LIMITATIONS TO INPUT-OUTPUT ANALYSIS OF CASCADE CONTROL OF UNSTABLE CHEMICAL REACTORS LOUIS P. RUSSO and B. WAYNE BEQUETTE Rensselaer Polytechnic Institute, Howard P. Isermann Department of Chemical Enineerin, Troy, NY 8-9, USA. Email: bequeb@rpi.edu Abstract. In this paper we show that the traditional methods of analyzin cascade control (based on series or parallel input/output relationships) should not be used when the primary and secondary processes are coupled, and the process is open-loop unstable. The connection between cascade control and traditional state feedback control is studied. In addition, we develop a cascade control system methodoloy which incorporates a specied inner-loop in the outer-loop desin. This cascade control desin procedure is compared to a conventional cascade control desin procedure and to a noncascade control stratey. Key Words. Cascade control; chemical s; unstable processes; state-space analysis; input/output analysis. INTRODUCTION Input/output (transfer function) analysis is often used to desin feedback control systems for chemical processes. Cascade control systems are usually desined by rst tunin a slave control loop based on the transfer function model of the secondary process. If the slave loop can be tuned \fast" enouh, then the master loop can be tuned based on the primary process transfer function, independent of the slave loop. If the bandwidth of the slave loop is close to that of the primary loop, then the loops must be iteratively tuned, until acceptable performance of the entire closedloop process is obtained. Implicit in the typical desin process is the assumption that the secondary and primary processes are decoupled. This is illustrated by the typical series cascade control block diaram, shown in Fiure. The output of the secondary process is the input to the primary process in a series cascade control block diaram. Similarly, a parallel cascade control structure (in which the manipulated input has a direct eect on both of the outputs) is shown in Fiure. The motivatin problem is cascade control of an exothermic continuous stirred tank (CSTR) operated at an open-loop unstable operatin point. Althouh many new nonlinear control techniques have been developed (Bequette, 99; McLellan et al., 99; Kravaris and Kantor, 99), we use linear control for two reasons: Linear feedback and cascade s are still the dominant strateies used in industry. The linear techniques are well-known and easy to use for comparison purposes.. CSTR MODELING EQUATIONS The standard two-state CSTR model (Uppal et al., 97) describin an exothermic diabatic irreversible rst-order reaction (A! B) is a set of two nonlinear ordinary dierential equations obtained from dynamic material and enery balances (with the assumptions of constant volume, perfect mixin, neliible coolin dynamics, and constant physical parameters). dc a Q?Ea dt V (C af? C a )? k exp C a () RT dt Q dt V (T f? T )? U A (T? T c ) V C p?h?ea + k exp C a () C p RT where C a and T are the concentration of component A and the in the, respectively. An additional enery balance around the coolin assumin perfect mixin yields dt c dt Q c (T cf? T c ) + U A (T? T c ) () V c V c c C pc
where T c is the coolin. Equations - can be written in dimensionless form dx d q(x f? x )? x (x ) () dx d q(x f? x )? (x? x ) +x (x ) () dx d [ (x f? x ) + (x? x )] () where x, x, x, and are the dimensionless concentration,, coolin, and coolin owrate, respectively. These dimensionless variables and parameters (for example:, q,, etc.) are dened in Table. Representative values of the parameters can be found in Russo and Bequette (99a, 99b). Case is open-loop stable over the entire reion of operation for the two-state CSTR model but open-loop unstable in a certain operatin reion for the three-state CSTR model, while Case exhibits inition/extinction behavior for the two and three-state models. It is well-known that the exponential relationship of reaction rate with respect to is one of the major nonlinearities of the CSTR. Russo and Bequette (99a, 99b) studied both the steady-state nonlinearities (output and input multiplicities, infeasible operation reions) and the dynamic nonlinearities (Hopf bifurcation behavior) of this three-state CSTR model. Sistu and Bequette (99) determined conditions under which input multiplicity behavior results in the open-loop system havin unstable zero dynamics (nonminimum-phase behavior).. CONTROL SYSTEM DESIGN A common control conuration in chemical processes is cascade control. In a cascade control con- uration we have one manipulated variable (in this case the dimensionless coolin owrate ) and more than one measurement (the dimensionless x and dimensionless coolin x ). Cascade control can also be thouht of as multiple state feedback. The major benet of cascade control is that action is taken to reject inner-loop disturbances before they eect