Full-Order Disturbance-Observer-Based Control for Singular Hybrid System

Abstract

The problem of the disturbance-observer-based control for singular hybrid system with two types of disturbances is addressed in this paper. Under the assumption that the system states are, unavailable, full-order observers (for both system states and the disturbance) and a nonlinear control scheme are constructed, such that the composite system can be guaranteed to be stochastically admissible, and the two types of disturbances can be attenuated and rejected, simultaneously. Based on the Lyapunov stability theory, sufficient conditions for the existence of the desired full-order disturbance-observer-based controllers are established in terms of linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the effectiveness of the proposed approaches.

1. Introduction

Singular systems, which are also referred to as implicit systems, descriptor systems, are widely used to model various engineering systems, such as electrical networks, power systems, networked control systems, and robotics, due to the fact that such systems can provide a more general representation than standard state-space systems in the sense of modeling [1]. A great number of fundamental results based on the theory of state-space systems have been successfully extended to singular systems. For some fundamental work on this subject, we refer the reader to [2–7].

Disturbance-observer-based control has been proven to be an effective strategy to reject the disturbance which can be modeled by an exogenous system [8–13]. Recently, its applications have been found in the robotic systems [8], table drive systems [12], missile system [11], and so on. The essential idea of the disturbance-observer-based control scheme is to design a disturbance observer to estimate the matched disturbance and cancel the effect of the matched disturbance by applying the estimation information into the control law. On another research front line, singular Markovian jump system which includes the dynamics of both singular system and Markovian jump system has attracted great attention from researchers, and recently some results are available in the publication: sliding control problem for continuous Markovian jump singular system is investigated in [14, 15], where the necessary and sufficient condition for the admissibility of the nominal system is presented. ℓ₂-ℓ_∞ filter problem is designed for discrete-time singular Markovian jump systems in [16]. Notice that in the above-mentioned publications, the disturbance considered in the plant has been assumed to be norm-bounded one. In this paper, we will consider a wider case: the plant is subject to multiple disturbances (one is norm-bounded disturbance, and the other is the disturbance that can be modeled by the exogenous system).

Based on the previous reasons, in this paper, we will investigate the disturbance-observer-based control problem for a class of singular systems with Markovian switching parameters and multiple disturbances. With the proposed nonlinear control scheme and by choosing a proper stochastic Lyapunov-Krasovskii functional, sufficient conditions for the existence of the desired controllers in terms of LMIs [17, 18] are presented, such that the composite system is stochastically admissible and meets certain performance requirements. Finally, a numerical example is used to illustrate the efficiency of the developed results.

The remainder of this paper is organized as follows. Section 2 describes the problem and preliminaries. Section 3 presents the main theoretical results. A numerical example is given in Section 4. Finally, we conclude the paper in Section 5.

2. Problem Statement and Preliminaries

Fix a probability space (Ω,ℱ,𝒫), where Ω is the sample space, ℱ is the 𝜎-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. Under this probability space, we consider the following singular MJLs:𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐺𝑟_𝑡𝑢(𝑡)+𝑑₁(𝑡)+𝐻𝑟_𝑡𝑑₂(𝑡),𝑦(𝑡)=𝐷₁𝑟_𝑡𝑥(𝑡)+𝐷₂𝑟_𝑡𝑑₁(𝑡), (1a)(1b)where 𝑥(𝑡)∈ℝ^𝑛 is the semistate vector, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑦(𝑡)∈ℝ^𝑠 is the output measurement and 𝑑₁(𝑡)∈ℝ^𝑚 is supposed to satisfy conditions described as Assumption 1, which can represent the constant and harmonic noises. 𝑑₂(𝑡)∈ℝ^𝑞 is another disturbance which is assumed to be an arbitrary signal in ℒ₂[0,∞). The matrix 𝐸∈ℝ^𝑛×𝑛 is singular with rank(𝐸)=𝑟<𝑛, and the matrices 𝐴_𝑖≜𝐴(𝑟_𝑡=𝑖),  𝐺_𝑖≜𝐺(𝑟_𝑡=𝑖),  𝐻_𝑖≜𝐻(𝑟_𝑡=𝑖),  𝐷_1𝑖≜𝐷₁(𝑟_𝑡=𝑖), and 𝐷_2𝑖≜𝐷₂(𝑟_𝑡=𝑖) are known real constant matrices of appropriate dimensions. {𝑟_𝑡} is a continuous-time Markov process with right continuous trajectories and taking values in a finite set 𝒮={1,2,…,𝒩} with transition probability matrix Π≜{𝜋_𝑖𝑗} given by Pr𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ+𝑜(Δ)if𝑗≠𝑖,1+𝜋_𝑖𝑖Δ+𝑜(Δ)if𝑗=𝑖, (2) where Δ>0,  lim_Δ→0(𝑜(Δ)/Δ)=0,  𝜋_𝑖𝑗≥0 is the transition rate from 𝑖 at time 𝑡 to 𝑗 at time 𝑡+Δ, and 𝜋_𝑖𝑖=−∑^𝒩_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗.

