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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY TABLE IV COMPARISON OF NUMERICAL SENSITIVITY FROM BOTH APPROACHES as (19), and copute the standard deviation of the values of the corresponding perturbed polynoial at the point ( 3 1 ;3 2). We perfor the test at ( 3 1 ;3 2)=(20; 20); (0; 20); (20; 0); (10; 10) and (0:6; 0:6) and copare the results of both approaches in Table IV. There are several reasons showing that our approach is proising. First, we can see fro Tables I and III that the nuber of variables in the conventional approach grows rapidly when the degrees of the onoial bases increase, while the nuber of variables in our approach grows linearly with respect to the nuber of subregions. Moreover, our approach attains the exact optial value with less coputational cost than the conventional SOS approach. Secondly, Table IV shows that the standard deviations of the perturbed polynoial fro the proposed approach are less than that fro the conventional approach, for all selected points. This iplies that the optial value of the proposed approach is less nuerically sensitive than that of the conventional approach. Strictly speaking, the asyptotic exactness of our schee is not guaranteed in the case of lowest-degree onoial bases. However, it is achieved apparently in this exaple. We expect that the asyptotic exactness can also be proved with the onoial bases of lowest degrees, and this is the direction of our further research. [13] J. Löfberg and P. A. Parrilo, Fro coefficients to saples: A new approach to SOS optiization, in Proc. 43rd IEEE Conf. Decision Control, Paradise Island, The Bahaas, Dec. 2004, pp [14] Y. Oishi, A region-dividing approach to robust seidefinite prograing and its error bound, in Proc. Aer. Control Conf., Minneapolis, MN, Jun. 2006, pp [15] Y. Oishi, Reduction of the nuber of constraints in the atrix-dilation approach to robust seidefinite prograing, in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp [16] Y. Oishi and Y. Isaka, Exploiting sparsity in the atrix-dilation approach to robust seidefinite prograing, in Proc. Aer. Control Conf., New York, NY, Jul. 2007, pp [17] A. Ohara and Y. Sasaki, On solvability and nuerical solutions of paraeter-dependent differential atrix inequality, in Proc. 40th IEEE Conf. Decision Control, Orlando, FL, Dec. 2001, pp [18] P. A. Parrilo, Seidefinite prograing relaxations for seialgebraic probles, Math. Progra. Series B, vol. 96, no. 2, pp , [19] S. Prajna, A. Papachristodoulou, and P. A. Parrilo, Introducing SOS- TOOLS: A general purpose su of squares prograing solver, in Proc. 41st IEEE Conf. Decision Control, Las Vegas, NV, Dec. 2002, pp [20] C. W. Scherer, Robust perforance analysis for paraeter dependent systes using tensor product splines, in Proc. 37th IEEE Conf. Decision Control, Tapa, FL, Dec. 1998, pp [21] C. W. Scherer, Relaxations for robust linear atrix inequality probles with verifications for exactness, SIAM J. Matrix Anal. Appl., vol. 27, no. 2, pp , [22] C. W. Scherer, LMI relaxations in robust control, Eur. J. Control, vol. 12, no. 1, pp. 3 29, [23] C. W. Scherer and C. W. J. Hol, Matrix su-of-squares relaxations for robust sei-definite progras, Math. Progra., Series B, vol. 107, no. 1 2, pp , [24] J. F. Stur, Using SeDuMi 1.02, a MATLAB toolbox for optiization over syetric cones, Opti. Methods Software, vol , pp , REFERENCES [1] A. Ben-Tal and A. Neirovski, Robust convex optiization, Math. Oper. Res., vol. 23, no. 4, pp , [2] A. Ben-Tal and A. Neirovski, Lectures on Modern Convex Optiization: Analysis, Algoriths, and Engineering Applications. Philadelphia, PA: SIAM, [3] A. Ben-Tal and A. Neirovski, On tractable approxiations of uncertain linear atrix inequality affected by interval uncertainty, SIAM J. Opti., vol. 12, no. 3, pp , [4] P.