Finite-Time 𝐻_∞ Control for a Class of Discrete-Time Markov Jump Systems with Actuator Saturation via Dynamic Antiwindup Design

Abstract

We deal with the finite-time control problem for discrete-time Markov jump systems subject to saturating actuators. A finite-state Markovian process is given to govern the transition of the jumping parameters. A controller designed for unconstrained systems combined with a dynamic antiwindup compensator is given to guarantee that the resulting system is mean-square locally asymptotically finite-time stabilizable. The proposed conditions allow us to find dynamic anti-windup compensator which stabilize the closed-loop systems in the finite-time sense. All these conditions can be expressed in the form of linear matrix inequalities and therefore are numerically tractable, as shown in the example included in the paper.

1. Introduction

It is well known that more and more attention has been paid to the study of actuator saturation due to its practical and theoretical importance. Therefore, various approaches were investigated to handle systems with actuator saturation and dynamic antiwindup approach which is one of the most effective ways to deal with it. To this end, a great number of results have been reported in the literature; see, for example, [1, 2]. Furthermore, the stabilization problem of singular Markovian jump systems with discontinuities and saturation inputs was presented in [3]. Via dynamic antiwindup fuzzy design, the robust stabilization problem of state delayed T-S fuzzy systems with input saturation was proposed in [4].

On the other hand, Markov jump is frequently encountered in many practical systems. Therefore, the study of Markov jump systems has been a hot research topic due to its importance, and many results have been proposed based on various control techniques, such as robust control [5–9], 𝐻_∞ control [10, 11], Passivity-based control [12–14], fuzzy dissipative control [15], and neural networks control [14, 16]. Furthermore, observer based finite-time 𝐻_∞ control problem of discrete-time Markov jump systems was studied [17].

As it is well known, when dealing with the stability of s system, a distinction should have been made between classical Lyapunov stability and finite-time stability (FTS). Conversely, a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed some bounds during this time-interval. Some results on FTS have been carried out; see, [18, 19]. Furthermore, finite-time 𝐻_∞ filtering problem of time-delay stochastic jump systems with unbiased estimation was proposed in [20]. By applying dynamic observer-based state feedback and the Lyapunov-Krasovskii functional approach, the finite-time 𝐻_∞ control problem for time-delay nonlinear jump systems was addressed in the work of He and Liu [21]. However, to the best of our knowledge, the problem of finite-time stabilization of discrete-time stochastic systems subject to actuator saturation has not been fully investigated and it is the main purpose of our study.

In this paper, the attention is focused on the finite-time 𝐻_∞ control problem of discrete-time Markov jump systems with actuator saturation based on dynamic antiwindup approach. A controller designed for unconstrained systems combined with a dynamic antiwindup compensator is given to ensure the stochastic finite-time boundedness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible disturbances. The desired compensator can be designed via solving a convex optimization problem. Finally, a numerical example is employed to show the effectiveness of the proposed method.

Notation. Throughout the paper, for symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). 𝐼 is the identity matrix with appropriate dimension. The notation 𝑁^𝑇 represents the transpose of the matrix 𝑁;𝜆_max(𝑀) (resp., 𝜆_min(𝑀)) means the largest (resp., smallest) eigenvalue of the matrix 𝑀; (Ω,ℱ,𝒫) is a probability space; Ω is the sample space; ℱ is the 𝜎-algebra of subsets of the sample space and 𝒫 is the probability measure on ℱ;ℰ{⋅} denotes the expectation operator with respect to some probability measure 𝒫. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symbol ∗ is used to denote a matrix which can be inferred by symmetry. 𝐻𝑒{𝐴}=𝐴^𝑇+𝐴.

