Dissipative Control for Singular Impulsive Dynamical

The aim of this work is to study the dissipative control problem for singular impulsive dynamical systems. We start by introducing the impulse to the singular systems, and give the definition of the dissipation for singular impulsive dynamical systems. Then we discuss the dissipation of singular impulsive dynamical systems, we obtain some sufficient and necessary conditions for dissipation of these systems by solving some linear matrix inequalities (LMIs). By using this method, we design a state feedback controller to make the closed-loop system dissipative. At last, we testify the feasibility of the method by a numerical example.


Introduction
Singular systems are found in engineering systems, such as electrical circuit network, power systems, aerospace engineering, and chemical processing, social systems, economic systems, biological systems, network analysis, time-series analysis, singular singularly perturbed systems with which the singular system has a great deal to do, etc [1].The concept of dissipation is of much interest in physics and engineering [2,3,4].And dissipative systems can be used as models for physical phenomena in which the energy or entropy exchanged with the environment plays a role.The storage function measures the amount of energy which is stored inside the system at any instant of time, and when the storage function is taken as the special form, the dissipation is changed into passivity.In practical systems, such as in the electrical circuit network systems, the impulse always exists, thus many scholars interest in the control problem of dynamical impulse systems, and this problem has been extensively studied and applied in many areas; see, e.g.[5,6,7,8].Now Hadded et al. have developed dissipation and exponential dissipation concepts for nonlinear impulse dynamical systems and left-continuous systems, see, e.g.[9,10,11,12].They have extended the notions of classical dissipative theory using generalized storage functions and supply rates for dynamical impulsive systems and left-continuous dynamical systems.And in [13], robust dissipativity for uncertain impulsive dynamical systems is discussed.Moreover singular systems and the impulse exists widely [14,15], in order to study the dynamical impulsive systems more widely, we discuss the robust dissipative control problem for singular dynamical impulsive system , and obtain the robust impulsive dissipative state feedback controller of singular dynamical impulsive systems by solving the linear matrix inequalities.
The contents of the paper are as follows.In Section 2 we establish definitions, notation and review some important results.In Section 3, we obtain sufficient and necessary conditions of the dissipation for singular dynamical impulse systems, and then give the sufficient and necessary conditions to prove the singular dynamical impulsive system is dissipative.In Section 4, by using the method we design a state feedback controller for singular dynamical impulsive systems.Then in Section 5, we testify the feasibility of the method by a numerical example.Finally we draw conclusions in Section 6.

Singular Impulsive Dynamical Systems
In this section we introduce the dynamical impulse into the singular systems, and a singular dynamical impulsive system consists of three elements: (i) A singular continuous-time dynamical equation, which governs the motion of the system between resetting events; (ii) A difference equation, which governs the way the states are instantaneously changed when a resetting event occurs; (iii) A criterion for determining when the states of the system are to be reset.
A singular dynamical impulsive system has the following form where x ∈ R n is the state, ω c ∈ R mc , ω d ∈ R m d the outside perturbation of the system whose square is integral, and and D d are known constant matrices of appropriate dimensions, and E is singular.
Definition 2. The singular impulsive dynamical system ( 1) is said to be dissipative with respect to supply rate (r c , r d ) if there exists a C r (r ≥ 0) nonnegative function V : R n → R with V (0) = 0, called storage function, such that, for all (ω c , ω d ) ∈ U, the following dissipation inequality holds: where x(t), t ≥ t 0 is a solution to system (1) with where S + 22 denotes the Moore-Penrose inverse of S 22 .

Dissipation of Singular Impulsive Dynamical Systems
From [9], we can get the trivial result of singular impulsive dynamical system as the following theorem: Theorem 1.A singular impulsive dynamical system given by ( 1) is dissipative with respect to the supply rate (r c , r d ) if and only if there exists a C 1 nonnegative definite function EJQTDE, 2012 No. 32, p. 3 Remark 1.In the theorem, where T is the set of times when jumps occur for x(t), then an equivalent statement for dissipation of the singular impulsive dynamical system (1) with respect to the supply rate (r c , r d ) is In the paper, we consider the following storage function and supply rate where E T P = P T E ≥ 0, and the matrices Q c , S c , R c , Q d , S d , and R d are given matrices with appropriate dimensions.
Without loss of the generality, we can assume that Q c and Q d are both negative definite.Then we will discuss the dissipation of the singular dynamical impulsive systems.Theorem 2. For system (1), if and only if the following inequalities have the feasible solution P ,    EJQTDE, 2012 No. 32, p. 4 system (1) is dissipative with respect to the supply rate (10), where " * " denotes the symmetric part of the matrix.
Proof.According to the storage function and the supply rate, we obtain then by Theorem 1, we obtain EJQTDE, 2012 No. 32, p. 5 where Then using Lemma 1, we get the the conclusion of theorem.

Dissipative Controller Design for Singular Impulsive Dynamical Systems
In this section, we will design a state feedback controller for the singular dynamical impulsive systems.The system considered is as the following: where u c ∈ R sc , u d ∈ R s d are the input.And the state feedback controller is as the following: Using Theorem 2 , we obtain the following results: EJQTDE, 2012 No. 32, p. 6 then system (1) has dissipative feedback controller (20), where K c = W c X −1 , and Proof.The closed-loop system of system (19) via the feedback controller (20) is Using Theorem 2, we obtain that where then multiply diag{X T , I , I} by the left side of (26) and its transfer by the right side, and then using Lemma 1, we can obtain (22), where X = P −1 .And by using the similar method, we can obtain the result.

Example
The parameters of system (1) are given as follows: EJQTDE, 2012 No. 32, p. 7 we get the value of V (t, x(t)) − r c (ω c (t), y c (t)) ≤ 0, ( t = t k ) for the system via the state feedback controller as Figure 1, and the value of V (t + k )−V (t k )−r d (ω d (t), y d (t)) ≤ 0, ( t = t k ) for the system via the state feedback controller as Figure 2. From the figures, we can make a conclusion that a singular impulsive dynamical system is dissipative with respect EJQTDE, 2012 No. 32, p. 8  to the supply rate.

Conclusion
We have studied the dissipation with respect to the quadratic supply rate for singular dynamical impulsive systems.By solving linear matrix inequalities, some sufficient and necessary conditions of dissipation for this kind of system are obtained.As the problem of the dissipation with respect to other supply rate for the singular dynamical impulsive systems exists, we will discuss it in future papers.

Figure 1 :
Figure1: The value of V (t, x(t)) − r c (ω c (t), y c (t)) ≤ 0, ( t = t k ) for the system via the state feedback controller.