Part 4 - Frequency Analysis; Correlation Analysis

7/25/2019 Part 4 - Frequency Analysis; Correlation Analysis

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System IdentificationControl Engineering B.Sc., 3rd yearTechnical University of Cluj-Napoca

Romania

Lecturer: Lucian Busoniu



Frequency analysis Correlation analysis Correlation analysis: Matlab example

Part IV

Frequency analysis. Correlation analysis




Table of contents

1 Frequency analysis

2 Correlation analysis

3 Correlation analysis: Matlab example




Classification

Recall Types of models from Part I:

1 Mental or verbal models

2 Graphs and tables (nonparametric)3 Mathematical models, with two subtypes:

First-principles, analytical models

Models from system identification

Frequency analysis produces a model in the form of the frequencyresponse – the Bode diagram .

C C




Response to sinusoidal input

Apply unit sine of frequency ω to system with transfer function H :

u (t ) = sin(ωt )

In steady-state, the output is a sine of the same frequency, possiblywith different amplitude and phase:

y ss(t ) = k (ω) sin[ωt + α(ω)]

Gain: k (ω) = |H ( j ω)|, where j is the imaginary unit, so H ( j ω)is a complex number.

Phase: α(ω) = arg(H ( j ω)).

F l i C l ti l i C l ti l i M tl b l




Frequency analysis

By walking through a grid of frequencies ω, and measuring the gains

k (ω) and phase shifts α(ω), a Bode diagram for the system isexperimentally created:

– a nonparametric model, which can be used to analyze the system

further or to perform control design.





Table of contents



Analytical development

A practical algorithm. FIR model

Accuracy guarantee






Context

Transient analysis of step and impulse responses gives a rough,heuristic model of the system.

The upcoming methods provide:

Fully implementable algorithms, which derive a solutionprogramatically.

Solution accuracy guarantees (under appropriate conditions).





Classification

Recall Types of models from Part I:

1 Mental or verbal models

2 Graphs and tables (nonparametric)3 Mathematical models, with two subtypes:

First-principles, analytical models

Models from system identification

Correlation analysis is still a nonparametric method; it produces animpulse response model .





Recall: discrete-time model




q y y y y p

Discrete-time impulse response

Discrete-time, unit impulse signal:

u I(k ) =

1 k = 0

0 k > 0

Discrete-time impulse response:

y I (k ) = h (k ), k ≥ 0

h (k ), k ≥ 0 is also called the weighting function of the system.




q y y y y p

Convolution

The (disturbance-free) response to an arbitrary signal u (k ) is the

convolution of the input and the impulse response:

y (k ) =∞

j =0

h ( j )u (k − j )

Intuition: Consider a signal u j (k ) equal to u ( j ) at k = j , and 0elsewhere; just a shifted and scaled unit impulse:

u j (k ) = u ( j )u I(k − j )

So, the response to u j (k ) is a shifted and scaled impulse response:

y j

(k ) = u ( j )h (k −

j )

Now, u (k ) is the superposition of all signals u j , and due to linearity:

y (k ) =

k

j =0

y j (k ) =

k

j =0

u ( j )h (k − j ) =

k

j =0

h ( j )u (k − j ) =∞

j =0

h ( j )u (k − j )

where zero initial conditions were assumed, i.e. u ( j ) = 0∀ j < 0.




Impulse-response model

y (k ) =∞

j =0

h ( j )u (k − j ) + v (k )

Includes, in addition to the ideal model, a disturbance term v (k ).




Assumptions

Assumptions

1 The input u (k ) is a stationary stochastic process.

2 The input u (k ) and the disturbance v (k ) are independent.

Recall:

Independence of random variables.

Stationary stochastic process: constant mean at every time step,

covariance only depends on difference between time steps andnot on absolute time.




Covariance functions

The covariance functions are defined as follows:

r yu (τ ) = E

{y (k + τ )u (k )}r u (τ ) = E {u (k + τ )u (k )}

Note: These quantities are true covariances only when the input andoutput are zero-mean.




Relationship of covariances and impulse response

If there were no disturbance, then:

r yu (τ ) = E {y (k + τ )u (k )}

= E∞

j =0

h ( j )u (k + τ − j )u (k )=

∞ j =0

h ( j )E {u (k + τ − j )u (k )} =

∞ j =0

h ( j )r u (τ − j )

The errors coming from the disturbance are dealt with later, usinglinear regression.




Impulse response identification

Writing the covariance relationship for all τ :

r yu (0) =∞

j =0

h ( j )r u (− j )

r yu (1) =

∞ j =0

h ( j )r u (1 − j )

. . .

we obtain (in principle) an infinite system of linear equations:

Coefficients r u (τ ), r yu (τ ).

Unknowns h (0),h (1), . . . , solution of the system.

Next, a practical algorithm working with finite data is given.




