Nonlinear Control Lecture # 13 Output Feedback Stabilization - PowerPoint PPT Presentation

Nonlinear Control Lecture # 13 Output Feedback Stabilization Nonlinear Control Lecture # 13 Output Feedback Stabilization

Passivity-Based Control In Section 9.6 we saw that if the system x = f ( x, u ) , ˙ y = h ( x ) is passive (with a positive definite storage function) and zero-state observable, it can be stabilized by y T φ ( y ) > 0 , u = − φ ( y ) , φ (0) = 0 , ∀ y � = 0 Suppose the system y = ∂h def = ˜ x = f ( x, u ) , ˙ ˙ ∂xf ( x, u ) h ( x, u ) is passive (with a positive definite storage function V ( x ) ) and zero state observable Nonlinear Control Lecture # 13 Output Feedback Stabilization

y + u ✲ ❥ ✲ ✲ Plant − ✻ z ✛ ✛ s φ ( · ) τs +1 y + u y ˙ ✲ ❥ ✲ ✲ ✲ s Plant − ✻ z ✛ 1 ✛ φ ( · ) τs +1 Nonlinear Control Lecture # 13 Output Feedback Stabilization

s τs + 1 τ ˙ w = − w + y, z = ( − w + y ) /τ MIMO systems τ i ˙ w i = − w i + y i , z i = ( − w i + y i ) /τ i , for 1 ≤ i ≤ m Note that τ i ˙ z i = − z i + ˙ y i Nonlinear Control Lecture # 13 Output Feedback Stabilization

Lemma 12.1 Consider the system x = f ( x, u ) , ˙ y = h ( x ) and the output feedback controller u i = − φ i ( z i ) , τ i ˙ w i = − w i + y i , z i = ( − w i + y i ) /τ i τ i > 0 , φ i (0) = 0 , z i φ i ( z i ) > 0 ∀ z i � = 0 Suppose the auxiliary system y = ˜ x = f ( x, u ) , ˙ ˙ h ( x, u ) is Nonlinear Control Lecture # 13 Output Feedback Stabilization

passive with a positive definite storage function V ( x ) V = ∂V u T ˙ y ≥ ˙ ∂x f ( x, u ) , ∀ ( x, u ) zero-state observable with u = 0 , ˙ y ( t ) ≡ 0 ⇒ x ( t ) ≡ 0 Then the origin of the closed-loop system is asymptotically stable. It is globally asymptotically stable if V ( x ) is radially � z i unbounded and 0 φ i ( σ ) dσ → ∞ as | z i | → ∞ Nonlinear Control Lecture # 13 Output Feedback Stabilization

Proof � z i m � W ( x, z ) = V ( x ) + τ i φ i ( σ ) dσ 0 i =1 m m z i ≤ u T ˙ z i φ i ( z i ) − u T ˙ W = ˙ ˙ � � V + τ i φ i ( z i ) ˙ y − y i =1 i =1 m ˙ � W ≤ − z i φ i ( z i ) i =1 ˙ W ≡ 0 ⇒ z ( t ) ≡ 0 ⇒ u ( t ) ≡ 0 and y ( t ) ≡ 0 ˙ Apply the invariance principle Nonlinear Control Lecture # 13 Output Feedback Stabilization

Example 12.2 ( m -link Robot Manipulator) M ( q )¨ q + C ( q, ˙ q ) ˙ q + D ˙ q + g ( q ) = u M = M T > 0 , ( ˙ M − 2 C ) T = − ( ˙ M − 2 C ) , D = D T ≥ 0 Stabilize the system at q = q r , e = q − q r , ˙ e = ˙ q M ( q )¨ e + C ( q, ˙ q )˙ e + D ˙ e + g ( q ) = u u = g ( q ) − K p e + v, [ K p = K p > 0] M ( q )¨ e + C ( q, ˙ q )˙ e + D ˙ e + K p e = v, y = e 2 e T K p e e T M ( q )˙ V = 1 e + 1 2 ˙ Nonlinear Control Lecture # 13 Output Feedback Stabilization

