Eciency of Prediction Error and Instrumental Variable Methods for Closed-loop Identication

(1)

Eciency of Prediction Error and Instrumental Variable Methods for Closed-loop Identication

U. Forssell and C.T. Chou Department of Electrical Engineering Linkping University, S-581 83 Linkping, Sweden

WWW:

http://www.control.isy.l iu.s e

Email:

ufo@isy.liu.se, chou@harding.et.tudelft.nl

February 26, 1998

REGLERTEKNIK

AUTOMATIC CONTROL LINKÖPING

Report no.: LiTH-ISY-R-2015 Submitted to CDC '98

Technical reports from the Automatic Control group in Linkping are available by anonymous ftp at the address

ftp.control.isy.liu.se

. This report is contained in the compressed postscript

le

^2015.ps.Z

.

(2)

Eciency of Prediction Error and Instrumental Variable Methods for Closed-loop Identication

Urban Forssell and C.T. Chou

^y

February 26, 1998

Abstract

We study the eciency of a number of closed-loop identication methods. Results will be given for methods based on the prediction error approach as well as those based on the instrumental variable approach. It is shown that all methods typically gives worse accuracy than a directly applied prediction error method. The key to this result is that in the direct method all the input signal power is utilized in reducing the variance while in the other methods only certain parts of the input spectrum is used, thus reducing the signal-to-noise ratio and consequently increasing the variance. Conditions for the instrumental variable method to give optimal accuracy are also given. Moreover, interesting insights in the properties of a recently suggested subspace method for closed-loop identication are obtained by exploring the links between this method and the instrumental variable method.

1 Introduction

Identication of systems operating under output feedback (i.e., in closed-loop) has been the topic of many papers since the sixties. Initially the focus was on identiability and accuracy issues 3].

With the increased interest in model based control the focus shifted towards problems concerning bias error, due to under-modeling, and especially on how to aect this bias error in the identied models with pre-lters.

It has been known for some time that a directly applied prediction error (PE) method gives consistency as long as the true system (including the noise properties) can be correctly described within the model class 5]. In practice this calls for exible parameterized noise models/pre-lters which implies that with this method the bias distribution cannot be manipulated at will. In order to circumvent the need for parameterized noise models, researchers have come up with alternative PE methods for closed-loop identication. We will study two of them here: the indirect method and the two-stage method 11]. These three methods will be introduced and studied in Section 3.

In the analysis we will exclusively discuss eciency. It will be shown that the latter two methods give sub-optimal accuracy, unless the signals are noise free.

The instrumental variable (IV) methods 7] form a dierent class of identication methods that is related, but not equivalent, to the prediction error methods. IV variants for closed-loop identication have also been suggested and the statistical properties of these methods been analyzed. The eciency of closed-loop IV methods has previously been studied in, e.g., 9] and we will re-use some results from there in this paper. Under certain circumstances the IV method can give the same level of accuracy as the direct PE method. In general though, the accuracy will be sub-optimal.

Division of Automatic Control, Department of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden. Email: ufo@isy.liu.se. U. Forssell is supported by the Swedish Research Council for Engineering Sciences.

y

Department of Electrical Engineering, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The

Netherlands. Email: chou@harding.et.tudelft.nl. C.T. Chou is supported by the Dutch Technology Foundation

(STW) under project number DEL55.3891.

(3)

Recently a new subspace method which can also be applied to closed-loop data was introduced in 1]. This scheme, which will referred to as the errors-in-variables (EIV) scheme as it solves an EIV problem, is based on IV and its eciency can be studied in the same vein as that in 13] where the statistical properties of IV based subspace identication methods are analyzed. However, the resulting variance expression, which has the same form as that appeared in 13], does not provide a useful platform for comparison with other identication methods. In this paper, we have chosen to analyze a modied, but yet representative, form of the EIV algorithm based on single-input single-output (SISO) ARMAX models. This provides a useful platform for cross comparison as well as valuable insights into the properties of this algorithm.

