1828
Ind. Eng. Chem. Res. 1989,28, 1828-1834
Ou, W. H. System Identification And Control With An Adjustable Identification Interval. Ph.D. Thesis, National Tsing Hua University, Hsinchu, Taiwan, R.O.C., 1989. Reddy, J. N.; Rasmussen, M. L. Advanced Engineering Analysis; Wiley: New York, 1982. Seraji, H. Design of Cascade Controllers for Zero Assignment in Multivariable Systems. Int. J. Control 1975,21, 485. Smith, C. A.; Corripio, A. B. Automatic Process Control. Wiley: New York, 1985; Chapter 6.
Wu, W. T.; Chu, Y. T.; Chen, K. C. Moving Identification via Weighted Least-squares Estimation. Int. J. System Sei. 1987,18, 477.
Yu, C. C.; Luyben, W. L. Conditional Stability in Cascade Controller. Ind. Eng. Chem. Fundam. 1986,25, 171.
Received for review September 16, 1988 Revised manuscript received May 1, 1989 Accepted August 31, 1989
Adaptive Optimizing Control of Multivariable Constrained Chemical Processes. 1. Theoretical Development Randall C. McFarlanet and David W. Bacon* Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6
An adaptive optimizing control system has been developed for multivariable constrained chemical processes. Previous optimizing control methods have had one or more drawbacks, for example, the need for mechanistic process models for steady-state optimization or the use of nonadaptive control systems. The optimizing control method proposed in this paper is based solely on an empirical dynamic process model, which is identified adaptively on-line. Dynamic information from the identified models is used for on-line updating of a multivariable Internal Model Controller. Constrained steady-state optimization is accomplished using steady-state information from the identified model in an on-line application of sectional linear programming. In the accompanying paper (part 2), the optimizing control strategy is tested on simulations of two nonlinear constrained processes which are multivariable and interacting: a CSTR system supporting a multiple reaction and a fluid catalytic cracker. Economic optimum performance of most industrial processes occurs at an intersection of constraint boundaries. Disturbances acting on a constrained process limit its economic potential in that steady-state operating points must be moved away from constraint boundaries in order to minimize the risk of constraint violations. A regulatory controller reduces the effect of these disturbances on a constrained output variable, permitting its setpoint to be placed closer t o the constrained optimum, thereby improving the economic return of the process. The effects of disturbances on a process, including those which are nonstationary, are often considered only in the context of this regulatory control problem. However, nonstationary disturbances may have additional impact on a constrained process; they are capable of shifting the positions of constraint boundaries sufficiently to relocate a constrained optimum to different intersections of constraint boundaries. In these situations, an optimizing control system can achieve significant gains in process profitability over that possible from simple regulatory controllers acting alone. In optimizing control, frequent steady-state reoptimization is performed on-line to track the constrained optimum as it shifts in response to nonstationary disturbance(s) acting on the process. Regulatory controllers counter the effects of any additional disturbances, allowing the setpoints of controlled variables to be placed closer to the current constrained optimum. Chemical processes that are candidates for optimizing control often involve complex reaction and separation systems and as such are typically multivariable, interacting, and nonlinear. For such processes, the design of an optimizing control strategy can be a complex undertaking
* Author to whom correspondence should be addressed. Current affiliation: Amoco Corp., Amoco Research Center, Naperville, IL 60566.
0888-588s189/2628-1828$0l.50J O
which may not be economically justified, particularly if the strategy requires prior development of mechanistic process models. The emphasis in this paper is on the development of a general approach to optimizing control using empirical rather than mechanistic process models. Due to the potentially complex nature of the process characteristics mentioned above, the design problem for servo and regulatory control in an optimizing control application can be a difficult one. Controller designs typically utilize process models that are approximate linear representations of local dynamic process behavior. These designs perform well if operation is restricted to a sufficiently small region around the operating point at which the model was identified, but a fixed-parameter controller cannot be expected to maintain satisfactory performance for a nonlinear process as steady-state optimization moves are made across a potentially large feasible operating region. An alternate approach, and the one adopted in this study, is to use an adaptive controller in which on-line updating of controller parameters is triggered by steady-state optimization moves. Interaction between process variables is particularly detrimental in an application where optimizing control is beneficial. During a period of on-line steady-state optimization, it can be expected that the constraints on some controlled outputs will be active while others will be inactive and that the status of any particular constraint might change as the optimization proceeds. To avoid constraint violations, setpoint changes to controlled outputs with inactive constraints must be accomplished with minimal disruption to the controlled outputs which are being held close to their constraint boundaries. A control system based on multiple single-input/single-output(SISO) control loops does not achieve decoupling of interacting variables and therefore in general will not be suitable for an optimizing control application. To avoid this 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 12,1989 1829
u
-4
SYSTEM
i-by'
Figure 1. Input/output representation of the system.
