Ind. Eng. Chem. Res. 1996, 35, 3547-3559
3547
Comparison of Model-Based and Conventional Control: A Summary of Experimental Results Hoshang Subawalla,† Venkat P. Paruchuri,‡ Amit Gupta,§ Hemant G. Pandit,§ and R. Russell Rhinehart* Department of Chemical Engineering, Texas Tech University, P.O. Box 43121, Lubbock, Texas 79409-3121
Nonlinear and linear model-based control (MBC) strategies including process model-based control (PMBC), model predictive control (MPC), internal model control (IMC), and advanced conventional control (ACC) have been experimentally compared for flow, temperature, pressure, and composition control. The test systems have nonlinear characteristics and include a pilot scale fluid flow and heat exchanger system, a commercial plasma reactor, and a laboratory scale distillation column. The evaluation criteria includes the effort required for model development and implementation, as well as conventional measures of controlled and manipulated variable activity. Introduction The inherent nonlinearity of chemical processes has been an ever present challenge for automatic control (Bequette, 1991); and a number of nonlinear modelbased control (MBC) strategies including process modelbased control (PMBC) (Rhinehart and Riggs, 1990), nonlinear internal model control (NLIMC) (Economou et al., 1986), nonlinear model predictive control (NLMPC) (Patwardhan et al., 1990), and nonlinear inferential control (NLIC) (Parrish and Brosilow, 1988) have been proposed and tested for such processes. The traditional IMC (Garcia and Morari, 1982) approach uses linear Laplace-transform-based models to represent the process, while NLIMC uses the process inverse as the controller model with a nonlinear search method to determine the desired manipulated variable action. PMBC and generic model control (GMC) (Lee and Sullivan, 1988; Lee, 1993) are reference system synthesis (RSS) techniques that use models developed from the laws of conservation, and integrate these with a closed loop control algorithm. A major advantage of these methods is that they can be tuned on-line with the same heuristic procedures that are practiced for conventional proportional integral (PI) control. Dynamic Matrix Control [DMC] (Culter and Ramaker, 1980) is a proprietary product of the Dynamic Matrix Control Corp. (DMCC) and is a widely used industrial linear MPC method. It uses a set of manipulated variable actions to minimize the sum of squared deviations between model predicted outputs and a desired process trajectory for a fixed number of control moves into the future (control horizon). Bequette (1991) reviews the advantages and disadvantages of these model-based control approaches but concludes that experimental applications of nonlinear control strategies and their comparison with the “best available linear systems are few.” While there is a growing number of reports on experimental applications of nonlinear controllers, there is no common basis for a unified comparison. Arkun et * Author to whom all correspondence should be addressed. † Current address: Department of Chemical Engineering, University of Texas at Austin, Austin, TX 78712-1062. ‡ Current address: Johnson Yokogawa Corp., 1201 West Crosby Rd., Carrollton, TX 75006-6905. § Current address: Aspen Technology, Inc., 9886 Bissonet, Houston, TX 77036.
S0888-5885(95)00686-5 CCC: $12.00
al. (1986) tested an IMC-based controller on a heat exchanger system. Mahuli et al. (1992, 1993) have employed PMBC for pH control, and Parekh et al. (1994) used fuzzy logic. Hutchinson and McAvoy (1973) developed a nonlinear, noninteracting control strategy for a multivariable pressurized stirred-tank heater system and compared it with time-optimal control. Marroquin and Luyben (1972) studied four different configurations of a temperature cascade control system for a batch reactor and verified the results of their simulations experimentally. Cheung and Luyben (1980) used the “wide-range” controller developed by Shunta and Fehervari (1976) for liquid level control. This controller is a controller whose gain and reset time are varied as functions of the error. Their simulation results were verified experimentally by Cheung (1978). Lee (1993) describes successful industrial applications of GMC to pH, distillation, particulate driers, cable jacketing, chemical reactor, and blast furnace control. However, the preceding applications do not provide a consistent basis for comparison of control strategies. In an attempt to assemble a unified and comprehensive comparison, this paper documents the results obtained from the installation and testing of model-based and conventional controllers on four experimental processes; fluid flow through pipes and devices, shell and tube heat exchange, reaction in a batch vessel, and distillation. These industrially important processes collectively represent the primary chemical process control problems (nonlinearity, nonstationary nature, unmeasured disturbance, multivariable interaction, and time delays and lags). The four control approaches were chosen either because they reflect standard industrial practice or because they are powerful nonlinear approaches which are relatively simple to implement. The advanced conventional controller (ACC) was a PI/PID controller with gain-scheduling, cascade, and feedforward control features. Both steady-state and dynamic models were employed for PMBC, while simple, linear first-orderplus-dead-time (FOPDT) transfer function models were used for IMC. Gain-scheduling features were added to the linear IMC and ACC controllers used in this work. Nonlinear process models used for PMBC account for disturbances that are likely to affect the system. While NLIMC includes a built-in feedforward option, the simple FOPDT models employed for the IMC controller in this work do not inherently include a feedforward feature. A standard lead-lag algorithm was hence © 1996 American Chemical Society
3548 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996
added for feedforward control in order to establish a feature equivalent comparison for both IMC and ACC. MPC integrates a linear dynamic model of how the manipulated and feedforward variables affect the process. Its control action accounts for present and future constraints. DMCC’s [DMC] was used. Controller performance was quantitatively compared using controlled variable performance criteria such as integral squared error (ISE) and integral absolute error (IAE). The magnitude of the change in manipulated variable is an important parameter that is often overlooked in controller performance comparisons. Manipulated variable movement was quantified using the relationship
Travel )
∑|oi - oi-1|
(1)
where oi is the controller output at the ith sampling interval. This parameter was used to compare the variation in manipulated variable for the different controllers. Other issues discussed in this comparative study include on-line model parameter adjustment and the engineering effort involved in the development of the models and their implementation. Computational issues such as the time required for calculating control action vis-a`-vis the sampling and control interval are also discussed. In general, MBC techniques share three common controller functions (Rhinehart, 1995), but the mechanics of performing these three functions vary considerably. The first MBC function is the use of a process model to predict the response of the process. The second common MBC function feeds back a correction that is based on the difference (residual) between the process and output. In IMC the filtered residual is used to bias the set point, while in MPC the difference is used to translate all future states. In PMBC the residual is used to adjust a model parameter which represents an unmeasured process characteristic (friction coefficient, tray efficiency, and ambient heat losses, for instance). The third common MBC function is the use of the model inverse to calculate the manipulated variable action which drives the process to meet a specified dynamic objective. In IMC the realizable model inverse is used to make the process output track a filtered biased set point, the reference trajectory. In the [DMC] MPC approach the approximate model is used within an optimization algorithm to balance concerns of manipulated variable movement and deviation from the set point while simultaneously avoiding constraints over the control horizon. In GMC and PMBC the model inverse is used to move the process to the set point in either a first- or a second-order manner. Experimental Systems Fluid-Flow and Heat Exchanger. This process consists of 1 in. nominal diameter BWG 16 copper tubes leading to the tube side of a heat exchanger and finally exiting at atmospheric conditions. The water flow is driven mainly by the header pressure of 55 psig (3.47 atm gauge) and partly by the potential head of 4.5 ft (1.4 m). The length of the flow line is about 100 ft (30 m) and consists of thirty-three elbows, two ball valves, six tees, five gate valves (normally full open), one rotameter, two orifices, one check valve, one flow control valve, twenty couplings, five thermowells, four pressure sensors, four sudden entrances, and three sudden exits (enclosed). The heat exchanger consists of eight BWG-
Figure 1. Inherent valve characteristics: (a, top) for the flow controller; (b, bottom) for the pressure controller.
