On Output Feedback Multiobjective Control for Singularly Perturbed Systems

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∞ andH2 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.


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 .

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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 H 2 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, H 2 performance, and singular perturbation bound design.For given an H ∞ performance bound γ or H 2 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 H 2 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 H 2 /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 H 2 , H ∞ and multiobjective H 2 /H ∞ design via output feedback control are presented in this section.Due to proposed theorems, robust H 2 , 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.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 * B C stands for A B T B C .

Problem Statement
Consider the following singularly perturbed system with slow and fast dynamics described in the standard "singularly perturbed" form: where x 1 t ∈ R n 1 and x 2 t ∈ R n 2 form the state vector, u t ∈ R p is the control input vector, y t ∈ R m 1 is the output, w t is a vector of exogenous inputs such as reference signals, disturbance signals, sensor noise and z 1 , z 2 are regulated outputs.By introducing the following notations: B w2 .

2.2
The system 2.1 can be rewritten into the following compact form:

2.3
By applying dynamic output feedback controller in the following form: x c2 ⎤ ⎦ D c y.

2.4
The controller 2.4 can be rewritten into the following compact form: where The closed loop system is where D cl 2 0 and

2.8
Note, following definitions and lemmas are useful in next sections.
Definition 2.1.For a linear time-invariant operator Lemma 2.4 see 21 .For stable system G : A, B, C the H 2 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:

6
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Main Result
Here, we address stability, H ∞ stability, H 2 performance, and multiobjective H 2 /H ∞ performance for a singularly perturbed system via dynamic output feedback control.

Stability Problem
Consider closed loop system 2.7 without disturbance w t : 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 , and X 22 satisfying the following LMIs where 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:

3.8
Proof.Choose the Lyapunov function as

V t x T
cl t E e P ε x cl t , 3.9 where

3.13
Now, define invert of E e P ε as follow and Q 11 has following structure: Using the following equality: we also have the constraint that 3.17 Here, we define new matrices Π 1 and Π 2 as follows: and Π 2 is obtained from 3.17 as follows: The derivative of the Lyapunov function 3.9 is where A cl is defined in 2.8 .

Sufficient condition to satisfy 3.21 is
Equation 3.22 will hold if and only if where Π 1 is defined in 3.18 .From 3.19 , equation 3.23 can be rewritten as With substituting Π 1 and A cl from 3.18 and 2.8 , respectively, we have Now, we define the new variables:

3.26
Then, from 3.13 , 3.15 , and 3.26 , equation 3.25 can be rewritten as Also, the condition 3.10 holds if and only if 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.

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For computing controller parameters we obtain P 11 and Q 11 from solving LMIs in 3.2 , 3.3 , 3.5 , and 3.6 .Then from constraint 3.17 with assumption Q 12 I we have where

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

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.

H ∞ Performance
Consider the closed loop system 2.7 with regulated output z 1 .In following theorem, we proposed a procedure for obtaining controller parameters such that the closed loop singular perturbed system 2.7 with regulated output z 1 becomes asymptotically stable and guarantees the H ∞ performance with attenuation parameter γ.

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 L 2 gain will be less or equal γ if 3.10 and following inequality are satisfied:

3.35
Also A cl , B cl , C cl 1 , and D cl 1 are defined in 2.8 .

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By pre-and postmultiplying with matrices diag Π 1 , I, I and diag Π T 1 , I, I , respectively, it is concluded that: where Π 1 is defined in 3.18 .According to proof of Theorem 3.1, are presented as follows:

3.37
From 3.36 , and 3.37 , we have where Υ 11 ε defined in 3.27 and

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.

H 2 Performance
Consider the closed loop system 2.7 with regulated output z 2 , 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 z 2 guarantees the H 2 performance with attenuation parameter v. First we propose the following lemma that is effective in proof of Theorem 3.5.

3.42
Proof.Consider the following closed-loop singular perturbed system:

3.43
where E e is defined in 2.8 .From Definition 2.3 we have the following equality: e t C T cl dt.

3.44
Now we define symmetric matrix W ε as follows:

3.45
G 2 H 2 is obtained from the following equality:

3.46
W e can be obtained from the following equation:

3.47
Suppose that there exist the matrix X ε with following structure: Mathematical Problems in Engineering that satisfies the following inequality: From 3.49 , equation 3.47 can be rewritten as follow:

3.50
Now, pre-postmultiplying 3.50 with E e we have Also, from 3.46 and 3.49 we have

3.52
This is equivalent to the existence of Z such that by using of Schur complement on 3.51 and 3.53 we can conclude 54 with assumption X −1 ε P ε and pre-and postmultiplying 3.54 by diag P T ε , I and diag P ε , I , respectively, we have

3.56
This completes the proof.

3.58
Then, for any ε ∈ 0, ε * , the closed-loop singularly perturbed system 2.7 is asymptotically stable and with an H 2 -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 H 2 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 Π 1 , I, I and diag Π T 1 , I, I , gives Mathematical Problems in Engineering 3.60 where: 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.

Multiobjective H 2 /H ∞ Performance
Now we have got the LMIs in Theorems 3.3 and 3.5 thus we can solve the multiobjective H 2 /H ∞ synthesis problem easily for closed-loop singularly perturbed system 2.7 with regulated outputs z 1 and z 2 .Theorem 3.6.Given an H ∞ performance bound γ, H 2 performance bound v and an upperbound ε * for the singular perturbation ε, if there exist matrices and X 22 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 H 2 -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.

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 H 2 performance.

H ∞ Performance
Consider the following two-dimensional system and performance index 10 From Theorem 3.

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 −7.4 −9.17

4.12
Figures 10 and 11 show input signal w t and regulated output z 2 .Based on the H ∞ /H 2 , considered controller is designed to minimize effect of signal w on regulated output z 2 .Also the ratio of the regulated output energy to the disturbance energy is shown in Figure 12.

Conclusions
In this paper, we addressed robust H 2 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 H 2 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.

Lemma 3 . 4 .
The closed-loop singularly perturbed G : A cl , B cl , C cl has the H 2 performance with attenuation parameter v if following LMIs hold
. Here, G is said to have L 2 gain less than or equal to γ > 0 if and only if 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 , Y 11 , Y 12 , Y 22 , X 11 , X 12 , and X 22 satisfying the following LMIs Theorem 3.5.Given an H 2 performance bound v and an upperbound ε * for the singular perturbation ε, if there exist matrices A k , B k , C k , D k , Y 11 , Y 12 , Y 22 , X 11 , X 12 , and X 22 such that trace Z < v andsatisfying the following LMIs: 3, for ε * 0.1 minimum of γ is obtained as 0.11942.Controller parameters are as follows: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: For upperband ε * 0.001, minimum v is calculated as 0.0002.As we expect v ε * 0.001 ≤ v ε * 0.1 .Figures5, 6, and 7 show input noise and regulated output z 2 , respectively.As a new example for H 2 performance, now consider singular perturbed system as By utilizing Theorem 3.5, minimum value of ν for ε * 0.1 is obtained as 0.003.The controller parameters are as follows: 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 Figures5, 8, and 9 show the regulated output z 2 , respectively.