Computers"chem. Engng, VoI. 21, Suppl., pp. S149-S154, 1997

Pergamon

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A Dynamical Supervisor Strategy for Multi-Product Processes Karlene A. Kosanovich* Michael J. Piovoso Department of Chemical Engineering Central Science & Engineering University of South Carolina DuPontCompany Columbia, SC 29208 Wilmington, DE 19880 Abstract

Competing in the global marketing place puts additional burden on multi-product manufacturing processes. Each transition to a different product grade must be made efficiently with the simultaneous objectives of minimal off-spec product and feed stock waste, and non-violation of safety and environmental constraints. In this work, a hybrid, dynamical supervisory control strategy is presented to address transition control of multi-product processes. The scheme involves the selection of the best setpoint controller, from an existing family of feedback controllers so as to cause the output to track the desired setpoint. The performance of the supervisory control strategy is demonstrated on an exothermic reactor.

INTRODUCTION To meet the growing demands of a global economy, many industries strive to remain competitive by producing multiple grades of a large volume, high value product using the same process equipment. One such example, is the production of different molecular weight grades of polyethylene, others include styrene and olefin production. Chemical reactors, in particular polymer reactors, are typically multivariable, exhibit nonlinear characteristics, and often have significant time delays. Lost production capacity and off-spec product associated with reactor grade transitions, are major costs associated with polymer production. It suffices to say, that because of the multi-product nature, the inherent complexity of chemical processes, and multiple objectives such as minimum off-spec production, feedstock waste, and energy consumption, and non-violation of safety and environmental constraints, the control of multi-product processes is difficult. This necessitates the design of a robust controller strategy that can regulate the process successfully, not only at the particular operating point but also in the regions of transition, since poor transition control leads to periods of transient operation that can be economically costly. Most of the research on multi-product chemical processes has been directed at batch or semi-continuous processes and at optimizing the production planning * author

to

whom

all c o r r e s p o n d e n c e

[email protected]

should

be

addressed,

and scheduling. In the case of multi-product continuous processes, the focus has been on acceptable control at an operating setpoint using adaptive control methods (on-line pole placement), gain scheduling, switching control, automatic tuning, and sliding-mode control. Recent research in transition control include the work by Banerjee et al. (Banerjee, Arkun, Pearson and Ogunnaike, 1994; Banerjee, Pearson, Ogunnaike and Arkun, 1995) and Morse (Morse, 1993; Morse, 1995). The former proposes the use of multiple linear models and a scheduling strategy. The effectiveness of this scheme is dependent on the number of models used, and the quality of the corresponding probability estimates. In the latter, the design of a supervisory control scheme is presented that selects the best feedback controller, from a set of linear controllers to cause a singleinput single-output linear system (SISO), to track a setpoint exactly even if bounded, constant process disturbances are present. The feedback controller strategy to be investigated in this proposal will build upon the works cited above and on the development of a dynamic, supervisory scheme for switching to appropriate controllers.

PROBLEM FORMULATION The objective of this work is to develop a methodology which will be able to switch properly into feedback control, a controller from a set of potential controllers, to maintain the process over a wide range of operating conditions. It is proposed to achieve this by,

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PSE '97-ESCAPE-7 Joint Conference

1. defining a set of appropriate linear dynamic timeinvariant systems, each of which models the process within a certain operating region, that together are sufficient to represent the process performance over the entire operating space; 2. designing a setpoint controller, one for each nominal model, that will control the process to a desired set of specifications within the corresponding valid region; 3. defining a shared state space representation, that is capable of simultaneously producing the controller output and an estimate of the plant/model mismatch for each of the models, so that control performance can be assessed without actually putting each controller in feedback with the process; and 4. defining an appropriate measure plant/model mismatch error.

of

the

Consider, that the multi-product process can be represented by a suitable set of local, linear models C(p) or linear in the parameter models centered at each operating point and that there is overlap between the receptive field of each local model. Let the process to be controlled be in the union of a number of subclasses defined over a finite, real-valued parameter set, P,

