An improved simplified model predictive control algorithm and its application to a continuous fermenter

An improved simplified model predictive control algorithm and its application to a continuous fermenter

W. H. Kwong

Department of Chemical Engineering , Federal University of São Carlos,

P. O. Box 676, CEP 13565-905, Phone [phone], Ext 228,

Fax [phone], São Carlos - SP, Brazil

E-mail: [e-mail]

(Received: March 26, 1999 ; Accepted: November 26, 1999)

Abstract - The development of a new simplified model predictive control algorithm has been proposed in this work. The algorithm is developed within the framework of internal model control, and it is easy to understanding and implement. Simulation results for a continuous fermenter, which show that the proposed control algorithm is robust for moderate variations in plant parameters, are presented. The algorithm shows a good performance for setpoint tracking.

Keywords: Model predictive control, simplified model predictive control, internal model control, multivariable systems, controllers.

INTRODUCTION

The past 20 years have witnessed the growing and widespread development of model predictive control (MPC). The general strategy of MPC algorithms is to use a model to predict the output in the future and to minimize the difference between this predicted output and that desired by computing the appropriate control actions. The early algorithms were developed independently in the 1970s: one, which is now referred to as Model Algorithmic Control (MAC) in France (Richalet et al., 1978) and the other, which was given the name Dynamic Matrix Control (DMC), in the United States (Cutler and Ramaker, 1979). At the time, both were developed on the basis of heuristics. The two methods use a similar approach, but they utilize different optimization routines for implementation. They have been used successfully in a large number of industrial processes because they explicitly handle constraints. Garcia and Morari (1982, 1985) published a series of papers wherein they showed that MAC and DMC are two of several possible manifestations of a single control strategy, which they referred to as Internal Model Control (IMC). Later, Arulalan and Deshpande (1987) published an algorithm denominated Simplified Model Predictive Control (SMPC) that retained many of the desirable characteristics of the model predictive control algorithms such as DMC and MAC, but was much easier to design and implement. A simple yet effective implementation of SMPC that significantly reduces the required computational effort and ensures zero offset was described by Kew et al. (1990). Machado et al. (1992) reported the implementation of SMPC and SMPC modified by Kew et al. (1990) in a multivariable pilot plant consisting of a mixing tank. Rubião and Lima (1990) proposed a rule for tuning SMPC controllers in multivariable systems where the matrix of gains is substituted by a constant multiplying of the inverse of the process gain matrix, reducing the problem of determining the tuning constants in the matrix of gains to a problem of determining a simple tuning parameter. Of the MPC algorithms, DMC is by far the most popular. A survey of model predictive control, including applied and theoretical papers, was conducted by Garcia et al. (1989).

Despite the vast number of studies on MPC, there are only a few on SMPC. The only focus of the present paper is the development of an alternative SMPC algorithm within the framework of IMC. First we present the SMPC algorithm proposed by Arulalan and Deshpande (1987). Then the alternative algorithm (IMC-SMPC) is presented and its closed-loop properties are examined. Finally, we demonstrate that IMC-SMPC can give a better closed-loop performance than SMPC.

THE SMPC METHOD

Single-Loop Systems

The block diagram of a typical sampled data control system is shown in Figure 1. The z-transform representation is assumed.

The method assumes that the system is open-loop stable and utilizes the specification that the control algorithm produces a setpoint response that is at least as good as the normalized open-loop response for design purposes. Thus,

(1)

Substituting for in the closed-loop transfer function of the system followed by some algebraic manipulations gives

(2)

or

(3)

Now the closed-loop response may be speeded up by replacing term 1/K by a tuning constant, a, giving

(4)

At this point, a z-domain impulse response model may be substituted for G(z) and the equation may be inverted into the time domain or alternatively an impulse response model may be employed, giving

(5)

Inversion of Eq. 5 gives the SMPC control law in the time domain as

(6)

Tuning constant a is the only tuning parameter in the algorithm which can be determined by simulation so it satisfies a suitable optimization criteria. A commonly used criteria is the IAE (Integral Absolute Error). Arulalan and Deshpande (1987) recommend that be selected from between 1 and ¥ . The inclusion of the constant a modifies the feedback controller to

(7)

Vaidya and Deshpande (1988) demonstrated that the SMPC algorithm, as given by Eq. 7, guarantees offset free performance in the presence of modeling errors.