2. Preliminaries and Problem Description

Consider the following discrete-time Markov jump system (Σ) in the probability space (Ω,ℱ,𝒫): 𝑥_𝑝(𝑘+1)=𝐴_𝑝(𝑟(𝑘))𝑥_𝑝(𝑘)+𝐵_𝑝,𝑢(𝑟(𝑘))sat(𝑢(𝑘))+𝐵_𝑝,𝑤(𝑟(𝑘𝑤(𝑘),𝑦(𝑘)))=𝐶_𝑝,𝑦(𝑟(𝑘))𝑥_𝑝(𝑘)+𝐷_{𝑝,𝑦𝑤}(𝑟(𝑘𝑤(𝑘),𝑧(𝑘)))=𝐶_𝑝,𝑧(𝑟(𝑘))𝑥_𝑝(𝑘)+𝐷_{𝑝,𝑧𝑢}(𝑟(𝑘))sat(𝑢(𝑘))+𝐷_{𝑝,𝑧𝑤}(𝑟(𝑘))𝑤(𝑘), (1) where 𝑥_𝑝(𝑘)∈ℝ^𝑛𝑝 is the state vector, 𝑢(𝑘)∈ℝ^𝑛𝑢 is the control input, and sat(𝑢(𝑘))∈ℝ^𝑛𝑢 is the saturated control input. 𝑤(𝑘)∈𝐿^𝑝₂[0+∞) is the external disturbances, 𝑦(𝑘)∈ℝ^𝑛𝑦 is the measurement output, and 𝑧(𝑘)∈ℝ^𝑛𝑧 is the performance output. {𝑟(𝑘)} is a discrete-time Markov process and takes values from a finite set 𝑆={1,2,…,𝒩} with transition probabilities given by Pr𝑟_𝑘+1=𝑗∣𝑟_𝑘=𝑖=𝜋_𝑖𝑗, (2) where 𝜋_𝑖𝑗≥0, for ∀𝑗,𝑖∈𝑆, and ∑_𝑗∈𝑆𝜋_𝑖𝑗=1. Moreover, the transition rates matrix of the system (Σ) is defined by 𝜋₁₁𝜋₁₂⋯𝜋_1𝒩𝜋₂₁𝜋₂₂⋯𝜋_2𝒩𝜋⋮⋮⋱⋮_𝒩1𝜋_𝒩2⋯𝜋_𝒩𝒩. (3) The plant inputs are supposed to be bounded as follows: −𝑢_0(𝑘)≤𝑢_(𝑘)≤𝑢_0(𝑘),𝑢_0(𝑘)>0,𝑘=1,…,𝑚. (4) For the system (Σ), to simplify the notation, we denote 𝐴_𝑝𝑖=𝐴_𝑝(𝑟(𝑘)) for each 𝑟(𝑘)=𝑖∈𝑆, and the other symbols are similarly denoted. Assume that a linear controller is designed for any 𝑟(𝑘)=𝑖∈𝑆; then, 𝑥_𝑐(𝑘+1)=𝐴_𝑐𝑖𝑥_𝑐(𝑘)+𝐵_{𝑐𝑦,𝑖}𝑦(𝑘)+𝐵_{𝑐𝑤,𝑖}𝑤(𝑘)+𝑣₁,𝑦_𝑐(𝑘)=𝐶_𝑐𝑖𝑥_𝑐(𝑘)+𝐷_{𝑐𝑦,𝑖}𝑦(𝑘)+𝐷_{𝑐𝑤,𝑖}𝑤(𝑘)+𝑣₂, (5) where 𝑥_𝑐(𝑘)∈ℝ^𝑛𝑐 is the controller state and 𝑦_𝑐(𝑘)∈ℝ^𝑛𝑢 is the controller output; 𝑣₁ and 𝑣₂ will be used for antiwindup augmentation. In absence of actuator saturation, the unconstrained closed-loop is formed by setting the following: 𝑢=𝑦_𝑐,𝑣₁=0,𝑣₂=0. (6)

Assumption 1. The unconstrained closed-loop system (1)–(5) is well posed and internally stable.

In the presence of actuator saturation, the relation between 𝑢 and 𝑦_𝑐 is that 𝑢=sat(𝑦_𝑐). To minimize performance degradation caused by saturation, the following antiwindup compensator is designed for the closed-loop systems: 𝑥_aw(𝑘+1)=𝐴_aw,𝑖𝑥_aw(𝑘)+𝐵_aw,𝑖𝜓𝑦_𝑐(𝑘),𝑣(𝑘)=𝐶_aw,𝑖𝑥_aw(𝑘)+𝐷_aw,𝑖𝜓𝑦_c(𝑘), (7) where 𝜓(𝑦_𝑐(𝑘))=sat(𝑦_𝑐(𝑘))−𝑦_𝑐(𝑘). The resulting nonlinear closed-loop system (1), (5), (7) is depicted in Figure 1 and can be represented in the following compact form: 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐵_𝑞𝑖𝜓𝑦_𝑐(𝑘)+𝐵_𝑤𝑖𝑤(𝑘),𝑦_𝑐(𝑘)=𝐾_𝑖𝑥(𝑘)+𝐾_𝜙,𝑖𝜓𝑦_𝑐(𝑘)+𝐾_𝑤𝑖𝑤(𝑘),𝑧(𝑘)=𝐶_𝑧𝑖𝑥(𝑘)+𝐷_{𝑧𝑞,𝑖}𝜓𝑦_𝑐(𝑘)+𝐷_{𝑧𝑤,𝑖}𝑤(𝑘), (8) where 𝐴_𝑖=𝐴_𝑝𝑖+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝐵_{𝑝𝑢,𝑖}𝐶_𝑐𝑖𝐵_{𝑝𝑢,𝑖}𝐼₂𝐶_{𝑎𝑤,𝑖}𝐵_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝐴_𝑐𝑖𝐼₁𝐶_{𝑎𝑤,𝑖}00𝐴_{𝑎𝑤,𝑖},𝐵_𝑞𝑖=𝐵_{𝑝𝑢,𝑖}𝐼₂𝐷_{𝑎𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝐼₁𝐷_{𝑎𝑤,𝑖}𝐵_{𝑎𝑤,𝑖},𝐵_𝑤𝑖=𝐵_{𝑝𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑦,𝑖}𝐷_{𝑝,𝑦𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑤,𝑖}𝐵_{𝑐𝑦,𝑖}𝐷_{𝑝,𝑦𝑤,𝑖}+𝐵_{𝑐𝑤,𝑖}0,𝐶_𝑧𝑖=𝐶_{𝑝𝑧,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖}𝐷_𝑐𝑖𝐶_{𝑝𝑦,𝑖}𝐷_{𝑝,𝑧𝑢,𝑖}𝐼₂𝐶_{𝑎𝑤,𝑖},𝐷_{𝑧𝑞,𝑖}=𝐼₂𝐷_{𝑎𝑤,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖},𝐷_{𝑧𝑤,𝑖}=𝐷_{𝑝,𝑧𝑤,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖}𝐷_{𝑐𝑤,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖}𝐷_𝑐𝑖𝐷_{𝑝,𝑦𝑤,𝑖},𝐾_𝑖=𝐷_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝐶_𝑐𝑖𝐼₂𝐶_{𝑎𝑤,𝑖},𝐾_𝜙,𝑖=𝐼₂𝐷_{𝑎𝑤,𝑖},𝐾_𝑤,𝑖=𝐷_{𝑐𝑤,𝑖}+𝐷_{𝑝,𝑦𝑤,𝑖},𝐼₁=𝐼0,𝐼₂=0𝐼. (9) For this system, we introduce the following definitions and assumption.