16 U-tubes of 0.625 in. (0.016 m) nominal diameter and has a shell side outer diameter of 5 in. (0.127 m). The length of the exchanger is 3 ft (0.914 m). Steam, on the shell side, comes from a 45 psig (400 kPa) header. The control valves are air-to-open, with a modified equal percentage trim (water, Cv ) 9; steam, Cv ) 2.5). The primary nonlinearity of the fluid flow process is illustrated by the inherent valve characteristic shown in Figure 1a. The process gain changes by a factor of 4 over the range of flow rates employed (5-17 gpm). The heat exchanger process is characterized by transport delays and higher order dynamics as well as nonlinearity due to both flow-to-valve and temperature-to-flow relationships. Plasma Reactor System. A plasma is a lowpressure (1-1000 mTorr), neutral, ionized gas consisting of an equal number of free positive and negative charges. Gas mixtures of CF4 and O2 were employed in this work to etch silicon and silicon dioxide surfaces. The experimental system consists of a commercial single slice plasma etch reactor, a radio frequency (RF) generator, and a vacuum pump. The process chamber has a diameter of 8 in. (20 cm) and consists of parallel plate electrodes spaced 1 in. (2.5 cm) apart. The reactor volume is 2750 cm3 with the glow volume, the volume between the two electrodes, comprising approximately 60% of the total. The gas enters through a “shower head” mesh at the top of the chamber and exits through an opening between the bottom plate and the lower electrode. The removal rate of the gases is controlled by a stepping motor powered butterfly type exhaust valve. The throttle valve controls chamber pressure by meter-
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3549
ing the flow to the vacuum pump in response to signals received by the software controlled stepping motor. The inherent valve characteristic, which is shown in Figure 1b, is clearly nonlinear. This characteristic was experimentally determined by varying both the valve position and the inlet flow rate and by measuring the pressure in the reactor. Additional sources of nonlinearity include the dependence of plasma phase dissociation on pressure, power, and feed gas composition. Process gain at higher pressures (800 mTorr is 6 times higher than that at lower pressures (200 mTorr)). The dynamics of this process are exceptionally fast with typical transient times being 5-6 s. The process gain changed by a factor of 7 for a 50% change in valve stem position. Distillation Column. This study was performed on a laboratory-scale distillation system. The unit consists of a 3 in. (0.0762 m) inside diameter, thermally insulated six-sieve tray glass column with a total condenser and an electrically heated reboiler. The condenser is a glass and stainless steel shell-and-tube type heat exchanger connected to a glass overhead receiver. The reboiler is a thermally insulated stainless steel tank with an approximate volume of 14 L, partially filled with glass marbles to give a liquid capacity of 9 L. It is equipped with a 2.4 kW sheathed bayonet type heating element. The column sieve trays are 1/8 in. (0.0031 m) thick with thirty-six 0.15 in. (0.0038 m) diameter holes on each and with weirs and downcomers adjusted such that the liquid height on the tray is 1/4 in. (0.0063 m). The feed and the reflux to the column are preheated using catridge type immersion preheaters rated at 250 W. Component and overall material balance closures at steady state are within (5%. The methanol-water column, which was operated with the reflux drum at atmospheric pressure (about 93.1 kPa), displayed nonideal thermodynamic behavior with the relative volatility changing from 2 to 6 along the length of the column. Other process characteristics include nonlinear multivariable interactions, unintentional disturbances, and disparate dynamics. The impurity changes, at the bottom of the column, in response to positive and negative reboiler heat input changes were in the ratio 5:1. The process time constant changed by a factor of 2 throughout the operating range. The experimentally determined relative gain array (RGA) has a value of 1.5 for the top composition-reflux rate element. Day-to-day ambient condition changes affect column behavior, and product compositions often differed by 2% (in absolute terms) with identical feed flow rates, feed compositions, reflux, and reboiler heat input. Advanced Conventional Control Advanced conventional control was applied to the column in an inferential, cascade strategy with PI controllers. The top and bottom compositions were accurately inferred from the temperatures on stage 6 (the top tray) and the reboiler, respectively. Per Luyben (1986), trays 5 and 1 were selected to manipulate the reflux flow and reboiler heat duty. The choice of these trays was based on the fact that the temperature on these trays is most sensitive to composition. The top composition inferred from the temperature in the condenser was compared with the composition set point, and the difference was used by the master controller to send a temperature set point signal to the tray 5 temperature controller. The slave controller then adjusts the reflux flow rate by comparing the temperature set point and the temperature on tray 5. A similar
strategy was employed to manipulate reboiler heat duty. Additionally, a standard lead-lag algorithm was used for feedforward control of feed-flow rate and composition disturbances. The transfer function models for these cases were developed from time-series models obtained using DMCC software. The heat exchanger ACC strategy included a master temperature controller which cascaded a steam flow set point to the slave flow controller. Gain scheduling was added to the flow controller to compensate for the equalpercentage characteristic of the flow control valve. Tube fluid flow rate was used for feedforward action. Nonlinear Model-Based Controllers Nonlinear control strategies combine a nonlinear representation of the process with a single-time-step control law. Consider a process which is approximately represented by a model given by
dy/dt ) f(y,u,d,p)
(2)
where y is the vector of outputs, u is the vector of manipulated variables, d is the vector of disturbances, and p is the vector of model parameters. A very simple control law is obtained by assuming that the process has to be moved from its current state, y, to a desired state, ysp, within a time period, τ. This specifies the desired value of the derivative in eq 2 and results in
(ysp - y)/τ ) K1(ysp - y) ) f(y,u,s,p)
(3)
Given ysp, y, d, and p, eq 3 can be used to determine u. For convenience K1 replaces 1/τ. Since no model is perfect, there would always be some steady-state offset. Feedback, in the PMBC approach, corrects the model by adjusting a coefficient to remove steady-state process model mismatch. In the GMC approach, an integral correction term is added to eq 3 to compensate for process-model mismatch.