Assumption 1. The disturbance 𝑑₁(𝑡) can be formulated by the following exogenous system: 𝜔̇(𝑡)=𝑊_𝑖𝜔(𝑡)+𝑀_𝑖𝑑₃(𝑡),𝑑₁(𝑡)=𝑉_𝑖𝜔(𝑡), (3) where 𝑊_𝑖, 𝑀_𝑖, and 𝑉_𝑖 are known matrices with proper dimensions. 𝑑₃(𝑡) is the additional disturbance belonging to ℒ₂[0,∞).

The following assumptions are necessary conditions for the disturbance-observer-based control problem.

Assumption 2. (𝐸,𝐴_𝑖,𝐷_1𝑖) is impulse observable [19].

Assumption 3. (𝐸,𝐴_𝑖,𝐺_𝑖) is impulse controllable, and (𝑊_𝑖,𝐺_𝑖𝑉_𝑖) is observable.

The free singular system with Markovian switching of (1a) and (1b) with 𝑢(𝑡)=0, 𝑑₁(𝑡)=0, and 𝑑₂(𝑡)=0 can be described as 𝐸𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡). (4) We give the following definition for the singular Markovian jump system (4).

Definition 4 (Dai [2]). The singular Markovian jump system (4) or the pair (𝐸,𝐴_𝑖) is said to be(i)regular if, for each 𝑖∈𝒮, det(𝑠𝐸−𝐴_𝑖) is not identically zero;(ii)impulse-free if, for each 𝑖∈𝒮, deg(det(𝑠𝐸−𝐴_𝑖))=rank(𝐸);(iii)stochastically admissible if it is regular, impulse-free, and stochastically stable.

In this section, we suppose that all of the states in (1a), (1b), and (3) are unavailable. Then, we need to estimate 𝑥(𝑡) and 𝜔(𝑡), respectively. Here, we construct full-order observers for the whole states, and then based on the estimated states, we design a composite controller such that the resulting composite system is stochastically admissible with ℋ_∞ performance 𝛾. For this purpose, Assumptions 2 and 3 are needed.

By augmenting the states of the system (1a) and (1b) by the disturbance dynamics (3), we obtain the following augmented model:𝐸𝜉̇(𝑡)=𝐴_𝑖(𝑡)𝜉(𝑡)+𝐻_𝑖𝑑(𝑡)+𝐺_𝑖𝑢(𝑡),𝑦(𝑡)=𝐷_𝑖𝜉(𝑡) (5a)(5b)with 𝜉(𝑡)≜𝑥(𝑡)^𝑇𝜔(𝑡)^𝑇𝑇, 𝑑(𝑡)≜𝑑₂(𝑡)^𝑇𝑑₃(𝑡)^𝑇𝑇, and 𝐴_𝑖≜𝐴_𝑖𝐺_𝑖𝑉_𝑖0𝑊_𝑖,𝐸≜,𝐸00𝐼𝐻_𝑖≜𝐻_𝑖00𝑀_𝑖,𝐷_𝑖≜𝐷_1𝑖𝐷_2𝑖𝑉_𝑖. (6) The full-order observer for both 𝑥(𝑡) and 𝜔(𝑡) is designed as𝐸̇̂𝜉(𝑡)=𝐴_𝑖𝜉̂(𝑡)+𝐺_𝑖𝑢(𝑡)+𝐿_𝑖𝑦(𝑡)(𝑡),𝑦(𝑡)=−𝑦𝐷_𝑖𝜉̂(𝑡) (7a)(7b)with 𝜉̂(𝑡)≜𝑥(𝑡)^𝑇𝜔(𝑡)^𝑇𝑇, and 𝐿_𝑖 is the observer gain to be determined.

