# On Output Feedback Multiobjective Control for Singularly Perturbed Systems

Authors: Mehdi Ghasem Moghadam; Mohammad Taghi Hamidi Beheshti
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

A new design procedure for a robustH2 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.


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## Body

## 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. Section2 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 AT+A>0, or in a symmetric matrix [A*BC] stands for [ABTBC].


## 2. Problem Statement

Consider the following singularly perturbed system with slow and fast dynamics described in the standard “singularly perturbed” form:(2.1)ẋ1=A1x1+A2x2+B1u+Bw1w,εẋ2=A3x1+A4x2+B2u+Bw2w,z1=Cz11x1+Cz12x2+Dzu1u+Dzw1wz2=Cz21x1+Cz22x2+Dzu2u,y=Cy1x1+Cy2x2+Dyww,,
where x1(t)∈ℜn1 and x2(t)∈ℜn2 form the state vector, u(t)∈ℜp is the control input vector, y(t)∈ℜm1 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:(2.2)x=[x1x2],A=[A1A2A3A4],B=[B1B2],Cy=[Cy1Cy2],Cz1=[Cz11Cz12],Cz2=[Cz21Cz22],Bw=[Bw1Bw2].
The system (2.1) can be rewritten into the following compact form:(2.3)Eεẋ=Ax+Bu+Bww,z1=Cz1x+Dzu1u+Dzw1w,z2=Cz2x+Dzu2u,y=Cyx+Dyww.
By applying dynamic output feedback controller in the following form:
(2.4)ẋc1=Ac1xc1+Ac2xc2+Bc1y,εẋc2=Ac3xc1+Ac4xc2+Bc2y,u=[Cc1Cc2][xc1xc2]+Dcy.
The controller (2.4) can be rewritten into the following compact form:
(2.5)Eεẋc=Acxc+Bcy,u=Ccxc+Dcy,
where(2.6)Ac=[Ac1Ac2Ac3Ac4],Bc=[Bc1Bc2],Cc=[Cc1Cc2].
The closed loop system is
(2.7)[Eε00I][ẋẋc]=[A+BDcCyBCcEε-1BcCyEε-1Ac][xxc]+[Bw+BDcDywEε-1BcDyw]w(t),zi=[Czi+DzuiDcCyDzuiCc][xxc]+(Dzwi+DzuiDcDyw)w(t),fori=1,2,
where Dcl2=0and
(2.8)Ee=diag⁡(Eε,I),Eε=diag⁡(I,εI),xcl⁡=[xTxcT]T,Acl⁡=[A+BDcCyBwCcEε-1BcCyEε-1Ac],Bcl⁡=[Bw+BDcDywEε-1BcDyw],Ccl⁡i=[Czi+DzuiDcCyDzuiCc],Dcl⁡1=(Dzw1+Dzu1DcDyw).
Note, following definitions and lemmas are useful in next sections.Definition 2.1.
For a linear time-invariant operatorG: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
(2.9)∫0T‖z(t)‖2dt≤γ2∫0t‖w(t)‖2dt
for all T∈ℜ+.Lemma 2.2 (see [21]).
For systemG:(A,B,C,D), the L2 gain will be less than γ>0 if there exist a positive definite matrix X=XT>0 such that
(2.10)[XA+ATXXBCTBTX-γIDTCD-γI]<0.Definition 2.3.
For a linear time-invariant operatorG:w→z, H2 norm G is defined by(2.11)‖G‖22=12πtrace∫-∞∞G(jω)G(jω)*dω.Lemma 2.4 (see [21]).
For stable systemG:(A,B,C) the H2 performance will be less than v if there exist a matrix Z and a positive definite matrix X=XT>0 such that the following LMIs are feasible:
(2.12)[ATX+XAXBBTX-I]<0,[XCTCZ]>0,trace(Z)<ν.Lemma 2.5 (see [22]).
For a positive scalarε* and symmetric matrices S1,S2, and S3 with appropriate dimensions, inequality
(2.13)S1+εS2+ε2S3>0
holds for all ε∈(0,ε*], if
(2.14)S1≥0,S1+ε*S2>0,S1+ε*S2+ε*2S3>0.