∫
f(y,u,d,p) ) K1(ysp - y) + K2 (ysp - y) dt
(4)
In either GMC or PMBC, the tuning parameter matrix, K1, determines the speed of the process response. Large values for the elements of K1, which correspond to smaller values of τ, would cause the process to be “pushed” to its new set point faster. The values in matrix K2 influence the rate of removal of steady-state offset due to process-model mismatch. When this formulation is used in conjunction with a steady-state model, a first-order response is preferentially assumed and a temporary steady-state target is calculated to give
∫
yss ) y + K1τp(ysp - y) + K2τp (ysp - y) dt (5) The nonlinear steady-state model, f(y,u,d,p) ) 0 is then used to determine u. Controller Models and Model Adjustment The controller models include nonlinear phenomenological models developed from first principles, FOPDT models, and time series models. The phenomenological models were used in conjunction with PMBC and GMC and were developed for all four processes. The FOPDT models were employed with IMC and ACC and were developed for the plasma reactor, fluid flow, and heat exchanger processes. A time series model was used for MPC for the distillation process.
3550 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 Table 1. Nominal FOPDT Parameters for All Processes process type
process gain (Kp)
process time constant (τp)
process dead time (θp)
flow temperature pressure top composn/reflux top composn/boil-up bottom composn/reflux bottom composn/boil-up
0.2 gpm/% 0.4 °C/% -43 mTorr/deg +0.0074 mole fraction/(mL/min) -0.0061 mole fraction/(mL/min) +0.0016 mole fraction/(mL/min) -0.0079 mole fraction/(mL/min)
3s 13.5 s 1.5 s 20 min 45 min 55 min 60 min
1s 11 s 0.2 s 0 min 0 min 10 min 10 min
FOPDT Modeling. The FOPDT models were experimentally generated from open loop step responses. Each of the four processes is nonlinear, noisy, and subject to uncontrolled environmental disturbances. Hence, the experimental procedure involved averaging the responses of four input steps about the operating point. The FOPDT model parameters were obtained by using a 1/3-2/3 parametric method similar to the 0.280.62 method suggested by Smith and Corripio (1985). FOPDT models were not updated on-line. Model parameter values are listed in Table 1. Time Series Modeling. The time series model for the distillation column was obtained from the Dynamic Matrix Identification [DMI] program under license from DMCC, and their [DMC] controller was used for MPC. The model-predictive controller was configured for four independent variables (feed flow rate, feed composition, reflux flow rate, and reboiler heat input) and two dependent variables (methanol composition in the top and bottom products). This model employed sixty coefficients for each of the eight responses. A steadystate time of 180 min was identified on the basis of the response of the slowest settling variable (bottom composition). The process tests required for the identification of the model followed the DMCC protocol and were performed over a 96 h period for step changes in the four independent variables. The time series models were not updated on-line. Phenomenological Modeling: Fluid Flow. A process model was obtained for the incompressible turbulent fluid flow system from a dynamic overall mechanical energy balance. On simplification this yields
(
)
d dv ) a∆P + bh - c + v2 dt f(x)2
(6)
where v is the average fluid velocity, ∆P is the static pressure drop of the system, h is the elevation difference across the system, and f(x) is the inherent valve characteristic. The coefficients a, b, and d are dependent on the transport properties of the fluid and on the physical dimensions of the flow system. This model assumes an ideal plug flow dependence on velocity (squared dependence) and ignores valve dynamics, calibration errors, and signal-time mismatch. On combining the nonlinear process model with the PMBC control objective of eq 3, we have
(
K1(v - vsp) + a∆P + bh - c +
)
d v2 ) 0 f(x)2
(7)
where K1 is the lone tuning parameter. The required valve position is then calculated from the inherent valve characteristic, f(x) (Figure 1a). The inherent valve characteristic was obtained experimentally by measuring the flow rate and valve assembly pressure drop. The coefficient c represents two friction loss mechanisms of the system: pipe friction losses which are flow dependent and device losses which are relatively inde-
pendent of flow rate. It is a parameter known with the least certainty, and it changes in response to manual valve positions, fluid path, screen blinding, etc. Hence, it was chosen to adjust the model to remove processmodel mismatch. Model parameter adjustment was done at every control interval on assuming the attainment of steady state. The steady-state assumption worked well in this case because the velocity attained 80% of its steady-state value, within the control interval of one second, in response to ideal step changes in valve stem position. The model parameter, c, was calculated from measured values of the pressure drop, flow rate, and valve stem position using the rearranged model equation given by
c)
d a∆P bh + 2 2 v v f(x)2
(8)
The actual valve stem position was used in the feedback parameter adjustment relationship given by eq 8. This is an unmeasured quantity and lags behind the controller output due to the dynamics of the pneumatic system. Open loop step tests indicated that the valve stem response to a step controller output was approximately first order with a 2 s time constant and hence a calculated lagged (filtered) value of the valve stem position, xf, given by
xfi ) 0.4xi + 0.6xfi-1
(9)
was used. In eq 9 xi is the hypothesized valve position (controller output), xfi is the filtered valve position used in the calculations at the ith time step, and xfi-1 is the filtered valve position used at the previous time step. Phenomenological Modeling: Heat Exchanger. The heat exchanger model incorporates a simple steadystate energy balance.