The rest of this paper is organized as follows: Section 2 denes the model and various notation to be used in this paper. Sections 3 and 4 analyze respectively the eciency of prediction error based and IV based closed-loop identication methods. Section 5 gives a discussion on the major results of this paper. Section 6 gives a simple but illustrative example and nally the conclusions are to be found in Section 7.

2 Preliminaries

We assume that the true system is given by

A

0

(

^q

)

^y

(

^t

) =

^B⁰

(

^q

)

^u

(

^t

) +

^C⁰

(

^q

)

^e

(

^t

) (1) where

A

0

(

^q

) = 1 +

^a¹⁰^q^;1

+

^:^:^:

+

^aⁿ^a⁰^q^;na

(2)

B

0

(

^q

) =

^q^;n^k

(

^b⁰⁰

+

^b¹⁰^q^;1

+

^:^:^:

+

^bⁿ^b⁰^q^;n^b

) (3)

C

0

(

^q

) = 1 +

^c¹⁰^q^;1

+

^:^:^:

+

^c^nc⁰^q^;n^c

(4) with

ⁿ^a ⁿ^bⁿ^c

ⁿ^k

0. The polynomial

^C⁰

(

^q

) is assumed to be Hurwitz, the noise

^e

(

^t

) is white with variance

⁰

. To identify this system we will work with models of the form

^

y

(

^tj

) =

^'^T

(

^t

)

(5)

where

=

^a¹^:^:^:^a^na^b⁰^:^:^:^b^nb^T

(6)

'

(

^t

) =

^;y

(

^t^;

1)

^:^:^:^;y

(

^t^;ⁿ^a

)

^u

(

^t^;ⁿ^k

)

^:^:^:^u

(

^t^;ⁿ^k^;ⁿ^b

)

^T

(7) The \true" parameter vector will be denoted by

⁰

. In some cases the model (5) will be applied together with a (possibly parameter-dependent) monic pre-lter

^L

that can be used to emphasize certain frequency regions. A standard result in identication is that the estimate ^

^N

, obtained using a prediction error method or instrumental variable method based on

^N

data samples, obeys

^

N

!

with probability 1 as

^N^!¹

(8)

p

N

(^

^N^;

)

²^AsN

(0

^P

) (9)

In this paper we will characterize the covariance matrix

^P

for a number of identication methods that guarantee

=

⁰

(i.e. that the parameter estimates converge to the true values).

We assume that there is a stabilizing LTI feedback between

^u

(

^t

) and

^y

(

^t

) given by

u

(

^t

) =

^r

(

^t

)

^;^K

(

^q

)

^y

(

^t

) =

^r

(

^t

)

^;^S

(

^q

)

R

(

^q

)

^y

(

^t

) (10)

with

^r

(

^t

) being a \reference signal" and

R

(

^q

) = 1 +

^r¹^q^;1

+

^:^:^:

+

^rⁿ^r^q^;nr

(11)

S

(

^q

) =

^s⁰

+

^s¹^q^;1

+

^:^:^:

+

^s^ns^q^;n^s

(12)

(4)

In addition the feedback loop is assumed to be well posed and

^r

(

^t

) is assumed to be independent of

^e

(

^t

).

By combining (1) and (10) it can be shown that the parts of the input and output that are due to the reference signal are

u

r

(

^t

) =

^A⁰

(

^q

)

^R

(

^q

)

A

0

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

)

^r

(

^t

) (13)

y

r

(

^t

) =

^B⁰

(

^q

)

^R

(

^q

)

A

0

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

)

^r

(

^t

) =

^B⁰

(

^q

)

A

0

(

^q

)

^u^r

(

^t

) (14) Similarly one can dene

^u^e

(

^t

) and

^y^e

(

^t

) as the parts of

^u

(

^t

) and

^y

(

^t

) that are due to the noise

^e

(

^t

).

