Nonlinear Control. Nonlinear Control Lecture # 25 State Feedback Stabilization

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1 Nonlinear Control Lecture # 25 State Feedback Stabilization

2 Backstepping η = f a (η)+g a (η)ξ ξ = f b (η,ξ)+g b (η,ξ)u, g b 0, η R n, ξ, u R Stabilize the origin using state feedback View ξ as virtual control input to the system η = f a (η)+g a (η)ξ Suppose there is ξ = φ(η) that stabilizes the origin of η = f a (η)+g a (η)φ(η) V a η [f a(η)+g a (η)φ(η)] W(η)

3 z = ξ φ(η) η = [f a (η)+g a (η)φ(η)]+g a (η)z ż = F(η,ξ)+g b (η,ξ)u V(η,ξ) = V a (η)+ 1 2 z2 = V a (η)+ 1 [ξ φ(η)]2 2 V = V a η [f a(η)+g a (η)φ(η)]+ V a η g a(η)z +zf(η,ξ)+zg b (η,ξ)u [ ] Va W(η)+z η g a(η)+f(η,ξ)+g b (η,ξ)u

4 ] Va V W(η)+z[ η g a(η)+f(η,ξ)+g b (η,ξ)u [ ] 1 Va u = g b (η,ξ) η g a(η)+f(η,ξ)+kz, k > 0 V W(η) kz 2

5 Example 9.9 ẋ 1 = x 2 1 x3 1 +x 2, ẋ 2 = u ẋ 1 = x 2 1 x3 1 +x 2 x 2 = φ(x 1 ) = x 2 1 x 1 ẋ 1 = x 1 x 3 1 V a (x 1 ) = 1 2 x2 1 V a = x 2 1 x 4 1, x 1 R z 2 = x 2 φ(x 1 ) = x 2 +x 1 +x 2 1 ẋ 1 = x 1 x 3 1 +z 2 ż 2 = u+(1+2x 1 )( x 1 x 3 1 +z 2)

6 V(x) = 1 2 x z2 2 V = x 1 ( x 1 x 3 1 +z 2) +z 2 [u+(1+2x 1 )( x 1 x 3 1 +z 2 )] V = x 2 1 x 4 1 +z 2 [x 1 +(1+2x 1 )( x 1 x 3 1 +z 2)+u] u = x 1 (1+2x 1 )( x 1 x 3 1 +z 2) z 2 V = x 2 1 x 4 1 z 2 2 The origin is globally asymptotically stable

7 Example 9.10 ẋ 1 = x 2 1 x 3 1 +x 2, ẋ 2 = x 3, ẋ 3 = u ẋ 1 = x 2 1 x3 1 +x 2, ẋ 2 = x 3 x 3 = x 1 (1+2x 1 )( x 1 x 3 1 +z 2) z 2 def = φ(x 1,x 2 ) V a (x) = 1 2 x z2 2, Va = x 2 1 x4 1 z2 2 z 3 = x 3 φ(x 1,x 2 ) ẋ 1 = x 2 1 x 3 1 +x 2, ẋ 2 = φ(x 1,x 2 )+z 3 ż 3 = u φ (x 2 1 x 3 1 +x 2 ) φ (φ+z 3 ) x 1 x 2

8 V = V a z2 3 V = V a x 1 (x 2 1 x3 1 +x 2)+ V a x 2 (z 3 +φ) +z 3 [ u φ x 1 (x 2 1 x3 1 +x 2) φ x 2 (z 3 +φ) ] V = x 2 1 x4 1 (x 2 +x 1 +x 2 1 [ )2 Va +z 3 φ (x 2 1 x 3 1 +x 2 ) φ ] (z 3 +φ)+u x 2 x 1 x 2 u = V a x 2 + φ x 1 (x 2 1 x3 1 +x 2)+ φ x 2 (z 3 +φ) z 3 The origin is globally asymptotically stable

