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### Keywords:

• nonlinear descriptor systems;
• input-to-state stability;
• linear matrix inequality (LMI)

### SUMMARY

This paper studies the input-to-state stability (ISS) of descriptor systems with exogenous disturbances. on the basis of the ISS theory of standard state-space nonlinear systems, a sufficient condition for a class of nonlinear descriptor system to be ISS is proved. Furthermore, a design method of the state feedback controllers is given to make the closed-loop system ISS. A numerical example is given to illustrate the effectiveness of the controller design.Copyright © 2012 John Wiley & Sons, Ltd.

### INTRODUCTION

Descriptor systems are also called singular systems, differential-algebraic equation systems, generalized state-space systems, semistate systems, implicit systems, or constrained systems ([1, 2]). In the past 30 years, descriptor systems have attracted much interest, and many fundamental system theories developed for standard state-space systems have been successfully generalized to descriptor systems ([3-8]).

Stability is a basic property of control systems. Until now, there have been many significant results on stability and stabilization of linear descriptor systems. However, investigation on stability of nonlinear descriptor systems is premature. The difficulties involved in the study of nonlinear descriptor systems stability include the following: (i) it is difficult to satisfy the existence and uniqueness of the system solution; (ii) there is jump or impulse in the solution; (iii) it is not easy to calculate the derivative of Lyapunov function by using Lyapunov theory.

Most of the practical control systems are subject to sensor noises and disturbances. For standard state-space nonlinear systems subject to disturbance inputs, Sontag ([9]) proposed the concept of input-to-state stability (ISS), which is an effective method to describe robust stability of nonlinear systems. There are two functions related to the ISS property. One function characterizes the asymptotic behavior of the system and is called the ISS gain. The other function characterizes the transient behavior of the system and is called the transient bound. ISS provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite L2 gains; this property takes account of initial states in a manner fully compatible with classical Lyapunov stability and replaces finite linear gains. ISS quickly became a foundational concept upon which much of modern nonlinear feedback analysis and design rests ([10-18]).

There have been many stability results of a standard state-space nonlinear systems, and a number of equivalent conditions of ISS have been obtained, see for instance [10-12]. On the contrary, the study on the ISS of nonlinear descriptor systems is very limited. For example, the ISS of nonlinear descriptor systems is studied, and some initial results on the analysis of ISS are obtained in [19]. In the results, for different nonlinear descriptor systems, different functions satisfying a certain condition need to be constructed such that the ISS property of the system can be tested. The analysis result is not suitable for the controller design to achieve closed-loop ISS.

In this paper, the ISS property of a class of descriptor systems with exogenous disturbances is studied. On the basis of the ISS theory of normal nonlinear systems, a sufficient condition for the ISS of the class of nonlinear descriptor system is obtained. Furthermore, a design method of the state feedback controllers is given to make the closed-loop system input-to-state stable (ISS). At last, one example is given to illustrate the effectiveness of the controller design.

The notations used in this paper are standard in general. We use and to denote the set of real numbers and the set of nonnegative real numbers, respectively. and denote the n-dimensional Euclidean space and the Cartesian product of n-dimensional and m-dimensional Euclidean spaces, respectively. Let I and Ir denote the identity matrix with appropriate dimension and dimension r, respectively. The notation A < B means that the matrix B − A is positive definite. λ(A) denotes the set of eigenvalues of A, and denotes the set of complex numbers. | | ⋅ | | denotes the usual Euclidean norm. The set of all such functions, endowed with the (essential) supremum norm , is denoted by .

### BASIC DEFINITIONS AND LEMMAS

First, we state some basic definitions related to ISS.

Definition 1. ([9]) A function is a K-function if it is continuous, strictly increasing and χ(0) = 0; it is a K ∞ -function if it is a K-function and also .

Definition 2. ([9]) A function is a KL-function if for each fixed t ⩾0, the function γ(s,t) is K-function, and γ(s,t)  0 as t ∞ for each fixed s ⩾0.

Consider the following nonlinear descriptor system:

• (1)

where is smooth enough and is the system perturbation input.

By [20], the knowledge of Ex(0) = Ex0 is sufficient to completely determine the solution of system (1). Therefore, we introduce the following assumptions.

Assumption 1. For any Ex0 with x0 being an initial condition and any being an input, system (1) has unique continuous solution over [0, + ∞ ).

Definition 3. ([21]) System (1) is ISS if there exist a KL-function γ and a K-function χ such that, for each input and each , it holds that

where x(t,Ex0,u) is the state trajectory of system (1) with initial condition Ex0 and perturbation input u.

Remark 1. If E = I, Definition 3 reduces to the classical concept of ISS for standard state-space systems given in [9].

The ISS property of standard state-space systems was studied in [10], and the following important result has been proved.

