# Robust Observer-Based Feedback Control for Lipschitz Singularly Perturbed Systems

Authors: Yanyan Wang; Wei Liu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The observer-based feedback control for singularly perturbed systems (SPSs) with Lipschitz constraint is addressed. A sufficient condition, independent of the perturbation parameter, for a full-order observer is presented in terms of linear matrix inequality (LMI) such that observation error is exponentially stable for all sufficiently small perturbation parameters. Then, for observer-based feedback control, a proper controller is constructed to guarantee the input-to-state stability of the system with regard to the observation error. Considering the convergence of observation error, the stability of the system can be obtained based on the input-to-state stability property. It is shown that the proposed method is simple and easy to operate. Moreover, the upper bound of the small perturbation parameter for stability of systems can be explicitly estimated with a workable computation way. Finally, two numerical examples show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Singularly perturbed systems or two-time-scale systems are usually described by state-space models in which a small parameter multiplies the time derivatives of some of the system states. During the past years, the robust stability of singularly perturbed system has been widely studied, and many different results have been reported; see [1–12] and the references therein. This is due not only to theoretical interests but also to the relevance of this topic in control engineering applications. Reference [2] presented a composite linear controller for robust stability of singularly perturbed linear systems with matching condition uncertainties. In [4], the robust stabilization problem of singularly perturbed systems with nonlinear uncertainties is studied; a control law is presented by the solutions of two independent Lyapunov equations. In addition, the stability bound is also derived via a state transformation and the constructive use of a Lyapunov function. The obtained result shows that the controller design approach is very complex and difficult to operate. More recently, the authors of [7] considered the robust stabilization problem for a class of singularly perturbed linear systems via a networked state feedback with the transmission time-delay. However, the computation of the maximum stability bound has not been involved.Most of the above results are obtained based on the assumption that the state variables of systems are available for direct measurement. However, in many control systems and applications, not all the state variables can be measured, or we may choose not to measure some of them due to technical or economic reasons. In this case, it is necessary to design a state observer used to reconstruct the states of a dynamic system and it has many important applications in practical systems such as system supervision and fault diagnosis. Recently, much effort has been devoted to the observer design and observer-based control, and there have been a few important results reported in the literatures; see [13–17] and the references therein. In [13], the observer design for a class of nonlinear systems with discrete-time measurements is considered. By using a continuous Newton method for the map inversion, the observer error was shown to converge to zero exponentially. Reference [14] proposes a separation principle for a class of nonlinear systems such that semiglobal stability can be achieved by means of dynamic output feedback. Jiang et al. [17] use a reduced-order high-gain observer to obtain semiglobal stabilization for a nonlinear benchmark example. However, to the best of authors’ knowledge, there is only little attention paid to the observer-based feedback control for singularly perturbed systems [18–22]. Reference [19] considers the robust stability of linear shift-invariant discrete-time singularly perturbed systems; a composite observer-based controller is given such that the closed-loop system is stable. In [21], an observer-based output feedback control for singularly perturbed linear systems is considered under the model-based framework. Based on the approximated slow and fast subsystems of the overall system on each sampling interval, a lower order test matrix for stability is obtained. Although these works present some methods for observer and observer-based controller design, many difficult and efficient nonlinear observer issues still need to be addressed, which still remains important and challenging.Motivated by the above works, we, in this paper, discuss robust observer-based feedback control for continuous singularly perturbed systems with Lipschitz constraint. Such a system has been studied without considering observer in [4]. Our concern is that if stabilization can still be achieved when considering observer. In this paper, we will first develop a new observer design approach based on linear matrix inequalities (LMIs) technique. It will be shown that, under the LMI condition, the design procedure for observer can be easily constructed and observation error between systems states and estimators is globally exponentially stable. Then, for observer-based control, a sufficient condition for control law, which is expressed in terms of an ε-independent LMI and, thus, well conditioned, is derived, under which the closed-loop system is made input-to-state stable (ISS) with observation error as the input. In addition, the maximum stability bound ε∗ that satisfies the previous LMI can be found by solving a generalized eigenvalue problem (GEVP), such that the system is asymptotically stable for any small parameter ε∈(0,ε∗]. Compared with the previous results, the newly developed method has the following advantages: (1) not only is the existence of the observer established, where observer gain matrix can be obtained easily by solving a simple LMI, but also an estimate for the stability bound of observation error is given; (2) a LMI based sufficient condition for input-to-state stability of system is provided, under which the control gain matrix can be solved efficiently and much more complex equation is not involved. In addition, the desired upper bound of singularly perturbed parameter subject to the stability of system can be obtained via solving GEVP, without any state transformation of the original and can be tested numerically efficiently using the LMI Toolbox. Thus, the effectiveness of the proposed method is clearly shown.


