Asymptotic Variance Expressions for Identied Black-box Models

Asymptotic Variance Expressions for Identied Black-box Models

Urban Forssell

Department of Electrical Engineering Linkping University, S-581 83 Linkping, Sweden

WWW:

E-mail:

28 December 1998

REGLERTEKNIK

AUTOMATIC CONTROL LINKÖPING

Report no.: LiTH-ISY-R-2089 Submitted to Systems & Control Letters

Technical reports from the Automatic Control group in Linkping are available

by anonymous ftp at the address

. This report is

contained in the compressed postscript le

.

Asymptotic Variance Expressions for Identied Black-box Models

Urban Forssell

Division of Automatic Control, Department of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden. E-mail: ufo@isy.liu.se.

Abstract

The asymptotic probability distribution of identied black-box transfer function models is studied. The main contribution is that we derive variance expressions for the real and imaginary parts of the identied models that are asymptotic in both the number of measurements and the model order. These expressions are considerably simpler than the corresponding ones that hold for xed model orders, and yet they frequently approximate the true covariance well already with quite modest model orders. We illustrate the relevance of the asymptotic expressions by using them to compute uncertainty regions for the frequency response of an identied model.

Key words: Identication Prediction error methods Covariance Uncertainty

1 Introduction

A general linear, discrete-time model of a time-invariant system can be written

y

(

) =

k

=1

g

(

)

(

) +

(

) (1) Here

(

)

is the output,

(

)

the input, and

(

)

an additive disturbance, whose character we will discuss later. With (1) we associate the transfer func- tion

G

(

) =

k

=1

g

(

)

(2)

?

This paper was not presented at any IFAC meeting. Corresponding author U.

Forssell. Tel. +46 13 282226. Fax +46 13 282622. E-mail ufo@isy.liu.se.

Preprint submitted to Elsevier Preprint 28 December 1998

In this paper we will discuss the statistical properties of transfer function mod- els of the form (2) when the impulse response coecients

(

) are determined or estimated using measured data

=

(1)

(1)

(

)

(

)

. The identication method we will consider is the classical prediction error method, e.g., 4].

The main goal of the paper is to derive explicit expressions for the covariance matrix

P

(

) =

2

6

4

Re ^

(

)

^

(

)]

Im ^

(

)

^

(

)]

3

7

5 2

6

4

Re ^

(

)

^

(

)]

Im ^

(

)

^

(

)]

3

7

5 T

(3) where ^

(

) denotes the identied model obtained using

measurements, that are asymptotic both in the number of observed data and in the model order. Similar results have previously been given in 3]. There, however, the focus was on expressions for

P



(

) =

^

(

)

^

(

)

² (= tr

(

)) (4) which is real-valued and hence does not bring any information about the phase of ^

(

)

^

(

). With an explicit expression for the covariance matrix

P

(

) dened in (3) we can construct condence ellipsoids for ^

(

) in the complex plane which is useful, e.g., for robust control design and analysis.

2 Preliminaries

Notation and Denitions

The delay operator is denoted by

¹ ,

q

;

1

u

(

) =

(

1) (5)

and the set of integers by

. The Kronecker delta function

(

) is dened as



k

=

8

<

:

1

= 0

0

= 0 (6)

3

As we shall work with signals that may contain deterministic components, we will consider generalized covariances and spectra of the form

R

yu

(

) = 

(

)

(

) (7)



(

) =

k

=

R

yu

(

)

(8)

In (7) the symbol 

is dened as



Ef

(

) = lim

N!1

1

N N

X

t

=1

Ef

(

) (9)

A lter

(

),

F

(

) =

k

=0

f

(

)

(10)

is said to be stable if

1

X

k

=0

jf

(

)

(11)

A set of lters

(

)

,

F

(

) =

k

=0

f

(

)

(12)

is said to be uniformly stable if

1

X

k

=0 sup

2D

M

jf

(

)

(13) Basic Prediction Error Theory

Consider the set of models

y

(

) =

(

)

(

) +

(

)

(

)

(14) where

is a compact and connected subset of

(

= dim

), and where

G

(

) =

k

=1

g

(

)

(15)

H

(

) = 1 +

k

=1

h

(

)

(16)

