Robust Impulsive Stabilization of Uncertain Nonlinear Singular Systems with Application to Transportation Systems

We consider the robust asymptotical stabilization problem for uncertain singular systems. We design a new impulsive control technique to ensure that the controlled singular system is robustly asymptotically stable andhence derive the corresponding stability criteria. These sufficient conditions are expressed in the form of algebra matrix inequalities and can be implemented numerically. We finally provide a numerical example of a transportation system to illustrate the effectiveness and usefulness of the proposed criteria.


Introduction
System stability is a fundamental issue for a nonlinear system.It not only relates to its system structure but also has relations with the exterior disturbance of the system.Singularity and parameter's uncertainty of the system can seriously affect its stability performance.For an unstable system, how to design a controller which stabilizes the system becomes a critical problem, which motivates the current research.The stabilization methods can be applied to a range of practical applications such as transportation systems [1,2].
As for general nonlinear systems, various design methods to control these dynamical systems have been proposed.Among them, impulsive control methods have attracted considerable attention because impulsive control laws have a fast response time, strong robustness and resistance to disturbances, and low energy consumption.They have been applied to many disciplines [3][4][5].The stabilization issue for the nonsingular certain system by use of the impulsive control method has been studied and many sufficient conditions of its asymptotical stability, such as [6][7][8][9], have been provided.Recent research work can be found in [10][11][12][13][14][15] and the references therein.
In this paper, we will develop the impulsive control method, the Lyapunov functional method, and the matrix inequality technology to solve the stabilizing issue of a class of singular systems with uncertainty, and design an impulsive controller of parts of state variables.Finally, the sufficient conditions of asymptotical stability will be given under our designed controller.This paper's notations are quite standard.Let   denote the n-dimensional Euclidean space,  + the set of nonnegative real numbers, and  × the set of all real  ×  matrices.The superscript "T" represents the matrix transposition operation and  ≥  (respectively,  > ), where  and  are symmetric matrices, indicates that  −  is positive semidefinite (respectively, positive definite).The symbol   is  × the identity matrix and ‖ ⋅ ‖ is the Euclidean norm in   .The   () and   () represent the largest and smallest eigenvalues of , respectively.Let  :  + →  denote the set of all continuous real-valued functions.We say that the function  :  + →  + is piecewise continuous if it is continuous on  + , except at the time points in the set {  }, is left-continuous, and has the right limit at   for all .The ( + ,  + ) is the set of all such piecewise continuous functions .
where () ∈   and () ∈   are the system state and output vectors, respectively.The parameter (, ()) is the norm-bounded external uncertainty described by a continuous vector-valued function.The matrices  ∈  × and  ∈  × are constant ones of appropriate dimensions.In the situation that 0 < rank() < , we say that system (1) is an uncertain singular system.Without loss of generality, we assume that where   ∈  × ( ∈  + ,  < ) is an  ×  identity matrix.We need the following assumptions for our later use.
Assumption .The uncertainty () is a norm-bounded nonlinear function and satisfies the following Lipschitz condition: where  > 0 is a known constant scalar.
To proceed, we need the following lemmas.
Lemma 4 (see [8]).Given a positive matrix  and a positive scalar , we have where  and  are real matrices with appropriate dimensions.

Application to Transportation Systems
In this section, we will consider a transportation application to illustrate our results obtained in Section 3. Consider a transportation system which is modelled by (1) with the following specifications: It is clear that  = 1.Then this transportation system is a singular 2-dimensional system with uncertainty.Now, we design an impulsive pair {  ,  1 (  )}, where  = −0.8 and  =  +1 −   = 0.02.Then, we have  = 1 > 0,  1 = 18,  2 = 0.04, and   ( 1 ) + ln   ( 2 ) = −3 < 0. Consequently, it follows from Theorem 5 that the transportation system is impulsively stabilizable under the following impulsive control pair: {  ,  1 (  ) = −0.8 1 (  )} . (26) From the example, it is concluded that the impulsive control method can effectively stabilize the transportation singular systems.

Conclusion
In this paper, we have proposed a design method for robust impulsive stabilizing control for a singular transportation system with uncertainty.Sufficient conditions are derived to guarantee the global asymptotical stability of the system.An application to transportation systems shows that our designed impulsive stabilizing control is effective and strongly robust.