Discrete Event Supervisory Control of Optimal Tracking Systems

Boby Philip

Research Scholar, Department of Electrical Engg. , Indian Institute of Technology Kharagpur, India.

E-mail:philipboby@yahoo.co.in

Abstract The Optimal tracking control under the supervision of a discrete event system is

considered here. A Discrete event system (DES) is a dynamic system, which evolves in accordance with the abrupt occurrence of events at possibly unknown irregular intervals of time. The Discrete Event Supervisory control is based on DES principles. In Optimal tracking system the final time is specified. But in Discrete Event Supervisory control, the time duration for the reference command is not known apriori. Hence the application of the conventional algorithm for the solution is not possible in the case of Discrete Event Supervisor controlled Optimal Tracking System. The algorithm for solution of the Optimal Tracking Problem in Discrete Event Supervisor controlled environment is developed in this work. 1. Introduction

The DES got the attention of control engineers in late 80’s and since then there have been a number of papers on the subject. There have been some papers on failure diagnosis, supervisory control of DES etc [1]-[4]. DES approach is most suitable for modular type of systems. There were some papers on Hybrid control system that consists of both discrete event and continuous time components. . In this work the Discrete Event Supervisory control of optimal tracking system (OTS) is developed. In supervisory control, a supervisor watches various events in the plant and environment and tells the controller to take corrective action. The DES supervisor analyses the plant dynamics and external events and generates the command to the tracking system. The command given to the tracking system is polynomial function of time in nature; the degree of such functions being lower at two or three. This happens due to the fact that signal issued by a Discrete Event supervisor will not have harsh variations and is smooth. The command will remain of a particular polynomial type until an event change occurs. After the event change, the supervisor changes the input signal to the tracking system. We take this discrete event supervisory command and generate the corresponding optimal control input to the plant. The advantage of the method adopted in this work is that we need not know the profile of the command apriori for calculation of optimal control. As we move toward more advanced intelligent control, the role of supervisory control becomes more important. Current work will be helpful in dealing with such situations. The supervisory command issued by the supervisor will be a polynomial function of time with the signal remaining as a particular polynomial till the supervisor notices some event changes. 2. Discrete event supervisory control of OTS

In Discrete event supervisor controlled system the tracking command to the plant is a fixed polynomial function of time for each event frame. However the command may change as and when an event occurs. The time instants at which these transitions occur are not predictable. Hence the conventional tracking control algorithm is not applicable here for each event frame.

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International Conference on Computational Intelligence and Multimedia Applications 2007

0-7695-3050-8/07 $25.00 © 2007 IEEEDOI 10.1109/ICCIMA.2007.151

557

When we move to optimal tracking control block, the cost state values denoted by s (t) must be calculated as required in the optimal tracking control. The problem for Discrete Event Supervisory control arises here. Even though the command function for each event frame is known at the starting of the frame, the time duration of the frame is not known apriori rendering the conventional procedure for computation of cost-states unsuitable here. Hence the steady state solution for the cost state function corresponding to the desired polynomial function command is derived directly as explained in the following section. It can be inferred from the equation for the calculation of optimal control that it also depends on K(t), the gain coefficients. But K(t) depends only on the system and its performance measure. 3. Calculation of s(t), the cost state values: New algorithm in the optimal tracking control

The values of cost state s (t) are found out from the equation [5]

(t)=-[AT-K (t) B R-1 BT]s(t)+Q r(t) (1)

It is observed from the equation that its Eigen values are negative of those for the plant under closed loop condition with steady state optimal gain matrix K. Since the steady state optimal controller guarantees closed loop stability, all Eigen values of the closed loop system will be in the RHS plane and hence unstable. Hence it is not possible to solve eq. (1) using forward integration. To apply backward integration, the fixed time for the corresponding event frame should be known apriori which is also not available. The method developed in the next section overcomes these difficulties for finite order polynomial signals 4. Direct solution of cost state for polynomial commands

.Let us denote the transfer function between s(t) and r (t) as represented by eq. (1) by G(s) so that