4

(In (14)

(

)

is supposed to be a sequence of independent, identically dis- tributed random variables of zero means, variances

0 , and bounded fourth order moments.) The corresponding one-step-ahead predictor is

^

y

(

) =

¹ (

)

(

)

(

) + (1

¹ (

))

(

) (17) The prediction error is

"

(

) =

(

)

^ (

) =

¹ (

)(

(

)

(

)

(

)) (18) The standard least-squares prediction error estimate is found as

^



N

= arg min

2D

M V

N

(

) (19)

V

N

(

) = 1

N N

X

t

=1

1 2

² (

) (20) We will denote the corresponding transfer function estimates, ^

(

) and

^

H

N

(

), respectively ^

(

) =

(

^

) and ^

(

) =

(

^

).

Under weak regularity conditions we have (see, e.g., 4])

^



N

!



with probability 1 as

(21)





= arg min

2D

M



V

(

) (22)



V

(

) = 

1

2

² (

) (23) Further,

p

N

(^

)

(0

) (24)

P



=

¹

¹ (25)

R

= 

(

) (26)

Q

= lim

N!1

E N V 0

N

(

)(

(

))

(27) Here prime and double prime denotes dierentiation once and twice, respec- tively, with respect to

.

Asymptotic Variances for Identied Models

The result (24)-(27) states that asymptotically, as

tends to innity, the estimate

^

will have a normal distribution with mean

and

5

covariance matrix

. Using the Taylor expansion

2

6

4

Re ^

(

)

(

)]

Im ^

(

)

(

)]

3

7

5

=

2

6

4

Re

(

)]

Im

(

)]

3

7

5

(^

) +

(

^

) (28) where

(

) is a 1

dimensional matrix being the derivative of

(

) with respect to

evaluated at

=

, we thus have

p

N 2

6

4

Re ^

(

)

(

)]

Im ^

(

)

(

)]

3

7

5

2AsN

(0

(

)) (29) with

P

(

) =

2

6

4

Re

(

)]

Im

(

)]

3

7

5P

 2

6

4

Re

(

)]

Im

(

)]

3

7

5 T

(30) The matrix

(

) in (30) gives an expression for the sought covariance ma- trix (3). However, evaluation of (30) is complicated and leads to intractable expressions, which limits the usefulness of the result. From, e.g., 3] we know that it is possible to compute variance expressions that are asymptotic in the model order, i.e., in the dimensionality of

. This type of expressions tend to be simpler and, hence, easier to work with and to interpret than those ob- tainable using the above technique. Furthermore, despite the fact that these results are asymptotic in both the number of measurements and the model order, they frequently give reasonable approximations of the variance even for xed model orders. In the next section we will derive the corresponding expressions for (3).

3 Main Result

Introduce

T

(

) =

2

6

4

G

(

)

H

(

)

3

7

5

(31)

and dene

0 (

) =

2

6

4 u

(

)

e

(

)

3

7

5

(32)

6

The spectrum of

0 (

)

is



(

) =

2

6

4



(

) 

(

)



(

)

0

3

7

5

(33)

Using (31) and (32) we can rewrite the model (14) as

y

(

) =

(

)

0 (

) (34) Suppose that the parameter vector

can be decomposed so that



=

₁

₂

dim

=

dim

=

(35) We shall call

the order of the model (34). Suppose also that

@

@

k

T

(

) =

⁺¹

@

1

(

)

⁺¹

(

) (36) where the matrix

(

) in (36) is of size 2

.

It should be noted that most polynomial-type model structures, like ARMAX, Box-Jenkins, etc., satisfy this shift structure. Thus (36) is a rather weak as- sumption. Consider, e.g., an ARX model

y

(

) =

(

)

A

(

)

(

) + 1

A

(

)

(

) (37) where

A

(

) = 1 +

1

1 +

+

(38)

B

(

) = 1 +

1

1 +

+

(39) In this case we have

=

and

Z

(

) =

¹

2

6

4

; B

(

)

A 2

(

) 1

A

(

)

;

1

A 2

(

) 0

3

7

5

(40)

From (36) it follows that the 2

dimensional matrix

(

), being the derivative of

(

) with respect to

, can be written

T 0



(

) =

(

)

(

) (41) where

W

n

(

) =

²

(42)

7

with

being an

identity matrix. The following lemma, which also is of independent interest, will be used in the proof of the main result, Theorem 2, below.