Figure 1 

The closed-loop systems with input saturation.

Assumption 2 (see [17]). The external disturbance 𝑤(𝑘) is varying and satisfies the following constraint condition: ^𝑇_𝑘=0𝑤(𝑘)^𝑇𝑤(𝑘)≤𝑑,𝑑≥0. (10)

Definition 3 (see [17]). The resulting closed-loop system (8) is stochastic finite-time stable (SFTB) with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝑑) with 0<𝛿_𝑥<𝜖,𝑅_𝑖>0, and 𝑁∈𝑍_𝑘≥0, if 𝐸𝑥^𝑇(0)𝑅_𝑖𝑥(0)≤𝛿²_𝑥⟹𝐸𝑥^𝑇(𝑘)𝑅_𝑖𝑥(𝑘)<𝜖²,∀𝑘∈{1,2,…,𝑁}. (11)

Definition 4 (see [17]). The resulting closed-loop system (8) is said to be stochastic 𝐻_∞ finite-time stable with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝛾,𝑑) with 0<𝛿_𝑥<𝜖, 𝑅_𝑖>0, 𝛾>0, and 𝑁∈𝑍_𝑘≥0, if the system (8) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝛾,𝑑), and under the zero-initial condition, the output 𝑧(𝑘) satisfies 𝐸^𝑁_𝑗=0𝑧^𝑇𝑗𝑧𝑗≤𝛾²𝐸^𝑁_𝑗=0𝑤^𝑇𝑗𝑤𝑗, (12) for any nonzero 𝑤(𝑘) which satisfies (10), where 𝛾 is a prescribed positive scalar.

3. Main Results

In this section, we investigate the stabilization analysis of the unconstrained systems and the antiwindup controller design of the resulting closed-loop system. Some sufficient conditions in terms of LMI are given. Before presenting the main results, we give some lemmas as follows.

Lemma 5 (see [4]). For the closed-loop systems (8) with the matrix 𝐾_𝑖, the appropriate matrix 𝐿_𝑖∈ℝ^𝑚×𝑛 is given, if 𝑥(𝑘) is in the set 𝐷(𝑢_𝑜), where 𝐷(𝑢_𝑜) is defined as follows: 𝐷𝑢_𝑜=𝑥(𝑘)∈ℝ^𝑛;−𝑢_0(𝑘)≤𝐾_𝑖(𝑘)+𝐿_𝑖(𝑘)𝑥(𝑡)≤𝑢_0(𝑘),𝑢_0(𝑘)>0,𝑘=1,…,𝑚, (13) then for any diagonal positive matrix 𝑇∈ℝ^𝑚×𝑚, one has the following: 𝜓(𝑢(𝑘))^𝑇𝑇𝜓(𝑢(𝑘))−𝐿_𝑖𝑥(𝑘)+𝐾_𝜙,𝑖𝜓𝑦_𝑐(𝑘)+𝐾_𝑤𝑖𝑤(𝑘)≤0. (14)

Lemma 6 (see [12]). For the given symmetric matrix 𝑆∈ℝ^{(𝑛+𝑚)×(𝑛+𝑚)}, 𝑆𝑆=¹¹𝑆₁₂𝑆^𝑇₁₂𝑆₂₂, (15) where 𝑆₁₁∈ℝ^𝑛×𝑛, 𝑆₁₂∈ℝ^𝑛×𝑚, and 𝑆₂₂∈ℝ^𝑚×𝑚, the following conditions are equivalent: (1)𝑆<0; (2)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0; (3)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

3.1. Design of Controller

In this section, we design the controller for the unconstrained systems with 𝑣₁=0 and 𝑣₂=0. Combining system (1) with controller (5), we have 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐵_𝑤𝑖𝑤(𝑘),𝑧(𝑘)=𝐶_𝑧𝑖𝑥(𝑘)+𝐷_{𝑧𝑤,𝑖}𝑤(𝑘), (16) where 𝐴_𝑖=𝐴_𝑝𝑖+𝐵_{𝑝𝑢,𝑖}D_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝐵_{𝑝𝑢,𝑖}𝐶_𝑐𝑖𝐵_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝐴_𝑐𝑖,𝐵_𝑤𝑖=𝐵_{𝑝𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑦,𝑖}𝐷_{𝑝,𝑦𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑤,𝑖}𝐵_{𝑐𝑦,𝑖}𝐷_{𝑝,𝑦𝑤,𝑖}+𝐵_{𝑐𝑤,𝑖}. (17)

Theorem 7. For each 𝑟(𝑘)=𝑖∈𝑆, the unconstrained system (16) is SFTB with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝑑) with 0<𝛿_𝑥<𝜖, if there exist scalars 𝜇≥0, 𝜎₁≥0, 𝜎₂≥0, and the given 𝜆>0, two sets of mode-dependent symmetric positive-defined matrices {𝑋_𝑖,𝑖∈𝑆} and {𝑄_𝑖,𝑖∈𝑆}, such that the following conditions hold: −𝜇𝜆𝐼0𝐿^𝑇_1𝑖∗−𝑄_𝑖𝐿^𝑇_2𝑖∗∗−𝑊<0,𝜎₂𝑑²−𝜇^−𝑁𝜖²∗𝛿_𝑥−𝜎₁<0,𝜆𝑋_𝑖<𝐼,𝜎₁𝑅_𝑖⁻¹<𝑋_𝑖<𝑅_𝑖⁻¹,0<𝑄_𝑖<𝜎₂𝐼, (18)(19)(20)(21)(22) where 𝑊=diag𝑋₁,𝑋₂,…,𝑋_𝑛,𝐿^𝑇_1𝑖=√𝜋_𝑖1𝐴^𝑇_𝑖√𝜋_𝑖2𝐴^𝑇_𝑖⋯√𝜋_𝑖𝑛𝐴^𝑇_𝑖,𝐿^𝑇_2𝑖=√𝜋_𝑖1𝐵^𝑇_𝑤𝑖√𝜋_𝑖2𝐵^𝑇_𝑤𝑖⋯√𝜋_𝑖𝑛𝐵^𝑇_𝑤𝑖. (23)

