Robust Stabilization of the Uncertain Linear Systems Based on Descriptor Form Representation t Toru ASAI* and Shinji HARA** This paper proposes a necessary and sufficient condition for the quadratic stabilization for a class of linear systems described by descriptor form representations. The system representation contains timevarying but bounded parameters representing the uncertainties. Similarly to the state-space form case, gives better performances as compared with the state-space method. 1. Introduction In recent years, robustness has been paid a lot of attention in control systems design. A lot of methods for the stabilization of systems with uncertainty have been proposed and efficient results have been achieved. Especially, a method of quadratic stabilization is one of powerful approaches for systems with the descriptor form is more suitable for the representation of real independent parametric Indeed, we can show that the descriptor uncertainties. form description is tighter than the state-space expression for representing real independent parametric perturbations. For example, let us consider a second order transfer function P(s) represented by bounded time-varying parametric uncertainties4),7)-9),14). The quadratic stabilization is based on the existence of a certain Lyapunov function of quadratic where T and K are assumed to be physical variables form which assures the global exponential and independent each other. A state-space representation asymptotic stability. It is well known that the qua- of P(s) is given by lem and the control law is constructed by solving a certain algebraic Riccati equation4). So far, the uncertainty have been considered based In the state space form, we have to consider the on the state-space form representation in the framework. perturbations of 1/T and K/T instead of T and K. However, in the process of modeling, a collec- Here, we have a question: Are there any differences tion of variables that is enough to describe the system between the perturbations of 1/T and K/T and those is defined and employed. Those variables generally of T and K? The answer is 'yes' and the difference have inherent meanings, or natural interpretations can be seen quite easily. Suppose that T and K vary within the context of the particular situation. Hence, over * Department of Control Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo ** Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama (Received May 31, 1994) (Reviced Octover 14, 1994) TR 0003/95/3108-1037 (C) 1994 SICE
The change of 1/T is equivalent to that of T. However, the change of K/T denotes a sector with the In Sec. 3, we introduce the uncertain linear system necessary and sufficient condition are shown in Sec. 2. angle from tan-1 (KL/TH) to tan-1(kh/tl) on the (T, based on the descriptor form representation and K) parameter space. Therefore, the region covered establish the quadratic stabilization. The effectiveness of the proposed method is illustrated by an The regions covered by (1.3) and (1.4) are depicted in Fig. 1 shaded darkly and shallowly, respectively. As seen so far, the uncertainty is enlarged unnecessary by considering the uncertainty in the state-space form and this enlargement would cause, the conservativeness in the robust control system design. Therefore, it is necessary to consider the uncertainty of (T,K) independently in order to avoid the enlargement of the uncertainty. As a way to consider the parameters independently, we can employ the following system representation : (1.5) is known to be a descriptor form representation of P(s)13). Using descriptor form representation, we can separately consider the parameters like mass, time-constant, etc. from other parameters. From the point of view discussed above, we consider the uncertainty represented in the descriptor form and establish the robust stabilization. In this paper, we propose a method of quadratic stabilization based on the descriptor form representation with parameter uncertainties. The approach is similar to the state-space form case. In other words, the problem of the quadratic stabilization can be The definition of the quadratic stability and the example in Sec. 4. The proof of Theorem 2.1 uses several lemmas in Appendix. The notation used in this paper is standard. For a given matrix A, AT and A* denotes its transpose and its conjugate transpose, respectively. If all eigenvalues of a matrix lie on the open left half side of the complex plane, we call the matrix Hurwitz. If eigenvalues of A, respectively. For any vector x, x denotes the standard Euclidean norm of x. For any matrix G partitioned as the linear fractional transformation of a matrix K is given as 5,(G,K):=G11+G12K(I-G22K)-1G21 (1.7) where K must have compatible sizes. 2. Definitions and Preliminary Results The stability of the closed-loop system is established through the analysis-of a quadratic Lyapunov function. In order to introduce the notion of quadratic stability, we employ an autonomous timevarying system. Let us first consider an uncertain system J represented by representing the uncertainty and assumed to be Lebesgue measurable function such that PT(t)P(t)<_I;dt>0 (2.2) Fig. 1 Regions covered by the uncertainty of (T,K) and (1/T, K/T)