## 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):(3.1)Eeẋcl⁡=[A+BDcCyBCcEε-1BcCyEε-1Ac]xcl⁡,where A, B, Cy 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 upperboundε* for the singular perturbation ε, if there exist matrices Ak,Bk,Ck,Dk,Y11,Y12,Y22,X11,X12, and X22 satisfying the following LMIs
(3.2)Υ11(0)≤0,(3.3)Υ11(ε*)<0,(3.4)[[Y11ε*Y12ε*Y12Tε*Y22][I00ε*I][I00ε*I][X11ε*X12ε*X12Tε*X22]]>0,(3.5)[Y11IIX11]>0,
where
(3.6)Υ11(ε*)=[A[Y11ε*Y12Y12TY22]+BCk+(*)A+BDkCy+AkT(*)[X11X12Tε*X12X22]A+BkCy+(*)].
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:
(3.7)Dc=Dk,Cc=(Ck-DcCQ11),Bc=ϑε-1(Bk-P11TBDc),Ac=ϑε-1(Ak-(P11T(A+BDcCy)Q11+ϑεBcCyQ11+P11TBCc)),
where
(3.8)P11=[X11εX12X12TX22],Q11=[Y11εY12Y12TY22],ϑε=[I-ϑ1-εϑ2-ϑ3-εϑ4I-εϑ2T-ϑ5],ϑ1=X11Y11,ϑ2=X12Y12T,ϑ3=X11Y12+X12Y22,ϑ4=X12TY11+X22Y12T,ϑ5=X22Y22.Proof.
Choose the Lyapunov function as(3.9)V(t)=xcl⁡T(t)EePεxcl⁡(t),
where
(3.10)EePε=PεTEe>0,(3.11)Pε=[P11P12P12TEεP22].
From (3.10), it is concluded that
(3.12)EePε=PεTEe⟹[EεP11EεP12P12TEεP22]=[P11TEεEεP12P12TEεP22T]
and from (3.12), P11 has following structure:
(3.13)EεP11=P11TEε⟹P11=[X11εX12X12TX22].
Now, define invert of EePε as follow
(3.14)(EePε)-1=Pε-1Ee-1,Pε-1=Qε=[Q11Eε-1Q12Q12TQ22]
and Q11 has following structure:
(3.15)Qε(Ee)-1=Ee-1Q11T⟹[Q11Eε-1Eε-1Q12Q12TEε-1Q22]=[Eε-1Q11TEε-1Q12Q12TEε-1Q22],Q11=[Y11εY12Y12TY22].
Using the following equality:
(3.16)[EεP11EεP12P12TEεP22][Q11Eε-1Eε-1Q12Q12TEε-1Q22]=[I00I],
we also have the constraint that
(3.17)Eε(P11Q11+P12Q12T)Eε-1=I⟹P11Q11+P12Q12T=I.
Here, we define new matrices Π1 and Π2 as follows:
(3.18)Π1=[Q11IQ12T0],(3.19)PεΠ1=Π2
and Π2 is obtained from (3.17) as follows:
(3.20)[P11P12P12TEεP22][Q11IQ12T0]=[P11Q11+P12Q12TP11P12TEεQ11+P22Q12TP12TEε],Π2=[IP110P12TEε].
The derivative of the Lyapunov function (3.9) is
(3.21)V̇=ẋcl⁡TEePεxcl⁡+xcl⁡TEePεẋcl⁡,V̇<0⟹xcl⁡T(Acl⁡TPε+PεTAcl⁡)xcl⁡<0,
where Acl is defined in (2.8).
Sufficient condition to satisfy (3.21) is
(3.22)Acl⁡TPε+PεTAcl⁡<0.
Equation (3.22) will hold if and only if
(3.23)Π1TAcl⁡TPεΠ1+Π1TPεTAcl⁡Π1<0,
where Π1 is defined in (3.18).
From (3.19), equation (3.23) can be rewritten as
(3.24)Π1TAcl⁡TΠ2+Π2TAcl⁡Π1<0.
With substituting Π1 and Acl  from (3.18) and (2.8), respectively, we have
(3.25)[I0P11TEεP12][A+BDcCBCcEε-1BcCEε-1Ac][Q11IQ12T0]+(*)<0.
Now, we define the new variables:
(3.26)Ck=DcCQ11+CcQ12T,Bk=P11TBDc+EεP12Eε-1Bc,Ak=P11T(A+BDcCy)Q11+EεP12Eε-1BcCyQ11+P11TBCcQ12T+EεP12Eε-1AcQ12T.
Then, from (3.13), (3.15), and (3.26), equation (3.25) can be rewritten as
(3.27)Υ11(ε)=[A[Y11εY12Y12TY22]+BCk+(*)A+BDkCy+AkT(*)[X11X12TεX12X22]A+BkCy+(*)]<0.
Also, the condition (3.10) holds if and only if
(3.28)Π1TEePεΠ1>0⟹[[Y11εY12εY12TεY22][I00εI][I00εI][X11εX12εX12TεX22]]>0.
According to Lemma2.5, 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 obtainP11 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
(3.29)P12=I-P11Q11=I-[X11εX12X12TX22][Y11εY12Y12TY22]=I-[ϑ1+εϑ2εϑ3ϑ4εϑ2T+ϑ5],EεP12Eε-1=[I-ϑ1-εϑ2-ϑ3-εϑ4I-εϑ2T-ϑ5],
where
(3.30)ϑ1=X11Y11,ϑ2=X12Y12T,ϑ3=X11Y12+X12Y22,ϑ4=X12TY11+X22Y12T,ϑ5=X22Y22.
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 limε→0+[AcBcCcDc] become
(3.31)Dc0=Dk,Cc0=(Ck-DcCQ110),Bc0=ϑε0-1(Bk-P110TBDc),Ac0=ϑε0-1(Ak-(P110T(A+BDcCy)Q11+ϑε0BcCyQ110+P110TBCc0)),
where ϑ1,ϑ3,ϑ5 are defined  in (3.30) and
(3.32)P110=[X110X12TX22],Q110=[Y110Y12TY22],ϑε0=[I-ϑ1-ϑ30I-ϑ5].
This completes the proof.Remark 3.2.
Throughout the paper, it is assumed that the singular perturbation parameterε 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 anH∞ performance bound γ and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, Y11, Y12, Y22, X11, X12, and X22 satisfying the following LMIs
(3.33)ψ11(0)=[Υ11(0)Υ12(0)(*)-γIDzw1+Dzu1DkDyw(*)-γI]≤0,ψ11(ε*)=[Υ11(ε*)Υ12(ε*)(*)-γIDzw1+Dzu1DkDyw(*)-γI]<0,
where Υ11(ε*) defined in (3.6) Υ12(ε*) is
(3.34)Υ12(ε*)=[Bw+BDkDyw[Y11Y12Tε*Y12Y22]CzT+CkTDzu1T[X11X12Tε*X12X22]Bw+BkDywCz1T+CyTDkTDzu1T].
