On Output Feedback Multiobjective Control for Singularly Perturbed Systems

Abstract

A new design procedure for a robust 𝐻₂ and 𝐻_∞ control of continuous-time singularly perturbed systems via dynamic output feedback is presented. By formulating all objectives in terms of a common Lyapunov function, the controller will be designed through solving a set of inequalities. Therefore, a dynamic output feedback controller is developed such that 𝐻_∞ and 𝐻₂ performance of the resulting closed-loop system is less than or equal to some prescribed value. Also, 𝐻_∞ and 𝐻₂ performance for a given upperbound of singular perturbation parameter 𝜀∈(0,𝜀^∗] are guaranteed. It is shown that the 𝜀-dependent controller is well defined for any 𝜀∈(0,𝜀^∗] and can be reduced to an 𝜀-independent one so long as 𝜀 is sufficiently small. Finally, numerical simulations are provided to validate the proposed controller. Numerical simulations coincide with the theoretical analysis.

1. Introduction

It is well known that the multiple time-scale systems, otherwise known as singularly perturbed systems, often raise serious numerical problems in the control engineering field. In the past three decades, singularly perturbed systems have been intensively studied by many researchers [1–8].

In practice, many systems involve dynamics operating on two or more time-scales [3]. In this case, standard control techniques lead to ill-conditioning problems. Singular perturbation methods can be used to avoid such numerical problems [1]. By utilizing the time scale properties, the system is decomposed into several reduced order subsystems. Reduced order controllers are designed for each subsystems.

In the framework of singularly perturbed systems, 𝐻_∞ control has been investigated by many researchers, and various approaches have been proposed in this field [9–14]. However, to the best of our knowledge, the problem of multiobjective control for linear singular perturbed systems is still an open problem. By multiobjective control, we refer to synthesis problems with a mix of time- and frequency-domain specifications ranging from 𝐻₂ and 𝐻_∞ performances to regional pole placement, asymptotic tracking or regulation, and settling time or saturation constraints.

In [12], the 𝐻_∞ control problem is concerned via state feedback for fast sampling discrete-time singularly perturbed systems. A new 𝐻_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminate the regularity restrictions attached to the Riccati-based solution. In [13], the 𝐻_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems is investigated. In fact, a new sufficient condition which ensures the existence of state feedback controllers is presented such that the resulting closed-loop system is asymptotically stable. In addition, the results were extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Presented condition, in terms of a linear matrix inequality (LMI), is independent of the singular perturbation parameter.

Undoubtedly, output feedback stabilization is one of the most important problems in control theory and applications. In many real systems, the state vector is not always accessible and only partial information is available via measured output. Furthermore, the reliability of systems and the simplicity of implementation are other reasons of interest in output control which is adopted to stabilize a system.

LMI’s have emerged as a powerful formulation and design technique for a variety of linear control problems. Since solving LMI's is a convex optimization problem, such formulations offer a numerically tractable means of attacking problems that lack an analytical solution. Consequently, reducing a control design problem to an LMI can be considered as a practical solution to this. Since the nonconvex formulation of the output feedback control, its conditions are restrictive or not numerically tractable [15]. It has been an open question how to make these conditions tractable by means of the existing software. Many research results on such a question have been reported in [15–20].

In [1, 2], the dynamic output feedback control of singular perturbation systems has been investigated. However, to the best of our knowledge, the design of dynamic output feedback for robust controller via LMI optimization is still an open problem. The main contribution of this paper is to solve the problem of multiobjective control for linear singularly perturbed systems. Considered problem consists of 𝐻_∞ control, 𝐻₂ performance, and singular perturbation bound design. For given an 𝐻_∞ performance bound 𝛾 or 𝐻₂ performance bound 𝜐, and an upperbound 𝜀^∗ for the singular perturbation, an 𝜀-dependent dynamic output feedback controller will be constructed, such that for all 𝜀∈(0,𝜀^∗]. Due to this, the closed-loop system is admissible and the 𝐻_∞ norm (𝐻₂ norm) of the closed-loop system is less than a prescribed 𝛾 (prescribed 𝜐). Sufficient conditions for such a controller are obtained in form of strict LMIs. A mixed control 𝐻₂/𝐻_∞ problem for singular systems is also considered in this paper. It is shown that the designed controller is well-defined for for  all  𝜀∈(0,𝜀^∗]. It is shown that if 𝜀 is sufficiently small, the controller can be reduced to an 𝜀-independent one. Numerical examples are given to illustrate the main results.

The paper is organized as follows. Section 2 gives the problem statement and motivations. Section 3 presents the main results. The theorems for 𝐻₂, 𝐻_∞ and multiobjective 𝐻₂/𝐻_∞ design via output feedback control are presented in this section. Due to proposed theorems, robust 𝐻₂, 𝐻_∞, and multiobjective performance of continuous-time singularly perturbed systems via dynamic output feedback for a linear systems will be accessible. Section 4 illustrates numerical simulations for the proposed theorems. Finally, conclusions in Section 5 close the paper.

Notation 1. A star (*) in a matrix indicates a transpose quantity. For example: (∗)+𝐴>0 stands for 𝐴^𝑇+𝐴>0, or in a symmetric matrix ^{𝐴∗𝐵𝐶} stands for ^𝐴𝐵𝑇_𝐵𝐶.

2. Problem Statement

Consider the following singularly perturbed system with slow and fast dynamics described in the standard “singularly perturbed” form: 𝜀𝑥𝑥̇₁̇₂=𝐴₁𝑥₁+𝐴₂𝑥₂+𝐵₁𝑢+𝐵_𝑤1𝑤,𝑧=𝐴₃𝑥₁+𝐴₄𝑥₂+𝐵₂𝑢+𝐵_𝑤2𝑤,₁=𝑧𝐶_𝑧112𝑥₁+𝐶_𝑧12𝑥₂+𝐷_𝑧𝑢1𝑢+𝐷_𝑧𝑤1𝑤𝑦==𝐶𝐶_𝑧21𝑦1𝑥₁+𝐶_𝑧22𝑥₂+𝐷_𝑧𝑢2𝑢,𝑥₁+𝐶_𝑦2𝑥₂+𝐷_𝑦𝑤𝑤,,(2.1) where 𝑥₁(𝑡)∈ℜ^𝑛1 and 𝑥₂(𝑡)∈ℜ^𝑛2 form the state vector, 𝑢(𝑡)∈ℜ^𝑝 is the control input vector, 𝑦(𝑡)∈ℜ^𝑚1 is the output, 𝑤(𝑡) is a vector of exogenous inputs (such as reference signals, disturbance signals, sensor noise) and 𝑧₁,𝑧₂ are regulated outputs.