S(s) = G(s) R(s) . (2) Where S(s) = L {s (t)} R(s) = L { (t)} The transfer function G(s) can be expanded as an infinite series as follows G(s) = C1+C2s+C3s2 (3) Therefore S(s) = C1R(s)+C2sR(s)+C3s2 R(s)+… ( 4) Taking inverse Laplace transforms, s(t)= C1 (t) +C2 (t)+C3 (t)+…. . ( 5) If the input command r(t) is a polynomial function of finite order, the higher order

derivative of r(t) will vanish and hence s(t) can be calculated by finite number of terms of the above equation. For example, if the command is a second order polynomial function, the s(t) can be given by

s(t)=C1 (t) +C2 (t)+C3 (t) (6) The steady state position coefficient C1 can be found as s(0) for unit position command r(t)=1. The steady state velocity coefficient C2 can be found as s(0) for a unit ramp command r(t)=t. The steady state acceleration coefficient can be found as s(0) for unit parabolic command r(t)=t2/2. The initial value s(0) can be obtain through simulation of differential equation (1) with the appropriate command as input (t) and by backward integration starting from a sufficiently large to which ensure steady state solution at t=0

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4. Estimation of commanLet assume that the output of for each event frame. If the po (t) = po (t) = p1+ (t) = 2pNote that as the DES supervis

directly obtained from (8) and 5. Simulation and Result

In order to rest the function taken. Simulation of the systewith step, ramp and parabolicunit ramp and unit parabolprevious section was used for

0 0 .2 0 .4 0 .6-3 0

-2 0

-1 0

0

1 0

2 0

3 0

4 0

time in seco nd s

ga

in c

oe

ffic

ien

ts/c

ost

ate

va

lue

s th e fee d b ack g a in coe ffic ie n ts

cos ta te va lu e s

Figure1 “Steady state values of gaK(t) & s(t) cost statevalues for uncommand”

Figure3 “Steady state values of gK(t) & s(t) cost state values for ucommand”

0 5 10-10

0

10

20

30

40

50

60

70

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90

time in seconds

inp

ut

refe

ren

ce

co

mm

an

d/o

utp

ut

nd derivatives discrete event supervisor be a finite order polynom

olynomial as degree is 2, it can be represented as,

o+p1t+ p2t2 (7) +2p2t (8) p2 (9) sor directly gives coefficients p0,p1 and p2

Rather (9).

ts of the developed algorithm, a linear time invariant

em was done using MATLAB software [6]. We tooc variations. The steady value of s (t) i.e. s(0) folic signals were calculated. The formula devr calculating the new values of cost state values.

0 0.2 0 .4 0 .6-40

-30

-20

-10

0

10

20

30

40

time in seconds

ga

in c

oe

ffic

ien

ts/c

ost

ate

va

lue

sthe feedback ga in coe ffic ien ts

costa te va lues

0 .8 1

ain coefficients nit step Figure 2 “Steady state feedback ga

K(t) and co state s(t) values for uncommand”

gain coefficients unit parabolic

Figure 4 “Plot of the reference comoutput versus time”

15

mial function

they can be

system was ok a signal

or unit step, veloped in The

0.8 1

ain coefficients nit ramp

mmand and

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system tracked perfectly and the result was compared with the conventional way of finding optimal tracking control where there was the prior knowledge of signal variations. The steady state values s(0) and K(0) can be noted from the plots for different cases of input signal. 6. Conclusions

The focus of the present work has been on the functioning of linear time invariant tracking system under Discrete Event Supervisory control. The degree of the polynomial reference command considered is two. The characteristics of the supervisory command do not vary fast in general. Hence a two-degree approximation of the same is justified. However this can be extended to higher degrees of polynomial function 7. References [1]Francois Charbonnier, Hassan Alla, Rene David “The supervised control of Discrete Event Dynamic Systems”, IEEE Transactions on control system technology, vol. 7,no. 2, , March 1999, pp175-187. [2] P.J. Ramadge W.M. Won ham “Supervisory control of a class of Discrete Event Processes”, SIAM journal of Control and optimization, vol .25,no .1 , January 1987. pp206-230 [3] James A. Stiver, Panos J. Antsaklis “Modeling and analysis of Hybrid Control Systems”, Proceedings of the 31st Conference on Decision and control, , December 1992 ,pp3748-3751. [4] X Cao, Y Ho, “Models of Discrete Event Dynamic Systems”, IEEE Control Systems magazine, June 1990 pp69-76. [5] Donald Kirk, Optimal Control Theory: An Introduction, PH Inc, Englewood Cliffs, New Jersey, 1970. [6]Boby Philip, “Discrete Event Supervisory Control of Optimal Tracking Systems”, MTech. thesis, University of Kerala. June 2001

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