (i)Υ˜ −1 (i)η˜ T (i) ϖ ˜ (i)Φ ˜ T (i) − −
≤E
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i).
≤Vx (i)f (i) +
2
− xT (t)C T (i)Σ1 (i)C (i)x(t)
n
=Vx (i)m(i)(G0 + G0 Ξ (t))u(t) β ≤Vx (i)m(i)G0 u(t) + uT (t)ΛT Λu(t)
T
τ2,i Vx (i)P2−1 VxT (i)
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
{∫
T
+
LV (xt , t , i) − J(y(t), v (t), i) ≤ 0.
Vx (i)m(i)ϕ (u(t))
T
Vx (i)P1−1 VxT (i)
where u˜ (i) ≜ u(t) + ˜ η˜ and v˜ (i) ≜ v (t) + ˜ ϖ ˜ (i). ˜ (i) ≤ 0, H1 (i) ≤ 0 and the control law given in Noting that Φ (8), we have that
j=1
+
πij V1 (x(t), j)
1 Υ −1 (i) T (i) 2
( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t)) −1
q ∑ j=1
1
2 Vx (i)P1 VxT (i)
(12)
Completing the squares of u(t) and v (t) in (12) yields
then, with the feedback control law
−1
3
)
− xT (t)C T (i)Σ1 (i)C (i)x(t)
− xT (t)C T (i)Σ1 (i)C (i)x(t)
{
–
+ 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
4
)
(
τ2,i Vx (i)P2−1 VxT (i) 4
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT −
˜ −1 (i)ϖ ϖ ˜ (i)Φ ˜ T (i) 4
Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
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( ) β + uT (t) −Σ3 (i) + α −1 I + ΛT Λ u(t) 4 ( ) + −2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 u(t) =H1 (i) ≤ 0.
)
–
(16)
Next, by defining ζ ≜ maxi∈M λmax (Σ (i)), we obtain know
ζ ≤ 0 and therefore,
LV (xt , t , i) < J(y(t), 0, i)
≤ ζ xT (t)C T (i)C (i)x(t) + ζ uT (t)u(t),
(17)
which means LV (xt , t , i) < ζ ρ xT (t)x(t)
(18)
Fig. 1. The dynamics of open-loop system.
where ρ ≥ 0 is defined as ρ ≜ maxi∈M λmax (C (i)C (i)). Subsequently, for T > 0, integrating both sides of (18) from 0 to T and taking mathematical expectation yield T
E{V (xT , T , rT )} − E{V (x0 , 0, r0 )}
≤ζ ρ
T
∫
E{xT (t)x(t)}dt
(19)
0
It is inferred from (19) and V (xT , T , rT ) ≥ 0 that T
∫
E{xT (t)x(t)}dt ≤ 0
1
−ζ ρ
E{V (x0 , 0, r0 )}.
(20)
Letting T → +∞, we arrive at +∞
∫
E{xT (t)x(t)}dt ≤ 0
1
ζρ
E{V (x0 , 0, r0 )}.
(21) Fig. 2. The dynamics of closed-loop system.
According to Barbalat’s Lemma (Slotine & Li, 1991), the following holds lim E{|x(t)|2 } = 0,
(22)
t →+∞
which means that the equilibrium is asymptotically stable in the mean square sense. The proof is now complete. 4. Simulation example Consider the following mechanical rotational cutting process taken from Wang, Sun, Shi, and Zhao (2013): z¨ + γ1 z˙ + γ2 (z + z 3 ) = −γ3 z(t − 1) where z represents the deflection of the machine tool, γ1 is the term proportional to the product of natural frequency, the damping ratio γ2 represents the tool stiffness and γ3 is the delay term proportional to effective cutting stiffness of the workpiece per unit of chip width. It should be pointed out that during the cutting process, the parameter γ3 may be increased and such an increment might be variant occasionally due to the abrupt change of circumstance and devices aging. As such, the cutting process can be modeled as a nonlinear time-delay system with a parameter varying according to a known Markov chain. Denote x1 = z, x2 = z˙ and x = [x1 x2 ]T . By taking the control input and disturbance into consideration, we have the following system:
[ x˙ =
]
[ ] 1 + ϕ (u) 0 −γ2 x1 − γ1 x2 − γ2 x31 [ ] [ ] 0 0 + + v, rt ∈ {1, 2}. −γ3 (rt )x1 (t − 1) 0.8 x2
We choose the following for the functional (6): P1 = I , V1 (x, 1) = V2 (x, 2) =
Π=
[ −0.5 0.5
γ2 4
x41 +
1 2
x22 ,
0.5 , Σ1 (i) = −0.05, i = 1, 2. −0.5
]
Fig. 3. The switching between two modes.
