# H∞ Observers Design for a Class of Continuous Time Nonlinear Singular Systems

Authors: Mohamed Zerrougui; Latifa Boutat-Baddas; Mohamed Darouach
Journal: Journal of Control Science and Engineering
 (2011)

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## Abstract

This paper considers
the problem ofH∞ observers
design for a class of Lipschitz continuous
nonlinear singular systems. The method is based
on the parameterization of the solution of the
generalized Sylvester equations obtained from
the estimation errors. Sufficient conditions for
the existence of the observers which guarantee
stability and the worst case observers error
energy over all bounded energy disturbances is
minimized are given. The approach also unifies
the full-order, the reduced-order, and the
minimal-order observers design. The solutions
are obtained through linear matrix inequalities
(LMIs) formulation. A numerical example is
given to illustrate our results.


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## Body

## 1. Introduction

Observers design for nonlinear systems has been a very active field during the last two decades. This is due to the fact that a state estimation is generally required for the control when all states of the system are not available. The observers are also used in the monitoring and fault diagnosis. For standard nonlinear systems, there exist several approaches for the observers design including one based on coordinate transformations which lead to a linear error dynamics [1–3] and one where the problem of the observers design can be treated without the need of these transformations [4]. An important class of standard nonlinear systems, the global Lipschitz, was considered by the authors in [5, 6], where the existence conditions for the observers are presented and constructive design methods were given for full-order and reduced-order cases.On the other hand, singular systems (known as generalized, descriptor, or differential algebraic (DA) systems) describe a large class of systems. They are encountered in chemical and mineral industries; for example, the dynamic balances of mass and energy are described by differential equations, while thermodynamic equilibrium relations constitute additional algebraic constraints. The problem of the state estimation for these practical applications arises in data reconciliation, for example, [7]. Singular systems are also frequently encountered in electronic and economics [8]. In recent years a great deal of work has been devoted to the analysis and design techniques for singular systems [9–11]. On the other hand, the problem of observer design for linear systems has been greatly treated for the standard and singular systems with or without unknown inputs (see [12–14] and references therein). In [15], extension to observers design for Lipschitz singular systems has been presented; however, the observer considered has a singular system form. Recently a new method for the observers design is presented for a class of singular systems, where the nonlinearity is assumed to be composed of a Lipschitz one and an arbitrary one; the latter can be considered as an unknown disturbance. The approach is based on the parameterization of the generalized Sylvester equations solutions and unifies the design of full-, reduced-, and minimal-order observers the observer presented is causal and has a standard system form [16]. However, only the case where the model and the measurement are free from noises was considered.The state estimation problem for linear singular systems in presence of noises has been the subject of several studies in the past decades. We can distinguish two approaches, the Kalman observering approach andH∞ approach. In the Kalman observering, the system and the measurement noises are assumed to be Gaussian with known statistics [17–19]. When the noises are arbitrary signals with bounded energy, the H∞ observering permits to guarantee a noise attenuation level [20]. Recently, a number of papers have appeared that deal with the H∞ observering for singular systems; see, for example, [21–23] and references therein. In all these works only full- or reduced-order observers were presented for the square singular systems.In this paper, we consider theH∞ observers design problem for a class of Lipschitz nonlinear singular systems. The approach extends the work [16] to the case where the model and the measurement are affected by noises. Sufficient conditions in terms of LMIs are given for this H∞ problem. The method is more general than the one considered in [16] since it assumes only the impulse observability of the linear part. It also unifies the design for full-order, reduced-order, and minimal-order observers. A numerical example is given to illustrate our results.


