Draft Event-triggered Control for Nonlinear Systems with Time-Varying Input Delay Erfan Nozari http://carmenere.ucsd.edu/erfan University of California, San Diego 55 th IEEE Conference on Decision and Control, Las Vegas, USA December 12, 2016 Joint work with Pavankumar Tallapragada and Jorge Cort´ es
Motivation Draft Time delay and bandwidth limitation are widespread in real-world implementations of networked control systems 2 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Motivation Draft We address bandwidth limitation using event-triggered (ET) control ? challenging due to interplay between ET and time delay Time delay ET Control No instantaneous Opportunistic control 3 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Outline Draft 1 Problem Statement 2 Event-Triggered Design and Analysis Predictor Feedback Event-Triggered Law Convergence Analysis 3 The Linear Case Communication-Convergence Trade-off 4 Numerical Results Compliant Nonlinear System Non-compliant Nonlinear System 4 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Outline Draft 1 Problem Statement 2 Event-Triggered Design and Analysis Predictor Feedback Event-Triggered Law Convergence Analysis 3 The Linear Case Communication-Convergence Trade-off 4 Numerical Results Compliant Nonlinear System Non-compliant Nonlinear System 4 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Problem Statement Draft Dynamics u ( φ ( t )) u ( φ ( t k )) ZOH Plant General nonlinear dynamics: � x ( t ) = f ( x ( t ) , u ( φ ( t ))) ˙ x ( t ) D ( t ) Plant: φ ( t ) = t − D ( t ) t k u ( t k ) Controller 5 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Problem Statement Draft Objective u ( φ ( t )) u ( φ ( t k )) ZOH Plant General nonlinear dynamics: � x ( t ) = f ( x ( t ) , u ( φ ( t ))) ˙ x ( t ) D ( t ) Plant: φ ( t ) = t − D ( t ) t k u ( t k ) Controller Assumptions • { u ( t ) | φ (0) ≤ t ≤ 0 } is given and bounded • No finite escape time • Delay bounds : 0 < t − φ ( t ) ≤ M 0 and 0 < m 2 ≤ ˙ φ ( t ) ≤ M 1 • Globally Lipschitz K : R n → R , K (0) = 0 exists s.t. x ( t ) = f ( x ( t ) , K ( x ( t )) + w ( t )) ˙ is ISS with respect to w 5 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Problem Statement Draft u ( φ ( t k )) u ( φ ( t )) Plant ZOH General nonlinear dynamics: � x ( t ) = f ( x ( t ) , u ( φ ( t ))) ˙ x ( t ) D ( t ) Plant: φ ( t ) = t − D ( t ) t k u ( t k ) Controller Design Objective 1. Event-triggered stabilization: closed-loop GAS using u ( t ) = u ( t k ) t ∈ [ t k , t k +1 ) , k ∈ Z ≥ 0 2. No Zeno behavior: k →∞ t k = ∞ lim 5 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Outline Draft 1 Problem Statement 2 Event-Triggered Design and Analysis Predictor Feedback Event-Triggered Law Convergence Analysis 3 The Linear Case Communication-Convergence Trade-off 4 Numerical Results Compliant Nonlinear System Non-compliant Nonlinear System 5 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Controller Structure Draft u ( φ ( t k )) u ( φ ( t )) Plant ZOH x ( t ) D ( t ) t k u ( t k ) Controller 6 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Controller Structure Draft u ( φ ( t k )) u ( φ ( t )) Plant ZOH x ( t ) D ( t ) t k u ( t k ) Controller K ( p ( t )) p ( t ) x ( t ) K ( · ) Predictor 6 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Controller Structure Draft u ( φ ( t k )) u ( φ ( t )) Plant ZOH x ( t ) D ( t ) t k t k u ( t k ) Controller K ( p ( t )) p ( t ) x ( t ) K ( · ) Predictor Predictor 6 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Predictor Feedback [Bekiaris-Liberis and Krstic, 2013] Draft • p ( t ) is the prediction of the future state of the plant: � φ − 1 ( t ) p ( t ) = x ( φ − 1 ( t )) = x ( t ) + � � f p ( φ ( τ )) , u ( φ ( τ )) dτ → s = φ ( τ ) t � t f ( p ( s ) , u ( s )) dφ − 1 ( s ) = x ( t ) + ds, t ≥ 0 ds φ ( t ) 7 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Predictor Feedback [Bekiaris-Liberis and Krstic, 2013] Draft • p ( t ) is the prediction of the future state of the plant: � φ − 1 ( t ) p ( t ) = x ( φ − 1 ( t )) = x ( t ) + � � f p ( φ ( τ )) , u ( φ ( τ )) dτ → s = φ ( τ ) t � t f ( p ( s ) , u ( s )) dφ − 1 ( s ) = x ( t ) + ds, t ≥ 0 ds