Robust stabilization of linear uncertain plants with polynomially time varying parameters

Transcription

1 Preprints of the 19th World Congress The International Federation of Automatic Control Robust stabilization of linear uncertain plants with polynomially time varying parameters L. Jetto V. Orsini R. Romagnoli Dipartimento di Ingegneria dell Informazione, Università Politecnica delle Marche, Ancona, Italy, Abstract: The robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics is considered. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. The stability conditions of the switching closed-loop system are derived defining a switched Lyapunov function and involving the dwell time of the system over each single mode. An important feature of the paper is that, unlie all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives. 1. INTRODUCTION In recent years analysis and synthesis of control systems for linear time-varying (LTV) plants with polytopic uncertainties have been widely investigating in different settings and from different points of view. Much attention has been recently devoting to the synthesis of control systems for linear parameter-varying (LPV) plants, (see e.g. Aparian et al. [1995],Daafouz et al. [2008], Jetto et al. [2010a], Heemels et al. [2010], Oliveira et al. [2009] and references therein). A wide literature also exists on the robust analysis and synthesis of LTV uncertain polytopic system with no on-line available information on physical parameters (see e.g. Daafouz et al. [2001], Dong et al. [2008], Geromel et al. [2006],Jetto et al. [2009],Jetto et al. [2010b], Mao [2003], Rugh et al. [2000],Trofino et al. [2001] and references therein). A common assumption of all the above papers is that the unnown parameters belong to a bounded (and generally small) uncertainty domain. The stability analysis reported in Jetto et al.[2009] showed that it is possible to state stability conditions under arbitrarily large time varying parametric uncertainties with possibly arbitrarily large variations rates. This is possible under the assumption of plants with a time-varying dynamical matrix whose elements are described by interval polynomial functions. The purpose of this paper is to extend the results of Jetto et al. [2009] to the controller synthesis problem for a plant with a dynamics switching among a finite number of modes. The physically meaningful modeling assumption allows us to transfer the uncertainty from the domain of the process parameter space to that of the relative polynomial coefficient space so that arbitrarily large uncertainty region can be obtained by increasing time. The stabilization of each single mode is obtained through an observer based controller with gain matrices polynomially depending on the time. A parameter dependent Lyapunov function whose matrix is itself polynomially depending on time is adopted and the stabilization problem is solved here defining a set of BMI s which reduce to a set of LMI s fixing two positive scalars. The main evident theoretical interest of the present paper is that, unlie all the other approaches, it allows the synthesis of a stabilizing controller for uncertain non uniformly bounded dynamical matrices. The overall controller is given by the switching among the family of observer-based controllers designed for each single mode. Closed-loop stability conditions are stated in terms of permanence time intervals of the plant dynamics over the same mode. This is accomplished by defining a suitable switched Lyapunov function. 