## 2. Problem Formulation

Consider the following nonlinear singular system:(1)Eẋ(t)=Ax(t)+Bu(t)+Df(t,FLx,u)+D1w(t),(2)y(t)=Cx(t)+D2w(t),
with the initial state x(0)=x0, where x(t)∈ℝn is the semi state vector, u(t)∈ℝm is the known input, w(t)∈ℝnw is the disturbance vector containing both system and measurement noises, and y(t)∈ℝp is the measurement output. Matrix E∈ℝn1×n, and, when n1=n, matrix E is singular. Matrices A∈ℝn1×n, B∈ℝn1×m, C∈ℝp×n, D∈ℝn1×nf, D1∈ℝn1×nw and D2∈ℝp×nw. The nonlinearity f(t,FLx,u) verifies the Lipschitz constraints:
(3)‖f(t,FLx1,u)-f(t,FLx2,u)‖≤λ(‖FL(x1-x2)‖),
where λ is a known Lipschitz constant and matrix FL is real with appropriate dimension.Letr be the rank of the matrix E, and let Φ∈ℝr1×n1 be a full row rank matrix such that ΦE=0, with r1=n1-r. Then, from (1), we obtain (4)ΦAx(t)+ΦD1w(t)+ΦDf(t,FLx,u)=-ΦBu(t).
In the sequel we assume the following.Assumption 1.
Considerrank[EΦAC]=n.Before presenting our main results, we can make the following remarks.Remark 2.
Whenn1=n,
Assumption1is exactly the impulse observability of the linear singular system(E,A,B,C).Remark 3.
ConditionΦE=0is more general than conditionΦ[ED]=0considered in [16]. In fact when the matrix[ED]is of full row rank, matrixΦ=0and Assumption1becomesrank[EC]=nwhich is more restrictive than the impulse observability condition.Now, let us consider the following reduced-order observer for system (1): (5)ζ̇(t)=Nζ(t)+Jy(t)+Hu(t)+TDf(t,FLx̂,u),x̂(t)=Pζ(t)-QΦBu(t)+Fy(t)-QΦDf(t,FLx̂,u),
with the initial condition ζ(0)=ζ0. Vector ζ(t)∈ℝq represents the state vector of the observer, and x̂(t)∈ℝn is the estimate of x(t). Matrices N, J, T, H, P, Q, and F are unknown matrices of appropriate dimensions, which must be determined such that,(1)
forw(t)=0, the error e(t)=x̂(t)-x(t) asymptotically converges to 0,(2)
forw(t)≠0, we solve the minsupω∈L2-{0}(∥e∥L2/∥ω∥L2).Let the error betweenζ(t) and TEx(t) be
(6)ε(t)=ζ(t)-TEx(t),
then we obtain the following dynamics of ε(t):(7)ε̇(t)=ζ̇(t)-TEẋ(t)=Nε+(NTE-TA+JC)x(t)+(H-TB)u(t)+TDΔf+(JD2-TD1)w(t),x̂(t)=Pε(t)+[PQF][TEΦAC]x(t)-QΦDΔf+(QΦD1+FD2)w(t),
where Δf=f(t,FLx̂(t),u)-f(t,FLx(t),u).Under Assumption1, if there exists a matrix T such that (8)NTE-TA+JC=0,(9)H=TB,[PQF][TEΦAC]=I,
then (7) becomes (10)ε̇(t)=Nε(t)+TDΔf+φ1w(t),(11)e(t)=Pε(t)-QΦDΔf+φ2w(t),
where φ1=JD2-TD1 and φ2=QΦD1+FD2.Now the problem of theH∞ observer design is reduced to find the matrices N,J,H,P,Q,F, and T such that (8) is satisfied and the worst case observers error energy over all bounded energy disturbances w(t) is minimized.Remark 4.
The observer given in (5) is more general than that presented in [16]; this can be seen from the fact that, for ΦD=0, and w(t)=0, we obtain the observer given in [16]. As in [16] this observer is general, and its design unifies the full-order (q=n), the reduced-order (q=n-p), and the minimal-order observers design.


## 3. Main Results

In this section we will present theH∞ observers design. The H∞ observer design problem can be formulated as follows: given the nonlinear singular system (1)-(2) and a prescribed level of noise γ>0, find a suitable observer of the form (5), such that,(1)
forw(t)=0, estimation error (11) converges asymptotically to 0, that is, e(t)→0 as t→∞,(2)
under the zero initial condition, the errore(t) satisfies ∥e(t)∥L2<γ∥w(t)∥L2 for any w(t)∈L2-{0}, where γ>0 is the prescribed constant.Now, the dynamics error (10)-(11) can be written in a singular form as (12)Eξ̇(t)=Aξ(t)+BΔf+Dw(t),
where ξ(t)=[ϵe], 𝔼=[I000], 𝔸=[N0P-I], 𝔹=[TD-QΦD], and D=[φ1φ2]. Then, the error e(t)→0 as t→∞, for w(t)=0 if system (12) is asymptotically stable.