2 e T K p e e T M ( q )˙ V = 1 e + 1 2 ˙ e T D ˙ e T v ˙ e T ( ˙ e T K p e + ˙ e T v + e T K p ˙ V = 1 2 ˙ M − 2 C )˙ e − ˙ e − ˙ e ≤ ˙ Is it zero-state observable? Set v = 0 e ( t ) ≡ 0 ⇒ ¨ ˙ e ( t ) ≡ 0 ⇒ K p e ( t ) ≡ 0 ⇒ e ( t ) ≡ 0 τ i ˙ w i = − w i + e i , z i = ( − a i w i + e i ) /τ i , for 1 ≤ i ≤ m u = g ( q ) − K p ( q − q r ) − K d z K d is positive diagonal matrix. Compare with state feedback u = g ( q ) − K p ( q − q r ) − K d ˙ q Nonlinear Control Lecture # 13 Output Feedback Stabilization

Observer-Based Control x = f ( x, u ) , ˙ y = h ( x ) State Feedback Controller: Design a locally Lipschitz u = γ ( x ) to stabilize the origin of x = f ( x, γ ( x )) ˙ Observer: ˙ ˆ x = f (ˆ x, u ) + H [ y − h (ˆ x )] x = x − ˆ ˜ x ˙ def ˜ x = f ( x, u ) − f (ˆ x, u ) − H [ h ( x ) − h (ˆ x )] = g ( x, ˜ x ) g ( x, 0) = 0 Nonlinear Control Lecture # 13 Output Feedback Stabilization

Design H such that ˙ x = g ( x, ˜ ˜ x ) has an exponentially stable equilibrium point at ˜ x = 0 and there is Lyapunov function V 1 (˜ x ) such that � � x � 2 , ∂V 1 ∂V 1 x � 2 ≤ V 1 ≤ c 2 � ˜ x � 2 , � � c 1 � ˜ x g ≤ − c 3 � ˜ � ≤ c 4 � ˜ x � � � ∂ ˜ ∂ ˜ x � u = γ (ˆ x ) Closed-loop system: ˙ x = f ( x, γ ( x − ˜ ˙ x )) , x = g ( x, ˜ ˜ x ) ( ⋆ ) Nonlinear Control Lecture # 13 Output Feedback Stabilization

Theorem 12.1 If the origin of ˙ x = f ( x, γ ( x )) is asymptotically stable, so is the origin of ( ⋆ ) If the origin of ˙ x = f ( x, γ ( x )) is exponentially stable, so is the origin of ( ⋆ ) If the assumptions hold globally and the system x = f ( x, γ ( x − ˜ ˙ x )) , with input ˜ x , is ISS, then the origin of ( ⋆ ) is globally asymptotically stable Nonlinear Control Lecture # 13 Output Feedback Stabilization

High-Gain Observers Example 12.3 x 1 = x 2 , ˙ x 2 = φ ( x, u ) , ˙ y = x 1 State feedback control: u = γ ( x ) stabilizes the origin of x 1 = x 2 , ˙ x 2 = φ ( x, γ ( x )) ˙ High-gain observer ˙ ˙ x, u ) + ( α 2 /ε 2 )( y − ˆ x 1 = ˆ ˆ x 2 + ( α 1 /ε )( y − ˆ x 1 ) , ˆ x 2 = φ 0 (ˆ x 1 ) φ 0 is a nominal model of φ , α i > 0 , 0 < ε ≪ 1 � b � � be − at/ε , ε 2 cM � εe − at/ε , εcM | ˜ x 1 | ≤ max , | ˜ x 2 | ≤ Nonlinear Control Lecture # 13 Output Feedback Stabilization

The bound on ˜ x 2 demonstrates the peaking phenomenon, which might destabilize the closed-loop system Example: x 2 = x 3 x 1 = x 2 , ˙ ˙ 2 + u, y = x 1 State feedback control: u = − x 3 2 − x 1 − x 2 Output feedback control: x 3 u = − ˆ 2 − ˆ x 1 − ˆ x 2 ˙ ˙ x 2 = (1 /ε 2 )( y − ˆ x 1 = ˆ ˆ x 2 + (2 /ε )( y − ˆ x 1 ) , ˆ x 1 ) Nonlinear Control Lecture # 13 Output Feedback Stabilization