msΛs ) mcp(Tout - Tin) + HL
(10)
In eq 10, ms is the shell-side steam flow rate, m is the cooling water flow rate, Cp is the specific heat capacity of the tube-side cooling water, Λs is the steam heat of vaporization, Tin and Tout are the tube-side temperatures at the inlet and outlet of the heat exchanger, and HL represents ambient heat losses. In actuality, HL is a lumped parameter which accounts for all model and measurement errors, including assumptions such as constant heat capacity and heat of vaporization, saturated steam at the entrance of the exchanger, no subcooling of the condensate, and calibration errors on measured flow rates and temperatures. HL is chosen as the adjustable parameter. Combining the controller model with the steady-state formulation of the PMBC control law, we have
ms ) (mcp(Tss - Tin) + HL)/Λs where
(11a)
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3551
Tss ) T + K11(Tsp - T)
(11b)
Tss represents the steady-state target temperature, Tsp is the set point temperature, and K11 is the master controller tuning parameter. The steam flow rate calculated from eq 11a was then cascaded to a slave controller which calculated the position of the final control element. The slave controller used a steadystate overall fluid mechanical energy balance,
ah + b∆P -
d 2 v )0 f(x)2
(12)
which relates the steam velocity (v) to the valve position (x). ∆P is the pressure drop across the system, h is the elevation difference, and f(x) is the isolated valve characteristic. Friction losses for steam in the short header are assumed negligible. The pressure drop and elevation difference remain relatively constant, and hence eq 12 can be simplified to
v ) f(x)/b1
(13)
where b1 is the adjustable parameter that accommodates calibration errors and process-model mismatch. Equation 13, when combined with the PMBC formulation, gives
f(x) ) b1[v + K21(vsp - v)]
(14)
where vsp is the velocity set point calculated from the steam flow rate and K21 is the slave controller tuning parameter. Model parameter adjustment was carried out using the steady-state relationship given by
HL ) msΛs - mcp(Tout - Tin)
(15)
However, this equation led to erroneous HL values during transient conditions as heat exchanger outlet temperature does not respond to changes in steam flow rate instantaneously and ignores heat exchanger dynamics. The lagged and delayed steam flow rate was then employed in eq 15. Phenomenological Modeling: Plasma Reactor. The controller model was developed from a dynamic mole balance and can be expressed in terms of the controlled variable (pressure) by using the ideal gas law, to give
dP QiPβ QoP ) dt V V
(16)
where P is the pressure in the reactor, Qi is the volumetric flow rate at the inlet of the reactor, Qo is the flow rate at the outlet of the reactor, V is the volume of the reactor, and β is the adjustable model parameter representing the net mole balance effects of plasma phase dissociation and reaction. On combining the controller model with the PMBC control law, we have
K1(P - Psp) +
QiPβ QoP )0 V V
(17)
the model parameter β is known with the least certainty, and, while it ostensibly describes net dissociation, it also incorporates process-model mismatch such as incorrect valve calibration, measurement errors, and leaks. This process did not attain steady conditions within 0.25 s, the control interval. Consequently, a
Figure 2. Process-model mismatch, distillation: (a, top) top composition; (b, bottom) bottom product composition (0) local values: (O) global values.
dynamic parametrization method known as incremental parametrization on-line (IMPOL) (Rhinehart and Riggs, 1991) was employed to update the model. The IMPOL method aims to find that value of the model parameter that reduces the difference between the model predicted output variable and the actual output variable to zero at each parametrization step. A single step adjustment to the model (Newton’s method), when combined with the controller model, leads to the dynamic form of the parametrization relationship given by
β ) βold +
V(P - Pm) QiP
(18)
where β is the current value of the adjustable parameter, βold is the previous value of the adjustable parameter, P is the measured value of pressure, and Pm is the pressure predicted by the model. Phenomenological Modeling: Distillation Column. A steady-state tray-to-tray approximate model based on component mass and energy balances and thermodynamic equilibrium was employed in conjunction with the GMC law along with the IMPOL method for model adjustment. This model employed two adjustable parameters. An adjustable tray efficiency was used to account for unknown tray efficiency, internal reflux, and other sources of process-model mismatch. The second adjustable parameter was a bias to correct the vapor boil-up rate for both ambient heat losses and errors in the empirical voltage and current measurements. Parts a and b of Figure 2 illustrate model accuracy by comparing model-predicted and actual top and bottom compositions for a wide operating range. Ideally model-predicted and measured compositions should be equal, and points should fall exactly on the diagonal line
3552 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 Table 2. Tuning Parameter Values for Temperature Controllers controller type PMBC controller
IMC controller advanced conventional controller
controller parameter
parameter value
steam flow (slave) controller gain, K1 temp (master) controller gain, K2 filter on the friction param, a1 filter on the heat loss param, a2 steam flow controller filter const, τf1 temp controller filter const, τf2 steam flow controller gain, Fc1 steam flow controller reset time, τi1 temp controller gain, Kc2 temp controller reset time, τi2 temp controller derivative time, τd2
1.2 1.8 s-1 1.0 0.8 2.5 s 17 s 450 %/(kg/s) 2.85 s-1 0.0008 (kg/s)/°C 1.143 s-1 0.019 s
Table 3. Test Conditions for Distillation Control controller function
case no.
