𝐻_∞ Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback

Abstract

This paper concentrates on 𝐻_∞ control problems of discrete-time singularly perturbed systems via static output feedback. Two methods of designing an 𝐻_∞ controller, which ensures that the resulting closed-loop system is asymptotically stable and meets a prescribed 𝐻_∞ norm bound, are presented in terms of LMIs. Though based on the same matrix transformation, the two approaches are turned into different optimal problems. The first result is given by an 𝜖-independent LMI, while the second result is related to 𝜖. Furthermore, a stability upper bound of the singular perturbation parameter is obtained. The validity of the proposed two results is demonstrated by a numerical example.

1. Introduction

Singularly perturbed systems widely exist in industrial processes, such as aircraft and racket systems, power systems, and nuclear reactor systems. These kind of systems usually embrace complicated dynamic phenomena which are characterized by slow and fast modes with multiple time-scales. This property causes high dimensionality and ill-conditioning problems. In control theory, a parameter-related state-space model is frequently used to describe a singularly perturbed system. With important practical meaning, the stability bounds of the singular perturbation parameter have been extensively studied by many researchers. In early times, a traditional method of decomposing the original system into fast and slow subsystems was frequently used, see [1, 2]. In [3], a method to testify the stability of singularly perturbed systems without the fast-slow decomposition was established. The stability bound was proved to have close relationship with the system matrix, which contributes to analyse some robust control problems. Furthermore, two algorithms to compute and improve the stability bound were developed in [4]. More stability problems are discussed in [5–10] and the references therein.

In recent years, computer science is increasingly applied to industrial processes. Therefore, discrete-time singularly perturbed systems have attracted much attention. An 𝐻_∞ control problem for uncertain discrete-time singularly perturbed systems via state feedback was studied in [11], where two methods of designing 𝐻_∞ controllers were given in terms of LMIs. Fast sampling discrete-time singularly perturbed systems were taken into consideration in [12], and the obtained results were generalized to robust controller design. In [13], a new sufficient condition which guaranteed the existence of state feedback controllers and made the closed-loop system asymptotically stable while satisfying a prescribed 𝐻_∞ norm requirement was proposed. This condition was also given in the form of LMIs but was proved to be less conservative than that in [11]. It will be more perfect if a theoretical proof was given. Interested readers can refer to [14–16] for more information of discrete-time singularly perturbed systems.

Though state feedback can achieve desired properties, it requires the availability of all state variables, which cannot be satisfied in most of the practical systems. On the other hand, dynamic output feedback usually increases the dimension of the original system. Therefore, static output feedback plays an important role in control theory considering that it is the simplest control technique in a closed-loop sense and can be easily realized with a cost not as high as that in state feedback case [17]. The primal point involved in static output feedback is the decoupling problem. A variety of approaches are developed to solve such issues. In [18], special inequality and some tuning scalars were used to transfer nonlinear matrix inequality to a linear one. A stabilizing state feedback controller gain and some matrix transformations were introduced to deal with the nonlinear inequalities in [19]. This method is effective and easy to implement. Thus, the same decoupling technique is adopted in this paper.

Based on those reasons and motivated by the above studies, we aim to design an 𝐻_∞ controller via static output feedback to stabilize a discrete-time singularly perturbed system and guarantee that the transfer function of the resulting closed-loop system satisfies a prescribed 𝐻_∞ norm bound.

The rest of this paper is organized as follows. Section 2 states the system description and some useful lemmas. In Section 3, two LMI-based methods are proposed to design a static output feedback controller for the system presented in Section 2. A numerical example is given to demonstrate the effectiveness of the proposed results in Section 4. Finally, conclusions are given in Section 5.

The following notation will be adopted throughout this paper. 𝐼 denotes an identity matrix with appropriate dimension. 𝐴^𝑇 denotes the transpose of matrix 𝐴. Sym{𝐴} denotes 𝐴+𝐴^𝑇. For a symmetric block matrix, (^∗) stands for the blocks induced by symmetry.

