Stability region estimation for systems with unmodeled dynamics

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1 Stability region estimation for systems with unmoele ynamics Ufuk Topcu, Anrew Packar, Peter Seiler, an Gary Balas Abstract We propose a metho to compute invariant subsets of the robust region-of-attraction for the asymptotically stable equilibrium points of systems with polynomial nominal vector fiels an unmoele ynamics. The effects of unmoele ynamics are accounte for as systems satisfying certain gain relations or issipation inequalities. The methoology is extene to systems with parametric uncertainties an an informal branch-an-boun type refinement proceure to reuce the conservatism is iscusse. We emonstrate the metho on a polynomial approximation of uncertain controlle short perio aircraft ynamics. I. INTRODUCTION We consier the problem of estimating the robust stability regions aroun stable equilibrium points of uncertain nonlinear ynamical systems. Two types of uncertainties are consiere: (1) boune uncertainties ue to unmoele ynamics in the feeback loop as shown in Figure 1, where Φ is consiere to be unknown with a known boun on the input-output gain an M is the known nonlinear part of the ynamics; (2) boune parametric uncertainties in M. Krstic et al. emonstrate that existence of unmoele ynamics may reuce the size of regions-of-attraction for nonlinear ynamical systems [1]. The approach for accounting for the unmoele ynamics with boune inuce L 2 L 2 norms is base on separate analyses of input-output gain properties of M an Φ. A local small-gain type argument, which uses the certificates from separate input-output gain analyses, is use to estimate stability regions for the closeloop ynamics. Following [2], [3], [4], we characterize upper bouns on local input-output gains for M ue to boune L 2 isturbances by Lyapunov/storage functions which satisfy certain local issipation inequalities [5]. We use polynomial Lyapunov/storage function caniates, simple generalizations of the S-proceure [6], an sum-ofsquares (SOS) relaxations for polynomial nonnegativity [7] an compute upper bouns on the input-output gains via (bilinear) SOS programming problems. The approach for the boune parametric uncertainties is similar to that evelope in [8], [9] in the context of region-of-attraction analysis for systems with parametric uncertainties (but with no unmoele ynamics). Namely, a U. Topcu is with Control an Dynamical Systems at California Institute of Technology, Pasaena, CA, 91125, USA. utopcu@cs.caltech.eu A. Packar is with the Department of Mechanical Engineering, The University of California, Berkeley, , USA. pack@jagger.me.berkeley.eu P. Seiler an G. Balas are with the Department of Aerospace Engineering an Mechanics, The University of Minnesota, Minneapolis, MN, 55455, USA. peter.j.seiler@gmail.com an balas@aem.umn.eu parameter-inepenent storage function is use to characterize input-output properties over the entire set of amissible values of uncertain parameters. The input-output relations characterize by a single parameter-inepenent certificate may be more conservative compare to those by parameterepenent certificates [10]. This potential conservatism is reuce by partitioning the uncertainty set into subregions following an informal branch-an-boun type refinement proceure [11] an computing parameter-inepenent certificates for each subregion. Although it is simplistic (compare to theories base on parameter-epenent Lyapunov functions), this approach offers certain computational avantages as iscusse in section III-C an in [9] in more etail. Notation: For x R n, x 0 means that x k 0 for k = 1,, n. For Q = Q T R n n, Q 0 (Q 0) means that x T Qx 0 (> 0) for all x R x. For x 1 R n1 an x 2 R n2, [x 1 ; x 2 ] R n1+n2 enotes the concatenation of x 1 an x 2. R[x] represents the set of polynomials in x with real coefficients. The subset Σ[x] := {π = π1 2 + π πm 2 : π 1,, π M R[x]} of R[x] is the set of SOS polynomials. For η > 0 an a