Proof. Define the following Lyapunov function for each 𝛿(𝑡)=𝑖∈𝑆: 𝑉(𝑘)=𝑥(𝑘)^𝑇𝑃_𝑖𝑥(𝑘). (24) It is readily obtained that 𝐸{𝑉(𝑘+1)}=𝐸^𝑛_𝑗=1𝜋_𝑖𝑗𝑥(𝑘+1)^𝑇𝑃_𝑗𝑥()(𝑘)𝑘+1=𝜉^𝑇𝐿_1𝑖𝐿_2𝑖^𝑇𝑊𝐿_1𝑖𝐿_2𝑖𝜉(𝑘), (25) where 𝜉(𝑘)=𝑥(𝑘)^𝑇𝑤(𝑘)^𝑇,𝑊=diag𝑃₁,𝑃₂,…,𝑃_ℎ. (26) By using of Schur complement lemma to (18), and note that 𝑃_𝑖⁻¹=𝑋_𝑖 and 𝜆𝑋_𝑖<𝐼, we derive 𝜆𝐼<𝑃_𝑖; then, we have 𝜉(𝑘)^𝑇−𝜇𝜆𝐼0∗−𝑄_𝑖𝜉(𝑘)+𝜉(𝑘)^𝑇𝐿_1𝑖𝐿_2𝑖^𝑇𝑊𝐿_1𝑖𝐿_2𝑖𝜉(𝑘)<0. (27) It follows that 𝐸{𝑉(𝑘+1)}<𝜇𝑥(𝑘)^𝑇𝑃_𝑖𝑥(𝑘)+𝑤(𝑘)^𝑇𝑄_𝑖𝑤(𝑘). (28) It is shown that 𝐸{𝑉(𝑘+1)}<𝜇𝑉(𝑘)+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝑤(𝑘)^𝑇𝑤(𝑘). (29) Then we have 𝐸{𝑉({𝑉(𝑘𝑘+1)}<𝜇𝐸)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝐸𝑤(𝑘)^𝑇𝑤(𝑘). (30) Since 𝜇≥1, it is easily found that 𝐸{𝑉(𝑘+1)}<𝜇𝐸{𝑉(0)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖×𝐸^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝑤𝑗^𝑇𝑤𝑗≤𝜇^𝑘𝐸{𝑉(0)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝜇^𝑘𝑑². (31) Letting 𝑃_𝑖=𝑅_𝑖^−1/2𝑃_𝑖𝑅_𝑖^−1/2, (32) and noting that 𝐸𝑥^𝑇(0)𝑅_𝑖𝑥(0)≤𝛿²_𝑥, (33) it can be verified that 𝐸{𝑉(0)}=𝐸𝑥^𝑇(0)𝑃_𝑖𝑥(0𝑥)=𝐸^𝑇(0)𝑅_𝑖^1/2𝑃_𝑖𝑅_𝑖^1/2𝑥(0)≤sup_𝑖∈𝑆𝜆_max𝑃_𝑖𝐸𝑥^𝑇(0)𝑅_𝑖𝑥(0)≤sup_𝑖∈𝑆𝜆_max𝑃_𝑖𝛿²_𝑥. (34) Similarly, for all 𝑖∈𝑆, we can obtain 𝐸{𝑉(𝑘)}=𝐸𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘𝑥)=𝐸^𝑇(𝑘)𝑅_𝑖^1/2𝑃_𝑖𝑅_𝑖^1/2𝑥(𝑘)≥inf_𝑖∈𝑆𝜆_min𝑃_𝑖𝐸𝑥^𝑇(𝑘)𝑅_𝑖𝑥(𝑘). (35) Then, it is not difficult to find that 𝐸𝑥^𝑇(𝑘)𝑅_𝑖𝑥(𝑘<)sup_𝑖∈𝑆𝜆_max𝑃_𝑖𝜇^𝑘𝛿²_𝑥+sup_{(𝑖∈𝑆)}𝜆_max𝑄_𝑖𝜇^𝑘𝑑²inf_𝑖∈𝑆𝜆_min𝑃_𝑖, (36) which implies that sup_𝑖∈𝑆𝜆_max𝑃_𝑖𝜇^𝑘𝛿²_𝑥+sup_{(𝑖∈𝑆)}𝜆_max𝑄_𝑖𝜇^𝑘𝑑²inf_𝑖∈𝑆𝜆_min𝑃_𝑖<𝜖². (37) Then, one can obtain that sup_𝑖∈𝑆𝜆_max𝑃_𝑖𝛿²_𝑥+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝑑²<inf_𝑖∈𝑆𝜆_min𝑃_𝑖𝜇^−𝑁𝜖². (38) Setting 𝑋_𝑖=𝑃_𝑖⁻¹,𝜎₁𝑅_𝑖⁻¹<𝑋_𝑖<𝑅_𝑖⁻¹,0<𝑄_𝑖<𝜎₂𝐼, (39) it is easy to see that 𝜎₁⁻¹𝛿²_𝑥+𝜎₂𝑑²<𝜇^−𝑁𝜖². (40) It is obvious that (40) is equivalent to (19).
This completes the proof.