### 3.1. Stability Analysis

Before giving theH∞ observers design method for system (1)-(2), let us deal with the stability analysis problem and derive a sufficient condition, in a strict LMI form, for system (12) to be asymptotically stable for w(t)=0. The following lemma gives this condition.Lemma 5.
Forw(t)=0, system (12) is asymptotically stable, if there exists a matrix Y, such that the following LMIs are satisfied:
(13)ETY=YTE⩾0,(14)[ATY+YTA+μρYTBBTY-μI]<0
with ρ=[000λ12FLTFL].Proof.
For stability analysis we construct the following Lyapunov candidate function:(15)V(t)=ξT(t)ETYξ(t)=ξT(t)YTEξ(t)
with 𝔼TY=YT𝔼⩾0. The derivative of V(t) along the solution of (12), for w(t)=0, is given by
(16)V̇(t)=ξ̇T(t)ETYξ(t)+ξT(t)YTEξ̇(t)=(Aξ(t)+BΔf)TYξ(t)+ξT(t)YT(Aξ(t)+BΔf)=ξT(t)(ATY+YTA)ξ(t)+ΔfTBTYξ(t)+ξT(t)YTBΔf.
Let u and v be two vectors of appropriate dimensions, then for all scalar μ>0 the following inequality holds:
(17)uTv+vTu⩽μuTu+1μvTv.
By using (17), it is not difficult to check that
(18)ΔfT(B)TYε(t)+εT(t)YT(B)Δf⩽1μεT(t)YTBBTYε(t)+μΔFTΔf.
In this case we have V̇(t)⩽ξT(t)(𝔸TY+YT𝔸)ξ(t)+μΔfTΔf+(1/μ)ξT(t)YT𝔹𝔹TYξ(t). On the other hand, by using the Lipschitz conditions and the fact that e(t)=[0I]ξ(t), we have μΔfTΔf⩽μξT(t)ρξ(t), where ρ=[000λ12FLTFL]. Then we obtain the following inequality:
(19)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+ξT(t)μρξ(t)+1μξT(t)YTBBTYξ(t)=ξT(t)(ATY+YTA+1μYTBTBY+μρ)ξ(t)
and V̇(t)<0, if 𝔸TY+YT𝔸+(1/μ)YT𝔹𝔹TY+μρ<0.
Using the Schur complement we obtain (14), then Lemma 5 is proved.From this lemma, one can see that the stability conditions (13) and (14) are nonstrict LMIs, which contain equality constraints; this may result in numerical problems when checking such conditions. Therefore, strict LMIs conditions are more desirable than nonstrict ones from the numerical point of view. The following lemma presents the stability conditions in a strict LMI formulation.Lemma 6.
Forw(t)=0, system (12) is asymptotically stable if there exist a positive definite matrix X1 and matrices X2,Q1, and Q2 and a scalar μ such that the following LMI is satisfied:(20)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2X1TTD-Q1TQΦD-X2QΦD*-Q2-Q2T+μλ2FLTFL-Q2TQΦD**-μI]<0.Proof.
LetY=X𝔼+𝔼⊥TQ̅, where 𝔼⊥ is the orthogonal matrix of 𝔼 satisfying 𝔼⊥𝔼=0 and 𝔼⊥𝔼⊥T>0, and X=[X1X2X2TX3], and Q̅=[Q1Q2]. It is easy to see that 𝔼TY=YTE⩾0,since X1>0.In this case (13)-(14) reduce to(21)[ATXE+ATE⊥TQ̅+ETXTA+Q̅TE⊥AETXTB+Q̅TE⊥BBTXE+BTE⊥TQ̅-μI]<0.On the other hand by inserting the values of 𝔼,𝔸,𝔹, and ρ into (21) we obtain (20).Remark 7.
ForΦD=0,inequality (20) becomes(22)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2X1TTD-X2T-Q1+Q2TP-Q2-Q2T+μλ12FLTFL0(TD)TX10-μI]<0.
By premultiplying (22) by [IPT000I] and postmultiplying it by [I0P00I], we obtain [NTX1+X1TN+μλ1FLTFLX1TTD(TD)TX1-μI]<0, which is exactly the inequality (15) of [16].


