Event-based Control: Theory and Application J AN L UNZE - - PowerPoint PPT Presentation

Event-based Control: Theory and Application

JAN LUNZE

Ruhr-Universität Bochum email: Lunze@atp.rub.de

Overview

1

Introduction to event-based control

2

Event-based state feedback

3

Analysis of the event-based control loop

4

Experimental evaluation

5

Event-based PI control

6

Conclusion and outlook

Introduction to event-based control

Event-based control

• Aim: Reduction of the network traffic.

Event-based control

Why is the theory of sampled-data control not applicable? No zero-order hold input. No periodic sampling → no discrete-time model available. Sampling theorem possibly violated. It is necessary to develop a new theory of event-based control.

Literature survey

• Deadband control

(OTANEZ, MOYNE, TILBURY, American Control Conf., 2002) Do not send new data as long as ||x(t) − x(tk)|| < ¯ e

• Quantised state feedback
• Self-triggered control

(TABUADA, IEEE Trans. Autom. Control, 2007) Estimate the next event time: tk+1 = h(x(tk), tk)

Literature survey

• Deadband control
• Quantised state feedback

(GRÜNE, JUNGE, Syst. Control Lett., 2005) (DE PERSIS, IFAC World Congress, 2008)

• Self-triggered control

(TABUADA, IEEE Trans. Autom. Control, 2007)

Literature survey

• Deadband control
• Quantised state feedback
• Self-triggered control

(TABUADA, IEEE Trans. Autom. Control, 2007) Estimate the next event time: tk+1 = h(x(tk), tk)

Literature survey

Evaluation of existing approaches: Many approaches result from an extension of sampled-data control and do not answer the three basic questions Many approaches do not consider any kind of disturbance and model uncertainties. Almost all approaches use a zero-order hold as control input generator.

Experiment

What is a typical behaviour of an event-based control loop?

Experiment

Discrete-time control:

2

Experiment

Event-based control:

• 2

2

Control aim

„Ultimate boundedness“ („practical stability“): Hold the state x(t) inside a set Ωd: x(t) ∈ Ωd, ∀t ≥ T(x0), x0 ∈ Ω1, d(t) ∈ [dmin, dmax]

d

1

Then the set Ωd is said to be robust positively invariant.

Event-based state feedback

Event-based state feedback

Why do we need information feedback? ... to stabilise an unstable plant, ... to compensate model uncertainties, ... to attenuate disturbances.

Event-based state feedback

Plant: ˙ x(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0 y(t) = Cx(t) Assumptions: Linear, asymptotically stable plant, no model uncertainties, synchronised clocks, no transmission delays, no computational restrictions.

Event-based state feedback

• Main idea: The event-based controller should mimic the

behaviour of a continuous state-feedback controller with adjustable precision. ||x(t) − xSF(t)||

!

≤ emax

Control input generator

• ˙

xSF(t) = (A − BK)

• ¯

A xSF(t) + Ed(t) u(t) = −KxSF(t) Behaviour of the continuous feedback loop after the state xSF(tk) at time tk is known: u(t) = −Ke

¯

A(t − tk)xSF(tk)− t

tk

Ke

¯

A(t − τ)Ed(τ) dτ, t ≥ tk

Control input generator

u(t) = −Ke

¯

A(t − tk)xSF(tk)− t

tk

Ke

¯

A(t − τ)Ed(τ) dτ, t ≥ tk ↓ Control input generator mimics the continuous state feedback: u(t) = −Ke

¯

A(t − tk)x(tk) − t

tk

Ke

¯

A(t − τ)Eˆ dk dτ, t ≥ tk

Control input generator

This input is generated by the following system: ˙ xs(t) = ¯ Axs(t) + Eˆ dk, xs(t+

k ) = x(tk),

t ≥ tk u(t) = −Kxs(t)

Event generator

• Event generator initiates an information exchange whenever

|| x(tk) − xs(tk

−)

• x∆(tk)

|| = ¯ e

Disturbance estimator

• For constant disturbances

d(t) = ¯ d, t ∈ [tk−1, tk) the state difference is x(tk) − xs(tk) = tk

tk−1

e A(t − τ)E (¯ d − ˆ dk−1) dτ = A−1 e A(tk+1 − tk) − I

• E (¯

d − ˆ dk−1).

