introduction - disc-cps15.imtlucca.itdisc-cps15.imtlucca.it/pdf/heemels.pdf · hybrid systems...

Where innovation starts

Resource-aware control

Maurice Heemels

Zandvoort, June 2015

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Standard digital control loop

Actuator SensorPhysical System

Controller

−→ All control tasks executed periodically and triggered by time

Introduction

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Resource-aware control

• Resource-constrained control systems

– Computation time on embedded systems

– Actuator limitations (strain)

– Network utilization in NCS

– Battery power in WCS

• Time-triggered periodic control: Inefficient usage of resources

Introduction4/52

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Periodic or Aperiodic: That’s the question!• Paradigm shift: Periodic control −→ Aperiodic control

• Only act when needed: bringing feedback in resource utilization


Controller

−→


Controller

Introduction

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Paradigm shift: Periodic control −→ Aperiodic control

• Event-triggered control:

u(t) = K(x(tk)), when t ∈ [tk, tk+1)

tk+1 = inf{t > tk | C(x(t), x(tk)) > 0}


Controller

[1] Arzen, IFAC WC’99 [2] Astrom & Bernhardsson, IFAC WC’99 [3] Heemels et al, CEP’99

Introduction5/52

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• Event-triggered control:


tk+1 = inf{t > tk | C(x(t), x(tk)) > 0}


Controller

• Example event-triggering condition

‖x(t)− x(tk)‖ > σ‖x(t)‖

Introduction

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• Event-triggered control: reactive


tk+1 = inf{t > tk | C(x(t), x(tk)) > 0}

• Self-triggered control: proactive


tk+1 = tk +M(x(tk))

Introduction7/52

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• Basic setup state-feedback ETC: ‖x(t)− x(tk)‖ > σ‖x(t)‖• Hybrid systems

• Challenges

– Performance/Robustness w.r.t. disturbances & Zeno-freeness– Output-based (& Decentralized)

• Alternative event-triggered controllers

– Relative, absolute and mixed event generators– Periodic event-triggered control– Time regularisation– Dynamic event generators

• Application to vehicle platooning

• Conclusions & What’s next?

Outline

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• Linear system

x(t) = Ax(t) + Bu(t)

• Linear state feedback

u(t) = Kx(t), t ∈ R>0


Controller

• Ideal loop: x(t) = (A + BK)x(t)

• Sampled-data control with execution times tk, k ∈ N (ZOH)

u(t) = Kx(t) = Kx(tk), t ∈ [tk, tk+1)

• Perturbation perspective: implementation-induced error

e(t) = x(tk)− x(t) for t ∈ [tk, tk+1)

x(t) = Ax(t) + BKx(tk) = (A + BK)x(t) + BKe(t)

[1] Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, TAC 2007

Basic ETC setup9/52

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• Perturbation perspective:


• SinceA + BK Hurwitz, quadratic Lyapunov function V (x) = x>Px s.t.

d

dtV 6 −a2‖x(t)‖2 + ‖e(t)‖2

• Crux: Guarantee ‖e(t)‖ 6 ρa · ‖x(t)‖ with 0 < ρ < 1 s.t.

d

dtV 6 −a2‖x(t)‖2 + ‖e(t)‖2 6 −(1− ρ2)a2‖x(t)‖2

• Guarantee for Global Exponential Stability

tk+1 = inf{t > tk | ‖x(tk)− x(t)︸︷︷︸=e(t)

‖ > ρa · ‖x(t)‖}

Basic ETC setup

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• Summary of event-triggered setup:

– Linear systemx(t) = Ax(t) + Bu(t)

– Execution times tk, k ∈ N

tk+1 = inf{t > tk | ‖x(tk)− x(t)‖ > ρa · ‖x(t)‖}– Control law:

u(t) = Kx(tk), t ∈ [tk, tk+1)

• Global exponential stability (GES)

Event-triggered control11/52

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• Summary of event-triggered setup:

– Linear systemx(t) = Ax(t) + Bu(t)

– Execution times tk, k ∈ N

tk+1 = inf{t > tk | ‖x(tk)− x(t)‖ > ρa · ‖x(t)‖}– Control law:

u(t) = Kx(tk), t ∈ [tk, tk+1)

• Global exponential stability (GES)

• Question: Which important issue should we still verify?