By introducing the following notations:𝑥=𝑥¹𝑥₂,𝐴=𝐴¹𝐴₂𝐵=𝐵₁𝐴₃𝐴₄,𝐵₂,𝐶^𝑦𝐶_𝑧1=𝐶_𝑦1𝐶_𝑦2,=𝐶_𝑧11𝐶𝐵^𝑧12_𝑤,𝐶_𝑧2=𝐶_𝑧21𝐶_𝑧22,=𝐵^𝑤1𝐵_𝑤2.(2.2) The system (2.1) can be rewritten into the following compact form:𝑧𝐸_𝜀𝑥̇=𝐴𝑥+𝐵𝑢+𝐵_𝑤𝑤,₁=𝑧𝐶_𝑧1𝑥+𝐷_𝑧𝑢1𝑢+𝐷_𝑧𝑤1𝑤,₂𝑦==𝐶𝐶_𝑧2𝑦𝑥+𝐷_𝑧𝑢2𝑢,𝑥+𝐷_𝑦𝑤𝑤.(2.3) By applying dynamic output feedback controller in the following form: 𝜀𝑥𝑥̇̇_𝑐1𝑐2=𝐴_𝑐1𝑥_𝑐1+𝐴_𝑐2𝑥_𝑐2+𝐵_𝑐1𝑦,𝑢==𝐶𝐴_𝑐3𝑐1𝑥_𝑐1+𝐴_𝑐4𝑥_𝑐2+𝐵_𝑐2𝑦,𝐶_𝑐2𝑥^𝑐1𝑥_𝑐2+𝐷^𝑐𝑦.(2.4) The controller (2.4) can be rewritten into the following compact form: 𝐸_𝜀𝑢𝑥=̇_𝑐=𝐶_𝑐𝐴_𝑐𝑥_𝑐+𝐵_𝑐𝑦,𝑥_𝑐+𝐷_𝑐𝑦,(2.5) where𝐴_𝑐=𝐴^𝑐1𝐴_𝑐2𝐴_𝑐3𝐴_𝑐4,𝐵^𝑐=𝐵^𝑐1𝐵_𝑐2,𝐶^𝑐=𝐶_𝑐1𝐶_𝑐2.(2.6) The closed loop system is 𝐸^𝜀0𝐼0𝑥𝑥̇_𝑐̇=𝐴+𝐵𝐷^𝑐𝐶_𝑦𝐵𝐶_𝑐𝐸^𝜀−1𝐵_𝑐𝐶_𝑦𝐸^𝜀−1𝐴_𝑐𝑥𝑥_𝑐+𝐵^𝑤+𝐵𝐷_𝑐𝐷_𝑦𝑤𝑧𝐸^𝜀−1𝐵_𝑐𝐷_𝑦𝑤𝑤(𝑡),_𝑖=𝐶_𝑧𝑖+𝐷_𝑧𝑢𝑖𝐷_𝑐𝐶_𝑦𝐷_𝑧𝑢𝑖𝐶_𝑐𝑥𝑥_𝑐+𝐷^𝑧𝑤𝑖+𝐷_𝑧𝑢𝑖𝐷_𝑐𝐷_𝑦𝑤𝑤(𝑡),for𝑖=1,2,(2.7) where 𝐷_cl2=0and 𝐸_𝑒=diag𝐸_𝜀,𝐼𝐴,_cl𝐸_𝜀=diag(𝐼,𝜀𝐼),𝑥_cl=𝑥^𝑇𝑥^𝑇𝑐^𝑇,=𝐴+𝐵𝐷^𝑐𝐶_𝑦𝐵_𝑤𝐶_𝑐𝐵_cl𝐸^𝜀−1𝐵_𝑐𝐶_𝑦𝐸^𝜀−1𝐴_𝑐,=𝐵^𝑤+𝐵𝐷_𝑐𝐷_𝑦𝑤𝐶_cl𝑖𝐸^𝜀−1𝐵_𝑐𝐷_𝑦𝑤,𝐷=_cl1𝐶_𝑧𝑖+𝐷_𝑧𝑢𝑖𝐷_𝑐𝐶_𝑦𝐷_𝑧𝑢𝑖𝐶_𝑐,=𝐷_𝑧𝑤1+𝐷_𝑧𝑢1𝐷_𝑐𝐷_𝑦𝑤.(2.8) Note, following definitions and lemmas are useful in next sections.

Definition 2.1. For a linear time-invariant operator 𝐺∶𝜔∈𝐿₂(ℜ⁺)→𝑧∈𝐿₂(ℜ⁺), 𝐺 is 𝐿₂ stable if 𝜔∈𝐿₂(ℜ) implies 𝑧∈𝐿₂(ℜ). Here, 𝐺 is said to have 𝐿₂ gain less than or equal to 𝛾>0 if and only if ^𝑇0‖𝑧(𝑡)‖²𝑑𝑡≤𝛾²^t₀‖𝑤(𝑡)‖²𝑑𝑡(2.9) for all 𝑇∈ℜ⁺.

Lemma 2.2 (see [21]). For system 𝐺∶(𝐴,𝐵,𝐶,𝐷), the 𝐿₂ gain will be less than 𝛾>0 if there exist a positive definite matrix 𝑋=𝑋^𝑇>0 such that ⎡⎢⎢⎣𝑋𝐴+𝐴^𝑇𝑋𝑋𝐵𝐶^𝑇𝐵^𝑇𝑋−𝛾𝐼𝐷^𝑇𝐶𝐷−𝛾𝐼⎤⎥⎥⎦<0.(2.10)

Definition 2.3. For a linear time-invariant operator 𝐺∶𝑤→𝑧, 𝐻₂ norm 𝐺 is defined by‖𝐺‖²²=12𝜋trace^∞−∞𝐺(𝑗𝜔)𝐺(𝑗𝜔)^∗𝑑𝜔.(2.11)

Lemma 2.4 (see [21]). For stable system 𝐺∶(𝐴,𝐵,𝐶) the 𝐻₂ performance will be less than 𝑣 if there exist a matrix 𝑍 and a positive definite matrix 𝑋=𝑋^𝑇>0 such that the following LMIs are feasible: 𝐴^𝑇𝑋𝐵+𝑋𝐴𝑋𝐵_𝑇𝑋𝑋𝐶_𝑇−𝐼<0,tr𝐶ace𝑍(𝑍)<>𝜈0.,(2.12)

Lemma 2.5 (see [22]). For a positive scalar 𝜀^∗ and symmetric matrices 𝑆₁,𝑆₂, and 𝑆₃ with appropriate dimensions, inequality 𝑆₁+𝜀𝑆₂+𝜀²𝑆₃>0(2.13) holds for all 𝜀∈(0,𝜀^∗], if 𝑆𝑆₁≥0,₁𝑆+𝜀^∗𝑆₂>0,₁+𝜀^∗𝑆₂+𝜀^∗2𝑆₃>0.(2.14)

3. Main Result

Here, we address stability, 𝐻_∞ stability, 𝐻₂ performance, and multiobjective 𝐻₂/𝐻_∞ performance for a singularly perturbed system via dynamic output feedback control.