Disturbance estimator

The „mean“ magnitude ¯ d of the disturbance in the time interval t ∈ [tk−1, tk] is used as disturbance estimate ˆ dk in the time interval t ≥ tk: Disturbance estimator ˆ d0 = ˆ dk = ˆ dk−1 +

• A−1

e A(tk − tk−1) − I

• E

+ x(tk) − xs(t−

k )

Event-based control algorithm

Summary of the event-based control algorithm Check the difference ||x(t) − xs(t)|| until an event is detected. Then:

1

Send the information x(tk) from the event generator to the control input generator.

2

Determine the disturbance estimate ˆ dk.

3

Reinitialise the control input generator: xs(t+

k ) = x(tk)

Event-based control algorithm

Behaviour of the event-based control loop:

x t

s( ),

( ) x t {

e t0 t1 t2 t

Summary

Three novelties of this methods with respect to literature: The control input generator is not a zero-order hold, but determines exponential inputs u(t). The event generator compares the behaviour of the event-based control loop with some reference system (model of the continuous feedback loop). A disturbance estimate is used to adapt the event-based loop to the unknown disturbance d(t).

Analysis of the event-based control loop

Analysis of the event-based control loop

Closed-loop system between two consecutive events t ∈ [tk, tk+1): ˙ x(t) ˙ xs(t)

• =

A −BK O ¯ A x(t) xs(t)

• +

E O

• d(t) +

O E

• ˆ

dk x(tk) xs(t+

k )

• =

x(tk) x(tk)

• y(t) = (C

O) x(t) xs(t)

• State transformation

x∆(t) xs(t)

• =

I −I O I x(t) xs(t)

Analysis of the event-based control loop

Transformed state-space model ˙ x∆(t) ˙ xs(t)

• =

A O O ¯ A x∆(t) xs(t)

• +

E O

• (d(t) − ˆ

dk)

• d∆(t)

+ O E

• ˆ

dk

• x∆(tk)

xs(t+

k )

• =
• x(tk)

Analysis of the event-based control loop

• d∆(t) = d(t) − ˆ

dk affects the (uncontrolled) plant. For a good approximation d(t) − ˆ dk ≈ 0, for t ≥ tk no communication is necessary.

Analysis of the event-based control loop

Event-based control loop for small disturbance ¯ d˜ d(t) There is no event for t > 0 if ||x(t) − xs(t)|| = || t e A(t − τ)E (¯ d˜ d(τ) − ˆ d0) dτ|| < ¯ e. Reformulation of the event condition: max

t≥0 ||

t e A(t − τ)E (¯ d˜ d(τ) − ˆ d0) dτ|| ≤ max

t≥0

t ||e A(t − τ)E|| dτ · max

t≥0 ||¯

d˜ d(t)|| ≤ ∞ ||e Aτ E|| dτ · |¯ d|.

Analysis of the event-based control loop

Theorem (Insensitivity to small disturbances) d(t) = ¯ d˜ d(t) ||˜ d(t)|| ≤ 1 For sufficiently small disturbance magnitude |¯ d| the event generator does not generate any event for t > 0. |¯ d| < ¯ e ∞

0 ||e Aτ E|| dτ

Analysis of the event-based control loop

Communication frequency described by Tmin = min

k (tk+1 − tk)

Assumption: ||d∆(t)|| ≤ d∆max for t ≥ 0 Theorem (Bounded sampling rate) Minimum time between two communication events: Tmin ≥ ¯ T with ¯

T

||e AτE|| dτ = ¯ e d∆max

Analysis of the event-based control loop

Event-based control vs. continuous state-feedback control For e(t) = x(t) − xSF(t) we get ˙ e(t) = ˙ x(t) − ˙ xSF(t) = Ax(t) − BKxs(t) + Ed(t) − ¯ AxSF(t) − Ed(t) = (A − BK)e(t) + BK (x(t) − xs(t))

• x∆(t)

. and ˙ e(t) = ¯ Ae(t) + BKx∆(t), e(0) = 0 ||e(t)|| ≤ ¯ e ∞ ||e ¯ Aτ BK|| dτ

Analysis of the event-based control loop

Theorem (Bounded approximation error) ||x(t) − xSF(t)|| ≤ emax with emax = ¯ e · ∞ ||e ¯ Aτ BK|| dτ The event-based control loop mimics the continuous loop with adjustable accuracy emax. x(t) ∈ Ωs(xSF(t)) = {x | ||x − xSF(t)|| ≤ emax}

Analysis of the event-based control loop

20 40 60 80 100

• 2
• 1

1 2

t x k ( ) x k

s( )

x k ( ) x k

SF( )

Analysis of the event-based control loop

Summary: The event-based control loop mimics the continuous state-feedback loop with adjustable precision Increase the precision emax by decreasing the event threshold ¯ e Consequence: Increase of communication frequency

1 Tmin

Experimental evaluation

Experimental evaluation

T1 T2 T3 T4 TS TW TB TM

Experimental evaluation

Thermofluid process

T3 T2 u1 d HW V1 V2 TB

L T

u2

Experimental evaluation

Design steps:

1

Design a state-feedback controller K with good disturbance attenuation properties.