Event-triggered control

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• Linear system


• Linear state feedback (ZOH)

u(t) = Kx(tk), t ∈ [tk, tk+1)


Controller

• Execution times: tk+1 = inf{t > tk | ‖x(tk)− x(t)︸︷︷︸=e(t)

‖ > σ‖x(t)‖}• Properties established in [1]:

– Global exponential stability (GES) when σ suff. small– Global positive lower bound on minimal inter-event time (MIET)

inf{tk+1 − tk | k ∈ N} > τmin > 0

• Improved designs for GES/L∞-gain via hybrid system analysis [2]

[1] Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, TAC 2007[2] Donkers, Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain ..., TAC 2012

Basic ETC setup12/52

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• Execution times tk, k ∈ N


‖ > σ‖x(t)‖}

• Hybrid system perspective [1,2] based on jump-flow models [3]:

d

dt

[xe

]=

[(A + BK)x + BKe−(A + BK)x−BKe

], when ‖e‖2 6 σ2‖x‖2

[x+

e+

]=

[x0

], when ‖e‖2 > σ2‖x‖2

[1] Donkers, Heemels, Output-Based Event-Triggered Control ..., TAC 2012 & CDC 2010[2] Postoyan, Anta, Nesic, Tabuada, A unifying Lyapunov-based framework ..., CDC-ECC 2011[3] Goebel, Sanfelice, Teel, Hybrid Dynamical Systems, Princeton, 2012.

Hybrid systems (side trip)

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Hybrid system perspective (side trip)

d

dt

[xe

]=

[(A + BK)x + BKe−(A + BK)x−BKe

]when ‖e‖2 6 σ2‖x‖2

[x+

e+

]=

[x0

]when ‖e‖2 > σ2‖x‖2

or compactly with ξ =

[xe

] {ξ = Φξ, when ξ>Qξ 6 0

ξ+ = Jξ, when ξ>Qξ > 0

ETC based on feedback14/52

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Hybrid system perspective (side trip)

ξ = Φξ when ξ>Qξ 6 0

ξ+ = Jξ when ξ>Qξ > 0

• Stability analysis using hybrid tools [1,2]: V (ξ) = ξ>Pξ

– ddtV (ξ) < 0 when ξ>Qξ 6 0

– V (Jξ) 6 V (ξ) when ξ>Qξ > 0

• Linear matrix inequalities: if there are α, β > 0 s.t.

– Φ>P + PΦ− αQ ≺ 0

– J>PJ − P + βQ � 0

• Guarantee for GES (extended ideas apply for L∞-gains)

• Never more conservative than perturbation approach [1]

[1] Donkers, Heemels, Output-based event-triggered control with guaranteed L∞-gain, TAC 2012 & CDC 2010[2] Goebel, Sanfelice, Teel, Hybrid Dynamical Systems: Modeling, Stability and Robustness, Princeton, 2012.

ETC based on feedback

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Example 1: State feedback control

• Consider x =[

0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖ MIET = 0.025

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTC

Illustrative Example15/52

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• Consider x =[

0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖ MIET = 0.025

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTC

Illustrative Example

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• Consider x =[

0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖ MIET = 0.025

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTCETC

Illustrative Example15/52

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• Consider x =[

0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖ MIET = 0.025

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 140

200

400

600

time t

number

ofevents

TTCETC

Illustrative Example

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Example 1: Comparison P and HS approach• Consider x =

[0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• Example taken from [1]

• We look for largest σ giving GES: ‖e‖2 6 σ2‖x‖2 [2]

σ2 MIETP: Results from [1] 0.0030 0.0318P: By minimising the L2-gain 0.0273 0.0840Hybrid System 0.0588 0.1136

• PS: via minimising L2-gain: maximise a (note σ = ρa)

V 6 −a2‖x(t)‖2 + ‖e(t)‖2 for x = (A + BK)x + BKe

• ETM:tk+1 = inf{t > tk | ‖x(tk)− x(t)︸︷︷︸

=e(t)

