Adaptive Event-triggered Control of a Uncertain Linear Discrete time

2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013

Adaptive Event-triggered Control of a Uncertain Linear Discrete time System Using Measured Input and Output Data Avimanyu Sahoo, Hao Xu, and S. Jagannathan Abstract -

In this paper, an adaptive model-based event­

feedback. Linear matrix inequalities (LMI) are used to prove

[9] introduced output

triggered control of an uncertain linear discrete time system is

the stability of the system. Authors in

developed.

feedback design for passive systems in the event trigger

Measured input and output vectors and their

history are utilized to express the unknown linear discrete-time system

as

an

autoregressive

Markov

representation

(ARMarkov). A novel adaptive model in the form of AR Markov is proposed and an update law is derived in order to estimate parameters of the ARMarkov model at triggered instants unlike periodic updates in standard adaptive control.

context. Both the system and controller are considered to be passive and the stability and performance of the event­ triggered system are studied. All these schemes

,[14]

dynamics. In order to accommodate system uncertainties in

Lyapunov method is used to derive the event trigger condition, convergence of the outputs and states. A simulation example is proposed with zero order hold (ZOH) and fixed model-based schemes is also discussed as part of simulation.

Key words: event-triggered, output feedback, adaptive control I.

[17], an

L, adaptive control technique is developed for the design and

prove boundedness of the parameter vector and asymptotic utilized to verify theoretical claims and a comparison of the

[6],[9],[13]

utilize output feedback by assuming known system

analysis of the event-triggered system. In this approach, the system dynamics are considered known with some uncertain parameters

are

being

estimated

through

the

projection

scheme as an adaptive law. A ZOH is utilized for maintaining the transmitted state and control input until the next update. In contrast, in this paper the need for known system

INTRODUCTION

dynamics is relaxed by using an adaptive model which

The congestion problem of a communication network can

provides the output estimates to the controller between any

be effectively solved by using an event trigger scheme

two triggered events. First, the uncertain linear time-invariant

[1],[3],[5],[9],

system

which

states

sampling

for

and

performance

[2]

employs

the

control

aperiodic

execution can

of

of

discrete-time system is transformed into an autoregressive

Aperiodic

Markov (ARMarkov) representation in the input-output form

transmission

control.

guarantee

better

system

in comparison to a conventional periodic

sampled control system.

In an event-triggered control

[7],[8].

Subsequently,

associated

controller

the

event

design

are

trigger

condition

introduced

by

and using

measured input and output sequence. A suitable aperiodic

system, a certain state or output dependent threshold is

law for updating the Markov parameters of the reference

designed to generate trigger instants while meeting stability

model is derived. Finally, stability analysis is included. A comparison with the ZOH and fixed model-based design

and performance. Out of the various schemes developed for event-triggered

[3]

control design, a zero order hold (ZOH) scheme

by using numerical example is included. It is demonstrated

is

that with the proposed adaptive scheme, the number of

proposed where the transmitted state and control input are

triggered instants will be fewer than the other methods. Next

held until the next transmission so that the system operates in

in Section II, development of the ARMarkov model and

an open-loop manner. To alleviate this issue, authors in

adaptive model-based trigger design is presented.

[5]

proposed a fixed model-based scheme wherein under the II.

assumption of small and bounded uncertainties, the control input between any two event-triggering instants is generated by

using

the

model

states.

However,

with

system

uncertainties, the selection of the model is not straightforward and the assumption on the model uncertainties to be small is stringent. In the above mentioned event trigger schemes

[3],[5],

states are considered to be measurable which may not be practical.

In most applications, available data is the output.

Among the earlier works on output feedback-based control, level crossing sampling by using a dynamic output feedback is presented in In

[6],

[13]

to mitigate the data rate constraint.

a dynamic controller for linear time invariant

system with guaranteed Loo gain is introduced by using output Research supported in part by NSF ECCS#1l28281 and lntelligent Systems Center. A. Sahoo, Hao Xu and S. Jagannathan are with the Dept. of Electrical and Computer Engineering, Missouri University of Science

DESIGN

[7], [8] is reproduced in brief and our proposed adaptive model­ In this section, the development of ARMarkov model

based scheme by using ARMarkov model is introduced. Consider a linear time-invariant discrete time system given by CXk Axk + Bup Yk 91" 91"' where xk E , Uk E , Xk+!