Table of contents





Accuracy guarantee





Covariances from data

Consider we are given signals u (k ), y (k ) with k = 1, . . . ,N .Denote by {a (k )}k =b ,c ,... the signal a (k ) at time steps b , c , . . . .

We have:r u (τ ) = E {u (k + τ )u (k )}

≈ mean{u (k + τ )u (k )}k =1,...,N −τ

=: r u (τ ), ∀τ ≥ 0

and r u (−τ ) = r u (τ ), due to u being a stationary process.

r yu (τ ) = E {y (k + τ )u (k )}

≈ mean{y (k + τ )u (k )}k =1,...,N −τ

=: r yu (τ ), ∀τ ≥ 0

Note: to allow negative τ in b r yu , the range of k would need to be

1 − min{τ, 0}, . . . , N − max{τ, 0}.




Finite impulse response model

Impose the condition h (k ) = 0 for k ≥ M . We obtain the finite impulseresponse (FIR) model:

y (k ) =M −1

j =0

h ( j )u (k − j ) + v (k )

The covariance relationship is similarly truncated:

r yu (τ ) =M −1 j =0

h ( j )r u (τ − j )

Note: M must be taken so that MT s dominant time constants!




Linear system

Using r yu , r u estimated from data, write the truncated equations forτ = 0, . . . ,T − 1:

r yu (0) =

M −1 j =0

h ( j )

r u (− j )

r yu (1) =M −1 j =0

h ( j ) r u (1 − j )

. . .

r yu (T − 1) =

M −1 j =0

h ( j ) r u (T − 1 − j )

This is a linear system of T equations in M unknownsh (0), . . . ,h (M − 1).




Linear system (continued)

In matrix form:

r yu (0)

r yu (1)..

.r yu (T − 1)

=

r u (0) r u (1) . . . r u (M − 1)

r u (1)

r u (0) . . .

r u (M − 2)

..

. r u (T − 1) r u (T − 2) . . . r u (T − M )

·

h (0)h (1)

..

.h (M − 1)

(keep in mind that

r u (−τ ) =

r u (τ )).

Taking T = M gives (if input is informative enough) an exact solution.

Due to disturbances and errors, it is a good idea to take T > M .

Then we can apply the machinery of linear regression (see Part 3) tosolve this problem.




Table of contents





Accuracy guarantee





Special case: White noise input

Assumptions

3 The input u (k ) is zero-mean white noise.

Then r u (τ ) = 0 whenever τ = 0 (since white noise is uncorrelated),

and r yu (τ ) = ∞ j =0 h ( j )r u (τ − j ) simplifies to:

r yu (τ ) = h (τ )r u (0)

leading to the easy algorithm:

h (τ ) = r yu (τ ) r u (0)




Simplified guarantee

Theorem

In the white-noise case, as the number of data points N grows to

infinity, the estimates h (τ ) converge to the true values h (τ ).

Remark: This type of property, where the true solution is obtained inthe limit of infinite data, is called consistency .




Table of contents







Experimental data

Consider we are given the following, separate, identification andvalidation data sets.

plot(id); and plot(val);

Note the identification input is white noise, while the validation input isnot. There are 2000 samples in the identification data.




Applying correlation analysis

fir = cra(id, M, 0); or fir = cra(id, M, 0, ’plot’);

Arguments:

1 Identification data.

2 FIR length M , here it is set to 40.

3 Third argument 0 means no input whitening is performed.

Dealing with non-ideal inputs:

If input is not zero-mean, pass the data through detrend to

remove the means.If input is not white noise, the third argument should be left todefault (by not specifying it or setting it to an empty matrix).




Applying correlation analysis (continued)

With the ’plot’ switch the covariance functions are shown, as well

as the FIR parameters with a 99% confidence interval.




Results on the identification data

yhat = conv(fir, id.u); yhat = yhat(1:length(id.u);

To simulate the FIR model, a convolution between the FIRparameters and the input is performed. The simulated output islonger than needed so we cut it off at the right length.




Validation of the FIR model

yhat = conv(fir, val.u); yhat = yhat(1:length(val.u);

Results OK, not great.


Al i f i



Alternative: impulse functionimpulse(id, ’sd’, 5);

This function produces a more complicated model than the one

studied in the lectures. Called as shown, it plots the impulseresponse with the 5-sigma confidence level.

Negative-time coefficients correspond to possible feedback in thedata (input depends on output); here the data is open-loop.Statistically significant coefficients appear to exist after the defaulttime interval. We therefore manually set a time interval [0,4].


f ti ( ti d)



impulse function (continued)

impulse(id, ’sd’, 5, [0 4]);

To obtain the actual model:mod = impulse(id, [0 4]);


V lid ti f i d l



Validation of impulse model

compare(mod, val);

The fit is very good.

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Part 4 - Frequency Analysis; Correlation Analysis

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