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CONVEXITY ON IN STABILIZATION OF NONLINEAR SYSTEMS Anders Rantzer Department of Automatic Control, Lund Institute of Technology Box 118, S-221 00 Lund, Sweden, Phone: +46 46 222 03 62 Email: rantzer@control.lth.se Pablo A. Parrilo Control and Dynamical Systems, California Institute of Technology Caltech 107-81, Pasadena, CA 91125-8100 Email: pablo@cds.caltech.edu Abstract A stability criterion for nonlinear systems, recently derived by the rst author, can be viewed as a dual to Lyapunov s second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and ows along the system trajectories towards the equilibrium. The new criterion has a remarkable convexity property, which in this paper is used for controller synthesis via convex optimization. Recent numerical methods for veri cation of positivity of multivariate polynomials are used. Keywords Stabilization, nonlinear systems, sums of squares, convexity 1. INTRODUCTION Lyapunov functions have long been recognized as one of the most fundamental analytical tools for analysis and synthesis of nonlinear control systems. See for example (Artstein, 1983; Brockett, 1983; Hahn, 1963; Isidori, 1995; Krstic et al., 1995; Ledyaev and Sontag, 1999). There has also been a strong development of computational tools based on Lyapunov functions. Many such methods are based on convex optimization and solution of matrix inequalities, exploiting the fact that the set of Lyapunov functions for a given system is convex. A serious obstacle in the problem of controller synthesis is however that the joint search for a controller u(x) and a Lyapunov function V (x) satisfying the condition V [ f (x) + (x)u(x)] < 0 x is not convex. In fact, for some systems the set of u and V satisfying the inequality is not even connected. Given the dif culties with Lyapunov based controller synthesis, it is most striking to nd that the new convergence criterion presented in (Rantzer, 2000b; Rantzer, 2000a) has much better convexity properties. Indeed, the set of ( , u ) satisfying [ ( f + u)] > 0 (1) is convex. In this paper, we will exploit this fact in the computation of stabilizing controllers for some example systems. For the case of polynomial (or rational) systems, the search for a candidate pair ( , u ) verifying the inequality (1) can be done using the methods introduced in (Parrilo, 2000). 2. THE CONVERGENCE CRITERION The main result of (Rantzer, 2000a) can be stated as follows: and with the notation t (z) = ( t (z)) t z (z) t THEOREM 1 Given the equation x(t) = f (x(t)), where f C1 (Rn , Rn ) and f (0) = 0, suppose there exists a non-negative C1 (Rn \ {0}, R) such that (x) f (x)/ x is integrable on {x Rn : x 1} and t t (z) t (z) t=0 = = = f + ( h f)= ( f ) (z) (z) t= h ( (z)) ( f ) ( (z)) z z h=0 (z) [ ( f )](x) > 0 for almost all x (2) Let ( ) be the characteristic function of Z. Then Then, for almost all initial states x(0) the trajectory x(t) exists for t [0, ) and tends to zero as t . Moreover, if the equilibrium x = 0 is stable, then the conclusion remains valid even if takes negative values. The proof is based on the following lemma, which can be viewed as a version of Liouville s theorem (Arnold, 1989; Mane, 1987). LEMMA 1 Let f C1 (D, Rn ) where D Rn is open and let C1 (D, R) be integrable. For x0 Rn, let t (x0 ) for t 0 be the solution x(t) of x = f (x), x(0) = x0 . For a measurable subset Z of D, let t (Z) = t (x) x Z . Then t (Z) (x)dx Z (z)dz (z)dz Z = = = Rn (x) ( 1(x))dx t ( t (z)) (z) t (z) z Rn dz Z (z)dz [ t (z) (z)] dz Z t Z0 t 0 = = [ ( f )] ( (z)) [ z (z) d dz (Z) ( f )] (x)dxd t (Z) (x)dx Z (z)dz Proof of Theorem 1, second statement. Here it is assumed that x = 0 is a stable equilibrium, while may take negative values. The proof for the other case is omitted from this conference manuscript. Rather than exploiting that f C1 (Rn, Rn ), we will actually prove the result under the weaker condition that f C1 (Rn \ {0}, Rn) and f (x)/ x is bounded near x = 0. Given any x0 Rn , let t (x0 ) for t 0 be the solution x(t) of x(t) = f (x(t)), x(0) = x0 . Assume rst that is integrable on {x Rn : x 1} and f (x) / x is bounded. Then t is well de ned for all t. Given r > 0, de ne Z = {x0 : t (x0 ) > r for some t > l} (3) l=1 Notice that Z contains all trajectories with