## 2. Problem Formulation

Consider the following singularly perturbed system with Lipschitz constraint:(1)Eεx˙t=Axt+But+Hft,xt,ut,(2)yt=Cxt+Dut,where x(t)=(x1T(t),x2T(t))T is the system state; x1∈Rn is the slow state; x2∈Rm is the fast state; u∈Rq is the input vector; y∈Rq is the output vector; A, B, H, C, and D are constant matrices with appropriate dimensions; the matrix Eε is given by(3)Eε=InOOεIm,where ε is the perturbation parameter which is small and positive but may be unknown, representing the response of the fast dynamics. f(t,x(t),u(t)) is vector-value time-varying nonlinear function with f(t,0,0)=0 for all t≥0, which is assumed to satisfy the following global Lipschitz condition for all (t,x,u),(t,x~,u~)∈R×Rn+m×Rq:(4)ft,x,u-ft,x~,u~≤Fx-x~+F1u-u~,where F and F1 are known constant matrices with appropriate dimensions.It is easy to verify thatf(t,x,u) satisfies the following condition for all (t,x,u)∈R×Rn+m×Rq:(5)ft,x,u≤Fx+F1u.Remark 1.
The Lipschitz property as formulated in (4) is capable of describing the structure of the uncertain term more accurately than the general one with equal weights in many practical situations, and thus it will help reduce the conservatism of the obtained results. Recently, this important formulation has been widely considered; see [3, 4, 23, 24] and the references therein. It is worth mentioning that the matched condition can be regarded as a special case of (4). However, little attention has been paid to the observer-based control of singularly perturbed systems.The objective of this paper is to design a suitable observer and observer-based feedback controller such that system (1) is made ISS with observation error as the input. Meanwhile, the observation error between systems states and estimators is globally exponentially stable. Thus, the asymptotical stability of system (1) can be guaranteed.


## 3. Observer Design

In this section, we propose a Luenberger-like observer as follows:(6)Eεx^˙=Ax^+Bu+Hft,x^,u+Ly^-y,y^=Cx^+Du,where x^ is the estimator of x, y^ is the observer output, and L is an observer gain matrix to be determined.Denote thate(t)=x^-x is the observation error between the controlled system and observer. Then the observation error system is given by(7)Eεe˙=A+LCe+Hft,x^,u-ft,x,u.The following result presents a sufficient condition via linear matrix inequalities which guarantees the existence of observer.Theorem 2.
If there exist scalarμ1>0, matrix Y1, and lower triangular matrix (8)P=P11OP21P22,where P11∈Rn×n and P22∈Rm×m are positive definite matrices, satisfying the linear matrix inequality(9)Φ=ATP+PTA+CTY1+Y1TC+μ1FTFPTHHTP-μ1I<0,then there exists ε1∗>0 such that the observation error e is exponentially asymptotically stable for ε∈(0,ε1∗]. Moreover, the observer gain matrix can be chosen as(10)L=P-TY1T.Proof.
SinceP11∈Rn×n and P22∈Rm×m are positive definite matrices, there exists scalar ε11>0 such that P11-εP12TP22-1P21≥0 for all ε∈(0,ε11]. By the Schur Complement Lemma, it yields(11)EεPε=P11εP21TεP21εP22>0,where Pε=P11εP21TP21P22. We choose the Lyapunov function candidate as follows:(12)V1e=eTEεPεe.Then, for any constant μ1>0, the derivative of V1 along trajectories of system (7) yields(13)V˙1e≤eTA+LCTPε+PεTA+LC+μ1FTF+2eTPεTHΔf-μ1ΔfTΔf=eTΔfTΦ+εΦ0eΔf,where (14)Φ0=ATP0+P0TA+CTLTP0+P0TLCP0THHTP0O<0,P0=OP21TOO,Δf=ft,x^,u-ft,x,u.
It follows from (9) that there exists sufficiently small scalar ε12>0 such that Φ+εΦ0<0 for any given ε∈(0,ε12], which implies that V˙1<0 for ε∈(0,ε12]. Let ε1∗=min⁡{ε11,ε12}, and then we have EεPε>0 and V˙1<0 for any given ε∈(0,ε1∗]. Therefore, the observation error e is globally asymptotically stable for any given ε∈(0,ε1∗]. Thus, the proof is completed.Remark 3.
Theorem2 presents a sufficient condition for the existence of observer. It is worth pointing out that derived condition (9) is independent of ε; therefore, the numerically stiff problem can be avoided.The estimation of the upper boundε1∗>0 to guarantee the existence of observer for all ε∈(0,ε1∗] is also an interesting topic. From the proof of Theorem 2, we can easily get the following approach to estimate the upper bound ε1∗.Theorem 4.
After observer gain matrixL has been obtained from (9)-(10), if there exist constant λ1>0, positive definite matrix W>0, and lower triangular matrix given in (8), satisfying the linear matrix inequalities(15)W<λP11,WP21TP21P22>0,Φ<0,Φ0<-λ1Φ,then observation error is exponentially stable for all ε∈(0,ε1∗] with ε1∗=λ1-1.Remark 5.
It follows from Theorem4 that upper bound of ε1∗ can be obtained by solving the following minimization problem [25]:(16)min⁡λ1subject  to15,which can be solved effectively by applying GEVP solver in LMI Control Toolbox.