2. Problem Formulation

In this paper, we consider a class of linear fast sampling discrete-time singularly perturbed systems of the following form: 𝑥_𝑘+1=𝐴_𝜖𝑥_𝑘+𝐵_1𝜖𝑤_𝑘+𝐵_2𝜖𝑢_𝑘,𝑧_𝑘=𝐶₁𝑥_𝑘+𝐷₁₁𝑤_𝑘+𝐷₁₂𝑢_𝑘,𝑦_𝑘=𝐶₂𝑥_𝑘, (1) where 𝑥_𝑘=^𝑥1𝑘_𝑥2𝑘, 𝐴_𝜖=^{𝐼𝑛1+𝜖𝐴11𝜖𝐴12}_𝐴21𝐴22, 𝐵_1𝜖=^𝜖𝐵11_𝐵12, 𝐵_2𝜖=^𝜖𝐵21_𝐵22, 𝐶₁=^𝐶𝑇11_𝐶𝑇12^𝑇, 𝐶₂=^𝐶𝑇21_𝐶𝑇22^𝑇, 𝑥_𝑘∈𝑅^𝑛 is the state vector, in which 𝑥_1𝑘∈𝑅^𝑛1, 𝑥_2𝑘∈𝑅^𝑛2, and 𝑛₁+𝑛₂=𝑛, 𝑢_𝑘∈𝑅^𝑚1 is the control input, 𝑤_𝑘∈𝑅^𝑚2 is the disturbance input which belongs to 𝐿₂[0,∞), 𝑦_𝑘∈𝑅^𝑞𝑙 is the measurement output, 𝑧_𝑘∈𝑅^𝑞2 is the controlled output. The scalar 𝜖>0 denotes the singular perturbation parameter.

In the rest of this paper, we will assume that system (1) is completely controllable and observable. We will also assume that not all of its state variables are available. Therefore, the following static output feedback control law is taken into consideration: 𝑢_𝑘=𝐹𝑦_𝑘, (2) then the resulting closed-loop system can be obtained as follows: 𝑥_𝑘+1=𝐴_𝜖𝑥_𝑘+𝐵_1𝜖𝑤_𝑘,𝑧_𝑘=𝐶₁𝑥_𝑘+𝐷₁₁𝑤_𝑘,𝑦_𝑘=𝐶₂𝑥_𝑘, (3) where 𝐴_𝜖=𝐼_𝑛1+𝜖𝐴₁₁𝜖𝐴₁₂𝐴₂₁𝐴₂₂,𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂=𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂+𝐵₂₁𝐵₂₂𝐹𝐶₂₁𝐶₂₂,𝐶₁=𝐶₁₁𝐶₁₂,𝐶₁₁𝐶₁₂=𝐶₁₁𝐶₁₂+𝐷₁₂𝐹𝐶₂₁𝐶₂₂. (4)

The 𝐻_∞ problem studied in this paper can be described as follows: given a scalar 𝛾>0 and a discrete-time singularly perturbed system (1), design a static output feedback controller in the form of (2) such that the closed-loop system (3) is asymptotically stable and its transfer function from 𝜔 to 𝑧, 𝐺(𝑧)=𝐶₁𝑧𝐼−𝐴_𝜖⁻¹𝐵_1𝜖+𝐷₁₁, (5) satisfies ‖𝐺‖_∞<𝛾.

The following lemmas will be used in establishing our main results.

Lemma 1 (see [19]). Consider a discrete-time transfer function 𝐺(𝑧)=𝐶(𝑧𝐼−𝐴)⁻¹𝐵+𝐷. The following statements are equivalent: (i)‖𝐺(𝑧)‖_∞<1 and 𝐴 are stable in the discrete-time sense (|𝜆_𝑖(𝐴)|<1);(ii) there exists 𝑋=𝑋^𝑇>0 such that 𝐴^𝑇𝑋𝐴−𝑋𝐴^𝑇𝑋𝐵𝐶^𝑇𝐵^𝑇𝑋𝐴𝐵^𝑇𝑋𝐵−𝐼𝐷^𝑇𝐶𝐷−𝐼<0 (6) holds.

Lemma 2 (see [20]). The following two statements are equivalent. (i) Let 𝐴, 𝐵, and 𝐶 be given such that the LMI 𝐵𝐴𝐵_T0+Sym𝑋𝑌𝐶^T−𝐼<0 (7) is feasible in 𝑋 and 𝑌. (ii) Let 𝐴, 𝐵, and 𝐶 be given such that the following LMI 𝐴+𝐵𝐶^T+𝐶𝐵^T<0 (8) holds.

Lemma 3 (see [20]). The following two statements are equivalent.(i) Let 𝐴, 𝐵, and 𝐶 be given such that the LMI 𝐴𝐵+𝐶𝐺^T𝐵^T+𝐺𝐶^T−𝐺−𝐺^T=𝐵𝐴𝐵_T0+Sym0𝐼𝐺𝐶^T−𝐼<0 (9) is feasible in 𝐺. (ii) Let 𝐴, 𝐵, and 𝐶 be given such that the LMIs 𝐴<0𝐴+𝐵𝐶^T+𝐶𝐵^T<0 (10) hold.

