Nonlinear stabilization when delay is a function of state Miroslav - PowerPoint PPT Presentation

Nonlinear stabilization when delay is a function of state Miroslav Krstic Sontagfest , May 2011

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Outline • LTI systems with time-varying delay • nonlinear systems with state-dependent delay • happy birthday slide lkj

LTI Systems w/ Constant Delay X ( t ) AX ( t )+ BU ( t − D ) ˙ = A - possibly unstable; D - arbitrarily large Assume: ( A , B ) controllable and matrix K found such that A + BK is Hurwitz.

LTI Systems w/ Constant Delay X ( t ) AX ( t )+ BU ( t − D ) ˙ = Predictor-based control law: Z t � � t − D e A ( t − θ ) BU ( θ ) d θ e AD X ( t )+ U ( t ) K = � �� � X ( t + D ) � P ( t )

Time-Varying Input Delay Basic idea introduced by Artstein (TAC, 1982) , but only conceptually (nor explicitly), for LT V systems with TV delays. Explicit design for LTI plants presented by Nihtila (CDC, 1991) , but no analysis of stability or of feasibility of the controller.

Time-Varying Input Delay AX ( t )+ BU ( φ ( t )) X ( t ) ˙ = φ ( t ) = t − D ( t ) : = “delayed time” Predictor feedback � Z t � � � � � U ( θ ) φ − 1 ( t ) − t φ − 1 ( t ) − φ − 1 ( θ ) e A φ ( t ) e A � d θ U ( t ) = K X ( t )+ B φ ′ � φ − 1 ( θ )

Need a Lyapunov functional. Construct one with a backstepping transformation of the actuator state : X ( φ − 1 ( θ ) � P ( θ ) � �� � Z θ � � � � � � U ( σ ) φ − 1 ( θ ) − t φ − 1 ( θ ) − φ − 1 ( σ ) e A φ ( t ) e A W ( θ ) = U ( θ ) − K � d σ X ( t )+ B φ ′ � φ − 1 ( σ ) φ ( t ) ≤ θ ≤ t

Need a Lyapunov functional. Construct one with a backstepping transformation of the actuator state : X ( φ − 1 ( θ ) � P ( θ ) � �� � Z θ � � � � � � U ( σ ) φ − 1 ( θ ) − t φ − 1 ( θ ) − φ − 1 ( σ ) e A φ ( t ) e A W ( θ ) = U ( θ ) − K � d σ X ( t )+ B φ ′ � φ − 1 ( σ ) φ ( t ) ≤ θ ≤ t b φ − 1 ( θ ) − t Z t φ − 1 ( t ) − t e V ( t ) = X ( t ) T PX ( t )+ a � W ( θ ) 2 d θ � � φ ′ � φ − 1 ( t ) − t φ − 1 ( θ ) φ ( t )

Theorem 1 ∃ G , g > 0 s.t. Z t Z 0 � � t − D ( t ) U 2 ( θ ) d θ ≤ G e − gt | X ( t ) | 2 + | X 0 | 2 + − D ( 0 ) U 2 ( θ ) d θ ∀ t ≥ 0 , , where G (but not g ) depends on the function D ( · ) .

Conditions on the delay function D ( t ) = t − φ ( t ) : • D ( t ) ≥ 0 (causality); • D ( t ) is uniformly bounded from above (all inputs applied to the plant eventually reach the plant); • D ′ ( t ) < 1 (plant never feels input values that are older than the ones it has already felt— input signal direction never reversed ); • D ′ ( t ) is uniformly bounded from below (delay cannot disappear instantaneously, but only gradually).

φ − 1 ( t ) > t > φ ( t ) Achilles heel: ! D ( t ) needs to be known sufficiently far in advance ⇒ method appears not to be usable for state-dependent delays

Nonlinear systems with state-dependent delay (with Nikolaos Bekiaris-Liberis)

• Control over networks • Driver reaction delay • Milling processes • Rolling mills • Engine cooling systems • Population dynamics

Nonlinear Systems with State-Dependent Input Delay � � �� X ( t ) = f X ( t ) , U t − D ( X ( t )) ˙ Challenge: P ( t ) value of the state at the time when the control applied at t reaches the system = � � X t + D ( P ( t )) = Z θ f ( P ( s ) , U ( s )) P ( θ ) t − D ( X ( t )) ≤ θ ≤ t X ( t )+ 1 − ∇ D ( P ( s )) f ( P ( s ) , U ( s )) ds , = t − D ( X ( t ))

Nonlinear Systems with State-Dependent Input Delay � � �� X ( t ) = f X ( t ) , U t − D ( X ( t )) ˙ Challenge: P ( t ) value of the state at the time when the control applied at t reaches the system = � � X t + D ( P ( t )) = Z θ f ( P ( s ) , U ( s )) P ( θ ) t − D ( X ( t )) ≤ θ ≤ t X ( t )+ 1 − ∇ D ( P ( s )) f ( P ( s ) , U ( s )) ds , = t − D ( X ( t ))

Controller (possibly time-varying) U ( t ) = κ (( t + D ( P ( t )) , P ( t ))

Example 1 (stabilizing, but not global even for linear systems) � t − X ( t ) 2 � X ( t ) = X ( t )+ U ˙ U ( θ ) = 0 , − X ( 0 ) 2 ≤ θ ≤ 0 . Simulations with input initial conditions For X ( 0 ) ≥ X ∗ = 1 √ 2 e = 0 . 43 , the controller never “kicks in” (dashed) 0.9 0.7 0.8 0.6 x ( t ) φ ( t ) 0.7 0.5 0.6 0.4 0.5 0.3 0.4 0.2 0.3 0.1 0.2 0 0.1 −0.1 0 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t t

Result not global because of feasibility condition “delay rate < 1” To keep the prediction horizon finite and control bounded, the initial conditions and solu - tions must satisfy ∇ D ( P ( θ )) f ( P ( θ ) , U ( θ )) < c , for all θ ≥ − D ( X ( 0 )) , F c : for some c ∈ ( 0 , 1 ] . We refer to F 1 as the feasibility condition of the controller.

