Nonlinear Control Lecture # 34 Output Feedback Stabilization - PowerPoint PPT Presentation

Nonlinear Control Lecture # 34 Output Feedback Stabilization Nonlinear Control Lecture # 34 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 | ≤ max Nonlinear Control Lecture # 34 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 # 34 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 # 34 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 # 34 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 # 34 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 # 34 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 # 34 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 # 34 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 # 34 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 ψ i satisfies a global Lipschitz condition. 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 # 34 Output Feedback Stabilization

In many problems, we can measure some state variables in addition to y Magnetic levitation system x 1 ˙ = x 2 4 cx 2 3 x 2 ˙ = − bx 2 + 1 − (1 + x 1 ) 2 1 � βx 2 x 3 � x 3 ˙ = − x 3 + u + T ( x 1 ) (1 + x 1 ) 2 Typical measurements are the ball position x 1 and the current x 3 Nonlinear Control Lecture # 34 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 # 34 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 # 34 Output Feedback Stabilization

Separation Principle Theorem 12.2 Suppose the origin of ˙ X = f ( X ) is asymptotically stable and R is its region of attraction. Let S be any compact set in the interior of R and Q be any compact subset of R ρ . Then, given any µ > 0 there exist ε ∗ > 0 and T ∗ > 0 , dependent on µ , such that for every 0 < ε ≤ ε ∗ , the solutions ( X ( t ) , ˆ x ( t )) of the closed-loop system, starting in S × Q , are bounded for all t ≥ 0 and satisfy �X ( t ) � ≤ µ and � ˆ x ( t ) � ≤ µ, ∀ t ≥ T ∗ �X ( t ) − X r ( t ) � ≤ µ, ∀ t ≥ 0 where X r is the solution of ˙ X = f ( X ) , starting at X (0) Nonlinear Control Lecture # 34 Output Feedback Stabilization

If the origin of ˙ X = f ( X ) is exponentially stable, then the origin of the closed-loop system is exponentially stable and S × Q is a subset of its region of attraction Nonlinear Control Lecture # 34 Output Feedback Stabilization