3. Main Results

In this section, two LMI-based methods of designing a static output feedback controller are proposed to ensure asymptotical stability of a closed-loop discrete-time singularly perturbed system (3). The first result is given in the form of an 𝜖-independent LMI, while the second result is presented by two LMIs which are related to the singular perturbation parameter 𝜖. Furthermore, we can obtain the stability upper bound of 𝜖.

Before giving the results, let 𝐹₀ be a stabilizing state feedback controller gain of the system (1). In other words, the matrix 𝐹₀ is chosen to make matrix 𝐴_𝜖0=𝐴_𝜖+𝐵_2𝜖𝐹₀ stable. Then, we introduce the following transformations which will be used in the rest of this paper: 𝐴₀=𝐴+𝐵_2𝜖𝐹₀,𝐶₂₀=𝐶₂+𝐷₁₂𝐹₀,𝐹𝐶₂=𝑆+𝐹₀, (11) where 𝐴₀=𝐴₁₁₀𝐴₁₂₀𝐴₂₁₀𝐴₂₂₀𝐴,𝐴=¹¹𝐴₁₂𝐴₂₁𝐴₂₂,𝐶₂₀=𝐶^𝑇₂₁₀𝐶^𝑇₂₂₀^𝑇𝑆,𝑆=^𝑇1𝑆^𝑇₂^𝑇,𝐹₀=𝐹^𝑇₀₁𝐹^𝑇₀₂^𝑇. (12)

Theorem 4. For a discrete-time singularly perturbed system in the form of (1), given a stabilizing state feedback controller with gain 𝐹₀, if there exist matrices 𝑃₁₁>0, 𝑃₂₂>0, and 𝐺>0 and matrices 𝑃₁₂, 𝐿, 𝑋_𝑖𝑗, 𝑌_𝑘𝑗, and𝑖=1,…,5, 𝑘,𝑗=1,2 with appropriate dimensions, such that the following LMI holds: Ξ−𝑋𝐹^𝑇₀𝑇−𝑋𝐹^𝑇₀𝑇^𝑇Φ−𝑋−𝑇^𝑇𝐹₀𝑌^T𝑋𝐶^𝑇₂+𝑇^𝑇𝐿∗−𝑌−𝑌^𝑇𝑌𝐶^𝑇₂∗∗−𝐺−𝐺^T<0, (13) where 𝑋=𝑋_𝑖𝑗,𝑌=𝑌_𝑘𝑗Δ,𝑖=1,…,5,𝑘,𝑗=1,2,₁₁₀=𝑃₁₁𝐴^𝑇₁₁₀+𝑃₁₂𝐴^𝑇₁₂₀+𝑃₁₁𝐴^𝑇₁₁₀+𝑃₁₂𝐴^𝑇₁₂₀^𝑇,Δ₁₂₀=𝑃₁₁𝐴^𝑇₂₁₀+𝑃₁₂𝐴^𝑇₂₂₀−𝑃₁₂,Φ=0𝑃₂₂𝑃₁₁𝑃₁₂,0𝐵000000𝑇=𝐵²¹₂₂𝐷₁₂0^𝑇,Ξ=−𝑃₂₂𝑃₂₂𝐴^T₁₂₀𝑃₂₂𝐴^T₂₂₀𝑃₂₂𝐶^T₁₂₀0∗Δ₁₁₀Δ₁₂₀𝑃₁₁𝐶^𝑇₁₁₀+𝑃₁₂𝐶^T₁₂₀𝐵₁₁∗∗−𝑃₂₂0𝐵₁₂∗∗∗−𝛾𝐼𝐷₁₁,∗∗∗∗−𝛾𝐼 (14) then there exists a scalar 𝜖^∗>0 such that for every 𝜖 ∈(0,𝜖^∗], the system (3) is asymptotically stable, and its transfer function satisfies ‖𝐺‖_∞<𝛾 with the static output feedback controller in the form of (2) whose control gain is given by 𝐹=𝐿𝐺^−𝑇.