particular problem, an adaptive multivariable controller is employed in this study. The need for fast on-line optimizing speeds is a function of the rate of movement of the optimum, which in turn depends upon the dynamic characteristics of the nonstationary disturbance(s) affecting the process. Direct optimization methods that rely on steady-state data or those that are based on on-line identification of a steady-state model are inherently slow. A t each point in the optimization (or identification) stage, the process must be allowed to reach steady state to obtain the necessary information. This steady-state approach would be too slow to track a rapidly moving optimum, particularly for processes with large time constants. An alternate approach described by Bamberger and Isermann (1978) has been employed in this study. Optimization information is obtained from the steady-state gain of an empirical dynamic process model identified on-line. The application of this approach for open-loop on-line optimization has been investigated by Garcia and Morari (1981) and Bhattacharya and Joseph (1982). Lee and Lee (1985) and Garcia and Morari (1984) demonstrated its use in optimizing control applications. The optimization procedure employed in the current study is an adaptive extension to closed-loop systems of the constrained multivariable approach based on sectional linear programming (SLP) described by Bhattacharya and Joseph (1982). In the next section, the optimizing control problem is described and an optimizing control system is developed which addresses the problems identified above. The proposed method overcomes many of the shortcomings of previous optimizing control systems, in particular the reliance on mechanistic process models for on-line steadystate optimization, and demonstrates the use of a multivariable adaptive controller in an optimizing control application. In an accompanying paper (McFarlane and Bacon, 1989),this optimizing control strategy is applied to simulations of a CSTR supporting a multiple reaction and a fluid catalytic cracker, both multivariable, interacting, and highly nonlinear systems.
Problem Definition The general form of the system to be optimized is depicted in Figure l. P is a known economic function reflecting the market values of the process outputs and the costs of the input streams. The outputs denoted by y 1 (dimension nI) are subject to specified inequality constraints arising from, for example, operability limits on equipment or limits on impurity levels in product streams (including limits on environmental contaminants in waste streams). The outputs denoted by y E (dimension nE)are subject to specified equality constraints which typically arise due to product quality specifications. For example, the mole fraction of a particular component in an output stream may have to be maintained at a given level within a specified tolerance.
It is assumed that all outputs are measurable at specified discrete equally spaced time intervals. The inputs u (dimension p ) may also be subject to inequality constraints. These constraints are referred to as hard constraints when they represent, for example, saturation limits on final control elements. Disturbances entering the system are assumed to be unmeasurable and are classified into two groups as suggested by Morari et al. (1980). Type 1 disturbances, denoted by d l in Figure l, are nonstationary and create the need for steady-state on-line optimization. Disturbances in this group are sufficiently large and persistent to cause significant movement in the position of a constrained optimum by shifting the positions of constraint boundaries. Typically these disturbances can be adequately modeled as randomly occurring deterministic steps (e.g., changes in feed composition with new feedstocks). Other type 1 disturbances may be autocorrelated. Examples of this type are ambient temperature fluctuations and slowly drifting properties of feedstreams from other processing units. The time constants of type 1 disturbances are large enough relative to the dominant time constant of the process that shifts in the positions of constraint boundaries are sufficiently permanent to warrant reoptimization. However, the time constants of type 1 disturbances are also sufficiently small that repositioning of the constrained optimum would occur too frequently for practical implementation of off-line