Lemma 1 ^Let

⁽

) be dened by (42) and let

(

)

be an

-dimensional process with invertible spectrum 

(

). Then

lim

n!1

1

n W

n

(

)

1 2

Z



;

W T

n

(

)

(

)

(

)

¹

(

)

=



(

1 )

¹

(

)mod2

(43)

PROOF. From Lemma 4.3 in 7] (see also 5], Lemma 4.2) we have that lim

n!1

1

n W

n

(

)

1 2

Z



;

W T

n

(

)

(

)

(

)

(

)

=



(

1 )

(

) (44)

(Note especially the transposes, which are due to dierent denitions of corre- lation functions and spectra than the ones used here.) The result (43) follows since 

(

) = 

(

) and

(

⁽

⁺²

⁾ ) =

(

)

.

For the proof of Theorem 2 below we additionally need a number of technical assumptions which we now list. Let





(

) = arg min

2D

M



V

(

) (45)

(If the minimum is not unique, let

(

) denote any, arbitrarily chosen mini- mizing element.) The argument

is added to emphasize that the minimization is carried out over

th order models. Dene

^



N

(

) = arg min

2D

M V

N

(

) (46)

V

N

(

) = 12

"

1

N N

X

t

=1

"

2 (

) +

(

)

²

#

(47) where

is a regularization parameter, helping us to select a unique minimizing element in (46) in case

= 0 leads to nonunique minima.

Assume that the true system can be described by

y

(

) =

0 (

)

(

) +

(

)

(

) =

0 (

)

(

) (48) where

(

)

is a sequence of independent, identically distributed random vari- ables with zero means, variances

0 , and bounded fourth order moments, and

8

where

0 (

) is stable and strictly causal and

0 (

) is monic and inversely sta- ble. From (48) it follows that the spectrum of the additive disturbance

(

)

is



(

) =

0

0 (

)

² (49) It will be assumed that there exists a

0

such that

G

(

0 ) =

0 (

) and

(

0 ) =

0 (

) (50) Further, suppose that the predictor lters

H

;

1 (

)

(

) and

¹ (

) (51) are uniformly stable for

along with their rst, second, and third order derivatives.

Let

T



n

(

) =

(

(

)) (52)

^

T

N

(

) =

(

^

(

)) (53)

T

0 (

) =

(

0 ) (54) Assume that

lim

n!1 n

2

E

(

(

))

(

)] ² = 0 (55) which implies that

(

) tends to

0 (

) as

tends to innity. Let

(

) dened in (36) be denoted by

0 (

) when evaluated for

=

0 . Assume that

Z

0 (

)

0

(

) (56) is invertible. Further assume that

(

) and

(

) exist and that

(

) = 0,

k <

0. Finally, assume that lim

N!1

1

p

N N

X

t

=1

"

d

d

"

2 (

(

))

#

= 0 (

xed) (57)

We can now state the main result of the paper.

Theorem 2 Consider the estimate ^

(

) under the assumptions (36) and (45)-(57). Then

p

N 2

6

4

Re ^

(

)

(

)]

Im ^

(

)

(

)]

3

7

5

2AsN

(0

 (

)) (58) as

for xed

,

]

9

where

lim

!

0

lim 1

n



P

(

) =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

1 2 

(

)

2

4

Re 

¹ (

)] Im 

¹ (

)]

;

Im 

¹ (

)] Re 

¹ (

)]

3

5



if

mod

= 0



(

)

2

4

Re 

¹ (

)] 0

0 0

3

5



if

mod

= 0

(59) The proof is given in Appendix A.

From (58)-(59) we conclude that, for

mod

= 0, the random variable

p

N

( ^

(

)

(

)) (60) asymptotically has a complex normal distribution (e.g., 1]) with covariance matrix 

(

) satisfying

lim

!

0

lim 1

n



P



(

) = 

(

)

¹ (

) (61) Compare also with the results in 3]. (In connection to this the author would like to point out that the matrix on the right hand side of equation (3.22) in 3] should be transposed.) For

mod

= 0 this is no longer true, since then the diagonal blocks of

lim

!