Proof. Based on Lemma 1, the closed-loop system (3) is asymptotically stable, and its transfer function satisfies ‖𝐺‖_∞<𝛾 if there exists a positive definite matrix 𝑃 such that the following LMI holds: −𝑃𝑃𝐴^𝑇_𝜖𝑃𝐶^𝑇₁0∗−𝑃0𝐵_1𝜖∗∗−𝛾𝐼𝐷₁₁∗∗∗−𝛾𝐼<0, (15) then for all 𝜖∈(0,𝜖^∗], let 𝑃(𝜖)=𝜖𝑃₁₁𝜖𝑃₁₂∗𝑃₂₂>0 (16) satisfy (15), and we have −𝜖𝑃₁₁−𝜖𝑃₁₂𝜖𝑃₁₁+𝜖²Δ₁₁𝜖Δ₁₂𝜖Γ₁0∗−𝑃₂₂Δ₂₁(𝜖)Δ₂₂(𝜖)Γ₂(𝜖)0∗∗−𝜖𝑃₁₁−𝜖𝑃₁₂0𝜖𝐵₁₁∗∗∗−𝑃₂₂0𝐵₁₂∗∗∗∗−𝛾𝐼𝐷₁₁∗∗∗∗∗−𝛾𝐼<0, (17) where Δ₁₁=𝑃₁₁𝐴^𝑇₁₁+𝑃₁₂𝐴^𝑇₁₂,Γ₁=𝑃₁₁𝐶^𝑇₁₁+𝑃₁₂𝐶^𝑇₁₂,Δ₂₁(𝜖)=𝜖𝑃^𝑇₁₂+𝑃₂₂𝐴^𝑇₁₂+𝜖²𝑃^𝑇₁₂𝐴^𝑇₁₁,Δ₁₂=𝑃₁₁𝐴^𝑇₂₁+𝑃₁₂𝐴^𝑇₂₂,Γ₂(𝜖)=𝜖𝑃^𝑇₁₂𝐶^𝑇₁₁+𝑃₂₂𝐶^𝑇₁₂,Δ₂₂(𝜖)=𝜖𝑃^𝑇₁₂𝐴^𝑇₂₁+𝑃₂₂𝐴^𝑇₂₂. (18) Using the Schur complement to (17), we conclude that−𝑃₂₂+𝜖Θ₁₁𝜖𝑃₂₂𝐴^𝑇₁₂+𝜖²Θ₁₂𝑃₂₂𝐴^𝑇₂₂+𝜖Θ₁₃𝑃₂₂𝐶^𝑇₁₂+𝜖Θ₁₄0∗𝜖²Δ₁₁+Δ^𝑇₁₁+𝜖²Θ₂₂𝜖Δ₁₂−𝑃₁₂+𝜖²Θ₂₃𝜖Γ₁+𝜖²Θ₂₄𝜖𝐵₁₁∗∗−𝑃₂₂+𝜖Θ₃₃𝜖Θ₃₄𝐵₁₂∗∗∗−𝛾𝐼+𝜖Θ₄₄𝐷₁₁∗∗∗∗−𝛾𝐼<0, (19)
where Θ₁₁=𝑃^𝑇₁₂𝑃⁻¹₁₁𝑃₁₂,Θ₁₂=−𝑃^𝑇₁₂𝑃⁻¹₁₁𝑃₁₂𝐴^𝑇₁₂,Θ₁₃=−𝑃^𝑇₁₂𝑃⁻¹₁₁𝑃₁₂𝐴^𝑇₂₂,Θ₁₄=−𝑃^𝑇₁₂𝑃⁻¹₁₁𝑃₁₂𝐶^𝑇₁₂,Θ₂₂=Δ^𝑇₁₁𝑃⁻¹₁₁Δ₁₁,Θ₂₃=Δ^𝑇₁₁𝑃⁻¹₁₁Δ₁₂,Θ₂₄=Δ^𝑇₁₁𝑃⁻¹₁₁Γ₁,Θ₃₃=Δ^𝑇₁₂𝑃⁻¹₁₁Δ₁₂,Θ₃₄=Δ^𝑇₁₂𝑃⁻¹₁₁Γ₁,Θ₄₄=Γ^𝑇₁𝑃⁻¹₁₁Γ₁. (20) Pre- and post-multiplying (19) by diag(𝐼,(1/𝜖)𝐼,𝐼,𝐼,𝐼), we have Π+𝜖Θ<0, (21) where Π=−𝑃₂₂𝑃₂₂𝐴^𝑇₁₂𝑃₂₂𝐴^𝑇₂₂𝑃₂₂𝐶^𝑇₁₂0∗Δ₁₁+Δ^𝑇₁₁Δ₁₂−𝑃₁₂Γ₁𝐵₁₁∗∗−𝑃₂₂0𝐵₁₂∗∗∗−𝛾𝐼𝐷₁₁,Θ∗∗∗∗−𝛾𝐼Θ=¹¹Θ₁₂Θ₁₃Θ₁₄0∗Θ₂₂Θ₂₃Θ₂₄0∗∗Θ₃₃Θ₃₄0∗∗∗Θ₄₄0.∗∗∗∗0 (22) because 𝜖 is small enough, we can simplify (21) as in the following form: Π<0. (23)
To turn (23) into an LMI, we will use the same idea proposed in [17]. By substituting the transformation defined before, we get another form of (23): Ξ+Φ𝑇^𝑇𝑆^𝑇+𝑇^𝑇𝑆Φ^𝑇<0, (24) where Ξ, Φ, 𝑆, and 𝑇 are given as before.
Applying Lemma 2 to (24), we get +ΞΦ∗0Sym𝑋𝑌𝑆^𝑇𝑇−𝐼<0. (25) It is obvious that (25) can be rewritten as follows: ΞΦ−𝑋∗−𝑌−𝑌^𝑇+Sym𝑋𝑆^𝑇𝑌𝑆^𝑇𝑇0<0. (26) Taking the expression of 𝑆 into account, we have Ξ−𝑋𝐹^𝑇₀𝑇−𝑋𝐹^𝑇₀𝑇^𝑇Φ−𝑋−𝑇^𝑇𝐹₀𝑌^𝑇∗−𝑌−𝑌^𝑇+Sym𝑋𝐶^𝑇₂𝑌𝐶^𝑇₂𝐹^𝑇𝑇0<0. (27) Using Lemma 3 to (27), we have Ξ−𝑋𝐹^𝑇₀𝑇−𝑋𝐹^𝑇₀𝑇^𝑇Φ−𝑋−𝑇^𝑇𝐹₀𝑌^𝑇𝑋𝐶^𝑇₂∗−𝑌−𝑌^𝑇𝑌𝐶^𝑇₂+∗∗0Sym00𝐼𝐺𝐹^𝑇𝑇0−𝐼<0. (28) By letting 𝐿=𝐹𝐺^𝑇, we can finally get (13). This completes the proof.