3.1. Stability Problem

Consider closed loop system (2.7) without disturbance 𝑤(𝑡):𝐸_𝑒𝑥̇_cl=𝐴+𝐵𝐷^𝑐𝐶_𝑦𝐵𝐶_𝑐𝐸^𝜀−1𝐵_𝑐𝐶_𝑦𝐸^𝜀−1𝐴_𝑐𝑥^cl,(3.1)where 𝐴, 𝐵, 𝐶_𝑦 were defined in (2.2). In the following theorem, we propose design procedure for obtaining controllers parameters such that the closed loop system (3.1) becomes asymptotically stable.

Theorem 3.1. Given an upperbound 𝜀^∗ for the singular perturbation 𝜀, if there exist matrices 𝐴_𝑘,𝐵_𝑘,𝐶_𝑘,𝐷_𝑘,𝑌₁₁,𝑌₁₂,𝑌₂₂,𝑋₁₁,𝑋₁₂, and 𝑋₂₂ satisfying the following LMIs ΥΥ₁₁(0)≤0,(3.2)₁₁⎡⎢⎢⎢⎢⎣𝑌𝜀^∗<0,(3.3)₁₁𝜀^∗𝑌₁₂𝜀^∗𝑌^𝑇12𝜀^∗𝑌₂₂𝐼0𝜀0^∗𝐼𝐼0𝜀0^∗𝐼𝑋¹¹𝜀^∗𝑋₁₂𝑌𝜀^∗𝑋^𝑇12𝜀^∗𝑋₂₂⎤⎥⎥⎥⎥⎦>0,(3.4)₁₁𝐼𝑋𝐼₁₁>0,(3.5) where Υ₁₁𝜀^∗=⎡⎢⎢⎢⎢⎣𝐴𝑌¹¹𝜀^∗𝑌₁₂𝑌^𝑇12𝑌₂₂+𝐵𝐶^𝑘+(∗)𝐴+𝐵𝐷_𝑘𝐶_𝑦+𝐴^𝑇𝑘(∗)𝑋¹¹𝑋^𝑇12𝜀^∗𝑋₁₂𝑋₂₂𝐴+𝐵^𝑘𝐶_𝑦+(∗)⎤⎥⎥⎥⎥⎦.(3.6) Then, for any 𝜀∈(0,𝜀^∗], the closed-loop singularly perturbed system (3.1) is asymptotically stable via dynamic output controller (2.4), also controller parameters are obtained from following equations: 𝐶_𝑐𝐷_𝑐=𝐷_𝑘,𝐵_𝑐=𝐶_𝑘−𝐷_𝑐𝐶𝑄₁₁,𝐴_𝑐=𝜗^𝜀−1𝐵_𝑘−𝑃^𝑇11𝐵𝐷_𝑐,=𝜗^𝜀−1𝐴_𝑘−𝑃^𝑇11𝐴+𝐵𝐷_𝑐𝐶_𝑦𝑄₁₁+𝜗_𝜀𝐵_𝑐𝐶_𝑦𝑄₁₁+𝑃^𝑇11𝐵𝐶_𝑐,(3.7) where 𝑃₁₁=𝑋¹¹𝜀𝑋₁₂𝑋^𝑇12𝑋₂₂,𝑄¹¹=𝑌¹¹𝜀𝑌₁₂𝜗_𝜀𝑌^𝑇12𝑌₂₂,=𝐼−𝜗¹−𝜀𝜗₂−𝜗₃𝜗₁−𝜀𝜗₄𝐼−𝜀𝜗^𝑇2−𝜗₅,=𝑋₁₁𝜗𝑌₁₁₄,𝜗₂=𝑋₁₂𝑌^𝑇12,𝜗₃=𝑋₁₁𝑌₁₂+𝑋₁₂𝑌₂₂,=𝑋^𝑇12𝑌₁₁+𝑋₂₂𝑌^𝑇12,𝜗₅=𝑋₂₂𝑌₂₂.(3.8)

