Stabilization of LPV systems: state feedback, state estimation and duality

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1 Stabilization of LPV systems: state feedback, state estimation and duality Franco Blanchini DIMI, Università di Udine Via delle Scienze 208, Udine, Italy tel: (39) fax: (39) Stefano Miani DIEGM, Università di Udine Via delle Scienze 208, Udine, Italy tel: (39) fax: (39) Abstract In this paper, the problem of the stabilization of Linear Parameter Varying systems by means of Gain Scheduling Control, is considered This technique consists on designing a controller which is able to update its parameters on line according to the variations of the plant parameters. We first consider the state feedback case and we show a design procedure based on the construction of a Lyapunov function for discrete time LPV systems in which the parameter variations are affine and occur in the state matrix only. This procedure produces a nonlinear static controller. We show that, differently from the robust stabilization case, we can always derive a linear controller, that is nonlinear controllers cannot outperform linear ones for the gain scheduling problem. Then we show that this procedure has a dual version which leads to the construction of a linear gain scheduling observer. The two procedures may be combined to derive an observer based gain scheduling compensator. Keywords LPV systems, scheduling, Lyapunov design. I. INTRODUCTION The gain scheduling approach for the control of Linear Parameter Varying (LPV) systems is a problem encountered in several applications in industrial world. when the plant parameter variations can be measured on line and the compensator may take advantage from this knowledge and improve its performances. However only quite recently techniques which have been heuristically applied became a subject of mathematical investigation. In a rigorous stability analysis of some general gain scheduled schemes is provided. A pole placement like technique for gain scheduling synthesis is proposed in 30. In 16 it is considered the problem of designing a nonlinear controller whose linearizations in several operating points match the linear controllers designed for these points. An LMI technique for the gain scheduled control design is proposed in 31. A µ analysis approach is proposed in 15. A more recent technique based on a set theoretic approach is presented in 35 for discrete time LPV systems with bounded variations. The reader is referred to 32 for a tutorial exposition. Although the knowledge of the plant parameters is an advantage for the compensator, an interesting exception has been investigated in the literature which concerns the state feedback case. Indeed for the class of control affine nonlinear continuous time systems in which the term associated with the control is certain 22 or the so called convex processes 8, the knowledge of the parameter is not an advantage for the compensator as far as it concerns its stabilization capability. As a particular case, gain scheduling design for state feedback LPV systems can be handled without restriction as a robust design, with the remarkable advantage that no parameter measurement are needed.. In the discrete time case this property does not hold. As we will see, trivial examples exist of LPV discrete time systems that can be stabilized via gain scheduling controller but not by means of state feedback controllers which ignore the parameter. This facts motivated us to derive a procedure to design a gain scheduling control in which the exploitation of the parameter measurement is allowed. In particular we consider a procedure for the robust stabilization case 6, which is in principle unsuitable for the gain scheduling design, and we show that it still comes into help if applied to a proper subsystem evidenced by a suitable transformation. Furthermore we show that in the gain scheduling case, nonlinear compensator cannot outperform linear ones. This is in contrast with the robust stabilization case in which nonlinear controller can outperform linear ones 9 (see also 21 for further implications). Thus limiting the attention to linear compensator is not a restriction for gain scheduling state feedback stabilization. Then we consider the dual problem of designing a gain scheduling state observer of the type considered in We show that necessary and sufficient conditions for the existence of a linear observer are the dual of those for the existence of a gain scheduling state feedback (linear or not) compensator. We show how this kind of duality does not hold for the robust state estimation problem, i.e. when the parameters are not available to the observer. Finally we show that the two procedures for state feedback stabilization and state detection can be combined together to solve the problem of stabilization by means of an observer based compensator. Due to lack of space the present version of the paper does not report any of the proofs.