### 3.2.H∞ Observes Design

In this section we shall present theH∞ observer design. The following lemma gives the sufficient conditions for system (12) to be stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0.Lemma 8.
The errore(t) given by (11) is asymptotically stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0,  if there exist a positive definite matrix X1 and matrices X2,Q1,Q2 and  scalars μ and γ such that the following LMI is satisfied:(23)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2(X1TTD-Q1TQΦD-X2QΦD)(X1Tφ1+X2φ2+Q1Tφ2)*-Q2-Q2T+I+μλ2FLTFL-Q2TQΦDQ2Tφ2**-μI0***-γ2I]<0.Proof.
Consider the following Lyapunov candidate functionV(t)=ξt𝔼TYξ=ξT(t)YT𝔼ξ(t),with 𝔼TY=YT𝔼.Then we have(24)V̇(t)=ξ̇T(t)ETYξ(t)+ξT(t)YTEξ̇(t)=(Aξ(t)+BΔf+Dw(t))TYξ(t)+ξ(t)TYT(Aξ(t)+BΔf+Dw(t))=ξT(t)(ATξ+YTA)ξ(t)+ΔfTBTYξ(t)+ξ(t)TYTBΔf+wT(t)DTYξ(t)+ξ(t)TYTDw(t).
From (17) we have the following inequality:
(25)ΔfT(B)TYε(t)+εT(t)YT(B)Δf⩽1μεT(t)YTBBTYε(t)+μΔfTΔf.
Then V̇(t) becomes
(26)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+μΔfTΔf+1μξT(t)YTBBTYξ(t)+wT(t)DTYξ(t)+ξT(t)YTDw(t).
On the other hand, we haveΔfTΔf⩽λ2eT(t)FLTFLe(t), and we know that e(t)=[0I]ξ(t) then ΔfTΔf⩽ξT(t)ρξ(t), where ρ=[000λ2FLTFL].Then
(27)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+μλ2ξT(t)ρξ(t)+1μξTvYTBBTYξ(t)+wTDTYξ(t)+ξT(t)YTDw(t).
Let η(t)=[ε(t)w(t)], then from (27) and (12) we obtain the following inequality:
(28)V̇+eT(t)e(t)-γ2wT(t)w(t)⩽ηT(t)Ση(t)
with
(29)Σ=[ATY+YTA+1μYTBBTY+ρ̅YTDDTY-γ2I],
where ρ̅=[000I+μλ2FLTFL].
IfΣ<0, we obtain
(30)V̇<γ2wT(t)w-eT(t)e(t).
Integrating the two sides of this inequality gives
(31)∫0∞V̇(τ)dτ<∫0∞γ2wT(τ)w(τ)dτ-∫0∞eT(τ)e(τ)dτ
or equivalently
(32)V(∞)-V(0)<γ2‖w(t)‖22-‖e(t)‖22,
under zero initial conditions, we obtain
(33)V(∞)<γ2‖w(t)‖22-‖e(t)‖22
which leads to
(34)‖e(t)‖22<γ2‖w(t)‖22.
Using Schur complement we obtain
(35)Σ=[ATY+YTA+ρ̅YTBYTDBTY-μI0DTY0-γ2I]<0.
Let Y=XE+𝔼⊥TQ̅ with
(36)Q̅=[Q1Q2],X=[X1X2X2TX3],
we obtain
(37)ETY=YTE⩾0.
And by substituting 𝔼,𝔸,𝔹 and 𝔻 by their values we obtain (23), which proves the lemma.Remark 9.
The results of Lemmas6 and 8 are independent of the choice of matrix 𝔼⊥; this can be seen from the fact that the general form of 𝔼⊥ is 𝔼⊥=[0M], where M is an arbitrary nonsingular matrix; in this case we have 𝔼⊥TQ̅=[0I][Q̅1Q̅2], where Q̅1=MQ1 and Q̅2=M, which shows that it suffices to choose 𝔼⊥=[0I].
Before giving the design method for the observer (5), let us consider (8) and let T̃=T+ΨΦ, where Ψ is an arbitrary matrix of appropriate dimension; they can be written as
(38)[NΨJ][T̃EΦAC]=T̃A,(39)[PQF][T̃EΦAC]=In.
Equations (38) and (39) have a solution if and only if
(40)rank⁡[T̃EΦACT̃AIn]=rank⁡[T̃EΦAC]=n.
Now, from Assumption 1 and (40), we have
(41)rank⁡[T̃EΦAC]=[EΦAC]=n.
Let R be any full row rank matrix such that
(42)rank⁡[RΦAC]=rank⁡[EΦAC]=rank⁡[T̃EΦAC]=n,
then there always exist matrices parameter K and T̃ such that
(43)T̃E=R-K[ΦAC]
or equivalently
(44)[T̃K][EΦAC]=R.
Then, under Assumption 1, there exists a solution to (43) given by
(45)[T̃K]=R[EΦAC]†.
In this case matrices T̃ and K are given by
(46)T̃=R[EΦAC]†[I0],K=R[EΦAC]†[0I].
Also, under Assumption 1, the general solution to (38) is given by
(47)[NΨJ]=T̃AΩ†-Z1(I-ΩΩ†),
here Ω=[T̃EΦAC] and Z1 is an arbitrary matrix of appropriate dimension.
And also under Assumption1 one solution to (39) is given by
(48)[PQF]=InΩ†.
Now, define the following matrices:
(49)ΛP=Ω†[I00],ΛQ=Ω†[0I0],ΛN=T̃A[I00],ΛΨ=T̃AΛQ,ΛJ=T̃AΛF,ΔN=(I-ΩΩ†)[I00],ΔΨ=(I-ΩΩ†)[0I0],ΔJ=(I-ΩΩ†)[00I],ΛF=Ω†[00I],
then we obtain
(50)N=ΛN-Z1ΔN,Ψ=ΛΨ-Z1ΔΨ,J=ΛJ-Z1ΔJ,P=ΛP,Q=ΛQ,F=ΛF,φ1=Λφ1-Z1Δφ1,φ2=Λφ2,TD=ΛTD-Z1ΔTD,
where
(51)Λφ1=(ΛJD2-T̃D1+ΛΨΦD1),Δφ1=(ΔJD2+ΔΨΦD1),ΔTD=-ΔΨΦD,Λφ2=(ΛQΦD1+ΛFD2),ΛTD=T̃D-ΛΨΦD.
Now, theH∞ observer design can be obtained from the following theorem.Theorem 10.
There exists an observer of the form (5) such that the error e(t) given by (10)-(11) is asymptotically stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0 if there exist a symmetric positive definite matrix X1 and matrices X2,ΩX1 and scalars μ and γ such that the following LMIs are satisfied:
(52)[(1,1)(1,2)(1,3)(1,4)(1,2)T(2,2)(2,3)(2,4)(1,3)T(2,3)T-μI0(1,4)T(2,4)T0-γ2I]<0,
where
(53)(1,1)=ΛNTX1-ΔNTΩX1+X1ΛN-ΩX1TΔN+X2ΛP+ΛPTX2T+ΛPTQ1+Q1TΛP,(1,2)=-X2-Q1T+ΛPTQ2,(1,3)=X1ΛTD-ΩX1TΔTD-Q1TΛQΦD-X2ΛQΦD,(1,4)=X1Λφ1-ΩX1TΔφ1+Q1TΛφ1+X2Λφ2,(2,2)=-Q2-Q2T+I+μλ2FLTFL,(2,3)=-Q2TΛQΦD,(2,4)=Q2TΛφ2.Proof.
LetΩX1=Z1TX1, and by substituting N,P,Q,φ1, and φ2 by their values in (23) we obtain (52).Procedure for the Observers Design
(1)
Under Assumption1, compute matrices ΛN, ΔN, ΛJ, ΔJ, ΛP, ΛQ,ΛF, ΛΨ, ΔΨ. Λφ1, Δφ1, Λφ2, ΛTD, ΔTD.(2)
Solve the LMI (52) to obtain the matrix parameter Z1.(3)
Compute the observers parametersN,J,T,H,P,Q, and F.


