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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online