Ind. Eng. Chem. Res. 1996,34, 209-215
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Online Optimization and Dual Composition Control of a Distillation Column Using a Nonlinear Controller Saibal Gangulyt and Deoki N. Saraf" Department of Chemical Engineering, Zndian Znstitute of Technology, Kanpur, 208 016, India
The nonlinear model predictive controller (NAMPC) reported earlier has been extensively tested on a pilot scale distillation column and its performance compared with other controllers such as generic model control (GMC) and proportional-integra1 (PI). The algorithm has also been extended for dual composition control of the distillation column. With the use of a transport phenomena based analytical model at the supervisory level, the disturbances in the secondary variables generate an automatic feedforward signal which significantly reduces their destabilizing effects. A comparison of NAMPC with GMC for a single input-single output system showed t h a t while the former is somewhat slow because of gradual change i n manipulated variable, the latter takes more drastic control action which may result in undershoots and overshoots. Dual composition control of distillation column using NAMPC was stable and smooth for both regulatory and servo problems for which PI was found to be oscillatory and unacceptably poor.
Introduction The term online optimization is used to indicate the continuous reevaluation of process parameters and alteration of operating conditions so that the economic productivity of the process is maximized subject to certain design and operational constraints. Most of the regulatory controllers use linear models because of the wide range of techniques available for identification and control action calculation with affordable computation time requirements. However, many chemical processes are highly nonlinear and involve distributed dynamics, a large number of variables, dynamic interactions, incomplete mixing, and nonideal thermodynamics. For such processes when operating conditions vary widely in the presence of severe disturbances or when large transitions in the state variables are desired, conventional proportional-integral-derivative(PID)controllers or linear model based controllers are often less than satisfactory. The shortcomings of the linearhnearized model based controllers have provided the impetus for the development of nonlinear model predictive controllers during the past decade. The nonlinear model predictive control (NMPC) of chemical process systems can be defined as an optimization problem to find the control vector trajectory that optimizes a performance objective over a future prediction horizon, subject to nonlinear dynamic state equations, operating limits, and variable bounds as constraints. This methodology is ideally suited for chemical processes that are best modeled by sets of nonlinear differential-algebraic equations, such as distillation. Single-level control is often found to be inadequate since optimization using highly nonlinear analytical models cannot be performed in the very short real times available between controller actions. Some limited details on multilayer structures with 1inearAinearized controllers are available in literature (Morari et al., 1980a,b; Stephanopoulos et al., 1980; Wright et al., 1987; Marques and Morari, 1988; Darby and White, 1988). Development of a hierarchical structure for online optimization and nonlinear analytical model predictive
* To whom correspondence should be addressed. E-mail: dnsaramitk.ernet.in. + Present address: Engineers India Limited, R&D Centre, Gurgaon, 122 001, India.
control (NAMPC) of a distillation column for a single input-single output (SISO) case has been reported recently (Gangulyet al., 1993; Ganguly and Saraf, 1992, 1993). In their scheme, they used a transport phenomena based nonlinear analytical model of the distillation column for online optimization at the supervisory level and a neighboring optimal control at the regulatory level. The present study is an extension of the earlier work which brings out (i) how the use of the analytical model provides automatic feedforward action t o compensate for any disturbance in the secondary varibles, (ii) a comparison between GMC, PI, and the present controller, and (iii) extension of the present controller to dual composition control of the distillation column for both servo and regulatory problems and its comparison with PI controllers. The experimental facility used in this study was same as that described by Ganguly and Saraf (1993). The top plate temperature was used to represent the distillate composition and the sixteenth-stage temperature a measure of the bottom product. However, in some of the experiments the sixth-stage temperature was used instead of the top plate to obtain enhanced sensitivity of the controlled variable to changes in the manipulated variable.
