DESIGN (OF OPTIMUM DYNAMIC CONTROL SYSTEMS FOR NONLINEAR PROCESSES C. A. J O N E S , ' E.
F. J O H N S O N ,
L E O N L A P I D U S , A N D
R . H . W I L H E L M
Dt-,bai/mer't GI Chemical Enyineering, Princeton L'nzierszty, Princeton. .Y. J .
A procedure ior designing optimum dynamic control systems for single- and multivariable linear and nonlinear processes by means of linear and nonlinear control elements i s presented. The procedure consists o f a set of well defined steps for establishing control system specifications and synthesizing optimum control systems. The synthesis procedure i s carried out by matching any given physical process or maihematical model with a sequence of control elements o f increasing complexity in a nonlinear sense, each such choice, in turn, being optimized through steepest ascent determination of the values o f the coefficients of the control function terms. SE
of the most interesting and important aspects of control
0 theory deals \vith the design of optimum dynamic control systems for chemical processes. Even when the objectives of control system design are clearly defined it is questionable as tn what is the best procedure to follow in designing optimum control systems. The design procedure that is selected depends to a large extent upon whether the process is linear or nonlinear. ho\v man!- 7;ariables are involved, and of major importance. whether a completely determined mathematical model of the process is available. If the mathematical model is available then there are a number of techniques-e.g.. dynamic programming ( 7 . I)-\vhich can be used to specify the time-optimal control sequence. In the present articli:! a systematic stepvise procedure is presented for designing optimum control systems for single- and multivariable linear and nonlinear processes by means of linear and nonlinear control elements. The procedure does not require a completely determined mathematical model of the process but only a gross knowledge of the mathematic structure of the governing process equations. The proposed synthesis procedure is carried out by matching a given process with a sequence of control elements of increasing complexity in a nonlinear sense, each such choice, in turn. being optimized through a steepest ascent determination of values of the coefficients cf the control function terms. The end result is a stable control system ivhich produces an optimal response to any process disturbance. TVhereas this Lvork relates to optimum design paths for the dynamics of processes. i t does not take into account the consideration lvhether a given path is technically or economically feasible in the light of process bounds. The optimum control system for a given process contains the optimum selection and arrangement of control functions. Since there are many different types of control systems and criteria of dynamic optimization available, the initial steps of the design procedure must rxplicitly specify these factors as well as the control functions that are to be used in the control system. Having once established these specifications, the synthesis procedure may be implemented to yield a n optimum selection and arrangement of the specified control functions, The control system that is synthesized by this procedure is therefore the best control system according to and within the above specifications. Since \ve are dealing with nonlinearities, a control system defigned to accommodate a particular forcing
' Present address, Shell Development Co., Emervville. Calif.
disturbance will not necessarily be dynamically optimum for another type of disturbance. In the present work, the procedure makes use of step forcing. which is a particularly severe kind of forcing. Furthermore, although the procedures herein suggested lead usefully to a dynamically optimum control system, it cannot be established that the system is unique. In this article, the sequential steps discussed below are used to obtain the desired control system. Control System Specification
Any feedback control system Type of Control System. must perform several basic functions. such as measurement of process variables, comparison of these with preset desired values, computation on the error. and then correction by manipulating the process input variables. A typical schematic representation of these basic functional elements arranged in closed loop fashion is shown in Figure 1. The problem of optimum control is that of manipulating the control variables of dynamic processes in the best possible manner Mithin the practical constraints of the process and control system. Since, depending upon the process and the economics associated with the process, the best performance of the process may be subject to many different criteria which may vary \vith time, it is necessary to prescribe some measure of the "goodness" of the control effort. Dynamic Optimization Criteria. All dynamic criteria are concerned \vith the shape and duration of the error or recovery curve because these properties are a measure of how far and how long the process is away from the desired steady state conditions. For this study, the simplest criterion of optimization \\'as selected that satisfied the requirements of being applicable to a real process and producing a very fast speed of response with negligible overshoot. This criterion minimizes the integral of the sum of the tveighted errors squared of each controlled variable Min
Sorn
(aeI2
+ be?2 + .
. ) dt
(1)
T h e Lveighting factors, a , b. , . . may be functions of physical constraints. economics associated \vith each variable, or in the absence of any weighting, they may be numerically equal to one. Other performance criteria are of course possible as alternates for Equation 1. Linear and Nonlinear Control Elements. The computation part of thr control system may consist of various linear VOL. 2
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81
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FINAL CONTROL ELEMENT
INPUT
CONTROLLED VA R I A B L E PROCESS
.-
OUTPUT
*
A
COMPUTATION I
I
MEASUREMENT I
I
ERROR
ERROR DETECTOR
SIGNAL COMPARISON
DESIRED VALUE
v
CONTROL
SYSTEM
Figure 1 . Relation of the four basic functional elements of an automatic control system to the closed loop of control
and nonlinear control function generators. Since there are many conceivable control functions a broad class of functions was developed (3) from ivhich a minimum number of control functions may be selected in a straightforivard manner to optimize the behavior of many different types of processes according to the specified performance criterion. In the present case) the class of linear control elements includes the conventional proportional. derivative and integral elements, plus second- and higher-order derivative and integral elements. The class of nonlinear control functions, selected by means of an elimination procedure, consists of the proposed linear control elements multiplied by the absolute value of the error signal expanded in power series. In the elimination procedure for selecting second order nonlinear control functions. such control functions as
de2 de J --edt, Jezdt, dt ' dt
,
.