Define 𝑒(𝑡)(𝑡)−𝜉̂(𝑡)≜𝑒≜𝜉^𝑥(𝑡)𝑒_𝜔(𝑡)≜𝑥(𝑡)−𝑥(𝑡)𝜔(𝑡)−𝜔(𝑡) (8) as the estimation error.

Based on (5a), (5b), (7a), and (7b), we obtain the estimation error dynamics as follows: 𝐸𝑒̇(𝑡)=𝐴_𝑖+𝐿_𝑖𝐷_𝑖𝑒(𝑡)+𝐻_𝑖𝑑(𝑡). (9) In the DOBC scheme, the control can be constructed as 𝑢(𝑡)=−𝑑₁(𝑡)+𝐾_𝑖𝑥(𝑡), (10) where 𝑑₁(𝑡)≜𝑉_𝑖𝜔(𝑡) is the estimation of 𝑑₁(𝑡) and 𝐾_𝑖 is the controller gain. Combining the estimation error equation (9) with system (1a) and (1b) yields 𝐸𝜂̇(𝑡)=𝐴_𝑖𝜂(𝑡)+𝐻_𝑖𝑑(𝑡) (11) with 𝜂(𝑡)≜𝑥(𝑡)^𝑇𝑒(𝑡)^𝑇𝑇 and 0𝐸≜𝐸0𝐸,𝐴_𝑖≜𝐴_𝑖+𝐺_𝑖𝐾_𝑖𝐵_𝑖0𝐴_𝑖+𝐿_𝑖𝐷_𝑖,𝐻_𝑖≜𝐻_𝑖00𝐻_𝑖,𝐵_𝑖≜−𝐺_𝑖𝐾_𝑖𝐺_𝑖𝑉_𝑖, (12) where 𝐸, 𝐴_𝑖, 𝐹_𝑖, 𝐻_𝑖, and 𝐷_𝑖 are defined in (5a) and (5b).

The reference output is set to be 𝑧(𝑡)=𝐶_𝑖𝑥(𝑡)≜𝐶_𝑖𝜂(𝑡) (13) with 𝐶_𝑖≜𝐶_𝑖0.

Therefore, the disturbance-observer-based control problem based on full-order observer (7a) and (7b) for system (1a) and (1b) with (3) can be formulated as follows.

Disturbance-Observer-Based Control Problem. Given the Markovian jump singular system (1a) and (1b) with (3), design full-order observer of the form (7a) and (7b) and controller of the form (10) such that the following requirements are satisfied: (R1) the composite system in (11) and (13) with 𝑑(𝑡)=0 is stochastically admissible; (R2) under the zero initial conditions, the following inequality holds: ‖𝑧(𝑡)‖₂<𝛾‖𝑑(𝑡)‖₂ (14)for all nonzero 𝑑(𝑡)∈ℒ₂[0,∞), where 𝛾>0 is a prescribed scalar and ‖𝑧(𝑡)‖₂=∫^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡.

3. Main Results

Under Assumptions 2 and 3, suppose that 𝐾_𝑖 and 𝐿_𝑖 are given, and we first present the bounded real lemma for the composite system in (11) and (13) in terms of LMIs.