Both the static output feedback gain matrix 𝐹 and the minimum 𝐻_∞ norm 𝛾 can be obtained by solving the following optimization problem: min_{𝑃𝑚𝑗,𝑋𝑖𝑗,𝑌𝑘𝑗,𝐿,𝐺,}𝛾, (OP1) where 𝑖=1,…,5, 𝑚≤𝑗, 𝑚,𝑘,𝑗=1,2, subject to 𝛾>0, 𝑃₁₁>0, 𝑃₂₂>0, and 𝐺>0 and the LMI in (13).

Note that the result in Theorem 4 is finally obtained by solving a 𝛾-related optimal problem. While in the following part, a different approach of designing a static output feedback controller is given by solving an optimal problem in which 𝜖 is involved.

Theorem 5. Given a scalar 𝛾>0 and a stabilizing state feedback controller with gain 𝐹₀, if there exist matrices 𝑃₁₁>0, 𝑃₂₂>0, 𝐺>0, and 𝑄_𝑚𝑚>0, 𝑚=1,…,4 and matrices 𝑃₁₂, 𝐿, 𝑌, 𝑋₁, 𝑋₂, 𝑌, 𝑋_𝑖, 𝑄_𝑚𝑛, 𝑛<𝑚, 𝑖,𝑚,𝑛=1,…,4, with appropriate dimensions, such that the following set of LMIs hold: 𝑄₁𝑄₂−𝑋₁𝑋₁𝐵₂₁+𝐶^𝑇₂𝐿^𝑇−𝐹^𝑇₀𝐺^T𝑄₃𝑄₄𝑝^T₁₂−𝑋₂𝑋₂𝐵₂₁−𝑋^T₁𝑝₁₂−𝑋^T₂−𝑌−𝑌^T𝑌𝐵₂₁𝐵^T₂₁𝑋^T₁+𝐿𝐶₂−𝐺𝐹₀𝐵^T₂₁𝑋^T₂𝐵^T₂₁𝑌^T−𝐺−𝐺^TΠ<0,₁₁Π∗∗∗∗∗0−𝛾𝐼∗∗∗∗₃₁Π₃₂Π₃₃Π∗∗∗₄₁Π₄₂Π₄₃Π₄₄∗∗−𝑋^T₁−𝑋^T₂Φ−𝑋^T₃−𝑋^T₄−𝑌−𝑌^T∗Ψ^T𝑋^T₁−𝐿𝐶₂+𝐺𝐹₀Ψ^T𝑋^T₂Ψ^T𝑋^T₃Ψ^T𝑋^T₄Ψ^T𝑌^T−𝐺−𝐺^𝑇<0, (29) where 𝑄₁=−𝑄₁₁−𝑄^T₂₁−𝑄₂₁−𝑄₂₂−𝑄₃₁−𝑄₃₂,𝑄₂=−𝑄^T₃₁−𝑄^T₄₁−𝑄^T₃₂−𝑄₄₂−𝑄₃₃−𝑄₄₃,𝑄₃=−𝑄₄₁+𝑃^T₁₂𝐴₁₁₀^T−𝑄^T₄₄^T,𝑄₄=−𝑄₄₃+𝑃^T₁₂𝐵₁₁^T−𝑄^T₄₄^T,Π₁₁=−𝜖^∗𝑃₁₁∗−𝑃^T₁₂−𝑃₂₂,Π₃₂=𝑃₂₂𝐵₁₂𝑃₁₁𝐵₁₁+𝑃₁₂𝐵₁₂𝐷₁₁,Π₃₃=−𝑃₂₂∗∗−𝑃₁₂−𝜖^∗𝑃₁₁∗,Π00−𝛾𝐼₄₁=𝑄₁₁𝑄^T₂₁𝑄₂₁𝑄₂₂𝑄₃₁𝑄₃₂𝑄₄₁𝑄₄₂,Π₄₃=𝑄^T₄₁𝑄00_T42𝑄00_T43𝑄00₄₄,Π00₃₁=𝑃^T₁₂+𝑃₂₂𝐴₂₁₀𝑃₂₂𝐴₂₂₀𝜖^∗𝑃₁₁+𝑃₁₁𝐴₁₁₀+𝑃^T₁₂𝐴₂₁₀𝑃₁₁𝐴₁₂₀+𝑃₁₂𝐴₂₂₀𝐶₁₁₀𝐶₁₂₀,𝑃Φ=𝑃²²00₁₂𝑃₁₁0,Π00𝐼₄₄=−𝜖^∗𝑄₁₁∗∗∗−𝜖^∗𝑄₂₁−𝜖^∗𝑄₂₂∗∗−𝜖^∗𝑄₃₁−𝜖^∗𝑄₃₂−𝜖^∗𝑄₃₃∗−𝜖^∗𝑄₄₁−𝜖^∗𝑄₄₂−𝜖^∗𝑄₄₃−𝜖^∗𝑄₄₄,Π₄₂=𝑄^𝑇₃₁𝑄^𝑇₃₂𝑄₃₃𝑄₄₃𝐵,Ψ=𝐵²²₂₁𝐷₁₂, (30) then for any singular perturbation parameter 𝜖∈(0,1/𝜖^∗], by introducing a static output feedback controller in the form of (2), the closed-loop system (3) is asymptotically stable, and its transfer function satisfies ‖𝐺‖_∞<𝛾 with the controller gain 𝐹=𝐺⁻¹𝐿.

