1. UFRGS
UNIVERSIDADE FEDERAL
DO RIO GRANDE DO SUL
Escola de Engenharia
Departamento de Engenharia Química
NONLINEAR MODEL PREDICTIVE CONTROL USING SUCCESSIVE LINEARIZATION APPROACH
PSE 091
R.G. Duraiski, J.O. Trierweiler and A.R. Secchi
{rduraisk, jorge,
Three different predictive controllers: the linear MPC Prett (1982), the extended DMC Peterson (1992) and the algorithm LLT were tested by a setpoint change in CB from 0.92 M to 1.12 M which is a little higher then maximum attainable value of 1.09 M (cf. Fig. 1). The simulation starting point is the steady-state corresponding to f=20 h-1, CAin=5.1 mol/L, and T=134.14 °C. According to Figure 1, the gain at the initial point is positive. Intuitively, the expected is the concentration of B will increase with increasing the feed flow rate. Indeed, all three controllers start increasing the control action (cf. Figures 2, 3, and 4), but the linear controller and the extended DMC cannot compensate the change of the gain sign which occurs at the maximum CB – value and, therefore, the closed loop turns unstable.
Fig, 1: CB vs. f corresponding to the steady-state solutions
Fig. 2: Simulation of the Extended DMC for a setpoint change in CB
Fig. 3: Simulation of the linear MPC for a setpoint change in CB
Fig. 4: Simulation of the LLT algorithm for a setpoint change in cB
1. Introduction
Nowadays, the industrial processes develop a quick progress, which requires new techniques for process control. This changes in industrial environment turns nonlinear predictive controllers more and more useful and necessary. This controller type, differently of the conventional controllers, determines the control actions in a more complex way. The movements applied to the manipulated variables are obtained by optimizing an objective function of control goals using an internal model to predict the future system outputs produced by the optimized inputs which are the optimization variables of the optimization problem.
The predictive controllers needs an internal model of the system to predict future outputs. A linear model is the most simple and common way to describe the dynamic behavior of a system, but it is known that a physical system rarely behaves totally in this way. To solve the control problem of the process with strong nonlinearities only linear models cannot completely describe the system behaviors. In this case, nonlinear models must be used. Unfortunately, the optimization problem in this case becomes a challenge problem. This paper presents a novel algorithm, which can efficiently work with these optimization problems based on nonlinear models.
2. Algorithm description
The LLT algorithm (Duraiski 2001) consists of the following iterative calculation steps:
1) The first solution is based on a linearized model at the current operating conditions. Using this trajectory it is possible to simulate the nonlinear model which is used to calculate a sequence of linear models that will be used in the next iteration step.
2) With the sequence of linearized models on the trajectory a new control action is calculated.
3) Based on the new control action, it is possible to determine a new set of linearized models in the same way as it is done in the first step. Then, this set of models is used in the next iteration step.
4) The steps 2 and 3 are sequentially carried out until the algorithm converges, i.e., when the last two trajectories do not differ too much to each other considering a given norm.
3.Van de Vusse Benchmark Control Problem
The Van de Vusse Benchmark Problem has been considered by several researchers as a benchmark problem for nonlinear process control algorithms (Engell and Klatt, 1993; Chen et al., 1995). The reactant A is feed into the reactor with concentration CAin and temperature Tin. Fin is the inlet volumetric flow through the reactor. The concentrations of substances A, B, C, and D are CA, CB, CC , and CD respectively. The reaction schem is given by these parallel reactions: and . The reaction is carried out in a isotherm CST-reactor. The model of the system for the isotherm case reduced to the following equations:
where f = Fin/Vr is the inverse of the residence time.
The product of interest of this reaction is component B. The component C and D are undesired subproducts. The plot of the concentration cB ( Figure 1 ) reveals an interesting behavior of the system. The reactor exhibits a change of the sign of the static gain at the peak of the reactor yield (i.e., where the concentration CB achieves its maximum value), and displays nonminimum phase behavior for operation to the left of this peak and minimum-phase behavior for operating points on the right.
4.Case Study: The Quadruple-Tank Process Process (Johansson 2000)
where
Ai : cross-section area of Tank i; Ri : outlet flow coefficients; hi : water level of Tank i; Fi: manipulated inlet flowrates; x1 and x2: valve distribution flow factors 0  xi  1
Fig. 5: Schematic diagram of the quadruple-tank process. The water levels in Tank 1 and Tank 2 are controlled by the flow rates F1 and F2.
4.3. Operating Points
Table 1: Definition of the Operating Points
.4. Simulations results
Fig. 6: LLT controller applied in the quadruple tank model when a disturbance in x1 carries the system from a minimum phase to a non minimum phase operating region.
Fig. 7: DMC controller applied on the quadruple tank model when a disturbance carries the system from a minimum phase to a non minimum phase operating point
5. Conclusions
As shown in this work, the LLT controller was effective in the control of highly nonlinear systems. It was the case of the Van de Vusse reactor where the change in the gain sign of the system turns its control difficult. In this case it was possible to control the system, even in the point of gain sign change, that is the most critical area. Besides, although the set point were a non feasible point, the controller shows the capacity of keeping the system stable in the nearest point which was feasible for the system.
References
Chen, H.; Kremling, A.; Allgöwer, F.; (1995) Nonlinear Predictive Control of a Benchmark CSTR, Proc. of 3rd ECC, Rome, Italy, pp. 
Duraiski, R.. G.; (2001) Controle Preditivo Não Linear Utilizando Linearizações ao Longo da Trajetória ; M. Sc. Thesis, Universidade Federal do Rio Grande do Sul, Brasil.
Engell, S.; Klatt, K.-U.; (1993) Nonlinear Control of a Non-Minimum-Phase CSTR, Proc. of American Control Conference, Los Angeles, pp
Johansson, K. H; (2000) The Quadruple-Ttank Process: A Multivariable Laboratory Process with an Ajustable Zero; IEEE Transactions on Control Systems Technology; v8; nº3; 456;
Peterson, T., Hernandez, E., Arkun, Y., Schork, F. J.; (1992) Nonlinear DMC Algoritm and its Application to a Semi-Batch Polimerization Reactor; Chemical Engineering Science; v.47, no 4; pp
Prett; D. M., Ramaker, B. L., Cutler, C.R.; (1982) Dynamic Matrix Control Method; US Patent Docment nº
Trierweiler, J. O., Farina, L. A., Duraiski, R. G.; (2001) RPN Tuning Strategy for Model Predictive Control; Dynamic Control Process Symposium.
Acknowledgment
UFRGS and OPP Química S/A.