# Full-Order Disturbance-Observer-Based Control for Singular Hybrid System

Authors: Xiuming Yao; Ze Dong; Dongfeng Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The problem of the disturbance-observer-based control for singular hybrid system with two types of disturbances is addressed in this paper. Under the assumption that the system states are, unavailable, full-order observers (for both system states and the disturbance) and a nonlinear control scheme are constructed, such that the composite system can be guaranteed to be stochastically admissible, and the two types of disturbances can be attenuated and rejected, simultaneously. Based on the Lyapunov stability theory, sufficient conditions for the existence of the desired full-order disturbance-observer-based controllers are established in terms of linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the effectiveness of the proposed approaches.


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## Body

## 1. Introduction

Singular systems, which are also referred to as implicit systems, descriptor systems, are widely used to model various engineering systems, such as electrical networks, power systems, networked control systems, and robotics, due to the fact that such systems can provide a more general representation than standard state-space systems in the sense of modeling [1]. A great number of fundamental results based on the theory of state-space systems have been successfully extended to singular systems. For some fundamental work on this subject, we refer the reader to [2–7].Disturbance-observer-based control has been proven to be an effective strategy to reject the disturbance which can be modeled by an exogenous system [8–13]. Recently, its applications have been found in the robotic systems [8], table drive systems [12], missile system [11], and so on. The essential idea of the disturbance-observer-based control scheme is to design a disturbance observer to estimate the matched disturbance and cancel the effect of the matched disturbance by applying the estimation information into the control law. On another research front line, singular Markovian jump system which includes the dynamics of both singular system and Markovian jump system has attracted great attention from researchers, and recently some results are available in the publication: sliding control problem for continuous Markovian jump singular system is investigated in [14, 15], where the necessary and sufficient condition for the admissibility of the nominal system is presented. ℓ2-ℓ∞ filter problem is designed for discrete-time singular Markovian jump systems in [16]. Notice that in the above-mentioned publications, the disturbance considered in the plant has been assumed to be norm-bounded one. In this paper, we will consider a wider case: the plant is subject to multiple disturbances (one is norm-bounded disturbance, and the other is the disturbance that can be modeled by the exogenous system).Based on the previous reasons, in this paper, we will investigate the disturbance-observer-based control problem for a class of singular systems with Markovian switching parameters and multiple disturbances. With the proposed nonlinear control scheme and by choosing a proper stochastic Lyapunov-Krasovskii functional, sufficient conditions for the existence of the desired controllers in terms of LMIs [17, 18] are presented, such that the composite system is stochastically admissible and meets certain performance requirements. Finally, a numerical example is used to illustrate the efficiency of the developed results.The remainder of this paper is organized as follows. Section2 describes the problem and preliminaries. Section 3 presents the main theoretical results. A numerical example is given in Section 4. Finally, we conclude the paper in Section 5.