function g : R n R, efine the η-sublevel set Ω g,η of g as Ω g,η := {x R n : g(x) η}. For a piecewise continuous map u : [0, ) R m, efine the L 2 norm as u 2 := u(t) T u(t)t. 0 In several places, a relationship between an algebraic conition on some real variables an state properties of a ynamical system is claime, an same symbol for a particular real variable in the algebraic statement as well as the state of the ynamical system is use. This coul be a source of confusion, so care on the reaer s part is require. II. PRELIMINARIES Consier the autonomous nonlinear ynamical system ẋ(t) = f(x(t)), (1) where x(t) R n is the state vector an f is an n-vector with entries in R[x] satisfying f(0) = 0, i.e., the origin is an equilibrium point of (1). Let φ(t; x 0 ) enote the solution to (1) at time t with the initial conition x(0) = x 0. The region-of-attraction (ROA) of the origin for the system (1) is { } x 0 R n : lim φ(t; x 0 ) = 0. t

2 A moification of a similar result in [12] provies a characterization of invariant subsets of the ROA in terms of sublevel sets of appropriately chosen Lyapunov functions. Lemma II.1. Let α R be positive. If there exists a continuously ifferentiable function V : R n R such that Ω V,α is boune, an (2) V (0) = 0 an V (x) > 0 for all x R n (3) Ω V,α \ {0} {x R n : V (x)f(x) < 0}, (4) then for all x 0 Ω V,α, the solution of (1) exists, satisfies φ(t; x 0 ) Ω V,α for all t 0, an lim t φ(t; x 0 ) = 0, i.e., Ω V,α is an invariant subset of the ROA. Our goal is to construct Lyapunov functions that estimate the regions-of-attraction for the close-loop ynamics in Figure 1 base on the input-output properties of M an Φ. To this en, consier the input-output system governe by ẋ(t) z(t) = f(x(t), w(t)) = h(x(t)), where x(t) R n, w(t) R nw, an f is a n-vector with elements in R[(x, w)] such that f(0, 0) = 0 an h is an n z -vector with elements in R[x] such that h(0) = 0. Let φ(t; x 0, w) enote the solution to (5) at time t with the initial conition x(0) = x 0 riven by the input/isturbance w. Lemma II.2. [3] If there exists a real scalar > 0 an a continuously ifferentiable function V such that (5) V (0) = 0 an V (x) 0 x R n, (6) V f(x, w) w T w 2 z T z (7) then it hols that for the system in (5) x Ω V,R 2 an w R nw, x(0) = 0 an w 2 R y 2 w 2. (8) In other wors, is a local upper boun for the input-output gain for the system in (5). We call to be a local upper boun because the upper boun on the norm of the output z is only suppose to hol whenever the norm of the input is boune by R. This is unlike the input-output gains for linear systems which hol for all values of input norms. Finally, the following lemma is a straightforwar generalization of the S-proceure [6] an is use to obtain algebraic sufficient conitions for certain set containment constraints. See [8] for a proof. Lemma II.3. Given g 0, g 1,, g m R[x], if there exist s 1,, s m Σ[x] such that m g 0 s i g i Σ[x], then i=1 {x R n : g 1 (x),..., g m (x) 0} {x R n : g 0 (x) 0}. z Φ M Fig. 1. Feeback interconnection of Φ an M. III. ROA ANALYSIS FOR SYSTEMS WITH UNMODELED DYNAMICS Consier the system interconnection in Figure 1. Let ẋ(t) z(t) = f(x(t), w(t)) = h(x(t)) be a realization of M, where x R n, f an h are vectors of polynomials satisfying f(0, 0) = 0 an h(0) = 0. A. Local small-gain type theorems Proposition III.1. Consier the system interconnection in Figure 1 with Φ stable linear time-invariant system satisfying Φ < 1. Let 0 < < 1, R > 0, an l be a positive efinite function with l(0) = 0. If there exists a continuously ifferentiable positive efinite function V satisfying V (0) = 0 an V f(x, w) w T w 1 2 z T z l(x) for all w R nw an x Ω V,R 2, w (9) (10) an Ω V,R 2 is boune, then, for all x(0) Ω V,R 2 an Φ starting from rest, x(t) Ω V,R 2 for all t 0 an x(t) 0 as t. Proof: Let ξ(t) = Aξ(t) + Bz(t) w(t) = Cξ(t) + Dz(t) be a realization of Φ with ξ R n ξ. By Boune Real Lemma [6], there exist Q 0 an ɛ > 0 such that [ A T Q + QA + C T C QB ] + C T D B T Q + D T C I + D T + ɛi 0 (11) D Let Q(ξ) = ξ T Qξ, then, for all ξ R n ξ an z R nz, the inequality (11) implies that t Q(ξ) = Q(ξ) ξ (Aξ + Bz) = ɛξ T ξ ɛz T z w T w + z T z z T z w T w ɛξ T ξ. (12) Now, let (x, ξ) Ω V +Q,R 2. By (10) an (12), ( ) t (V (x) + Q(ξ)) 1 1 z T z l(x) ɛξ T ξ 2 l(x) ɛξ T ξ. (13) Integrate (13) from 0 to T 0 to get V (x(t )) V (x(t )) + Q(ξ(T )) T 0 (l(x) + ɛξt ξ)t + V (x(0)) V (x(0)) R 2,