9 Strict-Feedback Form ẋ = f 0 (x)+g 0 (x)z 1 ż 1 = f 1 (x,z 1 )+g 1 (x,z 1 )z 2 ż 2 = f 2 (x,z 1,z 2 )+g 2 (x,z 1,z 2 )z 3. ż k 1 = f k 1 (x,z 1,...,z k 1 )+g k 1 (x,z 1,...,z k 1 )z k ż k = f k (x,z 1,...,z k )+g k (x,z 1,...,z k )u g i (x,z 1,...,z i ) 0 for 1 i k

10 Example 9.12 ẋ = x+x 2 z, ż = u ẋ = x+x 2 z z = 0 ẋ = x, V a = 1 2 x2 V a = x 2 V = 1 2 (x2 +z 2 ) V = x( x+x 2 z)+zu = x 2 +z(x 3 +u) u = x 3 kz, k > 0, V = x 2 kz 2 Global stabilization Compare with semiglobal stabilization in Example 9.7

11 Example 9.13 ẋ = x 2 xz, ż = u ẋ = x 2 xz z = x+x 2 ẋ = x 3, V 0 (x) = 1 2 x2 V = x 4 V = V (z x x2 ) 2 V = x 4 +(z x x 2 )[ x 2 +u (1+2x)(x 2 xz)] u = (1+2x)(x 2 xz)+x 2 k(z x x 2 ), k > 0 V = x 4 k(z x x 2 ) 2 Global stabilization

12 Passivity-Based Control ẋ = f(x,u), y = h(x), f(0,0) = 0 u T y V = V x f(x,u) Theorem 9.1 If the system is (1) passive with a radially unbounded positive definite storage function and (2) zero-state observable, then the origin can be globally stabilized by u = φ(y), φ(0) = 0, y T φ(y) > 0 y 0

13 Proof V = V x f(x, φ(y)) yt φ(y) 0 V(x(t)) 0 y(t) 0 u(t) 0 x(t) 0 Apply the invariance principle A given system may be made passive by or both (1) Choice of output, (2) Feedback,

14 Choice of Output V ẋ = f(x)+g(x)u, f(x) 0, x x No output is defined. Choose the output as y = h(x) def = [ ] T V x G(x) V = V V f(x)+ x x G(x)u yt u Check zero-state observability

15 Example 9.14 ẋ 1 = x 2, ẋ 2 = x 3 1 +u V(x) = 1 4 x x2 2 With u = 0 V = x 3 1 x 2 x 2 x 3 1 = 0 Is it zero-state observable? Take y = V x G = V x 2 = x 2 with u = 0, y(t) 0 x(t) 0 u = kx 2 or u = (2k/π)tan 1 (x 2 ) (k > 0)

16 Feedback Passivation Definition The system ẋ = f(x)+g(x)u, y = h(x) ( ) is equivalent to a passive system if u = α(x)+β(x)v such that ẋ = f(x)+g(x)α(x)+g(x)β(x)v, y = h(x) is passive Theorem [20] The system (*) is locally equivalent to a passive system (with a positive definite storage function) if it has relative degree one at x = 0 and the zero dynamics have a stable equilibrium point at the origin with a positive definite Lyapunov function

17 Example 9.15 (m-link Robot Manipulator) M(q) q +C(q, q) q +D q +g(q) = u M = M T > 0, (Ṁ 2C)T = ( M 2C), D = D T 0 Stabilize the system at q = q r e = q q r, ė = q M(q)ë+C(q, q)ė+dė+g(q) = u (e = 0,ė = 0) is not an open-loop equilibrium point u = g(q) K p e+v, (K p = Kp T > 0) M(q)ë+C(q, q)ė+dė+k p e = v

18 M(q)ë+C(q, q)ė+dė+k p e = v V = 1 M(q)ė+ 1 2ėT 2 et K p e V = 1 (Ṁ 2ėT 2C)ė ėt Dė ė T K p e+ė T v +e T K p ė ė T v y = ė Is it zero-state observable? Set v = 0 ė(t) 0 ë(t) 0 K p e(t) 0 e(t) 0 v = φ(ė), [φ(0) = 0, ė T φ(ė) > 0, ė 0] u = g(q) K p e φ(ė) Special case: u = g(q) K p e K d ė, K d = K T d > 0