We also set the parameters γ1 = 0.1, γ2 = 0.1, γ3 (1) = 0.2, γ3 (2) = 0.15 and τ1,1 = 1, τ1,2 = 0.8, τ1,3 = 0.8, τ2,1 = 0.4, τ2,2 = 0.6 and τ2,3 = 0.8. For the possible actuator failures, it is assumed that χ (1) = 0.6 and χ (1) = 1.4. Set α = 2 and β = 10. Then,
˜ (i) < 0 according to Theorem 1, we can verify that H1 (i) < 0 and Φ for i = 1,2. Therefore, the condition in Theorem 1 is satisfied and the desired controller is given by u(t) = −0.8x1 − 0.1x31 . The simulation figures are shown in Figs. 1–4, which indicate that the proposed control scheme is applicable. 5. Conclusion The dissipative control problem has been investigated for the nonlinear time-delay system subject to Markovian jump parameters and actuator failures. The mode-dependent time-delays under consideration include both discrete and distributed time-delays. In terms of Hamilton–Jacobi inequalities, sufficient conditions have been proposed for the existence of the required state feedback controller guaranteeing simultaneously the mean-square asymptotical stability and pre-specified dissipativity of the closed-loop
Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
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Fig. 4. The dynamics of closed-loop system without considering actuator failures.
system. A simulation example has been presented to show the correctness and applicability of the proposed control scheme. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61773209, 61773017 and 61873148, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the six talent peaks project in Jiangsu Province under Grant XYDXX-033, the Fundamental Research Funds for the Central Universities under Grant 30916011337, the Postdoctoral Science Foundation of China under Grant 2014M551598, and Alexander von Humboldt Foundation of Germany. References Aberkane, S., & Dragan, V. (2012). H∞ filtering of periodic Markovian jump systems: Application to filtering with communication constraints. Automatica, 48(12), 3151–3156. Aliyu, M. D. S. (1999). Dissipativity and stability of nonlinear jump systems. In Proceedings of the 1999 American control conference, San Diego, CA (pp. 795–799).
)
–
5
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Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays✩ Lifeng Ma a , Zidong Wang b,c, *, Qing-Long Han d , Yurong Liu e a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, UK d School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia e Department of Mathematics, Yangzhou University, Yangzhou 225002, China b c
article
info
Article history: Received 8 July 2008 Received in revised form 8 July 2018 Accepted 14 August 2018 Available online xxxx Keywords: Dissipative control Nonlinear Markovian jump systems Actuator failures Mixed time-delays Hamilton–Jacobi inequality
a b s t r a c t This paper addresses the dissipative control problem for nonlinear Markovian jump systems subject to actuator failures and mixed time-delays, where the mixed time-delays consist of both discrete and distributed time-delays and are mode-dependent. The purpose of the problem under investigation is to design a state feedback controller such that, in the presence of actuator failures and mixed timedelays, the closed-loop system is asymptotically stable in the mean square sense while achieving the prespecified dissipativity. By constructing a Lyapunov–Krasovskii functional and using a completing square approach, sufficient conditions are proposed for the existence of the desired controller in terms of the solvability of certain Hamilton–Jacobi inequalities. Finally, an illustrative numerical example is provided to demonstrate the effectiveness of the developed control scheme. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Markovian jump systems (MJSs) have been attracting everincreasing research interest within the systems science and control communities owing primarily to their ability in modeling random variations, see Aberkane and Dragan (2012), Bian, Jiang, and Jiang (2016), Boukas (2006), Terra, Ishihara, Jesus, and Cerri (2013), Vamvoudakis and Safaei (2017) and the references therein. On the other hand, in engineering practice, actuator failure is one of the most frequently encountered phenomena which could give rise to performance degradation or even instability (Lunze & Steffen, 2006; Ma, Wang, & Lam, 2017a; Seo & Kim, 1996). Moreover, since time-delays are ubiquitous in a variety of practical systems (e.g. chemical, biological and engineering systems), considerable attention has been paid to the analysis and synthesis issues for systems with time-delays. According to the ways they occur, timedelays can be categorized into discrete and distributed delays (Liu, Liu, Obaid, & Abbas, 2016; Scarciotti & Astolfi, 2016). So far, much effort has been devoted to an investigation on linear systems ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hiroshi Ito under the direction of Editor André L. Tits. Corresponding author at: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. E-mail addresses: [email protected] (L. Ma), [email protected] (Z. Wang), [email protected] (Q.-L. Han), [email protected] (Y. Liu).
*
with discrete and/or distributed delays, see e.g. Basin, RodriguezGonzalez, Fridman, and Acosta (2005), among which the most exploited algorithm is arguably the linear matrix inequality (LMI) framework. Nevertheless, when it comes to the nonlinear delayed systems (especially with Markovian jump parameters), the widely used LMI-based Lyapunov–Krasovskii functional method is no longer applicable. To date, the issues of stability analysis, control and filtering have not been adequately studied for nonlinear MJSs subject to both actuator failures and mixed time-delays, which constitute the first motivation of the current research. Ever since the seminal work in Willems (1972), the theory of dissipative systems has been playing a paramount role in the study of dynamical systems. In particular, the dissipative control/filtering problems have stirred an increasing research interest leading to a multitude of results reported in the literature. For linear systems, stabilization, control and filtering problems with desired dissipativity have been extensively investigated, see e.g. Feng, Lam, and Shu (2013) and Tan, Soh, and Xie (1999). It should be pointed out that, however, limited work has been done for nonlinear systems with or without Markovian jumping parameters. Those few available results include stability and dissipativity conditions established in (1) Aliyu (1999) for nonlinear Markovian jump systems in virtue of the Hamilton–Jacobi inequality (HJI) approach; and (2) Sheng, Gao, & Zhang (2014) for a class of nonlinear MJSs using an LMI-based method. When the nonlinear MJS with modedependent mixed time-delays is concerned, the corresponding dissipative control problem has not been fully examined due mainly
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Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
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to the technical difficulties stemming from the coupling between the nonlinear dynamics and the mode-dependent time-delays. As such, in this paper, we are motivated to design a state feedback controller for a class of nonlinear MJSs to ensure the expected stability and dissipativity in the presence of actuator failures and mode-dependent mixed time-delays. 2. Problem formulation
P {rt +∆ = j|rt = i} =
πij ∆ + o(∆), if i ̸ = j, 1 + πij ∆ + o(∆), if i = j,
where ∆ > 0,∑ and πij > 0 is the transition rate from i to j if j ̸ = i while πii = − j̸=i πij . On a probability space (Ω , F , {Ft }t ≥0 , P ), we consider the following class of nonlinear delayed systems:
⎧ x˙ (t) =f (x(t), rt ) + m(x(t), rt )ϕ (u(t)) + D(rt )v (t) ⎪ ⎪ ∫ t ⎪ ⎪ ⎪ ⎪ ⎪ h(x(s))ds + g(x(t − τ )) + 1 , r t ⎨ t −τ2,rt [ ] ⎪ C (rt )x(t) ⎪ ⎪ y(t) = ⎪ ⎪ u(t) ⎪ ⎪ ⎩ x0 (t) =φ (t), t ∈ [−τ , 0]
G(t) ≜ diag{χ1 (t), χ2 (t), . . . , χµ (t)}, 0≤χ
≤ χl (t) ≤ χ
(l)
≤ +∞,
l = 1, 2, . . . , µ
(2)
where χ (l) (0 ≤ χ (l) < 1) and χ (l) (1 ≤ χ (l) < +∞) are known scalars serving as the lower and upper bounds on χl (t), respectively. (l) (i) Denoting χ0 ≜ (χ (l) + χ (l) )/2, G0 ≜ diagµ {χ0 }, ξl (t) ≜ (l) 0 )
(l) 0 ,
(χl (t) − χ /χ Ξ (t) ≜ diagµ {ξi (t)}, λl ≜ (χ − χ )/(χ + χ ) and Λ ≜ diagµ {λi }, we can reformulate ϕ (u(t)) by ϕ (u(t)) = (G0 + G0 Ξ (t))u(t) with G0 > 0,
Ξ T (t)Ξ (t) ≤ ΛT Λ ≤ I .
