Model identification

Transcription

1 SE5 Prof. Davide Manca Politecnico di Milano Dynamics and Control of Chemical Processes Solution to Lab #5 Model identification Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 1 1

2 Model identification In defining a black-box model, the output data (y) of the system are calculated from the input data (u) and from the past history (y old, u old ): y f y, u old In general, in order to have a model as close as possible to the real system some adaptive parameters (p) are introduced: old y f y, u, p old It is also possible to introduce in the model the error (e) that is defined as y real -y: old y f y, u, e, p old old old Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5

3 Model identification The system to be identified has the following structure: y t f y1t y1t ny y 1 r t yr t ny r u1t u1t nu u 1 m t um t num e1 t 1,, e1 t ne,, er t 1,, er t ne 1,,,, 1,,, 1,,,, 1,,, 1 r The vector is the vector of the regressors: t y1t 1,, y1t ny,, y 1 r t 1,, yr t ny, r u1t u1t nu u 1 m t um t num T e1 t 1,, e1 t ne,, er t 1,, er t ne 1,,,, 1,,, 1 r Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 3

4 Model identification The function f, by means of the parameters p, maps the vector of the regressors in the output variables y: t t, y f p The simplest function f is: y t p t Row vector Column vector Mathematical models may have scalar, vector or mixed structure: SISO: Single Input Single Output MISO: Multiple Input Single Output MIMO: Multiple Input Multiple Output Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 4

5 Identification procedure 1. Determination of the system limits and necessary variables. Design of experiments 3. Selection of the model structure 4. Parameters evaluation 5. Simulation and validation Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 5

6 Identification procedure 1. Definition of the system limits and necessary variables The exact number of input (u) and output (y) variables is defined. The variability range of the variables is identified to create a sutiable sampling domain for the next identification step.. Design of experiments Once the variables are identified, the sampling frequency is assigned All the input variables must be disturbed Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 6

7 Identification procedure 3. Selection of the model structure We have to define: The length of the regressors vector (see the following point) The order of the model respect to every variable The linearity or non-linearity respect to the regressors and the parameters 4. Parameters evaluation We have to choose the numerical algorithm for the evaluation of the model parameters The models may be classified as: Deterministic (error minimization) Stochastic (maximum likelihood method) Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 7

8 Identification procedure 5. Simulation and validation Once the model is identified, it is required to test its predictive capability and its goodness by using a set of not formerly used data The validation procedure is based of a validation data set (cross-validation set), properly chosen a priori and kept separated from the learning set Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 8

9 Disturbance sequence generation The input-output data collection for the identification and validation procedures is obtained by disturbing the process input variables. The PRBS (Pseudo Random Binary Sequence) method is used: Two bounds are chosen, umin, umax, for the variability range of the disturbed variable u The variable quantity is varied randomly. The variable can assume only the bound values (i.e. umin and umax) The corresponding output vector is measured Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 9

10 PRBS sequences 1. Input variables can assume only two values, equal in amplitude but with opposite sign, ±Δu, respect to the stationary conditions. The shift from the positive condition to the negative one, and vice-versa, is made randomly in order to give the sequence a kind of white noise behaviour (i.e. null average) 3. The disturbance on the input variables is made every n sampling times (t s = sampling time) 4. Usually the range of nt s is equal to the 0% of the time needed by the system to end its transient 5. The amplitude of the disturbance Δu should be high enough to eliminate the measurement error due to the system noise Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 10

11 Disturbance sequences generation 0. Disturbo [-] Liquid level [m] Disturbance [-] Disturbance sequence Output variable Tempo Time [s] [s] 0.1 Altezza liquido [-] Tempo [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 11

12 Data pre-processing At the stage of field data collection it is possible to apply some appropriate mathematical operators able to damp the excessive oscillations (the moving average for example) It is possible to apply some high-cut, low-cut filters in order to remove sudden variations beyond the normal operating intervals It is possible to remove the so called outliers by means of appropriate techniques of statistical analysis DETREND: the average value is subtracted to the sampled data. By doing so, the sampled variables express the deviation from either the stationary conditions or the mean operating conditions. As a consequence, it is also possible to use the model (at the cost of lower quality results) even for other steady state conditions. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 1