3.2. Design of Dynamic Antiwindup Compensator

Theorem 8. For each 𝑟(𝑘)=𝑖∈𝑆, with antiwindup compensator (7), such that the resulting closed-loop system (10) is 𝑆𝐹𝑇𝐵 with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝑑) with 0<𝛿_𝑥<𝜖, if there exist scalars 𝜇≥0, 𝜎₁≥0, and 𝜎₂≥0, three sets of mode-dependent symmetric positive-defined matrices {𝑋_𝑖,𝑖∈𝑆}, {𝑄_𝑖,𝑖∈𝑆} and diag positive-defined matrices {𝑆_𝑖,𝑖∈𝑆}, and two sets of mode-dependent matrices {𝑌_𝑖,𝑖∈𝑆} and {𝐿_𝑖=𝐿_𝑖𝑋_𝑖,𝐴_{𝑎𝑤,𝑖}=𝐴_{𝑎𝑤,𝑖}𝑋_𝑖,𝐶_{𝑎𝑤,𝑖}=𝐶_{𝑎𝑤,𝑖}𝑋_𝑖,𝐵_{𝑎𝑤,𝑖}=𝐵_{𝑎𝑤,𝑖}𝑆_𝑖,𝐷_{𝑎𝑤,𝑖}=𝐷_{𝑎𝑤,𝑖}𝑆_𝑖𝑖∈𝑆}, such that the following conditions hold: −𝜇𝑋_𝑖0𝐿^𝑇_𝑖𝐿^𝑇_1𝑖∗−𝑄_𝑖𝐾^𝑇_𝑤𝑖𝐿^𝑇_2𝑖∗∗−2𝑆_𝑖−𝐻𝑒𝐾_𝜑,𝑖𝐿^𝑇_3𝑖∗∗∗−𝑊<0,𝜎₂𝑑²−𝜇^−𝑁𝜖²∗𝛿_𝑥−𝜎₁<0,𝑋_𝑖∗𝐾_𝑖+𝐿_𝑖𝑢²_0(𝑘)>0,𝑘=1,…,𝑚,𝜎₁𝑅_𝑖⁻¹<𝑋_𝑖<𝑅_𝑖⁻¹,0<𝑄_𝑖<𝜎₂𝐼, (41)(42)(43)(44)(45) where 𝑊=diag𝑋₁,𝑋₂,…,𝑋_𝑛,𝐿^𝑇_1𝑖=√𝜋_𝑖1𝐴^𝑇_𝑖√𝜋_𝑖2𝐴^𝑇_𝑖⋯√𝜋_𝑖𝑛𝐴^𝑇_𝑖,𝐿^𝑇_2𝑖=√𝜋_𝑖1𝐵^𝑇_𝑤𝑖√𝜋_𝑖2𝐵^𝑇_𝑤𝑖⋯√𝜋_𝑖𝑛𝐵^𝑇_𝑤𝑖,𝐿^𝑇_3𝑖=√𝜋_𝑖1𝐵^𝑇_𝑞𝑖√𝜋_𝑖2𝐵^𝑇_𝑞𝑖⋯√𝜋_𝑖𝑛𝐵^𝑇_𝑞𝑖, (46) with 𝐴_𝑖=𝐴_𝑝𝑖𝑋_𝑖+𝐵_{𝑝𝑢,𝑖}𝐷_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝑋_𝑖𝐵_{𝑝𝑢,𝑖}𝐶_𝑐𝑖𝑋_𝑖𝐵_{𝑝𝑢,𝑖}𝐼₂𝐶_{𝑎𝑤,𝑖}𝐵_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝑋_𝑖𝐴_𝑐𝑖𝑋_𝑖𝐼₁𝐶_{𝑎𝑤,𝑖}00𝐴_{𝑎𝑤,𝑖},𝐵_𝑞𝑖=𝐵_{𝑝𝑢,𝑖}𝐼₂𝐷_{𝑎𝑤,𝑖}+𝐵_{𝑝𝑢,𝑖}𝑋_𝑖𝐼₁𝐷_{𝑎𝑤,𝑖}𝐵_{𝑎𝑤,𝑖},𝐾_𝑖=𝐷_{𝑐𝑦,𝑖}𝐶_{𝑝𝑦,𝑖}𝑋_𝑖C_𝑐𝑖𝑋_𝑖𝐼₂𝐶_{𝑎𝑤,𝑖},𝐾_𝜙,𝑖=𝐼₂𝐷_{𝑎𝑤,𝑖}. (47)