the outer-loop. Therefore, the innerloop of the cascade control stratey is enerally tuned very tihtly in order to obtain a fast response. In addition, intuitively we expect the performance of a cascade control system to be superior (enerally) to a noncascade control system, since more system information (outputs) is bein fed back to the control system... Transfer Function Models The transfer functions between the deviation variables x (s), x (s), and (s) are: x (s) (s)q c(s) (c s + ) k (a s + a s + a s + ) c(s) (7) q x (s) (s)x (s) k (c s + ) (b s + b s + ) x (s) (8) x (s) (s)q c (s) k (b s + b s + ) (a s + a s + a s + ) (s) (9) where the ains and coecients of the transfer functions are dened in Table (the indicates deviation variable). We can view these equations in block diaram form, as shown in Fiure. The poles of the two-state model relationship (relatin x (s) to x (s)), (s), are the zeros of the x (s) - qc(s) relationship in the three-state model, (s). When (s) is unstable at a particular operatin point, then (s) will have riht-halfplane zeros. We would think that the performance of the \inner-loop" process would be reduced, since riht-half-plane zeros cannot be \inverted" for \perfect" control. We nd that this is not the case, because the real cascade system is actually a multiple state feedback control. Since the real measures two outputs and manipulates a sinle input, there are no problems with the secondary process riht-half-plane-zeros... Series and Parallel Cascade Control The two most typical cascade control conurations are the series and parallel structures, which were shown in ures and. Luyben (97) stressed the dierences between conventional series cascade control and parallel cascade control. The three-state CSTR transfer function models have a parallel cascade control structure, since the manipulated variable (dimensionless coolin owrate) has a direct inuence on x (dimensionless ) and x (dimensionless coolin ) throuh parallel transfer functions. In this subsection we show that the traditional methods of analyzin cascade control (based on series or parallel input/output relationships) should not be used when the primary and secondary processes are coupled, and the process is
open-loop unstable. Consider the series cascade control case, shown in Fiure. The state-space realization is: A series B @ C series a a a a a a a a a a a a B series B @ b C A C A where states? represent the secondary process and states? represent the primary process transfer functions, respectively. The coecients of the A series and B series matrices come from the oriinal open-loop A and B matrices. The rst (C series ) and second outputs (C series ) correspond to the dimensionless coolin and s, respectively. The statespace realization of the two open-loop transfer functions in series results in a state-space model with a non-minimal realization, because the primary and secondary processes are actually coupled. This state-space realization is not controllable and when the primary and secondary processes are unstable it is not stabilizable (Russo, 99). Consider the case of parallel cascade control (Fiure ) of the ed exothermic CSTR. The state-space realization is also not minimal when the parallel form is used. This conuration suffers from internal stability problems (Russo, 99) when the primary and secondary processes are unstable. Hence, the unstable modes cannot be stabilized and a process so-simulated will be unstable while the actual process may be stable. These non-minimal realizations can be a serious problem if one attempts to use a commercial simulation packae such as Simulink. The actual coupled process is best represented by Fiure, where it is shown clearly that the primary and secondary outputs are coupled throuh the process model... IMC-based Control System Desin Internal model control (IMC) (Morari and Zariou, 989) provides a clear framework for desin. One advantae to the IMC procedure is that it simplies tunin because a sinle tunin parameter,, is used as opposed to the three parameters (k c, I, D ) for a PID. Linear s are often not valid over a wide rane of operatin reions due to process nonlinearities. However, since we are concerned here with operation in a small reion where the stability characteristics do not chane, linear s enerally provide acceptable performance. It is well known that for open-loop unstable responses, the IMC must be implemented in conventional feedback form since the IMC form is internally unstable (Morari and Zariou, 989). c (s) q(s)? ~(s)q(s) () Here c (s) and ~(s) are the feedback and process model transfer functions, while q(s) is the IMC. The IMC is: q(s) ~?? (s)f(s) () where ~?? (s) is the invertible part of the process transfer function and f(s) is a lter transfer function which is added to make the IMC proper. Rivera et al. (98) demonstrate that for a lare number of sinle input sinle output (SISO) linear systems the IMC desin procedure yields PID-type s. There are many possible choices for the IMC lter structure; one possible structure for open-loop unstable systems is f(s) e ks k + e k? s k? + : : : + e s + e (s + ) n () Here k is the number of distinct poles in the rihthalf-plane (RHP), n is chosen to make the IMC q(s) proper, and is the lter time constant. The IMC lter is chosen such that its steady-state ain is unity; for open-loop unstable systems, the lter must also be unity at the k unstable poles (Morari and Zariou, 989) f(s) at s p i () where p i is the i th unstable pole of (s). Rotstein and Lewin (99) showed that the IMC desin procedure for some rst and second-order unstable transfer functions (without numerator dynamics) results in PID feedback s. The choice of the IMC lter f(s) and subsequently structure for the inner and outer-loops of the cascade control scheme depends on the number of riht-half-plane (RHP) zeros and poles... Cascade Control System Desin In subsection. we saw that traditional methods of analyzin cascade control (based on series and parallel input-output relationships) should not be used when the primary and secondary processes are coupled, and the process is open-loop unsta-
ble. In this case input-output relationships do not adequately describe the process dynamics, hence we need to incorporate the full state-space model in our control system desin procedure. Let us consider the ubiquitous case when a PI/P structure is used for the outer and inner loops, respectively. It is well-known that interal action in the outer-loop can be attained by appendin an additional state variable to the model: _x?x () The state feedback is iven by: u?kx () where the feedback ain row vector is: h K k kc k c k ck c c I i () As one can see, cascade control can be thouht of as partial state feedback. Note that the rst feedback ain is constrained to be zero with this structure (because we are assumin that the st state can not be measured or reconstructed with an observer). Hence arbitrary relocation of the \closed-loop" poles is not possible. One potential problem with this type of desin procedure is that a stabilizin feedback ain (K) may not exist, i.e. the constrained structure may not in eneral be able to stabilize the process. Understandin the eect of structure is the focus of current work. In some industrial applications the control structure is limited to P, PI, or PID. In order to develop the desin procedure, we use the followin state-space model, which assumes proportional control on the innerloop. _x Ax + Bx sp (7) y Cx (8) where A; B; C are iven by: A A? : : : B (9) B Bk c () C () where k c is the proportional ain on the \innerloop". The desin transfer function is: y(s) C? si? A? B xsp (s) (s) x sp (s) () The IMC-desin procedure can be used on (s), which in our case turns out to be minimum-phase and is of the followin form: (s) k ( n s + ) ( s + )( s + )(? u s + ) () For a perfect linear model and process, the closedloop response will be iven by: x (s) s + (s + ) x sp(s) () where is chosen to satisfy the IMC lter f(s) unity ain condition at the unstable pole... Control Case Studies The IMC-based feedback cascade is obtained usin equation with equations and. The resultin \outer-loop" is a PI with a double lead-la. Generally, one of the poles of this is fast hence a reduced-order PI with a lead-la can be used with very little performance deradation. We compare the robustness of standard noncascade and cascade feedback control with our new cascade desin procedure, applied to the nonlinear process. The \inner-loop" proportional ain is k c?: and the IMC lter time constant is : for both sets of simulations. The rst case study considers the inuence of the cascade control system desin. The parameters are Case conditions (Russo and Bequette; 99a, 99b); the nominal is x :7 (which is open-loop unstable). A lter ( F : ) is used to remove the overshoot which results from the ISE-optimal IMC desin procedure. Fiures and demonstrate that our cascade desin procedure is superior to the normal cascade control desin where the inner-loop desin is not incorporated in the \outer-loop" desin. The second case study deals with the eect of an unmeasured disturbance on the performance of cascade and noncascade s. The parameters are Case conditions (Russo and Bequette; 99a, 99b); the nominal is x : (which is open-loop unstable). An unmeasured disturbance in the dimensionless coolin feed (x f ) occurs at (x f chanes from - to -.). Fiures 7 and 8 demonstrate that the cascade control scheme does a better job of rejectin the unmeasured disturbance in the dimensionless coolin feed. An interestin point is that if we had analyzed a two-state CSTR model (nelectin coolin dynamics) one would think that it would be possible to arbitrarily detune the system, because Case is open-loop stable for the two-state CSTR model.
