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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

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Event-triggered Control for Nonlinear Systems with Time-Varying Input Delay

Erfan Nozari http://carmenere.ucsd.edu/erfan University of California, San Diego 55th IEEE Conference on Decision and Control, Las Vegas, USA December 12, 2016 Joint work with Pavankumar Tallapragada and Jorge Cort´ es
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Motivation

Time delay and bandwidth limitation are widespread in real-world implementations of networked control systems Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 2/16
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Motivation

We address bandwidth limitation using event-triggered (ET) control

? challenging due to interplay between ET and time delay

ET Control Opportunistic Time delay No instantaneous control Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 3/16
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Outline

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 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 4/16
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Outline

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 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 4/16
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Problem Statement

Dynamics General nonlinear dynamics: Plant:
  • ˙
x(t) = f(x(t), u(φ(t))) φ(t) = t − D(t) Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 5/16
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Problem Statement

Objective General nonlinear dynamics: Plant:
  • ˙
x(t) = f(x(t), u(φ(t))) φ(t) = t − D(t) Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Assumptions
  • {u(t) | φ(0) ≤ t ≤ 0} is given and bounded
  • No finite escape time
  • Delay bounds: 0 < t − φ(t) ≤ M0 and 0 < m2 ≤ ˙
φ(t) ≤ M1
  • Globally Lipschitz K : Rn → R, K(0) = 0 exists s.t.
˙ x(t) = f(x(t), K(x(t)) + w(t)) is ISS with respect to w Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 5/16
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Problem Statement

General nonlinear dynamics: Plant:
  • ˙
x(t) = f(x(t), u(φ(t))) φ(t) = t − D(t) Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Design Objective
  • 1. Event-triggered stabilization: closed-loop GAS using
u(t) = u(tk) t ∈ [tk, tk+1), k ∈ Z≥0
  • 2. No Zeno behavior:
lim k→∞ tk = ∞ Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 5/16
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Outline

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 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 5/16
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Controller Structure

Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 6/16
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Controller Structure

Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Predictor K(·) x(t) p(t) K(p(t)) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 6/16
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Controller Structure

Plant ZOH Controller u(φ(t)) x(t) u(tk) D(t) u(φ(tk)) tk Predictor K(·) x(t) p(t) K(p(t)) tk Predictor Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 6/16
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Predictor Feedback [Bekiaris-Liberis and Krstic, 2013]

  • p(t) is the prediction of the future state of the plant:
p(t) = x(φ−1(t)) = x(t) + φ−1(t) t f
  • p(φ(τ)), u(φ(τ))
  • dτ
→ s = φ(τ) = x(t) + t φ(t) f(p(s), u(s))dφ−1(s) ds ds, t ≥ 0 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 7/16
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Predictor Feedback [Bekiaris-Liberis and Krstic, 2013]

  • p(t) is the prediction of the future state of the plant:
p(t) = x(φ−1(t)) = x(t) + φ−1(t) t f
  • p(φ(τ)), u(φ(τ))
  • dτ
→ s = φ(τ) = x(t) + t φ(t) f(p(s), u(s))dφ−1(s) ds ds, t ≥ 0
  • 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
Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 7/16
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Event-Triggered Law

  • S(x(t)) = Lyapunov function for the delay-free system:
α1(|x|) ≤ S(x) ≤ α2(|x|) ∂S ∂x f(x, K(x) + w) ≤ −γ(|x|) + ρ(|w|) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 8/16
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Event-Triggered Law

  • S(x(t)) = Lyapunov function for the delay-free system:
α1(|x|) ≤ S(x) ≤ α2(|x|) ∂S ∂x f(x, K(x) + w) ≤ −γ(|x|) + ρ(|w|)
  • V (t) = Lyapunov function of the delayed system (b > 0)
V (t) = S(x(t)) + 2 b 2L(t) ρ(r) r dr, L(t) = sup t≤τ≤σ(t) |eb(τ−t)w(φ(τ))| w(t) = u(t) − K(p(tk)) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 8/16
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Event-Triggered Law