3 which implies that, for all t 0, x(t) Ω V,R 2 whenever x(0) Ω V,R 2 an ξ(0) = 0. Convergence of (x, ξ) an, in particular of x, follows from Lemma II.1 using the inequality in (13) (note that S := V + Q is a Lyapunov function for the close-loop system). Remarks III.1. Note that Proposition III.1 only states that Ω V,R 2 is invariant but oes not assure the invariance of the sublevel sets Ω V,r 2 for any 0 < r 2 < R 2. Hence, V may increase along the trajectories of the close-loop system. Nevertheless, if V (x(0)) R 2 an Φ starts from rest, then V cannot excee R 2 an converges to the origin along every trajectory with x(0) Ω V,R 2 an ξ(0) = 0. In the proof Proposition III.1, the linear time-invariance property of Φ is only use to establish the existence of a storage function Q satisfying the inequality (12). Therefore, if Φ is a ynamical system known to satisfy t Q(ξ) zt z w T w ɛξ T ξ for all ξ R n ξ an z R nz with ɛ > 0 where Q is a continuously ifferentiable, positive efinite function with boune sublevel sets an Q(0) = 0, then the conclusion of Proposition III.1 still applies. Proposition III.2. Consier the system interconnection in Figure 1. Let 0 < < 1, R > 0, an l : R n R an : R n ξ R be positive efinite functions with l(0) = an l(0) = 0, an let ξ(t) = f2(ξ(t), z(t)) w(t) = h 2 (ξ(t)) (14) be a realization of Φ with ξ R n ξ such that there exists a continuously ifferentiable positive efinite function Q that has boune sublevel sets an satisfies Q(0) = 0 an Qf 2 (ξ, z) z T z w T w l(ξ) for all z R nz an ξ R n ξ. (15) If there exists a continuously ifferentiable positive efinite function V satisfying V (0) = 0 an (10), an Ω V,R 2 is boune, then, for ξ(0) = 0 an all x(0) Ω V,R 2, (x(t), ξ(t)) Ω V +Q,R 2 for all t 0 an lim t (x(t), ξ(t)) = (0, 0), in particular, x(t) Ω V,R 2 for all t 0 an lim t x(t) = 0. Conitions Φ in Proposition III.2 can be relaxe to obtain the following result. Proposition III.3. Consier the system interconnection in Figure 1. Let 0 < < 1, R > R > 0, an l : R n R be a positive efinite function with l(0) = 0. Let (14) be a realization of Φ such that there exists a continuously ifferentiable positive efinite function Q that has boune sublevel sets an satisfies Q(0) = 0 an Qf 2 (ξ, z) z T z w T w for all z R nz an ξ R n ξ. (16) If there exists a continuously ifferentiable positive efinite function V satisfying V (0) = 0 an (10), an Ω V,R 2 is boune, then, for ξ(0) = 0 an all x(0) Ω V,, (x(t), ξ(t)) Ω V +Q, for all t 0 an, in particular, x(t) Ω V, for all t 0. Moreover, lim t x(t) = 0. Proof: Define S := V + Q. ts(x, ξ) = t (V (x) + Q(ξ)) l(x) x Ω V,R 2, ξ R n ξ. Let x(0) Ω V, an ξ(0) = 0, an integrate this last inequality to get S(x(t), ξ(t)) S(x(0), ξ(0)) V (x(t)) + Q(ξ(t)) V (x(0)) t 0 p(x(τ))τ, an x(t) Ω V, an (x(t), ξ(t)) Ω S, follow. With x(0) Ω V, an ξ(0) = 0, S is monotonically non-increasing an boune below (by zero). There- t fore, lim t Ṡ(τ)τ exists an is finite. Since S 0 is boune from below an positive efinite, S(x(t), ξ(t)) is uniformly boune for t 0, an (ẋ(t), ξ(t)) = (f(x(t), w(ξ(t))), f 2 (z(x(t)), ξ(t))) is uniformly boune for all t 0. Hence x(t), ξ(t) is uniformly continuous on [0, ). S is uniformly continuous in t on [0, ) because S(x, ξ) is uniformly continuous in (x, ξ) on the compact set Ω S,. Therefore, by Barbalat s Lemma [13], Ṡ(t) 0 as t. Consequently, l(x(t)) 0 an x(t) 0 as t. B. Estimating the region-of-attraction in the presence of unmoele ynamics By Proposition III.1, for positive scalars µ an with 1, an linear time-invariant Φ with Φ < 1 starting from rest, if there exists positive efinite V such