Proof. According to Lemma 1, we know that the closed-loop system (3) is asymptotically stable, and its transfer function is less than 𝛾 if there exists a positive definite matrix 𝑃 such that the following LMI holds: 𝑃𝐴−𝑃∗∗∗0−𝛾𝐼∗∗_𝜖𝑃𝐵_1𝜖𝐶−𝑃∗₁𝐷₁₁0−𝛾𝐼<0. (31) For every singular perturbation parameter 𝜖∈(0,1/𝜖^∗], let 𝑃(𝜖)=1𝜖𝑃₁₁∗𝑃^𝑇₁₂𝑃₂₂>0 (32) satisfy (31), and we get 1𝜖𝑃₁₁∗∗∗∗∗−𝑃^𝑇₁₂−𝑃₂₂1∗∗∗∗00−𝛾𝐼∗∗∗𝜖𝑃₁₁+𝑃₁₁𝐴₁₁+𝑃₁₂𝐴₂₁𝑃₁₁𝐴₁₂+𝑃₁₂𝐴₂₂𝑃₁₁𝐵₁₁+𝑃₁₂𝐵₁₂−1𝜖𝑃₁₁𝑃∗∗_𝑇12+𝜖𝑃^𝑇₁₂𝐴₁₁+𝑃₂₂𝐴₂₁𝜖𝑃₁₂𝐴₁₂+𝑃₂₂𝐴₂₂𝜖𝑃^𝑇₁₂𝐵₁₁+𝑃₂₂𝐵₁₂𝑃^𝑇₁₂𝑃₂₂∗𝐶₁₁𝐶₁₂𝐷₁₁00−𝛾𝐼<0. (33)
Pre- and post-multiplying (33) by 𝐼000000𝐼000000𝐼0000000𝐼0000𝐼0000000𝐼 (34) and its transpose, respectively, then for every 𝜖∈(0,1/𝜖^∗], we have Θ₁₁1𝜖∗Θ₂₁1𝜖Θ₂₂1𝜖+𝜖Ω000<0, (35) where 𝑃Ω=0∗∗∗00∗∗000∗_𝑇12𝐴₁₁𝑃^𝑇₁₂𝐴₁₂𝑃^𝑇₁₂𝐵₁₁0,Θ₁₁1𝜖=−1𝜖𝑃₁₁∗∗∗−𝑃^𝑇₁₂−𝑃₂₂𝑃∗∗00−𝛾𝐼∗_𝑇12+𝑃₂₂𝐴₂₁𝑃₂₂𝐴₂₂𝑃₂₂𝐵₁₂−𝑃₂₂,Θ₂₁1𝜖=1𝜖𝑃₁₁+Υ₁₁Υ₁₂Υ₁₃−𝑃₁₂𝐶₁₁𝐶₁₂𝐷₁₁0,Θ₂₂1𝜖=−1𝜖𝑃₁₁∗,Υ0−𝛾𝐼₁₁=𝑃₁₁𝐴₁₁+𝑃₁₂𝐴₂₁,Υ₁₂=𝑃₁₁𝐴₁₂+𝑃₁₂𝐴₂₂,Υ₁₃=𝑃₁₁𝐵₁₁+𝑃₁₂𝐵₁₂. (36) If there exists a positive definite matrix 𝑄=[𝑄_𝑖𝑗], 𝑖,𝑗=1,…,4 satisfies Ω−𝑄<0, (37) and for all 𝜖∈(0,1/𝜖^∗], Θ₁₁𝜖^∗Θ∗∗₂₁𝜖^∗Θ₂₂𝜖^∗∗𝑄0−𝜖^∗𝑄<0, (38) then we can easily conclude that Θ₁₁1𝜖Θ∗∗₂₁1𝜖Θ₂₂1𝜖∗1𝑄0−𝜖𝑄<0. (39) Applying the Schur complement to (39), we get Θ₁₁1𝜖∗Θ₂₁1𝜖Θ₂₂1𝜖+𝜖𝑄000<0. (40) Taking (37) into consideration, we have (35).
