Nonlinear Control Lecture # 14 Tracking & Regulation Nonlinear Control Lecture # 14 Tracking & Regulation
Normal form: η ˙ = f 0 ( η, ξ ) ˙ ξ i = ξ i +1 , for 1 ≤ i ≤ ρ − 1 ˙ ξ ρ = a ( η, ξ ) + b ( η, ξ ) u y = ξ 1 η ∈ D η ⊂ R n − ρ , ξ = col( ξ 1 , . . . , ξ ρ ) ∈ D ξ ⊂ R ρ Tracking Problem: Design a feedback controller such that t →∞ [ y ( t ) − r ( t )] = 0 lim while ensuring boundedness of all state variables Regulation Problem: r is constant Nonlinear Control Lecture # 14 Tracking & Regulation
Assumption 13.1 b ( η, ξ ) ≥ b 0 > 0 , ∀ η ∈ D η , ξ ∈ D ξ Assumption 13.2 η = f 0 ( η, ξ ) is bounded-input–bounded-state stable over ˙ D η × D ξ Assumption 13.2 holds locally if the system is minimum phase and globally if ˙ η = f 0 ( η, ξ ) is ISS Assumption 13.3 r ( t ) and its derivatives up to r ( ρ ) ( t ) are bounded for all t ≥ 0 and the ρ th derivative r ( ρ ) ( t ) is a piecewise continuous r, . . . , r ( ρ − 1) ) ∈ D ξ for all function of t . Moreover, R = col( r, ˙ t ≥ 0 Nonlinear Control Lecture # 14 Tracking & Regulation
The reference signal r ( t ) could be specified as given functions of time, or it could be the output of a reference model Example: For ρ = 2 ω 2 n , ζ > 0 , ω n > 0 s 2 + 2 ζω n s + ω 2 n y 2 = − ω 2 n y 1 − 2 ζω n y 2 + ω 2 y 1 = y 2 , ˙ ˙ n u c , r = y 1 r = y 2 , ˙ r = ˙ ¨ y 2 Assumption 13.3 is satisfied when u c ( t ) is piecewise continuous and bounded Nonlinear Control Lecture # 14 Tracking & Regulation
Change of variables: e 2 = ξ 2 − r (1) , e ρ = ξ ρ − r ( ρ − 1) e 1 = ξ 1 − r, . . . , η ˙ = f 0 ( η, ξ ) e i ˙ = e i +1 , for 1 ≤ i ≤ ρ − 1 a ( η, ξ ) + b ( η, ξ ) u − r ( ρ ) e ρ ˙ = Goal: Ensure e = col( e 1 , . . . , e ρ ) = ξ − R is bounded for all t ≥ 0 and converges to zero as t tends to infinity Assumption 13.4 r, r (1) , . . . , r ( ρ ) are available to the controller (needed in state feedback control) Nonlinear Control Lecture # 14 Tracking & Regulation
Feedback controllers for tracking and regulation are classified as in stabilization State versus output feedback Static versus dynamic controllers Region of validity local tracking regional tracking semiglobal tracking global tracking Local tracking is achieved for sufficiently small initial states and sufficiently small �R� , while global tracking is achieved for any initial state and any bounded R . Nonlinear Control Lecture # 14 Tracking & Regulation
Practical tracking: The tracking error is ultimately bounded and the ultimate bound can be made arbitrarily small by choice of design parameters local practical tracking regional practical tracking semiglobal practical tracking global practical tracking Nonlinear Control Lecture # 14 Tracking & Regulation