Optimization. The dynamic behavior of type 2 disturbances, denoted by d 2in Figure 1,is characterized by small time constants relative to the dominant time constant of the process. These disturbances are stationary, and because their effects are short-lived, they are irrelevant to the long-term optimization of the process. The long-term expected values of disturbances d 2 are zero. Variation in system outputs caused by disturbances d 2 can be reduced by regulatory feedback control. In an optimizing control application, it is imperative to achieve very small (preferably zero) steady-state offsets in controlled outputs, since the setpoints of some of these outputs may be very close to constraint boundaries. This can be a complex problem because optimal operation may require some inputs to reach their saturation limits. In an application of optimizing control to a fluid catalytic cracker, Prett and Gillette (1979) described one approach to this problem using a nonsquare (four inputs, two outputs) multivariable dynamic matrix controller (DMC). When the optimization called for a steady-state operating point that saturated an input, the DMC control law was modified on-line to include an equality constraint for that particular input. This forced the controller to maintain this input at a constant level as long as the optimizer predicted that its constraint remained active. Setpoint changes were accomplished without offset by manipulation of the three remaining free inputs. Steady-state optimization was accomplished by sectional linear programming based on a mechanistic process model. The problem considered in this paper is one in which the number of constrained outputs (including both inequality and equality constrained outputs) that could potentially benefit from regulatory control exceeds the number of available inputs for control manipulation. In this situation, offset in controlled outputs cannot be avoided if a nonsquare multivariable controller is employed. To circumvent this problem, an optimizing control strategy is proposed that is based on a square multivariable controller regulating a fixed subset of outputs equal in dimension to the number of inputs p . Since equality
1830 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 I
r
y EC sei
T + y
-
i
OPTIMIZER
IC Sel
MULTIVARIABLE CONTROLLER
SYSTEM
mI
I '
i:c MODEL INVERSION FOR IMC CONTROL LAW
Y Ec
I
I
I
CONVERSION OF IrO MODEL TO IMPULSE RESPONSE FORM
OPTIMIZATION PROBLEM
Figure 2. Optimizing control configuration.
RECURSIVE ESTIMATION OF SYSTEM MODEL A(q1yItI = BIq)uli) + c + e ( t )
LOCALLY VALID STEADY STATE MODEL A(liytv= B(i)ulv + c
Figure 3. Block diagram of the optimizing control strategy.
constraints are always active, they must be controlled as a first priority. Any remaining degrees of freedom among the inputs allow a selected subset of inequality constrained outputs, denoted by y I C , to be controlled. All other inequality constrained outputs y' are necessarily uncontrolled. This optimizing control problem is illustrated in Figure 2 . The variables adjusted by the on-line optimizer for steady-state optimization are the setpoints yIcpSet of the controlled inequality constrained outputs. Setpoints of are fixed at controlled equality constrained outputs yECvet the specified levels for these variables. With a square multivariable controller, steady-state offset in at least some of the controlled outputs will occur if the on-line optimizer requests setpoint changes (i.e., an optimization step), which would attempt to force one or more input levels, in a steady-state sense, beyond their saturation limits. Therefore, the optimizer must be designed to recognize input constraints and to maintain process operation at a suitable distance away from input constraint limits in order to be capable of responding to type 2 disturbances. Inclusion of input constraints in the optimizing control problem is facilitated if a mechanistic model is available to predict input/output behavior, as in the example described by Prett and Gillette (1979). The optimizing control strategy described below requires only empirical input/output process models and accommodates input constraints, as well as equality and inequality constraints on outputs. A general representation of the optimizing control problem illustrated in Figure 2 is the following:
outputs are considered. The method to be described is completely general, allowing upper and lower constraint limits as well as constraints on functions of system variables, as long as such functions are measurable. The overall optimizing control strategy is now described. Empirical Optimizing Control Algorithm The optimizing control algorithm is shown in Figure 3. The central component of the algorithm is an on-line estimation of an empirical multivariable dynamic input/ output model in the following form: A(q)y(t) = B ( q ) u ( t )+ c + e(t)
(6)
where
+ A 1 q l + A 2 q 2+ ... + A,q-" B(q) = B,q-' + B2q-2+ ... + BmqTm
A(q) = I
(7)
(8)
Ai and Bi are constant matrices, c is a vector of constants, e ( t ) is a vector sequence of random shocks assumed to satisfy i f j for all k (9) E[ei(t)e j ( t + k ) ] = 0
E [ e ( t ) ]= 0 and q-' is the backward shift operator,
(10)
q-'y(t) = y(t-1)
y(t) is the measured output vector at time t defined as
max P yIcs*t
subject to yEc,set
- yEc,spec = 0
yIc,set
- yIc,max 5 0
yI -
Y
1,max
5 u5
-< 0
(2) (3) (4)
(5) where P is the objective function to be optimized, yEc.sw is the vector of specified levels for controlled equality constrained outputs yEC (dimension nF), y E c v s e t is the setpoint vector for yEC (dimension nEC),y c,max is the vector of specified constraint limits for controlled inequality constrained outputs y I C (dimension nIc),y I c W t is the vector of setpoints for yIC(dimension nIc),yr*" is the vector of specified constraint limits for uncontrolled inequality constrained outputs y' (dimension nI), u is the vector of inputs (dimension p ) , and uminand u- are the vectors of specified lower and upper constraint (saturation) limits for the inputs u (dimension p ) . Without loss of generality, only the upper constraints on inequality constrained p i n
Umex
The steady-state gain of the estimated eq 6 is a source of information for on-line steady-state optimization. The resulting steady-state model represents a local linearization of the steady-state behavior of the process and is assumed to be valid in a local region around the current operating point. Since the identification vector y ( t ) includes all constrained outputs, this model provides local predictions of the positions of linearized constraint boundaries as well as the behavior of the objective function. This information forms the basis of a local linear programming problem, the solution of which defines a feasible optimization step in the region of validity of the identified model. Adaptive recursive estimation of the system model and repeated solution of the resulting local linear program at each new operating point defines the on-line optimization procedure. This approach is an extension to closed-loop systems of the empirical SLP method developed by Bhattacharya and Joseph (1982) for open-loop systems. This closed-loop
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1831 empirical SLP method recognizes all modeled constraints, including inequality constraints on inputs. Control action is executed at discrete equally spaced time intervals small enough to regulate effectively against disturbances d 2 . Optimization steps are executed less frequently, in response to disturbances d The number of discrete time intervals between optimization steps defines the adaptation interval Nad,which is selected as a compromise between the need for a fast optimizing speed and the need to allow sufficient time for adaptive updating of the system model after an optimization step. The demand for optimizing speed is a function of the dominant time constant of disturbances d l . At the end of an adaptation interval, updated steady-state information is supplied to the optimizer to enable it to determine the next optimization step. As described below, updated dynamic information is also provided to update the control law in anticipation of a move to a new operating level.
On-Line Estimation of the Input/Output Model Since the elements of the noise vector e ( t ) are assumed to be independent, the parameters in each row of the system model in eq 6 can be estimated independently of those in other rows. The ith row of eq 6 can be written as yi(t) = eT4i(t) + ei(t)
where
eT
=
[-ah...-at
...-a,l,;..-aO
8 4
b,ll...b$
(12)
... bjp...@ ci] (13)
@ = [yl(t-l).-yl(t-n)
y,(t-l)...y,,(t-n) ui(t-l)*..ui(t-m) uP(t-l)...~,(t-m)
11 (14)
nt = 1 + n1 + nI, + nEc ak. is the (ij)th element of Ak and b t is the (ij)th element
Bk.