Definition 2 (Aliyu, 1999). A function J(v (t), y(t), rt ) : V × Y × M ↦ → R is said to be a supply rate to system (1) if J(·, ·, ·) satisfies w
{∫ E
} ⏐ ⏐ ⏐J(v (t), y(t), rt )⏐dt < ∞,
∀w ≥ s ≥ 0.
(4)
s
In this paper, we consider a supply rate associated with system (1) of the following form:
(l)
+ v T (t)Φ (i)v (t),
(l)
(l)
(l)
(3)
Definition 1 (Liu et al., 2016). The equilibrium x(t) = 0 of the unforced system (1) (i.e., u(t) = 0 and v (t) = 0) is said to be asymptotically stable in the mean square sense if the solution x(t) to (1) satisfies limt →∞ E{|x(t)|2 } = 0.
i∈M
(5)
where Σ (i), Ω (i) and Φ (i) are real-valued constant matrices with Σ (i) and Φ (i) being symmetric. Here, we assume that Σ (i) ≤ 0 and Φ (i) ≥ 0. Moreover, for convenience of subsequent derivation, we assume that Σ (i) and Ω (i) can be decomposed as follows:
Σ1 (i) Σ (i) = Σ2T (i)
] Σ2 (i) , Σ3 (i)
[
[ ] Ω1 (i) Ω (i) = . Ω2 (i)
Definition 3. System (1) with supply rate J(v (t), y(t), rt ) defined in (5) is said to be dissipative on [s, +∞) if there exists a non-negative continuous function V : Rn × T × M ↦ → R+ such that
E{V (xw , w, rw )} − V (xs , s, is ) ≤ E (1)
where x(t) ∈ Rn is the system state, u(t) ∈ Rµ is the control input, v (t) ∈ Rν is the exogenous disturbance belonging to L2 [0, +∞) and y(t) ∈ Rp is the controlled output; f (·, ·) : Rn × R ↦ → Rn and m(·, ·) : Rn × R ↦ → Rµ are, respectively, known continuous vector-valued and matrix-valued functions with f (0, rt ) = 0 and m(0, rt ) = 0; g(·) : Rn ↦ → Rn and h(·) : Rn ↦ → Rn are known continuous vector-valued functions with g(0) = h(0) = 0; C (rt ) and D(rt ) are known real-valued matrices with compatible dimensions; τ1,rt stands for the discrete mode-dependent time-delay satisfying 0 < τ1,rt ≤ τM where τM is a known constant; τ2,rt describes the distributed mode-dependent time-delay; the parameter τ ≜ max{τij |i = 1, 2; j = 1, 2, . . . , q} and φ ∈ C ([−τ , 0], Rn ) is the initial condition. We also assume that nonlinear functions f (·), g(·) and h(·) are locally Lipschitz, which indicates that the existence of the solution to system (1) is guaranteed. In (1), the nonlinear function ϕ (·) : Rµ ↦ → Rµ with ϕ (0) = 0 is introduced to characterize the actuator failure phenomenon. ϕ (u(t)) is defined as ϕ (u(t)) ≜ G(t)u(t) where G(t) ∈ Rµ×µ is a time-varying matrix describing how the failures affect the control input. G(t) is assumed to have the following form:
(l)
–
J(y(t), v (t), i) ≜yT (t)Σ (i)y(t) + 2yT (t)Ω (i)v (t)
Let rt (t ≥ 0) be a right-continuous Markovian chain on a probability space (Ω , F , {Ft }t ≥0 , P ) taking values in a finite state space M = {1, 2, . . . , q} with generator Π = {πij } given by
{
)
w
{∫
J(v (t), y(t), rt )dt
}
s
for all w ≥ s ≥ 0, xs ∈ Rn and rs = is ∈ M. It is our objective in this paper to design a state-feedback control law u(t) = k(x(t), rt ) such that, the equilibrium x(t) = 0 of the closed-loop system with v (t) = 0 is asymptotically stable in the mean square sense, and meanwhile, the closed-loop system is dissipative on [s, +∞) with respect to supply rate (5). 3. Main results Defining xt ≜ x(t + θ ), we consider a functional V (xt , t , rt ) ∈ C 1 (Rn × T × M) associated with system (1) as follows: V (xt , t , rt ) ≜V1 (x(t), rt ) +