C(P) = U C ( P ) '

(1)

about which the subclass is centered and there exists for each subclass a controller,

- ~q(s) ~q- pq(s)

SHARED CONTROLLER REPRESENTATION Given a process that is to be operated at several setpoints, generate a linear, time-invariant transfer function model, one for each operating region. Each model, up(s), is a member of the class of admissible transfer functions, that is parameterized by p E P < c~. All up(s) are reduced and proper, ~(~p(s)) > ~(~p(s)) but not necessarily stable, rational functions, with

~p(S) and ap(s) coprime and each Bp(s) is a monic polynomial. The symbol ~(.) represents the degree of the polynomial.

pEP

where p is C(p)'s parameter space. Each subclass contains a nominal process model defined as ap(S)

The output of the controllers together with the actual process output are used by a suitably constructed supervisor controller to assess the potential closed-loop performance should the qth controller be switched into feedback with the process. Such a supervisor controller is designed to be a specially structured causal, dynamic system whose output is a switching signal, a, and whose inputs are the states, xc, of an expanded linear system. These states contain the information about the plant/model mismatch errors and the controller output. This intelligent design is intended to avoid an exhaustive or round robin search or intentional excitation to discover which setpoint controller is most appropriate during transitions at or near the new operating point.

For setpoint control, it is further assumed that the numerator of each transfer function in C(p) is nonzero at s = 0; because if it were then the system cannot be stabilized with a single, fixed-parameter, linear controller. Figure 1 illustrates the structure of the feedback control system. The integrator shown in this figure is required to avoid offset errors and may be included as part of the process or the controller.

(2)

which would solve the positioning control problem were the process sufficientlywell-modeled by a member of that subclass. The pair, {up, sq}, defines a system of two outputs: the pth model output, Ym, and the qth controller output, Vq. Define the norm of the pth plant/model mismatch error to be Ileplt---IlYp- Yll (3) where ~p is the filteredpredicted response, one for each value of p E P and y is the true nonlinear performance response. This error will be used to assess the appropriateness of the pth model to represent the current process operation.

Figure i.

FeedbackSystem

The controllers are designed using a pole-placement method (Astr6m and Wittenmark, 1984) where the control law (error feedback) is given as , ~q (s) , = (r - y) p - - ~

(4)

where r is the reference input, y is the controlled output, and 7(s) and p(s) are polynomials to be deter-

PSE '97-ESCAPE-7 Joint Conference mined so that a pre-specified closed-loop transfer function

B~(s) =

Am(s)

~(s)~(s) s~p(s)p~(s) + ~,(~)%(s)

(5)

is obtained. The control law given by Equation (4) will be proper if and only if

~(p~(s)) > 5(%(s)) In general, if Am(s) and Bin(s) are chosen arbitrarily, the only controller solution possible is one in which the plant's poles and zeros are canceled by the controller's zeros and poles. There will be unstable poles in the controller when there are unstable zeros in the plant transfer function. A realizable, pole-placement controller design with good disturbance rejection properties, places restrictions on the selection of the polynomials, 7(s) and p(s). Factor av(s ) into unstable and stable zeros respectively,

~ ( s ) = ~; (s)~.+(s) Observe that a ~ (s) must remain a factor of

polynomial must be sufficiently high if a realizable control law is to be found. The proof of the theorem can be found in (Kosanovich, Charboneau and Piovoso, 1997). In order to avoid an exhaustive procedure of trying all controllers in feedback to determine the best closedloop performance, a supervisory control strategy is employed. The strategy relies upon a shared controller state that is a minimal realization of an input/output mapping between [y veT]' and [~Tpvq]'. y is the controlled variable, v is the current controller output, eT is the tracking error defined as

eT = r

Making these substitutions, Equation (5) becomes

o~(~)D.,(s)

o~(s)~ +(s)%(s)

Am(s)

a+(s) (sflp(s)~q(s) + ap(S)Tq(s))

There may be other terms on the right hand side of this equation that may have common factors that will cancel. Let these terms be Ao(s), the observer polynomial (AstrSm and Wittenmark, 1984). The conditions on the choices of Am(s), Bin(s), and Ao(s) are given by the following theorem.

