Adaptive Control Chapter 5: Recursive plant model identification in - PowerPoint PPT Presentation

Adaptive Control Chapter 5: Recursive plant model identification in open loop 1 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Chapter 5: Recursive plant model identification in open loop 2 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Why to review “open loop system identification” ? • We need information upon the complexity of the plant model and its basic dynamical features for building an “adaptive control scheme” •Recursive identification algorithms are used in indirect adaptive control •It is an introduction to “identification in closed loop “ which is used for “Iterative identification in closed loop and controller redesign” (an adaptation technique) 3 Adaptive Control – Landau, Lozano, M’Saad, Karimi

OUTLINE Open loop system identification -Data acquisition -Model complexity -Parameter estimation -Validation 4 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Objective of system identification (for control) To extract from experimental data a dynamic model of the plant which will allow to design a controller in order to match the control specifications 5 Adaptive Control – Landau, Lozano, M’Saad, Karimi

System Identification Methodology I/0 Data Acquisition under anExperimental Protocol Model Complexity Estimation ( or Selection ) Choice of the Noise Model Parameter Estimation Model Validation No Yes Control Design 6 Adaptive Control – Landau, Lozano, M’Saad, Karimi

I/O Data Acqusition Signal : a P.R.B.S sequence Magnitude : few % of the input operating point f s = = = f ( 1 / p ) f ; p 1 , 2 , 3 ( sampling frequency ) Clock frequency : clock s Length : − = = − ( 2 N 1 1 ) pT ; N number of cells , T 1 / f s s s NpT Largest pulse : s Length : < allowed duration for the experiment ≥ Largest pulse : (rise time) t R P.R.B.S . NpT s t r 7 Adaptive Control – Landau, Lozano, M’Saad, Karimi