Proof. Define the following Lyapunov function for each 𝛿(𝑡)=𝑖∈𝑆: 𝑉(𝑘)=𝑥(𝑘)^𝑇𝑃_𝑖𝑥(𝑘). (48) It is readily obtained that 𝐸{𝑉(𝑘+1)}=𝐸^𝑛_𝑗=1𝜋_𝑖𝑗𝑥(𝑘+1)^𝑇𝑃_𝑗𝑥()(𝑘)𝑘+1=𝜉^𝑇𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖^𝑇𝑊𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖𝜉(𝑘), (49) where 𝜉(𝑘)=𝑥(𝑘)^𝑇𝑤(𝑘)^𝑇𝜓(𝑘)^𝑇,𝑊=diag𝑃₁,𝑃₂,…,𝑃_ℎ,𝐿^𝑇_1𝑖=√𝜋_𝑖1𝐴^𝑇_𝑖√𝜋_𝑖2𝐴^𝑇_𝑖⋯√𝜋_𝑖𝑛𝐴^𝑇_𝑖,𝐿^𝑇_2𝑖=√𝜋_𝑖1𝐵^𝑇_𝑤𝑖√𝜋_𝑖2𝐵^𝑇_𝑤𝑖⋯√𝜋_𝑖𝑛𝐵^𝑇_𝑤𝑖,𝐿^𝑇_3𝑖=√𝜋_𝑖1𝐵^𝑇_𝑞𝑖√𝜋_𝑖2𝐵^𝑇_𝑞𝑖⋯√𝜋_𝑖𝑛𝐵^𝑇_𝑞𝑖. (50) Then, by pre- and postmultiplying (41) by diag{𝑃_𝑖,𝐼,𝑇_𝑖,𝐼} with 𝑃_𝑖=𝑋_𝑖⁻¹,𝑇_𝑖=𝑆_𝑖⁻¹, we have −𝜇𝑃_𝑖0𝐿^𝑇_𝑖𝑇_𝑖𝐿^𝑇_1𝑖∗−𝑄_𝑖𝐾^𝑇_𝑤𝑖𝐿^𝑇_2𝑖∗∗−2𝑇_𝑖−𝐻𝑒𝑇_𝑖K_𝜙,𝑖𝐿^𝑇_3𝑖∗∗∗−𝑊<0. (51) By using of Schur complement lemma, we derive 𝜉(𝑘)^𝑇−𝜇𝑃_𝑖0𝐿^𝑇_𝑖𝑇_𝑖∗−𝑄_𝑖𝐾^𝑇_𝑤𝑖∗∗−2𝑇_𝑖−𝐻𝑒𝑇_𝑖𝐾_𝜙,𝑖𝜉(𝑘)+𝜉(𝑘)^𝑇𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖^𝑇𝑊𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖𝜉(𝑘)<0. (52) It follows that 𝐸{𝑉(𝑘+1)}<𝜇𝑥(𝑘)^𝑇𝑃_𝑖𝑥(𝑘)+𝑤(𝑘)^𝑇𝑄_𝑖𝑤(𝑘)+𝜓(𝑘)^𝑇2𝑇_𝑖+𝐻𝑒𝑇_𝑖𝐾_𝜙,𝑖𝜓(𝑘)−2𝜓(𝑘)^𝑇𝑇_𝑖𝐿_𝑖𝑥(𝑘)+2𝜓(𝑘)^𝑇𝑇_𝑖𝐾_𝑤𝑖𝑤(𝑘). (53) Since 𝜓(𝑘)^𝑇(2𝑇_𝑖+𝐻𝑒{𝑇_𝑖𝐾_𝜙,𝑖})𝜓(𝑘)−2𝜓(𝑘)^𝑇𝑇_𝑖𝐿_𝑖𝑥(𝑘)+2𝜓(𝑘)^𝑇𝑇_𝑖𝐾_𝑤𝑖𝑤(𝑘)≤0, we get 𝐸{𝑉(𝑘+1)}<𝜇𝑥(𝑘)^𝑇𝑃_𝑖𝑥(𝑘)+𝑤(𝑘)^𝑇𝑄_𝑖𝑤(𝑘). (54) It is shown that 𝐸{𝑉(𝑘+1)}<𝜇𝑉(𝑘)+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝑤(𝑘)^𝑇𝑤(𝑘). (55) Then, we have 𝐸{𝑉(𝑘+1)}<𝜇𝐸{𝑉(𝑘)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝐸𝑤(𝑘)^𝑇𝑤(𝑘). (56) Since 𝜇≥1, it is easily found that 𝐸{𝑉(𝑘+1)}<𝜇𝐸{𝑉(0)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖×𝐸^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝑤𝑗^𝑇𝑤𝑗≤𝜇^𝑘𝐸{𝑉(0)}+sup_{()𝑖∈𝑆}𝜆_max𝑄_𝑖𝜇^𝑘𝑑². (57) The following proof is similar to the process of Theorem 7.
Based on Lemma 5, it is easy to obtain that 𝑃_𝑖∗𝐾_𝑖+𝐿_𝑖𝑢²_0(𝑘)>0,𝑘=1,…,𝑚, (58) then pre- and post-multiply (58) by diag{𝑋_𝑖,𝐼} which implies (43). This completes the proof.