‖ > ρa · ‖x(t)‖}

[1] Tabuada, TAC ’07 [2] Donkers, Heemels, CDC10 & TAC12

Illustrative Examples17/52

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• Linear system


• Linear state feedback (ZOH)

u(t) = Kx(tk), t ∈ [tk, tk+1)


Controller

• Execution times: tk+1 = inf{t > tk | ‖x(tk)− x(t)︸︷︷︸=e(t)

‖ > σ‖x(t)‖}• Properties established in [1]:

– Global exponential stability (GES) when σ suff. small– Global positive lower bound on minimal inter-event time (MIET)

inf{tk+1 − tk | k ∈ N} > τmin > 0

• Improved designs for GES/L∞-gain via hybrid system analysis [2]

[1] Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, TAC 2007[2] Donkers, Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain ..., TAC 2012

Summary

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• Performance/Robustness w.r.t. disturbances

• Output-based (& Decentralized)

Challenges19/52

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Illustrative example

• Consider x =[

0 1−2 3

]x +

[01

]u and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTCETC

Borgers, Heemels, Event-Separation Properties of Event-Triggered Control Systems, TAC 2014

Disturbances in ETC

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• Consider x =[

0 1−2 3

]x +

[01

]u + w and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTCETC


Disturbances in ETC21/52

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• Consider x =[

0 1−2 3

]x +

[01

]u + w and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTCETC


Disturbances in ETC

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• Consider x =[

0 1−2 3

]x +

[01

]u + w and u(t) = [1 −4]x(tk)

• TTC: tk = k · 0.025

• ETC: tk = t ⇐⇒ ‖e(t)‖ > 0.05‖x(t)‖

0 2 4 6 8 10 12 140

0.5

1

1.5

time t

‖x(t)‖

TTCETC

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

time t

inter-eventtimeτ i

TTCETC



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Controller

• Consider{xp =

[1 −110 −1

]xp +

[11

]u

y = [1 0 ]xpu(t) = −2y(tk)

• ETM: ‖y(t)− y(tk)‖2 > σ2‖y(t)‖2

• Parameter: σ2 = 0.5

Output-based ETC

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• Minimal inter-event time (MIET) is zero! (Zeno behavior)

Donkers, Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain and Improved

and Decentralised Event-Triggering, TAC 2012

Output-based ETC24/52

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• Relative: ‖y − y‖ > σ‖y‖ [1]

• Absolute: ‖y − y‖ > δ [2-4]

• Mixed: ‖y − y‖ > σ‖y‖ + δ [5]

[1] Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, TAC 2007[2] Yook, Tilbury, Soparkar, Trading computation for bandwidth: Reducing communication in

distributed control systems using state estimators, TCST 2002[3] Miskowicz, Send-on-delta concept: An event-based data-reporting strategy, Sensors, 2006[4] Lunze and Lehmann, A state-feedback approach to event-based control, Automatica, 2010[5] Donkers, Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain ..., TAC 2012

Event-triggered control schemes

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Inverted pendulum

Movie ETC in action26/52

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Event-separation properties / Zeno-freeness

• Consider x = Ax + Bu + w and u(t) = Kx(tk) = K(x(t) + e(t))

• Execution times:


‖ > σ‖x(t)‖ + δ}

→MIET τ (x0, w) dependent on x0 and w: τ (x0, w) = infk∈N (tk+1 − tk)

• Event-separation properties (nominal)

– Global ESP: infx0∈Rn τ (x0, 0) > 0

– Semi-global ESP: for compact X0 ⊂ Rn: infx0∈X0 τ (x0, 0) > 0

– Local ESP: for each x0 ∈ Rn: τ (x0, 0) > 0


Disturbances in ETC

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Event-separation properties / Zeno-freeness

• Consider x = Ax + Bu + w and u(t) = Kx(tk) = K(x(t) + e(t))

• Execution times:


‖ > σ‖x(t)‖ + δ}

→MIET τ (x0, w) dependent on x0 and w: τ (x0, w) = infk∈N (tk+1 − tk)