## 4. Observer-Based Feedback Control

In this section, we will consider the observer-based state feedback control. We will give an observer-based feedback controller of the form(17)u=Kx^,where K is a controller gain, such that the resulting closed-loop system (18)Eεx˙=A+BKx+Hft,x,Kx+e+BKewith the constraint(19)ft,x,Kx+e≤Fx+F1Kx+F1Ke=F+F1Kx+F1Keis made ISS with observation error e as the input.Before giving the result, we first recall the following definition and lemma that will be used for the proof of the main result.Definition 6 (see [26]).
Consider the system(20)x˙=ft,x,u,where state x(t) is in Rn and control input u(t) in Rm. f:[0,∞)×Rn×Rm→Rn is continuous and locally Lipschitz in x and u. The input u is a bounded function for all t≥0. Then the system is said to be input-to-state stable (ISS) if there exist class KL function β and class K function γ such that, for any initial state x(t0), the solution x(t) exists for all t≥t0 and satisfies(21)xt≤βxt0,t-t0+γsupt0≤τ≤t⁡uτ.Remark 7.
The last inequality guarantees that, for any bounded inputu(t), the state x(t) will be bounded, and as t increases, the state x(t) will be ultimately bounded by class K function of u. Furthermore, the inequality also shows that if u(t) converges to zero as t→∞, so does x(t).Lemma 8 (see [26]).
LetV:[0,∞)×Rn→R be a continuously differentiable function such that (22)α1x≤Vt,x≤α2x,∂V∂t+∂V∂xft,x,u≤-W1x,∀x≥ρu>0,where α1, α2 are class K∞ functions, ρ is class K function, and W1(x) is a continuous positive definite function on Rn. Then, system (20) is input-to-state stable with γ=α1-1∘α2∘ρ.Theorem 9.
If there exist scalarμ2>0, matrices Y2,Q>0, and lower triangular matrix (23)X=X11OX21X22,where X11∈Rn×n and X22∈Rm×m are both positive definite matrices, satisfying the matrix inequality(24)Ω=Ω1FX+F1Y2Tμ2-1HFX+F1Y2-μ2-1IOμ2-1HTO-μ2-1I<0,where Ω1=AX+XTAT+BY2+Y2TBT+XTQX, then there exists ε2∗>0 such that closed-loop system (18) is ISS with respect to observation error e for ε∈(0,ε∗]. Moreover, the observer-based controller gain matrix can be chosen as(25)K=Y2X-1.Proof.
By substituting (25) into (24), we obtain that inequality (24) is equivalent to (26)Ω2XTFX+F1KTμ2-1HF+F1KX-μ2-1IOμ2-1HTO-μ2-1I<0,where Ω2=XT(A+BK)T+(A+BK)X+XTQX. By Schur’s Complement Lemma, inequality (26) is equivalent to (27)Ω3HHT-μ2I<0,where (28)Ω3=XTA+BKT+A+BKX+μ2XTF+F1KTF+F1KX+XTQX.Pre- and postmultiplying inequality (27) by diag⁡(X-T,I) and diag⁡(X-1,I), respectively, let X=P--1, Y2=KP--1; then (27) is equivalent to(29)Ω-=Ω4P-THHTP--μ2I<0,where(30)Ω4=A+BKTP-+P-TA+BK+μ2F+F1KTF+F1K+Q.We choose the Lyapunov function candidate as follows:(31)Vx=xTEεP-εx+∫0txTQxdt,where P-ε=P-+εP-0, P-=P-11OP-21P-22, and P-0=OP-21TOO.