2 II. PROBLEM STATEMENT AND BASIC RESULTS In this paper we consider LPV systems of the form x(k + 1) A(w(k))x(k) + Bu(k) y(k) Cx(k) where x(k) IR n is the state, u(k) IR q is the control input, y(k) IR p is the control output and w(k) is a time varying parameter. We assume that the state matrix A(w(k)) is constrained to belong to the matrix polytope: A(w(k)) m h1 (1) A w h (k) (2) where, for every k, w(k) W {w : w h 0, m h1 w h 1}, and A, h 1,2,...,m are assigned constant n n matrices. We consider the following assumption. Assumption 1: The matrices B and C have full column and row rank respectively. The basic problem considered in this paper is the stabilization of system (1) by means of a state observer and an estimatedstate feedback compensator which are scheduled on the parameter w(k). Definition 2.1: The system (1) is gain scheduling state feedback (GSSF) stabilizable if there exists a (possibly dynamic) continuous state feedback compensator whose equations are function of the time varying parameter w(k) z(k + 1) F(z(k),x(k),w(k)) u(k) G(z(k), x(k), w(k)) such that the resulting closed loop system is globally uniformly asymptotically stable. The next definition is essentially the dual Definition 2.2: The system (1) is gain scheduling detectable (GSDE) if there exists a (possibly dynamic) system whose equations are function of the time varying parameter w(k) z(k + 1) F(z(k),y(k),w(k)) (4) ˆx(k) G(z(k), y(k), w(k)) and such that for all x(0), z(0), w( ) the condition e(k). ˆx(k) x(k) 0 as k is assured. With obvious meaning, we will say that (1) is robustly stabilizable (RS) and robustly detectable (RD) if the equations (3) and (4) do not depend on the parameter w. Furthermore, we will distinguish the case in which (3) and (4) are linear with respect to x(k) and z(k) (respectively y(k) and z(k)). A. Robust stabilization and state detection One of the main points of the paper is to show that there is no apparent duality between the problem of robust detection and robust state feedback stabilization while, in turn, a kind of duality relationship exists between the gain scheduling stabilization and detection. Let us consider the following system x(k + 1) A(w(k))x(k) + Bu(k) y(k) x(k) (3) (5) and let us define the dual as follows x(k + 1) A T (w(k))x(k) + u(k) y(k) B T x(k) As it is well known, if A is a constant matrix, we have a duality relation which says that (5) is reachable iff (6) is observable. In our case such a relation does not exist as long as the parameter w is unknown as it can be shown by the next counterexamples. Consider the system x1 (k + 1) x 2 (k + 1) y(k) x 2 (k) w(k) 0 x1 (k) x 2 (k) with w(k) w. Such a system is such that there is no observer which can estimate asymptotically x 1 (k) ignoring w(k). Indeed, the second state equation x 1 (k) enters in the term ξ. (1 + w(k))x 1 (k). Even if ξ(k) x 2 (k + 1) can be determined with one step delay, it is impossible to estimate x 1 (k) from ξ(k), if w > 0, up to the fact it belongs to the uncertainty interval ξ/(1 + w) x 1 ξ/(1 w), which size grows arbitrarily if x 1 (0) 0. Conversely, the dual of this system x1 (k + 1) x 2 (k + 1) w(k) 0 0 x1 (k) x 2 (k) (6) u(k) with the control u(k) 4x 1 (k) 2x 2 (k) is stable provided that ŵ is sufficiently small. Thus, roughly, robust stabilizability does not imply the robust detectability of the dual. It can be easily shown that robust detectability does not imply the robust stabilizability of the dual, as well. To this aim it is sufficient to consider the following simple example with A(w(k)) w(k), B 1 and C 1 (say the system and is dual coincide) x(k + 1) w(k)x(k) + u(k) (7) y(k) x(k) (8) with w(k) 2 which can not be robustly stabilized by any state feedback, but is obviously detectable. Note in passing that the last example shows that robust state feedback stabilization and the gain scheduling state feedback stabilization are different problems for discrete time systems. Indeed the system can be gain scheduling stabilized (e.g. by u(k) w(k)x(k)), but not robustly stabilized. This fact was pointed out in 8 and will motivate the results of the next section in which we will present a procedure for the gain scheduling state feedback stabilization by means of the procedures already available for the robust control synthesis 6. B. Some preliminary results In this section we will remind some basic results concerning the stability of linear LPV systems. We denote by C-set a convex and compact set containing the origin as an interior point. Given x IR n the (dual) one and infinity norms are

