STABILITY RESULTS FOR CONTINUOUS AND DISCRETE TIME LINEAR PARAMETER VARYING SYSTEMS

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1 STABILITY RESULTS FOR CONTINUOUS AND DISCRETE TIME LINEAR PARAMETER VARYING SYSTEMS Franco Blanchini Stefano Miani Carlo Savorgnan,1 DIMI, Università degli Studi di Udine, Via delle Scienze 208, Udine - Italy DIEGM, Università degli Studi di Udine, Via delle Scienze 208, Udine - Italy Abstract: In this paper, stabilizability probles for discrete and continuous tie linear systes with switching or polytopic uncertainties are dealt with. Both state feedback and full-inforation controllers are considered, possibly with an integral action. Several cases will be analyzed, depending on the control, the uncertainty and the presence of uncertainties in the input atrix. A coplete overview of the stabilizability iplications aong these different cases will be given. Copyright c 2005 IFAC Keywords: Stabilizability, LPV systes, Lyapunov functions, uncertainties, switching. 1. INTRODUCTION Linear paraeter varying systes are an iportant class of systes fro a theoretical and practical point of view. In this paper, the stabilization proble of LPV systes is investigated by focussing on two ain factors: the uncertainty characterization and the class of adopted controllers. As far as the tie-varying paraeter is concerned, two cases are considered: the case of switched systes, in which the tie-varying paraeter is allowed to take values in a discrete set of points, and the polytopic case, in which the paraeter ranges in the convex hull of such points. A further relevant distinction that will coe into play is the presence or absence of uncertainty in the input atrices. As far as the class of controllers is concerned, depending on which are the variables easured for control purposes, different concepts of stabilizability will be defined. We will talk about robust stabilizability when no online inforation concerning the uncertainty is available to the controller. The case in which such inforation is available will be referred to as gain- 1 Corresponding author. Eail carlo.savorgnan@uniud.it scheduling or full inforation stabilizability. Also, a certain class of integral controllers, will be analyzed by eans of properly expanded systes. Introducing an integrator in a loop has several well-known advantages (such as that of iposing a null steady-state error or handling rate-bounded control probles). Furtherore, the resulting ad-hoc built expanded syste has no uncertainties on the control input atrix. This fact has several iplications (Barish (1983)), including the property that gradient-based controllers can be applied. The class of functions that will be used in this work to establish the several interconnections that hold aong the entioned stabilizability concepts is that of polyhedral Lyapunov functions. Such class is wide enough for our purposes, as it has been established that the existence of polyhedral Lyapunov functions is a necessary and sufficient condition for stability (Molchanov and Pyatnitskiĭ (1986)) and for stabilizability of LPV systes (Blanchini (1995); Blanchini and Miani (2003)). Recently these functions have been considered for the stabilization of switched systes (De Santis et al. (2004); Sun and Ge (2005)). We will review soe results already known in the literature

2 and we will introduce soe new stateents to coplete the scenario about the subject. 2. PRELIMINARIES AND DEFINITIONS The systes considered are of the for ẋ(t) = A(w(t))x(t) + B(w(t))u(t) (1) in the continuous tie case and x(t + 1) = A(w(t))x(t) + B(w(t))u(t) (2) in the discrete tie case, where x(t) IR n is the state variable, u(t) IR q is the control input, w(t) W IR is a tie varying paraeter. A and B are polytopes of atrices: A(w) = i=0 W = {w : w i A i, B(w) = i=0 w i B i w i 1, w i 0} (3) We distinguish different kinds of syste depending on the function w(t). Definition 2.1. The systes is said to be switched if A and B assue values only on the vertices, precisely A(w(t)) = A i and B(w(t)) = B i for all t (i.e. at each tie instant the coponents of the signal w(t) take only values w i = 1 and w j = 0 for i j, for soe i). In the following, w i (to be intended as w i(t) ) will represent this kind of signals. By default (i.e. without switching specification w), the syste is intended as a polytopic LPV, say w is an arbitrary piecewise-continuous function satisfying (3). As a special case, we will consider systes