Lemma 1. ([10]) System

• (2)

is ISS if and only if there exists a smooth function and K ∞ -functions Ki( ⋅ ),i = 1,2,3,4, such that

and

where V (x) is called ISS-Lyapunov function.

Remark 2. When u = 0, ISS is equivalent to globally asymptotic stability by Lemma 1.

The following definition and lemma will be used in the proof of the main results in this paper.

Definition 4. ([1, 2]) A pair (E,A) is called to be regular, if there exists satisfying det(sE − A) ≠ 0. A regular pair (E,A) is called impulsive-free, if deg(det(sE − A)) = rank(E). A pair (E,A) is called stable; if satisfied det(sE − A) = 0, then Re(s) < 0. A pair (E,A) is admissible, if it is regular and impulse free

Lemma 2. ([22]) The pair (E,A) is admissible if and only if there exists such that

### MAIN RESULTS

Consider a nonlinear descriptor system

• (3)

where is the system state, is the system perturbation input, and are constant matrices, E may be singular; without loss of generality, we shall assume that 0 < rank(E) = r ⩽ n. Function is smooth, and f(0,0) = 0.

Theorem 1. System (3) is ISS, if there exist α,β > 0 such that

1. nonlinear function f(x,u) satisfies the following Lipschitz condition for all :

• (4)
2. there exists matrix such that

• (5)
• (6)

Proof. For system (3), by Lemma 2, linear matrix inequalities (LMIs) (5) and (6) imply that (E,A) is admissible, and then there exist two nonsingular matrices , such that

where . By [23], we can always assume that . Let , by (5), we have

Let

where . Premultiplying (3) by M, we have

• (7)

where

Now system (7) is

• (8)
• (9)

By inequality (6), we have

• (10)

Inequality (10) implies that

• (11)

By inequality (11), we have

• (12)

which implies that

• (13)

Hence,

which is equivalent to

which implies that . Therefore, for any given and ,

• (14)

It follows from the fixed-point principle that (14) implies that there exists unique solution for any given and u from the second equation of (9). Next, we show that exists and is unique; by means of the Picard Lindelof theorem for the existence and uniqueness of solution to ordinary differential equations, we need to show that is Lipschitz with respect to , that is, we only need to show that is Lipschitz with respect to . In fact, for any given and ,

• (15)

Therefore,

• (16)

By (4) and (16), we have is Lipschitz with respect to .

Next premultiplying and postmultiplying (13) by and its transposition, respectively, we have

and hence,

• (17)

Notice that ; then from (9) and (17), we have

where f = f(x,u). By (4), we have

• (18)

Therefore,

which implies that

thereby

By inequality (12), we have , note that P1 0, and we can prove that P1 > 0. In fact, if min{λ(P1)} = 0, then there exists ξ such that P1ξ = 0; thus , which is impossible.

Now let and K1(ξ) = min{λ(P1)}ξ2 and K2(ξ) = max{λ(P1)} ξ2; then, K1,K2 are K ∞ -functions, and

From (3) and (5),

By (6), choose λ = max{λ(Θ)} < 0,λN = min{λ(NTN)} > 0; we have

Then, there exist K ∞ -function K3(ξ) = − λλNξ2 and K-function K4(ξ) = βξ2, such that

By Lemma 1, there exist KL-function γ1 and K-function χ1, such that

Denote , and then

Hence,

Then, there exist KL-function γ2 and K-function χ2, such that

which implies that there exist KL-function γ3 and K-function χ such that

Note that

Hence,

Consequently, there exist KL-function γ and K-function χ such that

thus, system (3) is ISS.□

Remark 3. In fact, from the proof of Theorem 1, the conditions of Theorem 1 can be replaced by

1. nonlinear function f(x,u) that satisfies the following condition:

2. there exists matrix such that (5) and (6) hold.

Remark 4. The sufficient condition of ISS property for nonlinear descriptor system in Theorem 1 implies that the matrix pair (E,A) is regular and impulse free (index one [24]). If the system (E,A) is impulse, then Theorem 1 cannot be applied. Up to now, most of the literature studying the stability of descriptor systems makes the impulse free assumption [6-8]. The study of the ISS for impulse systems (E,A) is more challenging because of the involvement of the nilpotent matrix. This will be the focus of our future research work.

Remark 5. One of the major difficulties involved in the proof of Theorem 1 is the estimation of . In our proof, we first estimate because | | f | | depends on both and and the estimating of is relatively easier. When estimating the bound of the matrix , we applied some matrix transformations. The same result can be obtained by using an approach similar to that used in the proof of Theorem 2.1 in [4], where the LMI is transferred into a non-LMI; then, Schur complement and singular value decomposition methods are used to first obtain an upper bound of . Comparing with the approach in [4], our proof is more straightforward and simpler.