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 Lemma2.2, 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:
(3.35)[Acl⁡TPε+PεTAcl⁡PεTBcl⁡Ccl⁡1T(*)-γIDcl⁡1T(*)(*)-γI]<0.
Also Acl, Bcl, Ccl1, and Dcl1 are defined in (2.8).
By pre- and postmultiplying with matricesdiag(Π1,I,I)anddiag(Π1T,I,I), respectively, it is concluded that:
(3.36)[Π1TAcl⁡TΠ2+Π2TAcl⁡Π1Π2TBcl⁡Π1TCcl⁡1T(*)-γIDcl⁡1T(*)(*)-γI]<0,
where Π1 is defined in (3.18). According to proof of Theorem 3.1, Π1TAclTΠ2+Π2TAclΠ1T is obtained. Also Π2TBcl and Π1TCcl1T are presented as follows:
(3.37)Π2TBcl⁡=[I0P11TEεP12][Bw+BDcDywEε-1BcDyw]=[Bw+BDkDywP11TBw+P11TBDkDyw+EεP12Eε-1BcDyw]=[Bw+BDkDywP11TBw+BkDyw],Π1TCcl⁡1T=[Q11TQ12I0][Cz1T+CyTDcTDzu1TCcTDzu1T]=[Q11Cz1T+Q11CyTDcTDzu1T+Q12CcTDzu1TCz1T+CyTDcTDzu1T]=[Q11Cz1T+CkTDzu1TCz1T+CyTDkTDzu1T].
From (3.36), and (3.37), we have
(3.38)ψ11(ε)=[Υ11(ε)Υ12(ε)Υ12T(ε)-γIDzw1+Dzu1DkDyw(*)-γI]<0,
where Υ11(ε) defined in (3.27) and
(3.39)Υ12(ε)=[Bw+BDkDyw[Y11Y12TεY12Y22]Cz1T+CkTDzu1T[X11X12TεX12X22]Bw+BkDywCz1T+CyTDkTDzu1T].
According to Lemma2.5, 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 Acl is stable, Dzw2=0 and Dyw=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 perturbedG:(Acl,Bcl,Ccl) has the H2 performance with attenuation parameter v if following LMIs hold
(3.40)[Acl⁡TPε+PεTAcl⁡PεTBcl⁡(*)-I]<0,(3.41)[PεTEεPεTCcl⁡T(*)Z]>0,(3.42)trace(Z)<v.Proof.
Consider the following closed-loop singular perturbed system:(3.43)ẋ(t)=Ee-1Acl⁡x(t)+Ee-1Bcl⁡v(t),z=Ccl⁡x(t),
where Ee is defined in (2.8). From Definition 2.3 we have the following equality:
(3.44)‖G‖H22=∫0∞trace(Ccl⁡eEe-1Acl⁡tEe-1Bcl⁡Bcl⁡TEe-1eAcl⁡TEe-1tCcl⁡T)dt.
Now we define symmetric matrixWε as follows:
(3.45)Wε=∫0∞(eEe-1Acl⁡tEe-1Bcl⁡Bcl⁡TEe-1eAcl⁡TEe-1t)dt.∥G∥H22 is obtained from the following equality:
(3.46)‖G‖H22=trace(Ccl⁡WεCcl⁡T).We can be obtained from the following equation:
(3.47)Ee-1Acl⁡Wε+WεAcl⁡TEe-1+Ee-1Bcl⁡Bcl⁡TEe-1=0.
Suppose that there exist the matrixXε with following structure:
(3.48)Xε=[X11′Eε-1X12′X12′TX22′]
that satisfies the following inequality:
(3.49)Wε<XεEe-1
From (3.49), equation (3.47) can be rewritten as follow:
(3.50)Ee-1Acl⁡XεEe-1+Ee-1XεTAcl⁡TEe-1+Ee-1Bcl⁡Bcl⁡TEe-1<0.
Now, pre- postmultiplying (3.50) with Ee we have
(3.51)Acl⁡Xε+XεTAcl⁡T+Bcl⁡Bcl⁡T<0.
Also, from (3.46) and (3.49) we have
(3.52)‖G‖H22=trace(Ccl⁡WεCcl⁡T)<trace(Ccl⁡XεEe-1Ccl⁡T)<v.
This is equivalent to the existence of Z such that
(3.53)Ccl⁡XεEe-1Ccl⁡T<Z,trace(Z)<v
by using of Schur complement on (3.51) and (3.53) we can conclude
(3.54)[XεTAcl⁡T+Acl⁡XεBcl⁡(*)-I]<0,(3.55)[EePεCcl⁡T(*)Z]>0
with assumption Xε-1=Pε and pre- and postmultiplying (3.54) by diag(PεT,I) and diag(Pε,I), respectively, we have
(3.56)[Acl⁡TPε+PεTAcl⁡PεBcl⁡(*)-I]<0,[EePεPεTCcl⁡T(*)Z]>0,trace(Z)<v.
This completes the proof.Theorem 3.5.
Given anH2 performance bound v and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, Y11, Y12, Y22, X11, X12, and X22 such that trace(Z)<v andsatisfying the following LMIs:
(3.57)ψ̃11(0)=[Υ11(0)Υ̃12(0)(*)-I]≤0,ψ̃11(ε*)=[Υ11(ε*)Υ̃12(ε*)(*)-I]<0,[[Y110I00000I0X1100000][Y11Y120Y22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T*Z]≥0,[[[Y11ε*Y12ε*Y12Tε*Y22][I00ε*I][I00ε*I][X11ε*X12ε*X12Tε*X22]][Y11Y12ε*Y12TY22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T(*)Z]>0,
where  Υ11(ε*) defined in (3.6) and Υ̃12(ε*) is
(3.58)Υ̃12(ε*)=[Bw[X11X12Tε*X12X22]Bw].
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 Theorem3.3. 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(Π1T,I,I), gives
(3.59)ψ̃11(ε)=[Υ11(ε)Υ̃12(ε)(*)-I]<0,(3.60)[[[Y11εY12εY12TεY22][I00εI][I00εI][X11εX12εX12TεX22]][Y11Y12εY12TY22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T(*)Z]>0,
where:
(3.61)Υ̃12(ε)=[Bw+BDkDyw[X11X12TεX12X22]Bw+BkDyw]
According to Lemma2.5, 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. MultiobjectiveH2/H∞ Performance