with no uncertainty on the input atrix B: ẋ(t) = A(w(t))x(t) + Bu(t) (4) x(t + 1) = A(w(t))x(t) + Bu(t) (5) in the continuous and discrete tie case respectively. We investigate the following concepts of stabilization. Definition 2.2. The state feedback u = Φ R (x) is robustly stabilizing (RS) for syste (1) if the closed loop syste ẋ = A(w(t))x(t) + B(w(t))Φ R (x) (6) is globally uniforly asyptotically stable (GUAS) with respect to the origin. Definition 2.3. The full inforation feedback u = Φ GS (x,w) is gain-scheduling stabilizing (GSS) for syste (1) if the closed loop syste ẋ = A(w(t))x(t) + B(w(t))Φ GS (x,w) (7) is GUAS with respect to the origin. Definition 2.4. The full inforation feedback u = Φ SGS (x,w i ) is switched gain-scheduling stabilizing (SGSS) for syste (1) if the closed loop switched syste ẋ = A(w(t))x(t) + B(w(t))Φ SGS (x,w) (8) is GUAS with respect to the origin. Siilar definitions can be given for discrete tie systes. We anticipate that, for robust state feedback controllers, the switched and the non switched case (i.e. w as in (3)) are indistinguishable, since they are strictly equivalent as later on will be explained. Let us now introduce the notation and soe basic results that will be used in the sequel. P 1 represents the one nor of the atrix P ( P 1 = ax j i p i j ); given a continuous function Ψ : IR n IR, we denote by N (Ψ,k) the set {x IR n : Ψ(x) k}; given a atrix X, conv(x) represents the convex hull of the vectors given fro the coluns of X; the tie dependency of the variables will be soeties oitted to siplify the notation (x(t + 1) will be indicated with x + ). Definition 2.5. The locally Lipschitz positive definite and radially unbounded function Ψ(x) is a Lyapunov function for syste (1) with the control Φ(x,w) if for all k > 0 there exist β > 0 such that D + Ψ(x,w) = lisup τ 0 + Ψ(x + τ[a(w)x + B(w)Φ(x,w)]) Ψ(x) β (9) τ w W and x / N (Ψ,k) Definition 2.6. The continuous positive definite and radially unbounded Ψ(x) is a Lyapunov function for syste (2) with the control Φ(x,w) if for all k > 0 there exists λ > 0 such that Ψ(x) Ψ(A(w(t))x(t) + B(w(t))Φ(x,w)) > λ (10) w W and x. Definition 2.7. A function Ψ : IR n IR is polyhedral if it is defined as follows Ψ(x) = Fx (11) where F is a full colun rank atrix. A crucial point for the considered class of systes is the following theore. Theore 2.1. Consider the following syste: ẋ(t) = f (x(t),w(t)), (12) where x(t) IR n is the state variable and w(t) W IR is the tie varying paraeter (W is a copact set and w(t) a piecewise continuous function). f is

3 continuous and locally Lipschitz on x uniforly in w. The following stateents are equivalent: (1) Syste (12) is GUAS with respect to the origin. (2) There exists a sooth Lyapunov function for (12). Proof 2.1. See Sontag et al. (1996). An earlier version was provided Meılakhs (1979) which holds for exponential stability only. A tailored version of this theore is stated below. Corollary 2.1. The following stateents are equivalent: (1) Syste (1) is GUAS with respect to the origin with the locally Lipschitz controller u(t) = Φ(x,w). (2) There exists a sooth Lyapunov function for ẋ(t) = A(w(t))x(t) + B(w(t))Φ(x,w). (13) Based on this fact, the following can be shown (Liberzon (2003)). Proposition 1. The closed loop syste (6) (or the corresponding discrete tie version) is GUAS if and only if its switched version ẋ = A(w i )x(t) + B(w i )Φ R (x) is GUAS. The sae property holds for discrete-tie systes. Indeed, any trajectory of the LPV syste is fored by vectors included in the convex hull of the vertices generated by the switching syste. This is why, when we deal with state feedback, we will not distinguish between switching and non-switching w. Lea 2.1. Assue there exists a Lyapunov function Ψ 1 (x) for syste (1) with the continuous control Φ 1 (x,w) locally Lipschitz uniforly with respect to w. Then there exists a polyhedral (and therefore convex) Lyapunov function Ψ 2 (x) and a control function Φ 2 : (vert{n (Ψ 2,1)},w) IR q for syste (1) such that any of the following equivalent conditions hold: there exists β > 0 such that D + Ψ 2 (x,φ 2 (x,w),w) β w W, x vert{n (Ψ,1)} (14) there exist τ > 0 and 0 λ < 1 such that Ψ 2 (x + τ[a(w)x + B(w)Φ 2 (x,w)]) λ w W, x vert{n (Ψ 2,1)} (15) Proof 2.2. It is basically the sae of that provided in Blanchini (1995). Reark 2.1. The first part of the previous lea states that there is no restriction in considering polyhedral Lyapunov functions when dealing