changes made
set point tracking
1
disturbance rejection disturbance rejection
2 3
disturbance rejection
4
top: methanol composition changed from 88 to 93% bottom: methanol composition changed from 1.4 to 4.2% feed flow disturbance: 220 to 180 mol/h feed composition disturbance: methanol composition changed from 20 to 35% (relative change ) +55%) feed composition disturbance: methanol composition changed from 30 to 20% (relative change ) -40%)
in this case. If there were no process-model mismatch, then measurement noise would cause data to be scattered randomly about this line. The data points indicated by circles represent model predictions with average adjustable parameter values. Evidence of processmodel mismatch is apparent from the bias and skew shown by the envelope of points. In contrast data points indicated by squares fall exactly on the diagonal. These points represent model predictions when tray efficiency and boil-up bias (adjustable parameters) are adjusted at each control interval. The two sets of data illustrate that local adjustment of the model makes the model true and useful for both servo and regulatory control. The required values of the manipulated variables were calculated from eqs 19 and 20, which are straightforward extensions of the GMC law to multivariable systems, where yss and xss represent steady-state com-
∫
yss ) y + K1,1τp,1(ysp - y) + K1,2τp,1 (ysp - y) dt (19)
∫
xss ) x + K2,1τp,2(xsp - x) + K2,2τp,2 (xsp - x) dt (20) position targets that the process aims to achieve. These equations specify yss and xss, which are in conjunction with the steady-state model to specify the reflux rate and vapor boil-up rate values. The product Kijτpi can be considered as as a lumped tuning parameter. This formulation is equivalent to the output linearization technique of Calvet and Arkun (1987). Controller Test Conditions and Implementation Issues Fluid Flow. A uniform test protocol was used to evaluate controller performance on each process. For the flow process, about 75 s after starting the controller, the set point was changed from 5 to 15 gpm. The process was returned to its original set point approximately 175 s after the initial set point change. The gain at the higher flow conditions was about four times that at the lower flow conditions. About 350 s after the controller was started, process disturbances were initiated by manually closing a bypass and forcing the fluid to go through a restricted path. This bypass was reopened 100 s later.
Both controllers had a sampling and control interval of 1 s. The PI controller was tuned for a quarteramplitude-damped (QAD) response at a flow of 5 gpm using a controller gain of 3%/gpm and an integral time of 4.3 s. A value of 5 s-1 was employed for K1, the lone tuning parameter for the model-based controller. Heat Exchanger. The sampling interval and the control period employed for all three controllers was 1 s. Set point changes were initiated at approximately 50, 450, 850, 1050, and 1400 s after start-up. Temperature set points were varied between 30 and 55 °C. Disturbances were initiated by manually opening and closing a water flow control valve by 20% (from 50 to 70%) and vice versa. These disturbances were created at 250, 400, 600, and 700 s after start-up. Both PMBC and IMC employed two tuning parameters, one each for the slave and master controllers. ACC requires three tuning parameters for the PID temperature controller and two tuning parameters for the slave PI controller. Tuning parameter values are listed in Table 2. Plasma Reactor. Controller evaluation was divided into two main stages. The initial stage included the disturbance caused by the ignition of the plasma. The second stage was used to test the controller in both servo and regulatory modes. A sampling and control interval of 0.25 s was employed in order to ensure a minimum of twenty control actions during the transient. The pressure set point was varied from 200 to 800 mTorr and the flow rate from 75 to 125 cm3(STP) min-1. A value of 1 s-1 was employed for the tuning constant K1 for the nonlinear process model-based controller for both servo and regulatory modes. The IMC filter time constant was initially chosen to be half that of the process time constant, or 0.75 s. Subsequently, it was reduced to a value of 0.5 s in order to make control action more aggressive. The PI controller was heuristically tuned (Smith and Corripio, 1985) for a quarter amplitude damped response using a gain, Kc ) -0.018°/ mTorr and an integral time τ1 ) 1.5 s. Valve movement was constrained to less than 1°/control action because of the rate of the servo motor. Distillation Column. The test conditions for the three control strategies are shown in Table 3. Controller execution frequency was fixed at 3 min. Parametrization of the nonlinear process model required the identification of steady-state conditions. It was as-
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3553 Table 4. PI Tuning Coefficients for Distillation Control controller parameter gain (Kc)
reset time (τi)
control loop identity
original
detuned
master controller (top) master controller (bottom) slave controller (top) slave controller (bottom) master controller (top) master controller (bottom) slave controller (top) slave controller (bottom)
454.5 720 5.0 0.9 600 1440 480 1920
180 700 1.8 0.8 2100 2400 2100 2400
sumed that steady-state conditions were attained when standard deviations of reboiler, top tray, feed tray, feed, and reflux temperatures were less than 0.5 °F and when the component material balance closure error was less than 5 g-mol %. Global values of the parameters (average values over the entire operating range and history) were used as initial values. The GMC tuning parameters used were K11 ) 1.6 h-1, K12 ) 2.9 h-1, K21 ) 1.2 h-1 and K22 ) 2.6 h-1. Initial MPC tuning parameters allowed smaller manipulated variable movements and used the same weights (“equal concern” error, scaling factors) for both the top and bottom compositions. Subsequent fine-tuning on-line gave more aggressive control and improved both servo and regulatory performance. The move suppression factors were reduced to make bigger manipulated variable movements, and the weighing for the bottom composition was increased for tighter control. The initial tuning parameters for ACC control were determine from the “first-order-response” method described previously. As both controllers were too aggressive, they were subsequently detuned by operator discretion. Both gain and reset rate of the top controller were changed, while only the reset rate of the bottom controller was altered. These values are shown in Table 4. DMCC procedures were used to determine initial tuning values for the MPC. It was then fine tuned online by operator discretion. For ACC, velocity-mode algorithms were used for the PI controllers for the top and bottom compositions. Feedforward correction was added to reflux flow rate and reboiler heat duty. Since this process had a very small dead time, as compared to either the sampling interval or the error in measuring the dead time, conventional PI controller tuning procedures such as Ziegler-Nichols, Cohen-Coon, and BLT (Luyben, 1986) could not be employed. A simple heuristic tuning procedure for first-order systems based on the method described by Smith and Corripio (1985) was used to determine controller gain and the integral time. The temperature-composition inference relationships were determined to be
Figure 3. PMBC flow control response to set point changes and disturbance.