To turn (37) and (40) into LMIs, we still adopt the same technique used in [17]. By introducing the transformations defined before, we can rewrite (37) and (40) into the following form: 𝑄₁𝑄₂𝑄₃𝑄₄+Sym0𝑃^𝑇₁₂𝐵₂₁Π𝑆0<0,₁₁Π∗∗∗0−𝛾𝐼∗∗₃₁Π₃₂Π₃₃∗Π₄₁Π₄₂Π₄₃Π₄₄+Sym00Φ0Ψ𝑆000<0. (41) Applying Lemma 2 to (41), we have 𝑄₁𝑄₂0𝑄₃𝑄₄𝑃^𝑇₁₂0𝑃₁₂0+Sym𝑋₁𝑋₂𝑌𝐵₂₁𝑆0−𝐼<0,Π₁₁0Π^𝑇₃₁Π^𝑇₄₁00−𝛾𝐼Π^𝑇₃₂Π^𝑇₄₂0Π₃₁Π₃₂Π₃₃Π^𝑇₄₃ΦΠ₄₁Π₄₂Π₄₃Π₄₄000Φ^𝑇+00Sym𝑋₁𝑋₂𝑋₃𝑋₄𝑌Ψ𝑆000−𝐼<0. (42)(43) It is obvious that (42) and (43) can be rewritten as 𝑄₁𝑄₂−𝑋₁𝑄₃𝑄₄𝑃^𝑇₁₂−𝑋₂−𝑋^𝑇₁𝑃₁₂−𝑋^𝑇₂−𝑌−𝑌^𝑇+Sym𝑋₁𝐵₂₁𝑋₂𝐵₂₁𝑌𝐵₂₁𝑆00<0,Π₁₁0Π^𝑇₃₁Π^𝑇₄₁−𝑋₁0−𝛾𝐼Π^𝑇₃₂Π^T₃₄−𝑋₂Π₃₁Π₃₂Π₃₃Π^𝑇₄₃Φ−𝑋₃Π₄₁Π₄₂Π₄₃Π₄₄−𝑋₄−𝑋^𝑇₁−𝑋^𝑇₂Φ^𝑇−𝑋^𝑇₃−𝑋^𝑇₁−𝑌−𝑌^𝑇+Sym𝑋₁Ψ𝑋₂Ψ𝑋₃Ψ𝑋₄Ψ𝑌Ψ𝑆0000<0, (44)(45) where Ψ is given as before.
Taking Lemma 3 into account, we know that (43) and (44) are equivalent to 𝑄₁𝑄₂−𝑋₁𝑋₁𝐵₂₁𝑄₃𝑄₄𝑃^𝑇₁₂−𝑋₂𝑋₂𝐵₂₁−𝑋^𝑇₁𝑃₁₂−𝑋^𝑇₂−𝑌−𝑌^𝑇𝑌𝐵₂₁𝐵^𝑇₂₁𝑋^𝑇₁𝐵^𝑇₂₁𝑋^𝑇₂𝐵^𝑇₂₁𝑌^𝑇0+Sym000𝐼𝐺₁𝑆00−𝐼<0,Π₁₁0Π^𝑇₃₁Π^𝑇₄₁−𝑋₁𝑋₁Ψ0−𝛾𝐼Π^𝑇₃₂Π^𝑇₃₄−𝑋₂𝑋₂ΨΠ₃₁Π₃₂Π₃₃Π^T₄₃Φ−𝑋₃𝑋₃ΨΠ₄₁Π₄₂Π₄₃Π₄₄−𝑋₄𝑋₄Ψ−𝑋^𝑇₁−𝑋^𝑇₂Φ^𝑇−𝑋^𝑇₃−𝑋^𝑇₄−𝑌−𝑌^𝑇Ψ𝑌Ψ_𝑇𝑋^𝑇₁Ψ^𝑇𝑋^𝑇₂Ψ^𝑇𝑋^𝑇₃Ψ^𝑇𝑋^𝑇₄𝑌Ψ^𝑇0+Sym00000𝐼𝐺₂𝑆0000−𝐼<0, (46)(47) respectively.
From the above analysis, we learn that if there exists 𝐺₁=𝐺₂=𝐺 which satisfies both (46) and (47), then (29) can be finally concluded by considering the expression of 𝑆. This completes the proof.