0.5 0 SFB OFB ε = 0.1 −0.5 x 1 OFB ε = 0.01 −1 OFB ε = 0.005 −1.5 −2 0 1 2 3 4 5 6 7 8 9 10 1 0 −1 x 2 −2 −3 0 1 2 3 4 5 6 7 8 9 10 0 −100 u −200 −300 −400 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t Nonlinear Control Lecture # 13 Output Feedback Stabilization

ε = 0 . 004 0.2 0 x 1 −0.2 −0.4 −0.6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 −200 x 2 −400 −600 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 2000 1000 u 0 −1000 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 t Nonlinear Control Lecture # 13 Output Feedback Stabilization

Closed-loop system under state feedback: � 0 � 1 x = Ax, ˙ A = − 1 − 1 � � 1 . 5 0 . 5 PA + A T P = − I ⇒ P = 0 . 5 1 Suppose x (0) belongs to the positively invariant set Ω = { V ( x ) ≤ 0 . 3 } | u | ≤ | x 2 | 3 + | x 1 + x 2 | ≤ 0 . 816 , ∀ x ∈ Ω Saturate u at ± 1 Nonlinear Control Lecture # 13 Output Feedback Stabilization

x 3 u = sat( − ˆ 2 − ˆ x 1 − ˆ x 2 ) SFB 0.15 OFB ε = 0.1 OFB ε = 0.01 0.1 OFB ε = 0.001 x 1 0.05 0 −0.05 0 1 2 3 4 5 6 7 8 9 10 0.05 0 x 2 −0.05 −0.1 0 1 2 3 4 5 6 7 8 9 10 0 u −0.5 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t Nonlinear Control Lecture # 13 Output Feedback Stabilization

Region of attraction under state feedback: 2 1 x 2 0 −1 −2 −3 −2 −1 0 1 2 3 x 1 Nonlinear Control Lecture # 13 Output Feedback Stabilization

Region of attraction under output feedback: 1 0.5 x 2 0 −0.5 −1 −2 −1 0 1 2 x 1 ε = 0 . 08 (dashed) and ε = 0 . 01 (dash-dot) Nonlinear Control Lecture # 13 Output Feedback Stabilization

Analysis of the closed-loop system: x 1 ˙ = x 2 x 2 ˙ = φ ( x, γ ( x − ˜ x )) ε ˙ η 1 = − α 1 η 1 + η 2 ε ˙ η 2 = − α 2 η 1 + εδ ( x, ˜ x ) η ✻ O (1 /ε ) q q ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ O ( ε ) ❉ ❉ ❲ ❲ ✲ ✛ ✲ Ω b x ✛ ✲ Ω c Nonlinear Control Lecture # 13 Output Feedback Stabilization

General case w ˙ = ψ ( w, x, u ) x i ˙ = x i +1 + ψ i ( x 1 , . . . , x i , u ) , 1 ≤ i ≤ ρ − 1 x ρ ˙ = φ ( w, x, u ) y = x 1 z = q ( w, x ) φ (0 , 0 , 0) = 0 , ψ (0 , 0 , 0) = 0 , q (0 , 0) = 0 The normal form and models of mechanical and electromechanical systems take this form with ψ 1 = · · · = ψ ρ = 0 Why the extra measurement z ? Nonlinear Control Lecture # 13 Output Feedback Stabilization

Stabilizing state feedback controller: ˙ ϑ = Γ( ϑ, x, z ) , u = γ ( ϑ, x, z ) γ and Γ are globally bounded functions of x Closed-loop system ˙ X = f ( X ) , X = col( w, x, ϑ ) Output feedback controller ˙ ϑ = Γ( ϑ, ˆ x, z ) , u = γ ( ϑ, ˆ x, z ) Nonlinear Control Lecture # 13 Output Feedback Stabilization

Observer x i , u ) + α i ˙ x i ˆ = x i +1 + ψ i (ˆ ˆ x 1 , . . . , ˆ ε i ( y − ˆ x 1 ) , 1 ≤ i ≤ ρ − 1 x, u ) + α ρ ˙ x ρ ˆ = φ 0 ( z, ˆ ε ρ ( y − ˆ x 1 ) ε > 0 and α 1 to α ρ are chosen such that the roots of s ρ + α 1 s ρ − 1 + · · · + α ρ − 1 s + α ρ = 0 have negative real parts Nonlinear Control Lecture # 13 Output Feedback Stabilization