Lecture 8 - Model Identification • • • • •
What is system identification? Direct pulse response identification Linear regression Regularization Parametric model ID, nonlinear LS
EE392m - Winter 2003
Control Engineering
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What is System Identification? Experiment Plant
Data
Identification
Model
• White-box identification – estimate parameters of a physical model from data – Example: aircraft flight model
• Gray-box identification
Rarely used in real-life control
– given generic model structure estimate parameters from data – Example: neural network model of an engine
• Black-box identification – determine model structure and estimate parameters from data – Example: security pricing models for stock market
EE392m - Winter 2003
Control Engineering
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Industrial Use of System ID • Process control - most developed ID approaches – all plants and processes are different – need to do identification, cannot spend too much time on each – industrial identification tools
• Aerospace – white-box identification, specially designed programs of tests
• Automotive – white-box, significant effort on model development and calibration
• Disk drives – used to do thorough identification, shorter cycle time
• Embedded systems – simplified models, short cycle time EE392m - Winter 2003
Control Engineering
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Impulse response identification • Simplest approach: apply control impulse and collect the data IMP ULS E RES P ONS E
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• Difficult to apply a short impulse big enough such that the response is much larger than the noise NOIS Y IMP ULS E RES P ONS E 1 0.5
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• Can be used for building simplified control design models from complex sims EE392m - Winter 2003
Control Engineering
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Step response identification • Step (bump) control input and collect the data – used in process control
S TEP RES P ONS E OF P AP ER WEIGHT 1.5 1
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• Impulse estimate still noisy: impulse(t) = step(t)-step(t-1) IMP ULS E RES P ONS E OF P AP ER WEIGHT 0.3 0.2 0.1 0 0
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Noise reduction Noise can be reduced by statistical averaging: • Collect data for mutiple steps and do more averaging to estimate the step/pulse response • Use a parametric model of the system and estimate a few model parameters describing the response: dead time, rise time, gain • Do both in a sequence – done in real process control ID packages
• Pre-filter data
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Control Engineering
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Linear regression • Mathematical aside – linear regression is one of the main System ID tools Data
Regression weights
Regressor
N
y (t ) = ∑θ jϕ j (t ) + e(t )
Error
y = Φθ + e
j =1
y (1) ϕ1 (1) K ϕ K (1) θ1 e(1) O M , θ = M , e = M y = M , Φ = M y ( N ) ϕ1 ( N ) K ϕ K ( N ) θ K e( N )
EE392m - Winter 2003
Control Engineering
8-7
Linear regression • Makes sense only when matrix Φ is tall, N > K, more data available than the number of unknown parameters. – Statistical averaging
• Least square solution:
||e||2
→ min
y = Φθ + e
θˆ = (Φ T Φ ) Φ T y −1
– Matlab pinv or left matrix division \
• Correlation interpretation: N N 2 K ∑ ϕ K (t )ϕ1 (t ) ∑ϕ 1 (t ) t =1 1 t =1 , M O M R= N N N 2 ∑ ϕ1 (t )ϕ K (t ) K ϕ ∑ K (t ) t =1 t =1 EE392m - Winter 2003
Control Engineering
θˆ = R −1c N ∑ ϕ1 ( t ) y ( t ) 1 t =1 M c= N N ∑ ϕ K (t ) y (t ) t =1 8-8
Example: linear first-order model y (t ) = ay (t − 1) + gu (t − 1) + e(t )
• Linear regression representation ϕ1 (t ) = y (t − 1) a θ = ϕ 2 (t ) = u(t − 1) g
−1 T T ˆ θ = (Φ Φ ) Φ y
• This approach is considered in most of the technical literature on identification Lennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999
• Matlab Identification Toolbox – Industrial use in aerospace mostly – Not really used much in industrial process control
• Main issue: – small error in a might mean large change in response EE392m - Winter 2003
Control Engineering
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Regularization • Linear regression, where Φ T Φ is ill-conditioned • Instead of ||e||2 → min solve a regularized problem 2 2 e + r θ → min y = Φθ + e r is a small regularization parameter • Regularized solution −1 T T ˆ θ = (Φ Φ + rI ) Φ y
• Cut off the singular values of Φ that are smaller than r
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Control Engineering
8-10
