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 1. Introduction and objectives 2. Description of the Model Predictive Controller 3. Optimal automatic tuning method 4. Results applied to the activated sludge process control 5. 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 1. Introduction and objectives 2. Description of the Model Predictive Controller 3. Optimal automatic tuning method 4. Results applied to the activated sludge process control 5. Conclusions

6 General MPC controller structure y 1,y 2 controlled (or constrained) u 1,u 2 manipulated variables 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 H p : Prediction horizon H c : Control horizon W u : Weights of the changes of manipulated variables Integer parameters (H p, H c ) Real parameters (W u )

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

9 Index 1. Introduction and objectives 2. Description of the Model Predictive Controller 3. Optimal automatic tuning method 4. Results applied to the activated sludge process control 5. 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) where f i * is the desired value for each objective function S’= output sensitivity M’= control sensitivity N = mixed sensitivity

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

13 Algorithm developed Algorithm steps Modified random search method for tuning MPC parameters 2. A random vector ξ(k) of Gaussian distribution is generated, with integer elements. 1. An initial point for horizons, variances and centre of gaussians (for random numbers generation) is chosen. 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 1. Introduction and objectives 2. Description of the Model Predictive Controller 3. Optimal automatic tuning method 4. Results applied to the activated sludge process control 5. Conclusions

15 Description of the process and control problem EffluentSettlerBioreactor Influent Recycling Non linear system Large disturbances Substrate control problem qr1 manipulated variable s 1 controlled x 1 constrained

16 Process disturbances: input flow and substrate Substrate concentration at the plant input (s i ) Flow rate at the plant input (q i ) Real data from a wastewater plant Benchmark disturbances

17 Tuning results (I) Weights considered and parameters of the MPC tuned automatically Substrate comparison for two weights (solid line – Wp1; dashed dotted line – Wp2) MPC constraints W u =[0.0023] H p =9, H c =2 Comparison of sensitivity functions for tuning with weights (Wp1; Wp2) W u =[0.0118] H p =8, H c =3 Fixed plant V 1 =7668 A= H  mixed sensitivity problem considering objectives f 1 and f 2 : Comparison of weights Wp Output variable: s 1

18 Tuning results (II) H  mixed sensitivity problem considering objectives f 1 and f 2 : Comparison of weights Wesf Substrate comparison for two weights (dashed dotted line – Wesf2; solid line – Wesf3) MPC constraints W u =[0.0011] H p =6, H c =2 Comparison of sensitivity functions to the control efforts s*M’ for tuning with two weights Wesf W u =[0.0118] H p =8, H c =3 Output variable: s 1 Weights considered and parameters of the MPC tuned automatically

19 Tuning Results (III) H  mixed sensitivity problem considering objectives f 1 and f 3 : Comparison of weights Wp TABLE II INDEXCase 4Case 5 Wu HpHp 910 HcHc 24 Max(qr1) Max(s1) WeightsWp1Wp2 Computational time (min) Comparison of substrate responses for two weights W p1 and W p2 Output variable: s 1 Wp1 is more restrictive than Wp2

20 Index 1. Introduction and objectives 2. Description of the Model Predictive Controller 3. Optimal automatic tuning method 4. Results applied to the activated sludge process control 5. 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.

22 H  norm, l1 norms (sensitivity, sensitivity to control, etc.), maximum output deviation, etc. Multiobjective Integrated Design Iterative methodology including predictive control and automatic tuning methods of the controllers, considering norm based indexes and dynamical constraints, for the resolution of the MINLP problem generated Construction and operational costs Automatic tuning method proposed EXAMPLE