-A. Blian, On robust seidefinite prograing, in Proc. 16th Int. Syp. Math. Theory Networks Syst. (MTNS 04), Leuven, Belgiu, Jul. 2004, [CD ROM]. [5] G. Calafiore and M. C. Capi, The scenario approach to robust control design, IEEE Trans. Autoat. Control, vol. 51, no. 5, pp , May [6] B. Chen and S. Lall, Degree bounds for polynoial verification of the atrix cube proble, in Proc. 45th IEEE Conf. Decision Control, San Diego, CA, Dec. 2006, pp [7] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, Polynoially paraeterdependent lyapunov functions for robust stability of polytopic systes: An LMI approach, IEEE Trans. Autoat. Control, vol. 50, no. 3, pp , Mar [8] G. Chesi, On the gap between positive polynoials and SOS of polynoials, IEEE Trans. Autoat. Control, vol. 52, no. 6, pp , Jun [9] L. El Ghaoui, F. Oustry, and H. Lebret, Robust solutions to uncertain seidefinite progras, SIAM J. Opti., vol. 9, no. 1, pp , [10] D. Henrion and J. B. Lasserre, Gloptipoly: Global optiization over polynoials with Matlab and SeDuMi, ACM Trans. Math. Software, vol. 29, no. 2, pp , [11] M. ojia, Su of Squares Relaxations in Polynoial Seidefinite Progras, Research Reports on Matheatical and Coputing Sciences Dept. Math. Coput. Sci., Tokyo Inst. Technol., Tokyo, Japan, [12] J. B. Lasserre, Global optiization with polynoials and the probles of oents, SIAM J. Opti., vol. 11, no. 3, pp , Robust Stability Analysis of Nonlinear Hybrid Systes Antonis Papachristodoulou and Stephen Prajna Abstract We present a ethodology for robust stability analysis of nonlinear hybrid systes, through the algorithic construction of polynoial and piecewise polynoial Lyapunov-like functions using convex optiization and in particular the su of squares decoposition of ultivariate polynoials. Several iproveents copared to previous approaches are discussed, such as treating in a unified way polynoial switching surfaces and robust stability analysis for nonlinear hybrid systes. Index Ters Hybrid systes, linear atrix inequality, su of squares, switched systes. I. INTRODUCTION Hybrid systes have dynaics that are described by a set of continuous (or discrete) tie differential equations in conjunction with a dis- Manuscript received July 02, 2007; revised March 21, Current version published May 13, This work was supported in part by the Engineering and Physical Sciences Research Council Grant EP/E05708X/1. Recoended by Guest Editors G. Chesi and D. Henrion. A. Papachristodoulou is with the Departent of Engineering Science, University of Oxford, Oxford OX1 3PJ, U.. (e-ail: antonis@eng.ox.ac.uk). S. Prajna was with Control and Dynaical Systes, California Institute of Technology, Pasadena, CA USA and is now with Credit Suisse, New York, NY USA (e-ail: prajna@cds.caltech.edu). 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2 1036 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009 crete-event (decision) process. Exaples include otion control systes and robotics [1], air traffic anageent [2], [3] etc. For the case of systes with continuous dynaics, stability properties are traditionally addressed using Lyapunov functions [4]. Extensions of these ideas to hybrid systes have appeared in, e.g., [5] [8]. See also [9] for a survey of the field, as well as the recent books [10] [12]. Piecewise quadratic Lyapunov functions have been introduced, which are constructed by concatenating several quadratic Lyapunov-like functions algorithically by solving a set of Linear Matrix Inequalities (LMIs) [13]. However, in soe cases such LMI conditions can be conservative or the nuber of quadratic Lyapunov-like functions needed is large, resulting in an increased coputational load. This technical note, which extends the work in [14] to treat robustness analysis in the presence of dynaic uncertainties, provides a ethodology for stability analysis of switched and hybrid systes. For proving stability, polynoial and piecewise polynoial Lyapunov functions are constructed using positive polynoials and the su of squares decoposition [15] [18], which can be efficiently coputed using seidefinite prograing, e.g., using the software [19]. An advantage of this ethod is that it provides a less conservative test for proving stability when switching between subsystes is arbitrary, provided that a finite nuber of switches occurs on every bounded tie interval. Moreover, we deonstrate that stability can be proven with a saller nuber of Lyapunov-like functions, eliinating the need of refining the state space partition in order to find quadratic such ultiple Lyapunov functions. The ethod can be readily applied to systes with nonlinear subsystes and nonlinear switching surfaces, therefore allowing uch richer syste descriptions. Finally, paraetric