with stability and the second part shows that such functions can be coputed by considering the discrete-tie Euler approxiating syste of (1), defined as x(t + 1) = (I + τa(w(t)))x + τb(w(t))u 3. EQUIVALENCES FOR CONTINUOUS TIME SYSTEMS 3.1 Matrix B with uncertainties It is known that for a continuous tie polytopic syste, the gain-scheduling stabilizability iplies robust stabilizability (and, of course, viceversa). Definition 3.1. An r r atrix H belongs to the set H if and only if it can be written as where P 1 < 1. H = τ 1 (P I), f orsoe τ > 0 (16) Theore 3.1. (Blanchini (2000)) The following stateents are equivalent. (1) There exists a locally Lipschitz stabilizing controller of the for Φ GS (x,w) for syste (1). (2) There exists a globally Lipschitz stabilizing controller of the for Φ R (x) for syste (1). (3) There exists r n and atrices X IR n r (full rank), U IR q r, and H i H such that, for k = 1,..., A i X + B i U = XH i (17) where A i, B i are the vertices of the polytopic syste. Fro the previous theore it follows that the knowledge of the disturbance acting on the syste is not necessary to achieve stability is the syste state is available for feedback. When switched gain-scheduling stabilizability is considered the scenario is different. Gain-scheduling (an therefore robust) stabilizability iplies switched gain-scheduling stabilizability, but the opposite iplications is not true in general as shown next. Exaple 3.1. Consider the uncertain syste ẋ = [α + 2(1 α)]x + [α (1 α)]u For α = 0 or α = 1 (switched case) Φ SGS = 3sgn[α (1 α)]x stabilizes the syste. When 0 α 1 the stabilizability is lost because for α = 0.5 the syste is unstable and not reachable. The next theore holds for switched systes. Theore 3.2. The following stateents are equivalent. (1) There exists a continuous stabilizing controller of the for Φ SGS (x,w i ) for (1).

4 (2) There exists r n and atrices X IR n r (full rank), U i IR q r, and H i H such that, for k = 1,..., A i X + B i U i = XH i. (18) Proof 3.1. Fro section 2 it follows that the first stateent is equivalent to the existence of a polyhedral Lyapunov function Ψ 2 (x) and a control Φ 2 (x,w i ). Therefore it just needs to be proved that (18) is equivalent to (15). Equation (15) iplies (see reark 2.1) that for every vertex x j of the unit ball Ω = N (Ψ 2,1) there exist u ji = Φ 2 (x ji,w i ) that assure (I + τa i )x j + τb i u ji λω (19) i = 1,...,. Equation (19) can be rewritten in the following way (I + τa i )x j + τb i u ji = X p ji (20) i = 1,...,, where p ji 1 1 and X represents the atrix X = [x 1 x 2... x r ]. Defining P i = [p 1i p 2i... p ri ] U i = [u 1i u 2i... u ri ] a copact version of (20) can be achieved: (I + τa i )X i + τb i U i = XP i, (21) that finally can be rewritten as A i X i + B i U i = X(P i I)/τ = XH i (22) obtaining (18). This procedure can be reversed to prove the existence of a polyhedral Lyapunov function starting fro the existence of the atrices H i. 3.2 Matrix B without uncertainties When the atrix B is not affected by uncertainty, the additional property that switched gain-scheduling stabilizability iplies gain-scheduling stabilizability holds. Theore 3.3. The next stateents are equivalent. (1) There exists a globally Lipschitz stabilizing controller of the for Φ R (x) for syste (4). (2) There exists a locally Lipschitz stabilizing controller of the for Φ GS (x,w) the syste (4). (3) There exists a locally Lipschitz stabilizing controller of the for Φ SGS (x,w) the syste (4). Proof 3.2. (1 2) follows fro Theore 3.1. (2 3) is trivial. (3 2) Stateent 3 iplies the existence of a sooth Lyapunov function Ψ(x) such that w i Ψ(x,w i ) = Ψ(A(w i )x + BΦ SGS (x,w i )) < 0 Choosing the following gain-scheduling controller (that is also locally Lipschitz) Φ GS (x,w) = Φ SGS (x,w i )w i the non-positivity of Ψ(x) holds: Ψ(x) = Ψ(A(w)x + BΦ GS (x,w)) = Ψ( = 3.3 Expanded systes A(w i )w i x + B Φ SGS (x,w i )w i ) w i Ψ(A(w i )x + BΦ SGS (x,w i )) < 0 The expanded syste is a concept already introduce in the quadratic stabilization fraework (Barish (1983)). In this subsection it will be shown that robust stabilizability iplies robust stabilizability of the expanded syste and viceversa. Theore 3.4. The following stateents are equivalent. (1) There exists a globally stabilizing controller of the for Φ R (x) for ẋ = A(w)x + B(w)u (23) (2) There exists a globally stabilizing controller of the for Φ R (x,u) for [ẋ ] [ ][ ] [ ] A(w) B(w) x 0 = + v (24) u 0 0 u