Lemma 5. Given the controller gains 𝐾_𝑖, the observer gains 𝐿_𝑖, parameters 𝜆_1𝑖>0,  𝜆_2𝑖>0, and 𝛾>0, the composite system in (11) and (13) is stochastically admissible and satisfies the ℋ_∞ performance inequalities (14) if there exist matrices 𝑃_𝑖 such that for 𝑖=1,2,…,𝒩,𝐸^𝑇𝑃_𝑖=𝑃^𝑇_𝑖Ξ𝐸≥0,_𝑖𝑃^𝑇_𝑖𝐻_𝑖⋆−𝛾²𝐼<0 (15a)(15b)with Ξ_𝑖≜𝐸^𝑇𝑃_𝑖+𝑃^𝑇_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝐶^𝑇_𝑖𝐶_𝑖,𝑃_𝑖≜^𝒩_𝑗=1𝜋_𝑖𝑗𝑃_𝑗. (16)

Proof. Define a Lyapunov functional candidate as follows: 𝑉𝜂(𝑡),𝑟_𝑡,𝑡≜𝜂^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝜂(𝑡). (17)
Let 𝒜 be the weak infinitesimal generator of the random process {𝜉(𝑡),𝑟_𝑡}. Then, for each 𝑟_𝑡=𝑖, 𝑖∈𝒮, it can be shown that 𝒜𝑉𝜂(𝑡),𝑖,𝑡=2𝜂^𝑇(𝑡)𝑃^𝑇_𝑖𝐴_𝑖𝜂(𝑡)+𝐻_𝑖𝑑(𝑡)+𝜂^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝜂(𝑡)=𝜁^𝑇(𝑡)Ξ_𝑖𝑃^𝑇_𝑖𝐻_𝑖𝜁(𝑡)⋆0 (18) with 𝜁(𝑡)≜𝜂^𝑇(𝑡)𝑑^𝑇(𝑡)^𝑇 and Ξ_𝑖≜𝐸^𝑇𝑃_𝑖+𝑃^𝑇_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖. (19)
Consider the following index: 𝐽(𝑇)≜𝐄^𝑇₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)𝑑𝑡. (20) Then, under the zero initial conditions, it follows from (13) and (18) that 𝐽(𝑇)=𝐄^𝑇₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)𝑑𝑡+𝐄𝑉𝜂(𝑇),𝑖,𝑇=𝐄^𝑇₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)+𝒜𝑉𝜂(𝑡),𝑟_𝑡=𝑖𝑑𝑡=𝐄^𝑇₀𝜁^𝑇(𝑡)Ω_𝑖𝜁(𝑡)𝑑𝑡 (21) with Ω_𝑖≜Ξ_𝑖+𝐶^𝑇_𝑖𝐶_𝑖𝑃^𝑇_𝑖𝐻_𝑖⋆−𝛾²𝐼. (22) Based on (15b), we can derive 𝐽(𝑇)≤0 by taking (21) into account. Thus, under the zero initial conditions and for any nonzero 𝑑(𝑡)∈ℒ₂(0,∞), letting 𝑇→∞, we obtain ‖𝑧(𝑡)‖₂≤𝛾‖𝑑(𝑡)‖₂. The proof is completed.

Now, we are in a position to present a solution to the composite DOBC and ℋ_∞ control problem formulated in this section.