t
∫
g T (x(s))P1 g(x(s))ds
t −τ1,rt
+ π¯
∫
τ1
τ1
τ2,rt
∫
t
∫
g T (x(θ ))P1 g(x(θ ))dθ ds t −s
∫
t
hT (x(θ ))P2 h(x(θ ))dθ ds
+ 0
+ π¯
∫
t −s
τ2 τ2
ϑ
∫ 0
∫
t
hT (x(θ ))P2 h(x(θ ))dθ dsdϑ
(6)
t −s
where V1 (0, rt ) = 0, V1 (x(t), rt ) > 0 for x(t) ̸ = 0, and P1 and P2 are positive definite matrices with compatible dimensions. Moreover, for V (xt , t , i) (i ∈ M), we define:
∂ V (xt , t , i) Vx (i) ≜ ∂ x1
[
∂ V (xt , t , i) ∂ x2
···
] ∂ V (xt , t , i) . ∂ xn
Before giving the main results, we first denote
τ i ≜ max{τi,j |i = 1, 2; j ∈ M}, τ i ≜ min{τi,j |i = 1, 2; j ∈ M}, π¯ ≜ max{|πii |}, β Υ˜ (i) ≜ − Σ3 (i) + α −1 I + ΛT Λ, 4
η˜ (i) ≜ − 2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 , ϖ ˜ (i) ≜ − 2xT (t)C T (i)Ω1 (i) + Vx (i)D(i), q ∑ H1 (i) ≜Vx (i)f (i) + πij V1 (x(t), j) j=1
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+
Vx (i)P1−1 VxT (i)
+
τ2,i Vx (i)P2−1 VxT (i)
2
2
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i) ˜ −1 (i)ϖ η˜ (i)Υ˜ −1 (i)η˜ T (i) ϖ ˜ (i)Φ ˜ T (i) − − . 4
Theorem 1. Consider the nonlinear time-delay system (1). For the supply rate (5), if there exist positive scalars α > 0, β > 0 and a series of functionals V (xt , t , i) (i ∈ M) solving the following set of HJIs: H1 (i) ≤ 0,
˜ (i) ≜ −Φ (i) + α Φ
Ω2T (i)Ω2 (i)
(7)
< 0,
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT ( ) β + uT (t) −Σ3 (i) + α −1 I + ΛT Λ u(t) 4 ( ) + −2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 u(t) ) ( + v T (t) −Φ (i) + α Ω2T (i)Ω2 (i) v (t) ( ) + −2xT (t)C T (i)Ω1 (i) + Vx (i)D(i) v (t). LV (xt , t , i) − J(y(t), v (t), i)
≤Vx (i)f (i) + +
(8) u(t) = k(x(t), i) = − Υ˜ −1 (i)η˜ T (i), 2 the equilibrium at the origin is asymptotically stable in the mean square sense, and meanwhile, the closed-loop system is dissipative with respect to the supply rate (5) over [s, +∞). Proof. First, let us investigate the dissipativity of system (1) with the feedback controller (8). By utilizing the completing square method shown in Ma, Wang, Liu, and Alsaadi (2017b), along the trajectory of system (1), LV (xt , t , i) (i ∈ M) satisfies LV (xt , t , i) ≤Vx (i)f (i) + Vx (i)m(i)ϕ (u(t)) q ∑ + Vx (i)D(i)v (t) + πij V1 (x(t), j)
+
τ2,i Vx (i)P2
VxT (i)
.
4 4 Next, for a given β > 0, it follows from (3) that
(9)
(10)
≤ − x (t)C (i)Σ1 (i)C (i)x(t) − 2x (t)C (i)Σ2 (i)u(t) T
T
Taking (9)–(11) into consideration, we acquire LV (xt , t , i) − J(y(t), v (t), i)
πij V1 (x(t), j)
(14)
}
J(y(t), v (t), rt )dt ,
(15)
which indicates that the closed-loop system is dissipative with respect to the supply rate (5) over [s, +∞). In the following, we shall proceed to examine the asymptotic stability of the equilibrium at the origin in the mean square sense. ˜ (i) < 0, we have First, letting v (t) = 0 and noticing α > 0 and Φ q ∑
πij V1 (x(t), j)
j=1
(11)
+
Vx (i)P1−1 VxT (i)
+
τ2,i Vx (i)P2−1 VxT (i)
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t)) 2
− xT (t)C T (i)Σ1 (i)C (i)x(t)
j=1
4
w
T
+ v T (t)(−Φ (i) + α Ω2T (i)Ω2 (i))v (t).