## 2. Problem Statement and Preliminaries

Fix a probability space(Ω,ℱ,𝒫), where Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. Under this probability space, we consider the following singular MJLs:(1a)Ex˙(t)=A(rt)x(t)+G(rt)[u(t)+d1(t)]+H(rt)d2(t),(1b)y(t)=D1(rt)x(t)+D2(rt)d1(t),where x(t)∈ℝn is the semistate vector, u(t)∈ℝm is the control input, y(t)∈ℝs is the output measurement and d1(t)∈ℝm is supposed to satisfy conditions described as Assumption 1, which can represent the constant and harmonic noises. d2(t)∈ℝq is another disturbance which is assumed to be an arbitrary signal in ℒ2[0,∞). The matrix E∈ℝn×n is singular with rank(E)=r<n, and the matrices Ai≜A(rt=i),  Gi≜G(rt=i),  Hi≜H(rt=i),  D1i≜D1(rt=i), and D2i≜D2(rt=i) are known real constant matrices of appropriate dimensions. {rt} is a continuous-time Markov process with right continuous trajectories and taking values in a finite set 𝒮={1,2,…,𝒩} with transition probability matrix Π≜{πij} given by
(2)Pr{rt+Δ=j∣rt=i}={πijΔ+o(Δ)ifj≠i,1+πiiΔ+o(Δ)ifj=i,
where Δ>0,  limΔ→0(o(Δ)/Δ)=0,  πij≥0 is the transition rate from i at time t to j at time t+Δ, and πii=-∑j=1,j≠i𝒩‍πij.Assumption 1.
The disturbanced1(t) can be formulated by the following exogenous system:
(3)ω˙(t)=Wiω(t)+Mid3(t),d1(t)=Viω(t),
where Wi, Mi, and Vi are known matrices with proper dimensions. d3(t) is the additional disturbance belonging to ℒ2[0,∞).The following assumptions are necessary conditions for the disturbance-observer-based control problem.Assumption 2.
(
E
,
A
i
,
D
1
i
) is impulse observable [19].Assumption 3.
(
E
,
A
i
,
G
i
) is impulse controllable, and (Wi,GiVi) is observable.The free singular system with Markovian switching of (1a) and (1b) with u(t)=0, d1(t)=0, and d2(t)=0 can be described as
(4)Ex˙(t)=Aix(t).
We give the following definition for the singular Markovian jump system (4).Definition 4 (Dai [2]).
The singular Markovian jump system (4) or the pair (E,Ai) is said to be(i)
regular if, for eachi∈𝒮, det(sE-Ai) is not identically zero;(ii)
impulse-free if, for eachi∈𝒮, deg(det(sE-Ai))=rank(E);(iii)
stochastically admissible if it is regular, impulse-free, and stochastically stable.In this section, we suppose that all of the states in (1a), (1b), and (3) are unavailable. Then, we need to estimate x(t) and ω(t), respectively. Here, we construct full-order observers for the whole states, and then based on the estimated states, we design a composite controller such that the resulting composite system is stochastically admissible with ℋ∞ performance γ. For this purpose, Assumptions 2 and 3 are needed.By augmenting the states of the system (1a) and (1b) by the disturbance dynamics (3), we obtain the following augmented model:(5a)E-ξ˙(t)=A-i(t)ξ(t)+H-id(t)+G-iu(t),(5b)y(t)=D-iξ(t)with ξ(t)≜[x(t)Tω(t)T]T, d(t)≜[d2(t)Td3(t)T]T, and
(6)A-i≜[AiGiVi0Wi],E-≜[E00I],H-i≜[Hi00Mi],D-i≜[D1iD2iVi].
The full-order observer for both x(t) and ω(t) is designed as(7a)E-ξ^˙(t)=A-iξ^(t)+G-iu(t)+Li(y^(t)-y(t)),(7b)y^(t)=D-iξ^(t)with ξ^(t)≜[x^(t)Tω^(t)T]T, and Li is the observer gain to be determined.Define(8)e(t)≜ξ(t)-ξ^(t)≜[ex(t)eω(t)]≜[x(t)-x^(t)ω(t)-ω^(t)]
as the estimation error.Based on (5a), (5b), (7a), and (7b), we obtain the estimation error dynamics as follows:
(9)E-e˙(t)=(A-i+LiD-i)e(t)+H-id(t).
In the DOBC scheme, the control can be constructed as
(10)u(t)=-d^1(t)+Kix^(t),
where d^1(t)≜Viω^(t) is the estimation of d1(t) and Ki is the controller gain. Combining the estimation error equation (9) with system (1a) and (1b) yields
(11)E~η˙(t)=A~iη(t)+H~id(t)
with η(t)≜[x(t)Te(t)T]T and
(12)E~≜[E00E-],A~i≜[Ai+GiKiB-i0A-i+LiD-i],H~i≜[Hi00H-i],B-i≜[-GiKiGiVi],
where E-, A-i, F-i, H-i, and D-i are defined in (5a) and (5b).The reference output is set to be(13)z(t)=Cix(t)≜C~iη(t)
with C~i≜[Ci0].Therefore, the disturbance-observer-based control problem based on full-order observer (7a) and (7b) for system (1a) and (1b) with (3) can be formulated as follows.Disturbance-Observer-Based Control Problem. Given the Markovian jump singular system (1a) and (1b) with (3), design full-order observer of the form (7a) and (7b) and controller of the form (10) such that the following requirements are satisfied:(R1)
the composite system in (11) and (13) with d(t)=0 is stochastically admissible;(R2)
under the zero initial conditions, the following inequality holds:(14)∥z(t)∥2<γ∥d(t)∥2for all nonzero d(t)∈ℒ2[0,∞), where γ>0 is a prescribed scalar and ∥z(t)∥2=∫0∞‍zT(t)z(t)dt.