Now we have got the LMIs in Theorems3.3 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 anH∞ performance bound γ, H2 performance bound v and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, 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 Theorems3.3 and 3.5 and omitted for save of brevity.


## 3.1. Stability Problem

Consider closed loop system (2.7)  without disturbance w(t):(3.1)Eeẋcl⁡=[A+BDcCyBCcEε-1BcCyEε-1Ac]xcl⁡,where A, B, Cy 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 upperboundε* for the singular perturbation ε, if there exist matrices Ak,Bk,Ck,Dk,Y11,Y12,Y22,X11,X12, and X22 satisfying the following LMIs
(3.2)Υ11(0)≤0,(3.3)Υ11(ε*)<0,(3.4)[[Y11ε*Y12ε*Y12Tε*Y22][I00ε*I][I00ε*I][X11ε*X12ε*X12Tε*X22]]>0,(3.5)[Y11IIX11]>0,
where
(3.6)Υ11(ε*)=[A[Y11ε*Y12Y12TY22]+BCk+(*)A+BDkCy+AkT(*)[X11X12Tε*X12X22]A+BkCy+(*)].
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:
(3.7)Dc=Dk,Cc=(Ck-DcCQ11),Bc=ϑε-1(Bk-P11TBDc),Ac=ϑε-1(Ak-(P11T(A+BDcCy)Q11+ϑεBcCyQ11+P11TBCc)),
where
(3.8)P11=[X11εX12X12TX22],Q11=[Y11εY12Y12TY22],ϑε=[I-ϑ1-εϑ2-ϑ3-εϑ4I-εϑ2T-ϑ5],ϑ1=X11Y11,ϑ2=X12Y12T,ϑ3=X11Y12+X12Y22,ϑ4=X12TY11+X22Y12T,ϑ5=X22Y22.Proof.
Choose the Lyapunov function as(3.9)V(t)=xcl⁡T(t)EePεxcl⁡(t),
where
(3.10)EePε=PεTEe>0,(3.11)Pε=[P11P12P12TEεP22].
From (3.10), it is concluded that
(3.12)EePε=PεTEe⟹[EεP11EεP12P12TEεP22]=[P11TEεEεP12P12TEεP22T]
and from (3.12), P11 has following structure:
(3.13)EεP11=P11TEε⟹P11=[X11εX12X12TX22].
Now, define invert of EePε as follow
(3.14)(EePε)-1=Pε-1Ee-1,Pε-1=Qε=[Q11Eε-1Q12Q12TQ22]
and Q11 has following structure:
(3.15)Qε(Ee)-1=Ee-1Q11T⟹[Q11Eε-1Eε-1Q12Q12TEε-1Q22]=[Eε-1Q11TEε-1Q12Q12TEε-1Q22],Q11=[Y11εY12Y12TY22].
Using the following equality:
(3.16)[EεP11EεP12P12TEεP22][Q11Eε-1Eε-1Q12Q12TEε-1Q22]=[I00I],
we also have the constraint that
(3.17)Eε(P11Q11+P12Q12T)Eε-1=I⟹P11Q11+P12Q12T=I.
Here, we define new matrices Π1 and Π2 as follows:
(3.18)Π1=[Q11IQ12T0],(3.19)PεΠ1=Π2
and Π2 is obtained from (3.17) as follows:
(3.20)[P11P12P12TEεP22][Q11IQ12T0]=[P11Q11+P12Q12TP11P12TEεQ11+P22Q12TP12TEε],Π2=[IP110P12TEε].
The derivative of the Lyapunov function (3.9) is
(3.21)V̇=ẋcl⁡TEePεxcl⁡+xcl⁡TEePεẋcl⁡,V̇<0⟹xcl⁡T(Acl⁡TPε+PεTAcl⁡)xcl⁡<0,
where Acl is defined in (2.8).
Sufficient condition to satisfy (3.21) is
(3.22)Acl⁡TPε+PεTAcl⁡<0.
Equation (3.22) will hold if and only if
(3.23)Π1TAcl⁡TPεΠ1+Π1TPεTAcl⁡Π1<0,
where Π1 is defined in (3.18).
From (3.19), equation (3.23) can be rewritten as
(3.24)Π1TAcl⁡TΠ2+Π2TAcl⁡Π1<0.
With substituting Π1 and Acl  from (3.18) and (2.8), respectively, we have
(3.25)[I0P11TEεP12][A+BDcCBCcEε-1BcCEε-1Ac][Q11IQ12T0]+(*)<0.
Now, we define the new variables:
(3.26)Ck=DcCQ11+CcQ12T,Bk=P11TBDc+EεP12Eε-1Bc,Ak=P11T(A+BDcCy)Q11+EεP12Eε-1BcCyQ11+P11TBCcQ12T+EεP12Eε-1AcQ12T.
Then, from (3.13), (3.15), and (3.26), equation (3.25) can be rewritten as
(3.27)Υ11(ε)=[A[Y11εY12Y12TY22]+BCk+(*)A+BDkCy+AkT(*)[X11X12TεX12X22]A+BkCy+(*)]<0.
Also, the condition (3.10) holds if and only if
(3.28)Π1TEePεΠ1>0⟹[[Y11εY12εY12TεY22][I00εI][I00εI][X11εX12εX12TεX22]]>0.
According to Lemma2.5, 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 obtainP11 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
(3.29)P12=I-P11Q11=I-[X11εX12X12TX22][Y11εY12Y12TY22]=I-[ϑ1+εϑ2εϑ3ϑ4εϑ2T+ϑ5],EεP12Eε-1=[I-ϑ1-εϑ2-ϑ3-εϑ4I-εϑ2T-ϑ5],
where
(3.30)ϑ1=X11Y11,ϑ2=X12Y12T,ϑ3=X11Y12+X12Y22,ϑ4=X12TY11+X22Y12T,ϑ5=X22Y22.
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 limε→0+[AcBcCcDc] become
(3.31)Dc0=Dk,Cc0=(Ck-DcCQ110),Bc0=ϑε0-1(Bk-P110TBDc),Ac0=ϑε0-1(Ak-(P110T(A+BDcCy)Q11+ϑε0BcCyQ110+P110TBCc0)),
where ϑ1,ϑ3,ϑ5 are defined  in (3.30) and
(3.32)P110=[X110X12TX22],Q110=[Y110Y12TY22],ϑε0=[I-ϑ1-ϑ30I-ϑ5].
This completes the proof.Remark 3.2.
Throughout the paper, it is assumed that the singular perturbation parameterε 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 anH∞ performance bound γ and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, Y11, Y12, Y22, X11, X12, and X22 satisfying the following LMIs
(3.33)ψ11(0)=[Υ11(0)Υ12(0)(*)-γIDzw1+Dzu1DkDyw(*)-γI]≤0,ψ11(ε*)=[Υ11(ε*)Υ12(ε*)(*)-γIDzw1+Dzu1DkDyw(*)-γI]<0,
where Υ11(ε*) defined in (3.6) Υ12(ε*) is
(3.34)Υ12(ε*)=[Bw+BDkDyw[Y11Y12Tε*Y12Y22]CzT+CkTDzu1T[X11X12Tε*X12X22]Bw+BkDywCz1T+CyTDkTDzu1T].
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 Lemma2.2, 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:
(3.35)[Acl⁡TPε+PεTAcl⁡PεTBcl⁡Ccl⁡1T(*)-γIDcl⁡1T(*)(*)-γI]<0.
Also Acl, Bcl, Ccl1, and Dcl1 are defined in (2.8).
By pre- and postmultiplying with matricesdiag(Π1,I,I)anddiag(Π1T,I,I), respectively, it is concluded that:
(3.36)[Π1TAcl⁡TΠ2+Π2TAcl⁡Π1Π2TBcl⁡Π1TCcl⁡1T(*)-γIDcl⁡1T(*)(*)-γI]<0,
where Π1 is defined in (3.18). According to proof of Theorem 3.1, Π1TAclTΠ2+Π2TAclΠ1T is obtained. Also Π2TBcl and Π1TCcl1T are presented as follows:
(3.37)Π2TBcl⁡=[I0P11TEεP12][Bw+BDcDywEε-1BcDyw]=[Bw+BDkDywP11TBw+P11TBDkDyw+EεP12Eε-1BcDyw]=[Bw+BDkDywP11TBw+BkDyw],Π1TCcl⁡1T=[Q11TQ12I0][Cz1T+CyTDcTDzu1TCcTDzu1T]=[Q11Cz1T+Q11CyTDcTDzu1T+Q12CcTDzu1TCz1T+CyTDcTDzu1T]=[Q11Cz1T+CkTDzu1TCz1T+CyTDkTDzu1T].
From (3.36), and (3.37), we have
(3.38)ψ11(ε)=[Υ11(ε)Υ12(ε)Υ12T(ε)-γIDzw1+Dzu1DkDyw(*)-γI]<0,
where Υ11(ε) defined in (3.27) and
(3.39)Υ12(ε)=[Bw+BDkDyw[Y11Y12TεY12Y22]Cz1T+CkTDzu1T[X11X12TεX12X22]Bw+BkDywCz1T+CyTDkTDzu1T].
