An Observer-Based Controller Design for a Markovian Jumping Singular System with Polytopic Uncertainties

In this paper, an observer-based controller design problem for a discrete-time Markovian jumping singular system (DMJSS) with polytopic uncertainties is considered, which is based on a set of necessary and sufficient conditions in terms of linear matrix inequalities (LMIs) for an admissibility analysis problem. A proposed mode dependent observer and a controller can be designed separately via two sets of derived LMI-based synthesis conditions. There are two principal contributions in this paper. One is that a set of strictly LMI-based necessary and sufficient conditions for the admissibility analysis problem for a DMJSS is extended to the case with uncertainties. Another one is that a suitable observer and a controller can be designed independently via two sets of strictly LMI-based synthesis conditions instead of a set of complex algorithm in terms of bilinear or nonconvex inequalities.


Introduction
A singular system description can simultaneously take into account a system's dynamics and its algebraic constraints, which is more general than a normal state-space one.The theoretical developments for a singular system are worthwhile and there are more challenges, because the essential consideration for a singular system is admissibility which includes stability, regularity and causality for a discrete-time system (or impulse elimination for a continuous-time system).Consequently, over past decades, many different practical systems and research areas (1)(2)(3)(4)(5) have been adopted the singular system description.Many fundamentals and advanced results for a normal state-space system have been successfully extended to a singular system, such as controllability and observability (6) , (robust) stability analysis and (robust) stabilization (7)(8)(9) , H∞ control and filtering (10)(11)(12) and so forth.In addition, in order to get less conservative results, singular system based methods have been adopted to study state-space systems (13,14) .
On the other hand, when a dynamic system experiences sudden change in its structure or parameters, it can be modelled as a Markovian jumping system (MJS).As a class of important stochastic hybrid systems, MJSs have attracted extensive and sustained research attention in control and filtering problems due to their theoretical importance and wide practical applications in manufacturing systems, economic systems, power systems, aerospace systems, and networked control systems (15,16) .Following the researching trends of a state-space system, in recent years, more notions and results in a normal MJS have been successfully extended to a singular Markovian jumping system, for example, stability and stabilization (17,18) , H∞ control and filtering problems (18,19) and so on.It should be pointed out that a discrete-time Markovian jumping singular system is more important than its continuous-time counterpart in our digital world.
For control systems, including singular and Markovian jumping systems, it has been recognized that the state variables are not often available or measurable due to technical or economical constraints in most practical situations.Therefore, a controller design that does not require complete access to the state vector is desirable.That is why observer-based control problems and observer design problems, which are involved with using the available information on inputs and outputs to reconstruct the unmeasured states of the concerned system, have been wide investigated in many practical applications (20)(21)(22) .However, only a few works have been carried out on singular systems (23,24) , not to mention on the ones simultaneously considering Markovian jumping cases (25,26) .Up to date and to the best of our knowledge, the observer-based control problem for a discrete-time Markovian jumping singular system (DMJSS) does not have clear investigation and still remains challenges.This observation motivates our studies.In this paper, we extend our recent results (27) consider a mode dependent observer and controller co-design problem for a DMJSS with polytopic uncertainties.The contributions of this paper are as follows.(a) We generalize problem formulation to a system with polytopic uncertainties and extend the resultant LMI-based conditions (27) for an admissibility analysis problem.(b) We unify a method to deal with two design tasks, and derive two sets of strictly LMI-based synthesis conditions for an observer and a controller.
The rest of this paper is organized as follows.Section 2 presents some preliminaries and problem description.Section 3 gives the main results.Section 4 provides a numerical example to show the feasibility of the proposed method, and Section 5 concludes this paper.Throughout this paper, the following notations are adopted.Matrices are assumed to have compatible dimensions for algebraic operations if their dimensions are not explicitly stated.For any matrix , sym() =  +  T , T is a transpose of a matrix. >  (< ) means that the symmetric matrix P is positive definite (negative definite).ℤ represents the set of non-negative integer numbers, and {⋅} denotes a block diagonal matrix.I and 0 are identity and zero matrices, respectively, with appropriate dimensions.