and dynaic robustness analysis can be perfored in a unified anner. Appropriate odeling for such systes, as well as the existence and uniqueness of solutions are iportant research topics [20], as any ties solutions ay not exist, ay not be unique (non-deterinis) or a Zeno behaviour is observed (infinite nuber of discrete transitions in finite tie). Here we will assue that the syste odels are such that these phenoena, including sliding odes and equivalent dynaics are avoided [21]. Moreover, we will say arbitrary switching to ean arbitrary switching in which only a finite nuber of switches is allowed in finite tie. The tools developed in this technical note can be extended to cover the case of systes with sliding odes if the additional ode that captures the sliding ode dynaics is added to the syste description and a condition that the Lyapunov function decreases along sliding-ode trajectories is iposed. The technical note is organized as follows. We first present soe preliinaries on hybrid and switched systes, as well as tools fro positive polynoials that we will be using to analyze the. In Section III we will forulate various algoriths for testing stability for hybrid and switched systes, giving exaples for the various cases. We then discuss robust stability analysis, before concluding the technical note. II. PRELIMINARIES A. Hybrid Systes We consider systes of the following for: _x = f l (x); l 2 L = f1;...;n Lg (1) where x 2 n is the continuous state, l is the discrete state (location), f l (x) is the vector field describing the dynaics of the l-th ode/subsyste (assued to be sufficiently sooth), and L is the finite index set. Executions (trajectories) of the syste are concatenations of a sequence of continuous flows and discrete transitions. During a continuous flow, the discrete location l is aintained and the continuous state evolves according to (1). The evolution of the discrete state l can be either tie-dependent or state-dependent. For tie-dependent switching we consider switching signals which are piecewise constant and continuous fro the right and which have a finite nuber of discontinuities on every bounded tie interval. For state-dependent switching, we assue that n is partitioned into operating regions X l, l =1;...;N L by guard sets (also called switching surfaces). These operating regions ay or ay not intersect, and their union is n. When the continuous state is x in location l and a guard set G(l; l 0 ) is et, a discrete transition to l 0 will occur and the continuous state will take the value x 0, which is prescribed by the single-valued reset ap R(l; l 0 )(x). Systes with infinitely fast switching, such as those that have sliding odes, are excluded fro our discussion, even though analysis in the case of sliding odes can still be perfored. We describe atheatically the regions X l by for soe g lk : given by X l = fx 2 n : g lk (x) 0; for k =1;...;X g (2) n!. A guard set between the l and l 0 odes is G(l; l 0 )=fx 2 n : h ll 0 (x) =0;h ll k (x) 0; for k =1;...; G(l;l ) g (3) for soe h ijk : n!. Lastly, the reset ap is given by R(l; l 0 )(x) = ll (x): (4) We assue that the origin is a coon equilibriu of the locations the stability of which we will investigate; this iplies that f l (0) = 0 for all l 2 L. 1 We will also consider systes of the for _x = f l (x; p); l 2 L = f1;...;n Lg (5) where p 2P denotes the uncertainty in the continuous flow which ay be tie-varying, in which case _p 2Q is bounded. Finally, we assue that the functions f l, g lk, h ll k and ll are polynoials. For the case in which any of these functions is nonpolynoial, see the coent at the end of Section III-D. B. Su of Squares Decoposition Our analysis is based on positive polynoials [15] [17] and the su of squares decoposition of ultivariate polynoials. A ultivariate polynoial p(x) is a su of squares if there exist polynoials p 1(x);...;p (x) such that p(x) = i=1 p2 i (x). This in turn is equivalent to the existence of a positive seidefinite atrix Q, and a properly chosen vector of onoials Z(x) such that p(x) =Z T (x)qz(x) [15]. What akes the su of squares decoposition attractive is the fact that it can be coputed using seidefinite prograing, since the coputation of Q is nothing but a search for a positive seidefinite atrix subject to soe affine constraints. Coupled with the property that p(x) being a su of squares iplies 2 p(x) 0, the su of squares decoposition provides a coputational relaxation for testing polynoial positivity, which belongs to the class of NP-hard probles. Three 1 Here we assue that 0 2X for all l 2 L. A relaxed assuption would be that f (0) = 0 for all l 2 L = fl 2 L j 0 2Xgand also that a transition can occur fro location l at state 0 only if l; l 2 L and (0) = 0. 