## 3. Main Results

Under Assumptions2 and 3, suppose that Ki and Li are given, and we first present the bounded real lemma for the composite system in (11) and (13) in terms of LMIs.Lemma 5.
Given the controller gainsKi, the observer gains Li, parameters λ1i>0,  λ2i>0, and γ>0, the composite system in (11) and (13) is stochastically admissible and satisfies the ℋ∞ performance inequalities (14) if there exist matrices Pi such that for i=1,2,…,𝒩,(15a)E~TPi=PiTE~≥0,(15b)[ΞiPiTH~i⋆-γ2I]<0with
(16)Ξi≜E~TP-i+PiTA~i+A~iTPi+C~iTC~i,P-i≜∑j=1𝒩‍πijPj.Proof.
Define a Lyapunov functional candidate as follows:(17)V(η(t),rt,t)≜ηT(t)E~TPiη(t).
Let𝒜 be the weak infinitesimal generator of the random process {ξ(t),rt}. Then, for each rt=i, i∈𝒮, it can be shown that
(18)𝒜V(η(t),i,t)=2ηT(t)PiT(A~iη(t)+H~id(t))+ηT(t)E~TP-iη(t)=ζT(t)[Ξ-iPiTH~i⋆0]ζ(t)
with ζ(t)≜[ηT(t)dT(t)]T and
(19)Ξ-i≜E~TP-i+PiTA~i+A~iTPi.
Consider the following index:(20)J(T)≜E{∫0T‍[zT(t)z(t)-γ2dT(t)d(t)]dt}.
Then, under the zero initial conditions, it follows from (13) and (18) that
(21)J(T)=E{∫0T‍[zT(t)z(t)-γ2dT(t)d(t)]dt}+EV(η(T),i,T)=E{∫0T‍[zT(t)z(t)-γ2dT(t)d(t)+𝒜V(η(t),rt=i)]dt}=E{∫0T‍ζT(t)Ωiζ(t)dt}
with
(22)Ωi≜[Ξ-i+C~iTC~iPiTH-i⋆-γ2I].
Based on (15b), we can derive J(T)≤0 by taking (21) into account. Thus, under the zero initial conditions and for any nonzero d(t)∈ℒ2(0,∞), letting T→∞, we obtain ∥z(t)∥2≤γ∥d(t)∥2. The proof is completed.Now, we are in a position to present a solution to thecomposite DOBC andℋ∞control problem formulated in this section.Theorem 6.
Consider system (1a) and (1b) with the disturbance (3) under Assumptions 2 and 3. Given parameters λ1i>0, λ2i>0, and γ>0, there exists a full-order observer in the form of (7a) and (7b) and there exists a controller in the form of (10) such that the augmented system in (11) and (13) is stochastically admissible and satisfies the ℋ∞ performance inequalities (14) if there exist parameters αi>0, matrices P1i>0, P2i, Qi>0, Xi, and Yi such that for i=1,2,…,𝒩,(23a)QiET=EQi≥0,(23b)QiET≤αiI,(23c)E-TP2i=P2iTE-≥0,(23d)[Π1i0Hi0QiCiTGiXiGiViWi⋆Π2iF-i0P2iTH-i0000⋆⋆000000⋆⋆000000⋆⋆-γ2I00000⋆⋆⋆-γ2I0000⋆⋆⋆⋆0000⋆⋆⋆⋆-I000⋆⋆⋆⋆⋆-Qi00⋆⋆⋆⋆⋆⋆-I0⋆⋆⋆⋆⋆⋆⋆-Λi]<0,(23e)P1iQi=I,where λi is defined in (15a) and (15b) and
(24)Wi≜[πi1Qi⋯πijQi⋯πi𝒩QiT]j≠i,Λi≜diag{2Q1-α1I,…,2Qj-αjI,…,2Q𝒩-α𝒩I}j≠i,Π1i≜AiQi+GiXi+(AiQi+GiXi)T+πiiQiET,Π2i≜P2iTA-i+YiD-i+(P2iTA-i+YiD-i)T+E-TP-2i+diag{P1i,I},P-2i≜∑j=1𝒩‍πijP2j.
Moreover, if the previous conditions are feasible, the gains of the desired observer in the form of (7a) and (7b) and the desired controller in the form of (10) are given by
(25)Ki=XiQi-1,Li=P2i-TYi.Proof.
Define(26)Pi≜[P1i00P2i]
with P1i>0.
SubstitutingA~i, H~i defined in (11), C~i defined in (13), and Pi defined in (26) into (15a) and (15b) of Lemma 5 and based on the process of the proof of Lemma 5, we can draw a conclusion that the system in (11) and (13) is stochastically admissible with ℋ∞ performance γ if (23c) and the following equalities and inequalities hold:(27a)ETP1i=P1iE≥0,(27b)ζT(t)[Γ1i0P1iHi0⋆Γ3i0P2iTH-i⋆⋆-γ2I0⋆⋆⋆-γ2I]ζ(t)+2xT(t)Γ2ie(t)<0,where P-2i is defined in (25) and
(28)Γ1i≜P1i(Ai+GiKi)+(Ai+GiKi)TP1i+ETP-1i+CiTCi,Γ2i≜P1i[-GiKiGiVi],Γ3i≜P2iT(A-i+LiD-i)+(A-i+LiD-i)TP2i+E-TP-2i,P-1i≜∑j=1𝒩‍πijP1j.
Note that
(29)2xT(t)Γ2ie(t)=-2xT(t)P1iTGiKiex(t)+2xT(t)P1iTGiVieω(t)≤xT(t)P1iGiKiP1i-1KiTGiTP1ix(t)+exT(t)P1iex(t)+xT(t)P1iGiViViTGiTP1ix(t)+eωT(t)eω(t).
Considering (27b) and (29) and using Schur complement, we can see that if the following equalities (30) hold, then (27b) holds as follows:
(30)[Γ1i0P1iHi0CiTP1iGiKiP1i-1P1iGiVi⋆Γ3i+R0P2iTH-i000⋆⋆-γ2I0000⋆⋆⋆-γ2I000⋆⋆⋆⋆-I00⋆⋆⋆⋆⋆-P1i-10⋆⋆⋆⋆⋆⋆-I]<0
with R≜diag{P1i,I}.
Define(31)Qi≜P1i-1,Xi≜KiP1i-1,Yi≜P2iLi.
Performing a congruence transformation to (30) by diag{Qi,I,I,I,I,I,I}, we obtain
(32)[Π-1i0Hi0QiCiTGiXiGiVi⋆Π2i0P2iTH-i000⋆⋆-γ2I0000⋆⋆⋆-γ2I000⋆⋆⋆⋆-I00⋆⋆⋆⋆⋆-Qi0⋆⋆⋆⋆⋆⋆-I]<0
with Π-1i≜AiQi+GiXi+(AiQi+GiXi)T+πiiQiET+∑j=1,j≠i𝒩‍πijQiETQj-1Qi.
Performing a congruence transformation to (23a) and (23b) by Qi-1, respectively, we can readily get (27a) and
(33)ETQi-1≤αiQi-1Qi-1.
By Schur complement to (32) and based on (33), we can conclude that if the following inequalities (34) hold, then (32) holds as follows:
(34)[Π~1i0Fi0Hi0QiUiTQiCiTGiXiGiVi⋆Π2i0P2iTF-i0P2iTH-i0000⋆⋆-λiI0000000⋆⋆⋆-λiI000000⋆⋆⋆⋆-γ2I00000⋆⋆⋆⋆⋆-γ2I0000⋆⋆⋆⋆⋆⋆-λ1i2I000⋆⋆⋆⋆⋆⋆⋆-I00⋆⋆⋆⋆⋆⋆⋆⋆-Qi0⋆⋆⋆⋆⋆⋆⋆⋆⋆-I]
with Π~1i≜AiQi+GiXi+(AiQi+GiXi)T+πiiQiET+∑j=1,j≠i𝒩‍πijQiαjQj-1Qj-1Qi.
By Schur complement to (34), we have
(35)[Π1i0Hi0QiCiTGiXiGiViWi⋆Π2i0P2iTH-i0000⋆⋆-γ2I00000⋆⋆⋆-γ2I0000⋆⋆⋆⋆-I000⋆⋆⋆⋆⋆-Qi00⋆⋆⋆⋆⋆⋆-I0⋆⋆⋆⋆⋆⋆⋆-Λ-i]<0
with
(36)Λ-i≜diag{α1-1Q12,…,αj-1Qj2,…,α𝒩-1Q𝒩2}j≠i.
By using the fact that
(37)αj-1Qj2≥2Qj-αjI,
we can show that if (23d) holds, then (34) holds, and thus (27b) holds. The proof is completed.Corollary 7.
Note that the conditions (23d) given in Theorem 6 are not strict LMI conditions due to (23e). However, with the result of [20], one can solve these nonconvex feasibility problems by formulating them into some sequential optimization problems subject to LMI constraints. By making the cone complementary linearization (CCL) [20], instead of dealing with the original nonconvex feasibility problem formulated in (23d) of Theorem 6, one may consider solving the following minimization problem involving LMI conditions (23d):
(38)mintrace{∑i=1𝒩QiP1i},s.t.(23a)–(23d)fori=1,2,…,𝒩,[P1iIIQi]≥0.Now, similar to Section3, we consider the following case: system (1a) and (1b) under Assumptions 1–3 is without jumping parameters (that is N=1), and thus the observer in (7a) and (7b) and the controller in (10) are mode independent. For such a case, the composite system in (11) and (13) becomes singular system effectively operating at one of the subsystems all the time, and it can be described by(39a)E~η˙(t)=A~η(t)+H~d(t),(39b)z(t)=C~η(t)with
(40)A~≜[A+GKB-0A-+LD-],A-≜[AGV0W],H~≜[H00H-],B-≜[-GKGV],C~≜[C0],H-≜[H00M],D-≜[D1D2V].Corollary 8.
Consider system (1a) and (1b) under Assumptions 1–3 without jumping parameters. Given parameters λ1>0, λ2>0, and γ>0, there exists a full-order observer in the form of (7a) and (7b) without jumping parameters and there exists a controller in the form of (10) without jumping parameters such that the composite system in (39a) and (39b) is admissible and satisfies the ℋ∞ performance inequalities (14) if there exist matrices P1>0, P2, Q>0, X, and Y such that
(41)mintrace{QP1},s.t.(42a),(42b),(42c),(42d)(42a)QET=EQ≥0,(42b)E-TP2=P2TE-≥0,(42c)[Π10H0QCTGXGV⋆Π20P2TH-000⋆⋆-γ2I0000⋆⋆⋆-γ2I000⋆⋆⋆⋆-I00⋆⋆⋆⋆⋆-Q0⋆⋆⋆⋆⋆⋆-I]<0,(42d)[P1IIQ]≥0with
(43)Π1≜AQ+GX+(AQ+GX)T,Π2≜P2TA-+YD-+(P2TA-+YD-)T+diag{P1,I}.
Moreover, if the previous conditions are feasible, the gains of the desired observer in the form of (7a) and (7b) without jumping parameters and the desired controller in the form of (10) without jumping parameters are given by
(44)K=XQ-1,L=P2-TY.Remark 9.
To the best of the authors' knowledge, this is also the first time that the full-order disturbance-observer-based control strategy is applied in the singular system with multiple disturbances.