φ ( t ) • Computing p ( t ) requires: 1. State feedback: x ( t ) 2. Control history: { u ( s ) | φ ( t ) ≤ s ≤ t } 3. Prediction history: { p ( s ) | φ ( t ) ≤ s ≤ t } • Either analytical or numerical integration is used 7 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft • S ( x ( t )) = Lyapunov function for the delay-free system : α 1 ( | x | ) ≤ S ( x ) ≤ α 2 ( | x | ) ∂S ∂x f ( x, K ( x ) + w ) ≤ − γ ( | x | ) + ρ ( | w | ) 8 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft • S ( x ( t )) = Lyapunov function for the delay-free system : α 1 ( | x | ) ≤ S ( x ) ≤ α 2 ( | x | ) ∂S ∂x f ( x, K ( x ) + w ) ≤ − γ ( | x | ) + ρ ( | w | ) • V ( t ) = Lyapunov function of the delayed system ( b > 0) � 2 L ( t ) V ( t ) = S ( x ( t )) + 2 ρ ( r ) | e b ( τ − t ) w ( φ ( τ )) | r dr, L ( t ) = sup b t ≤ τ ≤ σ ( t ) 0 w ( t ) = u ( t ) − K ( p ( t k )) 8 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft • S ( x ( t )) = Lyapunov function for the delay-free system : α 1 ( | x | ) ≤ S ( x ) ≤ α 2 ( | x | ) ∂S ∂x f ( x, K ( x ) + w ) ≤ − γ ( | x | ) + ρ ( | w | ) • V ( t ) = Lyapunov function of the delayed system ( b > 0) � 2 L ( t ) V ( t ) = S ( x ( t )) + 2 ρ ( r ) | e b ( τ − t ) w ( φ ( τ )) | r dr, L ( t ) = sup b t ≤ τ ≤ σ ( t ) 0 w ( t ) = u ( t ) − K ( p ( t k )) Proposition: Bound on ˙ V If e ( t ) = p ( t k ) − p ( t ) is the prediction error , ˙ V ( t ) ≤ − γ ( | x ( t ) | ) − ρ (2 L ( t )) + ρ (2 L K | e ( φ ( t )) | ) 8 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft Proposition: Bound on ˙ V If e ( t ) = p ( t k ) − p ( t ) is the prediction error , ˙ V ( t ) ≤ − γ ( | x ( t ) | ) − ρ (2 L ( t )) + ρ (2 L K | e ( φ ( t )) | ) 9 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft Proposition: Bound on ˙ V If e ( t ) = p ( t k ) − p ( t ) is the prediction error , ˙ V ( t ) ≤ − γ ( | x ( t ) | ) − ρ (2 L ( t )) + ρ (2 L K | e ( φ ( t )) | ) × θ ∈ (0 , 1) 9 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft Proposition: Bound on ˙ V If e ( t ) = p ( t k ) − p ( t ) is the prediction error , ˙ V ( t ) ≤ − γ ( | x ( t ) | ) − ρ (2 L ( t )) + ρ (2 L K | e ( φ ( t )) | ) × θ ∈ (0 , 1) Triggering Condition ρ (2 L K | e ( φ ( t )) | ) ≤ θγ ( | x ( t ) | ) ⇔ | e ( t ) | ≤ ρ − 1 ( θγ ( | p ( t ) | )) , θ ∈ (0 , 1) 2 L K 9 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Event-Triggered Law Draft Proposition: Bound on ˙ V If e ( t ) = p ( t k ) − p ( t ) is the prediction error , ˙ V ( t ) ≤ − γ ( | x ( t ) | ) − ρ (2 L ( t )) + ρ (2 L K | e ( φ ( t )) | ) × θ ∈ (0 , 1) Triggering Condition ρ (2 L K | e ( φ ( t )) | ) ≤ θγ ( | x ( t ) | ) ⇔ | e ( t ) | ≤ ρ − 1 ( θγ ( | p ( t ) | )) , θ ∈ (0 , 1) 2 L K ˙ V ( t ) ≤ − (1 − θ ) γ ( | x ( t ) | ) − ρ (2 L ( t )) 9 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Satisfaction of Design Objectives Draft 1. Event-triggered stabilization: Corollary There exists β ∈ KL s.t. for any x (0) ∈ R n and bounded { u ( t ) } 0 t = φ (0) , � � | x ( t ) | + sup | u ( τ ) | ≤ β | x (0) | + sup | u ( τ ) | , t , t ≥ 0 φ ( t ) ≤ τ ≤ t φ (0) ≤ τ ≤ 0 10 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Satisfaction of Design Objectives Draft 1. Event-triggered stabilization: Corollary There exists β ∈ KL s.t. for any x (0) ∈ R n and bounded { u ( t ) } 0 t = φ (0) , � � | x ( t ) | + sup | u ( τ ) | ≤ β | x (0) | + sup | u ( τ ) | , t , t ≥ 0 φ ( t ) ≤ τ ≤ t φ (0) ≤ τ ≤ 0 2. No Zeno behavior: Proposition • Solve ˙ r = M 2 (1 + r )( L f (1 + L K ) + L f L K r ) , r (0) = 0 • Define δ = r − 1 � 1 � 2 L γ − 1 ρ/θ L K Then: t k +1 − t k ≥ δ, k ≥ 1 10 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
Outline Draft 1 Problem Statement 2 Event-Triggered Design and Analysis Predictor Feedback Event-Triggered Law Convergence Analysis 3 The Linear Case Communication-Convergence Trade-off 4 Numerical Results Compliant Nonlinear System Non-compliant Nonlinear System 10 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
The Linear Case Draft Exponential Stability x ( t ) = f ( x ( t ) , u ( φ ( t ))) = Ax ( t ) + Bu ( φ ( t )) ˙ 11 / 16 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay
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