that V (0) = 0, Ω V,R 2 is boune, an V f(x, w) w T w 1 2 z T z µv (x) for all w R nw an x Ω V,R 2, (17) then Ω V,R 2 is invariant an all trajectories of the closeloop system with x(0) Ω V,R 2 converge to the origin. In orer to enlarge Ω V,R 2 by choice of V, we efine a variable size region Ω p,β = {x R n : l(x) β}, where p R[x] is a fixe positive efinite polynomial, an maximize β while imposing the constraint Ω p,β Ω V,R 2 along with the constraints V (0) = 0, that Ω V,R 2 is boune, an (17), i.e., max V V,β 0,R 0 β subject to V (x) > 0 for all x 0, V (0) = 0, Ω p,β Ω V,R 2, Ω V,R 2 is boune, V f(x, w) w T w 1 z T z µv (x) 2 x Ω V,R 2, w R nw, (18) where V is the family of functions over which the maximum in (18) is compute. Now, let l be a positive efinite polynomial (typically l(x) = ɛx T x with a small positive scalar ɛ), µ be positive an 0 < 1, an V poly V be a

4 prescribe subset of R[x]. Define β := max β subject to V V poly,β 0,R 0,s 1 S 1,s 2 S 2 V l Σ[x], (R 2 V ) s 1 (β p) Σ[x], V f + w T w 1 z T z µv s 2 2 (R 2 V ) Σ[(x, w)], (19) where S 1 an S 2 are prescribe subsets of R[x] an R[(x, w)], respectively. Then, by Lemma II.3, β is a lower boun for the maximum value of β in (18). Remark III.1. Note from (18) that computation of the set Ω V,R 2 oes not require the knowlege of the storage function for Φ but requires the extra assumption that Φ starts from rest. When a storage function for Φ is known, this extra assumption can be remove. Furthermore, even when only an upper boun Q 0 on the value of Q at the nonzero initial conition of Φ is known such that V (0) + Q R 2, then the conclusion of Proposition III.1 can be moifie such that x(t) Ω V,R2 Q for all t 0 whenever x(0) Ω V,R2 Q an x(t) 0 as t. By Willems [5], such bouns on the value of the storage functions at the initial value of Φ may theoretically be establishe using input-output tests, i.e., oes not require the knowlege of specific realization of Φ. Nevertheless, these bouns may not be easily etermine by numerical computation in practice. C. ROA analysis in the presence of parametric uncertainties We now generalize the evelopment in section III-B to the case where M is governe by orinary ifferential equations that contain unknown but fixe an boune parameters. Following the methoology iscusse in [9] in the context of robust stability analysis, we first restrict our attention to ẋ(t) = f(x(t), w(t), δ) := f 0 (x(t), w(t)) + m i=1 δ if i (x(t), w(t)) z(t) = h(x(t)), (20) where f 0, f 1,..., f m are n-vectors with elements in R[(x, w)] such that f 0 (0, 0, δ) = f 1 (0, 0, δ) =... = f m (0, 0, δ) = 0, for all δ R m, an is a known boune polytope. Let φ(t; x 0, w, δ) enote the solution of (20) for δ at time t with the initial conition x(0) = x 0 riven by the input/isturbance w an E enote the set of vertices of. Proposition III.4. If, for some V : R n R, an scalars, µ, an R, V f(x, w, δ) w T w 1 2 z T z µv x Ω V,R 2, w R nw (21) hols for each δ E, then (21) hols for each δ. Proposition III.4 follows from the affine δ epenence in (21) an the restriction that is a boune polytope. By Proposition III.4, for positive scalar µ, positive scalar with 1, an linear time-invariant Φ that starts from rest an satisfies Φ < 1, if there exists a positive efinite V such that V (0) = 0, Ω V,R 2 is boune, an V f(x, w, δ) w T w 1 2 z T z µv (x) for all w R nw, x Ω V,R 2, an δ E (22) then all trajectories of the close-loop system in (20) with x(0) Ω V,R 2 converge to the origin. Furthermore, SOS base sufficient conitions for those in (22) can be obtaine using Lemma II.3 an SOS relaxations. The approach propose here to account for parametric uncertainties is restrictive: (1) only affine epenence on δ an polytopic are allowe; (2) SOS relaxations for the conitions in (22) may inclue a large number of semiefinite programming (SDP) constraints - one for each δ E ; (3) single (δ-inepenent) Lyapunov/storage function is to certify properties for an entire family of systems. These limitations can be partially