2 Note that the converse iplication is true only in special cases. One such instance is when the polynoial is quadratic. 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3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY TABLE I THREE INDS OF POLYNOMIAL POSITIVITY (ON THE LEFT) AND THE CORRESPONDING SUM OF SQUARES CONDITIONS (ON THE RIGHT). CONDITIONS ON THE RIGHT ARE SUFFICIENT FOR THOSE ON THE LEFT. THE POLYNOMIAL DEGREE N IS ASSUMED TO BE EVEN, OTHERWISE THE POLYNOMIAL WILL BE NEGATIVE FOR SOME x. HERE >0 kinds of polynoial positivity and their corresponding su of squares coputational relaxations are shown in Table I. The su of squares decoposition has been exploited to algorithically construct Lyapunov functions for nonlinear systes [15], [18], [22], [23]. For this purpose, real coefficients c 1 ;...;c are used to paraeterize a set of Lyapunov functions in the following way: V = p(x) :p(x) =p 0 (x) i=1 c i p i (x) (6) where p i (x) are soe polynoials; for exaple they could be onoials of degree up to soe nuber. The search for a Lyapunov function V (x) 2V, or equivalently soe c i, such that V (x) is positive definite and dv =dt is negative definite can still be forulated as a su of squares proble and solved using seidefinite prograing. The results in the subsequent sections will be forulated in ters of inequalities such as V (x) 0 or V (x) > 0. However, if the coputation of Lyapunov functions is to be perfored using seidefinite prograing, then these inequalities have to be relaxed to su of squares constraints, in the way suarized in Table I. A. Stability of Hybrid Systes III. STABILITY ANALYSIS Stability of equilibria of hybrid systes has been addressed in [5], [6], [9], [24], [25]. We will use the following two Lyapunov theores in the sequel. The first theore concerns the case of what is known as a coon Lyapunov function: Theore 1: Consider a hybrid syste (1) with 0 an equilibriu point and let R(l; l 0 )(x) =x. Suppose that there exists an open set U n such that 0 2U. Let V : U! be a continuously differentiable function such that: 1) V (0) = 0 and V (x) > 0 for all x 2Unf0g, 2) (@V (x)=@x)f l (x) 0 for all x 2U, l 2 L. Then x =0is a stable equilibriu of the hybrid syste. If furtherore (@V (x)=@x)f l (x) < 0 for all x 2Unf0g, l 2 L then x =0is an asyptotically stable equilibriu. The proof of this theore can be found in, e.g., [10]. Global asyptotic stability can be obtained if U = n and V is radially unbounded. The above theore has the drawback that such a V ay be difficult to construct even if switching occurs between linear subsystes. But since the switching signal is not state-dependent, it is useful when investigating stability under arbitrary switching. A ore general theore that will lead our discussion on ultiple Lyapunov functions for deterining stability is stated below, which includes ipulsive jups, even if 0 is an equilibriu point. Recall that R(l; l 0 )(x) =x 0 is the reset ap when G(l; l 0 ) is et. Let us denote the switching ties by i so that R(l( i);l( i ))(x( i)) = x( i ), where i are the ties just after the switching ties. Theore 2: Consider a hybrid syste with 0 as an equilibriu point. For each l 2 L, suppose that there exists a continuously differentiable function V l : X l! such that: 1) V l (0) = 0 and V l (x) > 0 for all x 2X l n 0, 2) (@V l (x)=@x)f l (x) 0 for all x 2X l. If oreover, for all executions and for all switching ties i,wehave V l( ) (x( i )) V l( )(x( i)) then x =0is stable. The above theore (a proof of which can be found in [6]) considers any Lyapunov functions, each defined for each subsyste, satisfying the failiar Lyapunov conditions. The last condition ensures that during switches, the value of the Lyapunov function is non-increasing, even if the continuous state is reset. It can be relaxed