For a sequence of measurements ( u(t),y(t)],t = 1, 2, ..., n*, a general recursive estimator for Bi, which minimizes la*
v(ei)= t = l ~ ( t ) n * - te^?(t)
(15)
where si(t) = yi(t)
- eT+i(t)
is the residual equation error associated with eq 12 and X(t), 0 < X(t) 5 1,is a variable forgetting factor described below, is available in the following algorithmic form (Ljung and Soderstrom, 1983):
+ K(t)[yi(t) - dT(t-l)$i(t)]
(16)
1 p ( t ) = W(t-1) - K(t)@(t)P(t-l)l~(t)
(18)
di(t) = di(t-1)
where dT denotes an estimate of and P(0) and &(O) are specified. Choosing X ( t ) < 1 results in exponential discounting of previous data for adaptive estimation. A variable forgetting factor is employed here, which generates rapid adaptation immediately following an optimization step but progressively slower adaptation thereafter. Greater discounting is applied to past data from the previous operating condition than to data from the new operating condition resulting from an optimization step. The variable
forgetting factor is generated from the fmt-order difference equation X ( t ) = X,X(t-l) + X(f)(l- A,) (19)
Xu),
where X(O), and A,, 0 < A,, 5 1, are specified, t = 0 denotes the discrete time period immediately following an optimization step, and t = f denotes the last discrete time period of an adaptation interval. For X(f) > h(O), eq 19 generates an exponential rise in h(t)from X(0) to the rate of rise is governed by A., Typically X(0) = 0.95 and 0.98 I XU) I 1.0. Closed-loop identifiability of the system model is ensured by addition of an independent persistently exciting signal to each of the system inputs (Gustavsson et ai., 1977). PRBS (pseudorandom binary sequence) signals, denoted by u * ( t ) in Figure 3, which are sufficiently rich in frequency content to be persistently exciting for most industrial systems, are employed for this purpose in this study.
Xu);
Multivariable Control System Multivariable Internal Model Control (MIMC) (Garcia and Morari, 1985a,b) has been selected as the control system for the optimizing control strategy. The theoretical development of MIMC provides a convenient vehicle for analysis of closed-loop stability, dead time factorization, and quality of decoupling and control performance in multivariable systems. An adaptive version of MIMC has been developed in this paper by adaptive on-line estimation of a multivariable input/output system model (eq 6), conversion of this model to the impulse response form required in the MIMC synthesis procedure, and on-line updating of the MIMC control law. In MIMC, the multivariable system model G(q) is expressed in truncated impulse response form: G ( q ) = H1q-' + Hzq-2 + ... + H f l - N (20) where N is the specified truncation length and &, lz = 1, 2, ..., N , are the impulse response matrices. The MIMC control law is obtained by exact or approximate inversion of G ( q ) through solution of a predictive optimization problem. selection of parameters in the optimization problem constitutes controller tuning and allows considerable flexibility in altering the performance and robustness of the resulting controller (Garcia and Morari, 1985b). A complete derivation of the MIMC controller is provided by Garcia and Morari (1985a). A brief description of the pertinent controller tuning parameters is provided in an accompanying paper (McFarlane and Bacon, 1989).
Impulse Response Model for MIMC The model impulse response parameters relating controlled output i to input j , Qj, l = l , 2,..., N i , j = l , 2,...,p (21) where h)j is the (ij)th element of 8, in eq 20, are obtained from the estimated version of eq 6 written in the following first difference form: A ( q ) V y ( t )= B ( q ) V u ( t ) (22) where signifies an estimated quantity and VY(t) = Y(t) - y(t-1) v u ( t ) = u ( t ) - u(t-1) The impulse response parameters relating controlled output i to input j are calculated recursively from the appropriate row of the estimated model (22) by setting vui
1832 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989
to 1 at time t and 0 at all other times so that @j
= Vyi(t+l),
E
= 1, 2, ..., N
i = 1, 2, ..., p
(23)
for each input j = 1, 2, ..., p . Steady-State Optimization In sectional linear programming, a solution to a nonlinear programming problem is obtained as a sequence of solutions to local linear problems in which local system behavior is approximated with a linear(ized)process model (Griffith and Stewart, 1961). In the approach described below, fast optimizing speeds can be achieved, despite small optimization steps, since the method is based on steady-state predictions obtained from dynamic system models. One disadvantage of this approach is that excitation signals must be applied to the input signals (or elsewhere in the feedback loop) to ensure identifiability, necessarily producing additional variation in system outputs. In order to avoid transient contraint