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

Theproblemof the disturbance-observer-based control for singular hybrid systemwith 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 linearmatrix inequalities (LMIs). Finally, a numerical example is provided to show the effectiveness of the proposed approaches.


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][3][4][5][6][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][9][10][11][12][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.ℓ 2 -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.Section 2 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.

Problem Statement and Preliminaries
Fix a probability space (Ω, F, P), where Ω is the sample space, F is the -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: where () ∈ R  is the semistate vector, () ∈ R  is the control input, () ∈ R  is the output measurement and  1 () ∈ R  is supposed to satisfy conditions described as Assumption where Δ > 0, lim Δ → 0 ((Δ)/Δ) = 0,   ≥ 0 is the transition rate from  at time  to  at time  + Δ, and Assumption 1.The disturbance  1 () can be formulated by the following exogenous system: where   ,   , and   are known matrices with proper dimensions. 3 () is the additional disturbance belonging to The following assumptions are necessary conditions for the disturbance-observer-based control problem.Assumption 2. (,   ,  1 ) is impulse observable [19].Assumption 3. (,   ,   ) is impulse controllable, and (  ,     ) is observable.The free singular system with Markovian switching of (1a) and (1b) with () = 0,  1 () = 0, and  2 () = 0 can be described as  ẋ () =    () . ( We give the following definition for the singular Markovian jump system (4).
In this section, we suppose that all of the states in (1a), (1b), and (3) are unavailable.Then, we need to estimate () and (), 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 .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: with The full-order observer for both () and () is designed as , and   is the observer gain to be determined. Define as the estimation error.

Main Results
Under Assumptions 2 and 3, suppose that   and   are given, and we first present the bounded real lemma for the composite system in ( 11) and ( 13) in terms of LMIs.
Let A be the weak infinitesimal generator of the random process {(),   }.Then, for each   = ,  ∈ S, it can be shown that with () ≜ [  ()   ()] and Consider the following index: Then, under the zero initial conditions, it follows from ( 13) and ( 18 with Based on (15b), we can derive () ≤ 0 by taking (21) into account.Thus, under the zero initial conditions and for any nonzero () ∈ L 2 (0, ∞), letting  → ∞, we obtain ‖()‖ 2 ≤ ‖()‖ 2 .The proof is completed.Now, we are in a position to present a solution to the composite DOBC and H ∞ control problem formulated in this section.Theorem 6.Consider system (1a) and (1b) with the disturbance (3) under Assumptions 2 and 3. Given parameters  1 > 0,  2 > 0, and  > 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   > 0, matrices  1 > 0,  2 ,   > 0,   , and   such that for  = 1, 2, . . ., N, where   is defined in (15a) and (15b) and (24) 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 Proof.Define with  1 > 0.
Substituting Ã , H defined in (11), C defined in (13), and   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  if (23c) and the following equalities and inequalities hold: where  2 is defined in (25) and Note that Considering (27b) and (29) and using Schur complement, we can see that if the following equalities (30) hold, then (27b) holds as follows: with  ≜ diag{ 1 , }.Define Performing a congruence transformation to (30) by diag{  , , , , , , }, we obtain Performing a congruence transformation to (23a) and (23b) by  −1  , respectively, we can readily get (27a) and By Schur complement to (32) and based on (33), we can conclude that if the following inequalities (34) hold, then (32) holds as follows: By using the fact that 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] Now, similar to Section 3, we consider the following case: system (1a) and (1b) under Assumptions 1-3 is without jumping parameters (that is  = 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 Ẽ η () = Ã () + H () , (39a) with Corollary 8. Consider system (1a) and (1b) under Assumptions 1-3 without jumping parameters.Given parameters  1 > 0,  2 > 0, and  > 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  1 > 0,  2 ,  > 0, , and  such that min trace { 1 } , ..(42a) , (42b) , (42c) , (42d) (41) 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 Remark 9. To the best of the authors' knowledge, this is also the first time that the full-order disturbance-observerbased control strategy is applied in the singular system with multiple disturbances.

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) The transition probability matrix is assumed to be Π = [ −0.5 0.5 1.0 −1.0 ], and  is set to be  = 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 ] . (46)

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.