An I/O File inertia d.c. motor . u(t) power M amplifier tacho generator y(t) filter TG 8 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Spectral density of a P.R.B.S. N=8 a) p = 1 10 0 p=1 -10 dB f clock =f s -20 -30 -40 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5 b) p = 2 10 0 p=2 -10 dB f clock =f s /2 -20 -30 -40 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5 c) p = 3 10 0 p=3 -10 dB f clock =f s /3 -20 -30 -40 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5 f/f s 9 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Data pre-processing The I/O data files should be centered The use of non centered data files can cause serious errors 10 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Complexity Estimation from I/O Data Objective : To get a good estimation of the model complexity ( n , n , d ) A B directly from noisy data [ ] = + = = + n max( n , n d ) n ˆ min CJ min J ( n ˆ ) S ( n ˆ , N ) A B opt n n ˆ ˆ minimum CJ (n) complexity S(n,N) penalty term J (n) error term (should be unbiased) 0 n opt n To get a good order estimation, J should tend to the value for → ∞ noisy free data when (use of instrumental variables) N 11 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Parameter Estimation Discretized plant u(t) y(t) DAC + Plant ADC ZOH + ε (t) Adjustable - y(t) discrete-time model Parameter Estimated ˆ t θ estimation ( ) model algorithm parameters batch (non recursive) Parameter estimation alg.: recursive It does not exists a unique algorithm providing good results in all the situations encountered in practice 12 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Plant Model u y − − − − − q d B ( q 1 ) q d 1 B * ( q 1 ) = = G − G ( q 1 ) − − A ( q 1 ) A ( q 1 ) = + + + = + A ( q − ) 1 a q − ... a q − 1 q − A * ( q − ) 1 1 n 1 1 A 1 n A = + + = − − − − − B ( q 1 ) b q 1 ... b q n q 1 B * ( q 1 ) B 1 n B + = − + − = θ T ψ − − y ( t 1 ) A * ( q 1 ) y ( t ) B * ( q 1 ) u ( t d ) ( t ) [ ] T θ = a ,... a , b ,..., b 1 n 1 n A B [ ] ψ = − − − + − − − + ( t ) y ( t )... y ( t n 1 ), u ( t d )... u ( t d n 1 ) T A B 13 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Recursive Parameter Estimation Methods Plant Model + = − + − = θ T ψ − − y ( t 1 ) A * ( q 1 ) y ( t ) B * ( q 1 ) u ( t d ) ( t ) θ − ψ − parameter vector ; measuremen t vector Estimated model ˆ ° + = θ φ y ˆ ( t 1 ) T ( t ) ( t ) ˆ θ − φ − estimated parameter vector ; observatio n vector Prediction error (a priori) ˆ ε + = + − θ Φ = + − + 0 ( t 1 ) y ( t 1 ) T ( t ) ( t ) y ( t 1 ) y ˆ 0 ( t 1 ) Parameter adaptation algorithm (P.A.A.) ˆ ˆ θ + = θ + + Φ ε + ( t 1 ) ( t ) F ( t 1 ) ( t ) 0 ( t 1 ) + = λ + λ Φ Φ − − F 1 ( t 1 ) ( t ) F 1 ( t ) ( t ) ( t ) T ( t ) 1 2 < λ ≤ ≤ λ < 0 ( t ) 1 ; 0 ( t ) 2 1 2 [ ] Φ = φ ( t ) f ( t ) regressor vector 14 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Recursive Least Squares Plant Model + = − + − = θ T ψ − − y ( t 1 ) A * ( q 1 ) y ( t ) B * ( q 1 ) u ( t d ) ( t ) θ − ψ − parameter vector ; measuremen t vector Estimated model ˆ ° + = θ φ y ( t 1 ) ( t ) ( t ) ˆ T ˆ − − θ φ estimated parameter vector ; observatio n vector [ ] T θ = a ,... a , b ,..., b 1 n 1 n A B [ ] ; φ = ψ = − − − + − − − + ( t ) ( t ) y ( t )... y ( t n 1 ), u ( t d )... u ( t d n 1 ) T T A B Disturbance Regressor vector u(t) y(t) 0 t Φ = φ = ψ Plant ( t ) ( t ) ( t ) ε ( ) + y(t-1) P.A.A q − 1 - > y°(t) See functions: rls.sci(.m) Adjustable Predictor on the book website 15 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Effect of stochastic disturbances (noise measurement) •Identification algorithms operates at low signal to noise ratio (in order to disturb as little as possible a plant under operation) •This often causes an error on estimated parameters called “bias” •The reason for the existence of many identification algorithms is that it does not exist an unique algorithm which gives unbiased estimates in all practical situations 16 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Non recursive least squares N ∑ ˆ θ = φ − ( N ) F ( N ) y ( i ) ( i 1 ) (*) = i 1 N ∑ − = φ − φ − 1 T F ( N ) ( i 1 ) ( i 1 ) = i 1 See functions: nrls.sci(.m) on the book website 17 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Bias in Least Squares Parameter Estimation In the presence of measurement noise the estimation of parameters is “biased” when using least squares algorithm Plant output in the presence of noise: + = θ T ψ + + = θ T φ + + (**) y ( t 1 ) ( t ) w ( t 1 ) ( t ) w ( t 1 ) Bias for the least squares algorithm (replace y in (*) by (**) ) : − 1 ⎡ ⎤ ⎡ ⎤ 1 N 1 N ˆ T θ = θ + φ − φ − φ − ∑ ∑ ( N ) ( t 1 ) ( t 1 ) ( t 1 ) w ( t ) ⎢ ⎥ ⎢ ⎥ ⎣ N ⎦ ⎣ N ⎦ = = t 1 t 1 Condition for ⎡ − ⎤ N 1 1 ∑ { } asymptotic φ − = φ − = (***) ⎢ ⎥ lim ( t 1 ) w ( t ) E ( t 1 ) w ( t ) 0 unbiased N → ∞ ⎢ ⎥ N ⎣ ⎦ = i 1 estimation regressor (observation) vector noise It is necessary that φ (t-1 )(the regressor) and w(t) be uncorrelated For the least squares this implies : w(t) = e(t) (white noise). For all the other cases the estimated parameters will be biased 18 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Unbiased estimation in the presence of noise ˆ Suppose : and we want that the algorithm leaves unchanged this value θ = θ + θ = θ T φ ε + θ = + − + θ = + y ( t 1 ) ( t ) ( t 1 ) y ( t 1 ) y ( t 1 ) w ( t 1 ) ˆ ˆ Necessary condition for unbiased estimation: ⎡ ⎤ − N 1 1 ∑ { } (****) φ − θ ε θ = φ − θ ε θ = ⎢ ⎥ lim ( t 1 , ) ( t , ) E ( t 1 , ) ( t , ) 0 N → ∞ ⎢ ⎥ N ⎣ ⎦ = i 1 necessary { } ˆ φ ε + = θ ≡ θ E ( t ) ( t 1 ) 0 f or To eliminate the bias : condition One modifies the LS algorithm in order to obtain: ˆ ε (t+1) as a white noise for: θ = θ or: ˆ uncorrelated φ (t ) and ε (t+1) for: θ = θ 19 Adaptive Control – Landau, Lozano, M’Saad, Karimi