Preliminaries and Problem Formulation
Consider an uncertain discrete-time Markovian jumping singular system (DMJSS), whose plant is where  ∈ ℤ, () ∈ ℝ  is the state vector, () ∈ ℝ  is the input vector, and () ∈ ℝ  is the measured output vector.{(),  ∈ ℤ} is a discrete-time, discrete-state Markovian chain taking values in a finite set  = {1,2, … , }, with the following given transition probability from mode i at time k to mode j at time k+1,  ∈ ℤ : for ,  ∈  , and ∑    =1 = 1 .The matrix  ∈ ℝ × is singular with () =  < .Other system matrices are supposed to be uncertain but belong to a known convex compact set of polytopic type: ( ) where , , Similar to the definition (7) related to the admissibility for an unforced DMJSS, we can define the same admissibility for an uncertain unforced DMJSS in the following.
Similar to the Definition 2.1 (28) related to R-controllable and controllable (R-observable and observable) for a discrete-time singular system, we can define the same controllability/observability for DMJSS in (1), and suppose the system Σ  is controllable and observable.
Definition 2. The system and a control law for Σ  in (1), where  () and  () are observer gains and controller gains in connection with different operation modes to be designed.The corresponding augmented closedloop dynamics Σ  can be formulated as where The purpose of this paper is to design a set of suitable observer gains and controller gains in ( 5) and ( 6), so that the augmented closed-loop system in ( 7)-( 8) is stochastically admissible.In the following, we also denote  () ,  () as   and   to simplify notations for each () = ,  ∈ .
For our analysis and design tasks, the following three lemmas are required.Lemma 1 which is an extension of Lemma 3 (27) is essential for our resultant conditions related to the corresponding analysis problems.Lemma 2 which is an extension of Lemma 4 (27) enables us to consider the observer design and the controller design problems independently.Lemma 3 provides us a key idea so that we can unify our methods for the design tasks. Lemma and matrix  is a given full-column-rank matrix satisfying  = ;  T = [  − ] 0 −T where  0 is nonsingular satisfying  0  0 = {  , }.Lemma 2. Suppose both singular systems (,   ) and (,   ) are admissible, where ,   ,   ∈ ℝ × and () =  <  .Then the augmented singular system ( � ,  � ) is admissible, where ( , ) , , and   is a parameter matrix for the augmented system.
Proof.It is obvious that these two systems are restricted system equivalence (1) .Lemma 3 is straightforward according to the property that singular systems with restricted system equivalence have same admissibility (29) .

Stochastic Admissibility Analysis
Along with Lemma 1 and the convex properties of polyhedral domain, we can easily modified Lemma 1 to be Theorem 1 which could suitably apply to analyze the augmented system Σ  with polytopic uncertainties.

Controller Design and Observer Design
According to Lemma 2, we can separately consider two subsystems (,    +      ) and (,    −      ) in ( 7)-( 8) to design a set of feasible observer gains and controller gains instead of using augmented system Σ  directly.In the following, Theorem 2 and Theorem 3 respectively provide a set of necessary and sufficient strictly LMI-based conditions for the controller design and observer design problems.( ) where =    T +   , and matrices , ,  () ,   are defined in Lemma 1.If in this case, the controller gains in (6) can be found by 1 .( ) �  =  �   +  ̅  �  , and matrix  ̅ is a given full-column-rank matrix satisfying  T  ̅ = ;  � = [  − ] 0 −T where  0 is nonsingular satisfying  0  0 = {  , }.If in this case, the observer gains in (5) can be found by T T .
Sketch of Proof.We can easily derive ( 16)-( 17) from ( 13)-( 14) via replacing the quadruple �,    ,    ,   � with ( T ,   T , −  T ,   T ) and modifying related variables and parameter matrices.Remark 2. Comparing with our recent results (27) , there are two advantages and differences in this paper.One is that the assumption that C must be full-row-rank is unnecessary and is removed in this paper, which fits more practical cases.Another is that the resultant conditions are extended to be feasible for uncertain systems.In addition, we unify one method for these two design tasks.

A Numerical Example
Consider a two-mode jumping system (1) with the following matrices:

A B C A B C E
There are two vertexes in every operation mode for this system.The transition probability matrix is supposed to be Obviously, () = 2.The jumping system is observable and controllable according to Definition 2, but it is not stochastically admissible because there is no feasible solutions (  ,   ,   ) for the conditions ( 9)-( 10) in Lemma 1.In order to design the observer in ( 5) and the controller in (6) so that the uncertain closed-loop system Σ  in ( 7)-( 8) is stochastically admissible for all jumping modes, we apply the resultant LMI conditions in Theorem 3 and Theorem 2 to be feasibility problems (30) , respectively.
For the observer design problem, via Theorem 3, we get the following feasible solutions.
And the corresponding feasible observer gains in (5) are [ ]

O O
For the controller design problem, via Theorem 2, we have the following feasible solutions.
According to the given transition probability matrix, the jumping modes are presented in Figure 1.Taking these feasible gains to the closed-loop system in ( 7)-( 8) under the operating situation (shown in Figure 1), the simulation results of the augmented system's states ((), ()) for different (, ) pairs are shown in Figure 2 and Figure 3, respectively.Both Figure 2 and Figure 3 illustrate the stability of the controlled system and the efficiency of the proposed resultant conditions.In addition, Figure 2 and Figure 3 also show that uncertainties affect  3 () and  3 ().

Conclusions
In this paper, we investigate an observer-based control problem for a discrete-time Markovian jumping singular system with polytopic uncertainties.We not only generalize the problem formulation to a system with polytopic uncertainties, but also extend our recent resultant LMI-based conditions for an admissibility analysis problem.In addition, we propose a method to separately deal with the observer and controller design problems, and derive two sets of strictly LMI-based synthesis conditions.The simulation results of a numerical example show that the proposed conditions are feasible and efficient.
and [    ] = ; (c) R-observable, if  �  −       � =  ; (d) observable, if it is both R-observable and 1.The unforced uncertain DMJSS (,    ) is stochastically admissible for all  ∈  if and only if there are feasible matrices   and positive definite matrices   ,   satisfying the strict LMIs in (9).
=   (   T +   ).   −      ) which is a subsystem of Σ  is stochastically admissible for all  ∈  if and only if there exist matrices  �  ,  �  , and positive definite matrices  �  ,  �  , such that LMIs (16) hold.