NORM BASED APPROACHES FOR AUTOMATIC TUNING OF M ODEL BASED PREDICTIVE CONTROL

1. NORM BASED APPROACHES FOR AUTOMATIC TUNING OF MODEL BASED PREDICTIVE CONTROL Pastora Vega, Mario Francisco, Eladio Sanz University of Salamanca – Spain European Congress of Chemical Engineering (Copenhaguen, September 2007)

2. Index • Introduction and objectives • Description of the Model Predictive Controller • Optimal automatic tuning method • Results applied to the activated sludge process control • Conclusions

3. Introduction • Model based predictive control (MPC) is the most popular advanced controller for industrial applications, due to its simplicity for operators, the natural way of incorporating constraints and its easy application to multivariable systems. • MPC tuning parameters are real numbers (weights, etc.) and integer numbers (prediction and control horizons), determining closed loop system dynamics. • Usually these parameters are tuned by a trial and error procedure, taking into account general system behaviour and expert knowledge. There exist some optimization based methods for automatic tuning, but they are rather slow due to the simulations needed to evaluate dynamical indexes.

4. Objectives • Develop a method for optimal automatic tuning of Model Based Predictive Controllers (MPC) that considers both real and integer parameters, using norm based performance indexes, avoiding numerical simulations. • Validate this method using a simple reference model based on the activated sludge process of a wastewater treatment plant, particularly to minimize the output substrate variations considering typical process disturbances at the input. • Include this method in a further Integrated Design of wastewater treatment plants and their control systems.

5. Index • Introduction and objectives • Description of the Model Predictive Controller • Optimal automatic tuning method • Results applied to the activated sludge process control • Conclusions

6. General MPC controller structure u1,u2manipulated variables y1,y2controlled (or constrained) Standard linear multivariable MPC controller, using state space model for prediction and state estimators (MPC Toolbox MATLAB) MPC controller index MPC constraints

7. Tuning parameters Hp : Prediction horizon Hc : Control horizon Wu: Weights of the changes of manipulated variables Real parameters (Wu) Integer parameters (Hp, Hc)

8. General MPC controller structure MPC general structure for the linear case without constraints Block diagrams (linear control system): Particular formulation: Transfer functions used for Automatic Tuning: output sensitivity(S’),control sensitivity(M’)

9. Index • Introduction and objectives • Description of the Model Predictive Controller • Optimal automatic tuning method • Results applied to the activated sludge process control • Conclusions

10. Optimal automatic tuning of MPC Tuning procedure based on a Hmixed sensitivity problem where are suitable weights Constraints: Over disturbance rejection and based on l1 norms to avoid actuator saturation

11. Optimization problem Multiobjective approach Objective function F: x=(Wu, Hp, Hc) M’= control sensitivity N = mixed sensitivity S’= output sensitivity where fi* is the desired value for each objective function

12. Algorithm developed An iterative two steps optimization algorithm has been proposed due to the existence of real and integer parameters Step 1: Fis minimized by a random search method keeping real parameters constant Specific random search Step 2: Fis minimized using “Goal Attainment” method, keeping constant now the integer parameters (horizons) with the values obtained in step 1 Method “Goal Attainment” (MATLAB) The algorithm converges when changes in F are smaller than a certain bound

13. Algorithm developed Modified random search method for tuning MPC parameters Algorithm steps 1. An initial point for horizons, variances and centre of gaussians (for random numbers generation) is chosen. 2. A random vectorξ(k)of Gaussian distribution is generated, with integer elements. 3. Two new points are obtained by adding and removing this vector to the current point. 4. Cost function is evaluated at the original point and at new points, and the algorithm chooses the point with smallest cost. 5. If some convergence criteria is satisfied, stop the algorithm, otherwise return to step 2. Variances are decreased.

14. Index • Introduction and objectives • Description of the Model Predictive Controller • Optimal automatic tuning method • Results applied to the activated sludge process control • Conclusions

15. Description of the process and control problem Bioreactor Settler Effluent Influent Recycling Non linear system Large disturbances Substrate control problem qr1manipulated variable s1 controlled x1 constrained

16. Process disturbances: input flow and substrate Substrate concentration at the plant input (si) Flow rate at the plant input (qi) Real data from a wastewater plant Benchmark disturbances

17. Tuning results (I) H mixed sensitivity problem considering objectives f1 and f2: Comparison of weights Wp Output variable: s1 MPC constraints Comparison of sensitivity functions for tuning with weights (Wp1; Wp2) Substrate comparison for two weights (solid line – Wp1; dashed dotted line – Wp2) Wu=[0.0023] Hp=9, Hc=2 Weights considered and parameters of the MPC tuned automatically Fixed plant V1=7668 A=2970.88 Wu=[0.0118] Hp=8, Hc=3

18. Tuning results (II) H mixed sensitivity problem considering objectives f1 and f2: Comparison of weights Wesf Output variable: s1 MPC constraints Comparison of sensitivity functions to the control efforts s*M’ for tuning with two weights Wesf Substrate comparison for two weights (dashed dotted line – Wesf2; solid line – Wesf3) Wu=[0.0118] Hp=8, Hc=3 Weights considered and parameters of the MPC tuned automatically Wu=[0.0011] Hp=6, Hc=2

19. Tuning Results (III) H mixed sensitivity problem considering objectives f1 and f3: Comparison of weights Wp Output variable: s1 Comparison of substrate responses for two weights Wp1and Wp2 Wp1 is more restrictive than Wp2

20. Index • Introduction and objectives • Description of the Model Predictive Controller • Optimal automatic tuning method • Results applied to the activated sludge process control • Conclusions

21. Conclusions and future work • A new methodology has been develop to tune automatically all parameters of Model Based Predictive Controllers, considering simultaneously horizons and weights. • This method has been tested for the MPC tuning of the activated sludge process in a wastewater treatment plant. • The plant with the MPC tuned with this method is able to reject substrate disturbances in the influent. • This method has been designed to be straightforward included within an Integrated Design scheme of wastewater treatment plants together with MPC controllers. Future work: • Include some robust stability and robust performance indexes.