• Event-separation properties (robust): there is ε > 0

– Robust global: infx0∈Rn, ‖w‖∞<ε τ (x0, w) > 0

– Robust semi-global: compact X0: infx0∈X0, ‖w‖∞<ε τ (x0, w)> 0

– Robust local: for each x0 ∈ Rn and ‖w‖∞ < ε: τ (x0, w) > 0



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State-feedback caseETM robust global global robust semi-global semi-global robust local localrelative × X × X × Xabsolute × × X X X Xmixed X X X X X X

Output-feedback case

ETM robust global global robust semi-global semi-global robust local localrelative × × × × × ×absolute × × X X X Xmixed × × X X X X

• Relative triggering fragile. Zero robustness

• Mixed or absolute effective (semi-global)

• However, only practical stability / ultimate boundedness (no GAS)

• Challenge: What about robust global ESP and GAS/L2-gains?


Overview

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P

ETM

C

ZOH

uwz

x

x

• Guaranteed control performance (L2−gain) from disturbance w tooutput z = q(x,w):

‖z‖L2 6 β(|ξ0|) + γ‖w‖L2 with ‖z‖L2 =

√∫ ∞

0

‖z(t)‖2dt

• Global asymptotic stability (GAS) in absence of disturbances

• Robust positive “minimal inter-event time” (τmiet)

• Reduced communication w.r.t. time-triggered control

Objectives29/52

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Cooperative Adaptive Cruise Control

WiFi-p: Feedforward

Radar: Feedback

• String stability: disturbance attenuation along the vehicle stringγ 6 1

‖z‖L2 6 β(|ξ0|) + γ‖w‖L2 with ‖z‖L2 =

√∫ ∞

0

‖z(t)‖2dt

• Communication resources limited→ event-triggered communica-tion

Motivation

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Time regularisation:• Periodic Event-Triggered Control (PETC) [6-9]

tk+1 = inf{t > tk | ‖y(t)− y(t)‖ > σ‖y(t)‖ ∧ t = kh, k ∈ N}

• Enforcing minimal inter-event time [7,9-11]

tk+1 = inf{t > tk+T | ‖y(t)− y(t)‖ > σ‖y(t)‖}

[6] Arzen, A simple event-based PID controller, IFAC 1999[7] Heemels, Sandee, van den Bosch, Analysis of event-driven controllers for linear systems, IJC 2008[8] Heemels, Donkers, Teel, Periodic Event-Triggered Control for Linear Systems, TAC 2013[9] Henningsson, Johannesson, Cervin, Sporadic event-based control of first-order linear stochastic .., Aut. 2008[10] Tallapragada, Chopra, Event-triggered dynamic output feedback control for LTI systems, CDC 2012[11] Tallapragada, Chopra, Event-triggered decentralized dynamic output .. LTI systems, NECSYS 2012

Event-triggered control schemes30/52

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• Periodic Event-Triggered Control (PETC)

tk+1 = inf{t > tk | ‖y − y‖ > σ‖y‖ ∧ t = kh, k ∈ N}

• Enforcing minimal inter-event time

tk+1 = inf{t > tk+T | ‖y − y‖ > σ‖y‖}

Output-feedback case

ETM robust global global robust semi-global semi-global robust local localrelative × × × × × ×absolute × × X X X Xmixed × × X X X Xtime-regu X X X X X X

Time regularized ETC

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tk+1 = inf{t > tk | ‖y − y‖ > σ‖y‖ ∧ t = kh, k ∈ N}

• Hybrid system analysis: GAS & finite L2-gains [1,2]

• Implementation advantages:

– Guaranteed (reasonable) minimal inter-event time

– Only time-periodic verification of event-triggering conditions

– More in line with time-sliced architectures

[1] Heemels, Donkers, Teel, Periodic Event-Triggered Control for Linear Systems, TAC 2013[2] Heemels, Donkers, Model-based Periodic Event-Triggered Control for Linear Systems, Automatica 2013

