SEARCH

SEARCH BY CITATION

Keywords:

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

SUMMARY

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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 inline image and inline image to denote the set of real numbers and the set of nonnegative real numbers, respectively. inline image and inline image 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 inline image denotes the set of complex numbers. | | ⋅ | | denotes the usual Euclidean norm. The set of all such functions, endowed with the (essential) supremum norm inline image, is denoted by inline image.

BASIC DEFINITIONS AND LEMMAS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

First, we state some basic definitions related to ISS.

Definition 1. ([9]) A function inline image 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 inline image.

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

Consider the following nonlinear descriptor system:

  • display math(1)

where inline image is smooth enough and inline image 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 inline image 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 inline image and each inline image, it holds that

  • display math

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

  • display math(2)

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

  • display math

and

  • display math

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 inline image 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 inline image 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 inline image such that

  • display math

MAIN RESULTS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

Consider a nonlinear descriptor system

  • display math(3)

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

    • display math(4)
  2. there exists matrix inline image such that

    • display math(5)
    • display math(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 inline image, such that

  • display math

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

  • display math

Let

  • display math

where inline image. Premultiplying (3) by M, we have

  • display math(7)

where

  • display math

Now system (7) is

  • display math(8)
  • display math(9)

By inequality (6), we have

  • display math(10)

Inequality (10) implies that

  • display math(11)

By inequality (11), we have

  • display math(12)

which implies that

  • display math(13)

Hence,

  • display math

which is equivalent to

  • display math

which implies that inline image. Therefore, for any given inline image and inline image,

  • display math(14)

It follows from the fixed-point principle that (14) implies that there exists unique solution inline image for any given inline image and u from the second equation of (9). Next, we show that inline image 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 inline image is Lipschitz with respect to inline image, that is, we only need to show that inline image is Lipschitz with respect to inline image. In fact, for any given inline image and inline image,

  • display math(15)

Therefore,

  • display math(16)

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

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

  • display math

and hence,

  • display math(17)

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

  • display math

where f = f(x,u). By (4), we have

  • display math(18)

Therefore,

  • display math

which implies that

  • display math

thereby

  • display math

By inequality (12), we have inline image, 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 inline image, which is impossible.

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

  • display math

From (3) and (5),

  • display math

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

  • display math

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

  • display math

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

  • display math

Denote inline image, and then

  • display math

Hence,

  • display math

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

  • display math

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

  • display math

Note that

  • display math

Hence,

  • display math

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

  • display math

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:

    • display math
  2. there exists matrix inline image 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 inline image. In our proof, we first estimate inline image because | | f | | depends on both inline image and inline image and the estimating of inline image is relatively easier. When estimating the bound of the matrix inline image, 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 inline image. 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:

  • display math(19)

where inline image, matrix inline image, and inline image.

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

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

    • display math
  2. there exists a matrix inline image such that

    • display math

where inline image.

Proof. Let f(x,u) = − ϕ(y) + Bu, then

  • display math

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

  • display math(20)

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

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

    • display math(21)
  2. there exist nonsingular matrix inline image and matrix inline image such that

    • display math(22)
    • display math(23)

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

Proof. The closed-loop system is

  • display math(24)

By Theorem 1, if inequality matrix

  • display math(25)

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

  • display math(26)

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

  • display math(27)

By Schur complement, inequality (27) implies that

  • display math

Let inline image, and the above inequality can be rewritten as

  • display math

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

  • display math

namely,

  • display math

CALCULATING ISSUE

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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 inline image and inline image such that

  • display math(28)

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

Let inline image, where inline image, inline image, and then

  • display math
  • display math

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

  • display math

so we have the following corollary.

Corollary 3. System (3) is ISS, if

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

    • display math
  2. there exists matrix inline image and inline image such that

    • display math

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

  • display math(29)

where inline image.

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

  • display math

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

    • display math
  2. there exist matrix inline image and matrix inline image such that (29) holds.

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

NUMERICAL EXAMPLE

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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 inline image; by Kirchhoff's law, the following is obtained:

  • display math

Let inline image, and take

  • display math

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

  • display math

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

image

Figure 1. A circuit network.