Figure 4. ACC flow control response to set point changes and disturbance. Table 5. Comparison of Integral Square Errors of PMBC and PI Controllers for Flow Control time period (s)
system event
60-125 125-200 200-250 250-350 350-450
change in set point (5-15 gpm) at a set point of 15 gpm change in set point (15 to 5 gpm) at a set point of 5 gpm disturbance by closing/opening the manual valve disturbance by closing/opening the manual valve
450-550
x ) 3.814004 - 0.072428(T) + 0.000343(T2) y ) 2.667568 - 0.025919(T) where T is measured in degrees centigrade. Results Fluid Flow. ISE values are recorded in Table 5, and the results are shown in Figures 3 and 4. During the initial set point change both controllers overshot the required set point (15 gpm), but PMBC was less oscillatory than PI and hence returned to its set point faster. The same trend was observed when the set point was changed to 5 gpm; however, PMBC did not display any overshoot and was able to position the valve at the
°C/mole fraction °C/mole fraction counts/°C counts/°C s s s s
PMBC PI (m2/s) (m2/s) 48.3 0.5
100.9 2.7
0.4 2.9
8.1 8.1
23.1
22.8
desired position almost immediately. The regulatory performance of both controllers was essentially equivalent. It must be noted that the flow disturbance created by closing the bypass valve is an unmeasured and unmodeled disturbance and is hence not contained in the controller model given by eq 7. During this unmeasured disturbance the model parameter c moved from its nominal value of about 1 ft-1 to about 6.5 ft-1. Heat Exchanger. Figures 5-10 show the test results, and Table 6 presents the performance comparisons. Both model-based controllers respond to changes in the temperature set point immediately, while the PI controller displays a small overshoot in most cases. In general the IMC slave flow controller had minimal
3554 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996
Figure 5. Master PMBC temperature control response to set point change and disturbance.
Figure 8. Slave PMBC temperature response to set point change and disturbance.
Figure 6. Master IMC temperature response to set point change and disturbance.
Figure 9. Slave IMC temperature response to set point change and disturbance.
Figure 7. Master ACC temperature response to set point change and disturbance.
Figure 10. Slave ACC temperature response to set point change and disturbance.
overshoot, but the steam flow rate oscillated about the new set point. The response of the PI controller was slower than that of the model-based controllers. The servo performance of the model-based controllers was essentially equivalent over almost the entire region of operation. The PI controller performed as well as the model-based controllers when the set point was changed from 30 to 55 °C because the PI controller was tuned at 55 °C. The disturbance rejection characteristics of the nonlinear process model-based controller were superior to that of the IMC and PI controllers. In the case of the IMC and PI controllers the feedforward controller immediately reacts to changes in the cold water flow rate by either increasing or decreasing the steam flow rate as required. However, this controller often overcompensates or undercompensates, depending on the region of operation. The performance of all three controllers in the regulatory mode was equivalent for temperatures near 55 °C, this being the nominal tem-
perature at which the feedforward controller models were developed. Plasma Reactor. In the servo mode (Figures 1113) both model-based controllers were able to move the process to its new set point almost immediately, while the PI controller displayed considerable overshoot. The internal model controller pushed the process to its new set point faster at the expense of a greater variation in valve position. The region of operation near 200 mTorr is an extremely nonlinear region of operation, and gain changes by a factor of 6 in this region. The model parameter β identifies this change in gain and adjusts the model accordingly. The valve moved to its new position almost immediately. The IMC controller exhibited a first-order type of response, while the PI controller exhibited an extremely sluggish response at these low pressures. Servo performance results in this nonlinear region of operation clearly indicated that the built-in gain adjustment mechanism, as is present in
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3555 Table 6. ISE Values for Temperature Controllers time period (s)
system event
PMBC (C2‚s)
IMC (C2‚s)
ACC (C2‚s)
30-180 180-305 305-460 460-600 600-715 715-825 825-1035 1035-1150 1150-1200 1200-1365 1365-1470
change in set point (55-30 °C) disturbance by decreasing water flow rate disturbance by increasing water flow rate change in set point (30-55 °C) disturbance by decreasing water flow rate disturbance by increasing water flow rate change in set point (55-30 °C) change in set point (30-40 °C) disturbance by increasing water flow rate disturbance by decreasing water flow rate change in set point (40-50 °C)
9478.4 11.9 6.9 9167.0 40.9 69.0 10424.0 1091.1 42.4 25.8 1303.6
11725.0 436.8 841.0 12197.7 36.8 65.5 11392.0 1923.8 700.4 290.3 2054.6
14852.2 894.5 1357.3 9833.3 42.0 69.7 14399.0 1943.3 95.9 136.5 1618.9
Figure 11. PMBC pressure control response to set point changes.
Figure 14. PMBC pressure control response to disturbances
Figure 12. PI pressure control response to set point changes.
Figure 15. PI pressure control response to disturbances
Figure 13. IMC pressure control response to set point changes.