violations, constraint buffer boundaries are defined at selected distances from the actual constraint limits defined in problem 1. The steady-state optimization then proceeds along the buffer boundary, thereby minimizing intrusion upon the original constraints. In the region of a constrained optimum when no further improvement is possible, the excitation signals can be removed to allow the operating point to be moved closer to the actual constrained optimum. Buffer constraint boundaries are also defined for input variables to allow for fluctuations from the excitation signals and to ensure sufficient freedom for the controller to manipulate input levels to counter the effects of disturbances d2. A locally valid steady-state model is obtained by applying the final value theorem for z transforms to the z transform of eq 6 estimated on-line. Since E [ e ( t ) ]= 0,
A(l)yIV= B(1)u”
+c
(24)
ulJvand udvare the lower and upper bounds of the region of validity of eq 24, d : is the vector of predicted current steady-state levels of the inputs, and Au is a specified vector defining the size of the region of validity of eq 24. The amplitudes of the PRBS signals added to the system inputs constitute a good basis for establishing the size of the region of validity of the identified model; thus, Au =
UPrbs
d: is obtained by inversion of eq 24 as follows. Equation 24 can be partitioned as
In eq 29, ylc*setJv and yEc~“Jv are known (current setpoint levels defined by the optimizer) whereas PIv,ylJv, and u l V are unknown (dynamic levels of these variables are measured). By use of the standard rules for manipulation of partitioned matrices, the unknown quantities are obtained as
where
where
A*(l) = [-B(l)IA’(l) IAp(l)]
(31)
A prediction of u , is obtained from the first p rows of eq 30 as ug =
ulv
(32)
A local linear programming problem is defined as yIC&and yEcwtappear in eq 25 instead of yICand yEcsince
the controller maintains controlled outputs at their setpoints with zero steady-state offset. The use of eq 24 as an optimization tool can be visualized as follows. For small changes in yIcpset (which are being manipulated for steady-state optimization), the process will come to a steady-state condition that approximately satisfies eq 24. Because eq 24 represents the local behavior of the constrained variables and the objective function, it can be used to predict a small feasible optimization step. A local linear programming problem is formulated below that includes all system constraints, including those on inputs and, as described in the following paragraph, on the region of assumed validity of eq 24. The region of validity of eq 24 can be established as follows. Since eq 6 is identified with inputs u as independent variables, its region of validity is defined in terms of deviations from the current steady-state levels u , of the inputs. Because the process is never required to reach steady state during an adaptation period, u , must be predicted as described below, using eq 24. The region of validity is defined as ul,lv < ulvsa < uu.lv (26)
max ylcsstJv
PIv
(33)
subject to
Equality 34 is the system constraint (eq 24). Equality 34 and inequality 35 (defining the region of validity of the system model) result in implicit constraints on the sizes
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1833 of the changes in the setpoints However, for operability and safety reasons, it is desirable to place explicit constraints on the setpoints (inequality 36),limiting the size of an optimization step from the current condiand ylcvwtJvJ are defined as tions. ylc,set,iv,u yIc,set,lv,u = yIc,set + AYset (41) yrctSet.
yIc,set,lv,l
= yIc,set - AYset
(42)
where Ayset is a vector of specified permissible setpoint changes defining the optimization step size. Inequalities 37 and 39 ensure that optimization steps respect the constraint boundaries of controlled and uncontrolled inequality constrained outputs. Buffer constraint boundaries ylc'u and yl*u are defined as follows: yIc,u
= yIc,max - tIC
yL = yI,max - cI
(43) (44)
where dc and c1 are specified vectors of small positive constants. Again, the lower limits on inequality constrained variables can be included if required. Buffer boundaries on the inputs are defined analogously to those on inequality constrained outputs: uu = Umax - t u (45) u1
=
Umin
+
(46) where cu and c1 are specified vectors of small positive constan ts. Equality 38 is an important aspect of the on-line optimization procedure. The control system ensures that equality constrained outputs are maintained at specified levels without offset. However, when setpoint changes are imposed on other controlled outputs, the steady-state input levels that are required to accomplish those changes must be predicted, and clearly those predictions must include the effects of maintaining equality constrained outputs at their specified setpoints.