to the stateent that V l( ) (x( i )) V l( ) (x(j )) where j < i is the tie that location l( i ) was last active. This condition is difficult to ipose algorithically, so we will use the condition stated in the theore instead, i.e., that the Lyapunov functions are non-increasing when switches occur. Corollary 3: Consider a hybrid syste with 0 as an equilibriu point. For each l 2 L, suppose that there exists a continuously differentiable function V l : X l! such that: 1) V l (0) = 0 and V l (x) > 0 for all x 2X l n 0, 2) (@V l (x)=@x)f l (x) 0 for all x 2X l, 3) V l (x 0 ) 0 V l (x) 0 for all x 2 G(l; l 0 );x 0 = R(l; l 0 )(x). Then x =0is a stable equilibriu. B. Stability Under Tie-Dependent Switching Here switching is tie-dependent, and the switching signal is piecewise constant and continuous fro the right. A finite nuber of switches is allowed on every bounded tie interval in order to exclude arbitrarily fast switching. We also assue X l = n and consider (1) with G(l; l 0 )= n, and R(l; l 0 )(x) =x. A sufficient condition for the stability of the origin in this case is the existence of a global coon Lyapunov function, as suarized in the following theore. Theore 4: Suppose that for syste (1) there exists a polynoial V (x) such that V (0) = 0 and V (x) > 0 8x 6= 0;V(x) radially unbounded f l(x) < 0 8x 6= 0; l2 L (8) then the origin is globally asyptotically stable for arbitrary switching. Notice in particular that if the vector fields are linear, i.e., f l (x) = A l x, and if V (x) is chosen to be quadratic, say V (x) = x T Px, then the conditions in Theore 4 correspond to the well-known LMIs P > 0, A T l P PA l < 0 for all l, which prove quadratic stability of the syste but ay be conservative. Several researchers have considered the use of non-quadratic Lyapunov functions for such systes, e.g., polyhedral [26], [27], piecewise-quadratic [28] and polynoial [29]. See also [30] where a sufficient condition for the existence of a hoogeneous polynoial Lyapunov function is given, which is also necessary in soe cases. For the case of systes with tie-varying Authorized licensed use liited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on June 22, 2009 at 13:58 fro IEEE Xplore. Restrictions apply.

4 1038 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009 for soe a lk (x) 0. Since g lk (x) is nonnegative on X l, the above condition iplies that V l (x) is positive on X l. An analogous condition can be iposed on dv l =dt. Note that there is no requireent in this ethod that the ultipliers a lk (x) be constants (as in the S-procedure); they can also be polynoials of higher degree [15]. Thus, this condition is generally less conservative than the S-procedure. 1) Switched Systes: In this case, the guards between two locations G(l; l 0 ) and G(l 0 ;l) coincide. Such systes are categorized as switched systes. The transition between locations is unknown a priori, but will depend on the direction of the vector fields. Without characterizing the direction of switching, it is essential that the piecewise Lyapunov function used to prove stability be continuous on G(l; l 0 ). Iposing V l (x) c ll 0 (x)h ll 0 (x) 0 V l (x) =0 (10) Fig. 1. Trajectories of the syste in Exaple 5 under arbitrary, tie dependent switching. The switching signal is piecewise constant and continuous fro the right and switches arbitrarily between the two locations with a bounded nuber of discontinuities on every bounded tie interval. Dashed curves are level curves of the coon Lyapunov function. uncertainties with a bounded variation rate [31], paraeter-dependent hoogeneous Lyapunov functions can be eployed. This is related to the work of P. -A. Blian [32], [33], which provides sufficient conditions for robust stability of linear systes, using quadratic Lyapunov functions with polynoial dependence on the paraeters. 