Theorem 6. Consider system (1a) and (1b) with the disturbance (3) under Assumptions 2 and 3. Given parameters 𝜆_1𝑖>0, 𝜆_2𝑖>0, and 𝛾>0, there exists a full-order observer in the form of (7a) and (7b) and there exists a controller in the form of (10) such that the augmented system in (11) and (13) is stochastically admissible and satisfies the ℋ_∞ performance inequalities (14) if there exist parameters 𝛼_𝑖>0, matrices 𝑃_1𝑖>0, 𝑃_2𝑖, 𝑄_𝑖>0, 𝑋_𝑖, and 𝑌_𝑖 such that for 𝑖=1,2,…,𝒩,𝑄_𝑖𝐸^𝑇=𝐸𝑄_𝑖𝑄≥0,_𝑖𝐸^𝑇≤𝛼_𝑖𝐼,𝐸^𝑇𝑃_2𝑖=𝑃^𝑇_2𝑖Π𝐸≥0,_1𝑖0𝐻_𝑖0𝑄_𝑖𝐶^𝑇_𝑖𝐺_𝑖𝑋_𝑖𝐺_𝑖𝑉_𝑖𝑊_𝑖⋆Π_2𝑖𝐹_𝑖0𝑃^𝑇_2𝑖𝐻_𝑖0000⋆⋆000000⋆⋆000000⋆⋆−𝛾²𝐼00000⋆⋆⋆−𝛾²𝐼0000⋆⋆⋆⋆0000⋆⋆⋆⋆−𝐼000⋆⋆⋆⋆⋆−𝑄_𝑖00⋆⋆⋆⋆⋆⋆−𝐼0⋆⋆⋆⋆⋆⋆⋆−Λ_𝑖𝑃<0,_1𝑖𝑄_𝑖=𝐼, (23a)(23b)(23c)(23d)(23e)where 𝜆_𝑖 is defined in (15a) and (15b) and 𝑊_𝑖≜√𝜋_𝑖1𝑄_𝑖⋯𝜋_𝑖𝑗𝑄_𝑖⋯√𝜋_𝑖𝒩𝑄^𝑇_{𝑖𝑗≠𝑖},Λ_𝑖≜diag2𝑄₁−𝛼₁𝐼,…,2𝑄_𝑗−𝛼_𝑗𝐼,…,2𝑄_𝒩−𝛼_𝒩𝐼_𝑗≠𝑖,Π_1𝑖≜𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖+𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖^𝑇+𝜋_𝑖𝑖𝑄_𝑖𝐸^𝑇,Π_2𝑖≜𝑃^𝑇_2𝑖𝐴_𝑖+𝑌_𝑖𝐷_𝑖+𝑃^𝑇_2𝑖𝐴_𝑖+𝑌_𝑖𝐷_𝑖^𝑇+𝐸^𝑇𝑃_2𝑖+diag𝑃_1𝑖,𝐼,𝑃_2𝑖≜^𝒩_𝑗=1𝜋_𝑖𝑗𝑃_2𝑗. (24) Moreover, if the previous conditions are feasible, the gains of the desired observer in the form of (7a) and (7b) and the desired controller in the form of (10) are given by 𝐾_𝑖=𝑋_𝑖𝑄_𝑖⁻¹,𝐿_𝑖=𝑃^−𝑇_2𝑖𝑌_𝑖. (25)