Single-Loop SMPC With Robustness Filter

To enhance the robustness of the system in the presence of modeling errors, the value of the tuning parameter a may be simply reduced (Vaidya and Deshpande, 1988). In this way, this reduction of the parameter value will provide smooth changes in the manipulated variable. Alternatively, the robustness of the system can be ensured by inserting a first-order filter in the feedback path (Figure 2), and constraints can be accommodated during the optimization step (Arulalan and Deshpande, 1986, 1987). The filter constant can function as an adjustable parameter that addresses the degree of plant-model mismatch.

Single-Loop SMPC in the Framework of IMC

The Simplified Model Predictive Control algorithm can be examined within the framework of IMC. Figure 3 shows an IMC control structure. The block diagrams for conventional feedback control (Figure 1) and IMC are identical if feedback controller and IMC controller satisfy the relation

(8)

or Gc(z)

(9)

Substitution of from Eq. 2 into Eq. 8 yields the IMC controller

(10)

Eq. 10 is only true if . Note that the IMC controller is the inverse of process gain. Robustness in the presence of modeling errors is achieved by inserting a filter (Figure 4). The filter is usually the simple exponential filter

(11)

where b is the adjustable filter constant. Thus,

(12)

Substitution of from Eq. 12 into Eq. 9 results in

(13)

Rearrangement of Eq. 13 yields the result

(14)

Using the same argument as that for SMPC to speed up the response, a constant, a, is introduced into Eq. 14, giving

(15)

Substitution of an impulse response model for the process and the filter (Eq. 11) results in

(16)

or equivalently in the time domain

(17)

Eq. 17 is the final form of the IMC-SMPC algorithm. Constants a and b are the tuning constants of the algorithm. They are determined by a suitable off-line optimization procedure that minimizes a performance index, i.e., IAE. If tuning constant a is selected to be , then b is the single tuning parameter of this algorithm.

Steady State Offset with the Single-Loop IMC-SMPC Algorithm

It will be shown that the algorithm guarantees offset free performance in the presence of modeling errors. Consider the closed-loop transfer function of the system to load changes

(18)

The inclusion of constant a modifies the feedback controller to

(19)

Thus, substituting Eq. 19 into Eq. 18 yields

(20)

and rearrangement gives

(21)

Now

(22)

(23)

(24)

where the steady state gain of the actual process is K and the same gain of the model is . Eq. 24 shows that the algorithm guarantees offset free performance in the presence of modeling errors.

Stability Properties of the Single-Loop IMC-SMPC Algorithm

Assume (model is exact). If the control system is expressed in the form of the usual sampled data control system shown in Figure 1, then the requirement for loop stability is that the roots of

(25)

must lie inside the unit circle in the z-plane. There is no requirement, as far as loop stability is concerned, that must be open-loop stable.

The IMC-SMPC control algorithm can be examined within the framework of IMC. is given by Eq. 19, and using the relationship (Eq. 8) from the equivalence between the two structures, we obtain

(26)

Therefore observing Eqs. 26 and 20 we cocluded that the loop stability requirements demand that be open-loop stable (Garcia and Morari, 1982).

Multivariable Systems

The results developed in the previous section for single loop can be extended to a multiple-input-multiple output system. Using the same arguments as those for the SISO case, the multivariable feedback SMPC controller is given by

(27)

or

(28)

For convenience let’s make

(29)

Then Eq. 29 becomes

(30)

The algorithm can be speeded up by introducing a matrix of gains, a, in Eq. 30 to give

(31)

or equivalently in the time domain employing impulse response coefficients for a 2´2 system,

(32)

Eq. 32 is the final form of the multivariable SMPC algorithm. Constants , , and are the tuning constants of the algorithm. They are determined by a suitable off-line optimization procedure that minimizes the performance index. An alternative way of determining the tuning constants in the matrix of gains is given by Rubião and Lima (1990). They proposed that matrix a be substituted by a constant multiplying the inverse of the process gain matrix, thereby reducing the problem of determining the tuning constants to a problem of determining a single tuning parameter.