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(local u.a.s. in sup -norm of U ) Theorem 2 ∃ ψ RoA ∈ K , ρ ∈ K C , and β ∈ K L s.t. ∀ initial cond. that satisfy | U ( θ ) | < ψ RoA ( c ) B 0 ( c ) : | X ( 0 ) | + sup − D ( X ( 0 )) ≤ θ ≤ 0 for some 0 < c < 1 , � � � � | U ( θ ) | ≤ β ρ | U ( θ ) | , c | X ( t ) | + | X ( 0 ) | + , t ∀ t ≥ 0 . sup sup , t − D ( X ( t )) ≤ θ ≤ t − D ( X ( 0 )) ≤ θ ≤ 0 If U is locally Lipschitz on the interval [ − D ( X ( 0 )) , 0 ) , there exists a unique solution to the closed-loop system with X Lipschitz on [ 0 , ∞ ) , U Lipschitz on ( 0 , ∞ )

Assumption 1 D ∈ C 1 ( R n ; R + ) X = f ( X , ω ) is forward complete ˙ Assumption 2 X = f ( X , κ ( t , X )) is g.u.a.s. ˙ Assumption 3

Lemma 1 (infinite -dimensional backstepping transformation of the actuator state) W ( θ ) = U ( θ ) − κ ( σ ( θ ) , P ( θ )) , t − D ( X ( t )) ≤ θ ≤ t , transforms the closed-loop system into the “target system” f ( X ( t ) , κ ( t , X ( t ))+ W ( t − D ( X ( t )))) X ( t ) ˙ = W ( t ) ∀ t ≥ 0 . = 0 , Lemma 2 (u.a.s. of target system) ∃ ρ ∗ ∈ K C , β 2 ∈ K L s.t., for all solutions satisfying F c for 0 < c < 1 ,     | W ( θ ) | β 2  ρ ∗ | W ( θ ) | , c | X ( t ) | +  | X ( 0 ) | +  , t ≤ sup sup  , t − D ( X ( t )) ≤ θ ≤ t − D ( X ( 0 )) ≤ θ ≤ 0

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Lemma 3 (norm equivalence between the original system and target system) ∃ ρ 2 ∈ K C ∞ , α 9 ∈ K ∞ s.t., for all solutions satisfying F c for 0 < c < 1 , � � | U ( θ ) | α − 1 | W ( θ ) | | X ( t ) | + | X ( t ) | + ≤ sup sup 9 t − D ( X ( t )) ≤ θ ≤ t t − D ( X ( t )) ≤ θ ≤ t � � | W ( θ ) | ρ 2 | U ( θ ) | , c | X ( t ) | + | X ( t ) | + ≤ sup sup t − D ( X ( t )) ≤ θ ≤ t t − D ( X ( t )) ≤ θ ≤ t B around the origin and within the feasibility region) (finding a ball ¯ Lemma 4 ρ c ∈ K C ∞ s.t. F c ( 0 < c < 1 ) is satisfied by all solutions that satisfy ∃ ¯ | U ( θ ) | < ¯ ρ c ( c , c ) B ( c ) : | X ( t ) | + ∀ t ≥ 0 . ¯ sup t − D ( X ( t )) ≤ θ ≤ t (finding a ball B 0 of initial conditions s.t. all solutions are confined in ¯ B ⊂ F c ) Lemma 5 ∃ ψ RoA ∈ K s.t. for all initial conditions in B 0 ( c ) , the solutions remain in ¯ B ( c ) ⊂ F c for some 0 < c < 1 .

Examples

Example 2 Non -holonomic unicycle with D ( x , y ) = x 2 + y 2 A predictor-based version of Pomet’s (1992) time-varying controller: � 1 + 25cos ( 3 σ ( t )) 2 � ω − 5 P 2 cos ( 3 σ ( t )) − pq − Θ = − P + 5 Q ( sin ( 3 σ ( t )) − cos ( 3 σ ( t )))+ Q ω , v = where X cos ( Θ )+ Y sin ( Θ ) P = X sin ( Θ ) − Y cos ( Θ ) , Q = and the predictor is given by Z t σ ( s ) v ( s ) cos ( Θ ( s )) ds X ( t ) x ( t )+ = ˙ t − D ( x ( t ) , y ( t )) Z t σ ( s ) v ( s ) sin ( Θ ( s )) ds Y ( t ) y ( t )+ = ˙ t − D ( x ( t ) , y ( t )) Z t Θ ( t ) θ ( t )+ σ ( s ) ω ( s ) ds = ˙ t − D ( x ( t ) , y ( t )) σ ( t ) = t + D ( X ( t ) , Y ( t )) 1 σ ( s ) ˙ = 1 − 2 ( X ( s ) v ( s ) cos ( Θ ( s ))+ Y ( s ) v ( s ) sin ( Θ ( s )))

Trajectory of the robot for t ∈ [0 , 500 ] 1 Trajectory of the robot for t ∈ [0 , 15 ] 15 0.8 10 y ( t ) y ( t ) 5 0.6 0 0.4 −5 0.2 −10 0 −15 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −20 −10 0 10 20 x ( t ) x ( t ) 10 9 8 D ( t ) 7 6 5 4 3 2 1 0 0 5 10 15 t Solid: with delay compensation; dashed: without.