13 ARX models Features The ARX model is linear both in the regressors and in the parameters As such, it is not able to describe different steady states By definition it cannot describe non-linear behaviours Its identification is quite simple The computational time for one prediction is extremely low Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 13

14 ARX SISO models SISO models: 1 1 n n y t a y t a y t a y t n = b u t 1 b u t b u t n 1 y n u y u In order to make a prediction, n y values of the dependent variable (y) and n u values of the independent variable (u) are needed. Example: evaluation based on 3 previous times for both the independent and dependent variables = b u t 1 b u t b u t 3 y t a y t a y t a y t 1 3 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 14

15 ARX MIMO models MIMO models: Example: system characterized by: 3 independent variables (u) dependent variables (y) 4 previous times a5 y t 1 a6 y t a7 y t 3 a8 y t 4 b1u 1 t 1 bu1 t b3u1 t 3 b4u1 t 4 b5u t 1 b6u t b7u t 3 b8u t 4 b u t 1 b u t b u t 3 b u t 4 y t a y t a y t a y t a y t = 0 parameters Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 15

16 Parameters determination (SISO) Given the following measurements on the real system 1 y y t y t y t n 1 T u u t u t u t n yt y t p t Different sets of parameters can be derived p1 a1 a a 1 n b b b First parameters set y n u 1 y 1 y t y t y t n 1 1 T u u t u t u t n yt1 y t p t p a1 a a 1 n b b b Second parameters set y n u Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 16

17 Parameter determination Searching for the best parameters set, an objective function should be optimized Least squares method ny n s real fobj min yi t fi t, p i1 t1 p ny n s real fobj min yi t yi t p i1 t1 ARX Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 17

18 Parameters determination n p Dependent variables calculation y t p t ARX i Objective function value calculation ny n s real fobj min yi t yi t p i1 t1 ARX n 1 p f f i obj i 1 obj opt p Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 18

19 Practical Consider two interacting tanks Find the parameters of an ARX model considering that the independent variable is the inlet flowrate and the dependent variable is the level of the second tank In order to compute the dependent variable at time t, consider old values for both the independent and the dependent variables Consider that the independent variable oscillates around the steady state value (which is assigned) of 10% Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 19

20 System model F i A 1 A h1 r 1 h r dh h 1 1 h A1 Fi dt r1 dh h h h 1 A dt r 1 r Data: A r 1 1 F i 40 m 0.9 s m 3 10 m s Tank 1: Tank : A r 30 m.1 s m I.C.: h h 0 0 s 1 h1 s h Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 0

21 Solution procedure 1. Determine the steady state conditions A dh h h 0 F dt r i s 1 dh h s 1 h h h r F A 0 dt r 1 r. Give a disturbance to the system in order to assess the characteristic time (for example +10% F i ) h1 r1 r Fi i 3. Evaluate the system dynamics according to the PRBS method Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 1

22 34 3 Altezza liquido [m] Liquid level [m] Characteristic time assessment Step disturbance on the inlet stream (+10%) Tank Characteristic time ( ) 750 s Tank Tempo [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5

23 Determination of the times Interval between two disturbances( ): t d td s t s Sampling time ( ): t s t d 3 50 s Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 3

24 MatLab implementation for i = 1:nSteps cont = cont + 1; if(cont == 3) randnum = rand(); if(randnum <= 0.5) Fi = Fi0 * 1.1; else Fi = Fi0 * 0.9; end cont = 0; end system dynamics evaluation end Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 4

25 Disturbance sequence 0. Disturbo [-] Disturbance [-] Tempo Time [s] [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 5

26 Real system response 0.1 Altezza Liquid liquido level [-] [-] Tempo [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 6

27 ARX performance assessment Confronto con i dati con cui è stato istruito il modello 3 Altezza liquido [m] Liquid level [m] Comparison of the ARX model with the original identification data Real system ARX model REMARK: this is NOT the validation of the ARX model Tempo [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 7

28 Liquid level [m] ARX validation Altezza liquido [m] Validation Convalida Real system ARX model Tempo [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano SE5 8