Similar to the proof of Theorem2, there exists scalar ε21>0 such that EεP-ε>0 for any ε∈(0,ε21]. Thus, the positive definiteness of V is guaranteed for any ε∈(0,ε21]. Then, the derivative of V along closed-loop system (18) yields (32)V˙≤A+BKx+Hf+BKeTPεx+xTPεTA+BKx+Hf+BKe+xTQx+μ2xTeTF+F1KTF1KTF+F1KF1Kxe-fTf=xTfTΩ-+εΩ-0xf+2xTP-εTBKe+2μ2xTF+F1KF1Ke+μ2eTF1KTF1Ke,where Ω-0=(A+BK)TP-0+P-0T(A+BK)P-0THHTP-0O. It follows from (29) that there exists sufficiently small scalar ε22>0 such that Ω-+εΩ-0<0 for any given ε∈(0,ε22].
Letε2∗=min⁡{ε21,ε22} and α-=λinfε∈(0,ε2∗](-Ω--εΩ-0); then α->0 for ε∈(0,ε2∗]. Thus, for any ε∈(0,ε2∗], we have that(33)V˙≤-α-x2+βxe+δe2≤-α-1-θx2,∀x≥β+β2+4δα-θ2α-θe,where 0<θ<1, β=2supε∈(0,ε2∗]⁡P-εTBK+μ2(F+F1K)TF1K, and δ=μ2(F1K)TF1K. Hence, the conditions of Lemma 8 are satisfied, and we conclude that closed-loop system (18) is ISS with respect to the observation error e. This completes the proof.Similar to Theorem4, we have the following result which gives the method for solving the upper bound of input-to-state stability.Theorem 10.
After the control gain matrixK has been obtained from (25), if there exist scalar λ2>0, positive matrices W->0, Q>0, and lower triangular matrix (34)P-=P-11OP-21P-22with 0<P-11∈Rn×n and 0<P-22∈Rm×m, then upper bound ε2∗=λ2-1 for input-to-state stability of closed-loop system (18) can be found by solving the following GEVP:(35)min⁡λ2s.t.W-<λ2P-11,W-P-21TP-21P-22>0,Ω-<0,Ω-0<-λ2Ω-.Remark 11.
Theorem9 provides an input-to-state stability criterion for system (18). Note that the observation error is globally exponentially stable by Theorem 2. Let ε∗=min⁡{ε1∗,ε2∗}; then it follows from Lemma 8 and Theorem 9 that closed-loop system (18) is asymptotically stable for any ε∈(0,ε∗]. In addition, observe that (24) is not a LMI due to the existence of the nonlinearity term XTQX. Let Z=XTQX; then (24) can be transformed to a standard LMI which can be solved effectively by the LMI Toolbox. Compared with [4], the derived sufficient condition for control law does not involve much more complex equations. In addition, Theorem 10 gives a sufficient condition for calculating the upper bound of the small parameter subject to input-to-state stability of (18). It can be obtained via solving the GEVP (35) and can be tested numerically efficiently using the LMI Toolbox. However, in [3, 4], some coordinate transformations for the original system are required in order to investigate the robust stability of the system. Moreover, the derived sufficient condition for calculating the upper bound in [3, 4] involves much more complex inequalities, which is numerically inefficient.