Tracking � a ( η, ξ ) + b ( η, ξ ) u − r ( ρ ) � η = f 0 ( η, ξ ) , ˙ e = A c e + B c ˙ Feedback linearization: � − a ( η, ξ ) + r ( ρ ) + v � u = /b ( η, ξ ) η = f 0 ( η, ξ ) , ˙ e = A c e + B c v ˙ v = − Ke , A c − B c K is Hurwitz η = f 0 ( η, ξ ) , ˙ e = ( A c − B c K ) e ˙ A c − B c K Hurwitz ⇒ e ( t ) is bounded and lim t →∞ e ( t ) = 0 ⇒ ξ = e + R is bounded ⇒ η is bounded Nonlinear Control Lecture # 14 Tracking & Regulation
Example 13.1 (Pendulum equation) x 1 = x 2 , ˙ x 2 = − sin x 1 − bx 2 + cu, ˙ y = x 1 We want the output y to track a reference signal r ( t ) e 1 = x 1 − r, e 2 = x 2 − ˙ r e 1 = e 2 , ˙ e 2 = − sin x 1 − bx 2 + cu − ¨ ˙ r u = 1 c [sin x 1 + bx 2 + ¨ r − k 1 e 1 − k 2 e 2 ] K = [ k 1 , k 2 ] assigns the eigenvalues of A c − B c K at desired locations in the open left-half complex plane Nonlinear Control Lecture # 14 Tracking & Regulation
Simulation r = sin( t/ 3) , x (0) = col( π/ 2 , 0) Nominal: b = 0 . 03 , c = 1 Figures (a) and (b) Perturbed: b = 0 . 015 , c = 0 . 5 Figure (c) Reference (dashed) √ � � 1 1 Low gain: K = , λ = − 0 . 5 ± j 0 . 5 3 , (solid) √ � 9 3 � High gain: K = , λ = − 1 . 5 ± j 1 . 5 3 , (dash-dot) Nonlinear Control Lecture # 14 Tracking & Regulation
(a) (b) 2 2 1.5 1.5 Output Output 1 1 0.5 0.5 0 0 −0.5 −0.5 0 2 4 6 8 10 0 2 4 6 8 10 Time Time (c) (d) 2 5 1.5 Control Output 1 0 0.5 −5 0 −10 −0.5 0 2 4 6 8 10 0 2 4 6 8 10 Time Time Nonlinear Control Lecture # 14 Tracking & Regulation
Robust 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 ˙ = Sliding mode control: Design the sliding surface e i = e i +1 , ˙ 1 ≤ i ≤ ρ − 1 View e ρ as the control input and design it to stabilize the origin e ρ = − ( k 1 e 1 + · · · + k ρ − 1 e ρ − 1 ) λ ρ − 1 + k ρ − 1 λ ρ − 2 + · · · + k 1 is Hurwitz Nonlinear Control Lecture # 14 Tracking & Regulation
s = ( k 1 e 1 + · · · + k ρ − 1 e ρ − 1 ) + e ρ = 0 ρ − 1 � k i e i +1 + a ( η, ξ ) + b ( η, ξ ) u + δ ( t, η, ξ, u ) − r ( ρ ) ( t ) s = ˙ i =1 � ρ − 1 � 1 � a ( η, ξ ) − r ( ρ ) ( t ) u = v or u = − k i e i +1 + ˆ + v ˆ b ( η, ξ ) i =1 s = b ( η, ξ ) v + ∆( t, η, ξ, v ) ˙ � � ∆( t, η, ξ, v ) � � Suppose � ≤ ̺ ( η, ξ ) + κ 0 | v | , 0 ≤ κ 0 < 1 � � b ( η, ξ ) � � s � β ( η, ξ ) ≥ ̺ ( η, ξ ) v = − β ( η, ξ ) sat , (1 − κ 0 ) + β 0 , β 0 > 0 µ Nonlinear Control Lecture # 14 Tracking & Regulation
s ˙ s ≤ − β 0 b 0 (1 − κ 0 ) | s | , | s | ≥ µ ˙ ζ = col( e 1 , . . . , e ρ − 1 ) , ζ = ( A c − B c K ) ζ + B c s � �� � Hurwitz V 0 = ζ T Pζ, P ( A c − B c K ) + ( A c − B c K ) T P = − I ˙ V 0 = − ζ T ζ +2 ζ T PB c s ≤ − (1 − θ ) � ζ � 2 , ∀ � ζ � ≥ 2 � PB c � | s | /θ 0 < θ < 1 . For σ ≥ µ {� ζ � ≤ 2 � PB c � σ/θ } ⊂ { ζ T Pζ ≤ λ max ( P )(2 � PB c � /θ ) 2 σ 2 } ρ 1 = λ max ( P )(2 � PB c � /θ ) 2 , c > µ Ω = { ζ T Pζ ≤ ρ 1 c 2 } × {| s | ≤ c } is positively invariant Nonlinear Control Lecture # 14 Tracking & Regulation
