part 6 - instrumental variable methods

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

Romania

Lecturer: Lucian Busoniu


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Analytical development Matlab example Theoretical guarantees

Part VI

Instrumental variable methods


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Table of contents

1 Analytical development

2 Matlab example

3 Theoretical guarantees

We stay in the single-output, single-input case for this lecture.


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Classification

1 Mental or verbal models

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

First-principles, analytical models

Models from system identification

Like prediction error methods, instrumental variable methods produce

parametric, polynomial models.


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Table of contents


Starting point: ARX

Instrumental variables methods

Comparison: IV versus PEM

2 Matlab example


A l ti l d l t M tl b l Th ti l t


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Setting: discrete time



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Recall ARX model

A(q1)y(k) =B(q1)u(k) + e(k)

(1+a1q1 + + anaq

na)y(k) =

(b1q1 + + bnbq

nb)u(k) + e(k)

In explicit form:

y(k) + a1y(k 1) + a2y(k 2) +. . .+ anay(k na)

=b1u(k 1) + b2u(k 2) +. . .+ bnbu(k nb) + e(k)

where the model parameters are: a1, a2, . . . , anaandb1, b2, . . . , bnb.



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Linear regression representation

y(k) = a1y(k 1) a2y(k 2) . . . anay(k na)

b1u(k 1) + b2u(k 2) +. . .+ bnbu(k nb) + e(k)

= y(k 1) y(k na) u(k 1) u(k nb)

a1 ana b1 bnb

+ e(k)

=:(k)+ e(k)

Regressor vector: Rna+nb, previous output and input values.Parameter vector: Rna+nb, polynomial coefficients.



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Identification problem and solution

Given datasetu(k), y(k),k=1, . . . , N, find model parameters toachieve small equation errors(k)in:

y(k) =(k)+(k)

Formal objective: minimize the mean squared error:

V() = 1

N

Nk=1

(k)2

Solution:can be written in several ways, we use:

= 1N

Nk=1

(k)(k)

1 1

N

Nk=1

(k)y(k)



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Theoretical guarantee

Recall that for the guarantees, a true parameter vector0 is assumed

to exist: y(k) =(k)0+ v(k)

Analyze the parameter errors (a vector ofnelements):

0 = 1NN

k=1

(k)

(k)1

1NN

k=1

(k)y(k)

1

N

Nk=1

(k)(k)

1 1

N

Nk=1

(k)(k)

0

=1

N

Nk=1

(k)(k)1 1

N

Nk=1

(k)[y(k) (k)0]

= 1

N

N

k=1 (k)(k)

1

1

N

N

k=1 (k)v(k)



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y p p g

Further assumptions

We wish the algorithm to be consistent: the parameter errors tobecome 0 in the limit of infinite data.

AsN :

1

N

N

k=1 (k)(k) E (k)(k)1

N

Nk=1

(k)v(k) E {(k)v(k)}

For the error to be well-defined and equal to zero, we need:

E(k)(k)

invertible.

E {(k)v(k)}zero.



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y p p g

Table of contents


Starting point: ARX



2 Matlab example




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Motivation: ARX requires white noise

We have E {(k)v(k)}= 0 if the elements of (k)areuncorrelated withv(k)(note thatv(k)is assumed zero-mean).

But(k)includesy(k 1), y(k 2), . . . , which depend onv(k 1), v(k 2), . . . !

So the only option is to have v(k)uncorrelated withv(k 1), v(k 2), . . . v(k)must bewhite noise.

Instrumental variablesare a solution to remove this limitation to whitenoise.



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Intuition

0 =

1

N

Nk=1

(k)(k)

1 1

N

Nk=1

(k)v(k)

Idea:What if a different vector than (k)could be included in theparameter errors?

0 =

1

N

N

k=1Z(k)(k)

1 1

N

N

k=1Z(k)v(k)

where the elements ofZ(k)are uncorrelated withv(k). ThenE {Z(k)v(k)}= 0 and the error can be zero.

VectorZ(k)has nelements, which are calledinstruments.



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Instrumental variable method

In order to have:

0 = 1N

Nk=1

Z(k)(k)1 1

N

Nk=1

Z(k)v(k)

(6.1)

the estimated parameter must be:

= 1N

Nk=1

Z(k)(k)1 1N

Nk=1

Z(k)y(k) (6.2)This

is the solution to the system of nequations:

1N

Nk=1

Z(k)[(k) y(k)]= 0 (6.3)Constructing and solving this system comprises thebasicinstrumental variable (IV) method.

Exercise: Show that(6.3) implies(6.2), and that (6.2)implies (6.1).



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Simple instruments

So far the instrumentsZ(k)were not discussed. They are usuallycreated based on the inputs (including outputs would lead tocorrelation withvand so eliminate the advantage of IV).

Simple possibility:just include additional delayed inputs to obtain avector of the appropriate size,n=na+ nb:

Z(k) = [u(k nb 1), . . . u(k na nb), u(k 1), . . . , u(k nb)]

Compare to original vector:

(k) = [y(k 1), . . . , y(k na), u(k 1), . . . , u(k nb)]



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Generalization

Pass the input through a transfer function:

C(q1)x(k) =D(q1)u(k)

(1+c1q1 + + cnbq

nc)x(k) =

(d1q1

+ + dndqnd

)u(k)x(k) =c1y(k 1) c2y(k 2) . . . cncy(k nc)

d1u(k 1) + d2u(k 2) +. . .+ dndu(k nd)

and takenapast values from the outputx:

Z(k) = [x(k 1), . . . , x(k na), u(k 1), . . . , u(k nb)]

Remark: C(q1),D(q1)have different meanings than in PEM.