alleviate using techniques propose in [9]. For example, polynomial epenence on δ in the vector fiel an the output map can be hanle by replacing non-affine appearances of δ by artificial parameters an covering the graph of non-affine functions of δ (in the conitions in (22)) by boune polytopes in the lifte uncertain parameter space. Furthermore, the fact that constraints in the SOS relaxations for the conitions in (22) are only couple through the Lyapunov/storage functions (which inclue relatively small portion of all ecision variables in the associate SDPs 1.) can be exploite through a suboptimal two-step proceure, which effectively ecouples the large number of constraints enabling the use of trivial parallelization. Finally, conservatism (ue to using a single parameterinepenent Lyapunov function an ue to the suboptimal two-step proceure) can be reuce by an informal branchan-boun type refinement proceure where is partitione into smaller subregions an a ifferent Lyapunov/storage function is compute for each subregion. See [9] for etails. D. Implementation issues The SOS relaxations in (19) lea to bilinear SDPs ue to the multiplication between the ecision variables in V an the multipliers. Therefore, solution techniques for these problems are usually limite local search schemes such as PENBMI [14] or coorinate-wise affine search base on the observation that, for given V an R, constraints in these problems are affine in the ecision variables in the multipliers. For example, one can obtain a suboptimal solution for the problem in (19) by alternating between solving Problems 1 an 2 (state below) until a maximum number of iterations or an increase in the value of certifie R smaller than a pre-specifie tolerance is reache. Problem 1: For given V (known to be) feasible for (19), solve max R 2 subject to R 0,s 2 S 2 V f + w T w 1 z T z µ V s 2 2 (R 2 V ) Σ[(x, w)], (23) 1 Note that the SOS relaxation for each constraint in (22) contains ecision variables in the S-proceure multipliers an those introuce in the corresponing SDP to certify the SOS property [7]

5 an, with the optimal value R of R in (23), solve max β subject to β 0,s 1 S 1 ( R 2 V ) s 1 (β p) Σ[x], (24) Problem 2: For given s 1 an s 2 (known to be) feasible for (19), solve max β subject to V V poly,β 0,R 0 V l Σ[x], (R 2 V ) s 1 (β p) Σ[x], V f + w T w 1 z T z µv s 2 2 (R 2 V ) Σ[(x, w)], (25) Note that Problems 1 an 2 can be solve using linear SDP solvers, such as SeDuMi [15], through a line search on R an β in (23) an (24), respectively. IV. EXAMPLES Consier the controlle short perio aircraft ynamics shown in Figure 2 with ẋ p = c 01 (x p ) + δ 1 c 11 (x p ) + δ 2 1q 31 (x p ) q 02 (x p ) + δ 1 l T 12x p + δ 2 q 22 (x p ) + x 1 lt b x p + b 11 + b 12 δ 1 b 21 + b 22 δ 2 0 u, (26) where x p := [x 1 x 2 x 3 ] T, x 1, x 2, an x 3 enote the pitch rate, the angle of attack, an the pitch angle, respectively. Here, δ 1 [0.99, 2.05] moels variations in the center of gravity in the longituinal irection an δ 2 [ 0.1, 0.1] moels variations in the mass. Here, c 01 an c 11 are cubic polynomials, q 02, q 22, an q 31 are quaratic polynomials, l 12 an l b are vectors in R 3, b 11, b 12, b 21, an b 22 are real scalars (see [9] for the values of the missing parameters). The controller output v, the elevator eflection, is etermine by ẋ 4 = 0.864y y 2 v = 2x 4, (27) where x 4 is the controller state an the plant output y = [x 1 x 3 ] T. Define x := [ x T p x 4 ] T an let the system from the input w to the output z be governe by ẋ = f(x, w, δ) an z = h z (x). Using the proceure explaine in section III with p(x) = x T x, = 0.99, an µ = 10 6, we compute estimates of the ROA for two cases: (uncertain parameters set to the nominal values δ 1 = 1.52 an δ 2 = 0) Ω p,4.24 an Ω p,6.67 are certifie to be in the robust region-of-attraction with (V ) = 2 an (V ) = 4, respectively. (with parametric uncertainty in the plant δ 1 [0.99, 2.05] an δ 2 [ 0.1, 0.1]) We implemente a branch-an-boun type refinement proceure