3 defined as x 1 n i1 x i and x max i x i, respectively, and the corresponding induced norm for matrices are H 1 sup j n i1 H Hx i j and H sup x 0 x Hx sup 1 x 0 x 1 sup i n j1 H i j. We define (symmetric) polyhedral function the Minkowski functional of a 0-symmetric convex and compact set containing the origin as an interior point (in brief symmetric C-set). Such a function can be represented as Ψ(x) Fx, (9) where F IR s n is a full column rank matrix, or in its dual form Ψ(x) inf{ p 1, s.t. x p} (10) where now the matrix IR n l is full row rank. There is a duality relation between (9) and (10) in the sense that the two definitions give the same function if and only if the convex hull of all the column vectors of and its opposite is the unit ball of Fx. Such duality holds also for the stability of an autonomous systems, as per the following (duality) lemma. Lemma 2.1: The system x(k + 1) A(w(k))x(k), (11) where A(w) is as in (2), is robustly stable for all admissible w(k) if and only if any of the following holds: i) there exists a full row rank matrix IR n l and m matrices P IR l l such that P 1 λ < 1 and A P. ii) there exists a full column rank matrix F IR s n and m matrices H IR s s such that H λ < 1 and FA H F. The coefficient λ above turns out to be an index of the speed of convergence. The proofs of the above lemma follows immediately by the fact that if the system is stable, then it admits a polyhedral Lyapunov function The details can be found in 7. This also means that (11) is robustly if and only if the dual system is robustly stable. III. SOLUTION OF THE GAIN SCHEDULING STATE FEEDBACK STABILIATION PROBLEM In this section we consider the problem of determining a state feedback control of the gain scheduling type for the system (1). To this aim we introduce the following important notion of speed of convergence. Definition 3.1: We say that the system (1) is Gain Scheduling stabilizable with speed of convergence λ if there exists a control such that x cl (k) Cλ k x cl (0), where cl is the closed loop system state. Note that the definition above implies exponential convergence of the state to 0. We will see that any stabilizable system of the considered class can be indeed exponentially stabilized. A. Problem solvability To solve our problem we consider the class of the polyhedral functions as candidate Lyapunov functions. The next theorem motivates this choice by showing that the existence of such function is necessary and sufficient for the Gain Scheduling stabilizability of system (1). Theorem 3.1: The following statements are equivalent. i ii iii The system (1) is gain scheduling stabilizable. There exists a control Φ(x,w) such that the polyhedral function (10) is a Lyapunov function for the system (1). There exist a full row rank matrix IR n l, m matrices P IR l l and U IR q l such that A + BU P, with P 1 < λ < 1, (12) The previous theorem states that eq. (12) is crucial, since the existence of a solution in terms of, P and U is necessary and sufficient for the GS stabilization problem to be solvable. The next theorem states that as long as the parameter w is known to the controller, we can always implement a linear controller for the system, namely GS stabilizability implies GS stabilizability by means of a linear controller. This result extends that in 8 to the discrete time case. To introduce the theorem we augment the equation (12) as follows A B I U V P, (13) with P 1 < 1 and where, is an arbitrary matrix such that is square invertible and V. P. We are now able to state the following theorem which states that Gain Scheduling stabilizability is equivalent to Gain Scheduling stabilizability via linear control. Theorem 3.2: If system (1) is GS stabilizable, then it is GS stabilizable via linear control. A stabilizing control is given by where K(w) G(w) with u(k) K(w)x(k) + H(w)z(k) z(k + 1) G(w)x(k) + F(w)z(k), H(w) F(w) K H G F m h1 K H G U V F (14) w h (15) 1 (16)