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Generalized instruments: obtaining the simple case

In order to obtain:


setC=1, D=qnb.

Exercise: Verify the desiredZ(k)is indeed obtained.



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Generalized instruments: Initial model

Generalized instruments:

Z(k) = [x(k 1), . . . , x(k na), u(k 1), u(k 2), . . . , u(k nb)]

Compare to original vector:

(k) = [y(k 1), . . . , y(k na), u(k 1), . . . , u(k nb)]

Idea:Take instrument generator equal to an initial model,

C(q1) =

A(q1),D(q1) =

B(q1). This model can be obtained e.g.

with ARX estimation.

In the ideal case where the model is correct,A(q1) =A(q1),B(q1) =B(q1), we obtain noise-free outputsyat the appropriatepositions inZ(k). Otherwise, we have an approximation ofy:

Z(k) = [y(k 1), . . . y(k na), u(k 1), . . . , u(k nb)]

but without the disadvantage of correlation with the noise.



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Table of contents


Starting point: ARX



2 Matlab example




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Comparison

Both PEM and IV can be seen as extensions of ARX:

A(q1)y(k) =B(q1)u(k) + e(k)

to disturbancesv(k)different from white noisee(k).

PEMexplicitly include the disturbance model in the structure,e.g. in ARMAXv(k) =C(q1)e(k)leading toA(q1)y(k) =B(q1)u(k) + C(q1)e(k).

IV methodsdo notexplicitly model the disturbance, but are

designed to be resilient to non-white, colored disturbance (byusing instrumentsZ(k)uncorrelated with it).



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Comparison (continued)

Advantage of IV: Simple model structure, identification consists ofsimply solving a system of linear equations. In contrast, PEMrequired solving optimization problems with Newtons method,

susceptible to local minima etc.

Disadvantage of IV: In practice (for finite numberNof data), modelquality depends heavily the choice of instrumentsZ(k).

Methods exist to choose instrumentsZ(k)that are optimal in a certain sense,

but they will not be discussed here.



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Table of contents


2 Matlab example




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Experimental data

Separate identification and validation data sets:plot(id); and plot(val);

From prior knowledge, the system has order 2 and the disturbance iscolored (does not obey the ARX model structure).

Remarks: As before, the identification input is a pseudo-randombinary signal, and the validation input a sequence of steps.



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IV identification with custom instruments

Define the instruments by the generating transfer function, usingpolynomialsC(q1)and D(q1).

model = iv(id, [na, nb, nk], C, D);

Arguments:1 Identification data.

2 Array containing the orders ofA and Band the delaynk (like forARX).

3

PolynomialsCandD, as vectors of coefficients in increasingpower ofq1.



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Result with simple instruments

TakeC(q1) =1, D(q1) =qnb, leading to


.Compare to ARX.

Conclusions:

IV models arenot guaranteed to be stable!

Results very bad with this simple choice.



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Result with ARX-model instruments

TakeC(q1) =A(q1),D(q1) =B(q1)from the ARX experiment,leading to

Z(k) = [y(k 1), . . . y(k na), u(k 1), . . . , u(k nb)]

.

Conclusion:IV obtains better results. This is because the disturbanceis colored, and IV can deal effectively with this case (whereas ARXcannot).



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Result with automatic instruments

model = iv4(id, [na, nb, nk]);

Implements an algorithm that generates near-optimal instruments.

Conclusion:Works slightly better than just ARX instruments.



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Table of contents


2 Matlab example




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Assumptions

Assumptions (simplified)

1 The disturbancev(k) =H(q1)e(k)wheree(k)is zero-meanwhite noise,H(q1)is a transfer function satisfying certainconditions.

2 The input signalu(k)is sufficiently informative and does notdepend on the disturbance (the experiment is open-loop).

3 The real system is stable and uniquely representable by themodel chosen: there exists exactly one0 so that polynomialsA(q1; 0)and B(q

1; 0)are identical to those of the real

system.4 Matrix E

Z(k)Z(k)

is invertible.



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Discussion of assumptions

Assumption 1 shows the main advantage of IV over PEM: the

disturbance can be colored.Assumptions 2 and 3 are not very different from those made byPEM. Stability of a discrete-time system requires its poles to bestrictly inside the unit circle:

Assumption 4 is required to solve the linear system, and given asufficiently informative input boils down to an appropriateselection of instruments (e.g. not repeating the same delayed

inputu(k i)twice).


G


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Guarantee

Theorem 1

As the number of data points N , the solutionof IV estimationconverges to the true parameter vector 0.Remark:This is aconsistencyguarantee, in the limit of infinitelymany data points.


P ibl t i


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Possible extensions

Multiple-input, multiple-output systems.

Larger-dimension instrumentsZthan parameter vectorswith other modifications, calledextendedIV methods.

part 6 - instrumental variable methods

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