For all e (0) ∈ Ω , e ( t ) enters the positively invariant set Ω µ = { ζ T Pζ ≤ ρ 1 µ 2 } × {| s | ≤ µ } Inside Ω µ , | e 1 | ≤ kµ k = � LP − 1 / 2 �√ ρ 1 , � 1 0 � L = 0 . . . Nonlinear Control Lecture # 14 Tracking & Regulation
Example 13.2 (Reconsider Example 13.1) e 1 = e 2 , ˙ e 2 = − sin x 1 − bx 2 + cu − ¨ ˙ r r ( t ) = sin( t/ 3) , 0 ≤ b ≤ 0 . 1 , 0 . 5 ≤ c ≤ 2 s = e 1 + e 2 s = e 2 − sin x 1 − bx 2 + cu − ¨ ˙ r = (1 − b ) e 2 − sin x 1 − b ˙ r − ¨ r � � (1 − b ) e 2 − sin x 1 − b ˙ r − ¨ r � ≤ | e 2 | + 1 + 0 . 1 / 3 + 1 / 9 � � � � c 0 . 5 � � e 1 + e 2 � u = − (2 | e 2 | + 3) sat µ Nonlinear Control Lecture # 14 Tracking & Regulation
Simulation: µ = 0 . 1 , x (0) = col( π/ 2 , 0) b = 0 . 03 , c = 1 (solid) b = 0 . 015 , c = 0 . 5 (dash-dot) Reference (dashed) Nonlinear Control Lecture # 14 Tracking & Regulation
(a) (b) 2 1.2 1.5 1 0.8 Output 1 0.6 s 0.5 0.4 0.2 0 0 −0.5 −0.2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Time Time Nonlinear Control Lecture # 14 Tracking & Regulation
Robust Regulation via Integral Action 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 Disturbance w and reference r are constant Equilibrium point: η, ¯ 0 = f 0 (¯ ξ, w ) ¯ 0 = ξ i +1 , 1 ≤ i ≤ ρ − 1 η, ¯ η, ¯ 0 = a (¯ ξ, w ) + b (¯ ξ, w )¯ u ¯ r = ξ 1 Nonlinear Control Lecture # 14 Tracking & Regulation
Assumption 13.5 η, ¯ 0 = f 0 (¯ ξ, w ) has a unique solution ¯ η = φ η ( r, w ) η, ¯ u = − a (¯ ξ, w ) def ¯ = φ u ( r, w ) η, ¯ b (¯ ξ, w ) Augment the integrator e 0 = y − r ˙ e 1 ξ 1 − r e 2 ξ 2 z = η − ¯ η, e = = . . . . . . e ρ ξ ρ Nonlinear Control Lecture # 14 Tracking & Regulation
def ˜ z ˙ = f 0 ( z + ¯ η, ξ, w ) = f 0 ( z, e, r, w ) e i ˙ = e i +1 , for 0 ≤ i ≤ ρ − 1 e ρ ˙ = a ( η, ξ, w ) + b ( η, ξ, w ) u Sliding mode control: s = k 0 e 0 + k 1 e 1 + · · · + k ρ − 1 e ρ − 1 + e ρ λ ρ + k ρ − 1 λ ρ − 1 + · · · + k 1 λ + k 0 is Hurwitz ρ − 1 � s = ˙ k i e i +1 + a ( η, ξ, w ) + b ( η, ξ, w ) u � ρ − 1 � i =0 1 � u = v or u = − k i e i +1 + ˆ a ( η, ξ ) + v ˆ b ( η, ξ ) i =0 Nonlinear Control Lecture # 14 Tracking & Regulation
s = b ( η, ξ, w ) v + ∆( η, ξ, r, w ) ˙ � � ∆( η, ξ, r, w ) � � � ≤ ̺ ( η, ξ ) � � b ( η, ξ, w ) � � s � v = − β ( η, ξ ) sat , β ( η, ξ ) ≥ ̺ ( η, ξ ) + β 0 , β 0 > 0 µ Assumption 13.6 α 1 ( � z � ) ≤ V 1 ( z, r, w ) ≤ α 2 ( � z � ) ∂V 1 ˜ f 0 ( z, e, r, w ) ≤ − α 3 ( � z � ) , ∀ � z � ≥ α 4 ( � e � ) ∂z Assumption 13.7 z = 0 is an exponentially stable equilibrium point of z = ˜ ˙ f 0 ( z, 0 , r, w ) Nonlinear Control Lecture # 14 Tracking & Regulation
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