  • S(x(t)) = Lyapunov function for the delay-free system:
α1(|x|) ≤ S(x) ≤ α2(|x|) ∂S ∂x f(x, K(x) + w) ≤ −γ(|x|) + ρ(|w|)
  • V (t) = Lyapunov function of the delayed system (b > 0)
V (t) = S(x(t)) + 2 b 2L(t) ρ(r) r dr, L(t) = sup t≤τ≤σ(t) |eb(τ−t)w(φ(τ))| w(t) = u(t) − K(p(tk)) Proposition: Bound on ˙ V If e(t) = p(tk) − p(t) is the prediction error, ˙ V (t) ≤ −γ(|x(t)|) − ρ(2L(t)) + ρ(2LK|e(φ(t))|) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 8/16
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Event-Triggered Law

Proposition: Bound on ˙ V If e(t) = p(tk) − p(t) is the prediction error, ˙ V (t) ≤ −γ(|x(t)|) − ρ(2L(t)) + ρ(2LK|e(φ(t))|) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 9/16
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Event-Triggered Law

Proposition: Bound on ˙ V If e(t) = p(tk) − p(t) is the prediction error, ˙ V (t) ≤ −γ(|x(t)|) − ρ(2L(t)) + ρ(2LK|e(φ(t))|) ×θ ∈ (0, 1) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 9/16
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Event-Triggered Law

Proposition: Bound on ˙ V If e(t) = p(tk) − p(t) is the prediction error, ˙ V (t) ≤ −γ(|x(t)|) − ρ(2L(t)) + ρ(2LK|e(φ(t))|) ×θ ∈ (0, 1) Triggering Condition ρ(2LK|e(φ(t))|) ≤ θγ(|x(t)|) ⇔ |e(t)| ≤ ρ−1(θγ(|p(t)|)) 2LK , θ ∈ (0, 1) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 9/16
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Event-Triggered Law

Proposition: Bound on ˙ V If e(t) = p(tk) − p(t) is the prediction error, ˙ V (t) ≤ −γ(|x(t)|) − ρ(2L(t)) + ρ(2LK|e(φ(t))|) ×θ ∈ (0, 1) Triggering Condition ρ(2LK|e(φ(t))|) ≤ θγ(|x(t)|) ⇔ |e(t)| ≤ ρ−1(θγ(|p(t)|)) 2LK , θ ∈ (0, 1) ˙ V (t) ≤ −(1 − θ)γ(|x(t)|) − ρ(2L(t)) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 9/16
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Satisfaction of Design Objectives

  • 1. Event-triggered stabilization:
Corollary There exists β ∈ KL s.t. for any x(0) ∈ Rn and bounded {u(t)}0 t=φ(0), |x(t)| + sup φ(t)≤τ≤t |u(τ)| ≤ β
  • |x(0)| +
sup φ(0)≤τ≤0 |u(τ)|, t
  • ,
t ≥ 0 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 10/16
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Satisfaction of Design Objectives

  • 1. Event-triggered stabilization:
Corollary There exists β ∈ KL s.t. for any x(0) ∈ Rn and bounded {u(t)}0 t=φ(0), |x(t)| + sup φ(t)≤τ≤t |u(τ)| ≤ β
  • |x(0)| +
sup φ(0)≤τ≤0 |u(τ)|, t
  • ,
t ≥ 0
  • 2. No Zeno behavior:
Proposition
  • Solve ˙
r = M2(1 + r)(Lf(1 + LK) + LfLKr), r(0) = 0
  • Define δ = r−1
1 2Lγ−1ρ/θLK
  • Then:
tk+1 − tk ≥ δ, k ≥ 1 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 10/16
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Outline

