Nonlinear Control Lecture # 14 Input-Output Stability. Nonlinear Control

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1 Nonlinear Control Lecture # 14 Input-Output Stability

2 L Stability Input-Output Models: y = Hu u(t) is a piecewise continuous function of t and belongs to a linear space of signals The space of bounded functions: sup t u(t) < The space of square-integrable functions: u T (t)u(t) dt < Norm of a signal u : u and u = u = au = a u for any a > Triangle Inequality: u 1 +u 2 u 1 + u 2

3 L p spaces: L : u L = sup u(t) < t L 2 : u L2 = u T (t)u(t) dt < ( 1/p L p : u Lp = u(t) dt) p <, 1 p < Notation L m p : p is the type of p-norm used to define the space and m is the dimension of u

4 Extended Space: L e = {u u τ L, { τ [, )} u(t), t τ u τ is a truncation of u: u τ (t) =, t > τ L e is a linear space and L L e Example u(t) = t, u τ (t) = { t, t τ, t > τ u / L but u τ L e

5 Causality: A mapping H : L m e Lq e is causal if the value of the output (Hu)(t) at any time t depends only on the values of the input up to time t (Hu) τ = (Hu τ ) τ Definition 6.1 A scalar continuous function g(r), defined for r [,a), is a gain function if it is nondecreasing and g() = A class K function is a gain function but not the other way around. By not requiring the gain function to be strictly increasing we can have g = or g(r) = sat(r)

6 Definition 6.2 A mapping H : L m e L q e is L stable if there exist a gain function g, defined on [, ), and a nonnegative constant β such that (Hu) τ L g( u τ L )+β, u L m e and τ [, ) It is finite-gain L stable if there exist nonnegative constants γ and β such that (Hu) τ L γ u τ L +β, u L m e and τ [, ) In this case, we say that the system has L gain γ. The bias term β is included in the definition to allow for systems where Hu does not vanish at u =.

7 Example 6.1: Memoryless function y = h(u) Suppose h(u) a+b u, u R Finite-gain L stable with β = a and γ = b If a =, then for each p [1, ) h(u(t)) p dt (b) p u(t) p dt Finite-gain L p stable with β = and γ = b For h(u) = u 2, H is L stable with zero bias and g(r) = r 2. It is not finite-gain L stable because h(u) = u 2 cannot be bounded γ u +β for all u R

8 Example 6.2: SISO causal convolution operator y(t) = t h(t σ)u(σ) dσ, h(t) = for t < Suppose h L 1 h L1 = h(σ) dσ < y(t) t h(t σ) u(σ) dσ t h(t σ) dσsup σ τ u(σ) = t h(s) dssup σ τ u(σ) y τ L h L1 u τ L, τ [, ) Finite-gain L stable Also, finite-gain L p stable for p [1, ) (see textbook)

9 Small-signal L Stability Example 6.3 y = tanu The output y(t) is defined only when the input signal is restricted to u(t) < π/2 for all t ( ) tanr u(t) { u r < π/2} y u r ( ) tanr y Lp u Lp, p [1, ] r

10 Definition 6.3 A mapping H : L m e Lq e is small-signal L stable (respectively, small-signal finite-gain L stable) if there is a positive constant r such that the condition for L stability ( respectively, finite-gain L stability ) is satisfied for all u L m e with sup t τ u(t) r

11 L Stability of State Models ẋ = f(x,u), y = h(x,u), = f(,), = h(,) Case 1: The origin of ẋ = f(x,) is exponentially stable Theorem 6.1 Suppose, x r, u r u, c 1 x 2 V(x) c 2 x 2 V x f(x,) c 3 x 2, V x c 4 x f(x,u) f(x,) L u, h(x,u) η 1 x +η 2 u Then, for each x with x r c 1 /c 2, the system is small-signal finite-gain L p stable for each p [1, ]. It is finite-gain L p stable x R n if the assumptions hold globally [see the textbook for β and γ]