According to Lemma2.5, 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 Acl is stable, Dzw2=0 and Dyw=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 perturbedG:(Acl,Bcl,Ccl) has the H2 performance with attenuation parameter v if following LMIs hold
(3.40)[Acl⁡TPε+PεTAcl⁡PεTBcl⁡(*)-I]<0,(3.41)[PεTEεPεTCcl⁡T(*)Z]>0,(3.42)trace(Z)<v.Proof.
Consider the following closed-loop singular perturbed system:(3.43)ẋ(t)=Ee-1Acl⁡x(t)+Ee-1Bcl⁡v(t),z=Ccl⁡x(t),
where Ee is defined in (2.8). From Definition 2.3 we have the following equality:
(3.44)‖G‖H22=∫0∞trace(Ccl⁡eEe-1Acl⁡tEe-1Bcl⁡Bcl⁡TEe-1eAcl⁡TEe-1tCcl⁡T)dt.
Now we define symmetric matrixWε as follows:
(3.45)Wε=∫0∞(eEe-1Acl⁡tEe-1Bcl⁡Bcl⁡TEe-1eAcl⁡TEe-1t)dt.∥G∥H22 is obtained from the following equality:
(3.46)‖G‖H22=trace(Ccl⁡WεCcl⁡T).We can be obtained from the following equation:
(3.47)Ee-1Acl⁡Wε+WεAcl⁡TEe-1+Ee-1Bcl⁡Bcl⁡TEe-1=0.
Suppose that there exist the matrixXε with following structure:
(3.48)Xε=[X11′Eε-1X12′X12′TX22′]
that satisfies the following inequality:
(3.49)Wε<XεEe-1
From (3.49), equation (3.47) can be rewritten as follow:
(3.50)Ee-1Acl⁡XεEe-1+Ee-1XεTAcl⁡TEe-1+Ee-1Bcl⁡Bcl⁡TEe-1<0.
Now, pre- postmultiplying (3.50) with Ee we have
(3.51)Acl⁡Xε+XεTAcl⁡T+Bcl⁡Bcl⁡T<0.
Also, from (3.46) and (3.49) we have
(3.52)‖G‖H22=trace(Ccl⁡WεCcl⁡T)<trace(Ccl⁡XεEe-1Ccl⁡T)<v.
This is equivalent to the existence of Z such that
(3.53)Ccl⁡XεEe-1Ccl⁡T<Z,trace(Z)<v
by using of Schur complement on (3.51) and (3.53) we can conclude
(3.54)[XεTAcl⁡T+Acl⁡XεBcl⁡(*)-I]<0,(3.55)[EePεCcl⁡T(*)Z]>0
with assumption Xε-1=Pε and pre- and postmultiplying (3.54) by diag(PεT,I) and diag(Pε,I), respectively, we have
(3.56)[Acl⁡TPε+PεTAcl⁡PεBcl⁡(*)-I]<0,[EePεPεTCcl⁡T(*)Z]>0,trace(Z)<v.
This completes the proof.Theorem 3.5.
Given anH2 performance bound v and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, Y11, Y12, Y22, X11, X12, and X22 such that trace(Z)<v andsatisfying the following LMIs:
(3.57)ψ̃11(0)=[Υ11(0)Υ̃12(0)(*)-I]≤0,ψ̃11(ε*)=[Υ11(ε*)Υ̃12(ε*)(*)-I]<0,[[Y110I00000I0X1100000][Y11Y120Y22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T*Z]≥0,[[[Y11ε*Y12ε*Y12Tε*Y22][I00ε*I][I00ε*I][X11ε*X12ε*X12Tε*X22]][Y11Y12ε*Y12TY22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T(*)Z]>0,
where  Υ11(ε*) defined in (3.6) and Υ̃12(ε*) is
(3.58)Υ̃12(ε*)=[Bw[X11X12Tε*X12X22]Bw].
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 Theorem3.3. 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(Π1T,I,I), gives
(3.59)ψ̃11(ε)=[Υ11(ε)Υ̃12(ε)(*)-I]<0,(3.60)[[[Y11εY12εY12TεY22][I00εI][I00εI][X11εX12εX12TεX22]][Y11Y12εY12TY22]Cz2T+CkTDzu2TCz2T+CyTDkTDzu2T(*)Z]>0,
where:
(3.61)Υ̃12(ε)=[Bw+BDkDyw[X11X12TεX12X22]Bw+BkDyw]
According to Lemma2.5, 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. MultiobjectiveH2/H∞ Performance