Proof. Choose the Lyapunov function as 𝑉(𝑡)=𝑥^𝑇cl(𝑡)𝐸_𝑒𝑃_𝜀𝑥_cl(𝑡),(3.9) where 𝑃𝐸_𝑒𝑃_𝜀=𝑃^𝑇𝜀𝐸_𝑒>0,(3.10)_𝜀=𝑃¹¹𝑃₁₂𝑃^𝑇12𝐸_𝜀𝑃₂₂.(3.11) From (3.10), it is concluded that 𝐸_𝑒𝑃_𝜀=𝑃^𝑇𝜀𝐸_𝑒⟹𝐸^𝜀𝑃₁₁𝐸_𝜀𝑃₁₂𝑃^𝑇12𝐸_𝜀𝑃₂₂=𝑃^𝑇11𝐸_𝜀𝐸_𝜀𝑃₁₂𝑃^𝑇12𝐸_𝜀𝑃^𝑇22(3.12) and from (3.12), 𝑃₁₁ has following structure: 𝐸_𝜀𝑃₁₁=𝑃^𝑇11𝐸_𝜀⟹𝑃₁₁=𝑋¹¹𝜀𝑋₁₂𝑋^𝑇12𝑋₂₂.(3.13) Now, define invert of 𝐸_𝑒𝑃_𝜀 as follow 𝑃_𝜀−1𝐸_𝑒𝑃_𝜀⁻¹=𝑃^𝜀−1𝐸^𝑒−1,=𝑄_𝜀=𝑄¹¹𝐸^𝜀−1𝑄₁₂𝑄^𝑇12𝑄₂₂(3.14) and 𝑄₁₁ has following structure: 𝑄_𝜀𝐸_𝑒⁻¹=𝐸^𝑒−1𝑄^𝑇11⟹𝑄¹¹𝐸^𝜀−1𝐸^𝜀−1𝑄₁₂𝑄^𝑇12𝐸^𝜀−1𝑄₂₂=𝐸^𝜀−1𝑄^𝑇11𝐸^𝜀−1𝑄₁₂𝑄^𝑇12𝐸^𝜀−1𝑄₂₂,𝑄¹¹=𝑌¹¹𝜀𝑌₁₂𝑌^𝑇12𝑌₂₂.(3.15) Using the following equality: 𝐸^𝜀𝑃₁₁𝐸_𝜀𝑃₁₂𝑃^𝑇12𝐸_𝜀𝑃₂₂𝑄¹¹𝐸^𝜀−1𝐸^𝜀−1𝑄₁₂𝑄^𝑇12𝐸^𝜀−1𝑄₂₂=𝐼0𝐼0,(3.16) we also have the constraint that 𝐸_𝜀𝑃₁₁𝑄₁₁+𝑃₁₂𝑄^𝑇12𝐸^𝜀−1=𝐼⟹𝑃₁₁𝑄₁₁+𝑃₁₂𝑄^𝑇12=𝐼.(3.17) Here, we define new matrices Π₁ and Π₂ as follows: Π₁=𝑄¹¹𝑄_𝑇12𝐼𝑃0,(3.18)_𝜀Π₁=Π₂(3.19) and Π₂ is obtained from (3.17) as follows: 𝑃¹¹𝑃₁₂𝑃^𝑇12𝐸_𝜀𝑃₂₂𝑄¹¹𝑄_𝑇12𝐼0=𝑃¹¹𝑄₁₁+𝑃₁₂𝑄^𝑇12𝑃₁₁𝑃^𝑇12𝐸_𝜀𝑄₁₁+𝑃₂₂𝑄^𝑇12𝑃^𝑇12𝐸_𝜀,Π²=𝐼𝑃¹¹0𝑃^𝑇12𝐸_𝜀.(3.20) The derivative of the Lyapunov function (3.9) is 𝑉̇<0𝑉̇⟹=𝑥̇𝑥^𝑇cl𝐸_𝑒𝑇cl𝑃_𝜀𝑥_cl+𝑥^𝑇cl𝐸_𝑒𝑃_𝜀𝑥̇_cl,𝐴^𝑇cl𝑃_𝜀+𝑃^𝑇𝜀𝐴_cl𝑥_cl<0,(3.21) where 𝐴_cl is defined in (2.8).
Sufficient condition to satisfy (3.21) is 𝐴^𝑇cl𝑃_𝜀+𝑃^𝑇𝜀𝐴_cl<0.(3.22) Equation (3.22) will hold if and only if Π^𝑇1𝐴^𝑇cl𝑃_𝜀Π₁+Π^𝑇1𝑃^𝑇𝜀𝐴_clΠ₁<0,(3.23) where Π₁ is defined in (3.18).
From (3.19), equation (3.23) can be rewritten as Π^𝑇1𝐴^𝑇clΠ₂+Π^𝑇2𝐴_clΠ₁<0.(3.24) With substituting Π₁ and 𝐴_cl from (3.18) and (2.8), respectively, we have 𝑃𝐼𝐸_𝜀0^𝑇11𝑃₁₂𝐴+𝐵𝐷^𝑐𝐶𝐵𝐶_𝑐𝐸^𝜀−1𝐵_𝑐𝐶𝐸^𝜀−1𝐴_𝑐𝑄¹¹𝑄_𝑇12𝐼0+(∗)<0.(3.25) Now, we define the new variables: 𝐵_𝑘𝐶_𝑘=𝐷_𝑐𝐶𝑄₁₁+𝐶_𝑐𝑄^𝑇12,𝐴_𝑘=𝑃^𝑇11𝐵𝐷_𝑐+𝐸_𝜀𝑃₁₂𝐸^𝜀−1𝐵_𝑐,=𝑃^𝑇11𝐴+𝐵𝐷_𝑐𝐶_𝑦𝑄₁₁+𝐸_𝜀𝑃₁₂𝐸^𝜀−1𝐵_𝑐𝐶_𝑦𝑄₁₁+𝑃^𝑇11𝐵𝐶_𝑐𝑄^𝑇12+𝐸_𝜀𝑃₁₂𝐸^𝜀−1𝐴_𝑐𝑄^𝑇12.(3.26) Then, from (3.13), (3.15), and (3.26), equation (3.25) can be rewritten as Υ₁₁(𝜀)=⎡⎢⎢⎢⎣𝐴𝑌¹¹𝜀𝑌₁₂𝑌^𝑇12𝑌₂₂+𝐵𝐶^𝑘+(∗)𝐴+𝐵𝐷_𝑘𝐶_𝑦+𝐴^𝑇𝑘(∗)𝑋¹¹𝑋^𝑇12𝜀𝑋₁₂𝑋₂₂𝐴+𝐵^𝑘𝐶_𝑦+(∗)⎤⎥⎥⎥⎦<0.(3.27) Also, the condition (3.10) holds if and only if Π^𝑇1𝐸_𝑒𝑃_𝜀Π₁>0⟹⎡⎢⎢⎢⎣𝑌¹¹𝜀𝑌₁₂𝜀𝑌^𝑇12𝜀𝑌₂₂𝐼0𝜀0𝐼𝐼0𝜀0𝐼𝑋¹¹𝜀𝑋₁₂𝜀𝑋^𝑇12𝜀𝑋₂₂⎤⎥⎥⎥⎦>0.(3.28)
According to Lemma 2.5, the conditions (3.27) and (3.28) are valid for all 𝜀∈(0,𝜀^∗], If (3.2), (3.3), (3.4), and (3.5) are satisfied.
For computing controller parameters we obtain 𝑃₁₁ and 𝑄₁₁ from solving LMIs in (3.2), (3.3), (3.5), and (3.6). Then from constraint (3.17) with assumption 𝑄₁₂=𝐼 we have 𝑃₁₂=𝐼−𝑃₁₁𝑄₁₁=𝐼−𝑋¹¹𝜀𝑋₁₂𝑋^𝑇12𝑋₂₂𝑌¹¹𝜀𝑌₁₂𝑌^𝑇12𝑌₂₂=𝐼−𝜗¹+𝜀𝜗₂𝜀𝜗₃𝐸_𝜀𝜗₄𝜀𝜗^𝑇2+𝜗₅,𝑃₁₂𝐸^𝜀−1=𝐼−𝜗¹−𝜀𝜗₂−𝜗₃−𝜀𝜗₄𝐼−𝜀𝜗^𝑇2−𝜗₅,(3.29) where 𝜗₃𝜗₁=𝑋₁₁𝑌₁₁,𝜗₂=𝑋₁₂𝑌^𝑇12,=𝑋₁₁𝑌₁₂+𝑋₁₂𝜗𝑌₂₂₅,𝜗₄=𝑋^𝑇12𝑌₁₁+𝑋₂₂𝑌^𝑇12,=𝑋₂₂𝑌₂₂.(3.30)
Also From (3.29) and (3.26) we can obtain controller parameters from (3.7). Also from (3.7) controllers parameters are always well defined for all 𝜀∈(0,𝜀^∗] and lim_𝜀→0⁺^{𝐴𝑐𝐵𝑐𝐶}_𝑐^𝐷_𝑐 become 𝐶_𝑐0𝐷_𝑐0=𝐷_𝑘,𝐵_𝑐0=𝐶_𝑘−𝐷_𝑐𝐶𝑄₁₁₀,𝐴_𝑐0=𝜗^−1𝜀0𝐵_𝑘−𝑃^𝑇110𝐵𝐷_𝑐,=𝜗^−1𝜀0𝐴_𝑘−𝑃^𝑇110𝐴+𝐵𝐷_𝑐𝐶_𝑦𝑄₁₁+𝜗_𝜀0𝐵_𝑐𝐶_𝑦𝑄₁₁₀+𝑃^𝑇110𝐵𝐶_𝑐0,(3.31) where 𝜗₁,𝜗₃,𝜗₅ are defined in (3.30) and 𝑃₁₁₀=𝑋¹¹𝑋_𝑇120𝑋₂₂,𝑄¹¹⁰=𝑌¹¹𝑌_𝑇120𝜗_𝜀0𝑌₂₂,=𝐼−𝜗¹−𝜗₃0𝐼−𝜗₅.(3.32) This completes the proof.

