NORM BASED APPROACHES FOR AUTOMATIC TUNING OF MODEL BASED PREDICTIVE CONTROL Pastora Vega, Mario Francisco, Eladio Sanz University of Salamanca – Spain.

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

y1,y2 controlled (or constrained)

u1,u2 manipulated variables

12 2

0

ˆ ˆmin ( ) ( ( | ) ( | )) ( ( | ))Hp Hc

y uui Hw i

V k W y k i k r k i k W u k i k

Standard linear multivariable MPC controller, using state space model for prediction and state estimators (MPC Toolbox MATLAB)

MPC controller

index

Ref. y1

Ref. y2 Controller

u1 y1

y2

PROCESS u2

i i iylb y yub MPC constraints i i ilb u ub i i iulb u uub

7

Tuning parameters

Hp : Prediction horizonHc : Control horizonWu: Weights of the changes of manipulated variables

Integer parameters (Hp, Hc) Real parameters (Wu)

12 2

0

ˆ ˆmin ( ) ( ( | ) ( | )) ( ( | ))Hp Hc

y uui Hw i

V k W y k i k r k i k W u k i k

8

General MPC controller structure

MPC general structure for the linear case without constraints

Particular formulation: 2 1K K

-

r + y u

d

K1 G

Gd

K3

3

11dGK Gy

Sd GK

3 1

11dK K Gu

Md GK

Transfer functions used for Automatic Tuning: output sensitivity (S’), control sensitivity (M’)

Block diagrams (linear control system):

r y u

d

KMPC G

Gd

1 2 3 1 2 3

r

u K K K y K r K y K d

d

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

minx

f x f x N

p

esf

W SN

W s M , ,p c ux H H W

,p esfW W

1

pW S

max1M u

max1S y

Constraints:Over disturbance rejection and based on l1 norms to avoid actuator saturation

11

Optimization problem

1 2 3( ) ( ) ( ) ( )F x f x f x f x

Multiobjective approach

Objective function F:

2 1max

d possible

uf M

d

3 1max

d possible

yf S

d

x=(Wu, Hp, Hc)

1

pW S,

minx R

( )i i if x w f

where fi* is the desired value for each objective

function

1 f x N

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

INTEGER PARAMETERS

TUNING Horizons

REAL PARAMETERS

TUNING

Wu Hp,Hc

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.

( ) ,p cc k H H

( ) ( )c k k

( ) ( )c k k

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

EffluentSettlerBioreactorInfluent

Recycling

Ref. s1 MPC qr1

s1

x1

PROCESS

Non linear systemLarge disturbancesSubstrate control

problem

qr1 manipulated 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)

Weights considered and parameters of the MPC tuned automatically

Substrate comparison for two weights(solid line – Wp1; dashed dotted line – Wp2)

MPC constraints

Wu=[0.0023] Hp=9, Hc=21

26.6 32

0.0001

sWp

s

2

8 9.6

0.0001

sWp

s

Comparison of sensitivity functions for tuning with weights (Wp1; Wp2)

Wu=[0.0118] Hp=8, Hc=3

Fixed plantV1=7668

A=2970.88

H mixed sensitivity problem considering objectives f1 and f2:Comparison of weights Wp

10 1000qr

1

1

0 125

400 3000

s

x

10 3500qr

Output variable: s1

2 1f M

18

Tuning results (II)

H mixed sensitivity problem considering objectives f1 and f2:Comparison of weights Wesf

Substrate comparison for two weights(dashed dotted line – Wesf2; solid line – Wesf3)

10 1000qr

1

1

0 125

400 3000

s

x

MPC constraints

Wu=[0.0011] Hp=6, Hc=2

10 3500qr

Comparison of sensitivity functions to the control efforts s*M’ for tuning with two weights

Wesf

Wu=[0.0118] Hp=8, Hc=32 2

0.2 0.02

5 0.0001

sWesf

s s

3 2

0.02 0.002

5 0.0001

sWesf

s s

Output variable: s1

Weights considered and parameters of the MPC tuned automatically

19

Tuning Results (III)

H mixed sensitivity problem considering objectives f1 and f3:Comparison of weights Wp

TABLE II

INDEX Case 4 Case 5

Wu 0.0019 0.0091

Hp 9 10

Hc 2 4

Max(qr1) 1185.5 1096.6

Max(s1) 64.17 65.33

0.95 0.97

Weights Wp1 Wp2

Computational time (min)

10.8 5.89

pW S

Comparison of substrate responses for two weights Wp1 and Wp2

Output variable: s1

Wp1 is more restrictive than Wp2

3 1f S

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.