Norm based approaches for Automatic tuning of Model based Predictive Control
Systematic methods and tools for managing the complexity
Process Control (T4-8)
Keywords: Integrated Design, Model Based Predictive Control, Multi Objective Optimization , Mixed Integer Non linear Optimization
Motivation and objectives
The public view concerning wastewater treatment these days is fairly positive. The EU Urban Water Directive (91/271/EC) adopted years ago, together with the newly adopted EU Water Framework Directive (2000/60/EC), define stringent requirements for urban wastewater treatment and a time frame for the step-wise implementation by the member countries. The application of the directive has lead to the construction of new plants and redesigns of the existing ones with the aim of reducing as much as possible the environmental impact. The associated costs to these actions have been very huge.
In this sense, the norm imposes several objectives, to be achieved by engineers that should be pointed out. First, the design of more complex but more flexible plants considering the new environmental restrictions facilitating their adaptation to future legislations, avoiding redesigns. Second, a stricter operation and control at must be guaranteed. To achieve the above mentioned aims, its necessary the use of Integrated Design Techniques, combining Optimization and Advanced Control, together with Computer Aided Tools to allow for the simultaneous design of plants and control systems at lowest costs.
Traditionally, process design and control system design have been performed sequentially. It is only recently displayed, that a simultaneous approach to the design and control leads to significant economic benefits and improved dynamic performance during plant operation. The Integrated Design methodology allows for the evaluation of the plant parameters and control system at the same time, making the designed system more controllable [4],[9]. At design stage, controllability indicators and controllers are evaluated together with economic considerations, in order to give an optimum closed loop plant. This problem can be stated mathematically as a multi objective nonlinear programming problem with differential and algebraic constraints (NLP/DAE). Many works apply Integrated Design techniques, particularly to chemical process design, such as distillation systems or reactors, stressing the interactions of design and control [8],[11]. These works also tackle process structure selection by solving a synthesis problem. A comprehensive review of advances in the area is given by [12].
Some good examples of Integrated Design applied to the activated sludge process are given in [5, 15] where the plant dimensions, an optimal working point and the parameters of PI controllers were obtained simultaneously. Despite of the complicated dynamics of the process under design, works adding advanced controllers to the Integrated Design procedure have not been reported in the literature even though it could be a good way to improve control performance. The reason for that it could be related to the complexity of the resulting optimization problem and to the important but difficult issue of tuning advance control systems within this framework.
Nevertheless, in our opinion, the field of integrated process design and control has reached a maturity level that mingles the best from process knowledge and understanding control theory on one side, with the best from numerical analysis and optimization on the other. Consequently, direct implementation of integrated methods should soon become the mainstream design procedure.
Within this context, the aim of this work was to tackle problems directly related to the Integrated Design of Activated Sludge Processes and Advance Control Systems to support engineers during the complex task of designing and control Wastewater Treatment Plants. The integration of Numerical Optimization, Dynamical Model Simulation and Model Based Predictive Control, is the most relevant feature of the work and, in our opinion, the key point to success in the design of flexible processes reducing the operation costs while legal specifications on the quality of the treated water are fulfilled.
Model predictive control (MPC) has been selected as advanced control method because of the existence of several successful applications in activated sludge control ([14], [15]) and its easiness to deal with multivariable systems and constraints. The important issue of tuning the controller parameters is deeply studied here and different approaches for the automatic tuning of linear MPC, within an Integrated design framework are proposed and widely commented in this work.
Automatic tuning of Model Based Predictive Control within an integrated design framework
In this work, the Integrated Design problem is stated mathematically as a constrained non-linear multi-objective optimization problem, in which economic and control objectives are considered together with some constraints. The solution of the ID problem is obtained following a constrained numerical cost optimization to evaluate the plant dimensions, the optimal operation points and the control system parameters.
When tackling the integrated design mathematical problem, specific features of the process (non-linearities, different sensitivity for plant parameters and controller parameters, etc.) increase the complexity of the problem. For this reason, when solving closed loop integrated design, we already have proposed in [6] the use of a methodology consisting of an iterative two steps approach. The first step performs the plant design, and the second step the controller tuning. At every step, plant or controller parameters obtained are used as constant values for the following optimization step. The loop ends when a convergence criterion is reached.
This paper is mainly focused on the second step, related to the automatic tuning of Model Based Predictive Control Systems although the whole problem of Integrated Design is also stated and referred.
Usually, the tuning of the parameters in linear Model Based Predictive Control schemes has been performed using expert knowledge and a trial and error procedure. However, some works deal with automatic tuning of MPC. Reference [2] proposed an off-line procedure for tuning the algorithm parameters of a nonlinear predictive controller specifying time-domain performance criteria. Results are good, but the tuning of integer parameters such as horizons is performed using a non intelligent grid search. For linear MPC, [1] has developed an on-line tuning strategy based on the linear approximation between the closed-loop predicted output and the MPC tuning parameters, but without considering output constraints on the on-line optimization step.
