Is there a relation between sequential control and continuous control?

MarhI Comput. Modelling, Vol. 13, No. I, pp. 55-59, 1990 Printed in Great Britain. All rights reserved

0895-7177/90$3.00+ 0.00 Copyright 0 1990Pergamon Press plc

IS THERE A RELATION BETWEEN SEQUENTIAL CONTROL AND CONTINUOUS CONTROL? S. PARTHASARATHY CMC

Ltd, 115 Sarojini Devi Rd, Secunderabad

(Received

April 1989; received for publication

500 011, India July 1989)

Communicated by E. Y. Rodin Abstract-Process control theory has traditionally been divided into two branches: continuous domain and discrete domain. Each of these two branches has its own set of tools and techniques. It may, however, prove interesting to examine the commonalities between the two classes of control. Besides proposing some basic terminology for sequential control, this paper presents a recent approach to sequential control and highlights its equivalence with certain notions of continuous control theory.

1.

INTRODUCTION

Traditionally, the community of process control theorists has been divided into two categories: the continuous domain specialists and the discrete domain specialists. As in the case of all conflicts of this kind, there is a common (but elusive) binding force between the two points of view. In our case, we hope to find these commonalities in the underlying mathematical concepts. There have been some publications attempting to find equivalent notions of continuous control theory (CCT) for discrete event systems. We may cite, for instance, the works of Wonham [l] and related publications. The sequential control problem can be considered as a particular case of discrete event control, and has many notions directly found in CCT. This paper studies the sequential problem in particular, using the transition rules approach published previously in this journal [2], to highlight its similarities with the state space concepts of CCT. The paper also proposes certain terminology to be employed in sequential control. The author warns the reader that the present paper is only an intuitive study and that a considerable amount of work needs to be done to proclaim the birth of a unified theory of control. Continuous control (or regulatory control) concerns maintaining physical values at desired levels (set point) by continuous correction to the control signal sent to the process. Continuous control is an effective means of compensating for small perturbations which do not modify the process status. Sequential control, on the other hand, develops ordered sequences of commands to switch the process between different statuses. We shall illustrate these concepts using a simple process (see Fig. 1). The pump and valve together constitute a sequential process. They can be only switched ON (or OFF) and OPENED (or CLOSED), respectively. Only after opening the valve, can the regulatory control loop (shown in the box) be used effectively to maintain a specified flow (set point). Thus, we see that sequential control defines the limits within which regulatory control loops operate. In fact, in all practical situations, both modes of control are inextricably intertwined. Sequential control is used in a wide variety of contexts: startup, shutdown, change of operating strategy, batching, dosing, reconfiguration etc. It is also used for safety-related procedures like exception handling (e.g. alarms, emergency shutdown etc.). It encompasses a wide variety of processes (e.g. manufacturing systems, electricity networks, chemical processes, railway signalling, communication networks etc.). Rosenof and Ghosh [3] present an exhaustive treatment of the problems of automation of this class of processes. Although the majority of research effort in control theory has been and continues to be concentrated on the area of continuous domain of 55

S. PARTHASARATHY

56

SET POINT

Fig. 1. A simple process.

control, it is nevertheless admitted that discrete control has an important In fact, a report by an Expert Committee set up by the IEEE laments

practical role to play.

“For each of these important classes of (discrete event dynamic) systems, there is a dearth of elegant and succint models and there are no control techniques rivalling the economy and power of those described by differential equations.” [4]. We conclude that in view of its important practical applications, sequential control detailed study, and should exploit the well-known notions of CCT, if possible.