4 B. Computation of the Lyapunov function The results above are non constructive as long as we cannot provide algorithms to compute the matrices, P and U in equation (12). As observed in 7 such type of equations are not convenient to solve the problem, since they are bilinear as long as and P are both unknown. Note also that the same problem holds in the robust stabilization case, where we have to cope with the same equation with the difference that 7 the matrices U are all equal according to the following proposition. Proposition 1: 6. The system x(k + 1) A(w(k))x(k) + B(w(k))u(k) y(k) x(k) where A(w(k)) is as in (2) and B(w(k)) m h1 (17) B w h (k) (18) and B IR n q, h 1,2,...,m are assigned constant matrices, is robustly stabilizable if and only if there exist a full row rank matrix IR n l, m matrices P, and a single matrix U IR q l such that A + B U P, with P 1 λ < 1, (19) In 6 an iterative procedure is proposed to solve the problem of the determination of, P and U. Clearly this procedure may be applied in a conservative way to our problem since if a system is robustly stabilizable then it is also GS stabilizable. However, for discrete time systems this is a conservative way of proceeding, since, as we have seen, the knowledge of w can be an advantage for the compensator. The next result will allow us to exploit the existing procedures for the solution of the robust stabilization problem for GS stabilizability. In view of Assumption 1 we can consider the system in the following form: ˆx 1 (k + 1) Â 11 (w) ˆx 1 (k) + Â 12 (w) ˆx 2 (k) (20) ˆx 2 (k + 1) Â 21 (w) ˆx 1 (k) + Â 22 (w) ˆx 2 (k) + u(k) (21) It is immediately seen that this form is achieved by applying to all the generating matrices A the linear transformation T B B, where B is any matrix such that T is invertible. By means of this form, we can recast the GS stabilization problem in a robust stabilization problem for the pair (Â 11 (w(k)),â 12 (w(k))) according to the following theorem. Theorem 3.3: System (1) is GS stabilizable if and only if the system ˆx 1 (k + 1) Â 11 (w) ˆx 1 (k) + Â 12 (w) ˆx 2 (k), (22) (where ˆx 2 has now to be thought to as input), is robustly stabilizable. Thus the above result allows us to use existing algorithms (properly modified) to compute a solution of (12). IV. SOLUTION OF THE GAIN SCHEDULING STATE ESTIMATION PROBLEM In the previous section we have considered the problem of designing a state feedback gain scheduling compensator. Among the results we have seen that if a system is gain scheduling stabilizable, then is gain scheduling stabilizable via linear state feedback control. Now we cope with the dual problem of designing a state observer for the system x(k + 1) A(w(k))x(k) + v(k) y(k) Cx(k) For this system we consider a linear observer of the form: z(k + 1) P(w(k))z(k) L(w(k))y(k) + T 1 (w(k))v(k) ˆx(k) Q(w(k))z(k) + R(w(k))y(k) (23) with z IR s. Definition 4.1: The system (23) is an asymptotic observer if and only if i for all w(k), v(k) and x(0) and z(0) we have ˆx(k) x(k) 0 as k ; ii If x(0) 0, z(0) 0 then ˆx(k) x(k) for all w(k) W and v(k). It is known that, for a given constant w, the system (23) represents the most general form of a linear observer 26. Furthermore, for a given constant w, for (23) to be an observer the following necessary and sufficient conditions must hold and P(w)T 1 ( w) T 1 ( w)a( w) L( w)c Q( w)t 1 ( w) + R( w)c I (24) P( w) is a stable matrix (i.e. its eigenvalues are in the open unit disk); T 1 ( w) has full column rank The main problem is now to see what happens when w(k) is time varying as in our case. To this aim introduce the assumption that the family of matrices P(w) does not contain common invariant subspaces included in the kernel of Q(w). We say that a subspace S of IR n is P(w) invariant, if x S implies P(w)x S for all w W. Furthermore we call the kernel of Q(w) the set of all vectors r such that Q(w)r 0 for all w W. Assumption 2: There are no P(w) invariant subspaces included in the kernel of Q(w). The assumption below essentially means that the system cannot be reduced by the transformation S S, where S is a basis of S, the invariant subspace and S is a complement matrix, to the form ˆP(w) ˆP 11 (w) ˆP 12 (w) 0 ˆP 22 (w), ˆQ(w) 0 ˆQ 2 (w) Note that for known P and Q this is just an observability assumption. The restriction above is obviously non restrictive,