## 3.1. Stability Analysis

Before giving theH∞ observers design method for system (1)-(2), let us deal with the stability analysis problem and derive a sufficient condition, in a strict LMI form, for system (12) to be asymptotically stable for w(t)=0. The following lemma gives this condition.Lemma 5.
Forw(t)=0, system (12) is asymptotically stable, if there exists a matrix Y, such that the following LMIs are satisfied:
(13)ETY=YTE⩾0,(14)[ATY+YTA+μρYTBBTY-μI]<0
with ρ=[000λ12FLTFL].Proof.
For stability analysis we construct the following Lyapunov candidate function:(15)V(t)=ξT(t)ETYξ(t)=ξT(t)YTEξ(t)
with 𝔼TY=YT𝔼⩾0. The derivative of V(t) along the solution of (12), for w(t)=0, is given by
(16)V̇(t)=ξ̇T(t)ETYξ(t)+ξT(t)YTEξ̇(t)=(Aξ(t)+BΔf)TYξ(t)+ξT(t)YT(Aξ(t)+BΔf)=ξT(t)(ATY+YTA)ξ(t)+ΔfTBTYξ(t)+ξT(t)YTBΔf.
Let u and v be two vectors of appropriate dimensions, then for all scalar μ>0 the following inequality holds:
(17)uTv+vTu⩽μuTu+1μvTv.
By using (17), it is not difficult to check that
(18)ΔfT(B)TYε(t)+εT(t)YT(B)Δf⩽1μεT(t)YTBBTYε(t)+μΔFTΔf.
In this case we have V̇(t)⩽ξT(t)(𝔸TY+YT𝔸)ξ(t)+μΔfTΔf+(1/μ)ξT(t)YT𝔹𝔹TYξ(t). On the other hand, by using the Lipschitz conditions and the fact that e(t)=[0I]ξ(t), we have μΔfTΔf⩽μξT(t)ρξ(t), where ρ=[000λ12FLTFL]. Then we obtain the following inequality:
(19)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+ξT(t)μρξ(t)+1μξT(t)YTBBTYξ(t)=ξT(t)(ATY+YTA+1μYTBTBY+μρ)ξ(t)
and V̇(t)<0, if 𝔸TY+YT𝔸+(1/μ)YT𝔹𝔹TY+μρ<0.
Using the Schur complement we obtain (14), then Lemma 5 is proved.From this lemma, one can see that the stability conditions (13) and (14) are nonstrict LMIs, which contain equality constraints; this may result in numerical problems when checking such conditions. Therefore, strict LMIs conditions are more desirable than nonstrict ones from the numerical point of view. The following lemma presents the stability conditions in a strict LMI formulation.Lemma 6.
Forw(t)=0, system (12) is asymptotically stable if there exist a positive definite matrix X1 and matrices X2,Q1, and Q2 and a scalar μ such that the following LMI is satisfied:(20)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2X1TTD-Q1TQΦD-X2QΦD*-Q2-Q2T+μλ2FLTFL-Q2TQΦD**-μI]<0.Proof.
LetY=X𝔼+𝔼⊥TQ̅, where 𝔼⊥ is the orthogonal matrix of 𝔼 satisfying 𝔼⊥𝔼=0 and 𝔼⊥𝔼⊥T>0, and X=[X1X2X2TX3], and Q̅=[Q1Q2]. It is easy to see that 𝔼TY=YTE⩾0,since X1>0.In this case (13)-(14) reduce to(21)[ATXE+ATE⊥TQ̅+ETXTA+Q̅TE⊥AETXTB+Q̅TE⊥BBTXE+BTE⊥TQ̅-μI]<0.On the other hand by inserting the values of 𝔼,𝔸,𝔹, and ρ into (21) we obtain (20).Remark 7.
ForΦD=0,inequality (20) becomes(22)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2X1TTD-X2T-Q1+Q2TP-Q2-Q2T+μλ12FLTFL0(TD)TX10-μI]<0.
By premultiplying (22) by [IPT000I] and postmultiplying it by [I0P00I], we obtain [NTX1+X1TN+μλ1FLTFLX1TTD(TD)TX1-μI]<0, which is exactly the inequality (15) of [16].