In [6] we already have proposed a methodology for the on-line automatic tuning of the whole set of parameters of linear Model Based Predictive Control Systems was carried out but by minimizing the Integral Square Error (ISE) norm as performance index. The main tuning parameters are those affecting the behaviour of the closed loop combination of plant and MPC. (control efforts weights, prediction and control horizons, time constant for the reference trajectories,..).
An important drawback of this work is that within the optimization procedure a dynamical simulation has to be carried out as a means of computing the objective function or any other dynamical performance indexes including real data records as disturbances. The control strategy is applied during the simulation. This makes the procedure extremely slow.
At the view of previous works, we propose a new approach for the optimal automatic tuning of MPC. The more relevant aspects of the actual proposal are:
1) The proposed methodology is based on the minimization of a set of indexes based on the weighted sum of the norm (H, l1, H2) of different closed loop transfer functions matrices of the system subject to a set of constraints. In this sense no dynamical simulation is needed and therefore the algorithm is very fast
The methodology proposed here is a general one and the formulation is a multivariable one. The use of linear models for the control also allows for the specification of any other convex performance criteria within an LMI framework to state stability conditions and some desired closed-loop behaviour.
2) The optimal evaluation of the whole set of real (Weighting factors) and integer parameters (horizons) of the linear MPC control scheme is carried out by solving a MINLP/DAE optimization problem by using a random search based method (MOAM).
3) The use of the proposed automatic tuning approach within an Integrated Design framework is also stated. The resolution of the resulting optimization problem is useful to perform at the same time the design of the optimal plant for activated sludge process and the optimal linear MPC for this process.
The approach has been validated on a simulated example based on a real wastewater treatment plant. Real scenarios have been considered in the simulated model by means of real data records of the main disturbances to make a more realistic analysis of the results.
References
[1] Al-Ghazzawi,. A., E. Ali, A. Nouh and E. Zafiriou (2001). On-line tuning strategy for model predictive controllers. Journal of Process Control, 11, pp. 265-284.
[2] Ali, E. and E. Zafiriou (1993). On the tuning of Nonlinear Model Predictive Control Algoritms. Proceedings of the American Control Conference, pp. 786-790.
[3] Copp, J.B. (2002). The COST Simulation Benchmark: Description and Simulator Manual. Office for Official Publications of the European Community. ISBN 92-894-1658-0.
[4] Fisher, W. R., M. F. Doherty and J. M. Douglas (1988). The Interface Between Design and Control. 1. Process Controllability. Ind. Eng. Chem. Res., 27, pp. 597-605.
[5] Francisco, M., P. Vega, O. Pérez and M. Poch (2003). Dynamic Optimization for Activated Sludge Integrated Design. European Control Conference (UK)
[6] Francisco, M., P. Vega, O. Pérez (2005). Process Integrated Design within a Model Predictive Control framework. Proceedings of IFAC World Congress (Prague)
[7] Francisco, M., P. Vega (2006). Optimal automatic tuning of model predictive controllers for the activated sludge process. Proceedings of Chemical Process Control CPC7 (Canadá)
[8] Gil, A., P. Vega and M. Francisco (2001). Integrated Design of pH processes. IASTED MIC Conference, pp 226-229.
[9] Luyben, M. L. (1993). Analyzing the Interaction Between Process Design and Process Control. Ph.D. Thesis, Princeton University.
[10] Maciejowsky, J. M. (2002). Predictive Control with Constraints. Prentice Hall.
[11] Ross, R., J.D. Perkins, E. N. Pistikopoulos, G.L.M. Koot and J.M.G. van Schijndel. (2001). Optimal design and control of a high purity industrial distillation system. Computers and Chemical Engineering, 25, pp. 141-150.
[12] Sakizlis, V., J. D. Perkins and E. N. Pistikopoulos (2004). Recent advances in optimization-based simultaneous process and control design. Computers and Chemical Engineering, 28, pp. 2069-2086.
[13] Solis, F.J. and R. J-B. Wets (1981). Minimization by random search techniques. Mathematics of Operations Research, 6, pp. 19-30.
[14] Sotomayor, O. A. Z. and C. García (2002). Model-Based Predictive Control of a pre-denitrification plant: a linear state-space model approach. Proceedings of the IFAC World Congress, Barcelona (Spain).
[15] Vega, P. and G. Gutiérrez (1999). Optimal Design Control and Operation of wastewater treatment plants. European Control Conference (Germany).
See the full pdf manuscript of the abstract.
Presented Tuesday 18, 16:40 to 17:00, in session Process Control (T4-8).