+
T
With given xs ∈ R and rs = is ∈ M, integrating both sides of (14) from s to w (w ≥ s ≥ 0) and taking mathematical expectation lead to
T
+ u (t)(−Σ3 (i) + α I)u(t) − 2x (t)C (i)Ω1 (i)v (t)
Vx (i)P1−1 VxT (i)
(13) 1 Φ −1 (i) 2
LV (xt , t , i) − J(y(t), 0, i)
− (yT (t)Σ (i)y(t) + 2yT (t)Ω (i)v (t) + v T (t)Φ (i)v (t)) −1
4
s
Likewise, for a given α > 0, it is known from (5) that
+
4
˜ (i)v˜ (i) + u˜ T (i)Υ˜ (i)u˜ (i) + v˜ T (i)Φ ˜ (i)v˜ (i) =H1 (i) + u˜ T (i)Υ˜ (i)u˜ (i) + v˜ T (i)Φ
E{V (xw , w, rw )} − V (xs , s, is )
4
q ∑
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i) ˜ −1 (i)ϖ η˜ (i)Υ˜ −1 (i)η˜ T (i) ϖ ˜ (i)Φ ˜ T (i) − −
≤E
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT (i).
≤Vx (i)f (i) +
2
− xT (t)C T (i)Σ1 (i)C (i)x(t)
n
=Vx (i)m(i)(G0 + G0 Ξ (t))u(t) β ≤Vx (i)m(i)G0 u(t) + uT (t)ΛT Λu(t)
T
τ2,i Vx (i)P2−1 VxT (i)
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
{∫
T
+
LV (xt , t , i) − J(y(t), v (t), i) ≤ 0.
Vx (i)m(i)ϕ (u(t))
T
Vx (i)P1−1 VxT (i)
where u˜ (i) ≜ u(t) + ˜ η˜ and v˜ (i) ≜ v (t) + ˜ ϖ ˜ (i). ˜ (i) ≤ 0, H1 (i) ≤ 0 and the control law given in Noting that Φ (8), we have that
j=1
+
πij V1 (x(t), j)
1 Υ −1 (i) T (i) 2
( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t)) −1
q ∑ j=1
1
2 Vx (i)P1 VxT (i)
(12)
Completing the squares of u(t) and v (t) in (12) yields
then, with the feedback control law
−1
3
)
− xT (t)C T (i)Σ1 (i)C (i)x(t)
− xT (t)C T (i)Σ1 (i)C (i)x(t)
{
–
+ 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
4 4 ( ) + 1 + π¯ (τ 1 − τ 1 ) g T (x(t))P1 g(x(t)) ( ) 1 + τ2,i + π¯ (τ 22 − τ 22 ) hT (x(t))P2 h(x(t))
4
)
(
τ2,i Vx (i)P2−1 VxT (i) 4
+ β −1 Vx (i)m(i)G0 GT0 mT (i)VxT −
˜ −1 (i)ϖ ϖ ˜ (i)Φ ˜ T (i) 4
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( ) β + uT (t) −Σ3 (i) + α −1 I + ΛT Λ u(t) 4 ( ) + −2xT (t)C T (i)Σ2 (i) + Vx (i)m(i)G0 u(t) =H1 (i) ≤ 0.
)
–
(16)
Next, by defining ζ ≜ maxi∈M λmax (Σ (i)), we obtain know
ζ ≤ 0 and therefore,
LV (xt , t , i) < J(y(t), 0, i)
≤ ζ xT (t)C T (i)C (i)x(t) + ζ uT (t)u(t),
(17)
which means LV (xt , t , i) < ζ ρ xT (t)x(t)
(18)
Fig. 1. The dynamics of open-loop system.
where ρ ≥ 0 is defined as ρ ≜ maxi∈M λmax (C (i)C (i)). Subsequently, for T > 0, integrating both sides of (18) from 0 to T and taking mathematical expectation yield T
E{V (xT , T , rT )} − E{V (x0 , 0, r0 )}
≤ζ ρ
T
∫
E{xT (t)x(t)}dt
(19)
0
It is inferred from (19) and V (xT , T , rT ) ≥ 0 that T
∫
E{xT (t)x(t)}dt ≤ 0
1
−ζ ρ
E{V (x0 , 0, r0 )}.