PETC31/52

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Hybrid systems formulation

d

dt

[ξτ

]=

[Aξ +Bw

1

], when τ ∈ [0, h],

[ξ+

τ+

]=

[J1ξ

0

], when ξ>Qξ > 0, τ = h

[J2ξ

0

], when ξ>Qξ 6 0, τ = h

z = Cξ +Dw

PETC

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d

dt

[ξτ

]=

[Aξ +Bw

1

], when τ ∈ [0, h],

[ξ+

τ+

]=

[J1ξ

0

], when ξ>Qξ > 0, τ = h

[J2ξ

0

], when ξ>Qξ 6 0, τ = h

z = Cξ +Dw

• In case w = 0 and interested in stability only

• Discretize at kh, k ∈ N (just before jump) leading to discrete-time PWL system[1,2,3]

ξk+1 =

{eAhJ1ξk, when ξ>k Qξk > 0

eAhJ2ξk, when ξ>k Qξk 6 0

[1] Heemels, Donkers, Teel, Periodic Event-Triggered Control for Linear Systems, TAC 2013[2] Heemels, Donkers, Model-based Periodic Event-Triggered Control for Linear Systems, Automatica 2013[3] Heemels, Sandee, van den Bosch, Analysis of event-driven controllers for linear systems, IJC 2008

PETC33/52

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Hybrid systems formulation• Including intersample-behavior, e.g., for L2-gain analysis

d

dt

[ξ

τ

]=

[Aξ + Bw

1

], when τ ∈ [0, h],

[ξ+

τ+

]=

[J1ξ

0

], when ξ>Qξ > 0, τ = h

[J2ξ

0

], when ξ>Qξ 6 0, τ = h

z = Cξ + Dw

PETC

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d

dt

[ξ

τ

]=

[Aξ + Bw

1

], when τ ∈ [0, h]

[ξ+

τ+

]=

[φ(ξ)

0

], when τ = h

z = Cξ + Dw

PETC33/52

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d

dt

[ξ

τ

]=

[Aξ + Bw

1

], when τ ∈ [0, h]

[ξ+

τ+

]=

[φ(ξ)

0

], when τ = h

z = Cξ + Dw

• L2-contractive: There are γ0 ∈ [0, 1) and a K-function β s.t.

‖z‖L2 6 β(|ξ0|) + γ0‖w‖L2 with ‖z‖L2 =

√∫ ∞

0

‖z(t)‖2dt

PETC

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d

dt

[ξ

τ

]=

[Aξ + Bw

1

], when τ ∈ [0, h]

[ξ+

τ+

]=

[φ(ξ)

0

], when τ = h

z = Cξ + Dw

ξk+1 = Adφ(ξk) + Bdvk

rk = Cdφ(ξk)

Main result: The hybrid system is internally stable and L2-contractiveiff the discrete-time nonlinear system is internally stable and `2-contractive.

• `2-contractive: there is γ0 ∈ [0, 1) s.t.

‖r‖`2 6 β(|ξ0|) + γ0‖v‖`2 with ‖r‖2`2 =

∞∑

k=0

|rk|2

[1] Heemels, Dullerud, Teel, L2-gain Analysis for a Class of Hybrid Systems with Applications to Reset andEvent-triggered Control: A Lifting Approach

Lifting-based approach34/52

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d

dt

[ξ

τ

]=

[Aξ + Bw

1

], when τ ∈ [0, h]

[ξ+

τ+

]=

[φ(ξ)

0

], when τ = h

z = Cξ + Dw

ξk+1 = Adφ(ξk) + Bdvk

rk = Cdφ(ξk)

Main result: The hybrid system is internally stable and L2-contractiveiff the discrete-time nonlinear system is internally stable and `2-contractive.