Remark 3.2. Throughout the paper, it is assumed that the singular perturbation parameter 𝜀 is available for feedback. Indeed, in many singular perturbation systems, the singular perturbation parameter 𝜀 can be measured. In these cases, 𝜀 is available for feedback, which has attracted much attention. For example, 𝜀-dependent controllers were designed for singular perturbation systems in [22–24]. Since 𝜀 is usually very small, an 𝜀-dependent controller may be ill-conditioning as 𝜀 tends to zero. Thus, it is a key task to ensure the obtained controller to be well defined. This problem will be discussed later.

3.2. 𝐻_∞ Performance

Consider the closed loop system (2.7) with regulated output 𝑧₁. In following theorem, we proposed a procedure for obtaining controller parameters such that the closed loop singular perturbed system (2.7) with regulated output 𝑧₁ becomes asymptotically stable and guarantees the 𝐻_∞ performance with attenuation parameter 𝛾.

Theorem 3.3. Given an 𝐻_∞ performance bound 𝛾 and an upperbound 𝜀^∗ for the singular perturbation 𝜀, if there exist matrices 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, 𝐷_𝑘, 𝑌₁₁, 𝑌₁₂, 𝑌₂₂, 𝑋₁₁, 𝑋₁₂, and 𝑋₂₂ satisfying the following LMIs 𝜓₁₁(0)=⎡⎢⎢⎣Υ¹¹(0)Υ₁₂(0)(∗)−𝛾𝐼𝐷_𝑧𝑤1+𝐷_𝑧𝑢1𝐷_𝑘𝐷_𝑦𝑤𝜓(∗)−𝛾𝐼⎤⎥⎥⎦≤0,₁₁𝜀^∗=⎡⎢⎢⎣Υ¹¹𝜀^∗Υ₁₂𝜀^∗(∗)−𝛾𝐼𝐷_𝑧𝑤1+𝐷_𝑧𝑢1𝐷_𝑘𝐷_𝑦𝑤(∗)−𝛾𝐼⎤⎥⎥⎦<0,(3.33) where Υ₁₁(𝜀^∗) defined in (3.6) Υ₁₂(𝜀^∗) is Υ₁₂𝜀^∗=⎡⎢⎢⎢⎣𝐵^𝑤+𝐵𝐷_𝑘𝐷_𝑦𝑤𝑌¹¹𝑌^𝑇12𝜀^∗𝑌₁₂𝑌₂₂𝐶^𝑇𝑧+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢1}𝑋¹¹𝑋^𝑇12𝜀^∗𝑋₁₂𝑋₂₂𝐵^𝑤+𝐵_𝑘𝐷_𝑦𝑤𝐶^𝑇𝑧1+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢1}⎤⎥⎥⎥⎦.(3.34) Then, for any 𝜀∈(0,𝜀^∗], the closed-loop singularly perturbed system (2.7) is asymptotically stable and with an 𝐻_∞-norm less than or equal to 𝛾, also parameters controller are obtained from (3.7).

Proof. According to Lemma 2.2, the closed-loop singularly perturbed system (2.7) is asymptotically stable and the 𝐿₂ gain will be less or equal 𝛾 if (3.10) and following inequality are satisfied: ⎡⎢⎢⎣𝐴^𝑇cl𝑃_𝜀+𝑃^𝑇𝜀𝐴_cl𝑃^𝑇𝜀𝐵_cl𝐶^𝑇cl1(∗)−𝛾𝐼𝐷^𝑇cl1(∗)(∗)−𝛾𝐼⎤⎥⎥⎦<0.(3.35) Also 𝐴_cl, 𝐵_cl, 𝐶_cl1, and 𝐷_cl1 are defined in (2.8).
By pre- and postmultiplying with matrices diag(Π₁,𝐼,𝐼)anddiag(Π^𝑇1,𝐼,𝐼), respectively, it is concluded that: ⎡⎢⎢⎣Π^𝑇1𝐴^𝑇clΠ₂+Π^𝑇2𝐴_clΠ₁Π^𝑇2𝐵_clΠ^𝑇1𝐶^𝑇cl1(∗)−𝛾𝐼𝐷^𝑇cl1(∗)(∗)−𝛾𝐼⎤⎥⎥⎦<0,(3.36) where Π₁ is defined in (3.18). According to proof of Theorem 3.1, Π^𝑇1𝐴^𝑇clΠ₂+Π^𝑇2𝐴_clΠ^𝑇1 is obtained. Also Π^𝑇2𝐵_cl and Π^𝑇1𝐶^𝑇cl1 are presented as follows: Π^𝑇2𝐵_cl=𝑃𝐼0_𝑇11𝐸_𝜀𝑃₁₂𝐵^𝑤+𝐵𝐷_𝑐𝐷_𝑦𝑤𝐸^𝜀−1𝐵_𝑐𝐷_𝑦𝑤=𝐵^𝑤+𝐵𝐷_𝑘𝐷_𝑦𝑤=𝑃𝐵^𝑇11_𝑤𝐵_𝑤+𝑃^𝑇11𝐵𝐷_𝑘𝐷_𝑦𝑤+𝐸_𝜀𝑃₁₂𝐸^𝜀−1𝐵_𝑐𝐷_𝑦𝑤+𝐵𝐷_𝑘𝐷_𝑦𝑤Π_𝑇1𝑃^𝑇11𝐵_𝑤+𝐵_𝑘𝐷_𝑦𝑤,𝐶^𝑇cl1=𝑄^𝑇11𝑄₁₂𝐼0𝐶^𝑇𝑧1+𝐶^𝑇𝑦𝐷^𝑇𝑐𝐷^{𝑇𝑧𝑢1}𝐶^𝑇𝑐𝐷^{𝑇𝑧𝑢1}=𝑄¹¹𝐶^𝑇𝑧1+𝑄₁₁𝐶^𝑇𝑦𝐷^𝑇𝑐𝐷^{𝑇𝑧𝑢1}+𝑄₁₂𝐶^𝑇𝑐𝐷^{𝑇𝑧𝑢1}=𝑄₁₁𝐶^𝑇𝑧1+𝐶^𝑇𝑦𝐷^𝑇𝑐𝐷^{𝑇𝑧𝑢1}𝐶^𝑇𝑧1+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢1}𝐶^𝑇𝑧1+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢1}.(3.37) From (3.36), and (3.37), we have 𝜓₁₁(𝜀)=⎡⎢⎢⎣Υ¹¹(𝜀)Υ₁₂(𝜀)Υ^𝑇12(𝜀)−𝛾𝐼𝐷_𝑧𝑤1+𝐷_𝑧𝑢1𝐷_𝑘𝐷_𝑦𝑤(∗)−𝛾𝐼⎤⎥⎥⎦<0,(3.38) where Υ₁₁(𝜀) defined in (3.27) and Υ₁₂(𝜀)=⎡⎢⎢⎢⎣𝐵^𝑤+𝐵𝐷_𝑘𝐷_𝑦𝑤𝑌¹¹𝑌^𝑇12𝜀𝑌₁₂𝑌₂₂𝐶^𝑇𝑧1+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢1}𝑋¹¹𝑋^𝑇12𝜀𝑋₁₂𝑋₂₂𝐵^𝑤+𝐵_𝑘𝐷_𝑦𝑤𝐶^𝑇𝑧1+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢1}⎤⎥⎥⎥⎦.(3.39)
According to Lemma 2.5, the condition (3.38) is satisfied for all 𝜀∈(0,𝜀^∗], if (3.33) is satisfied. (3.4) and (3.5) compute from procedure similar to proof of Theorem 3.1 and this completes the proof.