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 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 10/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
S(x) = xT Px
  • (A + BK)T P + P(A + BK) = −Q,
Q > 0 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
S(x) = xT Px
  • (A + BK)T P + P(A + BK) = −Q,
Q > 0 Triggering condition: |e(t)| ≤ λmin(Q) √ θ 4|P B||K| |p(t)|, θ > 0 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
S(x) = xT Px
  • (A + BK)T P + P(A + BK) = −Q,
Q > 0 Triggering condition: |e(t)| ≤ λmin(Q) √ θ 4|P B||K| |p(t)|, θ > 0 Closed-loop GES with rate µ = (2−θ)λmin(Q) 4λmax(P ) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
S(x) = xT Px
  • (A + BK)T P + P(A + BK) = −Q,
Q > 0 Triggering condition: |e(t)| ≤ λmin(Q) √ θ 4|P B||K| |p(t)|, θ > 0 Closed-loop GES with rate µ = (2−θ)λmin(Q) 4λmax(P ) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Exponential Stability ˙ x(t) = f(x(t), u(φ(t))) = Ax(t) + Bu(φ(t)) K(x) = Kx, globally Lipschitz
  • LK = |K|
S(x) = xT Px
  • (A + BK)T P + P(A + BK) = −Q,
Q > 0 Triggering condition: |e(t)| ≤ λmin(Q) √ θ 4|P B||K| |p(t)|, θ > 0 Closed-loop GES with rate µ = (2−θ)λmin(Q) 4λmax(P ) Question: How to balance communication cost (∼ δ) and convergence speed (∼ µ)? Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 11/16
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The Linear Case

Communication-Convergence Trade-off Using Q = qIn, P1 = q−1P, we have a multi-objective optimization: J1(θ) = δ = 1 a − c ln c + √ θ |P1B||K|a c + √ θ |P1B||K|c , J2(θ) = µ = 2 − θ 4λmax(P1) Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 12/16
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The Linear Case

Communication-Convergence Trade-off Using Q = qIn, P1 = q−1P, we have a multi-objective optimization: J1(θ) = δ = 1 a − c ln c + √ θ |P1B||K|a c + √ θ |P1B||K|c , J2(θ) = µ = 2 − θ 4λmax(P1) √ θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ(θ) ×10-3 1 2 µ(θ) 0.1 0.2 δ(θ) µ(θ) The Pareto front is the entire domain θ ∈ [0, 1] Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 12/16
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Outline

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 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 12/16
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Numerical Results

Compliant Nonlinear System f(x, u) =
  • x1 + x2
tanh(x1) + x2 + u
  • ,
t − φ(t) = D(t) = 1 + t 1 + 2t
  • 6
  • 3
3 6 x1(t) x2(t) p1(t) p2(t) t
  • 1
1 2 3 4 5 6 7 8 9 10 V (t) 10-6 100 106 1012 Triggering condition: |e(t)| ≤ ρ|p(t)| Analytically ρ ≃ 0.015, but stability remains until ρ ≃ 0.9 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 13/16
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Numerical Results

Non-compliant Nonlinear System f(x, u) =
  • x1 + x2
x2 + x3 1 + u
  • ,
t − φ(t) = D + a sin(t) D = 0.5 is known but the perturbation magnitude a = 0.05 is not
  • 14
  • 8
  • 2
4 x1(t) x2(t) p1(t) p2(t) t
  • 0.5 0
1 2 3 4 5 6 7 V (t) 10-4 100 104 108 Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 14/16
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Conclusions and Future Work

In this talk, we designed a predictor-based event-triggered GAS control law for arbitrary, known time-varying delays uniformly lower bounded the inter-event times proved GES in the linear case analyzed the communication-convergence trade-off for linear systems Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 15/16
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Conclusions and Future Work

In this talk, we designed a predictor-based event-triggered GAS control law for arbitrary, known time-varying delays uniformly lower bounded the inter-event times proved GES in the linear case analyzed the communication-convergence trade-off for linear systems Future work includes the extension of this approach to

? systems with disturbances ? systems with unknown time delays ? networked control scenarios with multiple agents

Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 15/16
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Questions and Comments

Erfan Nozari (UCSD) Event-triggered Control with Time-Varying Delay 16/16