There exists a causal solution to the poleplacement design if Theorem

(1) 5(Am(s)) - 5(Bm(s)) >_5(st3p(s)) - 5(av(s)) (2) ~(Ao(s)) > 25(s/3p(s)) - 5(Am(s)) - 6(a+(s)) - 1 where Am(s) and Bm(s) define the desired closed loop poles and the zeros, respectively. The zeros of /3p(s) are the process poles, those of ap(s) are the process zeros, and a+(s) defines the stable zeros to be canceled. Intuitively, the theorem implies that the pole excess for the closed-loop has to be at least as great as that of the plant plus integrator, and the degree of the observer

-

y

(6)

!Tp is a filtered model prediction of the pth model, and This realization is accomplished by constructing a transfer function such that, after cancellation of common factors, % will be the feedback connection between the tracking error, aT, and the qthcontroller output, vq. Let Tpq(s) be that transfer function matrix defined as 1 -

p~(s) = %+(s)~(s)

-

vq is the qth candidate control signal.

s13p(s)Ov(s)

ap(S)Op(S)

.~(s)

Bin(s) = ap(s)Bm(s) to avoid generating unstable poles in the controller. Since a + ( s ) will be canceled by the poles of the controller, it must be a factor of

S151

o

]

~(s)

o

~(s)-pq(s)¢q(s) ~(s)

%(s)¢~(s) ~(s)

The entries of Tvq(S ) are assumed to be in reduced form such that the denominator polynomial is monic and is given by the least common multiple (LCM) of the denominator polynomials of Tp,q(S) to guarantee that the entries are proper. For a unique realization, w~(s) and w~(s) are chosen to be stable, monic polynomials such that

~(~.(s)) > ~(sn~Op) ~(w~(s)) > ~ ( p ~ ) Op(S) and Cq(s) are the observer polynomials required to obtain a proper transfer function. A minimal realization of Tpq(S) is obtained by first selecting SISO controllable pairs {A~, b~}, { A ~ , b,~}, and {A~,b~} in such a way that wv(s) and w~(s) are the characteristic polynomials of A . and A~, respectively; while the LCg({w~(s),w~(s)}) is the characteristic polynomial of A,~. Define

A~

to

be

a

block

diagonal

matrix,

{ A ~ , A ~ , A ~ } , and Bc to be a three column matrix Bc =

b~,,~ 0 0 b,~

This defines a realization

Xc = Acxc + Bc [y v aT]'

~p = cpx~ Vq --- fpXc Jr gpeT

(7)

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with [y v eT]' as inputs and [~p vq]' as outputs, and xc is a shared control state. Row vectors [c~ fq] and scalar gp are found as unique solutions to

(cf: ) (sI - Ac)-lBe + (O1 0 :p ) = for each process and associated controller (Kosanovich et al., 1997). DYNAMICAL SUPERVISOR The overall goal is to develop a finite dimensional, dynamical controller, whose inputs are xc and y and whose output is a piecewise-constant switching signal, aq. The value of aq at each instant of time indicates which controller should be used as the feedback controller. The supervisor controller generates a signal that rates the expected closed-loop performance of the qth controller, one for each q E P. This is accomplished by determining which of the process models is most closely associated with the current plant performance. Define the scalar-valued performance signal to be a function of the filtered plant/model mismatch error

~p = fot e-~(t-r) Ep(T)d~"

(8)

where ~ is a positive constant and represents the rate at which the past performance is forgotten, and Ep is a vector of the square of the integrated plant/model mismatch errors. Note that ep(t) is given by a filtered version of the plant/model mismatch error and this filter has a zero at s = 0. Thus, ep(t) is zero for all the models whenever the plant is at steady state. By integrating ev(t ) the zero is removed and model mismatches are more clearly seen.

event time, AT (the elapsed time between Tc and TD) to be the time when the closed-loop performance is evaluated and used to find a candidate aq that would provide the best closed-loop performance during the time interval AT < t < TD. Two possible outcomes may result. First, if a different controller produces an improvement over the current controller then this new controller is switched into feedback to replace the existing controller. Second, if there is no improvement, initiate another evaluation of the closed-loop performance.