## 4. Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed approaches. Consider the systems in (1a), (1b), and (3) with the following parameters:
(45)A1=[-2.21.2-0.9-0.2],G1=[-0.10.1],H1=[0.20.1],E=[1.0000],C1=[0.10.1],D11=[0.1000.1],D21=[0.10.1],W1=[00.2-0.20],V1=[3.00],M1=[0.20.4],A2=[-1.20.50.2-0.8],G2=[0.10.3],C2=[0.20.1],D12=[0.05000.1],D22=[0.10.1],H2=[0.21.0],W2=[00.5-0.50],V2=[1.00],M2=[0.10.3].
The transition probability matrix is assumed to be Π=[-0.50.51.0-1.0], and γ is set to be γ=1. Our intention here is to design reduced-order-observer-based controller in the form of (7a), (7b), and (10), such that the composite system is stochastically admissible and satisfies prescribed ℋ∞ performance. We resort to the LMI Toolbox in MATLAB to solve the problems established in (38), and the gains of the desired observer and controller are given by
(46)K1=[0.5657-9.2561],K2=[-0.0461-3.2981],L1=[-1.70202.8579-4.23662.7401-3.9409-4.4407-5.3114-10.7753],L2=[35.2258-19.4542119.1428-95.2410-22.2709-0.0573-42.1632-0.2238].


## 5. Conclusion

The problem of disturbance-observer-based control for Markovian jump singular systems with multiple disturbance has been studied. Full-order observer- (both disturbance and system states) based controller has been constructed. The explicit expression of the desired disturbance-observer-based controller has also been presented. Finally, the proposed methods have been verified by a numerical example.



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