Theorem 9. For each 𝑟(𝑘)=𝑖∈𝑆, with antiwindup compensator (7), such that the resulting closed-loop system (10) is said to be Stochastic 𝐻_∞ finite-time stable via state feedback with respect to (𝛿_𝑥,𝜖,𝑅_𝑖,𝑁,𝛾,𝑑), if there exist three scalars 𝜇≥0, 𝜎₁≥0, and 𝛾≥0, two sets of mode-dependent symmetric positive-defined matrices {𝑋_𝑖,𝑖∈𝑆} and diag matrices {𝑆_𝑖,𝑖∈𝑆}, and two sets of mode-dependent matrices {𝑌_𝑖,𝑖∈𝑆} and {𝐿_𝑖=𝐿_𝑖𝑋_𝑖,𝑖∈𝑆}, such that the following conditions hold: −𝜇𝑋_𝑖0𝐿^𝑇_𝑖𝐿^𝑇_1𝑖𝐶^𝑇_𝑧𝑖∗−𝛾²𝜇^−𝑁𝐼𝐾^𝑇_𝑤𝑖𝐿^𝑇_2𝑖𝐷^𝑇_𝑧𝑤,i∗∗−2𝑆_𝑖−𝐻𝑒𝐾_𝜙,𝑖𝐿^𝑇_3𝑖𝐷^𝑇_{𝑧𝑞,𝑖}∗∗∗−𝑊0∗∗∗∗−𝐼<0,𝜇^−𝑁𝑑²𝛾²−𝜖²∗𝛿_𝑥−𝜎₁<0,𝑋_𝑖∗𝐾_𝑖+𝐿_𝑖𝑢²_0(𝑘)>0,𝑘=1,…,𝑚,𝜎₁𝑅_𝑖⁻¹<𝑋_𝑖<𝑅_𝑖⁻¹, (59)(60)(61)(62) with 𝐶_𝑧𝑖=𝐶_{𝑝𝑧,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖}𝐷_𝑐𝑖𝐶_{𝑝𝑦,𝑖}𝑋_𝑖𝐷_{𝑝,𝑧𝑢,𝑖}𝑋_𝑖𝐼₂𝐶_{𝑎𝑤,𝑖},𝐷^𝑇_{𝑧𝑞,𝑖}=𝐼₂𝐷_{𝑎𝑤,𝑖}+𝐷_{𝑝,𝑧𝑢,𝑖}. (63)

Proof. Choose the similar Lyapunov function as Theorem 7 and denote Π(𝑥(𝑘)(𝑘)(𝑘)){𝑉((𝑘)(𝑘),𝑤,𝑟=𝑖=𝐸𝑘+1)}−𝜇𝑉+𝑧^𝑇𝑧(𝑘)−𝛾²𝜇^−𝑁𝑤(𝑘)^𝑇𝑤(𝑘). (64) Thus, in light of Lemma 5, we have Π𝑥(𝑘),𝑤(𝑘),𝑟_𝑘(𝑘)=𝑖≤𝜉^𝑇𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖^𝑇𝑊𝐿_1𝑖𝐿_2𝑖𝐿_3𝑖𝜉(𝑘)+𝜉(𝑘)^𝑇𝐶_𝑧𝑖𝐷_{𝑧𝑤,𝑖}𝐷_{𝑧𝑞,𝑖}^𝑇𝐶_𝑧𝑖𝐷_{𝑧𝑤,𝑖}𝐷_{𝑧𝑞,𝑖}𝜉(𝑘)+𝜉^𝑇(𝑘)−𝜇𝑋_𝑖0𝐿^𝑇_𝑖𝑇_𝑖∗−𝛾²𝜇^−𝑁𝐼𝐾^𝑇_𝑤𝑖∗∗−2𝑇_𝑖−𝐻𝑒𝑇_𝑖𝐾_𝜙,𝑖𝜉(𝑘). (65) Then pre- and postmultiply (59) by diag{𝑃_𝑖,𝐼,𝑇_𝑖,𝐼}, and considering Schur complement lemma and (65), we derive that Π(𝑥(𝑘),𝑤(𝑘),𝑟(𝑘)=𝑖)<0 (66) holds for all 𝑟_𝑘=𝑖∈𝑆. According to (66), one can obtain that 𝐸{𝑉(𝑘+1)}<𝜇𝐸{𝑉(𝑘)}−𝐸𝑧(𝑘)^𝑇𝑧(𝑘)+𝛾²𝜇^−𝑁𝐸𝑤(𝑘)^𝑇𝑤(𝑘). (67) Then, we have 𝐸{𝑉(𝑘)}<𝜇^𝑘𝐸{𝑉(0)}−^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝐸𝑧𝑗^𝑇𝑧𝑗+𝛾²𝜇^−𝑁𝐸^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝑤(𝑗)^𝑇𝑤𝑗. (68) Under the zero-value initial condition and noting that 𝑉(𝑘)≥0, for all 𝐾∈𝑍_𝑘≥0, it is shown that ^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝐸𝑧𝑗^𝑇𝑧𝑗<𝛾²𝜇^−𝑁𝐸^𝑘−1_𝑗=0𝜇^{𝑘−𝑗−1}𝑤𝑗^𝑇𝑤𝑗. (69) Since 𝜇≥1 and from (69), we have 𝐸^𝑁_𝑗=0𝑧𝑗^𝑇𝑧𝑗=^𝑁_𝑗=0𝐸𝑧𝑗^𝑇𝑧𝑗≤^𝑁_𝑗=0𝐸𝜇^𝑁−𝑗𝑧𝑗^𝑇𝑧𝑗≤𝛾²𝜇^−𝑁𝐸^𝑁_𝑗=0𝜇^𝑁−𝑗𝑤𝑗^𝑇𝑤𝑗≤𝛾²𝐸^𝑁_𝑗=0𝑤𝑗^𝑇𝑤𝑗. (70) The following proof is similar to the process of Zhang and Liu [17].
Since 𝜀(𝑃_𝑖,1)⊂𝐷(𝑢₀), it follows that 𝑃_𝑖∗𝐾_𝑖+𝐿_𝑖𝑢²_0(𝑘)>0,𝑘=1,…,𝑚, (71) and then pre- and post-multiply (71) by diag(𝑋_𝑖,𝐼) and its transpose, respectively; we derive condition (61). This completes the proof.

4. Illustrative Examples

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed method. Consider the following systems with four operation modes.