Figure 16. IMC pressure control response to disturbances
PMBC, is superior to empirical additions such as “tackon” gain scheduling mechanisms. Regulatory control results are shown in Figures 1416. The disturbance caused by the ignition of the plasma was efficiently rejected by both model-based controllers. However, the model parameter β was able to respond to the creation of new species, and hence this disturbance was rejected faster by the nonlinear modelbased controller. In the regulatory mode the controlled variable performance of the gain scheduled IMC con-
troller with built-in feedforward action was as good or better than PMBC. It should be noted that the IMC controller was tuned for conditions at 500 mTorr. Further, model parameters for the FOPDT feedforward model were also determined by making positive and negative step changes from 500 mTorr. While local performance of a linear IMC controller may be better in a specific region of operation for which it is tuned, global performance of the PMBC controller is superior
3556 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 Table 7. IAE and Valve Travel for Reactor Pressure Controllers system event
IAE (mTorr‚s)
valve travel (deg of arc)
start-up, power disturbance, and set point changes start-up, power disturbance, and increase in flow rate start-up, power disturbance, and decrease in flow rate start-up, power disturbance, and set point changes start-up, power disturbance, and increase in flow rate start-up, power disturbance, and decrease in flow rate start-up, power disturbance, and set point changes start-up, power disturbance, and increase in flow rate start-up, power disturbance, and decrease in flow rate start-up, power disturbance, and set point changes start-up, power disturbance, and increase in flow rate start-up, power disturbance, and decrease in flow rate start-up, power disturbance, and set point changes start-up, power disturbance, and increase in flow rate start-up, power disturbance, and decrease in flow rate
8481 5194 5335 8771 5185 5099 10325 5731 5783 8930 5635 5750 9192 5789 5535
201 133 79 279 128 129 153 73 68 189 68 63 286 121 144
controller identification nonlinear PMBC (dynamic model) IMC with gain-scheduling and feedforward action PI control (position mode) nonlinear PMBC (steady-state model) IMC without gain-scheduling and feedforward action
because the model adjusts itself for different regions of operation. Moreover, as seen from Figures 14-16, control valve (manipulated variable) movement is greatest for the IMC controller. Thus, good controlled variable performance is obtained at the expense of greater wear-andtear of the valve. The PMBC and PI controllers show a much smaller amount of valve travel. Other controller algorithms developed and tested included PMBC in conjunction with a steady-state model and IMC without gain-scheduling and feedforward action. Table 7 summarizes controller performance for all of the pressure controllers. Distillation Column. In the servo mode (Figures 17-19) both the model-based controllers (PMBC and MPC) reached the new set point in a shorter time than that required by the ACC. No significant overshoot was observed for any controller. However, the PI controller showed a small amount of oscillation in the top composition at the new set points (it was still slightly aggressive). While both of the model-based controllers exhibited smooth manipulated variable responses, the advanced conventional controller had a noiselike character. The sudden drop in composition that is present in some of the results shown in Figures 17-19 was attributed to weeping on the top tray and was observed frequently. These changes were confirmed by visual inspection of the top tray. In the regulatory mode all three controllers did a fairly good job of maintaining the top and bottom compositions at their set points. The manipulated variable responses were similar to those observed in the servo mode with the advanced PI controller displaying a noisy response. A detailed quantitative comparison of controller performance is given in Table 8. IAE and ISE were recorded for 2 h after a set point change or a disturbance was introduced. Discussion SISO Systems. The results for the SISO systems displayed some common performance characteristics. Servo performance of the model-based controllers (PMBC and IMC) was superior to that of the ACC for all three systems. A change in set point causes the manipulated variable to move to its new position almost immediately in the case of the model-based controllers, while ACC often either oscillates or is sluggish. The performance of the model-based controllers was generally equivalent except in nonlinear regions of operation, where the inherent gain-scheduling characteristics of the nonlinear model-based controller considerably improved set point tracking. However, when the gain-scheduling
Figure 17. . PMBC distillation control to set point changes.
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3557
Figure 18. [DMC] distillation control to set point changes.
Figure 19. ACC distillation control to set point changes.
characteristics were incorporated in IMC and ACC, the servo control performance was equivalent to nonlinear PMBC. Though useful, gain scheduling is a tack-on feature to compensate for the inherent nonlinearity of the process that linear model-based controlers are unable to represent. We feel that a controller which unifies all compensations will be easier to implement and maintain. The use of steady-state models for the heat exchanger and flow controller model adjustment required an empirical dynamic compensation. While steady-state phenomenological modeling has the advantage of sim-
plicity over dynamic modeling, dynamic compensation is required at times. It was not required with the distillation steady-state model, and we suspect that the reason is grounded in the fact that the dynamics were first-order-ish with similiar time constants. The regulatory performance of all three SISO controllers was essentially equivalent, with temperature control for the heat exchanger being the sole case where the nonlinear model-based controller performed better than the other two controllers. In certain cases the disturbance rejection characteristics of the IMC controller were better than that of the nonlinear model-based
3558 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 Table 8. ISE, IAE, and IAT Values for Distillation Controllers set point tracking controller/control loop identity ACC (top) ACC (bottom) DMC (top) DMC (bottom) PMBC (top) PMBC (bottom)
disturbance rejection
overshoot (mole fract.)
Rise time (min)
IAE (min)
travel (counts)
ISE (min)
max dev (mole fract.)