• Lifting with verifiable conditions without linearity

• For PETC piecewise linear system −→ contractivity/stability anal-ysis via LMIs using piecewise quadratic Lyapunov functions

[1] Heemels, Dullerud, Teel, L2-gain Analysis for a Class of Hybrid Systems with Applications to Reset andEvent-triggered Control: A Lifting Approach

Lifting-based approach

35/52

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P

ETM

C

ZOH

ua = uw

x

xc

P : x =

[0 1

−2 3

]x +

[0

1

]u + w

C : u =[

1 −4]xc


tk+1 = inf{t > tk | ‖xc(t)− x(t)‖ > σ‖x(t)‖ ∧ t = kh, k ∈ N}

• Enforcing minimal inter-event time

tk+1 = inf{t > tk+T | ‖xc(t)− x(t)‖ > σ‖x(t)‖}

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

time t

‖x(t)‖

time reg.PETC

0 2 4 6 8 10 12 14 16 18 2010−2

10−1

100

time t

inter-eventtimeτk

time reg.PETC

Time regularisation: Example36/52

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• Static event generator: tk+1 := inf{t > tk + T |C(x(t), e(t)) > 0}

Dynamic event generator [1,2,3]

η = Ψ(x, e, η)

tk+1 := inf{t > tk + T | η(t) < 0}

• How to find Ψ and T?

[1] Postoyan et al., “Event-triggered and self-triggered stabilization ...,” CDC 2011[2] Girard, “Dynamic triggering mechanisms for event-triggered control,” TAC 2015[3] Dolk, Borgers, Heemels, “Dynamic Event-triggered Control...,” CDC 2014

Dynamic event-triggered control

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• SinceA + BK Hurwitz, quadratic Lyapunov function V (x) = x>Px

d

dtV 6 −a2‖x(t)‖2 + ‖e(t)‖2

• Crux: Guarantee ‖e(t)‖ 6 ρa · ‖x(t)‖ with 0 < ρ < 1 s.t.

d

dtV 6 −a2‖x(t)‖2 + ‖e(t)‖2 6 −(1− ρ2)a2‖x(t)‖2

• Guarantee for Global Exponential Stability


‖ > ρa · ‖x(t)‖}

• Zeno-free: There is T > 0 such that tk+1 − tk > T for all k ∈ N.

Recap: Design relative triggering38/52

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• SinceA + BK Hurwitz, quadratic Lyapunov function V (x) = x>Px

d

dtV 6 −a2‖x(t)‖2 + ‖e(t)‖2

• Now consider η = Ψ(x, e, η) and LF U(x, η) = V (x) + η [2] :

d

dtU 6 −a2‖x‖2 + ‖e‖2 + Ψ

• To get ddtU 6 −(1− ρ2)a2‖x‖2 − εη for some ε > 0 we require

−a2‖x‖2 + ‖e‖2 + Ψ = −(1− ρ2)a2‖x‖2 − εηand thus η = Ψ(x, e, η) = ρ2a2‖x‖2 − εη − ‖e‖2• Now tk+1 := inf{t > tk + T | η(t) < 0}, η(0) = 0 and t0 = 0 :

– η(t) > 0 for t ∈ R>0 and thus U positive definite

– ddtU 6 −(1− ρ2)a2‖x‖2 − εη and thus GES

• Never triggers before the static version!!

Basic design dETM

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• Static event generator: tk+1 := inf{t > tk + T |C(x(t), e(t)) > 0}

Dynamic event generator [1,2,3]

η = Ψ(x, e, η)

tk+1 := inf{t > tk + T | η(t) < 0}

• [1,2] design for w = 0 (no disturbances)

• Recently, [3] new design methodology for output-based decentral-ized triggering under disturbances (Lp-gain)

[1] Postoyan et al., “Event-triggered and self-triggered stabilization ...,” CDC 2011[2] Girard, “Dynamic triggering mechanisms for event-triggered control,” TAC 2015[3] Dolk, Borgers, Heemels, “Dynamic Event-triggered Control...,” CDC 2014 and TAC?