2. SINGLE MODE PLANT Consider the following uncertain polynomially time varying plant Σ {ẋ(t) = A(t,α)x(t)+Bu(t) Σ = (1) y(t) = Cx(t) l with A(t,α) = A 0 (α 0 )+ A i (α i )t i, t 0, (2) where: A i (α i ) = n i j=1 α i,ja i,j, n i j=1 α i,j = 1, α i,j 0, and A i,j,i = 0,,l;j = 1,,n i, are nown square constant matrices. Remar 1.TheassumptionofconstantmatricesB andc is notalossofgeneralitybecause,asshowninaparianetal. [1995], it can always be satisfied if proper LTI filters are applied to the original signals u(t) and y(t). This implies a controller of increased order. The robust stabilization problem defined in this section consists in finding (if it exists) a dynamic output feedbac controller Σ c which guarantees the exponential γ-stability (as defined in Jetto et al. [2009]) of the closed-loop uncertain polynomially time varying system Σ f given by the Copyright 2014 IFAC 11734

2 feedbac connection of Σ c with Σ. The two following preliminary results will be exploited to solve the aforementioned problem. PR1: If G is a positive definite matrix, X and Y are matrices of appropriate dimensions and ε is a positive scalar, then the following direct consequence of the Schur complement holds [ ] [ 0 XY T εxgx T YX T 0 0 ] 0 1 ε YG 1 Y T. PR2, Song et al. [2011]: Let Φ be a symmetric matrix and N, M be matrices of appropriate dimensions. The following statements are equivalent: (1) Φ < 0 and Φ+NM[ T +MN T < 0; ] Φ M +NF (2) The LMI problem M T +F T N T F F T < 0 is feasible with respect to F. Consider the following observer based controller Σ c { ξ(t) = A(t)ξ(t)+Bu(t) L(t)(y(t) Cξ(t)) Σ c = (3) u(t) = K(t)ξ(t) wherea(t)definesasortof nominalcentralplant andis given by A(t) = A 0 + ni l A it i j=1, t 0, A i = Ai,j n i. The gains K(t) and L(t) are polynomially time varying matrices defined as K(t) = K l t l L(t) = L l t l, (4) where K l and L l, are constant matrices to be computed. The assumed form of K(t) and L(t) will be justified in the light of Remar 3 reported later. Applying the usual transformation matrix, the state space representation of the uncertain time varying closed loop system Σ f = ( Ĉ f,âf(t,α)) is ] A(t,α)+BK(t) BK(t) ˆx f (t)=[ ˆx A(t, α) A(t) + L(t)C f (t) (5) y(t)=[c 0]ˆx f (t) (6) [ T, where: ˆxf (t) = ẋ T (t), ẋ( T t) ξ (t)] T A(t,α) = A(t,α) Ā(t) = l A i(α i )t i, with A i (α i ) (A i (α i ) A i ) = n j j=1 α i,j A i,j, A i,j = Ai,j Āi, nj j=1 α i,j = 1, α i,j 0. To investigate the stability of Σ f the following parameter dependent Lyapunov function is considered where V(ˆx f (t),α) = ˆx f (t) T R(t,α)ˆx f (t), α S (7) R(t,α) = [ ] P(t,α) W(t,α) W T = (t,α) Q(t,α) = l R i (α i )t i (8) is a symmetric positive definite matrix t 0, α S, with Pi (α R i (α i )= i ) W i (α i ) Wi T,i = 0,,l 1, (α i ) Q i (α i ) Pl (α and R l (α l )= l ) 0. (9) 0 Q l (α l ) The time derivative of V(ˆx f (t),α) is V(ˆx f (t),α) = ˆx T f (t)h(t,α)ˆx f(t) (10) with H(t,α) = ÂT f (t,α)r(t,α)+r(t,α)âf(t,α)+ṙ(t,α). By(5)and(8),H(t,α) isthesymmetricpolynomiallytime varying matrix given by H(t,α) = (t,α) H (1,2) (t,α) (t,α) H (2,2). (11) (t,α) The form of each single bloc H (i,j) (t,α) is reported in (12) at the top of page 3. Exploiting the polynomial timedependence of matrices on the r.h.s. of (12) (see page 3), an equivalent representation of (10) is ( V(ˆx f (t),α) = ˆx T f(t) H 0 (α)+ H (α)t )ˆx f (t)(13) =1 with H (α) = (α) H (1,2) (α) (α) H (2,2), = 0,,. (α) Exploiting (10)-(13), it can be shown that each single bloc H (i,j) (t,α), i,j = 1,2 of H(t,α) in (11) has the form: H (i,j) (t,α) = H (i,j) 0 (α) + =1 H(i,j) (α)t. Each term H (i,j) (α) has the form reported in Appendix. Remar 2. A conceptually simple but algebraically tedious generalization of the results reported in the Appendix shows that even assuming K(t) = l K it i, L(t) = l L it i, K i,l i 0, in any case H (α) only depends on K l and L l. As neither R(t,α), nor A(t,α) are uniformly bounded, the stability analysis of Σ f requires the two following lemmas. Lemma 1. Jetto et al. [2009] If there exists a finite t 0 such that V(ˆx f (t),α) < 0, α S, t t, (14) then Σ f is exponentially γ-stable. Lemma 2. Jetto et al. [2009] If [0,1,,] such that α S, one has H (α) < 0, H +j (α) 0, 1 j, (15) then condition (14) holds t t = t( ) 0, where t( ) is the minimum t such that 1 H (α)t ( ) < H (α)t ( ). (16) =0 = Remar 3.TheimportanceofLemma2isthatthestability of Σ f can be guaranteed with no constraint on H (α), for <. Hence a robust stabilizing output dynamic controller can be found by simply imposing the fulfillment of (15) for =. This drastically reduce the numerical 11735

3 (t,α) = P(t,α)(A(t,α)+BK(t))+(A(t,α)+BK(t)) T P(α)+ P(t,α)+W(t,α) A(t,α)+ A T (t,α)w T (t,α) (t,α) = W T (t,α)(a(t,α)+bk(t))+q(t,α) A(t,α) K T (t)b T P(t,α)+(A(t)+L(t)C) T W T (t,α)+ẇt (t,α) H (2,2) (t,α) = Q(t,α)(A(t)+L(t)C) +(A(t)+L(t)C) T Q(t,α)+ Q(t,α) W T (t,α)bk(t) K T (t)b T W(t,α) (12) complexity of the synthesis procedure because H (α) = H (α l ) < 0 only involves ( the following ) matrices: A l (α l ), Pl (α A l (α l ), R l (α l ) = l ) 0, K 0 Q l (α l ) l and L l. The diagonalstructureofr l (α l ) introduces someconservatism but on the other hand allows the simultaneous design of K l and L l under the assumption of parametric uncertainties arbitrarily increasing with time. The counterpart is that V(ˆxf (t),α) < 0, α S, is not satisfied for t 0 and hence both a slower convergence to the null equilibrium point and a more conservative estimate of the dwell time are obtained. Nevertheless, as shown later, the conservatism can be greatly reduced through an iterative constrained minimization procedure(icmp) based on semidefinite programming. The ICMP exploits the degrees of freedom introduced by the full bloc matrices R (α ), = 0,,l 1, of R(t,α) to minimize the time instant t = t( ) = t() such that (16) for = is satisfied (or equivalently (14) hold). Remar 4. In the light of Remar 3, Remar 2 justifies the assumption on the form of K(t) and L(t) given by (4). Theorem 1. The condition (15) of Lemma 2 holds for = if there exist positive definite symmetric matrices P l,j, Q l,j, j = 1,,n l, Z, Jl and matrices V l, X l, Y l, such that for some fixed positive scalars β l, γ l, conditions (17)-(20)(reported at the top of page 4) are satisfied. Proof of Theorem 1. Not reported for brevity. If the set of LMIs (17)-(20) admits a solution for a fixed pair (γ l,β l ) then (15) holds for =. By Lemmas 1 and 2, the closed loop system Σ f results to be exponentially γ-stable for some γ > 0. The gain matrices characterizing the dynamic output controller (3)-(4) are: K l = Y l Z 1 L l = V T l X l. (21) The respective( Lyapunov function ) (7)-(9) is characterized Pl (α l ) 0 with vertices R 0 Q l (α l ) l,j = ), j = 1,,n 0 Q l and by the full l,j bloc matrices R i (α i ) s, i = 0,, 1 with vertices by R l (α l ) = ( Z T Pl,j Z 1 0 R i,j = Pi,j W i,j Wi,j T Q i,j still to be determined. According to Remar 3, they are degrees of freedom exploited by the ICMP for minimizing t = t(), as explained beneath, and hence to reduce the introduced conservatism imposing V(ˆx f (t),α) < 0 for t t > Minimization of t As =, by Lemma 2, t is the minimum t such that 1 H i (α) t i < H (α). (22) Taing into account that: 1) H (α) = H (α l ) < v I for some nown positive scalar v, 2) for each H i (α), i = 0,, 1, there exists a scalar v i, to be determinated, such that H i (α) < v i I, inequality (22) maesitevidentthattoreduce titisnecessarytominimize themaximumeigenvaluev i ofeachh i (α),i = 0,, 1, α S. Unfortunately, the l degrees of freedom R i (α i ), i = 0,,l 1, allow us to only solve l problems of minimization with respect to the maximum eigenvalues of l matrices H i (α) with i [0,, 1]. Inequality (22) suggests that a more significant reduction of t is obtained minimizing the maximum eigenvalues of the l matrices H l+ (α), for = 0,,l 1. As the l matrices H l+ (α), for = 0,,l 1, are functions of the unnown R i (α i ), i = 0,,l 1, according to the following relation for = 0,,l 1: H l+ (α) = f l+ (R (α ),R +1 (α +1,,R l 1 (α l 1 )), the ICMP has the following structure: (1) put = l 1 (2) min v l+ subject to H l+ (α) < v l+ I (23) Constraint (23) corresponds to the finite set of LMIs whose unnowns to be determinated are v l+ and the vertices of the polytopic R (α ) (3) put = 1, if 0 go to step (2) otherwise end. At the end of the ICMP, all the vertices of R i (α i ) s, i = 1,,l 1, have been computed. This means that all available degrees of freedom have been used and, as a consequence, the matrices H i (α), i = 0,,l 1, (namely those ones which have not been taen into account in the minimization procedure) are automatically determinated at the vertices as well as their respective maximum eigenvalues v i. In conclusion, by (22) one has that the finite t = t() such that V(ˆx f (t),α) < 0, t t() is obtained in the following way: t = t() = min t : 1 v i t i < v. (24) 3. SWITCHING MODE PLANTS The results of Section 2 are here exploited to investigate stabilizability conditions for the class of switching systems Σ σ(t) that, according to the introductory considerations, are described by {ẋ(t) = Aσ(t) (t,α Σ σ(t) = (σ(t)) )x(t)+bu(t), y(t) = Cx(t) (25) where σ(t) : [0, ) P = [1,, p] is a piecewise constant function which represents the switching signal, 11736