we can rewrite the above inequality as While many researcher have defined the quadratic stability for several uncertain linear systems4),9),7), we define the quadratic stability for a linear timeinvariant system with a bounded time-varying matrix. However, our definition is equivalent to that of Petersen et al.4),9),7), since our uncertain linear system includes those systems as special cases. easily checked by constructing output of the time-varying that. This fact is the system from the uncertainty to the input of If the inequality (2.3) is satisfied, it is straight forward to verify that the time-varying system J is uniformly, exponentially and asymptotically stable. The quadratic stability of the system J has a close relation to the small-gain theorem measured with the Proof: (Sufficiency) Consider the quadratic form V(x(t)):=xT(t)Px(t), where P is a positive definite matrix satisfying (A.1) in Appendix. The derivative of V(x(t)) is written by Furthermore, the bound is replaced with xtqx, where (Necessity) Suppose that the system J is quadratically stable. From the definition of the quadratic stability, (x,t): the following inequality holds for all pairs Since I-DTD is positive definite from Lemma A.3, Since the RHS of the above inequality is non-positive, we have the following inequality :
3. Main Result 3.1 Descriptor Form Representation As stated in the introduction, we want to robustly stabilize the uncertainty in the descriptor form. Firstly, let us consider a nominal plant with descriptor form representation expressed as vector, the control input and the output, respectively. Usually, in the context of the descriptor system, the matrix E is assumed to be singular in general. However, it is assumed in Sections 3.1, 3.2 and 3.3 that E is non-singular, since the focus is only on investigating the properties of the robust stability in terms of the representations of real parameters. A slight extension to the singular case will be briefly examined in Section 3.4. Corresponding to the above system representation, we consider a linear system described by the descriptor form equation:
ble iff there exists a linear time-invariant system K which satisfies the following two statements concerned as the closed-loop system Fl(G,K): (S1) The closed-loop system Fl(G,K) is internally stable. Fig. 3 Relations of system
4. Numerical Illustrations Here we illustrate the effectiveness of the stabilizing method proposed in the previous section. Let us consider a transfer function of a motor given by (1.1) and uncertain parameters (T,K) given by (1.3). The system is described in the descriptor form as (1.5). In order to express the perturbation of (T,K) with time-varying admissible uncertainties, let us Here we define GDF as the time-invariant (3.3) whose values are given by (4.4). On the other hand, in the state-space system is described as (1.2). (4.4) system form, the In this case, we must consider the perturbation of parameters (1/T, K/T) instead of (T,K). Let us define the time-varying
T. SICE Vol. 31 No. 8 August 1995 5. Conclusion A necessary and sufficient condition for the qua- dratic stabilization based on the descriptor form representation has been provided in this paper. It was shown that the stability of the system can be estab- References
Robust and Nonlin. Contr., 1, 153/169 (1991) 6) A. Packard and J. Doyle: Quadratic Stability with Real and Complex Perturbations, IEEE Trans., AC-35-2, 198/201 (1990) 7) I. R. Petersen: A Stabilization Algorithm for a Class of Uncertain Linear Systems, Syst. Contr. Lett., 8, 351/357 (1987) 8) I. R. Petersen: Stabilization of an Uncertain Linear System in which Uncertain Parameters Enter into the Input Matrix, SIAM J. Contr. Optimiz., 26-6, 1257/ 1264 (1988) 9) I. R. Petersen and C. V. Hollot: A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems, Automatica, 22-4, 397/411 (1986) 11) T. Shiotsuki and S. Kawaji: A Model Reduction Algorithm of Descriptor Systems with Purely Static Part, Trans. of SICE, 24-10, 1056/1063 (1988) (In Japanese) 12) E. Uezato and M. Ikeda: Quadratic Stabilization of Linear Descriptor Systems, Preprints of the 16th SICE Symposium on Dynamical System Theory, 271/ 276 (1993) (In Japanese) 13) G. C. Verghese, B. C. Levy and T. Kailath: A Generalized State-space for Singular Systems, IEEE Trans., AC-26-4, 811/831 (1981) Lemma A. 4 Suppose that I-DTD>0 holds and that a positive definite matrix P is given. Then, for the system Y defined by (2.1), there exists an admissible uncertainty xt(t)pb(i-dtd)-1(w(t) satisfying the following equation: -DTz(t))
T. SICE Vol. 31 No. 8 August 1995 Also note that (A.11) and (A.12) imply Torn ASAI (Student Member) He received the B. E. and M. E. degrees in control engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1991, and 1993, respectively. He is currently a doctor student in Tokyo Institute of Technology. His current research interests are in robust control theory and its application. Shinji HARA (Member)