## 3.2.H∞ Observes Design

In this section we shall present theH∞ observer design. The following lemma gives the sufficient conditions for system (12) to be stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0.Lemma 8.
The errore(t) given by (11) is asymptotically stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0,  if there exist a positive definite matrix X1 and matrices X2,Q1,Q2 and  scalars μ and γ such that the following LMI is satisfied:(23)[NTX1+X1TN+X2P+PTX2T+PTQ1+Q1TP-X2-Q1T+PTQ2(X1TTD-Q1TQΦD-X2QΦD)(X1Tφ1+X2φ2+Q1Tφ2)*-Q2-Q2T+I+μλ2FLTFL-Q2TQΦDQ2Tφ2**-μI0***-γ2I]<0.Proof.
Consider the following Lyapunov candidate functionV(t)=ξt𝔼TYξ=ξT(t)YT𝔼ξ(t),with 𝔼TY=YT𝔼.Then we have(24)V̇(t)=ξ̇T(t)ETYξ(t)+ξT(t)YTEξ̇(t)=(Aξ(t)+BΔf+Dw(t))TYξ(t)+ξ(t)TYT(Aξ(t)+BΔf+Dw(t))=ξT(t)(ATξ+YTA)ξ(t)+ΔfTBTYξ(t)+ξ(t)TYTBΔf+wT(t)DTYξ(t)+ξ(t)TYTDw(t).
From (17) we have the following inequality:
(25)ΔfT(B)TYε(t)+εT(t)YT(B)Δf⩽1μεT(t)YTBBTYε(t)+μΔfTΔf.
Then V̇(t) becomes
(26)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+μΔfTΔf+1μξT(t)YTBBTYξ(t)+wT(t)DTYξ(t)+ξT(t)YTDw(t).
On the other hand, we haveΔfTΔf⩽λ2eT(t)FLTFLe(t), and we know that e(t)=[0I]ξ(t) then ΔfTΔf⩽ξT(t)ρξ(t), where ρ=[000λ2FLTFL].Then
(27)V̇(t)⩽ξT(t)(ATY+YTA)ξ(t)+μλ2ξT(t)ρξ(t)+1μξTvYTBBTYξ(t)+wTDTYξ(t)+ξT(t)YTDw(t).
Let η(t)=[ε(t)w(t)], then from (27) and (12) we obtain the following inequality:
(28)V̇+eT(t)e(t)-γ2wT(t)w(t)⩽ηT(t)Ση(t)
with
(29)Σ=[ATY+YTA+1μYTBBTY+ρ̅YTDDTY-γ2I],
where ρ̅=[000I+μλ2FLTFL].
IfΣ<0, we obtain
(30)V̇<γ2wT(t)w-eT(t)e(t).
Integrating the two sides of this inequality gives
(31)∫0∞V̇(τ)dτ<∫0∞γ2wT(τ)w(τ)dτ-∫0∞eT(τ)e(τ)dτ
or equivalently
(32)V(∞)-V(0)<γ2‖w(t)‖22-‖e(t)‖22,
under zero initial conditions, we obtain
(33)V(∞)<γ2‖w(t)‖22-‖e(t)‖22
which leads to
(34)‖e(t)‖22<γ2‖w(t)‖22.
Using Schur complement we obtain
(35)Σ=[ATY+YTA+ρ̅YTBYTDBTY-μI0DTY0-γ2I]<0.
Let Y=XE+𝔼⊥TQ̅ with
(36)Q̅=[Q1Q2],X=[X1X2X2TX3],
we obtain
(37)ETY=YTE⩾0.
And by substituting 𝔼,𝔸,𝔹 and 𝔻 by their values we obtain (23), which proves the lemma.Remark 9.
The results of Lemmas6 and 8 are independent of the choice of matrix 𝔼⊥; this can be seen from the fact that the general form of 𝔼⊥ is 𝔼⊥=[0M], where M is an arbitrary nonsingular matrix; in this case we have 𝔼⊥TQ̅=[0I][Q̅1Q̅2], where Q̅1=MQ1 and Q̅2=M, which shows that it suffices to choose 𝔼⊥=[0I].