Download figure to PowerPoint

Take inline image, and then inline image. Hence,

  • display math

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

  • display math

Solving LMI (29), we obtain

  • display math

Then, the following controller is obtained:

  • display math

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

image

Figure 2. Taking u(t) = 100 sin 7t, the state x(t) is bounded.

Download figure to PowerPoint

image

Figure 3. Taking u(t) = 5e − 3t, the state x(t) tends to zero.

Download figure to PowerPoint

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

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES

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.

REFERENCES

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. BASIC DEFINITIONS AND LEMMAS
  5. MAIN RESULTS
  6. CALCULATING ISSUE
  7. NUMERICAL EXAMPLE
  8. CONCLUSIONS
  9. ACKNOWLEDGEMENT
  10. REFERENCES
  • 1
    Campbell SL. Singular Systems of Differential Equations. Pitman: London, 1980.
  • 2
    Dai L. Singular Control Systems. Springer: Berlin Germany, 1989.
  • 3
    Zhang LQ, Huang B, Lam J. LMI synthesis of H2 and mixed H2 ∕ H ∞ controllers for singular systems. IEEE Transactions on Circiuts and Systems II 2003; 50(9):615626.
  • 4
    Lu GP, Ho DWC. Full-order and reduced-order observers for lipschitz descriptor systems: the unified LMI approach. IEEE Transactions on Circuits and Systems II 2006; 53:563567.
  • 5
    Xu SY, Paul VD, Radu S, Lam J. Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Transactions on Automatic Control 2002; 47(7):11221128.
  • 6
    Gao ZW, Ho DWC. State/noise estimator for descriptor systems with application to sensor fault diagnosis. IEEE Transactions on Signal Processing 2006; 54(4):13161326.
  • 7
    Xu TQ, Chen YP, Zhang JX, Liu HB, Zhou ZD. H ∞  filter design for uncertain descriptor systems. 2008 Chinese Control and Decision Conference, Yantai, Shangdong, China, July 2-4, 2008; 43934398.
  • 8
    Chen N, Gui WH. Robust decentralized generalized H2 control of multi-channel uncertain descriptor systems with time-delays. Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, China, July 16-18, 2008; 730734.
  • 9
    Sontag ED. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control 1989; 34:435443.
  • 10
    Sontag ED, Wang Y. On characterizations of the input-to-state stability property. Systems and Control Letters 1995; 24:351359.
  • 11
    Sontag ED, Wang Y. New characterizations of input-to-state stability. IEEE Transactions on Automatic Control 1996; 41(9):12831293.
  • 12
    Sontag ED. Further facts about input-to-state stabilization. IEEE Transactions on Automatic Control 1990; 35(4):473476.
  • 13
    Jiang ZP, Teel AR, Praly L. Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems 1994; 7:95120.
  • 14
    Sontag ED, Teel A. Changing supply function in input/state stable systems. IEEE Transactions on Automatic Control 1995; 40:14761478.
  • 15
    Sontag ED. Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer-Verlag: New York, 1998.
  • 16
    Sontag ED. Comments on integral variants of ISS. System Control Letters 1998; 34:93100.
  • 17
    David A. Input-to-state stability of PD-controlled robotic systems. Automatica 1999; 35:12851290.
  • 18
    Xiang ZR, Xiang WM. Input-state stability analysis for a class of switched nonlinear systems. Journal of Jishou University (Natural Science Edition) 2007; 28(2):5155.
  • 19
    Ma HB, Kang HG. Input-to-state stability of nonlinear descriptor systems. Journal of Anhui University Natural Science Edition 2009; 33(2):911.
  • 20
    Wu H, Yung C, Chang F. H ∞  control for nonlinear descriptor systems. IEEE Transactions on Automatic Control 2002; 47(11):19191925.
  • 21
    Yang CY. Stability analysis and design for some classes of nonlinear descriptor systems. Ph.D. Dissertation, Northeastern University, Shenyang, China, 2009.
  • 22
    Masubuchi I, Kamitane Y, Ohara A, Suda N. H ∞  control for desciptor systems: a matrix inequalities approach. Automatica 1997; 33(4):669773.
  • 23
    Lu GP, Ho DWC. Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation. IEEE Transactions on Automatic Control 2006; 51(5):818823.
  • 24
    Kunkel P, Mehrmann V. Differential-algebraic Equations. Analysis and Numerical Solution. EMS Publishing House: Zürich, Switzerland, 2006.