travel (counts)
0.013 0.003 0.012 0.004 0.010 0.002
28 87 17 24 31 70
1.188 0.911 1.104 0.523 0.880 1.070
V ) 1498 R ) 3160 V ) 1066 R ) 940 V ) 647 R ) 914
0.00709 0.000214 0.00905 0.000488 0.00762 0.000426
0.0337 0.00383 0.0227 0.00407 0.0143 0.00517
V ) 1519 R ) 1964 V ) 544 R ) 1134 V ) 693 R ) 1487
controller. This result was normally observed in the region of operation where the feedforward model for the disturbance was developed. The nonlinear model used for PMBC incorporates feedforward action, as the term which represents the disturbance is included in the model equation(s) used. This framework is more global and is effective for any modeled disturbance that affects the system. Additionally, for IMC and ACC separate feedforward models have to be developed for each disturbance that affects the system. Valve travel quantified for the plasma reactor system indicated that ACC had minimal variations in manipulated variable. In general it was found that the variation in the manipulated variable was greatest for IMC. The difference between the model-predicted and process output is used to bias the set point in the case of IMC. In PMBC this difference is used to adjust the model parameter. When the process is at its set point ACC takes no action as the difference between the process value and set point value is negligible. However, even at set point, other process measurements can lead to PMBC model adjustment. A simple variable that could be employed as an indicator of control performance is the product of either IAE or ISE and the valve travel. This product would account for both controlled and manipulated variable performance. When this quantity was employed for the plasma reactor, we found that ACC gave as good a performance as model-based control. However, this result was not observed for either the fluid-flow or heat exchange results. For the fluid flow system it was observed that the model parameter, c, which represented frictional losses in the system increased with decreasing flow rate and vice versa. This result closely mirrors what happens in the actual process. An increase in this parameter could indicate the possibility of an obstruction in flow such as a plugged filter or screen or inadvertently closed valves. A decrease could indicate a failed filter or a sudden line opening. A similar observation was made for the model parameter, β, in the plasma reactor system. This parameter which represents net dissociation of all species in the plasma showed a sharp increase when power was initially coupled into the plasma. This increase reflects the sudden creation of new species in the reactor. In each case, the adjustable model parameter was chosen to have a phenomenological significance, and it can be used for such supervisory functions as process diagnostics and identifying constraints. PI controllers require two tuning parameters, while model-based controllers normally require only one, the filter factor for IMC and the parameter K1 for PMBC, both of which affect the dynamics of the response. Model parametrization in the case of nonlinear process model-based control normally makes the integral correction unnecessary as the process-model mismatch is removed. However, for a SISO system, in addition to the filter factor, IMC requires three model parameters for each of the feedback and feedforward controller
models. This is in addition to the parameters required for gain-scheduling. The ACC controller also requires parameters for feedforward action and gain-scheduling. We found that the engineering and experimental effort involved in developing these empirical models is equivalent to that required to develop a simple nonlinear process model from first principles. Additionally, gainscheduling is limited to applications where process dynamics depend on known variables. Often, therefore, engineering process knowledge is required to choose empirical models. It seems that if used, process knowledge is most completely utilized in the PMBC or GMC strategies. IMC is likely the easiest model-based controller to implement in commercial equipment as the lead, lag, delay, and summation functions are all preprogrammed. For SISO systems, PMBC strategies calculate the required manipulated action explicitly by sequentially solving a set of equations. These equations are all algebraic equations if a steady-state model is employed. When a dynamic model is employed the model predicted output variable is obtained by integration with the rest of the equations remaining unchanged. The manipulated variable is related to the output variable using a cascaded approach, as has been done for the three systems shown here, and does not have to appear in the equation representing the model, as suggested by Henson et al. (1989). MIMO Systems. The performance of both PMBC and MPC is equivalent for both servo and regulatory control for this system. The performance of ACC also compares well with that of the model-based controllers, especially in the case of disturbance rejection. MPC is more aggressive with the fastest rise time and settling time among the three controllers. The open loop settling time for both top and bottom compositions is about 3-4 h. The settling times for the top composition for MPC, PMBC, and ACC controllers are approximately 30, 40, and 60 min, respectively, while those for the bottom composition are 75, 80, and 95 min, respectively. Both PMBC and MPC have inherent decoupling characteristics. PMBC provides nonlinear decoupling. ACC displayed the greatest (least desirable) amount of variation in the manipulated variable for both servo and regulatory modes, on account of inherent coupling. MPC strategies such as [DMC] are most useful for processes that have constraints and/or ill-behaved dynamics (inverse responses or dead time). The simple PMBC or ACC formulations of this work cannot handle processes which have large dead times or inverse responses, as the GMC law is a single-time-step law. We experienced that constraint handling with nonlinear model-based controllers required extensive computational time and an expert system supervisor to ensure feasible initial guesses for the decision variables. While model-based controllers have demonstrated economic advantage over ACC approaches, they require greater process knowledge, engineering effort, and additional hardware and software. The engineering
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3559
effort required to implement PMBC on the distillation system was greater than that required for either MPC or ACC. Conclusions Experimental control results on four nonlinear systems shows that nonlinear model-based controllers give better servo performance than advanced conventional controllers. Perfect modeling is unnecessary because feedback of some form is continously updating the model for the local conditions. It seems that gain prediction is more important to control than state prediction. Accurate gain scheduling of the linear model-based controllers makes their servo performance equivalent to PMBC. Little distinction can be made between regulatory performance of model-based and advanced conventional controllers at conditions for which the advanced conventional controller was tuned and when features such as gain-scheduling and feedforward action were included in order to ensure a functionally equivalent comparison between the controllers. IMC was computationally efficient and easy to implement. In the case of MIMO systems model development and implementation for PMBC required more knowledge and effort. MPC model development was considerably simplified by use of the commercial software. IMC with feedforward action performs well even for mildly nonlinear SISO loops for disturbance rejection. For nonlinear SISO processes GMC and PMBC outperform the linear controllers. GMC, PMBC, and MPC may be used for plantwide multivariable control where the economic benefits are larger. For the latter case, MPC methods have proven to be successful for a variety of applications, and are especially efficient at handling constraints and ill-behaved dynamics. Acknowledgment The authors would like to thank Texas Instruments, Inc., for equipment donation and the following members of the Texas Tech Process Control and Optimization Consortium for their technical guidance and financial support: Albemarle Corp., Amoco Oil Co., Arco Exploration and Production, Conoco Inc., Diamond Shamrock, Dow Chemical Co., Dynamic Matrix Control Corp., Exxon Co. USA, Hyprotech Inc., Johnson Yokogawa Corp., Phillips Petroleum Co., and Setpoint Inc. A part of this work was supported by the Texas Advanced Technology Program under Grant No. 003644-058. Literature Cited Arkun, Y.; Hollet, J.; Canney, W. M.; Morari, M. Experimental Study of Internal Model Control. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 102. Bequette, B. W. Nonlinear Control of Chemical Processes: A Review. Ind. Eng. Chem. Res. 1991, 30, 1391. Calvet, J. P.; Arkun, Y. Feedback and Feedforward Linearization of Nonlinear Systems and Its Implementation Using Internal Model Control. Ind. Eng. Chem. Res. 1987, 27, 1822. Cheung T. F. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1978. Cheung, T. F.; Luyben, W. L. Nonlinear and Nonconventional Liquid Level Controllers. Ind. Eng. Chem. Fundam. 1980, 19, 93.
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Received for review November 13, 1995 Revised manuscript received May 30, 1996 Accepted May 30, 1996X IE950686H
X Abstract published in Advance ACS Abstracts, July 15, 1996.