Mode 1 is 𝐴_𝑝1=0.75−0.751.5−1.5,𝐵_𝑝𝑢,1=10,𝐵_𝑝𝑤,1=10,𝐶_𝑝𝑦,1=−0.1−0.2,𝐶_𝑝𝑧,1=,𝐷10_{𝑝𝑦𝑤,1}=1,𝐷_{𝑝𝑧𝑢,1}=1,𝐷_{𝑝𝑧𝑤,1}=0.8. (72) Mode 2 is 𝐴_𝑝2=0.154.52.10−0.4,𝐵_𝑝𝑢,2=10,𝐵_𝑝𝑤,2=10,𝐶_𝑝𝑦,2=−0.1−0.1,𝐶_𝑝𝑧,2=,𝐷10_{𝑝𝑦𝑤,2}=1,𝐷_{𝑝𝑧𝑢,2}=0.9,𝐷_{𝑝𝑧𝑤,2}=0.8. (73) Mode 3 is 𝐴_𝑝3=0.242.501.2−2.1,𝐵_𝑝𝑢,3=00.9,𝐵_𝑝𝑤,3=10,𝐶_𝑝𝑦,3=−0.10,𝐶_𝑝𝑧,3=,𝐷10_{𝑝𝑦𝑤,3}=0.8,𝐷_{𝑝𝑧𝑢,3}=1,𝐷_{𝑝𝑧𝑤,3}=1.2. (74) Mode 4 is 𝐴_𝑝4=1−0.251.5−1.5,𝐵_𝑝𝑢,4=10,𝐵_𝑝𝑤,4=10,𝐶_𝑝𝑦,4=10,𝐶_𝑝𝑧,4=,𝐷10_{𝑝𝑦𝑤,4}=1,𝐷_{𝑝𝑧𝑢,4}=0.5,𝐷_{𝑝𝑧𝑤,4}=1. (75) With the given designed controllers, 𝐴_𝑐1=−5.5,𝐵_𝑐𝑦,1=−1,𝐵_𝑐𝑤,1𝐶=1,_𝑐1=−1,𝐷_𝑐𝑦,1=−0.1,𝐷_𝑐𝑤,1𝐴=0.5,_𝑐2=−5,𝐵_𝑐𝑦,2=−0.9,𝐵_𝑐𝑤,2𝐶=1,_𝑐2=−1,𝐷_𝑐𝑦,2=5.9,𝐷_𝑐𝑤,2𝐴=1,_𝑐3=−4.5,𝐵_𝑐𝑦,3=−1,𝐵_𝑐𝑤,3𝐶=1,_𝑐3=−1.5,𝐷_𝑐𝑦,3=5.1,𝐷_𝑐𝑤,3𝐴=1,_𝑐4=−7,𝐵_𝑐𝑦,4=−1,𝐵_𝑐𝑤,4𝐶=1,_𝑐4=−1.5,𝐷_𝑐𝑦,4=−2,𝐷_𝑐𝑤,4=1. (76) The transition rate matrix is given by the following: .0.30.30.20.20.40.30.20.10.20.10.40.30.20.30.10.4 (77) In this case, we choose the initial values for 𝑅_𝑖=𝐼₂,𝑖=1,2,3,4, 𝛿_𝑥=1,𝑁=5,𝛼=10⁻¹⁰,𝜇=2.5,𝑑=1, and 𝑤(𝑘)=0.5(1+cos𝑥(𝑘)); Theorem 7 yields to 𝜖=36.2671,𝜎₁=0.4906,𝜎₂=13.7421, and the bounds of the input saturation 𝑢₀=0.08.

Based on Theorem 9, we derive 𝐴_𝑎𝑤,1=−2.67,𝐴_𝑎𝑤,2𝐴=−1.86,_𝑎𝑤,3=−1.88,𝐴_𝑎𝑤,4𝐵=−2.59,_𝑎𝑤,1=−0.02,𝐵_𝑎𝑤,2𝐵=−0.01,_𝑎𝑤,3=−0.01,𝐵_𝑎𝑤,4𝐶=−0.02,_𝑎𝑤,1=17.270.68,𝐶_𝑎𝑤,2=,𝐶68.44−48.31_𝑎𝑤,3=68.74−49.29,𝐶_𝑎𝑤,4=,𝐷17.270.66_𝑎𝑤,1=0.210.1,𝐷_𝑎𝑤,2=,𝐷0.18−0.1_𝑎𝑤,3=00.19,𝐷_𝑎𝑤,4=.0.20.1 (78)

Remark 10. Figures 2, 3, and 4 are given on the last page. Figure 1 is 𝑟_𝑘 of the jump rates, Figure 2 and Figure 3 are state response of open and closed-loop system. Based on the figures provided, the controller and the compensator we designed guarantee that the resulting closed-loop systems are mean-square locally asymptotically finite-time stabilizable.

Figure 2 

𝑟𝑘 of jump rates.

Figure 3 

𝑥(𝑘) of the system (1)–(5).

Figure 4 

𝑥(𝑘) of the closed-loop system (1)–(7).

5. Conclusions and Future Work

In this paper, the finite-time 𝐻_∞ stabilization problem for a class of discrete-time Markov jump systems with input saturation has been investigated. Based on stochastic finite-time stability analysis, a controller designed for the unconstrained system with a dynamic antiwindup compensator subject to actuator saturation is given to guarantee the stochastic finite-time boundedness and stochastic finite-time stabilization of the considered closed-loop system for all admissible disturbances. Finally, the effectiveness of the proposed approach has been illustrated by simulation results. The finite-time stabilization problem of Markov jump systems with constrained input and time-delay will be considered in the future work.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61203047, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, the Key Foundation of Natural Science for Colleges and Universities in Anhui province under Grant KJ2012A049.