3.3. 𝐻₂ Performance

Consider the closed loop system (2.7) with regulated output 𝑧₂, assume 𝐴_cl is stable, 𝐷_𝑧𝑤2=0 and 𝐷_𝑦𝑤=0. In following theorem, we proposed a procedure for obtaining controller parameters such that the closed loop singularly perturbed system (2.7) with regulated output 𝑧₂ guarantees the 𝐻₂ performance with attenuation parameter 𝑣. First we propose the following lemma that is effective in proof of Theorem 3.5.

Lemma 3.4. The closed-loop singularly perturbed 𝐺∶(𝐴_cl,𝐵_cl,𝐶_cl) has the 𝐻₂ performance with attenuation parameter 𝑣 if following LMIs hold 𝐴^𝑇cl𝑃_𝜀+𝑃^𝑇𝜀𝐴_cl𝑃^𝑇𝜀𝐵_cl𝑃(∗)−𝐼<0,(3.40)_𝑇𝜀𝐸_𝜀𝑃^𝑇𝜀𝐶^𝑇cl(∗tr)ace(𝑍𝑍)<𝑣>.0,((33..4412))

Proof. Consider the following closed-loop singular perturbed system: 𝑥̇(𝑡)=𝐸^𝑒−1𝑧𝐴=_cl𝑥𝐶(𝑡_cl)+𝐸^𝑒−1𝐵_cl𝑣(𝑡),𝑥(𝑡),(3.43) where 𝐸_𝑒 is defined in (2.8). From Definition 2.3 we have the following equality: ‖𝐺‖^2𝐻₂=^∞0trace𝐶_cl𝑒^{𝐸𝑒−1𝐴cl𝑡}𝐸^𝑒−1𝐵_cl𝐵^𝑇cl𝐸^𝑒−1𝑒^{𝐴𝑇cl𝐸𝑒−1𝑡}𝐶^𝑇cl𝑑𝑡.(3.44)
Now we define symmetric matrix 𝑊_𝜀 as follows: 𝑊_𝜀=^∞0𝑒^{𝐸𝑒−1𝐴cl𝑡}𝐸^𝑒−1𝐵_cl𝐵^𝑇cl𝐸^𝑒−1𝑒^{𝐴𝑇cl𝐸𝑒−1𝑡}𝑑𝑡.(3.45)‖𝐺‖^2𝐻₂ is obtained from the following equality: ‖𝐺‖^2𝐻₂=trace𝐶_cl𝑊_𝜀𝐶^𝑇cl.(3.46)𝑊_𝑒 can be obtained from the following equation: 𝐸^𝑒−1𝐴_cl𝑊_𝜀+𝑊_𝜀𝐴^𝑇cl𝐸^𝑒−1+𝐸^𝑒−1𝐵_cl𝐵^𝑇cl𝐸^𝑒−1=0.(3.47)
Suppose that there exist the matrix 𝑋_𝜀 with following structure: 𝑋_𝜀=𝑋^11𝐸^𝜀−1𝑋^12𝑋^𝑇12𝑋^22(3.48) that satisfies the following inequality: 𝑊_𝜀<𝑋_𝜀𝐸^𝑒−1(3.49) From (3.49), equation (3.47) can be rewritten as follow: 𝐸^𝑒−1𝐴_cl𝑋_𝜀𝐸^𝑒−1+𝐸^𝑒−1𝑋^𝑇𝜀𝐴^𝑇cl𝐸^𝑒−1+𝐸^𝑒−1𝐵_cl𝐵^𝑇cl𝐸^𝑒−1<0.(3.50) Now, pre- postmultiplying (3.50) with 𝐸_𝑒 we have 𝐴_cl𝑋_𝜀+𝑋^𝑇𝜀𝐴^𝑇cl+𝐵_cl𝐵^𝑇cl<0.(3.51) Also, from (3.46) and (3.49) we have ‖𝐺‖^2𝐻₂=trace𝐶_cl𝑊_𝜀𝐶^𝑇cl<trace𝐶_cl𝑋_𝜀𝐸^𝑒−1𝐶^𝑇cl<𝑣.(3.52) This is equivalent to the existence of 𝑍 such that 𝐶_clt𝑋ra_𝜀c𝐸e(𝑍^𝑒−1𝐶)<^𝑇cl<𝑣𝑍,(3.53) by using of Schur complement on (3.51) and (3.53) we can conclude 𝑋^𝑇𝜀𝐴^𝑇cl+𝐴_cl𝑋_𝜀𝐵_cl𝐸(∗)−𝐼<0,(3.54)_𝑒𝑃_𝜀𝐶^𝑇cl(∗)𝑍>0(3.55) with assumption 𝑋^𝜀−1=𝑃_𝜀 and pre- and postmultiplying (3.54) by diag(𝑃^𝑇𝜀,𝐼) and diag(𝑃_𝜀,𝐼), respectively, we have 𝐴^𝑇cl𝑃_𝜀+𝑃^𝑇𝜀𝐴_cl𝑃_𝜀𝐵_cl𝐸(∗)−𝐼<0,_𝑒𝑃_𝜀𝑃^𝑇𝜀𝐶^𝑇cl(t∗r)ace(𝑍𝑍)<𝑣>.0,(3.56) This completes the proof.

Theorem 3.5. Given an 𝐻₂ performance bound 𝑣 and an upperbound 𝜀^∗ for the singular perturbation 𝜀, if there exist matrices 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, 𝐷_𝑘, 𝑌₁₁, 𝑌₁₂, 𝑌₂₂, 𝑋₁₁, 𝑋₁₂, and 𝑋₂₂ such that trace(𝑍)<𝑣 andsatisfying the following LMIs: 𝜓₁₁(0)=Υ¹¹𝜓₁₁(∗()0)Υ−₁₂𝐼(0)≤0,𝜀^∗=Υ¹¹⎡⎢⎢⎢⎢⎢⎣⎡⎢⎢⎢⎣𝑌(∗)𝜀^∗Υ₁₂−𝐼𝜀^∗<0,₁₁𝐼0000𝑋𝐼0₁₁0000000⎤⎥⎥⎥⎦𝑌¹¹𝑌₁₂0𝑌₂₂𝐶^𝑇𝑧2+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢2}𝐶^𝑇𝑧2+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢2}⎡⎢⎢⎢⎢⎢⎣⎡⎢⎢⎢⎢⎣𝑌∗𝑍⎤⎥⎥⎥⎥⎥⎦≥0,₁₁𝜀^∗𝑌₁₂𝜀^∗𝑌^𝑇12𝜀^∗𝑌₂₂𝐼0𝜀0^∗𝐼𝐼0𝜀0^∗𝐼𝑋¹¹𝜀^∗𝑋₁₂𝜀^∗𝑋^𝑇12𝜀^∗𝑋₂₂⎤⎥⎥⎥⎥⎦𝑌¹¹𝑌₁₂𝜀^∗𝑌^𝑇12𝑌₂₂𝐶^𝑇𝑧2+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢2}𝐶^𝑇𝑧2+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢2}(∗)𝑍⎤⎥⎥⎥⎥⎥⎦>0,(3.57) where Υ₁₁(𝜀^∗) defined in (3.6) and Υ₁₂(𝜀^∗) is Υ₁₂𝜀^∗=⎡⎢⎢⎣𝐵^𝑤𝑋¹¹𝑋^𝑇12𝜀^∗𝑋₁₂𝑋₂₂𝐵^𝑤⎤⎥⎥⎦.(3.58) Then, for any 𝜀∈(0,𝜀^∗], the closed-loop singularly perturbed system (2.7) is asymptotically stable and with an 𝐻₂-norm less than or equal to 𝑣, also parameters controller are obtained from (3.7).