We thus have

^u

(

^t

) =

^u^r

(

^t

) +

^u^e

(

^t

) and

^y

(

^t

) =

^y^r

(

^t

) +

^y^e

(

^t

). In the same vein the regression vector

'

(

^t

) can be split up into two parts:

'

(

^t

) =

^'^r

(

^t

) +

^'^e

(

^t

) (15) where

^'^r

(

^t

) and

^'^e

(

^t

) are the parts of

^'

(

^t

) that are due to

^r

(

^t

) and

^e

(

^t

), respectively.

3 Prediction Error Methods

3.1 General

Methods that aim at minimizing the prediction errors are perhaps the most common among all identication methods. With measured data

^Z^N

=

^fy

(1)

^u

(1)

^:^:^:^y

(

^N

)

^u

(

^N

)

^g

and a parameterized model structure ^

^y

(

^tj

) the prediction error estimate is found as the straightforward t 5]:

^

N

= argmin

1

N N

X

t=1

"

2

F

(

^t

) (16)

"

F

(

^t

) =

^L

(

^q

)(

^y

(

^t

)

^;^y

^ (

^tj

)) (17) Here

^L

(

^q

) is a monic pre-lter. Alternatively

^L

(

^q

) can be viewed as an inverse noise model used to whiten the prediction errors.

This approach can also be applied to closed-loop identication. Several dierent \methods"

and parameterizations have been suggested. See, e.g., 12, 2]. Here we will discuss three of them, namely the so called direct, indirect, and two-stage methods.

3.2 The Direct Method

The most basic one is simply to ignore the feedback and apply the method directly to measured input-output data. This is called the direct method and coincides with the standard prediction error method for open-loop identication, albeit some care has to be exercised in the analysis in case of closed-loop data.

The direct method gives consistency and optimal accuracy provided that the parameterization is exible enough. The requirement is that there exists a

⁰

in the parameter set such that

"

(

^t⁰

) =

^e

(

^t

). Under fairly mild additional conditions we also have

p

N

(^

^N^;⁰

)

²^AsN

(0

^P

) (18)

P

=

⁰^h^E

(

^t⁰

)

^T

(

^t⁰

)

ⁱ^;1

(19)

(

^t⁰

) =

^;^d

d

"

F

(

^t

)

=0

(20) By the chain rule, the gradient vector

(

^t

) can be written (cf. (5) and (17))

(

^t

) =

^L⁰

(

^q

)(

^y

(

^t

)

^;^'^T

(

^t

)

) +

^L

(

^q

)

^'

(

^t

) (21)

(5)

Since

^L

(

^q⁰

) = 1

^=C⁰

(

^q

) and

^y

(

^t

)

^;^'^T

(

^t

)

⁰

=

^C⁰

(

^q

)

^e

(

^t

) we may split

(

^t

) into two parts, one due to

^r

(

^t

) and one due to

^e

(

^t

):

(

^t⁰

) =

^r

(

^t⁰

) +

^e

(

^t⁰

) (22) where

r

(

^t⁰

) = 1

C

0

(

^q

)

^'^r

(

^t

) (23)

e

(

^t⁰

) =

^L⁰

(

^q⁰

)

^C⁰

(

^q

)

^e

(

^t

) + 1

C

0

(

^q

)

^'^e

(

^t

) (24) Moreover, since

^r

(

^t

) and

^e

(

^t

) are independent we get

P

=

⁰^h^E

^r

(

^t⁰

)

^T^r

(

^t⁰

) +

^E^e

(

^t⁰

)

^T^e

(

^t⁰

)

ⁱ^;1

(25) With

R

e

=

^E^e

(

^t⁰

)

^e^T

(

^t⁰

) (26) we thus have, for the direct method,

P D

=

⁰^h^E

1

C

0

(

^q

)

^'^r

(

^t

) 1

C

0

(

^q

)

^'^T^r

(

^t

) +

^R^eⁱ^;1

(27) The key observation here is that

^R^e

0 regardless of the parameterization of

^L

(

^q

) (as long as it is suciently exible), so the noise actually helps reducing the variance in the direct method.

Furthermore,

P D

0 h^E

1

C

0

(

^q

)

^'

(

^t

) 1

C

0

(

^q

)

^'^T

(

^t

)

ⁱ^;1^,^P^OD

(28) with equality for

^L

(

^q

) =

^L

(

^q

) = 1

^=C⁰

(

^q

).