deserves a more

A few definitions are necessary at this point. A sequential control process consists of decision elements (or simply: elements). An element may either be a physical object or an abstract object. These elements can take either of two possible binary states, e.g. TRUE/FALSE, OPEN/CLOSE, HIGH/LOW, READY/NOT READY etc. The term element may include elements which are controlled explicitly (e.g. pump, valve etc.), as well as those which evolve independently due to external factors (e.g. limit switches, thermostats, circuit breakers etc.). An elementary command (or simply: command) results in the change of state of a controlled element. An elementary command consists of a verb denoting an operation and an argument denoting the particular element on which this operation is to be performed, e.g. the “transition” of an element from one state to its complementary state results in the evolution of the process from one “status” to another. The status of a process, as determined by the state of each of its elements, may be thought of as a snapshot of the process. We define “sequential control” as the problem of developing an ordered set of elementary commands to conduct the process from a given status to a desired status, respecting a certain number of constraints which may forbid certain transitions or certain statuses (or both).

2. AN

APPROACH

TO

SEQUENTIAL

CONTROL

The method of transition rules offers a compact, consistent and complete model for sequential process control. The transition rules approach to sequential control [2] consists of defining for each element of the process the conditions under which it can change states. For an element k the transition rule may be defined as

Tk = [element

k is in a TRUE

state

AND (condn

1 for change

to a FALSE

state is valid

OR condn 2 for change to a FALSE state is valid

Sequential control and continuous control

57

OR

condn n for change to a FALSE state is valid)] OR

[element k is in a FALSE state AND (condn 1 for change to a TRUE state is valid OR condn 2 for change to a TRUE state is valid OR

condn n for change to a TRUE state is valid)]. Tk = TRUE implies that at the present status, the element k can change states. The transition rule therefore gives only a necessary but not sufficient condition for the change of state of an element. The transition rules are constructed, for one element at a time, and progressively for each possible condition of permissible transition. The r.h.s. of the above equation can be represented by a Boolean function defined on the status of the process. In the expert systems approach for process control (expert control), the 1.h.s and the r.h.s. would correspond to the consequent and the antecedent, respectively, of the transition rule. For each element, we can enumerate the statuses where the elements are authorized to change states. All the permissible statuses are grouped to fit into the above definition and the transition rule is thus created. Graph theory based techniques have often been employed for solving control related problems, e.g. Petri nets, signal flow graphs, bond graphs, state transition diagrams etc. It is not surprising that the problem of sequential control can be translated into that of finding a path in an oriented graph (called the Partha graph). A sequential controller using this technique is described in Ref. [5]. In the paragraphs that follow, the operations “ . “, “+” and “ - ” shall denote the Boolean AND, OR and NOT operations, respectively (both on scalar and on vector operands). In the case of vector operands, these operations will be assumed to be executed between corresponding elements of each vector. Let us assume a sequential process of n elements. Let B be the set of two Boolean states {TRUE, FALSE}. Let B” denote the n-dimensional state space of vectors of Boolean elements, i.e. B”=BxBxBxBxB...ntimes. Let I,, be the set of the first n positive integers, i.e. 1,2, . . . , n. Let sijE B Vi E Z,, represent the state of the n th element at the jth instant, such that sj=

{slj, s2j, s3j,. *

* 3snj}*

Sj E B”, the ordered set of states at a given instant of time, as defined above, will be called a “status”.

Let S’ = initial status of the process and S* = desired status of the process.

S. PARTHASARATHY

58

The sequential control problem consists of finding an ordered set E of statuses S E B”, such that E={S,,S2,S,,...,SN},

iEIN,

where s, =S’ and s, = 9. For each element k, we can define a manoeuvrability function pk(S1) E B. pk(Si) = TRUE implies that the kth element can change states at the status S,. The function pk corresponds to the antecedent part (r.h.s.) of the transition rule described earlier. The evolution of the process is provoked by the elementary command ukj~ B. (By common practice, x
Vk Ez,,,Ukj,pkjEB.