1) Exaple 5: Consider the syste _x = f i(x), x = [x1;x 2] T under arbitrary switching, with f 1 (x) = 05x 1 0 4x 2 0x 1 0 2x 2 ; f 2 (x) = 02x 1 0 4x 2 20x 1 0 2x 2 : It can be proven using a dual seidefinite progra that no global quadratic Lyapunov function exists for this syste [7]. Nevertheless, a global sextic Lyapunov function V (x) =19:861x :709x 5 1x 2 14:17x 4 1x 2 2 4:2277x 3 1x 3 2 8:3495x 2 1x :2117x 1 x 5 2 1:0421x 6 2 exists, and therefore the syste is asyptotically stable under arbitrary switching (cf. Fig. 1). For higher degree polynoial vector fields and Lyapunov functions, the search for V (x) can also be perfored using seidefinite prograing by forulating the conditions as su of squares conditions. C. Piecewise Polynoial Lyapunov Functions For state-dependent switching, the analysis ethod presented in Section III-B will be too conservative. Stability can be proven in a ore effective way using piecewise polynoial Lyapunov functions. Such functions are patched fro several polynoial functions V l (x) (also tered Lyapunov-like functions), typically corresponding to the regions X l. Theore 2 requires that the Lyapunov-like function V l (x) and its tie derivative along the trajectory of the l-th location need only be positive and negative respectively within X l. The conditions in the previous paragraph can be accoodated using a ethod siilar to the S-procedure [13] as follows. To incorporate the fact that V l (x) only needs to be positive on X l, where X l is described by (2), we ipose the relaxed condition V l (x) 0 a lk (x)g lk (x) > 0 (9) where c ll 0 (x) is an arbitrary polynoial, will guarantee the continuity of V (x) on G(l; l 0 ) (equivalently G(l 0 ;l)). This results in the following Theore for switched systes. Theore 6: Consider a switched syste with G(l; l 0 )=G(l 0 ;l) for all l; l 0 2 L. Assue that there exist polynoials V l (x), c ll (x), with V l (0) = 0 and a lk (x) 0, b lk (x) 0, such that V l (x) 0 a lk (x)g lk (x) > 0 8x 6= 0;l 2 L; f l(x) b lk (x)g lk (x) < 0 8x 6= 0;l 2 L; (12) V l (x) c ll 0 (x)h ll 0 (x)0v l (x) =0 8l; l 0 : (13) Then the origin of the state space is asyptotically stable. A Lyapunov function that proves this is the piecewise polynoial function V (x) defined by V (x) =V l (x); if x 2X l : (14) Moreover, if each V l (x) is radially unbounded in the invariant X l and [ l X l = n then the result holds globally. Even though the switched syste is stable, low degree (e.g., quadratic) V l (x) that satisfy the above conditions ay not exist, as those conditions are only sufficient for stability. In this case, an iproved test can be perfored by dividing the continuous state space into a ore refined partition than the original X l, and then constructing a piecewise Lyapunov function (of the sae degree as before) based on this new partition. For systes with ore than two state variables, this refineent is obviously not an easy atter. A sipler way for obtaining an iproved test is to use a higher degree Lyapunov function based on the original partition, as illustrated by the following exaple. 2) Exaple 7: Consider the switched syste _x = f l (x) with four state variables and two odes f 1(x) = f 2(x) = 0x x 2 12x 3 0 2x 4 00:5x 1 8:5x 2 0 6x 3 0:5x 4 0:5x 1 26x 2 0 9:5x 3 5x 4 03x x 2 12x 3 0 6x 4 01:4x :6x 2 8x 3 0 1:6x 4 00:3x 1 8:3x 2 0 4x 3 1:3x 4 1:7x 1 20:6x 2 0 5:7x 3 3:6x 4 03:4x :6x 2 8x 3 0 4:6x 4 X 1 = fx 2 4 : g(x) 0g; X 2 = fx 2 4 : g(x) 0g where g(x)=(x 1 0:5x 2 1:5x 3 0:5x 4 )(x 1 00:5x 2 0:5x 3 00:5x 4 ).No piecewise quadratic Lyapunov function (using the original state space partition) exists for this syste. Refining the partition for this syste is not easy, thus we resort to higher order Lyapunov function instead. A hoogeneous piecewise quartic Lyapunov function can be found by ; ; Authorized licensed use liited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on June 22, 2009 at 13:58 fro IEEE Xplore. Restrictions apply.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY solving the optiization proble corresponding to the conditions in Theore 6. This proves that the origin of the state space is globally asyptotically stable. 