,
e.e.
,
. ~Jedt
were eliminated because thev do not respond to changes in the sign of the error signal. Since a criterion of optimization which iveights large error signals more than small error signals
is important, a function of the form el ~
de
-
dt
.
is selected over others
as the best second-order derivative control function. In much the same manner, the singular selection from among many possible second-order nonlinear gain elements is e P while e l J'edt is the best selection for the nonlinear integral control functions. The optimum class of second-order nonlinear control functions therefore is the following:
1 1
I
Following an identical set of elimination rules, the optimum class of third-order nonlinear control functions becomes :
Table I summarizes the final optimum class of linear and nonlinear control functions. When all the coefficients? a,, bi, , . ,, i = 1, 2, . . . , of the tabulated control functions are zero, the control system is linear-i.e., ,f(c) = aoc, g ( L ) = bot: , . . . \$'hen these coefficients are not zero. the size of the ,
82
l&EC
FUNDAMENTALS
Table I.
Optimum Class of linear and Nonlinear Control Functions I
m
\
Figure 2. Structure of continuous stirred tank reactor with an exothermic first-order reaction
disturbances determines, the form of the control function. Thus if the disturbances are very small the control system functions linearly because the square of the error signal and all of the higher degree terms are negligible, compared with the error signal itself. (311the other hand, large disturbances force the inclusion of the higher degree terms? and thus the control function becomes nonlinear. As a n illustration. f ( i ) = uoc for ver>-small dismrbances and f ( c ) = (uo ai e . . ) I - for large disturbances. A suitable abstraca21e l tion of elements from Table I can lead to almost any desired control function.
+
+
DESCRIPTIVE EXAMPLE 1. T h e physical problem chosen lor a descriptive example has received considerable attention in the literature (2). Consider a perfectly mixed reactor with exothermic chemical reaction B + C: an Arrhenius-type reaction constant and a cooling coil inserted in the reactor to control the reaction temperature. Mass balances on reactant B and product C and a n enthalpy balance yield
+ 1 1
Optimum Confrol System Synthesis
Determination of Process Structure. I n the present contest. the structure of a process is taken as a schematic diagram representing this solution by successive integrations of the differential equations \vhich describe the behavior of the process. This is analogous to the manner of preparation of process equations for analog computation. T h e relationship of a structure to the differential equations is quite important in the synthesis of control systems and thus the perrinent points are summarized beloiv:
Every term in a differential equation that is equated to the highest derivative and contains at least one dependent variable that is the same as the dependent variable of the highest derivative is represented by a direct loop. The signs of direct loops are the same as the signs of the corresponding terms in the differential equation. Every trrni in a differential equation that is equated to the highest derivative and contains a dependent variable that is different from the dependent variable of the highest deri\,ative is represented by a simple connection. An indirect loop is represented by two simple connections of two interconnected differential equations. The signs of simple connections are the same as the signs of the correspondine. terms in the differential equation. The sign of an indirect loop may be determined by the product of the signs of simple connrctions. Terms of two differential equations containing dependent variables that are common to both equations are represented by a mutual connection. A mutual connection consists of two direct loops and one indirect loop. The signs of the direct and indirect loops are determined in the same manner as outlined before.
where pc C,, (1 - e--h ~ ~ ~ / ~ ~ , P ~ ~ P C ) (5) a function of cooling coil fluid flow rate and physical parameters T h e basic structure of this reactor as described by Equations 2 to 4 are shown in Figure 2 in a form equivalent to a n analog computer schematic. Five negative direct loops. one positive direct loop, one simple connection, and one negative indirect loop are involved. Equation 2 has two negative direct loops; Equation 3 has one negative direct loop and one simple connection; Equation has two negative and one positive direct loops with the negative indirect loop coming from the connection of Equations 2 and 4.
f ( F c ) = F, f(F,)
=
Determination of Basic Control System Structure. The basic structure of the control system is formed by providing signal paths from the controlled output variables to their corresponding manipulated variables. T h e following properties of any structure are important to the control of the system.
