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# Lect_13 - Nonlinear Systems and Control Lecture 13...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 13 Perturbed Systems p.1/15 Nominal System: x = f (x), f (0) = 0 Perturbed System: x = f (x) + g(t, x), g(t, 0) = 0 Case 1: The origin of the nominal system is exponentially stable c1 x 2 V (x) c2 x 2 V x f (x) -c3 x V x c4 x 2 p.2/15 Use V (x) as a Lyapunov function candidate for the perturbed system V (t, x) = V x f (x) + V x g(t, x) Assume that g(t, x) x , 0 V x 2 V (t, x) -c3 x -c3 x 2 2 + g(t, x) + c4 x p.3/15 < c3 c4 2 V (t, x) -(c3 - c4 ) x The origin is an exponentially stable equilibrium point of the perturbed system p.4/15 Example x1 = x2 x2 = -4x1 - 2x2 + x3 , 2 x = Ax + g(x) A= 0 1 -4 -2 , g(x) = 0 x3 2 0 The eigenvalues of A are -1 j 3 P A + AT P = -I P = 3 2 1 8 1 8 5 16 p.5/15 For the quadratic Lyapunv function V (x) = xT P x, min (P ) x =:c1 2 2 V (x) max (P ) x =:c2 2 2 LAx V = V x Ax = -xT Qx - min (Q) x =:c3 2 2 V x 2 = 2xT P 2 P 2 x 2 = 2max (P ) x =:c4 2 p.6/15 V (x) = x P x, c3 = 1, c4 = 2 P T V x Ax = -xT x = 2max (P ) = 2 1.513 = 3.026 2 2 g(x) = |x2 |3 k2 |x2 | k2 x , |x2 | k2 g(x) satisfies the bound g(x) x over compact sets of x. Consider the compact set c = {V (x) c} = {xT P x c}, xT P xc xT P xc c>0 k2 = max |x2 | = max |[0 1]x| p.7/15 Fact: xT P xc max Lx = c LP -1/2 Proof x Px c T 1 c x Px 1 T 1 c xT P 1/2 P 1/2 x 1 xT P xc max 1 y = P 1/2 x c -1/2 Lx = max L c P y = c LP -1/2 y T y1 p.8/15 k2 = max |[0 1]x| = xT P xc c [0 1]P -1/2 = 1.8194 c g(x) c (1.8194)2 x , g(x) x , < c3 c4 x c , 3.026 1 x c 0.1 c 2 = c (1.8194)2 (1.8194)2 c < < 0.1/c V (x) -(1 - 10c) x Hence, the origin is exponentially stable and c is an estimate of the region of attraction p.9/15 Alternative Bound on V (x) = - x 2 2 2 + 2xT P g(x) + 1 x3 ([2 5]x) 2 8 + 29 x2 2 8 2 = - x - x x Over c , x2 (1.8194)2 c 2 V (x) - 1 - = - 1- 29 (1.8194)2 c 8 x 2 c 0.448 x 2 If < 0.448/c, the origin will be exponentially stable and c will be an estimate of the region of attraction p.10/15 Remark: The inequality < 0.448/c shows a tradeoff between the estimate of the region of attraction and the estimate of the upper bound on The smaller the upper bound on , the larger the estimate of RA p.11/15 Case 2: The origin of the nominal system is asymptotically stable V (t, x) = V x f (x)+ V x g(t, x) -W3 (x)+ V x g(t, x) Under what condition will the following inequality hold? V x g(t, x) < W3 (x) Special Case: Quadratic-Type Lyapunov function V x f (x) -c3 (x), 2 V x c4 (x) p.12/15 (x) : Rn R is positive definite and continuous V (t, x) -c3 2 (x) + c4 (x) g(t, x) If g(t, x) (x), with < c3 c4 V (t, x) -(c3 - c4 )2 (x) p.13/15 Example V (x) = x4 is a quadratic-type Lyapunov function for the nominal system x = -x3 V x (-x ) = -4x , 3 6 x = -x3 + g(t, x) V x = 4|x|3 c4 = 4 (x) = |x|3 , c3 = 4, Suppose |g(t, x)| |x|3 , x, with < 1 V (t, x) -4(1 - )2 (x) Hence, the origin is a globally uniformly asymptotically stable p.14/15 Remark: A nominal system with asymptotically, but not exponentially, stable origin is not robust to smooth perturbations with arbitrarily small linear growth bounds Example x = -x3 + x The origin is unstable for any > 0 (can be easily seen via linearization) p.15/15 ...
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