12 Proof V = V V f(x,)+ x x [f(x,u) f(x,)] V c 3 x 2 +c 4 L x u c 3 V + c 4L u V c 2 c1 W(x) = V(x) Ẇ aw +b u(t), a = c 3 2c 2, b = c 4L 2 c 1 U(t) = e at W(x(t)) U = e at Ẇ +ae at W be at u U(t) U()+ t be aτ u(τ) dτ

13 W(x(t)) e at W(x())+ x(t) t c1 x W(x) c 2 x c2 c 1 x() e at + c 4L 2c 1 e a(t τ) b u(τ) dτ t y(t) η 1 x(t) +η 2 u(t) e at u(τ) dτ t y(t) k x() e at +k 2 e a(t τ) u(τ) dτ +k 3 u(t)

14 Example 6.4 ẋ = x x 3 +u, y = tanhx+u V = 1 2 x2 V = x( x x 3 ) x 2 c 1 = c 2 = 1 2, c 3 = c 4 = 1, L = η 1 = η 2 = 1 Finite-gain L p stable for each x() R and each p [1, ] Example 6.5 ẋ 1 = x 2, ẋ 2 = x 1 x 2 atanhx 1 +u, y = x 1, a V(x) = x T Px = p 11 x p 12 x 1 x 2 +p 22 x 2 2

15 V = 2p 12 (x 2 1 +ax 1tanhx 1 )+2(p 11 p 12 p 22 )x 1 x 2 2ap 22 x 2 tanhx 1 2(p 22 p 12 )x 2 2 p 11 = p 12 +p 22 the term x 1 x 2 is canceled p 22 = 2p 12 = 1 P is positive definite V = x 2 1 x2 2 ax 1tanhx 1 2ax 2 tanhx 1 V x 2 +2a x 1 x 2 (1 a) x 2 a < 1 c 1 = λ min (P),c 2 = λ max (P),c 3 = 1 a,c 4 = 2c 2 L = η 1 = 1, η 2 = For each x() R 2, p [1, ], the system is finite-gain L p stable γ = 2[λ max (P)] 2 /[(1 a)λ min (P)]

16 Case 2: The origin of ẋ = f(x,) is asymptotically stable Theorem 6.2 Suppose that, for all (x,u), f is locally Lipschitz and h is continuous and satisfies h(x,u) g 1 ( x )+g 2 ( u )+η, η for some gain functions g 1, g 2. If ẋ = f(x,u) is ISS, then, for each x() R n, the system is L stable ẋ = f(x,u), y = h(x,u)

17 Proof { ( )} x(t) max β( x,t),γ sup u(t) t τ ( { y(t) g 1 max β( x,t),γ ( sup t τ u(t) )}) +g 2 ( u(t) )+η g 1 (max{a,b}) g 1 (a)+g 1 (b) g = g 1 γ +g 2 y τ L g( u τ L )+β and β = g 1 (β( x,))+η

18 Theorem 6.3 Suppose f is locally Lipschitz and h is continuous in some neighborhood of (x =, u = ). If the origin of ẋ = f(x,) is asymptotically stable, then there is a constant k 1 > such that for each x() with x() < k 1, the system is small-signal L stable Proof ẋ = f(x,u), y = h(x,u) Use Lemma 4.7 (asymptotic stability is equivalent to local ISS)

19 Example 6.6 ẋ = x 2x 3 +(1+x 2 )u 2, y = x 2 +u ISS from Example 4.13 g 1 (r) = r 2, g 2 (r) = r, η = L stable

20 Example 6.7 ẋ 1 = x 3 1 +x 2, ẋ 2 = x 1 x 3 2 +u, y = x 1 +x 2 V = (x 2 1 +x2 2 ) V = 2x 4 1 2x4 2 +2x 2u x 4 1 +x x 4 V x 4 +2 x u = (1 θ) x 4 θ x 4 +2 x u, < θ < 1 ) 1/3 (1 θ) x 4, x ISS ( 2 u θ g 1 (r) = 2r, g 2 =, η = L stable

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