Proof. Define 𝑃_𝑖≜𝑃_1𝑖00𝑃_2𝑖 (26) with 𝑃_1𝑖>0.
Substituting 𝐴_𝑖, 𝐻_𝑖 defined in (11), 𝐶_𝑖 defined in (13), and 𝑃_𝑖 defined in (26) into (15a) and (15b) of Lemma 5 and based on the process of the proof of Lemma 5, we can draw a conclusion that the system in (11) and (13) is stochastically admissible with ℋ_∞ performance 𝛾 if (23c) and the following equalities and inequalities hold:𝐸^𝑇𝑃_1𝑖=𝑃_1𝑖𝜁𝐸≥0,_𝑇(𝑡)Γ_1𝑖0𝑃_1𝑖𝐻_𝑖0⋆Γ_3𝑖0𝑃^𝑇_2𝑖𝐻_𝑖⋆⋆−𝛾²𝐼0⋆⋆⋆−𝛾²𝐼𝜁(𝑡)+2𝑥^𝑇(𝑡)Γ_2𝑖𝑒(𝑡)<0, (27a)(27b)where 𝑃_2𝑖 is defined in (25) and Γ_1𝑖≜𝑃_1𝑖𝐴_𝑖+𝐺_𝑖𝐾_𝑖+𝐴_𝑖+𝐺_𝑖𝐾_𝑖^𝑇𝑃_1𝑖+𝐸^𝑇𝑃_1𝑖+𝐶^𝑇_𝑖𝐶_𝑖,Γ_2𝑖≜𝑃_1𝑖−𝐺_𝑖𝐾_𝑖𝐺_𝑖𝑉_𝑖,Γ_3𝑖≜𝑃^𝑇_2𝑖𝐴_𝑖+𝐿_𝑖𝐷_𝑖+𝐴_𝑖+𝐿_𝑖𝐷_𝑖^𝑇𝑃_2𝑖+𝐸^𝑇𝑃_2𝑖,𝑃_1𝑖≜^𝒩_𝑗=1𝜋_𝑖𝑗𝑃_1𝑗. (28) Note that 2𝑥^𝑇(𝑡)Γ_2𝑖𝑒(𝑡)=−2𝑥^𝑇(𝑡)𝑃^𝑇_1𝑖𝐺_𝑖𝐾_𝑖𝑒_𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃^𝑇_1𝑖𝐺_𝑖𝑉_𝑖𝑒_𝜔(𝑡)≤𝑥^𝑇(𝑡)𝑃_1𝑖𝐺_𝑖𝐾_𝑖𝑃⁻¹_1𝑖𝐾^𝑇_𝑖𝐺^𝑇_𝑖𝑃_1𝑖𝑥(𝑡)+𝑒^𝑇_𝑥(𝑡)𝑃_1𝑖𝑒_𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_1𝑖𝐺_𝑖𝑉_𝑖𝑉^𝑇_𝑖𝐺^𝑇_𝑖𝑃_1𝑖𝑥(𝑡)+𝑒^𝑇_𝜔(𝑡)𝑒_𝜔(𝑡). (29) Considering (27b) and (29) and using Schur complement, we can see that if the following equalities (30) hold, then (27b) holds as follows: Γ_1𝑖0𝑃_1𝑖𝐻_𝑖0𝐶^𝑇_𝑖𝑃_1𝑖𝐺_𝑖𝐾_𝑖𝑃⁻¹_1𝑖𝑃_1𝑖𝐺_𝑖𝑉_𝑖⋆Γ_3𝑖+𝑅0𝑃^𝑇_2𝑖𝐻_𝑖000⋆⋆−𝛾²𝐼0000⋆⋆⋆−𝛾²𝐼000⋆⋆⋆⋆−𝐼00⋆⋆⋆⋆⋆−𝑃⁻¹_1𝑖0⋆⋆⋆⋆⋆⋆−𝐼<0 (30) with 𝑅≜diag{𝑃_1𝑖,𝐼}.
Define 𝑄_𝑖≜𝑃⁻¹_1𝑖,𝑋_𝑖≜𝐾_𝑖𝑃⁻¹_1𝑖,𝑌_𝑖≜𝑃_2𝑖𝐿_𝑖. (31) Performing a congruence transformation to (30) by diag{𝑄_𝑖,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼}, we obtain Π_1𝑖0𝐻_𝑖0𝑄_𝑖𝐶^𝑇_𝑖𝐺_𝑖𝑋_𝑖𝐺_𝑖𝑉_𝑖⋆Π_2𝑖0𝑃^𝑇_2𝑖𝐻_𝑖000⋆⋆−𝛾²𝐼0000⋆⋆⋆−𝛾²𝐼000⋆⋆⋆⋆−𝐼00⋆⋆⋆⋆⋆−𝑄_𝑖0⋆⋆⋆⋆⋆⋆−𝐼<0 (32) with Π_1𝑖≜𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖+(𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖)^𝑇+𝜋_𝑖𝑖𝑄_𝑖𝐸^𝑇+∑^𝒩_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗𝑄_𝑖𝐸^𝑇𝑄_𝑗⁻¹𝑄_𝑖.
Performing a congruence transformation to (23a) and (23b) by 𝑄_𝑖⁻¹, respectively, we can readily get (27a) and 𝐸^𝑇𝑄_𝑖⁻¹≤𝛼_𝑖𝑄_𝑖⁻¹𝑄_𝑖⁻¹. (33) By Schur complement to (32) and based on (33), we can conclude that if the following inequalities (34) hold, then (32) holds as follows: Π_1𝑖0𝐹_𝑖0𝐻_𝑖0𝑄_𝑖𝑈^𝑇_𝑖𝑄_𝑖𝐶^𝑇_𝑖𝐺_𝑖𝑋_𝑖𝐺_𝑖𝑉_𝑖⋆Π_2𝑖0𝑃^𝑇_2𝑖𝐹_𝑖0𝑃^𝑇_2𝑖𝐻_𝑖0000⋆⋆−𝜆_𝑖𝐼0000000⋆⋆⋆−𝜆_𝑖𝐼000000⋆⋆⋆⋆−𝛾²𝐼00000⋆⋆⋆⋆⋆−𝛾²𝐼0000⋆⋆⋆⋆⋆⋆−𝜆²_1𝑖𝐼000⋆⋆⋆⋆⋆⋆⋆−𝐼00⋆⋆⋆⋆⋆⋆⋆⋆−𝑄_𝑖0⋆⋆⋆⋆⋆⋆⋆⋆⋆−𝐼 (34) with Π_1𝑖≜𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖+(𝐴_𝑖𝑄_𝑖+𝐺_𝑖𝑋_𝑖)^𝑇+𝜋_𝑖𝑖𝑄_𝑖𝐸^𝑇+∑^𝒩_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗𝑄_𝑖𝛼_𝑗𝑄_𝑗⁻¹𝑄_𝑗⁻¹𝑄_𝑖.
By Schur complement to (34), we have Π_1𝑖0𝐻_𝑖0𝑄_𝑖𝐶^𝑇_𝑖𝐺_𝑖𝑋_𝑖𝐺_𝑖𝑉_𝑖𝑊_𝑖⋆Π_2𝑖0𝑃^𝑇_2𝑖𝐻_𝑖0000⋆⋆−𝛾²𝐼00000⋆⋆⋆−𝛾²𝐼0000⋆⋆⋆⋆−𝐼000⋆⋆⋆⋆⋆−𝑄_𝑖00⋆⋆⋆⋆⋆⋆−𝐼0⋆⋆⋆⋆⋆⋆⋆−Λ_𝑖<0 (35) with Λ_𝑖≜diag𝛼₁⁻¹𝑄²₁,…,𝛼_𝑗⁻¹𝑄²_𝑗,…,𝛼_𝒩⁻¹𝑄²_{𝒩𝑗≠𝑖}. (36) By using the fact that 𝛼_𝑗⁻¹𝑄²_𝑗≥2𝑄_𝑗−𝛼_𝑗𝐼, (37) we can show that if (23d) holds, then (34) holds, and thus (27b) holds. The proof is completed.