4 2γ l Pl,j + J l Pl,j +Z T A T l,j +Yl T B T +γ l Z T Pl,j P l,j +A l,j Z +BY l +γ l Z Z Z T 0 < 0, j = 1,,n l, (17) P l,j 0 Z T [ ] Jl Z T A T l,j < 0, j = 1,,n A l,j Z I l, (18) [ ] Z Y T l B T < 0, (19) BY l Z 2β l Q l,j Q l,j +A T l V l +C T Xl T +β lv l Q l,j I Q l,j +Vl T A l +X l C +β l Vl T V l Vl T 0 0 Q l,j 0 I 0 < 0, j = 1,,n l. (20) I 0 0 Z T its value σ(t) identifies the particular mode acting at time t. Any interval over which a particular mode p is active is denoted by T (p) = [t (p) ;t(q) m ),,m Z +. This notation means that over T (p) the p-th mode occurs for the -th time (sincet = 0)andit isfollowedbythem-thoccurrence (since t = 0) of the q-th mode. Hence, t T (p), each A p (t,α (p) ) is of the same ind of A(t,α) given by (2) and can be written as lp A p (t,α (p) ) = A p,0 (α (p) 0 )+ A p,i (α (p) i )(t t (p) )i (26) where A p,i (α (p) i ) = n pi j=1 α(p) i,j A(p) i,j. The following assumption is made: A1). Both the switching instant and the new configuration assumed by the switched plant are assumed to be nown. The Lyapunov function associated to each mode is defined as : V p (ˆx f (t),α (p) ) = ˆx T f (t)r p(t,α (p) )ˆx f (t), p P, where, accordingly to (26), the time is reset at every switching instant defining R p (t,α (p) ), t T (p) as lp R p (t,α (p) ) = R p,0 (α (p) 0 )+ R p,i (α (p) i )(t t (p) )i.(27) Each R p,i (α (p) i ),p P, is of the same ind of R i (α i ) given by (9). Moreover it is assumed : A2). R p,0 (α (p) 0 ) = R 0(α 0 ), p P,forasuitablydefined constant vector α 0 R n0 where n 0 = max p {n p 0 }. Assumption A2) implies that the degree of freedom R 0 (α 0 ) is common to all the p Lyapunov matrices R p (t,α (p) ). As it will be explained later, this hypothesis will allow us to derive stabilizability conditions for the switching plant Σ σ(t). The derivative of each V p (ˆx f (t),α (p) ) is V p (ˆx f (t),α (p) ) = ˆx T f (t)h p(t,α (p) )ˆx f (t),whereanalogouslyto(13), t T (p), H p (t,α (p) ) is given by p H p (t,α (p) ) = H p,0 (α (p) )+ H p,i (α (p) )(t t (p) )i.(28) Let Σ f,p be the feedbac connection of the mode Σ p with the corresponding stabilizing observer based controller Σ c,p computed as explained in section 2. By Lemma 2, each V p (ˆx f (t),α (p) ) is negative definite t t p ( p ). By A2), the p ICMP s are independent of each other until = 1. For = 0, the p constraints of the ind of (23) must be simultaneously satisfied. Hence putting all the v (p) l p s, p = 1,, p, equal to v one has min v subject to H 1,l1 (α (1) ) < vi. H p,l p (α ( p) ) < vi (29) where the unnowns are the scalar v and the vertices of R 0 (α 0 ). Analogously to (24), each t p = t p ( p ) > 0, p P, can be obtained as: t p ( p ) = min t : l p 1 v (p) i t p i + v t lp + p 1 i=l p+1 v (p) i < v(p) p i p. t (30) 3.1 Stabilizability conditions for the switching mode plant Provided that the conditions of Theorem 1 are satisfied, each controller Σ c,p stabilizing the corresponding Σ p guarantees that inside each T (p), p P, there exists a t (p), with t (p) t (p) = t p independent of, such that V p (ˆx f (t),α (p) ) < 0, t t (p), t T(p). In practice t (p) has the same meaning of t p in (30) for a single mode of the plant. Stability conditions for the switching closed loop system Σ f,σ(t) are stated in terms of minimum dwell time, as stated in the following Theorem. Theorem 2. Provided that each Σ f,p is exponentially γ p stable for some γ p > 0, the switching system Σ f,σ(t) is asymptotically stable if the length L (p) of each T (p) = m ), p P,,m Z + is such that: [t (p) ;t(q) L (p) τ p > t p. where τ p is the minimum time(31) such that t (p) +τp t (p) V p (ˆx f (t),α (p) )dt < 0. (32) 11737