Before giving the design method for the observer (5), let us consider (8) and let T̃=T+ΨΦ, where Ψ is an arbitrary matrix of appropriate dimension; they can be written as
(38)[NΨJ][T̃EΦAC]=T̃A,(39)[PQF][T̃EΦAC]=In.
Equations (38) and (39) have a solution if and only if
(40)rank⁡[T̃EΦACT̃AIn]=rank⁡[T̃EΦAC]=n.
Now, from Assumption 1 and (40), we have
(41)rank⁡[T̃EΦAC]=[EΦAC]=n.
Let R be any full row rank matrix such that
(42)rank⁡[RΦAC]=rank⁡[EΦAC]=rank⁡[T̃EΦAC]=n,
then there always exist matrices parameter K and T̃ such that
(43)T̃E=R-K[ΦAC]
or equivalently
(44)[T̃K][EΦAC]=R.
Then, under Assumption 1, there exists a solution to (43) given by
(45)[T̃K]=R[EΦAC]†.
In this case matrices T̃ and K are given by
(46)T̃=R[EΦAC]†[I0],K=R[EΦAC]†[0I].
Also, under Assumption 1, the general solution to (38) is given by
(47)[NΨJ]=T̃AΩ†-Z1(I-ΩΩ†),
here Ω=[T̃EΦAC] and Z1 is an arbitrary matrix of appropriate dimension.
And also under Assumption1 one solution to (39) is given by
(48)[PQF]=InΩ†.
Now, define the following matrices:
(49)ΛP=Ω†[I00],ΛQ=Ω†[0I0],ΛN=T̃A[I00],ΛΨ=T̃AΛQ,ΛJ=T̃AΛF,ΔN=(I-ΩΩ†)[I00],ΔΨ=(I-ΩΩ†)[0I0],ΔJ=(I-ΩΩ†)[00I],ΛF=Ω†[00I],
then we obtain
(50)N=ΛN-Z1ΔN,Ψ=ΛΨ-Z1ΔΨ,J=ΛJ-Z1ΔJ,P=ΛP,Q=ΛQ,F=ΛF,φ1=Λφ1-Z1Δφ1,φ2=Λφ2,TD=ΛTD-Z1ΔTD,
where
(51)Λφ1=(ΛJD2-T̃D1+ΛΨΦD1),Δφ1=(ΔJD2+ΔΨΦD1),ΔTD=-ΔΨΦD,Λφ2=(ΛQΦD1+ΛFD2),ΛTD=T̃D-ΛΨΦD.
Now, theH∞ observer design can be obtained from the following theorem.Theorem 10.
There exists an observer of the form (5) such that the error e(t) given by (10)-(11) is asymptotically stable for w(t)=0 and ∥e(t)∥L2<γ∥w(t)∥L2 for w(t)≠0 if there exist a symmetric positive definite matrix X1 and matrices X2,ΩX1 and scalars μ and γ such that the following LMIs are satisfied:
(52)[(1,1)(1,2)(1,3)(1,4)(1,2)T(2,2)(2,3)(2,4)(1,3)T(2,3)T-μI0(1,4)T(2,4)T0-γ2I]<0,
where
(53)(1,1)=ΛNTX1-ΔNTΩX1+X1ΛN-ΩX1TΔN+X2ΛP+ΛPTX2T+ΛPTQ1+Q1TΛP,(1,2)=-X2-Q1T+ΛPTQ2,(1,3)=X1ΛTD-ΩX1TΔTD-Q1TΛQΦD-X2ΛQΦD,(1,4)=X1Λφ1-ΩX1TΔφ1+Q1TΛφ1+X2Λφ2,(2,2)=-Q2-Q2T+I+μλ2FLTFL,(2,3)=-Q2TΛQΦD,(2,4)=Q2TΛφ2.Proof.
LetΩX1=Z1TX1, and by substituting N,P,Q,φ1, and φ2 by their values in (23) we obtain (52).Procedure for the Observers Design
(1)
Under Assumption1, compute matrices ΛN, ΔN, ΛJ, ΔJ, ΛP, ΛQ,ΛF, ΛΨ, ΔΨ. Λφ1, Δφ1, Λφ2, ΛTD, ΔTD.(2)
Solve the LMI (52) to obtain the matrix parameter Z1.(3)
Compute the observers parametersN,J,T,H,P,Q, and F.