Corollary 7. Note that the conditions (23d) given in Theorem 6 are not strict LMI conditions due to (23e). However, with the result of [20], one can solve these nonconvex feasibility problems by formulating them into some sequential optimization problems subject to LMI constraints. By making the cone complementary linearization (CCL) [20], instead of dealing with the original nonconvex feasibility problem formulated in (23d) of Theorem 6, one may consider solving the following minimization problem involving LMI conditions (23d): mintrace^𝒩_𝑖=1𝑄_𝑖𝑃_1𝑖,()()𝑃𝑠.𝑡.23a--23d𝑓𝑜𝑟𝑖=1,2,…,𝒩,_1𝑖𝐼𝐼𝑄_𝑖≥0. (38)

Now, similar to Section 3, we consider the following case: system (1a) and (1b) under Assumptions 1–3 is without jumping parameters (that is 𝑁=1), and thus the observer in (7a) and (7b) and the controller in (10) are mode independent. For such a case, the composite system in (11) and (13) becomes singular system effectively operating at one of the subsystems all the time, and it can be described by𝐸𝜂̇(𝑡)=(𝑡)+(𝑡),𝑧(𝑡)=(𝑡)𝐴𝜂𝐻𝑑𝐶𝜂 (39a)(39b)with 𝐴≜𝐴+𝐺𝐾𝐵0𝐴+𝐿𝐷,,0𝐴≜𝐴𝐺𝑉0𝑊𝐻≜𝐻0𝐻,𝐵≜−𝐺𝐾𝐺𝑉,𝐶≜𝐶0,𝐻≜,𝐻00𝑀𝐷≜𝐷₁𝐷₂𝑉. (40)