Supervisor ~p~ Perf ormance ~ Switchin~.~ Wei g ht Y Generator Logic g

cq

Figure 3. The components of the supervisor. Observe that this process involves both continuous and discrete time scales; hence the hybrid nature of the finite dimensional, dynamical controller. Figure 2 is a block diagram of the overall supervisory strategy, Figure 3 illustrates the components of the supervisor, and Figure 4 is the switching signal algorithm.

I initialize ak

Supervisor

yp xc = A c X c + B e [ y v ~p----CpX c

r

Vq----fpXc+ gpe T ep= yp- y

?y Figure 2. Block diagram of the supervisory strategy The switching logic that is employed is a function of two additional parameters, a dwell time, TD, and a computation time, Tc, that is at most TD. Define the

Figure 4. Switching signal algorithm EXOTHERMIC

REACTOR

The open-loop behavior of a 2-state exothermic continuous stirred tank reactor (CSTR) is known to exhibit output multiplicities (Uppal, Ray and Poore, 1974).

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PSE '97-ESCAPE-7 Joint Conference Parameters kl k2 k3 k4 k5

In more recent work, Russo et al. (Russo and Bequette, 1992) have shown the importance of including jacket dynamics for this system. With this additional state, the reactor has been observed to exhibit chaotic behavior. This work uses the 3-state system (ECSTR) studied by Russo et al. to demonstrate the proof-ofconcept of this transition control approach.

-

y

xi)

x2 =/~¢xlK:(x2) - (q + ~)x2 + ~x3 "~-q x 2 f X3 ---- ~ l U ( X 3 f -- X3) -]- ~ 1 ~ 2 ( X 2 -- X3) X2

2 -0.2692 -0.5110 1.6077 -0.1128 -0.7205

3 -0.4160 -2.0331 3.4094 4.9280 2.5185

Table 1. Parameter Values of the linear ECSTR Models

The dimensionless component and energy balances that describe this system are given by xl = -¢xlK:(x2) + q(xly

1 -0.1030 -0.1149 2.9245 2.6810 0.7566

(9)

~(x2) = exp(i--~)

6

4

~2

o

where xl is the conversion, and x2 and x3 are the di- 3 10 20 10 20 mensionless reactor and jacket temperatures, respectively. The manipulated variable is u, the cooling water Figure 5. Scheduling change 10~o ---, 80~o conversion. flowrate. Left panel: the output (temperature) and the setpoint, In this work, the parameters values are right panel: switching signal and the controller output. ¢ = 0 . 0 7 2 ~f--20 /~=8 ---- 0,3

xlf--1

~1 ---- 1

~2 :

x2f =O

x3f = - 1

1

Three conversions 10%, 47%, and 80% can be obtained for the same parameter values; the first and the third are stable points while the second is unstable. The corresponding dimensionless reactor temperature values are (0.48, 3.0, 5.0}. From a conversion viewpoint, the third operating point is attractive, however, this operation is at or very near the safety design constraints on the cooling system. Hence, the middle operating point is desirable, even though it is open-loop unstable. This 3 system can be placed in the framework of transition control if one considers the three different conversions, as three different operating regimes 2 The transformed linearized system about the three operating points yields third order transfer functions, 1

klS + k2 lip(S) = 83 q- k382 "Jrk4s "~ k5 Values of the kjs at each operating point are provided in the table below. Using pole placement techniques, appropriate linear controllers are designed. Controllers 1 and 3 are designed to have 0% overshoot, settling times of 7 and 8 dimensionless units of time, and stability margins of 0.54 and 0.78, respectively. Controller 2 has a 40% overshoot, a settling time of 20 dimensionless time units, and a stability margin of 1.38. The hybrid dynamical supervisor control strategy is demonstrated for a scheduling policy of 10% ~ 80% and 10% ~ 47%. Figure 5 illustrates the process output and controller performance for the 10% --* 80% setpoint change. To

accomplished the transition, the temperature setpoint is ramped from one operating point to the next (left panel). We observe (right panel) that the supervisor switches from the controller designed for the 10% conversion to one designed for the 47% conversion and finally to the controller designed for the 80% conversion. The selection of controller 2 occurs because the dynamics during the transition approached the regime where this controller produced the smallest tracking error. '