3.3 The Indirect Method

If the regulator polynomials

^R

(

^q

) and

^S

(

^q

) are known, one can use the indirect method to get consistent estimates (but not of optimal accuracy) of the open-loop system without having to use a parameter-dependent pre-lter. The idea is to estimate the closed-loop system using a standard (prediction error) method and from this estimate determine the open-loop parameters using the knowledge of the regulator. To circumvent the last step one can parameterize the closed-loop system in terms of the open-loop parameters directly. If both the reference signal

^r

(

^t

) and the input

^u

(

^t

) are measurable one can show that the indirect method with a xed pre-lter

L

(

^q

) =

^L¹

(

^q

) is equivalent to the direct method, given that we parameterize the pre-lter as

L

(

^q

) =

^L¹

(

^q

)

^R

(

^q

)

A

(

^q

)

^R

(

^q

) +

^B

(

^q

)

^S

(

^q

) (29) The prediction errors are

"

F

(

^t

) =

^L

(

^q

)(

^y

(

^t

)

^;^'^T

(

^t

)

) =

^L¹

(

^q

)(

^y

(

^t

)

^; ^B

(

^q

)

^S

(

^q

)

A

(

^q

)

^R

(

^q

) +

^B

(

^q

)

^S

(

^q

)

^r

(

^t

)) (30) By straightforward calculations it can be shown that

(

^t

) =

^L

(

^q

)

^'^r

(

^t

) (31)

(6)

where

^L

(

^q

) is given by (29). At

⁰

we have

L

(

^q⁰

) =

^L¹

(

^q

)

^R

(

^q

)

A

0

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

) (32) Let

F

(

^q

) =

^L¹

(

^q

)

^C⁰

(

^q

)

^R

(

^q

)

A

0

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

) =

^C⁰

(

^q

)

^L

(

^q⁰

) =

^X¹

i=0 f

i q

;i

(33)

and introduce

~

'

r

(

) =

^X¹

i=0 f

i '

r

(

+

ⁱ

) (34)

Then

^P^ID

, the covariance matrix for the indirect method, is

P ID

=

⁰^h^E

^L

(

^q⁰

)

^'^r

(

^t

)

^L

(

^q⁰

)

^'^T^r

(

^t

)

ⁱ^;1^h^E

^L

(

^q⁰

)~

^'^r

(

^t

)

^L

(

^q⁰

)~

^'^T^r

(

^t

)

ⁱ

h^E

^L

(

^q⁰

)

^'^r

(

^t

)

^L

(

^q⁰

)

^'^T^r

(

^t

)

ⁱ^;1

(35) For all choices of the lter

^L¹

(

^q

),

P ID

0 h^E

1

C

0

(

^q

)

^'^r

(

^t

) 1

C

0

(

^q

)

^'^T^r

(

^t

)

ⁱ^;1^,^P^OID

(36) Equality holds if

^L¹

(

^q

) = (

^A⁰

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

))

⁼

(

^C⁰

(

^q

)

^R

(

^q

)).

The particular indirect method exemplied here is perhaps the simplest one that can be used.

However, the eciency is not optimal clearly

^P^OID ^P^D

. With some more eort it is possible to do better: If the closed-loop system is identied using an ARMAX model and the open-loop parameters are solved for using the Markov estimate it can actually be shown that this method is equivalent, as far as the accuracy is concerned, to the direct method. Or, in other words that

P ID

ARMAX

=

^P^D^ARMAX

. The details can be found in 6]. See also 10, 2].

3.4 The Two-stage Method

The two-stage method 11] is known to be quite robust and give good estimates of the open-loop system. No knowledge of the regulator parameters is required, but the regulator must be linear.

This method gives consistent estimates of the open-loop system regardless of the pre-lter used, but the eciency is sub-optimal.