Conclusions An adaptive optimizing control strategy has been developed for nonlinear, multivariable, constrained processes. The procedure is based on on-line adaptive identification of a multivariable dynamic input/output model. The steady-state gain of this model provides information for on-line optimization using sectional linear programming. With this approach, all modeled process constraints are recognized, including hard inequality constraints on inputs and equality and inequality constraints on process outputs. Dynamic information from the identified model is used for on-line updating of the control law of a multivariable internal model controller. The performance of this explicit adaptive controller in an on-line optimization application is expected to be superior to that of a fixed parameter controller because model error is minimized by model updating at each optimization step. In the approach described in this paper, impulse response parameters are derived indirectly by on-line identification of a parametric input/output model. From statistical and computational points of view, this approach is preferred over direct estimation of the impulse response parameters because a low-order parametric input/output model requires far fewer parameters to describe a system. Acknowledgment Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Nomenclature A(q) = matrix polynomial in q-' in the input/output model A . = component of A(q) u$ = (ij)th element of Ak B(q) = matrir polynomial in q-' in the inputfoutput model B. = component of B(q) b$ = (ij)th element of Bk c = vector of constants in the input/output model d = type 1disturbances d 2 = type 2 disturbances e = vector of random shocks in the inputfoutput model ei = ith element of e G ( q ) = system transfer function matrix H = impulse response matrix h = element of H
I = identity matrix K = gain matrix in recursive estimation algorithm m = degree of B(q) N = truncation length Nad= number of discrete time steps defining an adaptation interval n = degree of A(q) nE = number of equality constrained outputs nEc= number of controlled equality constrained outputs nI = number of uncontrolled inequality constrained outputs nIc = number of controlled inequality constrained outputs n, = 1 + n1 + n1, + na n* = number of measurements of a variable P = variance-covariance matrix of parameter estimates P = objective function p = number of inputs t = discrete time f = current time u = vector of inputs U P r b = vector of amplitudes of PRBS excitation signals u 85 = vector of steady-state levels of inputs V = objective function in the least-squares estimation y ( t ) = output vector defined in eq 11 yE= vector of equality constrained outputs yEC = vector of controlled equality constrained outputs y1= vector of inequality constrained outputs yIC= vector of controlled inequality constrained outputs Greek Symbols Au = vector defining the size of the region of local validity Aywt = vector of permissible setpoint changes e = vector of specified positive constants 0 = vector of parameters to be estimated, defined in eq 13 A(t) = forgetting factor in recursive estimation A, = constant in definition of A ( t ) (eq 19) 4 = memory vector, defined in eq 14 Operators E [ ] = expectation operator q-' = backward shift operator, q-'y(t) = y(t-1) T = transpose v = first difference operator, v y ( t ) = y ( t ) - y(t-1) Superscripts 1 = lower bound lv = locally valid model max = upper constraint boundary of an inequality constrained variable min = lower constraint boundary of an inequality constrained variable set = setpoint spec = specified level of an equality constrained variable u = upper bound = estimate = parameter (or matrix) in truncated impulse response model
-
A
1834
Ind. Eng. Chem. Res. 1989,28, 1834-1845
Abbreviations
DMC = dynamic matrix controller MIMC = multivariable internal model control SISO = single-input single-output SLP = sectional linear programming Literature Cited Bamberger, W.; Isermann, R. Adaptive On-line Steady-State Optimization of Slow Dynamic Processes. Automatica 1978, 14, 223-230. Bhattacharya, A,; Joseph, B. On-line Optimization of Chemical Processes. Presented at the First American Control Conference, Arlington, VA, 1982; paper MP5. Garcia, C. E.; Morari, M. Optimal Operation of Integrated Processing Systems. Part I: Open-loop On-line Optimizing Control. AIChE J. 1981,27,960-968. Garcia, C. E.; Morari, M. Optimal Operation of Integrated Processing Systems. Part 11: Closed-loop On-line Optimizing Control. AIChE J. 1984, 30, 226-234. Garcia, C. E.; Morari, M. Internal Model Control 2. Design Procedures for Multivariable Systems. Ind. Eng. Chem. Process Des. Deu. 1985a, 24, 472-484. Garcia, C. E.; Morari, M. Internal Model Control 3. Multivariable Control Law Computation and Tuning Guidelines. Ind. Eng. Chem. Process Des. Dev. 1985b,24, 484-494.