We can obtain the static output feedback controller gain matrix 𝐹, as well as the stability upper bound of the singular perturbation parameter which we record as 1/𝜖^∗ by solving the following optimal problem: min_{𝑃𝑖𝑗,𝑄𝑚𝑛,𝑋𝑘𝑋,𝑌,𝑙,𝑌,𝐿,𝐺}𝜖^∗, (OP2) where 𝑖≤𝑗, 𝑖,𝑗,𝑙=1,2, 𝑚,𝑛,𝑘=1,…,4, subject to 𝜖^∗>0, 𝑃_𝑖𝑖>0, 𝑄_𝑛𝑛>0, and𝐺>0 and the LMIs in (29).

Remark 6. The result in Theorem 4 finally turns into (OP1), which is solved by optimizing 𝛾, the minimum 𝐻_∞ norm of the closed-loop system (3). While results in Theorem 5 are obtained by solving a different optimal problem (OP2), which optimizes the stability upper bound 1/𝜖^∗ with a fixed performance index 𝛾.

4. A Numerical Example

In this section, a numerical example is presented to illustrate the effectiveness of the proposed results.

Example 7. Consider a discrete-time singularly perturbed system described by (1) with 𝐴_𝜖=,𝐵1+0.2129𝜖1.8140𝜖−0.18140.8179_1𝜖=00.1,𝐵_2𝜖=,𝐶0.1874𝜖0.1812₁=101100,𝐷₁₁=00,𝐷0.01₁₂=000.31,𝐶₂=0.43940.1372^𝑇. (48) First of all, by solving the optimal problem (OP1), we can get the introduced static output feedback controller gain 𝐹=−0.1078 (49) and the minimum 𝐻_∞ norm of the closed-loop system 𝛾=0.8557. (50) Then, referring to (OP2), we can solve the corresponding LMIs with 𝛾=1. The obtained static output feedback controller gain is 𝐹=−23.3354, (51) while the stability upper bound of the singular perturbation parameter is 1𝜖^∗=0.2775. (52)

Remark 8. Though the performance of the closed-loop system is not as good as the state feedback case, static output feedback controller plays more important role in implemental sense with proper performance.

5. Conclusion

In this paper, 𝐻_∞ control problems for fast sampling singularly perturbed systems via static output feedback have been discussed. Rather than adopting the traditional design method of decomposing the original system into fast and slow subsystems, two LMI-based sufficient conditions have been given to guarantee the existence of static output feedback controllers and the asymptotical stability of the closed-loop system with a transfer function whose 𝐻_∞ norm is less than 𝛾. With LMI toolbox in matlab platform, the obtained LMI results can be solved easily. The proposed methods simplify the controller design procedure.

Acknowledgments

This work is supported by the National 973 Program of China (2012CB821202) and the National Natural Science Foundation of China (61174052, 90916003).