Proof. The proof is similar to proof of Theorem 3.3. From Lemma 3.4, the closed-loop singularly perturbed system (2.7) has 𝐻₂ performance less than or equal 𝑣 if (3.40), (3.41) and (3.42) are satisfied.
From (3.18) and (3.37), multiplying inequalities (3.40) and (3.41), by the matrices diag(Π₁,𝐼,𝐼) and diag(Π^𝑇1,𝐼,𝐼), gives 𝜓₁₁(𝜀)=Υ¹¹⎡⎢⎢⎢⎢⎢⎣⎡⎢⎢⎢⎣𝑌(∗()𝜀)Υ−₁₂𝐼(𝜀)<0,(3.59)₁₁𝜀𝑌₁₂𝜀𝑌^𝑇12𝜀𝑌₂₂𝐼0𝜀0𝐼𝐼0𝜀0𝐼𝑋¹¹𝜀𝑋₁₂𝜀𝑋^𝑇12𝜀𝑋₂₂⎤⎥⎥⎥⎦𝑌¹¹𝑌₁₂𝜀𝑌^𝑇12𝑌₂₂𝐶^𝑇𝑧2+𝐶^𝑇𝑘𝐷^{𝑇𝑧𝑢2}𝐶^𝑇𝑧2+𝐶^𝑇𝑦𝐷^𝑇𝑘𝐷^{𝑇𝑧𝑢2}(∗)𝑍⎤⎥⎥⎥⎥⎥⎦>0,(3.60) where: Υ₁₂(𝜀)=⎡⎢⎢⎣𝐵^𝑤+𝐵𝐷_𝑘𝐷_𝑦𝑤𝑋¹¹𝑋^𝑇12𝜀𝑋₁₂𝑋₂₂𝐵^𝑤+𝐵_𝑘𝐷_𝑦𝑤⎤⎥⎥⎦(3.61)
According to Lemma 2.5, the inequalities (3.59) and (3.60) are valid for all 𝜀∈(0,𝜀^∗], if (3.57) is satisfied and proof is complete.

3.4. Multiobjective 𝐻₂/𝐻_∞ Performance

Now we have got the LMIs in Theorems 3.3 and 3.5 thus we can solve the multiobjective 𝐻₂/𝐻_∞ synthesis problem easily for closed-loop singularly perturbed system (2.7) with regulated outputs 𝑧₁ and 𝑧₂.

Theorem 3.6. Given an 𝐻_∞ performance bound 𝛾, 𝐻₂ performance bound 𝑣 and an upperbound 𝜀^∗ for the singular perturbation 𝜀, if there exist matrices 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, 𝐷_𝑘, 𝑌₁₁, 𝑌₁₂, 𝑌₂₂, 𝑋₁₁, 𝑋₁₂, and 𝑋₂₂ such that trace(𝑍)<𝑣 satisfying the LMIs (3.4), (3.5), (3.33), and (3.57). Then, for any 𝜀∈(0,𝜀^∗], the closed-loop singularly perturbed system (2.7) is asymptotically stable and with an 𝐻_∞-norm less than or equal to 𝛾, an 𝐻₂-norm less than or equal to 𝑣, also parameters controller are obtained from (3.7).

Proof. It is from the proof of Theorems 3.3 and 3.5 and omitted for save of brevity.

4. Numerical Example

In this section, we present a numerical results to validate the designed dynamic output feedback controller for singularly perturbed systems with 𝐻_∞ or 𝐻₂ performance.

4.1. 𝐻_∞ Performance

Consider the following two-dimensional system and performance index [10] 𝑥̇¹𝜀𝑥̇₂(𝑡)(𝑡)=𝑧−21−12𝑥𝑥¹₂+21𝑢(𝑡)+13𝑤(𝑡),₁=20.1𝑥¹𝑦(𝑡)=10𝑥₂𝑥₁+0.1𝑤(𝑡),𝑥₂+𝑤(𝑡).(4.1) From Theorem 3.3, for 𝜀^∗=0.1 minimum of 𝛾 is obtained as 0.11942. Controller parameters are as follows:𝐴_𝑐=𝐵−9110.83.589−53.210.41_𝑐2,𝐶=−0..025_𝑐2,=𝐷−1793.2−536.1_𝑐,=−4.7658.(4.2) For upperband 𝜀^∗=0.001, minimum 𝛾 is calculated as 0.1. As we expect 𝛾_{𝜀∗=0.001}≤𝛾_𝜀∗=0.1.

From 𝜀=0, we obtain reduced order dynamic as follows: 𝑧𝑥̇₁(𝑡)=1.5𝑥₁(𝑡)+2.5𝑢(𝑡)+2.5𝑤(𝑡),₁=1.95𝑦𝑥(₁𝑡)(𝑡=)+𝑥₁0.05𝑢(𝑡)+0.15𝑤(𝑡),(𝑡)+𝑤(𝑡).(4.3) Here, minimum value 𝛾 is calculated as 0.029. For 𝑤=𝑒^−0.1𝑡sin(10𝑡), Figures 1, 2, and 3 exhibit ∫^t₀𝑧^𝑇1(𝑠)𝑧₁(𝑠)𝑑𝑠/∫^𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠, respectively.

Figure 1 

Simulation for full-order system (4.1) with 𝜀∗=0.1.

Figure 2 

Simulation for full-order system (4.1) with 𝜀∗=0.001.

Figure 3 

Simulation for reduced order system (4.3).