where v IR q is the input variable of the expanded syste. Proof 3.3. (2 1) Fro theore (3.1) it follows that [ ][ ] [ ] [ ] Ai B i X 0 X + V = H i 0 0 U U The equation given fro the first row guarantees the stability of (23). (1 2) As a consequence of theore 3.1, there exists r n and atrices X IR n r (full rank), U IR q r, and H i H such that, for i = 1,..., A i X + B i U = XH i (25) An equivalent gain-scheduling for for (24) is now sought. The above equation can be written as follows: [ ][ ] [ ] [ ] Ai B i X 0 X + V i = H i 0 0 U U with V i = UH i. Theore 3.2 can be used to proof the switched gain-scheduling stability of (24), but this is not guaranteed if [X T U T ] T is not a full row rank atrix. In this case q zero coluns are added to the vector X obtaining the following equation (equivalent to (25)) [ ] Hi 0 A i [X 0] + B i [U 0] = [X 0] 0 0 and its expanded version [ ][ ] [ ] Ai B i X U 0 V i = [ X 0 U 0 ][ ] Hi 0 0 0

5 where V i = [V i 0]. A perturbation to the above equation is now introduced [ ][ ] [ ] [ ] Ai B i X 0 0 X 0 + Z i = Ĥ i 0 0 U γi U γi where γ > 0, V i = [V i V i ] and [ Hi H ] i Ĥ i = 0 H i It is alway possible to find V i and Ĥ i such that the perturbed equation holds. The full row rank condition is now satisfied. The last thing to be proved is the existence of γ such that Ĥ i H. Since H i H, it follows that H i = τ 1 (P i I), where P i 1 < 1. Using the sae value of τ, the following conditions can be checked: [H i T (H i τi) T ] T 1 < τ B i γi = XH i It is always possible to find H i to satisfy the last equation since X is full row rank (finding such atrices corresponds to solve n q linear equations with r q variables). Decreasing the value of γ also H i 1 becoes saller and therefore a suitable H i to satisfy the first inequality can be found. So far only the switched gain-scheduling stabilizability of (24) has been proved. Due to theore 3.1, the expanded syste is also gain-scheduling and robustly stabilizable. 4. EQUIVALENCES FOR DISCRETE TIME SYSTEMS 4.1 Matrix B with uncertainties In this subsection it will be shown that for discrete tie systes not all the iplications that hold in the continuous tie case are still valid. Anyway, soe theores that reseble the results of the previous section can be stated. In Blanchini (1995) the following theore has been proved. Theore 4.1. The following stateents are equivalent: (1) There exists a stabilizing controller of the for Φ RS (x) for syste (2). (2) There exists r n and atrices X IR n r (full rank), P i IR q r and P 1 < 1 such that, for k = 1,..., A i X + B i U = XP i. (26) For the gain-scheduling case there is a siilar result only when the syste is switched. Theore 4.2. The following stateents are equivalent: (1) There exists a stabilizing controller of the for Φ SGS (x,w) for syste (2). (2) There exists r n and atrices X IR n r (full rank), P i IR q r and P 1 < 1 such that, for k = 1,..., A i X + B i U i = XP i. (27) Proof 4.1. The proof is siilar to one given for theore 3.1 in Blanchini and Miani (2003) and it will be oitted for brevity. For a discrete tie syste robust stabilizability obviously iplies gain-scheduling stabilizability (RS GS). Unfortunately for discrete tie systes the equivalence does not hold, as it can be easily shown by eans of a counterexaple. Exaple 4.1. Consider the following syste with one state and one input variable x(t + 1) = w(t)x(t) + u(t) where w(t) 3. It is easy to show that the syste is gain-scheduling stabilized by u(t) = w(t)x(t) but it is not robustly stabilizable. Also for discrete tie systes switched gain-scheduling stabilizability does not iply gain-scheduling stabilizability as it can be shown with the following exaple. Exaple 4.2. Consider the uncertain syste x(t + 1) = 2x(t) + [α (1 α)]u(t) For α = 0 or α = 1 (switched case) it is possible to find Φ SGS to stabilize the syste. When 0 α 1 the stabilizability is lost because for α = 0.5 the syste is unstable and not reachable. 4.2 Matrix B without uncertainties When there are no uncertainties on the atrix B the following theore can be stated. Theore 4.3. The following stateents are equivalent. (1) There exists a controller of the for Φ GS (x,w) for (5). (2) There exists a controller of the for Φ SGS (x,w) for (5). (3) There exists r n and atrices X IR n r (full rank), P i IR q r and P 1 < 1 such that, for k = 1,..., A i X + BU i = XP i. (28) Proof 4.2. (1) (3) was proved in Blanchini and Miani (2003). (2) (3) is a particular case of Theore 4.2.