. CONCLUSION We have shown in this work that traditional methods of analyzin cascade control should not be used when the primary and secondary processes are coupled (throuh the state-space model) and the process is open-loop unstable. The parallels between traditional full state feedback control and cascade control were demonstrated. The eectiveness of the new cascade control procedure over conventional cascade and noncascade control was studied. APPENDIX Table. Dimensionless variables and parameters for the three-state CSTR model. x Ca x Caf E a RTf x Tc?Tf q Tf c T?T f Tf Qc Q (x ) exp( x + x ) (?H)C af UA C pt f C pq. REFERENCES V Q k e? q Q Q Bequette, B.W. (99). Nonlinear control of chemical processes: A review. Ind. En. Chem. Res.,, 9-. Q V t V V c Cp ccpc x f Caf Caf Kravaris, C. and J.C. Kantor (99). Geometric methods for nonlinear process control. -. Ind. En. Chem. Res., 9, 9-. Luyben, W.L (97). Parallel cascade control. Ind. En. Chem. Fund.,, -7. McLellan, P.J., T.J. Harris, and D.W. Bacon (99). Error trajectory descriptions of nonlinear desins. Chem. En. Sci,, 7-. Morari, M. and E. Zariou (989). Robust process control. Prentice Hall, Enlewood Clis, NJ. Rivera, D.E., M. Morari, and S. Skoestad (98). Internal model control.. PID Controller desin. Ind. En. Chem. Proc. Des. Dev.,, -. Rotstein, G.E. and D.R. Lewin (99). Simple PI and PID tunin for open-loop unstable systems. Ind. En. Chem. Res.,, 8-89. Russo, L.P. and B.W. Bequette (99a). Impact of process desin on the multiplicity behavior of a ed exothermic CSTR. AIChE J., in press. Russo, L.P. and B.W. Bequette (99b). Eect of process desin on the open-loop behavior of a ed exothermic CSTR. Comp. Chem. En, in press. Russo, L.P. (99). Ph.D. Thesis. Rensselaer Polytechnic Institute, Troy, NY, in proress. Sistu, P.B. and B.W. Bequette (99). Model predictive control of processes with input multiplicities. Chem. En. Sci., in press. Uppal, A., W.H. Ray, and A.B. Poore (97). On the dynamic behavior of continuous stirred tank s. Chem. En. Sci., 9, 97-98. x f Tf?Tf x T f f Table. Transfer function models. Tcf?Tf T f The transfer functions (s), (s), and (s) are: (s) k (c s+) (a s +a s +a s+) (s) k (c s+) (b s +b s+) (s) k (b s +b s+) (a s +a s +a s+) where k, k, and k and c - a are: k (x f? x s ) C k C k (x f? x s ) B c C b B B b B a A a A a C - are dened in terms of the CSTR parameters. B qxf x s (xf?xs) + (q + )? q (+ x s ) B (q+)qxf x s? q (x f?x s) (+ x s ) A B + ( + s ) A B + [( + s )B? ] [( + s )B? qx f x s ]
x sp c x sp c coolin coolant flowrate x x...8. Fi.. Series cascade control structure for a CSTR.. x sp c x sp c coolin x coolant flowrate x.. Includes inner-loop desin Doesn't include inner-loop desin Fi.. Comparison of the closed-loop dimensionless coolin owrate performance of the two cascade control strateies. Fi.. Parallel cascade control structure for a CSTR. x x Fi.. Relationship between the open-loop transfer functions. x x q sp sp c x x. x f(x,u) C + c + c - - x x C x........9.8 Cascade Control Noncascade Control Fi. 7. Comparison of the closed-loop dimensionless performance of the cascade and noncascade strateies to a disturbance in the coolin feed. 7 8 Fi.. Coupled Process Formulation usin a State Space Representation. The desin is based on a linear state-space model. Actual implementation is on a nonlinear process. x.8.8.8.78.7.7.7.7.8 Includes inner-loop desin Doesn't include inner-loop desin Fi.. Comparison of the closed-loop dimensionless performance of the two cascade control strateies...8.....8. Cascade Control Noncascade Control Fi. 8. Comparison of the closed-loop dimensionless coolin owrate performance of the cascade and noncascade strateies to a disturbance in the coolin feed. 7 8