Multivariable SMPC within the Framework of IMC

In the presence of model inaccuracies, the robustness

of the IMC loop is improved by introducing a filter, F(z). A diagonal filter of the exponential type

(33)

can guarantee robustness for arbitrarily large modeling errors (Garcia and Morari, 1985). F(z) can also be used to make the input actions less severe and to shape the output response. The equivalence of Eq. 15 for a multivariable system is

(34)

In the case of a system with two inputs and two outputs, the final form of the algorithm is

The choice for tuning constant is a= K. This reduces the problem of determining the tuning constants to a problem of tuning a single parameter, b.

SIMULATION STUDY: CONTINUOUS FERMENTER

In this section, the performance of the proposed algorithm (IMC-SMPC) is compared with that of SMPC and SMPC with a robustness filter (SMPC-RF). A SISO and a MIMO example for a continuous fermenter are considered. The first example consists in the control of productivity under uncertain parameters. This problem has been used by Henson and Seborg (1992) to study the performance of nonlinear control strategies. In the second example, the control of a two input-two output system, where the control objective is to move the system from a given initial condition to the optimum operating point for the continuous fermenter, is studied. This problem was considered by Patwardhan and Madhavan (1993) for testing a nonlinear model predictive control algorithm using second-order model approximation.

Example 1. Single-Loop System

We assume that the fermenter has a constant volume, its contents are well-mixed and the feed is sterile. This simple microbial culture involves a single biomass, X, growing on a single substrate, S, and yielding a single product, P. The bioprocess is supposed to be continuous with a dilution rate, D, and an input substrate concentration, S_f. The mathematical dynamic model of the process is formed by the following three balance equations associated with X, S and P (Henson and Seborg, 1992), respectively:

(36)

(37)

(38)

Using a substrate and product inhibition model, specific growth rate m is given as

(39)

Also, am + b is the specific product formation rate and Y_X/S is the yield of cell mass with respect to the limiting substrate. For most of the continuous fermentation processes, the control objective is to maximize productivity per unit time. If the biomass and substrate are of a negligible value compared to the product, productivity Q can be defined as the amount of product cells produced per unit of time

(40)

The optimum steady-state operating point can be obtained by maximizing the productivity rate using the dilution rate and feed substrate concentration as independent variables with steady-state model equations as constraints. In the simulation the nominal model parameters and operating conditions near the optimum operating point are those given by Henson and Seborg (1992). Parameters Y_X/S and m_m are especially sensitive to changes in operating conditions. Thus from a process control perspective, these two model parameters can be viewed as unmeasured disturbances because they may exhibit significant time-varying behavior. For the simulations shown in this paper, we examine the control of productivity using dilution rate as the manipulated input. Note that for this pairing, the process exhibits unusual dynamic behavior for a step change in dilution rate, as can be observed in Figure 5.

Figures 6-8 show the effect of tuning parameters a and b on the regulatory performance of the controllers for a -0.02 h^-1 unmeasured step disturbance in the maximum growth rate in terms of the IAE index in productivity. It can be observed that for the IMC-SMPC controller the range of values of acceptable performance for these parameters is larger than it is for the other two controllers, indicating that the tuning task can be easier. Note that for the IMC-SMPC controller b = 1 corresponds to a nonfiltering case.

Initially, to maintain the same base of comparison between these controllers, only one parameter will be adjusted and the other will be fixed for the SMPC-RF and IMC-SMPC controllers. In Figure 9 we compared the regulatory performance for the SMPC, SMPC-RF and IMC-SMPC controllers, manipulating the dilution rate for a -0.02 h^-1 unmeasured step disturbance in the maximum growth rate. A tuning parameter of a = 0.318 was chosen for the SMPC controller, which corresponds to a minimum IAE of 0.5825. For the SMPC-RF controller, tuning constant a was 0.273 (= 1/K) and 0.932, which corresponds to a minimum IAE of 0.6778 was chosen as the robustness filter constant, b. Finally, for the IMC-SMPC controller, tuning constant a was also maintained at 0.273 and 0.962, which corresponds to a minimum IAE of 0.2614 was chosen as the filter constant b. The SMPC controller provides a very oscillatory response while the SMPC-RF controller yields a much smoother response. Both controllers yield large deviations from the setpoint, while the IMC-SMPC provides a smooth response and a very effective attenuation of the disturbance. Figure 10 gives the productivity response of the SMPC-RF and IMC-SMPC controllers when the IAE index is minimized with respect to both parameters. This result demonstrates that the IMC-SMPC controller can exhibit good robustness to modeling errors.