5 Proof of Theorem 2. Not reported for brevity. 4. NUMERICAL RESULTS Consider the following dynamical plant Σ σ(t) dependent on the switching signal σ(t) : [0, ) P = [1, 2], described by the following triplet (C,A p (t,α (p) ),B): C = [ ] [ ] θ (p) (t), ]. The dynamical matrix is borrowed from [0 1], A p (t,α (p) ) = [ 1 0 B = Montagner et al. [2003], where the stability is proved in the uncertainty range θ(t) [0,1]. The parameter θ(t) is here assumed to be a switching interval polynomial function θ (p) (t) of degree l p = 2, more precisely: θ (1) (t) = [0, 0.2] + [0.01, 0.02]t+ [0.005, 0.01]t 2 and θ (2) (t) = [0, 0.2]+[ 0.02, 0.01]t+[ 0.002, 0.001]t 2. Unlie all the existing literature both the parameter and its derivative are allowed to vary over theoretically unbounded uncertainty sets. For each mode p P, by varying both γ lp = γ 2 and β lp = β 2 with a logarithmic scale inside [10 1, 10 1 ], conditions of Theorem 1 result to be satisfied for γ 2 = β 2 = 1. This means that H p,p (α (p) ) < 0, for each p P. More precisely the solution is given by: H 1,4 (α (1) ) < v (1) 4 I = I (p = 1) and H 2,4 (α (2) ) < v (2) 4 I = I (p = 2). By Lemma 2, Vp (ˆx f (t),α (p) ) < 0, t t p ( p ) for some t p ( p ) > 0. By Lemma 1, each Σ f,p, p P is exponentially stable and the gain matrices, obtained through (21), of the respective controller Σ c,p are: K 2 =, L =, (p = 1), K 2 =, L = (p = 2) By Remar 3, as l p = 2, p P, two degrees of freedom R p,1 (α (p) 1 ) and R p,0(α (p) 0 ) = R 0(α 0 ) are available for minimizing t p ( p ). Each ICMP consists of two iterations described as reported at the top of page 6. By A2), the two ICMP s are simultaneously solved in the last iteration. The solution is given by: v (1) 3 = v (2) 3 = 1 and v = As mentioned in Section 2, the maximum eigenvalues v (p) 1 and v (p) 0 of H p,1 (α (p) ) s and H p,0 (α (p) ) s respectively p P, are automatically determinated and their values are v (1) 1 = 479.5, v (2) 1 = 11,112, v (1) 0 = 29,415 and v (2) 0 = 29,433. The Lyapunov functions R p (t,α (p) ) s, p P are not reported to save space. Applying (30) one has: V 1 (ˆx f (t),α (1) ) < 0, t t 1 = 20 and V 2 (ˆx f (t),α (2) ) < 0, t t 2 = 26. Assuming to now that x(0) 1, some calculations (not reported for brevity) show that condition (32) is satisfied for τ 1 = 29, (p = 1) and τ 2 = 41, (p = 2). Recalling that L p is the length of T(p), by (31), the switching closed loop system Σ f,σ(t) is asymptotically stable if L 1 29 and L A simulation has been performed starting from x(0) = [0.1,0.1,0,0] T. The trajectories of θ (1) (t) and θ (2) (t) have been generated according to θ (1) (t) = t+0.01 t 2 (p = 1) and θ (2) (t) = t t 2 (p = 2) respectively. Over a simulation interval of amplitude 80, the switching plant Σ σ(t) is such that: Σ σ(t) = Σ 1 (t), t [0,30] = T (1) 1 and Σ σ(t) = Σ 2 (t), t (30,80] = T (2) 1. The output response of the switching closed loop system Σ f,σ(t) practically converges to zero for t 1. The plot is not reported for brevity. 