couple with a generalization of the sequential suboptimal solution technique (as mentione in section III-C an etaile in [9]). Ω p,2.39 an Ω p,4.14 are certifie to be in the robust region-of-attraction with (V ) = 2 an (V ) = 4, respectively. On the other han, in case there is no unmoele ynamics, the following was proven in [16]. (uncertain parameters set to the nominal values) Ω p,9.38 an Ω p,16.11 are certifie to be in the region-ofattraction with (V ) = 2 an (V ) = 4, respectively. (with parametric uncertainty in the plant) Ω p,5.45 an Ω p,7.93 are certifie to be in the region-of-attraction with (V ) = 2 an (V ) = 4, respectively. V. CONCLUSIONS We propose a metho to compute invariant subsets of the robust region-of-attraction for the asymptotically stable equilibrium points of systems with polynomial nominal vector fiels an unmoele ynamics. The effects of unmoele ynamics were accounte for as systems satisfying certain gain relations or issipation inequalities. The methoology was extene to systems with parametric uncertainties an an informal branch-an-boun type refinement proceure to reuce the conservatism is iscusse. We emonstrate the metho on a polynomial approximation of uncertain controlle short perio aircraft ynamics. VI. ACKNOWLEDGEMENTS This work was sponsore by the Air Force Office of Scientific Research, USAF, uner grant/contract number FA The views an conclusions containe herein are those of the authors an shoul not be interprete as necessarily representing the official policies or enorsements, either expresse or implie, of the AFOSR or the U.S. Government. U. Topcu acknowleges partial support from the Boeing Corporation. A. Packar an G. Balas acknowlege partial support from NASA uner the contract NNX08AC80A. REFERENCES [1] M. Krstic, J. Sun, an P. V. Kokotovic, Robust control of nonlinear systems with input unmoele ynamics, Automatic Control, IEEE Transactions on, vol. 41, no. 6, pp , Jun [2] Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, an A. Packar, Control applications of sum of squares programming, in Positive Polynomials in Control, D. Henrion an A. Garulli, Es. Springer- Verlag, 2005, pp [3] W. Tan, A. Packar, an T. Wheeler, Local gain analysis of nonlinear systems, in Proc. American Control Conf., Minneapolis, MN, 2006, pp [4] W. Tan, U. Topcu, P. Seiler, G. Balas, an A. Packar, Simulationaie reachability an local gain analysis for nonlinear ynamical systems, in Proc. Conf. on Decision an Control, 2008, pp [5] J. C. Willems, Dissipative ynamical systems I: Genaral theory, Archive for Rational Mechanics an Analysis, vol. 45, pp , [6] S. Boy, L. E. Ghaoui, E. Feron, an V. Balakrishnan, Linear Matrix Inequalities in Systems an Control Theory. Philaelphia: SIAM, [7] P. Parrilo, Semiefinite programming relaxations for semialgebraic problems, Mathematical Programming Series B, vol. 96, no. 2, pp , [8] U. Topcu an A. Packar, Local stability analysis for uncertain nonlinear systems, IEEE Transactions on Automatic Control, vol. 54, pp , 2009.

6 z Φ w ẋ 4 = A cx 4 + B cy v = C cx u + ẋ p = f p(x p, δ) + B(x p, δ)u y y = [x 1 x 3 ] T Fig. 2. Controlle short perio aircraft ynamics with unmoele ynamics (δ = (δ 1, δ 2 )). [9] U. Topcu, A. Packar, P. Seiler, an G. Balas, Robust regionof-attraction estimation, 2009, to appear in IEEE Transaction on Automatic Control. [10] G. Chesi, Estimating the omain of attraction of uncertain polynomial systems, Automatica, vol. 40, pp , [11] E. Lawler an D.E.Woo, Branch-an-boun methos: a survey, Operations Research, vol. 14, no. 4, pp , [12] M. Viyasagar, Nonlinear Systems Analysis, 2 n e. Prentice Hall, [13] H. K. Khalil, Nonlinear Systems, 3 r e. Prentice Hall, [14] M. Ko cvara an M. Stingl, PENBMI User s Guie (Version 2.0), available from [15] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methos an Software, vol , pp , 1999, available at [16] U. Topcu, Quantitative local analysis of nonlinear systems, Ph.D. Dissertation, UC, Berkeley, 2008, available at utopcu/issertation.