Now we have got the LMIs in Theorems3.3 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 anH∞ performance bound γ, H2 performance bound v and an upperbound ε* for the singular perturbation ε, if there exist matrices Ak, Bk, Ck, Dk, 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 Theorems3.3 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 withH∞ or H2 performance.


### 4.1.H∞ Performance

Consider the following two-dimensional system and performance index [10]
(4.1)[ẋ1(t)εẋ2(t)]=[21-1-2][x1x2]+[21]u(t)+[13]w(t),z1=[20.1][x1x2]+0.1w(t),y(t)=[10][x1x2]+w(t).
From Theorem 3.3, for ε*=0.1 minimum of γ is obtained as 0.11942. Controller parameters are as follows:(4.2)Ac=[-108.89-31.4291.355.201],Bc=[-.220.05],Cc=[-1793.2-536.1],Dc=-4.7658.
For upperband ε*=0.001, minimum γ is calculated as 0.1. As we expect γε*=0.001≤γε*=0.1.Fromε=0, we obtain reduced order dynamic as follows:
(4.3)ẋ1(t)=1.5x1(t)+2.5u(t)+2.5w(t),z1=1.95x1(t)+0.05u(t)+0.15w(t),y(t)=x1(t)+w(t).
Here, minimum value γ is calculated as 0.029. For w=e-0.1tsin(10t), Figures 1, 2, and 3 exhibit ∫0tz1T(s)z1(s)ds/∫0twT(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 forH∞ performance, now consider an F-8 aircraft model [25]:
(4.4)[ẋ1εẋ2]=([A1A2A3A4])[x1x2]+([B1B2])u+[Bw1Bw2]w,z=[Cw1Cw2][x1x2],y=[Cy1Cy2][x1x2]+[11]w
with
(4.5)A1=[-0.01357-0.06440.060],A2=[-0.00308700.0404670],A3=[-0.045377500.071250],A4=[-0.030550.075-0.075083-0.01674],B1=[-0.00043330.0697],B2=[-0.0522750.019712],Bw1=[-0.4636.07],Bw2=[-4.5525-11.262499],Cy1=[0010],Cy2=[00.0200].
From Theorem 3.3, for ε*=0.025 minimum of γ is obtained as 1.908. Controller parameters are as follows:
(4.6)Ac=[-17.007841.1022-62.09670.15894800.6192-2017.73070.05-5.964-1964.94954.9-754115.03339387-109682.3165444.1-407.4100],Bc=[-0.59660.589329.39-28.98-72.24970.97831586.7-1406.2],Cc=[-3643.87991-6482336.66],Dc=[-76.567-4.3010].
For upperband ε*=0.1, minimum γ is calculated as 3.96. As we expect γε*=0.025≤γε*=0.1.Forw=e-0.1tsin(10t), Figure 4 exhibits∫0tz1T(s)z1(s)ds/∫0twT(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
(4.7)[ẋ1(t)εẋ2(t)]=[-21-1-2][x1(t)x2(t)]+[21]u(t)+[13]w(t),z2=[40][x1(t)x2(t)]+2u(t),y(t)=[10][x1(t)x2(t)].By utilizing Theorem3.5, minimum value of ν for ε*=0.1 is obtained as 0.074. The controller parameters are as follows:(4.8)Ac=[-2558.6-4.2892566620-195.16],Bc=[1.62-163.03],Cc=[-0.0340.011],Dc=-4.1.
For upperband ε*=0.001, minimum v is calculated as 0.0002. As we expect  vε*=0.001≤vε*=0.1. Figures 5, 6, and 7 show input noise and regulated output z2, respectively.Figure 5
Band-limited white noise.Figure 6
Regulated outputz2  for ε*=0.001.Figure 7
Regulated outputz2 for ε*=0.1.As a new example forH2 performance, now consider singular perturbed system as
(4.9)[ẋ1(t)εẋ2(t)]=[-10.51-2][x1(t)x2(t)]+[21]u(t)+[13]w(t),z2=[30][x1(t)x2(t)]+2u(t),y(t)=[11][x1(t)x2(t)].By utilizing Theorem3.5, minimum value of ν for ε*=0.1 is obtained as 0.003. The controller parameters are as follows:(4.10)Ac=[5243.9-14292.829566.58-80500.28],Bc=[-0.167-9.4145],Cc=[-77312.66829.2],Dc=-3.8×10-9.
For upperband ε*=0.001, minimum v is calculated as 0.0004. As we expect  vε*=0.001≤vε*=0.1. Input noise is shown in Figures 5, 8, and 9 show the regulated output z2, respectively.Figure 8
Regulated outputz2  for ε*=0.001.Figure 9
Regulated outputz2 for ε*=0.1.


### 4.3.H∞/H2 Performance

Now, consider singular perturbed system (3.60) as
(4.11)[ẋ1εẋ2]=[21-1-2][x1x2]+[21]u+[13]w,z1=[20.1][x1x2]+0.1w,z2=[40][x1x2]+2u,y=[10][x1x2]+w.
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 whenε*=0.1:(4.12)Ac=[-7.4-9.17817.41-0.5],Bc=[4.16-52.18],Cc=[-0.15×10-40],Dc=-3.206.
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 signalw(t).Figure 11
Regulated outputz2 with ε*=0.1.Figure 12
Simulation for∫0tz1T(s)z1(s)/∫0twT(s)w(s) with ε*=0.1.