## 5. Numerical Examples

In this section, we present two examples to illustrate the effectiveness of our results.Example 1.
Consider the following DC-Motor plant with matched condition in [2]:(36)diadt=-RaLaia-KbLaωm+1Laet;d2θmdt2=-KiJmia-BmJmdθmdt-KiJmθm-θl;d2θldt2=-KlJlθm-θl+TfJl,where Jl=5(1+δl), δl≤0.5; Tf=(1+δf)u1, δf≤0.2. Let x1=θm-θl, x2=θ˙m=ωm, x3=θ˙l=ωl, and x4=ia; u2=et; ε=La>0. Then, we have the state-space form of the uncertain DC-Motor model given by (37)x˙1x˙2x˙3εx˙4=01-10-KlJm-BmJm0-KiJm-KlJl0000-Kb0-Rax1x2x3x4+00001Jl001Tfu2.Substituting the system parameters (see Table 1 in [2]) and noticing the uncertainties Jl and Tf, we can transfer (37) into the following form:(38)x˙1x˙2x˙3εx˙4=01-10-2-0.2022000052×10-50-2+ΔAtx1x2x3x4+00000.2001+ΔBtu1u2,y=Cx+Du,with(39)ΔAt=HΔtF,ΔBt=HΔtF1,where (40)H=0.20.20.20.2T,F=1110,F1=11,C=I4,D=00100001T,and the matrix Δ(t) is a time-varying uncertainty satisfying(41)ΔTtΔt≤I.Let f(t,x,u)=Δ(t)(Fx+F1u); then it is easy to verify that f(t,x,u) satisfies constraint (5).We find the feasible solutions of LMIs (9) as follows:(42)P=1.084200001.084200001.08420-1.0842-1.0842-1.08421.0039,μ1=0.6947,Y1=-0.8894-10.5746-12.2089-4.890110.9646-0.6720-44.7469-5.072610.430044.0528-0.88940.23562.72180.7355-2.40401.4656.Hence, observer gain (10) is obtained as follows:(43)L=-5.69185.06039.85513.9705-14.6251-5.673040.86772.1384-16.1325-46.3264-0.5857-0.7573-4.8714-5.05320.23471.4600.For observer-based feedback control, utilizing the LMI Toolbox, we get some parameters from (24):(44)X=3.8833-1.64411.35110-1.64415.45880.379901.35110.37994.228700.8891-3.2438-2.14684.4639,Q=0.48250.2659-0.2904-0.02060.26590.3309-0.12260.1079-0.2904-0.12260.42660.0971-0.02060.10790.09710.2154,Y2=-4.498310.7055-6.9933-4.80931.1984-15.07280.86094.6366,μ2=1.3619.Thus, we can obtain control gain matrix from (25):(45)K=Y2X-1=0.71321.7155-2.5827-1.0774-1.6030-2.73041.48841.0387.Therefore, ε∗>0 exists such that singularly perturbed system (18) is asymptotically stable for all ε∈(0,ε∗]. Then solve the generalized eigenvalue minimization problems (15) and (35). We get that the upper bound is ε∗=0.6709.Example 2.
Consider the following singularly perturbed systems in [4]:(46)100εx˙=-321-1x+01u+1001f1f2,y=12x1x2+u,where fi≤αix1+βix2(i=1,2) with α1=1,α2=0.15, and β1=β2=0.15.
Note that(47)fTf=f12+f22≤2α12+α22x12+2β12+β22x22=x1Tx2T2α12+α22002β12+β22x1x2.Thus, we have(48)F=2α12+α22002β12+β22=1.43000.3,F1=0.Then, by solving LMI (9), the obtained parameters are given by(49)P=0.313600.02130.3443,μ1=0.7845,Y1=-0.4167-0.1099.Hence, observer gain (10) is obtained as follows:(50)L=-1.3069-0.3193.For observer-based feedback control, we find the feasible solutions of LMIs (24) as follows:(51)X=0.37240-0.33611.5293,Q=13.53370.69650.69650.7716,Y2=-3.7671-0.2753,μ2=1.0218.Thus, we can obtain control gain matrix from (25):(52)K=Y2X-1=-10.2789-0.1800.Then, by solving GEVP (15) and (35), we get that the upper bound is ε∗=0.7296. However, when using the method in [3, 4], the maximum stability bound is ε∗=0.3326, which is slightly smaller.From these numerical studies, it is clear that our observer design approach is simple and it is easy to compute. In addition, it can be easily seen that our controller design method is much simpler than the one adopted in [4]. The effectiveness of the proposed method is shown clearly.


## 6. Conclusion

This paper has presented the observer design and observer-based control for continuous singularly perturbed system with Lipschitz constraint. A sufficient condition for existence of observer has been obtained in terms of LMI under which the observation error is exponentially stable for all sufficiently small perturbation parameters. For observer-based control, based on the Lyapunov stability theory and LMI technique, a proper feedback controller for input-to-state stability of the system with respect to observation error has been presented. Therefore, the stability of the system is guaranteed based on the input-to-state stability property. Moreover, the problem of searching the upper bound can be obtained by solving the GEVP. However, in practice, nonlinear singularly perturbed system is more realistic; a future research will be devoted to observer-based control of nonlinear singularly perturbed systems.



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