Griffith, R. E.; Stewart, R. A. A Nonlinear Programming Technique for the Optimization of Continuous Processing Systems. Manage. S C ~1961, . 7, 379-392. Gustavsson, I.; Ljung, L.; Soderstrom, T. Identification of Processes in Closed Loop-Identifiabilityand Accuracy Aspects. Automatica 1977, 13, 59-75. Lee, K. S.; Lee, W. On-line Optimizing Control of a Nonadiabatic Fixed Bed Reactor. AZChE J. 1985, 31, 667-675. Ljung, L.; Soderstrom, T. Theory and Practice of Recursive Estimation; MIT Press: Cambridge, MA, 1983. McFarlane, R. C.; Bacon, D. W. Adaptive Optimizing Control of Multivariable Constrained Chemical Processes. Part 2: Application Studies. Znd. Eng. Chem. Res. 1989, following paper in this issue. Morari, M.; Arkun, Y.; Stephanopolous, G. Studies in the Synthesis of Control Structures for Chemical Processes. Part I: Formulation of the Problem. Decomposition and the Classification of the Control Tasks. AIChE J . 1980, 26, 220-232. Prett, D. M.; Gillette, R. D. Optimization and Constrained Multivariable Control of a Catalytic Cracking Unit. Presented at the AIChE National Meeting, Houston, TX, 1979; paper WP5-C. Received for review October 5, 1988 Revised manuscript received July 11, 1989 Accepted August 15, 1989
Adaptive Optimizing Control of Multivariable Constrained Chemical Processes. 2. Application Studies Randall C. McFarlanet and David W. Bacon* Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6
The adaptive optimizing control strategy described in an accompanying paper (part 1)was successfully applied to simulations of two highly nonlinear, multivariable interacting processes. These simulated applications demonstrated the feasibility of using on-line identification of multivariable dynamic input/output behavior as a basis for employing sectional linear programming for on-line optimization. In addition, dynamic information from the identified input/output model was used for on-line updating of a control model for a multivariable internal model controller. Adaptive on-line identification of the system model ensured that the effects of model error on control and optimization performance were minimized.
As described in an accompanying paper (McFarlane and Bacon, 1989), chemical processes that are candidates for optimizing control often involve complex reaction and separation systems. For such systems, development of an optimizing control system based on a mechanistic process model may not be economically justified due to the expense of developing the model. In process control, the lack of adequate mechanistic process models has been overcome by research that has focused on the use of empirical input/output models. The availability of the resulting control designs has led to a dramatic increase in the number of industrial applications. A similar focus on systematic development of empirical methods for on-line optimization and optimizing control might be expected to produce similar results. The primary drawback of empirical steady-state approaches to on-line optimization is slowness of optimizing speed. Optimization methods that utilize direct search approaches (e.g., those based on simplex pattern searches) require that the process reach steady state at each operating condition in the search pattern, resulting in slow optimizing speeds. Similar criticisms apply to approaches
* Author to whom correspondence should be addressed. Current affiliation: Amoco Corp., Amoco Research Center, Naperville, IL 60566. 0888-5885 f 8912628-l834$0l.50/0
based on response surface methodology (e.g., Biles and Swain, 1980;Box and Draper, 1987). While these methods could be applied on-line, they would be useful only in situations where the movement in the location of the constrained optimum was sufficiently slow to allow it to be tracked by a relatively slow optimizer. However, the need for on-line optimization is more often motivated by a rapidly shifting optimum requiring faster optimizing speeds. The optimizing control strategy developed in our accompanying paper (McFarlane and Bacon, 1989) utilizes on-line identification of dynamic empirical process models to develop fast optimizing speeds. The steady-state gain of a dynamic input/output model provides information for optimization without actually requiring that the process ever reach steady state during the search. A related approach has been reported by Cutler and Hawkins (1987) in an application to a hydrotreater reactor. In that study, steady-state predictions from a step response model of the process provided information for on-line application of sectional linear programming. The step response parameters were identified off-line using test process data. No on-line updating of the process model was performed. The optimizing control strategy developed in the present study has a number of advantages over the approach described by Cutler and Hawkins. The use of a parsimonious 0 1989 American Chemical Society