Corollary 8. Consider system (1a) and (1b) under Assumptions 1–3 without jumping parameters. Given parameters 𝜆₁>0, 𝜆₂>0, and 𝛾>0, there exists a full-order observer in the form of (7a) and (7b) without jumping parameters and there exists a controller in the form of (10) without jumping parameters such that the composite system in (39a) and (39b) is admissible and satisfies the ℋ_∞ performance inequalities (14) if there exist matrices 𝑃₁>0, 𝑃₂, 𝑄>0, 𝑋, and 𝑌 such that mintrace𝑄𝑃₁,𝑠.𝑡.(42a),(42b),(42c),(42d) (41)𝑄𝐸^𝑇=𝐸𝑄≥0,𝐸^𝑇𝑃₂=𝑃^𝑇₂Π𝐸≥0,₁0𝐻0𝑄𝐶^𝑇𝐺𝑋𝐺𝑉⋆Π₂0𝑃^𝑇₂𝐻000⋆⋆−𝛾²𝐼0000⋆⋆⋆−𝛾²𝑃𝐼000⋆⋆⋆⋆−𝐼00⋆⋆⋆⋆⋆−𝑄0⋆⋆⋆⋆⋆⋆−𝐼<0,₁𝐼𝐼𝑄≥0 (42a)(42b)(42c)(42d)with Π₁≜𝐴𝑄+𝐺𝑋+(𝐴𝑄+𝐺𝑋)^𝑇,Π₂≜𝑃^𝑇₂𝐴+𝑌𝐷+𝑃^𝑇₂𝐴+𝑌𝐷^𝑇+diag𝑃₁,𝐼. (43) Moreover, if the previous conditions are feasible, the gains of the desired observer in the form of (7a) and (7b) without jumping parameters and the desired controller in the form of (10) without jumping parameters are given by 𝐾=𝑋𝑄⁻¹,𝐿=𝑃₂^−𝑇𝑌. (44)

Remark 9. To the best of the authors' knowledge, this is also the first time that the full-order disturbance-observer-based control strategy is applied in the singular system with multiple disturbances.

4. Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed approaches. Consider the systems in (1a), (1b), and (3) with the following parameters: 𝐴₁=−2.21.2−0.9−0.2,𝐺₁=,𝐻−0.10.1₁=,𝐶0.20.1,𝐸=1.0000₁=0.10.1,𝐷₁₁=,𝐷0.1000.1₂₁=0.10.1,𝑊₁=,𝑉00.2−0.20₁=3.00,𝑀₁=,𝐴0.20.4₂=−1.20.50.2−0.8,𝐺₂=,𝐶0.10.3₂=0.20.1,𝐷₁₂=,𝐷0.05000.1₂₂=0.10.1,𝐻₂=,𝑊0.21.0₂=00.5−0.50,𝑉₂=1.00,𝑀₂=.0.10.3 (45) The transition probability matrix is assumed to be Π=^−0.50.5_1.0−1.0, and 𝛾 is set to be 𝛾=1. Our intention here is to design reduced-order-observer-based controller in the form of (7a), (7b), and (10), such that the composite system is stochastically admissible and satisfies prescribed ℋ_∞ performance. We resort to the LMI Toolbox in MATLAB to solve the problems established in (38), and the gains of the desired observer and controller are given by 𝐾₁=0.5657−9.2561,𝐾₂=,𝐿−0.0461−3.2981₁=,𝐿−1.70202.8579−4.23662.7401−3.9409−4.4407−5.3114−10.7753₂=.35.2258−19.4542119.1428−95.2410−22.2709−0.0573−42.1632−0.2238 (46)

5. Conclusion

The problem of disturbance-observer-based control for Markovian jump singular systems with multiple disturbance has been studied. Full-order observer- (both disturbance and system states) based controller has been constructed. The explicit expression of the desired disturbance-observer-based controller has also been presented. Finally, the proposed methods have been verified by a numerical example.

Acknowledgments

This work was supported in part by the Major State Basic Research Development Program of China (973 Program) under Grant 2012CB720003, by the National Natural Science Foundation of China under Grants 61203041, 61127007, and 61121003, by the Chinese National Postdoctor Science Foundation under Grants 2011M500217 and 2012T50036, by the Doctoral Fund of Ministry of Education of China under Grant 20120036120013, and by the Fundamental Research Funds for the Central Universities under Grant 11QG70.