Hierarchical Online Optimization For practical implementation of hierarchical online optimization and control, the following steps are involved. (i) The first step was nonlinear parameter estimation and generation of a nominal control trajectory at the supervisory level. The semirigorous nonlinear model reported earlier (Ganguly and Saraf, 1992) was used for this purpose. The parameters of the nonlinear model were estimated online using the available measurements (Ganguly et al., 1993). The nominal control trajectory was obtained using the procedure discussed by Ganguly and Saraf (1993). This information was periodically passed on to the regulatory level. (ii) The second step was generation of a neighboring optimal control and its superposition with the nominal trajectory for its implementation through cascade controllers. Method of Neighboring Optimal Solution. While large disturbances or large setpoint changes are ac-
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counted for in the generation of the nominal trajectory, at the regulatory level control action is generated t o eliminate fast-occurring small disturbances. Here locally linearized dynamic models can be used in conjunction with a quadratic cost function. The neighboring optimal solution is then approximated as the sum of the nominal plus the linear optimal solution (Ganguly, 1993). For implementation of neighboring optimal control using microcomputers,the discrete formulation and the numerical solution of the discrete Riccati equations is essential. The continuous time domain deduction for this theory is available in literature (Bryson and Ho, 1975; Stengel, 1986). The discrete quadratic function can be written as N- 1
where
and control (u(k))vectors. 4 and r are the coefficients of transformation. T,(k) is linearized using
(9) The subscript i has been dropped in eqs 8 and 9 for simplicity. T,*is a function of u*, the nominal trajectory. The values of 4, r, Ts*, and aT&u are calculated a t the supervisory level (PCIAT 80386) and communicated to the regulatory level (PCKT 1) at a slow sampling rate (every 30-120 s). It is important to note that in the hierarchical scheme, the Au*(t) of a continuous time domain equation gets replaced by AT,*(k) in the discrete form. AT,*(k) represents the difference between Ts(k) and the last available T,* from the supervisory level. The discrete Riccati equation is solved to obtain AT,*(k), which is used to calculate Au*(k) from eq 9. The discrete form of the following linear perturbation equation provides the setpoint for the cascade controller residing in PCKT 2.
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u(k)= u*(k) Au*(k) = u*(k) - G ( k ) [ x ( k )- x*(k)l (10)
(3) The model equation is given by
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k ( k + l ) = @ k ( k ) OAu(k)
(4)
The step-by-step discrete Riccati solution algorithm is as follows: Step 1: From problem conditions identify 8,4, M , R, Q, S, and N . (For the present study N has been taken to be 8.) Step 2: Assign K(N) = S as the boundary condition. Step 3: For k = N - 1 to 0 calculate the following:
The calculated manipulated variables provide setpoints to the PI-type cascade controllers, which are required to eliminate boiler output pressure fluctuations, piston pump pulses, valve nonidealities, etc. The cascade controllers work a t a fast sampling rate of 1-2 s per sample. Since the measured signal may show large fluctuations and spikes, a filter is often required. The filtered signal is then used for generating the cascade controller outputs of reflux flow and steam pressure, respectively.