2

Implement the event-based controller (control input generator, event generator, disturbance estimator)

3

Choose the event threshold ¯ e and evaluate the performance

• f the event-based control loop.

Experimental evaluation

Simulation results

t0=0 t0

, , , 4

Experimental evaluation

Simulation results emax = ¯ e · ∞

0 ||e ¯

Aτ BK|| dτ = 2.26 dmax =

¯ e ∞ ||eAτE|| dτ = 0.0114

Experimental evaluation

Experimental results

Experimental evaluation

Evaluation: Considerable reduction of the communication. Robustness against model uncertainties. Necessary extensions:

... for set-point tracking ... for quantitative robustness evaluation ... for nonlinear plants.

Event-based PI control

Event-based PI control

Constant external signals: d(t) = ¯ d, w(t) = ¯ w Set-point tracking: lim

t→∞ ||y(t) − ¯

w|| = 0.

Event-based PI control

Plant: ˙ x(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0 y(t) = Cx(t) Continuous PI controller: ˙ xr(t) = y(t) − w(t), xr(0) = xr0 u(t) = −KIxr(t) − KPx(t) Known result: If the closed-loop system is asymptotically stable, set-point tracking occurs.

Event-based PI control

PI-control input generator:

Model

• K=- K K

( )

P I

u( ) t dk xs( ) t w( ) t u( ) t x( ) tk dk Disturbance estimator xs( ) t ys( ) t xsr( ) t

Event-based PI control

˙ xs(t) ˙ xsr(t)

• =
• A − BKP

−BKI C O xs(t) xsr(t)

• xsI(t)

+

• O

−I

• w(t) +
• E

O

• ˆ

dk xs(t+

k )

= x(tk) u(t) = −(KP KI)xsI(t) Only the state xs(t) is updated at event time tk. Event generator and disturbance estimator remain the same.

Set-point tracking properties

For large disturbance ¯ d, an event is generated at time t1 and the disturbance is estimated correctly: ˆ d1 = ¯ d. Hence, set-point tracking occurs: limt→∞ ||y(t) − ¯ w|| = 0. For small disturbance ¯ d, no event occurs: ||x∆(t)|| = ||A−1 e At − I

• E¯

d|| < ¯ e, t ≥ 0. Hence, the output remains in the set lim

t→∞ y(t) ∈ {||y − ¯

w|| ≤ ||C||¯ e}

Set-point tracking properties

2000 0.1 0.2

• 1

1 2 3

• 1

1 t0=0 t1 t0 1000 4000 events x2 in K x1 in cm d d ,

k

^ t0 t1 t1 t2 t2 t3 t3 t4 t4 t5 t6 t in s dk ^ d _ d _ d _ x x x x x x xs xs xs xs xs xs dk ^

Further extensions

(Lehmann, PhD Thesis 2011): Event-based output feedback using a state observer Robustness against communication delay and packet loss Data-rate constraints → communication of quantised information Discrete-time implementation

Conclusion and outlook

Conclusions and outlook

Main idea: Mimic the continuous feedback system by the event-based control loop with adjustable precision

Conclusions and outlook

In which situation should information be transmitted? ... if ||x(t) − xs(t)|| = ¯ e Which information should be transmitted? x(tk) How should the control input be generated? ... use the model of the continuous loop

Conclusions and outlook

Interesting open problems: What happens if the network introduces time delay and package loss? How should event-based control be implemented without synchronous clocks? How can event-based control be extended to multiloop systems? Further extensions to unstable plants, nonlinear systems, output feedback etc.

Conclusions and outlook

To read:

• J. Lunze, D. Lehmann: A state-feedback approach to

event-based control, Automatica (2010)

• D. Lehmann, J. Lunze: Extension and experimental

evaluation of an event-based state-feedback approach, Control Engineering Practice 19 (2011)

• J. Lunze: Event-based control: A tutorial introduction,

SICE Journal 11 (2010).

• L. Grüne, F. Müller, S. Jerg, O. Junge, M. Post, D.

Lehmann, J. Lunze: Two complementary approaches to event-based control, Automatisierungstechnik 58 (2010).

• D. Lehmann: Event-Based State-Feedback Control, PhD

Thesis, Ruhr-Universität Bochum, 2011.