=

=

Yk E

91P

(l)

states, input and 91nxn 91'lXm , BE output of the system respectively, and A E xn p 91 are system state, input and output matrices. and C E 91nxn 91nxII1 E and BE Here, A are considered unknown while their

upper

bounds

including

are

the

output

matrix C is

considered known. Next, the standard assumption is stated. Assumption

1

The linear discrete time system in

(1)

is

considered reachable and observable, i.e., reachability matrix

and Technology, USA. ({asww6,hx6h7, sarangapj@mst.,edu).

978-1-4799-0178-4/$31.00 ©2013 AACC

ADAPTIVE MODEL-BASED EVENT-TRIGGERED CONTROL

5672

C N (A,

B) and observability matrix

0 N (C,

A) are full rank.

is updated continuously by using the estimated output Yk

Next, the ARMarkov representation is used to transfer the original linear system (1) from the state space into the input­ output form.

generated by the adaptive model. Thus, the adaptive model­ based scheme overcomes the drawbacks of both fixed model-based [5] and ZOH schemes [3].

A. Development ofARMarkov model [7],[8] For current time instant k , the linear system

Plant

Xk+l =.4xk + BUk

dynamics from (1) can be written by using time history of system input and outputs in a time horizon [k -N, k ] as ARMarkov model

Y k .--------, 1----tio>I Sensor/

yk=CXk

(2)

Xk = lviYk-l.k-N +(CN -MTN )Uk-1.k-N where T is Toeplitz matrix, CNand N

ON

represent

controllability and observability matrix respectively, and M satisfies AN -MO N = O. In terms of the system output dynamics, (2) can be written by multiplying and forwarding one time step as

C

on both sides

Yk+l =ClviYk.k-N+l +C(CN -MTN)Uk.k_N+l

ByUk

with �+l = [yJ+l

T

h

Uk = Uk.k-N+1 =[uJ A

y

I

In addition, the sensor and triggering mechanism senses the system output Yk' update the output sequence � with

(4)

the current system output Yk and compares the output

=

0

o

CM

I

0

I

o

sequence with the model output Yk

,

,

and

EO

0

output

and

input

sequences,

C ( CN -MTN )

9iNpxNp 'B

y

o

=

0

o

0

o

0

0

0

EO

and

0

in the controllable canonical form which means that, there exists a feedback matrix gain matrix, K , which can place the system poles at desired locations. Next, the adaptive model-based event-triggered control design is introduced by using the input-output representation described bye 4).

Adaptive model- based event-triggered control design

, , �+l =Ay,� + By, Uk

where � E mNp ,and

=

and input vectors, and Ay, EmNpxNp ,By, E mNpxNm are the estimates of the parameters of the Markov model of (4). The model parameters are updated at the triggering instants by using an aperiodic adaptive law compared to a traditional adaptive control system. The controller input k

U

; Event is not triggered

Again, the event trigger error

(6)

;MB is reset to zero. At the

e

same time, the model parameters are also tuned for the next instant by the update law. A filter is used to separate the current control input Uk from the augmented control input vector

Uk =[uJ

UJ_l

. . . UJ-N+l

r

which

is

a

combination of the present and past control inputs. Next the representation of system in a parametric form and the design of the update law are discussed

P arametric form and design of the update law

The linear dynamic system in ARMarkov model (4) can be written in the parametric form as (7) �+l =AYYk + =

ByUk lj/T r;k

Similarly, the adaptive model dynamics (5) at the event trigger instants can be represented in parametric form as ,

,

,

where lj/

y, Uk =lj/k r;k = [Ay By r and,yk = [ Ay,

target

and

�+l =Ay, �

(5)

Uk EmNmare augmented model output

the estimated output Yk exceeds the output

� ; Event is triggered

(i)

In this proposed adaptive model-based scheme, as shown in Figure 1, a communication network is considered between the sensor and the controller. An adaptive model in terms of the ARMarkov model is proposed to estimate the unknown system dynamics and generate the estimate of the output vector. The adaptive model dynamics is defined as