The relation < in the above equation implies that the transition rule gives a necessary but not a sufficient condition for the change of state of an element. uk,= TRUE implies that the kth element effectively changes state at the jth instant. The next state of an element of the process will therefore be defined by

which gives rise to the following expression for the evolution of the process status: S~+~=Sj’Dj+3,‘Uj* By appropriately choosing the command vectors U,, we can arrive at the safe sequence of statuses to lead the process from a given status to a desired status, respecting constraints on intermediate statuses and transitions. 3. COMPARISON

WITH

CONTINUOUS

CONTROL

We notice the striking similarity between the above equation and the classical state difference equations used in the linear state space theory of continuous control. The major difference, however, lies in the fact that the above equation is in the Boolean space. All variables belong to B and all operations are Boolean operations. Earlier studies have been reported where there is a strong mathematical basis for defining a differential calculus in the Boolean space. The application of this technique to switching theory is reported in Ref. [6]. In fact, we surmise that sequential control could be greatly influenced by the application of the technique of Boolean differential calculus. The notion of a state variable is central to CCT. In the case of sequential control, we replace this by the state of an element. The state vector of CCT is equivalent to the status of the sequential control process. The states constituting a state vector are expected to be linearly independent in much the same way as the status is made of logically independent states. In a canonical representation of state vectors, a state variable can be defined as the derivative of another state vector. In the case of sequential processes, the transition of an element can be taken as another element in the status. The transition corresponds to the partial derivative of the status. The minimum number of elements necessary for representing a process by its state vector is known as the dimension of the process. The dimension of the status is measured by the minimum number of elements needed. The control input in the case of sequential control is represented by the elementary commands sequence. As in the case of element states, the elementary commands are also members of the Boolean space. In sequential control, exceptions are used for triggering control commands. Exceptions are Boolean variables which are related to one or more states or transitions. They are a generalization of the measurand vector commonly used in conventional control. For instance, in the sequential

Sequential

control

and continuous

control

59

process in Fig. 1, we can use the states of the pump and valve as measurands. Alternatively, we may use the line pressure P and the current Z drawn by the pump to indicate the states of the valve and the pump, respectively. Rather than use the actual value of these variables, they are compared with a threshold, giving rise to the term exception. It may therefore be stated that exceptions are a Boolean transformation of the states, in just the same way as an output vector is a transformation of the state vector in CCT. It is conjectured that many more interesting properties can be found in the above formulation. In particular, it would be interesting to explore the notions of stability, controllability and observability in sequential control. In view of the Boolean nature of the variables and operations, we will have to exploit Boolean differential calculus techniques, to arrive at comparable notions. 4. CONCLUSIONS

This paper has a paper published sequential control calculus to model scope for research

outlined certain notions of CCT as applicable to sequential control, based on previously in this journal and work done by the author subsequently. Since operates on a state space of Boolean variables, the use of Boolean differential and analyse sequential control systems seems promising. There is considerable to find equivalent notions between these two important branches of control.

Acknowledgemenfs-The presentation of this paper has been possible due to the facilities provided by the Ecole des Mines de St Etienne, France, where the author was recently on a sabbatical visit. The author thanks the Ecole des Mines de St Etienne and Professors P. Ladet and A. Mathon for their support.

REFERENCES 1. W. M. Wonham, A control theory for discrete-event systems. In NATO ASI Series, Vol. F47 (Edited by M. J. Denham and A. J. Laub). Springer-Verlag, Berlin (1988). 2. S. Parthasarathy, The transition rules model for real time process control. Math1 Mode&g 7, 163-171 (1986). 3. H. P. Rosenof and A. Ghosh, Batch Process Automation-Theory and Practice. Van Nostrand-Reinhold, New York (1987). 4. IEEE, Challenges to control: a collective view. IEEE Trans. autom. Control AC32(4), 275-285 (1987). 5. S. Parthasarathy, AUTO-SAFE: an expert controller for sequential and batch sequential processes. Engng Applic. artif. Intellig. 1, 25&257 (1988). 6. A. Thayse and M. Davio, Boolean differential calculus and its application to switching theory. IEEE Trans. Comput. C-22(4), 409420 (1973).