3) Hybrid Systes: Directions of transitions in ost hybrid systes are characterized a priori. Because of this, a piecewise Lyapunov function for a hybrid syste need not be continuous, and it is enough to have V l (x) V l (x) on G(l; l 0 ). This is taken into account in condition (15) of the theore below. Theore 8: Consider a hybrid syste and assue that there exist polynoials V l (x), c ll 0 (x; x 0 ), d ll (x; x 0 ), and a lk (x) 0, b lk (x) 0, c ll k (x; x 0 ) 0 such that V l (0) = 0 and V l (x) f l(x) V l (x 0 )c ll 0 (x; x 0 )h ll 0 (x) a lk (x)g lk (x) > 0 8x 6= 0;l 2 L; b lk (x)g lk (x) < 0 8x 6= 0;l 2 L; c ll k (x; x 0 )h ll k (x) d ll (x; x 0 )(x 0 0 ll (x)) 0 V l (x) 0 8l; l 0 : (15) Then the origin is asyptotically stable. Moreover, if each V l (x) is radially unbounded in the invariant X l and [ l X l = n then the result holds globally. We reark that the last condition iplies that when h ll k (x) 0, h ij0 (x) =0and x 0 = ll (x) we have V l (x 0 ) V l (x), which is the last condition of Corollary 3. D. Nonlinear Vector Fields and Switching Surfaces/Transition Sets The sae ethodology can be applied to systes with nonlinear vector fields and nonlinear switching surfaces or transition sets. To illustrate this, consider the following exaple. 1) Exaple 9: Let the hybrid syste _x = f i(x) be coposed of two subsystes f 1 (x)= 02x 1 0 x x 2 0 x 3 2 6x 1 x x 2 0 x 3 2 with a guard set ;f 2 (x)= x 2 x x 3 1 4x 1 2x 2 (16) G(1; 2) = fx 2 2 jx 1 0andx 2 =0g; (17) G(2; 1) = fx 2 2 jx 2 2 = x 3 1g: (18) Fig. 2 depicts soe trajectories of the syste, when the syste is initialized with subsyste 1. Using Theore 8, the origin can be proven globally asyptotically stable with a sextic piecewise polynoial Lyapunov function given by V (x(t)) = V l (x(t)) if l is active, for soe V l (x) s that are oitted for brevity. This way we have deonstrated how ore coplicated switching rules can be taken into account when analyzing a hybrid syste. We note here that systes with rational or nonpolynoial vector fields can still be analyzed using the su of squares decoposition. This has been presented in [22] and will not be discussed in this technical note. The sae technique can also be applied to nonpolynoial guard sets. IV. ROBUST STABILITY ANALYSIS Uncertainty in a switched or hybrid syste can be present in the vector fields describing the flow of the syste and/or in the switching schee/transition law. The uncertainty can be of paraetric nature, or caused by tie-varying perturbations of the vector field, switching delays, etc. A ethod for robustness analysis has been proposed in [5]. The approach is based on bounding the guard sets by an uncertain switching Fig. 2. Trajectories of the syste in Exaple 9. Dash-dotted line and dashed curves show G(1; 2) and G(2; 1) respectively. The switching rule is given by (17) (18). set, and the subsyste invariants by a bigger set where the corresponding Lyapunov-like function is decreasing. Since this analysis is carried out using conditions siilar to those given in Section III, it can be iediately generalized to ake use of polynoial functions. The ethod is well-suited for robustness analysis with respect to nonparaetric uncertainty, but unfortunately, although in principle paraetric uncertainty can be handled in a siilar fashion, it is not treated in a direct and efficient way. Instead, in this section we present an analysis technique for handling paraetric and dynaic uncertainty in a direct way, based on paraeter dependent Lyapunov-like functions and ultipliers. Coputation of paraeter dependent quadratic Lyapunov-like functions using LMIs had been previously difficult, since such functions are nonquadratic polynoials in the state and paraeter variables. Using the su of squares decoposition, coputation of even higher degree functions becoes straightforward. Recall the description of the continuous flow field introduced in Section II-A. Let the set of adissible paraeters be given by P = fp 2 : r 1k (p) 0;k 1 =1;...; 1 ; r 2k (p) =0;k 2 =1;...; 2 g (19) and in the case they are tie-varying, let the set Q be given by Q = fq 2 : s 1k (q) 0;k 3 =1;...; 3 ; s 2k (q) =0;k 4 =1;...; 4 g (20) for soe polynoials r 1k (p), r 2k (p), s 1k (q) and s 2k (q). Furtherore, assue that the vector fields f i and the polynoials describing the invariants (2), guards (3) and reset aps (4) are dependent on p. Theores 4, 6, and 8 can be odified to accoodate paraeter dependent Lyapunov functions and ultipliers. For brevity, we only present the paraeter dependent version of Theore 8, for the case the paraeters are tie-varying. Theore 10: Consider a hybrid syste in which f l (x; p) has unknown paraeters p 2 P, where P is as in (19) so that their tie-variation is in Q, described by (20). Assue that there exist polynoials V l (x; p), a lk (x; p) 0, ^a lk (x; p), ~a lk (x; p) 0, b lk (x; p; q) 0 ^b lk (x; p; q), ~ b lk (x; p; q) 0, b lk (x; p; q), b lk (x; p; q) 0, c ll 0 (x; x 0 ;p), c ll k (x; x 0 ;p) 0, ^c ll k (x; x 0 ;p), Authorized licensed use liited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on June 22, 2009 at 13:58 fro IEEE Xplore. Restrictions apply.