5. CONCLUSIONS The main novelty of this paper is a controller synthesis method for uncertain plants whose parameters are allowed to vary inside arbitrarily large domains. Another interesting by-product of the paper is the way each single-mode stabilizing controller is derived. The dynamic outputfeedbac controller is obtained through a set of LMI s by fixing two positive scalars. If the set is feasible, it provides both the observer and the static gains in only one step for parametric uncertainty arbitrarily increasing with time. Appendix Explicit expression of H (i,j) (α): Exploiting (10)-(13), it can be shown that each term H (i,j) (α) has the form reported on page 6. REFERENCES P.Aparian,P.Gahinet andg. Becer. Self-scheduledH control of linear parameter varying systems: a design example. Automatica, 135: , J. Daafouz, J.Bernussou. Parameter-dependent Lyapunov functions for discrete-time systems with time varying parametric uncertainties. Syst. & Contr. Lett., 43: ,2001. J. Daafouz, J. Bernussou, and J.C.Geromel. On inexact LPV control design of continuous-time polytopic systems. IEEE TAC, 53: , J. Dong, G.H. Yang. Robust static output feedbac control for linear discrete-time systems with time-varying uncertainties. Syst & Contr. Lett., 57: ,2008. J.G.Geromel and P. Colaneri. Robust stability of time varying polytopic systems. Syst. & Contr. Lett., 55: 81 85,2006. L. Jetto, V. Orsini. LMI conditions for the stability of linear uncertain polynomially time varying systems. IEEE TAC, 54: , L.Jetto, V. Orsini. 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6 ICMP s (p=1) (p=2) i) min v (1) 3 subject to H 1,3 (α (1) ) < v (1) 3 I min v(2) 3 subject to H 2,3 (α (2) ) < v (2) 3 I ii) min v subject to { H1,2 (α (1) ) < vi H 2,2 (α (2) ) < vi H i,j (α), i = j = 1 (α) = [P i (α i )A j (α j )+A T j (α j )P i (α i )+W i (α i ) A j (α j )+ A T j (α j )Wi T (α i )]+( +1)P +1 (α +1 ), 0 l 1 l+ 0 l 1 0 l 1 i+j= i,j 0 (α) = P (α )(A l (α l )+BK l )+(A l (α l )+BK l ) T P (α )+P l (α l )A (α )+A T (α )P l (α l ) + W (α ) A l (α l )+ A T l (α l )W T (α ) + [P i (α i )A j (α j )+A T j (α j)p i (α i )+W i (α i ) A j (α j )+ A T j (α j)wi T (α i)], +1 i l 1 i+j=l+ (α) = P l (α l )(A l (α l )+BK l )+(A l (α l )+BK l ) T P l (α l ) H i,j (α), i = 2, j = 1 (α) = { [Wi T (α i)a j (α j )+Q i (α i ) A j (α j )+A T ( +1)W j WT i (α T i)]+ +1 (α +1 ), = 0,,l 2 0, = l 1 l+ 0 l 1 i+j= i,j 0 (α) = W T (α )(A l (α l )+BK l )+Q (α ) A l (α l )+Q l (α l ) A (α ) + (A l +L l C) T W T (α ) Kl T B T P (α ) + [Wi T (α i)a T j (α j)+a j Wi T (α i)+q i (α i ) A j (α j )] +1 i l 1 i+j=l+ (α) = Q l (α l ) A l (α l ) Kl T B T P l (α l ) H i,j (α), i = j = 2 (α) = H (2,2) 0 l 1 H (2,2) l+ 0 l 1 i+j= i,j 0 [Q i (α i )A j +A T j Q i(α i )]+( +1)Q +1 (α +1 ), (α) = Q (α )(A l +L l C)+(A l +L l C) T Q (α )+Q l (α l )A +A T Q l(α l ) W T (α )BK l Kl T B T W (α )+ [Q i (α i )A j +A T j Q i (α i )] +1 i l 1 i+j=l+ H (2,2) (α) = Q l (α l )(A l +L l C)+(A l +L l C) T Q l (α l ) R.C.L.F. Oliveira, M.C. de Oliveira, P.L.D. Peres. Special time-varying Lyapunov function for robust stability analysis of linear parameter varying systems with bounded parameter variation. IET Control Theory Appl., 3: , W.J.Rugh and J.S. Shamma. A survey of research on gainscheduling. Automatica, 36: , M. Sato. Gain-scheduled output-feedbac controllers using inexactly measured scheduling parameters. 49th CDC, Atlanta,GA,USA, L.Song, J. Yang. An improved approach to robust stability analysis and controller synthesis for LPV systems. Int. J. Robust and Nonlinear Control, 21: , A. Trofino, and C.E. De Souza. Biquadratic stability of uncertain linear systems. IEEE TAC, 46: ,