X11 negative loops are self-correcting and tend to restore the system to steady state. .411 positive loops are regenerative or tend to move the system away from steady state. .4 direct effect is the summed effects of all direct loops that originate from terms containing dependent variables of the same order. When a loop is formed by the cross product of two or more dependent variables, the loop is associated with the direct effect of the corresponding higher order term. VOL. 2
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Indirect effects may be produced only by the structure of a system represented by two or more coupled differential equations. An indirect effect is the summed effects of all indirect loops that are formed by terms containing dependent variables of the same order. Since the minimum requirement for a n optimum control system is to stabilize the process over its entire range of operation, and all positive direct effects and all indirect effects are possible sources of nonstable operation, all output variables associated with positive direct effects and a t least one output variable associated with every significant indirect effect must be controlled. T o stabilize the system over its entire range of operation, the signs of all the direct effects of each control loop should remain negative, and the product of the magnitudes of the indirect effects should remain less than the product of the magnitudes of the corresponding direct effects of each control loop. A41though the exact differential equations describing the behavior of the process may not be known, usually enough basic information about a process is available from which a reasonable guess can be made regarding the basic control structure. If the structure is not satisfactory, the procedure can be repeated with a different one. This is somewhat analogous to methods of steepest ascent which also depend upon the starting point in the region of realizable operation. because more than one peak of steady state optimum performance may occur in the region. However. over most practical ranges of operation. only one peak of performance is likely to occur, and only one basic control structure is suitable.
DESCRIPTIVE EXAMPLE 2. T o investigate the stability of the nonlinear system of Equations 2 to 4 near steady state, it is a relatively simple matter to expand the functions B and 7' in Taylor series about a n equilibrium point to yield a linearized set'of equations. Let . a = B s - B -1 =
T,
-
The result is:
=
Q + Ae-E/R?' V
I n order for the linear system described by these equations to be stable near a steady state, the roots of the characteristic equation must be negative real or the following inequalities must hold : a11
alia??
+
>0 - alpa?1 > 0 a~
I n terms of reactor quantities, the inequalities are:
84
18EC FUNDAMENTALS
RT,
>0
(18)
and
This condition automatically satisfies the requirement of stability for the indirect loop, Equation 17, because the mutual connection is common to both sources of instability. Since Equation 18 is always satisfied, the stability of the reactor is only dependent upon Equation 19. If Q i V is maintained constant: then f ( F , ) must be controlled. T h e basic control structure for stabilizing the reactor then consists of the reactor temperature, T , as the controlled output variable and the cooling water flow rate! F,, as the control variable. This portion of the system is referred to as the temperature control loop. Furthermore, consideration must be given to dynamic optimization of the reactor. T h e portion of the system involved here is referred to as the concentration control loop and consists of the product concentration, C. as controlled output variable and the feed reactant concentration, Bi, as control variable. T h e determination of a basic control system structure therefore depends only upon the form of the equations which describe process behavior and the variables Jvhich are to be optimized. Determination of t h e Optimum Linear Control Functions. T h e linear control functions of the control system are selected from the class of Table I-i.e.. from bot, . , . . d,
......
where all
2 + Ae-E V
aoc,
T
Lvhere subscript J denotes steady state
Note that the only factor that can be negative and eflect stability near the steady state is the heat of reaction which occurs in both inequalities. If the process is stabilized by the requirements previously proposed, one condition eliminates both sources of instability. T h e condition is that the direct effects of each subsystem must be negative at all times or the following inequalities must hold
. .
fcdt ..., ,,
k O ~Pop, : . . . . , ni,, J i d t
T h e end result is to damp critically the cont:.olled output variables of each control loop of the basic structure when the corresponding reference inputs are disturbed by small step changes. A small step change is taken as one ivhich only affects the process to the extent that the response of the process is a linear function of the disturbance. Such a criterion is suitable for use on both linear and nonlinear processes. T h e linear control functions are selected to give critical damping in the control loops because this condition is easily identifiable and because it ensures system stability for small perturbations near the operating point. Critical damping is taken to be the tightest control which does not result in detectable overshoot in the response to step forcing. For linear systems, this condition exists when the roots of the characteristic equation for the system are all real and negative and at least two of the predominant roots are equal. T h e subsequent addition of nonlinear control functions is made using the optimization criterion that the error squared integral be minimized. These nonlinear functions esert a predominating influence except when the error is very small. T h e necessary linear control functions for critically damping each control loop of a process subject to small disturbances may be determined by a n iterative procedure. This procedure involves changing the magnitudes of the coefficients of the specified class of linear control functions in a prescribed manner starting with a proportional element and no derivative elements and progressing from the lower-order to the higher-order derivatives until each control loop is observed to be critically
I
-
ITC,
lil
mo
Figure 3. Structure of continuous stirred tank reactor coupled with its linear control system
damped. ‘ I h e effect of changing the magnitudes of each coefficient of the linear control functions assigned to a control loop is observed \