6 1040 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009 Fig. 3. Trajectories of the syste in Exaple 11 for different values of p. Dashed lines represent the guard sets. ~c ll k (x; x 0 ;p) 0, and d ll (x; x 0 ) such that V i (0;p) = 0 and conditions (21) (23) are satisfied. Then the origin of the state space is robustly asyptotically stable with respect to the unknown paraeters p 2Pwhen they vary inside _p 2Q. TABLE II RELATION BETWEEN THE DEGREE OF V (x), k =1,2,AND THE VALUE OF C FOR WHICH ROBUST STABILITY CAN BE PROVEN. RECALL THAT THE SYSTEM IS STABLE FOR p > 2:165 V l (x; p) 0 l (x; a lk (x; p)g lk (x; p) 0 ~a lk (x; p)r 1k (p) ^a lk (x; p)r 2k (p) > 0 8x 6= 0;i 2 I; (21) f l (x; l(x; p) b lk (x; p; q)g lk (x; p) ~ blk (x; p; q)r 1k (p) ^blk (x; p; q)r 2k (p) blk (x; p; q)s 1k (q) blk (x; p; q)s 2k (q) < 0 8x 6= 0;i 2 I; (22) V l (x 0 ;p)c ll 0 (x; x 0 ;p)h ll 0 (x; p) c ll k (x; x 0 ;p)h ll k (x; p) d ll (x; x 0 )(x 0 0 ll (x)) 0V l (x; p) ~c ll k (x; x 0 ;p)r 1k (p) ^c ll k (x; x 0 ;p)r 2k (p) 0 8l; l 0 : (23) 1) Exaple 11: Let us consider the hybrid syste _x = f i(x), with vector fields and guard sets f 1 (x) = 0x (1 1(t))x2 ; 10x 1 0 x 2 f 2(x) = x 1 10(1 2 (t))x x 1 x 2 Notice the dependence of the first guard set on the unknown paraeter p 2, and the tie-varying uncertainties in the vector field, 1 (t) and 2(t). Let us first concentrate on the case 1 (t) = 2 (t) =0. Obviously, stability of the syste depends on the value of p. In this exaple, we have deliberately chosen a syste with linear subsystes, so that robust stability of the syste can also be analyzed in a purely analytical way for coparison purposes. By coputing the flows of the subsystes when the syste is initialized in subsyste 1, it can be proven that the syste is stable for p > 2:165 and unstable for p < 2:163. Atp ' 2:164 it exhibits a liit cycle (see Fig. 3). With paraeter dependent Lyapunov-like functions of the for V i(x; p) =Vi;1(x) pv i;2(x) (24) robust stability of the syste with respect to p 2 P = fp : p0c 0g, where C is a constant, can be proven. Using quadratic V i;1(x) and V i;2 (x), we can prove robust stability for C =5:86. Tighter robustness bounds can be obtained by increasing the degree of the Lyapunov-like functions, as depicted in Table II. Now, let us fix p =8, and consider the robust stability of the syste under the presence of dynaic uncertainties 1(t) and 2(t). We assue that j 1 (t)j 1 ; j 2 (t)j 2 (25) for soe nubers 1 ; 2 > 0 (these inequalities define the set P), and we consider different variation levels j _ 1(t)j v; j _ 2(t)j v: (26) G(1; 2) = fx 2 G(2; 1) = fx 2 2 j0px 1 (t) 0 x 2 (t) =0g; 2 j2x 1 (t) 0 x 2 (t) =0g: for soe nuber v 0. Fig. 4 shows robust stability regions that are obtained using the sae quadratic Lyapunov function structure, for different variation rates v. Authorized licensed use liited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on June 22, 2009 at 13:58 fro IEEE Xplore. Restrictions apply.

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY Fig. 4. Robust stability regions verified by a quadratic ultiple Lyapunov function for Exaple (11) with p =8. Here, and v are given by (25) (26). V. CONCLUSION A ethod for stability analysis of switched and hybrid systes has been presented. The ethod is based on polynoial and piecewise polynoial Lyapunov functions, whose coputation can be efficiently perfored using the su of squares decoposition and seidefinite prograing. Using this approach, higher degree Lyapunov functions can be constructed, thus reducing the conservatis of searching for only quadratic candidates. In the sae way paraetric uncertainty can be incorporated in the search. Several exaples have been provided to illustrate the benefits of this approach. At the present tie, the largest syste that we could analyze was a polynoial hybrid syste with 10 continuous states and 6 locations, for which the safety properties were assessed [34]. 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