The idea is to rst estimate the noise-free part of the input. Call this estimate ^

^u^r

(

^t

). Ideally

^

u

r

(

^t

) =

^u^r

(

^t

). Now consider the model

^

y

(

^tj

) =

^B

(

^q

)

A

(

^q

) ^

^u^r

(

^t

) (37)

and a pre-lter

^L¹

(

^q

). The prediction errors are

"

F

(

^t

) =

^L¹

(

^q

)(

^y

(

^t

)

^;^B

(

^q

)

A

(

^q

) ^

^u^r

(

^t

)) (38) With a pre-lter parameterized as

^L

(

^q

) =

^L¹

(

^q

)

⁼^A

(

^q

) the prediction errors can alternatively be written

"

F

(

^t

) =

^L

(

^q

)(

^y

(

^t

)

^;^'

^

^T

(

^t

)

) (39)

(7)

with

as in (6) and

^

'

(

^t

) =

^;y

(

^t^;

1)

^:^:^:^;y

(

^t^;ⁿ^a

)

^u

^

^r

(

^t^;ⁿ^k

)

^:^:^:^u

^

^r

(

^t^;ⁿ^k^;ⁿ^b

)

^T

(40) With

^

y

r

(

^t

) =

^B

(

^q

)

A

(

^q

) ^

^u^r

(

^t

) (41)

^

'

r

(

^t

) =

^;^y

^

^r

(

^t^;

1)

^:^:^:^;^y

^

^r

(

^t^;ⁿ^a

)

^u

^

^r

(

^t^;ⁿ^k

)

^:^:^:^u

^

^r

(

^t^;ⁿ^k^;ⁿ^b

)

^T

(42) the gradient vector

(

^t

) becomes

(

^t

) =

^L

(

^q

)^

^'^r

(

^t

) (43) Suppose that

"

(

^t⁰

) =

^L¹

(

^q

)(

^y

(

^t

)

^;^B⁰

(

^q

)

A

0

(

^q

) ^

^u^r

(

^t

)) =

^H

(

^q

)~

^e

(

^t

) =

^X¹

i=0 h

i^e

~ (

^t^;ⁱ

) (44) for some zero mean, white noise signal ~

^e

(

^t

) with variance ~

⁰

. With

~^

'

r

(

) =

^X¹

i=0 h

i^'

^

^r

(

+

ⁱ

) (45)

we then have

P T

= ~

⁰^h^E

^L

(

^q⁰

)^

^'^r

(

^t

)

^L

(

^q⁰

)^

^'^T^r

(

^t

)

ⁱ^;1^h^E

^L

(

^q⁰

)~^

^'^r

(

^t

)

^L

(

^q⁰

)~^

^'^T^r

(

^t

)

ⁱ

h^E

^L

(

^q⁰

)^

^'^r

(

^t

)

^L

(

^q⁰

)^

^'^T^r

(

^t

)

ⁱ^;1

(46) Denote the spectrum of a signal

^s

(

^t

) by !

^s

(

^!

). Typically, !

^ur^{^}

(

^!

)

^<

!

^ur

(

^!

) and ~

⁰ ^>⁰

which implies that the signal-to-noise ratio tend to be worse than for the indirect method. Hence

P T

P

ID

in general. If ^

^u^r

(

^t

) =

^u^r

(

^t

) then, e.g., ~

⁰

=

⁰

and

P T

0

h^E

^A⁰

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

)

A

0

(

^q

)

^C⁰

(

^q

)

^R

(

^q

)

^'^r

(

^t

)

^A⁰

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

)

A

0

(

^q

)

^C⁰

(

^q

)

^R

(

^q

)

^'^T^r

(

^t

)

ⁱ^;1 ^, ^P^OT

(47) Equality holds if

^L¹

(

^q

) = (

^A⁰

(

^q

)

^R

(

^q

) +

^B⁰

(

^q

)

^S

(

^q

))

⁼

(

^C⁰

(

^q

)

^R

(

^q

)).