Regularization • Analysis through SVD (singular value decomposition) Φ = USV T ; V ∈ R n ,n ;U ∈ R m ,m ; S = diag{s j }nj =1 • Regularized solution n s − 1 θˆ = (Φ T Φ + rI ) Φ T y = V diag 2 j U T y s j + r j =1 • Cut off the singular values of Φ that are smaller than r REGULARIZED INVERS E 2 1.5
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Linear regression for FIR model P RBS EXCITATION S IGNAL
• Identifying impulse response by 1 0.5 applying multiple steps 0 -0.5 • PRBS excitation signal -1 • FIR (impulse response) model 0 K
y ( t ) = ∑ h ( k )u ( t − k ) + e( t ) k =1
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PRBS = Pseudo-Random Binary Sequence, see IDINPUT in Matlab
• Linear regression representation
ϕ1 (t ) = u(t − 1) M
ϕ K (t ) = u (t − K ) EE392m - Winter 2003
h (1) θ = M h ( K ) Control Engineering
−1 T T ˆ θ = (Φ Φ + rI ) Φ y
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Example: FIR model ID P RBS e xcita tion
• PRBS excitation input
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S YS TEM RES P ONS E
• Simulated system 1 output: 4000 0.5 samples, random 0 -0.5 noise of the -1 amplitude 0.5 0
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Control Engineering
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Example: FIR model ID FIR es tima te
• Linear regression 0.2 estimate of the FIR0.15 0.1 model 0.05 0 -0.05
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Impuls e Re s pons e
• Impulse response for the simulated system:
0.2 0.15 0.1 0.05 0
-0.05 T=tf([1 .5],[1 1.1 1]); 0 P=c2d(T,0.25);
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Time (s ec)
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Nonlinear parametric model ID • Prediction model depending on the unknown parameter vector θ u (t ) → MODEL(θ ) → yˆ (t | θ )
Loss Index V
V = ∑ y (t ) − yˆ (t | θ )
• Loss index
J = ∑ y (t ) − yˆ (t | θ )
θ
Optimizer
2
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• Iterative numerical optimization. Computation of V as a subroutine
y (t ) u (t )
Model including the parameters θ sim
Lennart Ljung, “Identification for Control: Simple Process Models,” IEEE Conf. on Decision and Control, Las Vegas, NV, 2002 EE392m - Winter 2003
Control Engineering
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Parametric ID of step response τ
• First order process with deadtime • Most common industrial process model • Response to a control step applied at tB g (1 − e( t −tB −TD ) /τ ), for t > tB − TD y (t | θ ) = γ + 0, for t ≤ tB − TD
γ g θ = τ TD
g TD
Example: Paper machine process
EE392m - Winter 2003
Control Engineering
8-16
Gain estimation • For given τ , TD , the modeled step response can be presented in the form y (t | θ ) = γ + g ⋅ y1 (t | τ , TD ) • This is a linear regression 2
w1 = g
k =1
w2 = γ
y (t | θ ) = ∑ wkϕ k (t )
ϕ1 (t ) = y1 (t | τ , TD ) ϕ 2 (t ) = 1
• Parameter estimate and prediction for given τ , TD wˆ (τ , TD ) = (Φ Φ ) Φ T y T
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yˆ (t | τ , TD ) = γˆ + gˆ ⋅ y1 (t | τ , TD )
Control Engineering
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Rise time/dead time estimation • For given τ , TD , the loss index is N
V = ∑ y (t ) − yˆ (t | τ , TD )
2
t =1
• Grid τ , TD and find the minimum of V = V (τ , TD )
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Control Engineering
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Examples: Step response ID • Identification results for real industrial process data • This algorithm works in an industrial tool used in 500+ industrial plants, many processes each P roce s s pa ra me te rs : Ga in = 0.134; Tde l = 0.00; Tris e = 119.8969
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Nonlinear Regression ID
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Linear Regression ID of the first-order model
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Nonlinear Regression ID
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Linear filtering • A trick that helps: pre-filter data • Consider data model y = h *u + e
• L is a linear filtering operator, usually LPF Ly = L( h * u ) + { Le { e yf
f
L( h * u ) = ( Lh ) * u = h * ( Lu )
• Can estimate h from filtered y and filtered u • Or can estimate filtered h from filtered y and ‘raw’ u • Pre-filter bandwidth will limit the estimation bandwidth EE392m - Winter 2003
Control Engineering
8-20
Multivariable ID • Apply SISO ID to various input/output pairs • Need n tests - excite each input in turn • Step/pulse response identification is a key part of the industrial Multivariable Predictive Control packages.
EE392m - Winter 2003
Control Engineering
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