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: QuadraticType 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 quadratictype Lyapunov function for the nominal system x = x3 V x (x ) = 4x ,
3 6 x = x3 + g(t, x) V x = 4x3 c4 = 4 (x) = x3 , c3 = 4, Suppose g(t, x) x3 , 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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 Spring '08
 CHOI
 #, ax, Stability theory, xT P xc, c3 c4, 3.026 2 2 g

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