Dynamic event-triggered control40/52

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P

ETM

C

ZOH

uwz

x

x

P : x =

[0 1

−2 3

]x +

[0

1

]u + w

C : u =[

1 −4]xc

Case study: L2-gain θ = 4 from input w to state x: τmiet = 9.1 · 10−3

• Dynamic event generator tk+1 := inf{t > tk + τmiet | η(t) < 0}• Static event generator: tk+1 := inf{t > tk + τmiet | Ψ(x, e, τ, η) < 0}

Dynamic ETC: Example

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P

ETM

C

ZOH

uwz

x

x

P : x =

[0 1

−2 3

]x +

[0

1

]u + w

C : u =[

1 −4]xc

Case study: L2-gain θ = 4 from input w to state x: τmiet = 9.1 · 10−3

• Dynamic event generator tk+1 := inf{t > tk + τmiet | η(t) < 0}• Static event generator: tk+1 := inf{t > tk + τmiet | Ψ(x, e, τ, η) < 0}

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

time t

|x(t)|

static ETMdynamic ETM

0 2 4 6 8 10 12 14 16 18 20

10−2

10−1

100

time t

inter-eventtimeτk

staticdynamic

Dynamic ETC: Example42/52

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0 1 2 3 4 5 6 7 8

·10−2

2

4

6

8

10

τmati / τmiet / τavg

L 2-gainθ

τmatiτmietτavg,staticτavg,dynamic

Dynamic ETC: Example

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Cooperative Adaptive Cruise Control

WiFi-p: Feedforward

Radar: Feedback

• String stability: disturbance attenuation along the vehicle string

– Lp-gain6 1

• Communication resources limited→ event-triggered communica-tion

Motivation44/52

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45/52

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• Headway time: 0.6 seconds

• MIET: 0.07 seconds

−→MOVIE

CACC46/52

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CACC

47/52

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CACC48/52

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• Event-triggered control: A new resource-aware control paradigm

• Several ETC algorithms discussed with their own tools (hybrid)

• Challenges

– Performance / Robustness w.r.t. disturbances– Output-based & decentralized event generators– Constrained systems (MPC)– Implementation and Applications– Better than periodic time-triggered control– Improved analysis and design tools: MIET, average inter-execution

times, Lp-gain, etc.

• Many interesting practical and theoretical issues open in this appealingresearch field

• More info: http://www.heemels.tue.nl

Conclusions

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• Collaborators

– Duarte Antunes, Niek Borgers, Florian Brunner, Victor Dolk, TijsDonkers, Tom Gommans, Heico Sandee, ...

– Frank Allgöwer, Adolfo Anta, Geir Dullerud, Kalle Johansson,Dragan Nesic, Romain Postoyan, Paulo Tabuada, Andy Teel,Paul van den Bosch, ...

• Financial support

• More info: http://www.heemels.tue.nl

Acknowledgements50/52

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• Arzen, A simple event-based PID controller, IFAC World Congress, 1999.

• Astrom, Bernhardsson Comparison of periodic and event based sampling for first order stochastic systems,IFAC World Congress 1999

• D.P. Borgers and W.P.M.H. Heemels, Event-separation properties of event-triggered control systems, IEEETransactions on Automatic Control, 59(10), p. 2644-2656, 2014.

• V.S. Dolk, D.P. Borgers, W.P.M.H. Heemels, Dynamic Event-triggered Control: Tradeoffs Between Transmis-sion Intervals and Performance, IEEE Conference on Decision and Control (CDC), pp. 2764-2769, 2014.

• V.S. Dolk, D.P. Borgers, W.P.M.H. Heemels, Output-based and Decentralized Dynamic Event-triggered Controlwith Guaranteed Lp-gain Performance and Zeno-freeness, prov. accepted in IEEE Transactions on AutomaticControl, 2015.

• M.C.F. Donkers and W.P.M.H. Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain andImproved and Decentralised Event-Triggering, IEEE Transactions on Automatic Control, 57(6), p. 1362-1376,2012.

• M.C.F. Donkers and W.P.M.H. Heemels, Output-Based Event-Triggered Control with Guaranteed L∞-gain andImproved Event-Triggering, IEEE Conference on Decision and Control (CDC) 2010, Atlanta, USA, p. 3246-3251.

• A. Girard, Dynamic triggering mechanisms for event-triggered control, IEEE Transactions on Automatic Con-trol, To appear, 2015.

• Goebel, Sanfelice, Teel, Hybrid Dynamical Systems, Princeton, 2012.

