Nonlinear Control Lecture # 38 Tracking & Regulation Nonlinear - PowerPoint PPT Presentation

Nonlinear Control Lecture # 38 Tracking & Regulation Nonlinear Control Lecture # 38 Tracking & Regulation

Output Feedback Tracking: η ˙ = f 0 ( η, ξ ) e i ˙ = e i +1 , 1 ≤ i ≤ ρ − 1 a ( η, ξ ) + b ( η, ξ ) u + δ ( t, η, ξ, u ) − r ( ρ ) ( t ) e ρ ˙ = Regulation: η ˙ = f 0 ( η, ξ, w ) ˙ ξ i = ξ i +1 , 1 ≤ i ≤ ρ − 1 ˙ ξ ρ = a ( η, ξ, w ) + b ( η, ξ, w ) u y = ξ 1 Design partial state feedback control that uses ξ Use a high-gain observer Nonlinear Control Lecture # 38 Tracking & Regulation

Tracking sliding mode controller: � k 1 e 1 + · · · + k ρ − 1 e ρ − 1 + e ρ � u = − β ( ξ ) sat µ Regulation sliding mode controller: � k 0 e 0 + k 1 e 1 + · · · + k ρ − 1 e ρ − 1 + e ρ � u = − β ( ξ ) sat µ e 0 = e 1 = y − r ˙ β is allowed to depend only on ξ rather than the full state vector. On compact sets, the η -dependent part of ̺ ( η, ξ ) can be bounded by a constant Nonlinear Control Lecture # 38 Tracking & Regulation

High-gain observer: e i +1 + α i ˙ e i ˆ = ˆ ε i ( y − r − ˆ e 1 ) , 1 ≤ i ≤ ρ − 1 α ρ ˙ ˆ e ρ = ε ρ ( y − r − ˆ e 1 ) λ ρ + α 1 λ ρ − 1 + · · · + α ρ − 1 λ + α ρ Hurwitz e → ˆ e ξ → ˆ ξ = ˆ e + R β (ˆ ξ ) → β s (ˆ ξ ) (saturated) Nonlinear Control Lecture # 38 Tracking & Regulation

Tracking: � k 1 ˆ e 1 + · · · + k ρ − 1 ˆ e ρ − 1 + ˆ e ρ � u = − β s (ˆ ξ ) sat µ Regulation: � k 0 e 0 + k 1 ˆ � e 1 + · · · + k ρ − 1 ˆ e ρ − 1 + ˆ e ρ u = − β s (ˆ ξ ) sat µ We can replace ˆ e 1 by e 1 Special case: When β s is constant or function of ˆ e rather than ˆ ξ , we do not need the derivatives of r , as required by Assumption 13.4 Nonlinear Control Lecture # 38 Tracking & Regulation

The output feedback controllers recover the performance of the partial state feedback controllers for sufficiently small ε . In the regulation case, the regulation error converges to zero Relative degree one systems: No observer � y − r � � k 0 e 0 + y − r � u = − β ( y ) sat , u = − β ( y ) sat µ µ Nonlinear Control Lecture # 38 Tracking & Regulation

Example 13.5 (Revisit Example 13.2 and 13.4) Use the high-gain observer e 2 + 2 e 2 = 1 ˙ ˙ ˆ e 1 = ˆ ε ( e 1 − ˆ e 1 ) , ˆ ε 2 ( e 1 − ˆ e 1 ) to implement the tracking controller � e 1 + e 2 � u = − (2 | e 2 | + 3) sat (Example 13.2) µ and the regulating controller � e 0 + 2 e 1 + e 2 � u = − (2 | e 1 | +4 | e 2 | +4) sat (Example 13.4) µ Replace e 2 by ˆ e 2 but keep e 1 Nonlinear Control Lecture # 38 Tracking & Regulation

Saturate | ˆ e 2 | in the β function over a compact set of interest. There is no need to saturate ˆ e 2 inside the saturation For Example 13.2, Ω = {| e 1 | ≤ c/θ } × {| s | ≤ c } , c > 0 , 0 < θ < 1 is positively invariant. Take c = 2 and 1 /θ = 1 . 1 Ω = {| e 1 | ≤ 2 . 2 } × {| s | ≤ 2 } Over Ω , | e 2 | ≤ | e 1 | + | s | ≤ 4 . 2 . Saturate | ˆ e 2 | at 4 . 5 � � | ˆ � � � e 1 + ˆ � e 2 | e 2 u = − 2 × 4 . 5 sat + 3 sat 4 . 5 µ Nonlinear Control Lecture # 38 Tracking & Regulation

For Example 13.4, � 0 � e 0 � � � 0 � 1 ˙ ζ = Aζ + Bs, ζ = , A = B = e 1 − 1 − 2 1 PA + A T P = − I, 0 < θ < 1 , ρ 1 = λ max ( P )(2 � PB � /θ ) 2 , c > 0 Ω = { ζ T Pζ ≤ ρ 1 c 2 } × {| s | ≤ c } is positively invariant. Take c = 4 and 1 /θ = 1 . 003 Ω = { ζ T Pζ ≤ 55 } × {| s | ≤ 4 } Over Ω , | e 0 + 2 e 1 | ≤ 22 . 25 ⇒ | e 2 | ≤ | e 0 + 2 e 1 | + | s | ≤ 26 . 25 Nonlinear Control Lecture # 38 Tracking & Regulation

Saturate | ˆ e 2 | at 27 � � | ˆ � � � e 0 + 2 e 1 + ˆ � e 2 | e 2 u = − 2 | e 1 | + 4 × 27 sat + 4 sat 27 µ Simulation; (a) Tracking, (b) regulation � 0 . 05 (dashed) ε = 0 . 01 (dash-dot) State feedback (solid) Nonlinear Control Lecture # 38 Tracking & Regulation

(a) (b) 1.6 2 1.4 1.5 1.2 Output Output 1 1 0.8 0.5 0.6 0 0 1 2 3 4 0 2 4 6 Time Time Nonlinear Control Lecture # 38 Tracking & Regulation