5 since, if the decomposition above can be achieved, then we can eliminate the non observable part of the system. Consider now a new variable r(k) such that z(k) r(k) + T 1 (w(k))x(k) (25) With simple computations if we replace (25) in (23) in view of (24) we get the following equation r(k + 1) P(w(k))r(k) + T 1 (w(k + 1)) T 1 (w(k))x(k + 1) ˆx(k) x(k) Q(w(k))r(k) (26) We have now the following result that concerns the structure of (23). Lemma 4.1: The system (23) is an observer only if the matrix T 1 (w) T 1 is a constant. Given that T 1 must be constant, without restriction we can consider observer of the form z(k + 1) P(w(k))z(k) L(w(k))y(k) + T 1 v(k) ˆx(k) Q(w(k))z(k) + R(w(k))y(k) whose associated error equation, derived from (26), is r(k + 1) P(w(k))r(k) e(k). ˆx(k) x(k) Q(w(k))r(k) (27) (28) The previous equation together with Assumption 2 leads us immediately to the following basic result. Lemma 4.2: The system (27) under Assumption 2 is an observer only if the system r(k + 1) P(w(k))r(k) is stable and there exists a full row rank matrix T 1 P(w)T 1 T 1 A(w) L(w)C (29) Q(w)T 1 + R(w)C I The proof of the above Lemma is immediate and thus it is omitted. We are in the position now to prove the main result of this section. Theorem 4.1: The following statements are equivalent. i ii iii The system (27) is an observer. There exists full column rank matrix F, m matrices H and m matrices Y such that FA +Y C H F, with H < 1, (30) The dual system x(k + 1) A T (w(k))x(k) +C T u(k) is gain scheduling stabilizable. The previous result is constructive, since, by duality, we can always apply the procedure to design a gain scheduling state feedback stabilizer to the dual system (A T (w),c T ) to design the observer. V. A SEPARATION PRINCIPLE FOR DESIGN We briefly describe a way to stabilize the system by means of a linear observer and gain scheduling estimated state feedback. The next theorem is a simple consequence of the results of the previous section. It shows that one can always synthesize a stabilizing compensator by separately designing an observer and a state feedback control if the system is state feedback stabilizable and detectable. Theorem 5.1: Assume that (A(w), B) is gain scheduling stabilizable and (C,A(w)) is gain scheduling detectable. Then the dynamic controller (we dropped the time-dependence in w(k) for clarity) z c (k + 1) G(w) ˆx(k) + F(w)z c (k) z o (k + 1) P(w)z o (k) L(w)y(k) + T 1 Bu(k) ˆx(k) Q(w)z o (k) + R(w)y(k) u(k) K(w) ˆx(k) + H(w)z c (k), (31) where the matrices in the above equations are as in theorem 3.2 and lemma 4.2, asymptotically stabilizes the plant. VI. CONCLUDING DISCUSSIONS In this paper we focused our attention on the stabilization of Linear Parameter Varying (LPV) discrete-time systems. We showed by very simple examples that the robust stabilization and the gain scheduling stabilization problems, differently from the continuous-time case, are rather different problems. We also showed the existence of a duality relation between the gain scheduling control and the gain scheduling observation problem for LPV. Finally a separation principle to derive output feedback stabilizing controllers was derived. VII. REFERENCES 1 N. E. Barabanov, Lyapunov indicator of discrete inclusion, Autom. Rem. Contr., parts I,II,III, Vol. 49, no. 2, pp ; no. 4, pp ; no. 5, pp , R. K. Brayton and C. H. Tong, Constructive stability and asymptotic stability of dynamical systems, IEEE Trans. Circ. Syst., Vol. 27, no. 11, pp , B. R. Barmish, I. Petersen, and A. Feuer, Linear Ultimate Boundedness Control of Uncertain Systems, Automatica, Vol. 19, no. 5, pp , B. R. Barmish, M. Corless, and G. Leitmann, A new class of stabilizing controllers for uncertain dynamically systems, SIAM J. Contr. Optim., Vol. 21, no. 2, B. R. Barmish and A. R. Galimidi, Robustness of Luemberger Observers: Linear Systems Stabilized via Non-linear control, Automatica, Vol. 22, no. 4, pp , F. Blanchini, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, IEEE Trans. Aut. Contr., Vol. 39, no. 2, pp , 1994.

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