## 4. Numerical Example

Let us consider the following continuous nonlinear singular system of the form (1) with (54)E=[010001000],A=[100010101],B=[111],D1=[111-111],D=[111],C=[100],D2=[11] and u(t)=sin(2t). The nonlinearity f(x,u,t)=sin(x2(t)). For this system, the matrix Φ=[001]. In this case it is easy to see that Assumption 1 is verified. We will design a reduced-order observer of dimension q=2; let R=[010001], then rank[RΦAC]=3. For γ2=9.762, from Section 3 we obtain the following results: (55)ΩX1=102[-4.1830.5-0.331-0.11-0.5670.276-0.0450.627],X1=[3.919-2.628-2.6285.139].
The H∞ observer is given by the following model:(56)ζ̇(t)=[0-1.9960.666-2.121]ζ(t)+[-0.331-1.414]y(t)+[-0.331-0.747]u(t)+[-0.331-0.747]sin⁡(x̂2(t)),x̂(t)=[0-0.3521000.705]ζ(t)-[0.23500.529]u(t)+[0.7640-0.529]y(t)-[0.23500.529]sin⁡(x̂2(t)).Simulation results are presented in Figures1–3. Figure 1 presents the noises w1(t) and w2(t). Figures 2 and 3 show the estimation of the states x1, x2, and x3. It can be seen that the observer performs as expected.Perturbationw(t).
(a)(b)xi (blue) and x̂i (red).
(a)(b)(c)Errorsei=x̂i-xi.
(a)(b)(c)


## 5. Conclusion

In this paper a new method for theH∞ observers design for a class of Lipschitz nonlinear singular systems has been developed. The obtained results unify the observers design of full, reduced, and minimal orders. Sufficient conditions for the existence of these observers are given in terms of LMIs. The advantage of these LMIs conditions is that they can be performed by using convex optimization techniques available in Matlab LMI toolbox, for example. A numerical example has been presented to show the applicability of our approach. The extension of our work to more general nonlinear singular systems is under study.



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*Source: 967571-2011-10-04.xml*