Figures 11-13 show practically the same analyses in respect to the tuning parameters for the case of a -0.04 g/g unmeasured step disturbance in the yield of cell mass.

The regulatory performance of the SMPC, SMPC-RF and IMC-SMPC controllers, manipulating the dilution rate for a -0.04 g/g unmeasured step disturbance in the yield of cell mass, is shown in Figure 14 where only one tuning parameter is used to minimize the IAE index. The value of 0.346, which corresponds to a minimum IAE of 1.9532 was chosen as the tuning constant a of the SMPC controller. For the SMPC-RF controller, the tuning constant was a =0.346 and the robustness filter constant was b = 0.897, which corresponds to a minimum IAE of 1.9531. In this case, the use of a tuning constant, a = 1/K, for the SMPC-RF controller yields an unstable response. Conversely, the IMC-SMPC controller with a = 0.273 and filter constant b = 0.959, which corresponds to the minimum IAE=0.8603, provides a smooth response. Now, both parameters are used to minimize the IAE index, and the productivity response of the SMPC-RF and IMC-SMPC controllers is given by Figure 15. Tables 1 and 2 summarize the simulation results for the example.

Example 2. Multivariable System

The optimum conditions for the nominal parameters are presented in Table 3. The control objective is to move the system from the given initial condition to the optimum operating point.

Neither the SMPC nor the SMPC-RF controller with matrix of gains a = k was able to track the step changes in the setpoint yielding unstable responses. It was necessary to include a reference model to reduce further the magnitude of the control action for step changes in the setpoint (Ogunnaike and Ray, 1994). The behavior of these controllers is shown in Figure 16, where the reference model was selected as a first-order exponential trajectory with parameter l = 0.7. For the SMPC-RF controller, the robustness filter used was

For step changes in the setpoint, the IMC-SMPC controller with a = k provides a tracking behavior similar to that of the other two controllers, but without the use of a reference model, as shown in Figure 17, which clearly demonstrates the advantage of the proposed algorithm.

Table 4 summarizes the simulation results for the example. The b values are those that minimize the IAE index.

CONCLUSIONS

An improved SMPC (IMC-SMPC) algorithm is presented. It is derived from the IMC concept, and it has been shown to be a simple and efficient algorithm for multivariable control. Simulation results for a continuous fermenter demonstrate the superiority of the IMC-SMPC compared with the original SMPC. IMC-SMPC maintains a high level of closed-loop performance in both servo and regulatory problems, despite appreciable variations in process dynamics and strong interactions between the manipulated variables. In this work we do not attempt to study the influence of tuning constant a on closed-loop performance for the controllers in the multivariable case. In the example, a was fixed as the inverse of the process gain matrix. It is clear that to improve the performance of the controller, it is necessary to consider the tuning constant in optimization of the IAE criterion.

NOMENCLATURE

a Yield parameter for the product b Yield parameter for the product C Controlled variable C Controlled variable vector D Dilution rate E Error E Error vector F Filter F Filter matrix G Process open-loop transfer function Feedback controller IMC controller Model open-loop transfer function G Matrix of process open-loop transfer function Feedback controller matrix Matrix of model open-loop transfer function h_i Impulse response coefficient I Identity matrix k Element of process gain inverse matrix k Inverse of process gain matrix K Steady state gain in the process K_i Substrate inhibition constant K_m Substrate saturation constant Steady state gain in the model K Process gain matrix L Load disturbance L Load disturbance vector M Manipulated variable M Manipulated variable vector N Number of samples P Product concentration P_m Product saturation constant P Closed-loop transfer function matrix Q Productivity R Reference R Reference vector S Substrate concentration S_f Feed substrate concentration t Time X Biomass concentration Y_X/S Yield of cell mass

Greek letters

a Constant (Eq. 11) b Filter constant l Reference model parameter

m

Specific growth rate m_m Maximum specific growth

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Publication Dates

• Publication in this collection
  06 July 2000
• Date of issue
  June 2000

History

• Received
  26 Mar 1999
• Accepted
  26 Nov 1999