As a new example for 𝐻_∞ performance, now consider an F-8 aircraft model [25]: 𝜀𝑥𝑥̇¹̇₂=𝐴¹𝐴₂𝐴₃𝑧𝐴=₄𝐶𝑥𝑥¹_2𝑤1+𝐵𝐵¹₂𝑢+𝐵𝐵^𝑤1_𝑤2𝑤,𝐶_𝑤2𝑥¹𝑦=𝐶_𝑦1𝑥₂,𝐶_𝑦2𝑥¹𝑥₂+11𝑤(4.4) with 𝐴𝐴₁=−00.0.016357−0.00644,𝐴₂=−00.0.0400340687700₃,=𝐵−00..00475132757500,𝐴₄=−−00..00735005853−00..00176574₁,𝐵=−00.0.0060947333,𝐵₂=−00.0.0159272172_𝑤15,=𝐶−60..04763,𝐵_𝑤2=−1−14.2.565224599_𝑦1,=0100,𝐶_𝑦2=000.002.(4.5) From Theorem 3.3, for 𝜀^∗=0.025 minimum of 𝛾 is obtained as 1.908. Controller parameters are as follows: 𝐴_𝑐=⎡⎢⎢⎢⎣−8−01310799..36608140𝐵79.7928−−4141209.0951164078.29.227.3−136−6052774.50044.904165.17−04−1.015575..0.984369134400_𝑐⎤⎥⎥⎥⎦,𝐶=⎡⎢⎢⎢⎣−−120759.285..36929.64769−7−001.25.498807.996838.32_𝑐⎤⎥⎥⎥⎦,=𝐷−3643.87991−6482336.66_𝑐,=−76.567−4.3010.(4.6) For upperband 𝜀^∗=0.1, minimum 𝛾 is calculated as 3.96. As we expect 𝛾_{𝜀∗=0.025}≤𝛾_𝜀∗=0.1.

For 𝑤=𝑒^−0.1𝑡sin(10𝑡), Figure 4 exhibits∫^t₀𝑧^𝑇1(𝑠)𝑧₁(𝑠)𝑑𝑠/∫^𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠.

Figure 4 

Simulation for full-order system (4.4) for 𝜀∗=0.1,0.025.

4.2. 𝐻₂ Performance

Consider singular perturbed system (3.60) as 𝑥̇¹𝜀𝑥̇₂(𝑡)(𝑡)=−−21−12𝑥¹𝑥(𝑡)₂𝑧(𝑡)+21𝑢(𝑡)+13𝑤(𝑡),₂=40𝑥¹𝑥(𝑡)₂𝑦(𝑡)=10(𝑡)𝑥₁+2𝑢(𝑡),𝑥(𝑡)₂(𝑡).(4.7)

By utilizing Theorem 3.5, minimum value of 𝜈 for 𝜀^∗=0.1 is obtained as 0.074. The controller parameters are as follows:𝐴_𝑐=𝐵2−526565682.60−−149.258.196_𝑐,𝐶=−116.632.03_𝑐,=𝐷−0.0340.011_𝑐,=−4.1.(4.8) For upperband 𝜀^∗=0.001, minimum 𝑣 is calculated as 0.0002. As we expect 𝑣_{𝜀∗=0.001}≤𝑣_𝜀∗=0.1. Figures 5, 6, and 7 show input noise and regulated output 𝑧₂, respectively.

Figure 5 

Band-limited white noise.

Figure 6 

Regulated output 𝑧2 for 𝜀∗=0.001.

Figure 7 

Regulated output 𝑧2 for 𝜀∗=0.1.

As a new example for 𝐻₂ performance, now consider singular perturbed system as 𝑥̇¹𝜀𝑥̇₂(𝑡)(𝑡)=−110−.25𝑥¹𝑥(𝑡)₂𝑧(𝑡)+21𝑢(𝑡)+13𝑤(𝑡),₂=30𝑥¹𝑥(𝑡)₂𝑦(𝑡)=11(𝑡)𝑥₁+2𝑢(𝑡),𝑥(𝑡)₂(𝑡).(4.9)

By utilizing Theorem 3.5, minimum value of 𝜈 for 𝜀^∗=0.1 is obtained as 0.003. The controller parameters are as follows:𝐴_𝑐=𝐵259254636..958−−8104520902..288_𝑐,𝐶=−−90..4116475_𝑐,=𝐷−77312.66829.2_𝑐,=−3.8×10⁻⁹.(4.10) For upperband 𝜀^∗=0.001, minimum 𝑣 is calculated as 0.0004. As we expect 𝑣_{𝜀∗=0.001}≤𝑣_𝜀∗=0.1. Input noise is shown in Figures 5, 8, and 9 show the regulated output 𝑧₂, respectively.

Figure 8 

Regulated output 𝑧2 for 𝜀∗=0.001.

Figure 9 

Regulated output 𝑧2 for 𝜀∗=0.1.

4.3. 𝐻_∞/𝐻₂ Performance

Now, consider singular perturbed system (3.60) as 𝜀𝑥𝑥̇¹̇₂=𝑧−21−12𝑥𝑥¹₂+21𝑢+13𝑤,₁=20.1𝑥¹𝑧𝑥₂+0.1𝑤,₂=40𝑥¹𝑦=10𝑥𝑥₂₁+2𝑢,𝑥₂+𝑤.(4.11) According to Theorem 3.6, due to minimization of 𝜈+𝛾 for 𝜀^∗=0.1, the values of 𝜈=0.14 and 𝛾=0.233 are calculated. By similar calculation, due to minimization of 𝜈+𝛾 for 𝜀^∗=0.001, the values of 𝜈 and 𝛾 are calculated as 0.088 and 0.2018, respectively.

Here, controller parameters are as the follows when 𝜀^∗=0.1:𝐴_𝑐=𝐵8−177..441−−90.1.5_𝑐7,𝐶=−45.21.618_𝑐,=𝐷−_𝑐0.15×10⁻⁴0,=−3.206.(4.12) Figures 10 and 11 show input signal 𝑤(𝑡) and regulated output 𝑧₂. Based on the 𝐻_∞/𝐻₂, considered controller is designed to minimize effect of signal 𝑤 on regulated output 𝑧₂. Also the ratio of the regulated output energy to the disturbance energy is shown in Figure 12.

Figure 10 

Disturbance signal 𝑤(𝑡).

Figure 11 

Regulated output 𝑧2 with 𝜀∗=0.1.

Figure 12 

Simulation for ∫𝑡0𝑧𝑇1(𝑠)𝑧1(𝑠)/∫𝑡0𝑤𝑇(𝑠)𝑤(𝑠) with 𝜀∗=0.1.

5. Conclusions

In this paper, we addressed robust 𝐻₂ and 𝐻_∞ control via dynamic output feedback control for continuous-time singularly perturbed systems. By formulating all objectives in terms of a common Lyapunov function, the controller was designed through solving a set of inequalities. A dynamic output feedback controller was developed such that first, the 𝐻_∞ and 𝐻₂ performances of the resulting closed-loop system is less than or equal to some prescribed values, and furthermore, these performances are satisfied for all 𝜀∈(0,𝜀^∗]. Apart from our main results, Theorems 3.1 to 3.6 show that the 𝜀-dependent controller is well defined for all 𝜀∈(0,𝜀^∗] and can be reduced to an 𝜀-independent providing 𝜀 is sufficiently small. Finally, numerical simulations were provided to verify the proposed controller.