## 4.1.H∞ Performance

Consider the following two-dimensional system and performance index [10]
(4.1)[ẋ1(t)εẋ2(t)]=[21-1-2][x1x2]+[21]u(t)+[13]w(t),z1=[20.1][x1x2]+0.1w(t),y(t)=[10][x1x2]+w(t).
From Theorem 3.3, for ε*=0.1 minimum of γ is obtained as 0.11942. Controller parameters are as follows:(4.2)Ac=[-108.89-31.4291.355.201],Bc=[-.220.05],Cc=[-1793.2-536.1],Dc=-4.7658.
For upperband ε*=0.001, minimum γ is calculated as 0.1. As we expect γε*=0.001≤γε*=0.1.Fromε=0, we obtain reduced order dynamic as follows:
(4.3)ẋ1(t)=1.5x1(t)+2.5u(t)+2.5w(t),z1=1.95x1(t)+0.05u(t)+0.15w(t),y(t)=x1(t)+w(t).
Here, minimum value γ is calculated as 0.029. For w=e-0.1tsin(10t), Figures 1, 2, and 3 exhibit ∫0tz1T(s)z1(s)ds/∫0twT(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 forH∞ performance, now consider an F-8 aircraft model [25]:
(4.4)[ẋ1εẋ2]=([A1A2A3A4])[x1x2]+([B1B2])u+[Bw1Bw2]w,z=[Cw1Cw2][x1x2],y=[Cy1Cy2][x1x2]+[11]w
with
(4.5)A1=[-0.01357-0.06440.060],A2=[-0.00308700.0404670],A3=[-0.045377500.071250],A4=[-0.030550.075-0.075083-0.01674],B1=[-0.00043330.0697],B2=[-0.0522750.019712],Bw1=[-0.4636.07],Bw2=[-4.5525-11.262499],Cy1=[0010],Cy2=[00.0200].
From Theorem 3.3, for ε*=0.025 minimum of γ is obtained as 1.908. Controller parameters are as follows:
(4.6)Ac=[-17.007841.1022-62.09670.15894800.6192-2017.73070.05-5.964-1964.94954.9-754115.03339387-109682.3165444.1-407.4100],Bc=[-0.59660.589329.39-28.98-72.24970.97831586.7-1406.2],Cc=[-3643.87991-6482336.66],Dc=[-76.567-4.3010].
For upperband ε*=0.1, minimum γ is calculated as 3.96. As we expect γε*=0.025≤γε*=0.1.Forw=e-0.1tsin(10t), Figure 4 exhibits∫0tz1T(s)z1(s)ds/∫0twT(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
(4.7)[ẋ1(t)εẋ2(t)]=[-21-1-2][x1(t)x2(t)]+[21]u(t)+[13]w(t),z2=[40][x1(t)x2(t)]+2u(t),y(t)=[10][x1(t)x2(t)].By utilizing Theorem3.5, minimum value of ν for ε*=0.1 is obtained as 0.074. The controller parameters are as follows:(4.8)Ac=[-2558.6-4.2892566620-195.16],Bc=[1.62-163.03],Cc=[-0.0340.011],Dc=-4.1.
For upperband ε*=0.001, minimum v is calculated as 0.0002. As we expect  vε*=0.001≤vε*=0.1. Figures 5, 6, and 7 show input noise and regulated output z2, respectively.Figure 5
Band-limited white noise.Figure 6
Regulated outputz2  for ε*=0.001.Figure 7
Regulated outputz2 for ε*=0.1.As a new example forH2 performance, now consider singular perturbed system as
(4.9)[ẋ1(t)εẋ2(t)]=[-10.51-2][x1(t)x2(t)]+[21]u(t)+[13]w(t),z2=[30][x1(t)x2(t)]+2u(t),y(t)=[11][x1(t)x2(t)].By utilizing Theorem3.5, minimum value of ν for ε*=0.1 is obtained as 0.003. The controller parameters are as follows:(4.10)Ac=[5243.9-14292.829566.58-80500.28],Bc=[-0.167-9.4145],Cc=[-77312.66829.2],Dc=-3.8×10-9.
For upperband ε*=0.001, minimum v is calculated as 0.0004. As we expect  vε*=0.001≤vε*=0.1. Input noise is shown in Figures 5, 8, and 9 show the regulated output z2, respectively.Figure 8
Regulated outputz2  for ε*=0.001.Figure 9
Regulated outputz2 for ε*=0.1.


## 4.3.H∞/H2 Performance

Now, consider singular perturbed system (3.60) as
(4.11)[ẋ1εẋ2]=[21-1-2][x1x2]+[21]u+[13]w,z1=[20.1][x1x2]+0.1w,z2=[40][x1x2]+2u,y=[10][x1x2]+w.
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 whenε*=0.1:(4.12)Ac=[-7.4-9.17817.41-0.5],Bc=[4.16-52.18],Cc=[-0.15×10-40],Dc=-3.206.
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 signalw(t).Figure 11
Regulated outputz2 with ε*=0.1.Figure 12
Simulation for∫0tz1T(s)z1(s)/∫0twT(s)w(s) with ε*=0.1.


## 5. Conclusions

In this paper, we addressed robustH2 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.



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*Source: 903126-2011-03-28.xml*