4 Instrumental Variable Methods

4.1 General

Consider the linear regression (5). Let

(

^t

) denote an IV vector (of the same size as

^'

(

^t

)) and

L

(

^q

) be a pre-lter, then with the standard IV method the parameter estimate is computed as

^

N

=

^h

1

N N

X

t=1

(

^t

)

^L

(

^q

)

^'^T

(

^t

)

ⁱ^;1

1

N N

X

t=1

(

^t

)

^L

(

^q

)

^y

(

^t

) (48) given that the indicated inverse exists. Consistency requires the instruments be chosen such that

E

(

^t

)

^L

(

^q

)

^C⁰

(

^q

)

^e

(

^t

)] = 0 (49) Depending on the choices of instruments

(

^t

) and pre-lter

^L

(

^q

) dierent instrumental variable

\methods" result.

(8)

Suppose that (49) holds. Dene

F

(

^q

) =

^L

(

^q

)

^C⁰

(

^q

) =

^X¹

i=0 f

i q

;i

(50)

and

~

(

) =

^X¹

i=0 f

i

(

+

ⁱ

) (51)

Then the covariance matrix

^P

for the IV method is given by 8, 5]:

P IV

=

⁰^h^E

(

^t

)

^L

(

^q

)

^'^T

(

^t

)

ⁱ^;1^h^E

~ (

^t

)~

^T

(

^t

)

^ih^E

(

^t

)

^L

(

^q

)

^'^T

(

^t

)

ⁱ^;T

(52) For all choices of

(

^t

) and

^L

(

^q

),

P IV

0 h^E

1

C

0

(

^q

)

^'

(

^t

) 1

C

0

(

^q

)

^'^T

(

^t

)

ⁱ^;1^,^P^OIV

(=

^P^OD

) (53) Equality holds in the last equation if

(

^t

) = 1

^=C⁰

(

^q

)

^'

(

^t

) and

^L

(

^q

) = 1

^=C⁰

(

^q

) (given that these choices satisfy (49)). However, this means we need exact knowledge of the true noise polynomial

C

0

(

^q

) and therefore optimal accuracy cannot be achieved in practice.

If the instruments are chosen as ltered versions of

^r

(

^t

) then (49) will be automatically satised since

^r

(

^t

) and

^e

(

^t

) are independent. The resulting method will here be denoted RIV ('R' for reference signal). Another possibility is to use delayed versions of the regression vector,

(

^t

) =

'

(

^t^;^k

). To satisfy (49), the delay

^k

has to be larger than the maximal delay in

^L

(

^q

)

^C⁰

(

^q

). This method will be referred to as PIV ('P' for past regressors).

4.2 RIV

For this method it can be shown that (see 9]):

P RIV

=

⁰^h^E

(

^t

)

^L

(

^q

)

^'^T^r

(

^t

)

ⁱ^;1^h^E

~ (

^t

)~

^T

(

^t

)

^{i h}^E

(

^t

)

^L

(

^q

)

^'^T^r

(

^t

)

ⁱ^;T

(54) Moreover, for any pre-lter

^L

(

^q

) and any

(

^t

) constructed from

^r

(

^t

) through ltering we have that

P RIV

0 h^E

1

C

0

(

^q

)

^'^r

(

^t

) 1

C

0

(

^q

)

^'^T^r

(

^t

)

ⁱ^;1^,^P^ORIV

(=

^P^OID

) (55) Equality holds if

(

^t

) =

^C0(q)¹ ^'^r

(

^t

) and

^L

(

^q

) =

^C0(q)¹

.

4.3 PIV

For this method, the instruments are chosen to be past regressors. Suppose that

L

(

^q

) =

^Xⁿ^l

i=0 l

i q

;i

(56)

In order to satisfy the consistency requirement, admissible instruments are

(

^t

) =

^'

(

^t^;^k

) where

k n

l

+

ⁿ^c

+ 1. Introduce

F

(

^q

) =

^L

(

^q

)

^C⁰

(

^q

) =

^nl+nc^X

i=0 f

i q

;i

(57)

(9)

and

~

'

(

) =

ⁿ^X^l^+nc

i=0 f

i

'

(

+

ⁱ

) (58)

Then

P PIV

=

⁰

^E^'