lim supt x(t) > r. The set Z, being the intersection of a countable number of open sets, is measurable. Moreover, t (Z) = t (x) x Z is equal to Z for every t. By stability of the equilibrium x = 0, there is a positive lower bound on the norm of the elements in Z, so Lemma 1 with D = {x : x > } gives 0= = 0 t (Z) [ ( f )] (x)dxd Proof. Note that for every C1 matrix function M (t) with M (0) = I det M (t) t=0 t = trace M (0) This follows by direct expansion of the determinant, since the rst order terms in t correspond to the diagonal elements of M (t). Let M (t) = zt (z) and use to denote determinant. The differentiability of f gives that t(z) is of class C1 in z and C2 in t (Lefschetz, 1977)page 40. Hence t t z (z) t=0 = trace = trace 2 tz t (z) t=0 t (Z) t 0 (x)dx Z (z)dz (4) (5) f (z) = z f (z) = (Z) [ ( f )] (x)dxd By the assumption (2), this implies that Z has measure zero. Consequently, lim supt x(t) r for almost all trajectories. As r was chosen arbitrarily, this proves that limt x(t) = 0 for almost all trajectories. When f (x) / x is unbounded, there may not exist any nonzero t such that t (z) is well de ned for all z. We then introduce e x 0 (x) = 1 + (x) f (x) (x) f 0 (x) = 0 (x) f (x) + x2 2 1/2 backstepping) use either implicitly of explicitly a sum of squares approach. As shown in (Parrilo, 2000), the problem of checking if a given polynomial can be written as a sum of squares can be solved using semide nite programming. We refer the reader to that work for a discussion of the speci c algorithms. For our purposes, however, it will enough to know that while the standard LMI machinery can be interpreted as searching for a positive de nite element over an af ne family of quadratic forms, the new tools provide a way of nding a sum of squares, over an af ne family of polynomials. The former problem is clearly a special case of the latter (in fact, they are equivalent). To apply these tools to the stabilization problem analyzed in the paper, consider the parameterized representation for and u : 2 (x) Then 0 f (x) / x is bounded and 0 is integrable on {x Rn : x 1}, so the argument above can be applied to f 0 together with 0 to prove that lim y( ) = 0 for almost all trajectories of the system d y/d = f 0 ( y( )). However, modulo a transformation of the time axis t= 0 ( y(s)) ds 0 ( y(s)) (x) = a(x) , b(x) u(x) (x) = c(x) , b(x) the trajectories are identical: x(t) = y( ). This, together with the boundedness of f (x)/ x near x = 0, also shows that x(t) exists for t [0, ) and tends to zero as t provided that lim y( ) = 0. Hence the proof of the second statement in Theorem 1 is complete. where a, b, c are polynomials, b(x) is positive, and is chosen to satisfy the integrability constraint. In this case, the condition (1) can be written as: [ ( f + u)] = = 1 [b b +1 [ 1 ( f a + c)] b b (a f + c)]. ( f a + c) 3. A COMPUTATIONAL APPROACH In order to understand the possibilities and limitations of computational approaches to nonlinear stability, an issue that has to be addressed is how to deal numerically with functional inequalities such as the standard Lyapunov one, or the divergence inequality (1). Even in the restricted case of polynomial functions, it is well-known that the problem of checking global nonnegativity of a polynomial of quartic (or higher) degree is computationally hard. For this reason, we need tractable suf cient conditions that guarantee nonnegativity, and that are not signi cantly conservative. A particularly interesting suf cient condition is given by the existence of a sum of squares decomposition: can the polynomial P(x) be writ2 ten as P(x) = i pi (x), for some polynomials pi(x)? Obviously, if this is the case, then P(x) takes only nonnegative values. Notice that in the case of quadratic forms, for instance, the two conditions (positivity and sum of squares) are equivalent. In this respect, it is interesting to notice that many methods used in control theory for constructing Lyapunov functions (for example, Since b is positive, we only need to satisfy the inequality: b ( f a + c) b (a f + c) > 0. (6) For xed b, , the inequality is linear in a, c. If instead of checking positivity, we check that the left-hand side is a sum of squares, for the case of polynomial (or rational) vector elds, the problem can be solved using LMI methods. 