0 lim

n!1

1

n



P

(

) (62)

are not equal. That the covariance for the imaginary part is zero in this case is very natural, since the transfer function is real-valued for

mod

= 0.

Further, with basically only notational dierences the result in Theorem 2 also holds for multivariable models. The extension of the results in 3] to the multivariable case was given in 8]. We can also prove the result for polyno- mial models where the dierent polynomials are of unequal orders, or if other criteria than the quadratic one is used. In open loop the situation simplies and, e.g., to prove the corresponding results for the model ^

(

) in this case the conditions in Theorem 2 can be relaxed so that a xed, arbitrary noise model

(

) =

(

) can be used. See 3] for all this.

Let us return to the result in Theorem 2. A more explicit expression for (59)

10

can be obtained if we note that



¹ (

) =

2

6

4

1



(

)



(

)

(

0 

(

))

;



(

)

(

0 

(

)) 1



(

)

3

7

5

(63)



(

) = 

(

)



(

)

²

0 (64)



(

) =

0



(

)

²



(

) (65)



(

) can be interpreted as the \noise-free" part of the input spectrum, i.e., that part of the total input spectrum 

(

) that does not originate from

fe

(

)

but from some external reference or set-point signal. In open loop we have 

(

) = 0, which, e.g., implies that 

(

) = 

(

), and the expression for 

¹ (

) simplies to



¹ (

) =

2

6

4

1



(

) 0 0 1

0

3

7

5

(66)

If we return to the general, closed loop situation we see from (63) that Re 

¹ (

)] =

2

6

4

1



(

)

Re 

(

)

(

0 

(

))]

;

Re 

(

)

(

0 

(

))] 1



(

)

3

7

5

(67) Im 

¹ (

)] =

2

6

4

0

Im 

(

)

(

0 

(

))]

;

Im 

(

)

(

0 

(

))] 0

3

7

5

(68) Using (67) and (68) we can thus easily prove the following consequence of The- orem 2, dealing with the asymptotic distribution of the estimate ^

(

).

Corollary 3 Consider the situation in Theorem 2. Then

p

N 2

6

4

Re ^

(

)

(

)]

Im ^

(

)

(

)]

3

7

52AsN

(0

(

)) (69) as

for xed

,

]

where

lim

!

0

lim 1

n

P

(

) =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

1 2 

(

)

2

4

1



(

) 0 0 1



(

)

3

5



if

mod

= 0



(

)

2

4

1



(

) 0

0 0

3

5



if

mod

= 0

(70)

11

From (69)-(70) we see that, for

mod

= 0, the random variable

p

N

( ^

(

)

(

)) (71) asymptotically has a complex normal distribution with covariance matrix

P



(

) satisfying lim

!

0

lim 1

n P



(

) = 

(

)



(

) (72)

This does not hold for

mod

= 0, but for all

we nevertheless have that tr

lim

!

0 lim

n!1

1

n

P

(

)

= 

(

)



(

) (73)

which ties in nicely with the results in 3], cf. (3) and (4).

An intuitive, but not formally correct, interpretation of the result (69)-(70) is that it gives the following convenient expression for the covariance matrix (3) (for the case

mod

= 0):

P

(

)

N

1 2

(

)

2

6

4

1



(

) 0 0 1



(

)

3

7

5

as

(74) From (74) we see that asymptotically, as both

and

tend to innity, the real and imaginary parts of the

-estimate are uncorrelated with equal variance proportional to the number-of-parameters-to-number-of-measurements ratio (

) and to the noise-to-signal ratio (

(

)



(

)).

4 Example

If we assume that the random variable

2

6

4

Re ^

(

)]

Im ^

(

)]

3

7

5

(75)

has a normal distribution with covariance matrix

(

), we may compute an



% condence ellipsoid for ^

(

) at a particular frequency

,

mod

= 0, as the set of all

(

) such that

2

6

4

Re

(

)

^

(

)]

Im

(

)

^

(

)]

3

7

5 P

;

1 (

)

2

6

4

Re

(

)

^

(

)]

Im

(

)

^

(

)]

3

7

5 T

C



(76)

12