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UFRGS
NONLINEAR MODEL PREDICTIVE
CONTROL USING SUCCESSIVE
LINEARIZATION APPROACH
UNIVERSIDADE FEDERAL
DO RIO GRANDE DO SUL
Escola de Engenharia
Departamento de Engenharia Química
www.enq.ufrgs.br
PSE 091
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.
R.G. Duraiski, J.O. Trierweiler and A.R. Secchi
{rduraisk, jorge, arge}@enq.ufrgs.br
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.
4.Case Study: The Quadruple-Tank Process
4.1. 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
(1-x1).F1
(1-x2).F2
h3
h4
x1.F1
F1
h1
x2.F2
T1
T2
F2
h2
V1
V2
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
Fig, 1: CB vs. f corresponding to the steady-state solutions
Variables
h1, h2 [cm]
h3, h4 [cm]
MOP
12.26, 12.78
1.63, 1.41
NMOP
12.44, 13.16
4.73, 4.99
F1,F2 [cm3/s]
x1, x2 [-]
9.99 , 10.05
0.7, 0.6
9.89, 10.36
0.43, 0.34
.4. Simulations results
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.
Fig. 2: Simulation of the Extended DMC for a setpoint change in CB
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.
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:

dC A
 f  C Ain C A  k1C A  k3C A 2
dt
Fig. 3: Simulation of the linear MPC for a setpoint change in CB

dC B
 f  C B k1C A k 2 C B 
dt
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.
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
Fig. 4: Simulation of the LLT algorithm for a setpoint change in cB
References
Chen, H.; Kremling, A.; Allgöwer, F.; (1995) Nonlinear Predictive Control of a Benchmark CSTR, Proc.
of 3rd ECC, Rome, Italy, pp. 3247-3252.
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. 2041-2045.
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 737-753.
Prett; D. M., Ramaker, B. L., Cutler, C.R.; (1982) Dynamic Matrix Control Method; US Patent Docment
nº 4.349.869.
Trierweiler, J. O., Farina, L. A., Duraiski, R. G.; (2001) RPN Tuning Strategy for Model Predictive
Control; Dynamic Control Process Symposium.
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.
Acknowledgment
UFRGS and OPP Química S/A.