• W.P.M.H. Heemels, M.C.F. Donkers, Model-based periodic event-triggered control for linear systems, Auto-matica 49(3), pp. 698-711, 2013.

• W.P.M.H. Heemels, M.C.F. Donkers, and A.R. Teel, Periodic Event-Triggered Control for Linear Systems, IEEETransactions on Automatic Control , 58(4), p. 847-861, 2013.

Literature

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• W.P.M.H. Heemels, G. Dullerud, A.R. Teel, L2-gain Analysis for a Class of Hybrid Systems with Applicationsto Reset and Event-triggered Control: A Lifting Approach, prov. accepted IEEE Transactions on AutomaticControl.

• W.P.M.H. Heemels, R.J.A. Gorter, A. van Zijl, P.P.J. v.d. Bosch, S. Weiland, W.H.A. Hendrix, M.R. Vonder, Asyn-chronous measurement and control: a case study on motor synchronisation, Control Engineering Practice,7(12), 1467-1482, (1999)

• W.P.M.H. Heemels, J.H. Sandee, P.P.J. van den Bosch, Analysis of event-driven controllers for linear systems,International Journal of Control, 81(4), pp. 571-590 (2008).

• Henningsson T, Johannesson E, Cervin A, Sporadic event-based control of first-order linear stochastic sys-tems, Automatica 44, pp. 2890-2895, 2008.

• M. Miskowicz, Send-on-delta concept: An event-based data-reporting strategy, Sensors 6, pp. 49-63, 2006.

• J. Lunze and D. Lehmann, A state-feedback approach to event-based control, Automatica 46, pp. 211-215,2010.

• R. Postoyan, A. Anta, D. Nesic and P. Tabuada, A unifying Lyapunov-based framework for the event-triggeredcontrol of nonlinear systems, CDC (IEEE Conference on Decision and Control), pp 2559?2564, 2011.

• R. Postoyan, P. Tabuada, D. Nesic and A. Anta, Event-triggered and self-triggered stabilization of distributednetworked control systems, CDC (IEEE Conference on Decision and Control), 2011.

• P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control,vol. 52, no. 9, pp. 1680-1685, 2007.

Literature52/52

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• Tallapragada P, Chopra N, Event-triggered decentralized dynamic output feedback control for LTI systems,IFAC workshop on distributed estimation and control in networked systems, pp 31-36, 2012.

• Tallapragada P, Chopra N, Event-triggered dynamic output feedback control for LTI systems, IEEE 51st annualconference on decision and control (CDC), Maui, pp 6597-6602, 2012.

• J.K. Yook and D.M. Tilbury and N.R. Soparkar, Trading Computation for Bandwidth: Reducing Communicationin Distributed Control Systems Using State Estimators, IEEE Trans. Control Systems Technology, 10(4), pp.503-518, 2002.

Recent overviews:

• W.P.M.H. Heemels, K.H. Johansson, and P. Tabuada, An introduction to event-triggered and self-triggeredcontrol, 51st IEEE Conference on Decision and Control 2012, Hawaii, USA, p. 3270-3285

• W.P.M.H. Heemels, K.H. Johansson, and P. Tabuada, Event-Triggered and Self-Triggered Control, Encyclope-dia of Systems and Control, Springer-Verlag London 2014.

Pointers for “better than periodic time-triggered control:”

• D. Antunes and W.P.M.H. Heemels, Rollout Event-Triggered Control: Beyond Periodic Control Performance,IEEE Transactions on Automatic Control 59(12), p. 3296-3311, 2014.

• Astrom, Bernhardsson Comparison of periodic and event based sampling for first order stochastic systems,IFAC World Congress 1999

• T.M.P. Gommans, D. Antunes, M.C.F. Donkers, P. Tabuada, W.P.M.H. Heemels, Self-Triggered Linear QuadraticControl, Automatica 50(4), p. 1279-1287, 2014.

−→More info: http://www.heemels.tue.nl

Literature

introduction - disc-cps15.imtlucca.itdisc-cps15.imtlucca.it/pdf/heemels.pdf · hybrid systems...

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