4. AN EXAMPLE A simple numerical example is the following: x = y x3 + x2 =u y The function b(x) is chosen based on the linearization of the system. We picked b(x) := 3x2 + 2x y + 2 y2, which is a control Lyapunov function for the linearized system, and therefore, b(x) (for some ) will be a good choice for a -function near the origin. Since we will be using cubic polynomials in x, y for c (a is taken x =y x +x y =u 6 3 2 u = 1.22 x 0.57 y .129 y3 are used for parameterization and positivity is veri ed using the ideas in (Parrilo, 2000). The numerical example should be viewed as a rst attempt to demonstrate the power of the approach. However, many modi cations are possible and much research in the area remains to be done. 4 2 0 y 2 4 ACKNOWLEDGMENT 6 4 2 0 x 2 4 6 6 Fig. 1 Phase plot of the closed-loop system for the Example. The collaboration between the two authors was supported by an exchange grant from the Swedish Foundation for International Cooperation in Research and Higher Education. to be a constant), we choose = 4 to satisfy the integrability condition. In this case, after solving the LMIs corresponding to the condition that the left-hand side of (6) is a sum of squares, we obtain an explicit expression for the controller, as a third order polynomial in x and y. The optimization criterion chosen is the 1 norm of the coef cients. This way, we approximately try to minimize the number of nonzero terms. The expression for the nal controller is: u(x, y) = 1.22x 0.57 y 0.129 y3 A phase plot of the closed-loop system in presented in Figure 4. This example has been chosen for its relative simplicity: in this particular case, it is possible to solve it directly using other methodologies. For instance, it can be noted that in this particular case b(x) is actually a control Lyapunov function for the system, and from that obtain a controller (e.g., using Sontag s formula). There is no requirement in the present framework that forces b(x) to be a clf. The main difference would be in terms of the computational dif culty of approximating the controller when the choice of the denominator b(x) is not optimal. Further research is needed in order to fully understand the practical implications. 6. REFERENCES Arnold, V. (1989): Mathematical Methods of Classical Mechanics, second edition. Graduate Texts in Mathematics. Springer. Artstein, Z. (1983): Stabilization with relaxed controls. Nonlinear Analysis TMA, 7, pp. 1163 1173. Brockett, R. (1983): Asymptotic stability and feedback stabilization. In Brockett et al., Eds., Differential Geometric Control Theory, vol. 27 of Progress in Mathematics. Birkhauser, Boston. Hahn, W. (1963): Theory and Applications of Lyapunov s Direct Method. Prentice-Hall, Englewood Cliffs, New Jersey. Isidori, A. (1995): Nonlinear Control Systems. Springer-Verlag, London. Krstic, M., I. Kanellakopoulos, and P. Kokotovich (1995): Nonlinear and Adaptive Control Design. John Wiley & Sons, New York. Ledyaev, Y. S. and E. D. Sontag (1999): A Lyapunov characterization of robust stabilization. Nonlinear Analysis, 37, pp. 813 840. Lefschetz, S. (1977): Differential Equations: Geometric Theory. Dover Publications, New York. Mane, R. (1987): Ergodic Theory and Differentiable Dynamics, english edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag. Parrilo, P. A. (2000): Structured semide nite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology. Rantzer, A. (2000a): A dual to Lyapunov s second theorem. To appear in Systems and Control Letters. 5. CONCLUDING REMARKS A new computational approach to nonlinear control synthesis has been introduced. The basis is a recent convergence criterion introduced by the rst author. The new criterion makes it possible to state the synthesis problem in terms of convex optimization and has earlier been exploited for optimal control problems in (Young, 1969; Vinter, 1993). Polynomials Rantzer, A. (2000b): On the dual of lyapunov s second theorem. In Proceedings of American Control Conference, pp. 1186 1189. Chicago. Vinter, R. (1993): Convex duality and nonlinear optimal control. SIAM J. Control and Optimization, 31:2, pp. 518 538. Young, L. C. (1969): Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders Company, Philadelphia, Pa.

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