Pilot Plant Experiments: Results and Discussion
Single Variable Control: Performance of NAMPC. Several experiments were performed using the pilot plant facility described elsewhere (Gangulyand Saraf, 1993). Linear controls are often adequate for regulatory as well as servo problems for small setpoint G(k)= {R @TK(k+l)O}-l{fl OTK(k+l)@} (5) changes. However, nonlinear controls are better suited for tracking large transitions, where linearity assumpK(K) = Q @TK(k+l)@ - {MT OTK(k+l)@}TG(k) tions become invalid. Figure 1 shows the performance (6) of the hierarchical NAMPC when the setpoint of the sixth stage temperature was suddenly changed from Step 4: Using the calculated values of feedback gain 97.5 to 84 "C and later subjected to another small G(k)for k = 0, ... ,N - 1 and the known value of initial setpoint change from 84 to 81.5 "C. The transition was state h x ( O ) , the entire set of control and state vectors found to be reasonably smooth in both cases. When the can be computed using the feedback gain matrix (eq 7) nonlinear controller alone was used, its performance was less than satisfactory particularly near the final Au(k) = -G(K) A d k ) (7) steady state because of the delayed control action owing to large computation time. This caused the disturThus the entire control vector for the future horizon is bances to persist, which, in turn, led to inaccuracies in calculated, and the value in immediate future is used process parameter estimation, adding to deterioration for implementation by the controllers. of control (Figure 2). On the other hand, when neighFor practical implementation of neighboring optimal boring optimal control was used in conjunction with control, the values of M and R are experimentally tuned. NAMPC, faster actions were taken thereby eliminating The discrete version of the model equation should be the disturbances effectively (Ganguly and Saraf, 1993). used. The expression for the particular plate temperaEffect of Changes in Secondary Variables on ture, being controlled, can be written as Controller Performance. A nonlinear model predictive control (NMPC) is said t o be analytical (NAMPC) T(lt+l) = @ T ( k ) TT,(k) (8) when the embedded model is also able to predict, from first principles, the primary variable interactions and where T&k)is the steady state solution of the nonlinear model based on the current values of measured (T(k)) the effect of the secondary variable disturbances on the
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Figure 1. Distillation column control using NAMPC coupled with neighboring optimal control: (a)sixth-stage temperature vs time; (b)manipulated variable vs time.
formed. The feed flow rate to the distillation column was suddenly changed from 180 to 120 L/h. Also, simultaneously, the feed temperature controller on the feed preheater was switched off to freeze the steam flow to the preheater a t a value existing at that instant of time. The decrease in feed flow rate with constant steam flow to the feed preheater results in a sudden increase in the feed temperature, which is another secondary variable. The amounts of secondary disturbances are shown in Figure 3a,b. When the filtered secondary measurements were utilized in the model calculations, an automatic feedforward effect on the final control resulted in an improved controller performance. Figure 3c,d shows the controller performance with and without changes in secondary variables feedforwarded. As seen in Figure 3c, when the changes in the secondaryvariables were accounted for, through the analytical model, in terms of feedforward signals, the resultant controller performance was much better as compared to the case when these disturbances were overlooked. The manipulated variable in the two cases is compared in Figure 3d. Comparison of NAMPC with GMC and PI Control. The comparison between the performances of the SISO NAMPC controller with other types of controllers has been presented in this section. PID types are still the most popular class of controllers in the process industry with multivariable processes using such a controller configured for each loop. Simulation studies showed that the PI controller could track small load and servo changes satisfactorily. However, in the presence of process model mismatch or large disturbances, the system became unstable and the performance was not acceptable (Ganguly and Saraf, 1993). Figure 4 shows the performance of an experimentally tuned PI controller to changes in setpoint and load and the corresponding control action, respectively. Though the PI controller works reasonably well around the operating zone, where it is tuned, it needs retuning when the operating zone shifts. Large changes caused valve saturation, oscillations, and other operating problems. Generic Model Control. Generic model control is a class of advanced nonlinear controllers based on the algorithm developed by Lee and Sullivan (1988a) and adapted in the present form by Riggs (1990). It directly embeds a nonlinear process model in the control algorithm. The nonlinear model is described by transport phenomenon equations and is given by
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where y is the output variable, u represents the manipulated variable, d represents the measured disturbances, and p is the vector of the process parameters. Several application studies of GMC are available in the literature (Lee and Sullivan, 1988b; Rhinehart and Riggs, 1990; Riggs, 1990; Pandit et al., 1992). Using the model equation in conjunction with the GMC law, the equation for the manipulated variable is obtained as
where K I J and K I , are ~ the lumped tuning constants. For practical implementation of SISO GMC (SGMC), using digital computers, in the presence of both fastacting (stationary)and slow-acting (nonstationary) disturbances, a hierarchical structure has been used. The solution of the parameter estimation algorithm and the
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