S

30

10

/f-'~

Y 20

1o

0

10

20

Figure 6. Scheduling change 10~ ~ 47~o conversion. Left panel: the output (temperature) and the setpoint, right panel: switching signal and the controller output. In the second case (see Figure 6), we observe an offset in the controlled variable of ~6%. However, the supervisor correctly switches to the controller designed for the 47% conversion case. There is noticeable chatter (left panel) in the manipulated variable which may be removed by improving the linear controller design.

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PSE '97-ESCAPE-7 Joint Conference SUMMARY

Lack of automatic transition control is a significant problem for the chemical process industry resulting in yield losses. Reliable automatic transition is difficult because of the wide swing in the dynamic behavior as the process transitions from one production schedule to another. This study addresses this important problem by proposing (a) the development of a set of linear models for the various operating regions of the process, (b) design of linear time-invariant controllers one for each nominal model, and (c) the construction of a hybrid dynamical controller strategy that switches between members of the family of fixed parameter controllers. This is accomplished by using a shared state space model that generates the controller outputs and an estimated specific normed output error between the true process output and that predicted by the models. The actual controller used is determined by a dynamical system, that produces a switching signal based on the estimated errors. The idea of a switching strategy is not new and builds upon the concepts reported by Morse (Morse, 1993) and Mayne (Mayne, 1988). What is new is the extension and interpretation of the filtered error and the application to a nonlinear, non-minimum phase chemical process. Research currently being pursued include a transition framework for SISO systems with time-delays and the accompanying analyses of stability and robustness of the framework (Sun and Kosanovich, 1996; Sun and Kosanovich, 1997). References Cited AstrSm, K. J. and Wittenmark, B. (1984). ComputerControlled Systems: Theory and Design, 2rid edn, Prentice-Hall, Engelwood Cliffs, NJ. Banerjee, A., Arkun, Y., Pearson, R. K. and Ogunnaike, B. A. (1994). Robust nonlinear control by scheduling multiple model based controllers, AIChE Nat. Mtg. paper no. 230a, San Francisco, CA. Banerjee, A., Pearson, R. K., Ogunnaike, B. A. and Arkun, Y. (1995). Robust nonlinear control by scheduling multiple model based controllers, Proc. Eur. Cont. Conf. Rome, Italy. Kosanovich, K. A., Charboneau, J. G. and Piovoso, M. J. (1997). Operating regime-based controller strategy for multi-product processes, J. Proc. Control 7(1): 43-56. Mayne, D. E. (1988). Design issues in adaptive control, IEEE Trans. Auto. Cont. 33(1): 50-58. Morse, A. S. (1993). Supervisory control of linear set-point controllers - part 1: exact matching. Submitted to IEEE Trans. on Auto. Contr. Morse, A. S. (1995). Control using logic-based switching, in A. Isidori (ed.), Trends in Control: A European Perspective, Springer-Verlag, London, pp. 69-113.

Russo, L. P. and Bequette, B. W. (1992). CSTR performance limitations due to cooling jacket temperature dynamics, DYCORD+'92 IFAC Syrup. College Park, MD, June. Sun, D. and Kosanovich, K. A. (1996). A new transition control structure for siso time-delay systems, submitted to J. Proc. Contr. Sun, D. and Kosanovieh, K. A. (1997). A robust transition control structure for time-delay systems, submitted to Int. J. of Control. Uppal, A., Ray, W. H. and Poore, A. B. (1974). On the dynamic behavior of continuous stirred tank reactors, Chem. Eng. Sci. 29: 967.