6 4.3 Expanded systes Theore 4.4. The following stateents are equivalent. (1) There exists a globally Lipschitz stabilizing controller of the for Φ R (x) for x + = A(w)x + B(w)u (29) (2) There exists a globally Lipschitz stabilizing controller of the for Φ GS (x,w) for [ ] x + [ ][ ] [ ] A(w) B(w) x 0 = + v (30) 0 I u u + where v(t) IR q is the input variable of the expanded syste. Proof 4.3. Alost identical to that of Theore 3.4. For the discrete tie case, the expanded syste achieved fro a robustly stabilizable syste is only gain-scheduling stabilizable but it ay fail to be robustly stabilizable as in the case of the next exaple. Exaple 4.3. Consider the following syste: x(t + 1) = 2x(t) + (1 + w(t))u(t) (31) where w(t) 0.4. The controller u(t) = 2x(t) stabilizes the syste, but there is no stabilizing controller for the expanded syste. Note that the expansion of a syste correspond to the insertion of a delay. If the expanded robust stabilizability would be aintained, the theore, recursively applied, would iply robust stabilizability with an arbitrary delay on the control input. 5. SUMMARY OF THE IMPLICATIONS AND DISCUSSION Let us denote by RS = robust stabilizability; GSS = gain-scheduling stabilizability; SGSS = switched gain-scheduling stabilizability; ERS = expanded robust stabilizability; EGSS = expanded gain-scheduling stabilizability; ESGSS = expanded switched gain-scheduling stabilizability. In the continuous tie case, the iplications aong these concepts are reported next. B with uncertainties RS GSS = SGSS B without uncertainties RS GSS SGSS Equivalences for the expanded syste RS ERS EGSS ESGSS In the corresponding discrete-tie table, reported below, several of the becoe =. B with uncertainties RS = GSS = SGSS B without uncertainties RS = GSS SGSS Equivalences for the expanded syste RS ERS = EGSS ESGSS In conclusion, we have seen how several options in the uncertainty specification and in the controller class can be crucial in the stabilization of LPV systes. We have also seen how continuous and discrete tie probles are differently affected by these options. REFERENCES B. R. Barish. Stabilization of uncertain systes via linear control. IEEE Trans. Autoat Control, 28(8): , F. Blanchini. Non-quadratic lyapunov function for robust control. Autoatica J. IFAC, 31(3): , F. Blanchini. The gain scheduling and the robust state feedback stabilization probles. IEEE Trans. Autoat Control, 45(11): , F. Blanchini and S. Miani. Stabilization of LPV systes: state feedback, state estiation and duality. SIAM J. Control Opti., 42(1):76 97, E. De Santis, M. D. Di Benedetto, and L. Berardi. Coputation of axial safe sets for switching syste. IEEE Trans. Autoat Control, 49(2): , D. Liberzon. Switching in Systes and Control. Birkauser, Boston, USA, A. M. Meılakhs. Design of stable control systes subject to paraetric perturbation. Autoat. Reote Control, 39(10): , A. P. Molchanov and E. S. Pyatnitskiĭ. Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systes. I. Autoat. Reote Control, 47(3): , E. D. Sontag, Y. Wang, and Y. Lin. A sooth converse lyapunov theore for robust stabylity. SIAM J. Control Opti., 34(1): , Z. Sun and S.S. Ge. Analysis and synthesis of switched linear control systes. Autoatica J. IFAC, 41(2): , 2005.