(20)
Letting T → +∞, we arrive at +∞
∫
E{xT (t)x(t)}dt ≤ 0
1
ζρ
E{V (x0 , 0, r0 )}.
(21) Fig. 2. The dynamics of closed-loop system.
According to Barbalat’s Lemma (Slotine & Li, 1991), the following holds lim E{|x(t)|2 } = 0,
(22)
t →+∞
which means that the equilibrium is asymptotically stable in the mean square sense. The proof is now complete. 4. Simulation example Consider the following mechanical rotational cutting process taken from Wang, Sun, Shi, and Zhao (2013): z¨ + γ1 z˙ + γ2 (z + z 3 ) = −γ3 z(t − 1) where z represents the deflection of the machine tool, γ1 is the term proportional to the product of natural frequency, the damping ratio γ2 represents the tool stiffness and γ3 is the delay term proportional to effective cutting stiffness of the workpiece per unit of chip width. It should be pointed out that during the cutting process, the parameter γ3 may be increased and such an increment might be variant occasionally due to the abrupt change of circumstance and devices aging. As such, the cutting process can be modeled as a nonlinear time-delay system with a parameter varying according to a known Markov chain. Denote x1 = z, x2 = z˙ and x = [x1 x2 ]T . By taking the control input and disturbance into consideration, we have the following system:
[ x˙ =
]
[ ] 1 + ϕ (u) 0 −γ2 x1 − γ1 x2 − γ2 x31 [ ] [ ] 0 0 + + v, rt ∈ {1, 2}. −γ3 (rt )x1 (t − 1) 0.8 x2
We choose the following for the functional (6): P1 = I , V1 (x, 1) = V2 (x, 2) =
Π=
[ −0.5 0.5
γ2 4
x41 +
1 2
x22 ,
0.5 , Σ1 (i) = −0.05, i = 1, 2. −0.5
]
Fig. 3. The switching between two modes.
We also set the parameters γ1 = 0.1, γ2 = 0.1, γ3 (1) = 0.2, γ3 (2) = 0.15 and τ1,1 = 1, τ1,2 = 0.8, τ1,3 = 0.8, τ2,1 = 0.4, τ2,2 = 0.6 and τ2,3 = 0.8. For the possible actuator failures, it is assumed that χ (1) = 0.6 and χ (1) = 1.4. Set α = 2 and β = 10. Then,
˜ (i) < 0 according to Theorem 1, we can verify that H1 (i) < 0 and Φ for i = 1,2. Therefore, the condition in Theorem 1 is satisfied and the desired controller is given by u(t) = −0.8x1 − 0.1x31 . The simulation figures are shown in Figs. 1–4, which indicate that the proposed control scheme is applicable. 5. Conclusion The dissipative control problem has been investigated for the nonlinear time-delay system subject to Markovian jump parameters and actuator failures. The mode-dependent time-delays under consideration include both discrete and distributed time-delays. In terms of Hamilton–Jacobi inequalities, sufficient conditions have been proposed for the existence of the required state feedback controller guaranteeing simultaneously the mean-square asymptotical stability and pre-specified dissipativity of the closed-loop
Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.
L. Ma et al. / Automatica (
Fig. 4. The dynamics of closed-loop system without considering actuator failures.
system. A simulation example has been presented to show the correctness and applicability of the proposed control scheme. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61773209, 61773017 and 61873148, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the six talent peaks project in Jiangsu Province under Grant XYDXX-033, the Fundamental Research Funds for the Central Universities under Grant 30916011337, the Postdoctoral Science Foundation of China under Grant 2014M551598, and Alexander von Humboldt Foundation of Germany. References Aberkane, S., & Dragan, V. (2012). H∞ filtering of periodic Markovian jump systems: Application to filtering with communication constraints. Automatica, 48(12), 3151–3156. Aliyu, M. D. S. (1999). Dissipativity and stability of nonlinear jump systems. In Proceedings of the 1999 American control conference, San Diego, CA (pp. 795–799).
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Please cite this article in press as: Ma, L., et al., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays. Automatica (2018), https://doi.org/10.1016/j.automatica.2018.09.028.