Zum Relativgrad zeitvarianter Systeme

Transkript

1 Elgersburg, 16. Februar 2006

2 Relative degree for linear time-invariant systems Definition n(s) d(s) = c(si n A) 1 b = cb s 1 + cab s ca r 2 b s r 1 + ca r 1 b s r +... : has relative degree r {1,..., n} r = deg d deg n k = 0,..., r 2 : ca k b = 0 ca r 1 b 0

3 Normal form for linear time-invariant systems Proposition ẋ = A x + b u has relative degree r {1,..., n} y = c x y c y (1) ca. = U x, U =. invertible y (r 1) ca r 1 η N y y y (1). d y (1) dt. = y (r 1) r 1... r r s y (r 1) η p Q η 0. 0 ca r 1 b 0 u

4 Relative degree for nonlinear time-invariant systems Definition ẋ = f (x) + g(x)u y = h(x) has relative degree r in X R n open : x X k = 0,..., r 2 : L g L k f h(x) = 0 L g L r 1 f h(x) 0.

5 Normal form for nonlinear time-invariant systems Proposition ẋ = f (x) + g(x)u y = h(x) has relative degree r in X R n open local diffeomorphism Φ(x) = (y,..., y (r 1), η T ) T = (ξ T, η T ) T ξ 1 = ξ 2. ξ r 1 = ξ r ξ r = L r f h(φ 1 (ξ, η)) + L g L (r 1) f h(φ 1 (ξ, η)) u η = q(ξ, η) y = ξ 1

6 System classes ẋ = A x + b u y = c x ẋ = A(t) x + b(t) u y = c(t) x ẋ = f (x) + g(x) u y = h(x) ẋ = F (t, x, u) y = H(t, x)

7 The upside down definition ẋ = F (t, x, u) y = H(t, x) Definition (Liberzon, Morse, Sontag (2002) IEEE T-AC) y(t) = H(t, x(t)) =: H 0 (t, x(t)) ẏ(t) = t H 0(t, x(t)) + x H 0(t, x(t)) F (t, x(t), u(t)) =: H 1 (t, x(t), u(t)) ÿ(t) = t H 1(t, x(t), u(t)) + x H 1(t, x(t), u(t)) F (t, x(t), u(t)) + u H 1(t, x(t), u(t)) u(t) =: H 2 (t, x(t), u(t), u(t)).

8 The correct definition ẋ = F (t, x, u) y = H(t, x) H k+1 (t, x, u 0,..., u k ) := H k t + H k x F (t, x, u 0) + k 1 j=0 H k u j u j+1. Definition (F, H) has relative degree r in T X R R n open k = 1,..., r 1 i = 0,..., k 1 (t, x, u 0,..., u k 1 ) T X R k : H k u i (t, x, u 0,..., u k 1 ) = 0 and H r u 0 (t, x, u 0,..., u k 1 ) 0.

9 Proposition: Relative degree for nonlinear systems For ẋ = f (x) + g(x) u y = h(x) we have: (f, g, h) has relative degree r by the classical Isidori-definition (f, g, h) has relative degree r by the upside down definition.

10 System classes ẋ = A x + b u y = c x ẋ = A(t) x + b(t) u y = c(t) x ẋ = f (x) + g(x) u y = h(x) ẋ = F (t, x, u) y = H(t, x)

11 For A( ) C (R, R n n ), c( ) C (R, R m n ) set, for all t R, ( d dt + A(t) ) r (c(t)) := ċ(t) + c(t)a(t) ( d dt + A(t) ) k ( r (c(t)) := d dt + A(t) ) ( ( d r dt + A(t) ) k 1 ) r (c(t)) k 1. Lemma H k+1 (t, x, u 0,..., u k ) = ( d dt + A ) k+1 ( ) k r c x + j=0 k i=j ( i ) ( d ) i j [ ( d j dt dt + A ) k i ( ) ] r c b u j.

12 Remark For ẋ = Ax + bu y = c x we have ( d dt + A ) k r (c) b = c A k b H k+1 (t, x, u 0,..., u k ) = c A k x + k c A k j b u j. j=0

13 Theorem: Relative degree of time-varying linear systems ẋ = A(t) x + b(t) u y = c(t) x has relative degree r in T R open t T k = 0,..., r 2 : ( d dt + A(t) ) k r (c(t)) b(t) = 0 t T : ( d dt + A(t) ) r 1 r (c(t)) b(t) 0

14 Lemma ẋ = A(t) x + b(t) u y = c(t) x c ( d dt + A ) ( ) r c. ( d dt + A ) r 1 ( ) r c } {{ } =: C [ has relative degree r in T R open b,..., ( d dt A) r 1 ( b ) ] } {{ } =: B 0 ( 1) r 1 Γ = Γ... Gl r (R where Γ(t) := ( d dt + A(t) r ) r 1 (c(t)) b(t).

15 Doležal s Theorem (1967) Suppose M C l (R, R n n ) with t 0 : rk M(t) = r. = T C l (R, Gl n (R)) : t 0 = M(t) T (t) = [ F (t), 0 n (n r) ] β > 0 t 0 = T (t) β ε > 0 t 0 = ε T T (t) T (t) 1/ε

16 Theorem: Normal form for time-varying linear systems ẋ = A(t) x + b(t) u y = c(t) x smooth N, V : d dt y y (1). y (r 1) η = [ C N has relative degree r in T R open ] [B(CB) 1, V ] = I n, r 1... r r s p Q y y (1). y (r 1) η y y (1). y (r 1) η [ ] C = x, N ca r 1 b 0 u.

17 Proposition The zero dynamics of ẋ = A(t) x + b(t) u y = c(t) x } ( ) are given by, and satisfy, ZD T := = { } (x, u) solves ( ) (x, u) C 1 (T, R n ) C(T, R) and y 0 on T { (Vη, Γ 1 sη ) η = Q η}, if ( ) has relative degree r.

18 Theorem For ẋ = A(t) x + b(t) u y = c(t) x with relative degree r on R we have (x, u) ZD R : lim t x(t) = 0 η = Q η is asymptotically stable.

19 Conclusions Upside down definition of relative degree. Linear time-varying systems: nice characterization of relative degree. Linear time-varying systems: linear coordinate transformation for normal form. Linear time-varying systems: parametrization of zero dynamics. Linear time-varying systems: characterization of stable zero dynamics.