� to

dependent threshold (Jk II� II (Jkto be designed latter), the sensor transmits the output vector through the network. Upon receiving the output vector the model output is reset to the measured output and can be written as 1':k

By

When the event trigger

; which is defined as the difference between the

e

, {�

9iNpxNm are

augmented system and control coefficient matrices of the ARMarkov representation. Remark 1: The ARMarkov model matrices Ay and are

B.

error

•

MB

measured

N.

system

0

N.p

N.p

� = Yk.k-N+l = [yJ augmented

T T Yk-N+2 J Em T T Yk-N+l J Em T JT Em mare Uk-N+1

I Adaptive model-based event-triggered control system.

(3)

In a compact form, equation (3) can be written as

�+l =Ay� +

Figure

+B

(8)

'T

estimated

r;k =[Y: UJ r denotes

By,

parameters

r represent

the

respectively

with

the augmented state vector or

regression vector in the input-output form. In the next paragraph the parameter update by using the parameter projection algorithm is discussed.

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Parameter identification of the unknown system by using projection scheme requires prior knowledge on the bound of the a convex set where the unknown parameters [10],[15],[16] belong. This prior knowledge of the system is stated in the following assumption. Assumption 2: The unknown parameters lie in a controllable subspace, that is, there exists a compact convex set Q c mn such that the upper bound of the unknown parameters is known. In addition, the order of the system n is assumed to be known. From Assumption 2, the target parameters of the ARMarkov model in (4) are bounded in a compact convex set, i.e.IIlf/II::; If/ E Q. To force the estimated unknown max

parameters to stay within the set Q c mn a parameter projection algorithm [10],[17] is used. Now define the error between the system and model output vectors as event trigger error as

ekAMB

'

=� -�

(9)

The dynamics of the event trigger error at the event trigger instants can be formulated by using (7) and (8) as If/T (10) e:�B !;k - rfJ!;k IfJ!;k

=

=

with parameter estimation error given by

V/k=lf/-rfk=[Ay, BhT

(11)

The update law by using a parameter projection [10] algorithm is defined as (12) rfk+1 pro)

=

(ij/k)

where If/k -

=If/k A

+ rOk

!;k

AMBT and

; otherwise r

= {o

as tuning parameter and O k is an indicator for the

event trigger condition given by event is not triggered ok 1 event is triggered

(13)

Ifk-rOk!;ke:�BT + ll- �;J J k _ 1f/k+1-

I

-

If/k

5 - yo

k

!;k

C

+ II!;k

ij/

I

AMBT 112 ek+1

; ifIlij/k II > If/

max

-II.·fllIf/k < If/ ,1

(14)

wh",",

�

!;k eAMBT 2 hi C + II!;k 11

VI_ lf/max ij/ 1� J II"', II

'

+

h

'

_

{-

Kk � ; event is not triggerd K - k � ; event is triggerd

(16)

where the control gain matrix is updated based on estimated parameters Ay, and By, at the trigger instant. Next the closed loop stability and event trigger condition are derived.

(ii) design

Closed loop stability and event trigger condition

The closed loop system by using (9) and (16) can be represented as

�+I

= ( Ay-ByKk ) �

MB

+ ByKke:

(17)

This can be represented in parametric form as By K e: MB

�+I =rfJLk� + lfJLk� + where Lk= [I -KJ r

k

(18)

represents the augmented time­

varying gain matrix which satisfies the recursive algebraic Lyapunov equation [12] of the adaptive estimator given by

(AYk -13Yk Kk )T P(AYk -13Yk Kk )-P=-Qk

Remark 3:

(19)

From Remark 2, the controllability of the

estimated parameters of Ay, and By, ensures the existence of gain

matrix Kk.

Hence,

the

eigenvalues

of

within the unit disc. In addition, it was shown in [11],[12] that a positive definite solution P is guaranteed for the above recursive Lyapunov equation (19) given a positive definite Qk matrix. Next, the main result is introduced.

(event trigger condition and stability): Given a linear discrete time system (1) expressed in ARMarkov model (4) along with an adaptive estimator (5) serving as the

k

; if Ilij/k II > If/

model. Given a positive constant r satistying 0