Remark 6. The results in this paper is more general than that in [4]. When considering ISS, one important issue is to find a proper condition on the function f(x,u) such that it is easy to check and compatible with ISS. In this paper, condition (4) is less strict than that used in [4]. In [4], the bound of the norm of f(x,u) only depends on the state x; it does not depend on the input u. In our results, the norm of f(x,u) is assumed to be bounded by the sum of a function on x and another function on u. Thus, the model used in this paper is more general and will have more applications than that in [4].

Next, we consider the following special descriptor system:

• (19)

where , matrix , and .

Corollary 1. System (19) is ISS, if the following conditions are satisfied:

1. nonlinear function ϕ(y) satisfies the following condition:

2. there exists a matrix such that

where .

Proof. Let f(x,u) = − ϕ(y) + Bu, then

By Theorem 1, we can complete the proof of Corollary 1.□

Now, we consider the design of state feedback controllers such that the closed-loop system is ISS.

Corollary 2. For system

• (20)

where is the system control input, is a constant matrix. If

1. nonlinear function f(x,u) satisfies the following condition:

• (21)
2. there exist nonsingular matrix and matrix such that

• (22)
• (23)

Then, there exists a state feedback controller ω = Kx, where , such that the closed-loop system is ISS.

Proof. The closed-loop system is

• (24)

By Theorem 1, if inequality matrix

• (25)

hold, then closed-loop system (24) is ISS. Next, we prove that inequality (25) is equivalent to (23). Inequality (25) is equivalent to

• (26)

Premultiplying and postmultiplying (26) by P − T and P − 1, respectively, we have

• (27)

By Schur complement, inequality (27) implies that

Let , and the above inequality can be rewritten as

Premultiplying and postmultiplying ETP = PTE by P − T and P − 1 gives

namely,

### CALCULATING ISSUE

In Theorem 1 and Corollary 2, we need to deal with the nonstrict LMIs (5) and (22); this difficulty can be resolved by using the result in [3].

For matrix E, there exists orthogonal matrix and such that

• (28)

where E1 = diag(e1,e2, … ,er),ei > 0,i = 1,2 … ,r. By (28), we have .

Let , where , , and then

We substitute the aforementioned for the matrix P in inequality (6); the following equality is obtained:

so we have the following corollary.

Corollary 3. System (3) is ISS, if

1. nonlinear function f(x,u) satisfies the following condition:

2. there exists matrix and such that

Let , where , . Then, nonstrict LMIs (22) and (23) regarding and Q are equivalent to the following strict LMI regarding X > 0,Y and Q:

• (29)

where .

If matrix is singular, we choose σ > 0 and let . Because

we can choose a sufficiently small scalar σ such that is nonsingular and satisfies (29). Thus, without loss of generality, we assume that is nonsingular and then the state feedback controller is , so we have the following corollary.

Corollary 4. Consider system (20), if it satisfies the following conditions:

1. nonlinear function f(x,u) satisfies the following condition:

2. there exist matrix and matrix such that (29) holds.

Then, there exists a state feedback controller ω = Kx, where such that the closed-loop system is ISS.

### NUMERICAL EXAMPLE

In this section, we use an example to illustrate Corollary 4, that is, achieving closed-loop ISS by designing a state feedback controller.

Consider the circuit network model, see Figure 1, where V s(t) represents the circuit source voltage, R,L and C represent resistance inductance and capacitance, respectively, and their voltages V R(t),V L(t) and V C(t). u(t) are the disturbed signal. V R(t) = m(I(t)) satisfy the restriction condition ; by Kirchhoff's law, the following is obtained:

Let , and take

then, the network model is described as the following system:

Now, we design the state feedback controller by Corollary 4 to make the closed-loop system ISS.

Take , and then . Hence,

Applying the singular value decomposition on matrices E, we obtain U1,U2,V 1,V 2, and E1 as

Solving LMI (29), we obtain

Then, the following controller is obtained:

Figures 2 and 3 show that the state response of the closed-loop system with .

It can be seen that the state of the system is bounded when the disturbance input is bounded. When the disturbance input approaches to 0, the state of the system approaches to 0. This is the practical meaning of ISS.

### CONCLUSIONS

In this note, a sufficient condition for a class of nonlinear descriptor system to be ISS is obtained on the basis of the ISS theory of standard state-space nonlinear systems. Furthermore, the design method of the state feedback controllers is given to make the closed-loop system input to state stable. Finally, one example is given to illustrate the effectiveness of the controller design.

### ACKNOWLEDGEMENT

This work was supported by the Natural Science Foundation of China under Grant No. 60974004, the Nature Science of Foundation of Liaoning Province under Grant 201202063.

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