(

^t^;^k

)

^L

(

^q

)

^'^T

(

^t

)]

^;1

^E^'

~ (

^t^;^k

)~

^'^T

(

^t^;^k

)]

^E^'

(

^t^;^k

)

^L

(

^q

)

^'^T

(

^t

)]

^;T

(59) This can be better or worse than for the RIV method depending, e.g., on how

^L

(

^q

) and the delay

^k

are chosen. To gain more insight in how

^P^ORIV

and

^P^PIV

compare we will use the following standard result on positive denite matrices 4]: let

^X

,

^Y

and

^Z

be matrices of compatible dimension and also let

^Y

and

^Z

be invertible then

Y

;1

XY

;T

Z

;1

,

X Y

Y T

Z

0 (60)

Therefore, we have

P PIV

P

ORIV

,

(61)

E

^'

~ (

^t^;^k

)~

^'^T

(

^t^;^k

)

^'

(

^t^;^k

)

^L

(

^q

)

^'^T

(

^t

)

L

(

^q

)

^'

(

^t

)

^'^T

(

^t^;^k

)

^C0(q)¹ ^'^r

(

^t

)

^C0(q)¹ ^'^T^r

(

^t

)

0 (62)

The left-hand-side of equation (62) equals

E

^'

~ (

^t^;^k

)~

^'^T

(

^t^;^k

)

^'

(

^t^;^k

)

^L

(

^q

)

^'^T

(

^t

)

L

(

^q

)

^'

(

^t

)

^'^T

(

^t^;^k

)

^C0¹ (q)

'

r

(

^t

)

^C0¹ (q)

' T

r

(

^t

)

=

^E

"

~

'

(

^t^;^k

)~

^'^T

(

^t^;^k

)

^'

~ (

^t^;^k

)

^C0(q)¹ ^'^T

(

^t

)

1

C

0 (q)

'

(

^t

)~

^'^T

(

^t^;^k

)

^C0¹ (q)

'

r

(

^t

)

^C0¹ (q)

' T

r

(

^t

)

#

=

^E

"

~

'

r

(

^t^;^k

)~

^'^T^r

(

^t^;^k

) ~

^'^r

(

^t^;^k

)

^C0(q)¹ ^'^T^r

(

^t

)

1

C0(q) '

r

(

^t

)~

^'^T^r

(

^t^;^k

)

^C0(q)¹ ^'^r

(

^t

)

^C0(q)¹ ^'^T^r

(

^t

)

#

+

^E

"

~

'

e

(

^t^;^k

)~

^'^T^e

(

^t^;^k

) ~

^'^e

(

^t^;^k

)

^C0¹ (q)

' T

e

(

^t

)

1

C0(q) '

e

(

^t

)~

^'^T^e

(

^t^;^k

) 0

#

(63) Note that the rst matrix in the right-hand-side of equation (63) is positive semi-denite while the second is indenite. Thus the accuracy for PIV can be better or worse than that of RIV.

Equation (63) also tells us when it is likely that PIV gives better accuracy than RIV: when the feedback loop is noisy. Under all circumstances

^P^PIV ^P^OIV

=

^P^OD

.

5 Discussion

So far we have reviewed and in some cases extended results for the accuracy of a number of PE

and IV methods. We have seen that the direct PE method gives the smallest variance of all

methods, which is natural if one think of the relation of this method to the maximum likelihood

(ML) method. A key point in understanding the dierences between this method and the other

PE methods is to realize that in the direct method the whole input spectrum helps in reducing

the variance, whereas in the alternative methods only the noise-free part of the input spectrum

contributes. This is natural when you think of the reason why the other methods give consistent

estimates despite the correlation between the noise and the input: they neutralize this complicating

fact by not using the noisy part of the input! As an example this has the advantage that we may

use output error models together with xed noise models to shape the bias in the PE setting. The

drawback is of course that the eective signal-to-noise ratio will be sub-optimal, thus deteriorating

Eciency of Prediction Error and Instrumental Variable Methods for Closed-loop Identication