Identification of the Input-Output- and Noise-Dynamics of Linear Multivariable Systems

IDENTIFICA TION OF THE INPUT-OUTPUT- AND NOISE-DYNAMICS OF LINEAR MULTIV ARIABLE SYSTEMS P. Blessing Institut fur Regelungstechnik, Fachgebiet Regelsystemtechnik, Technische Hochschule Darmstadt, Darmstadt, F.R. G. Abstract. A method for the identification of the input-output-dynamics and noise-dynamics of a linear time-discrete multi variable system contaminated by coloured noise admitting a Markov representation is described using a state-space form. The considered system is equivalent to a system in an augmented model driven by white noise disturbances. Based on an observable canonic system-representation the final result is obtained as follows : perturbing the inputs by test-sequences the input-output-dynamics are identified first by an estimation of correlation-functions and a least squares fit. The residual between L,e output and its estimate calculated with the input-output model are then used to identify the noise-dynamics in an innovation representation. Utilizing results of time series analysis and evaluating the correlation functions of the residuals the parameters of a noise model are estimated by a least squares approach and an iterative gradient algorithm. Finally the models are combined to a global model with common coordinate bases. Keywords.

Identification; correlation methods; multivariable systems.

In this paper a method is proposed which separately identifies the input-output- and noise-dynamics of a multivariable system contaminated by coloured noise using the same data set. The resulting models are then combined in a global model.

INTRODUCTION The field of identification multi variable systems has developed rapidly over the past decade. The proposed methods may generally be divided into methods for the identification of the input-output-dynamics by evaluating input-output-sequences and approaches which yield to a noise-dynamic-model by analyzing stochastic output sequences. The pioneering work in the identification of the input-output-dynamics was done by Ho and Kalman (1966), who deduced a state-space model from a Markovparameter sequence. This work was continued by Gopinath (1969) and Budin (1971) using selector matrices. These approaches however attempt to estimate many more parameters than the system actually needs. Using a canoni c al form o f the state-space representation of the system the total number of unknown parameters are reduced. Canonical forms are the basis of several identification approaches (Ackerrnann, Bucy, 1971; Glover, Willems, 1974; Rosza, Sinha, 1975). A very promising approach is described in Bingulac and Farias (1977). Unfortunately the practical application of these methods is limited due to the assumption of noi s e-free input-output measurements. Stochastical perturbations in the identification of the input-output-dynamics are considered by Guidorzi (1975) and Sinha, Kwong (1977). On the other hand there are several publications dealing with the identification of the noisedynami c s of stochastic multivariable systems, e.g . Mehra (1971); Tse, Weinert (1975).

FORMULATION OF THE PROBLEM A linear time-invariant discrete multivariable stochastic system is described by the following state-space equations: x*(k+1)=A*x*(k)+B*U(k)+F*W (k)+v*(k) -

--

-

-

--'l{

-

(1)

=C*x* (k) +w (k) --

-y

where k takes intege r values. The statevector ~* ( .) has dimension m, the input ~ ( . ) and output ~(.) have dimensions q and p respectively. ~ ( .) and ~y( ' ) are zero-mean white Gaussian processes of dimensions p and r respectively and covariances E{w (k)w -'l{

E{w (k)w -y

T

-x

T

-y

(l)}=W 6(k,l) -'l{

(2)

(l)}=W 6(k,l) -y

where E{.} denotes the expectation and 6 (k,l) characteri zes the Kronecker-delta functi o n . ~(.) and ~y(') will be assumed uncorrelated. ~*(.) is a stationary coloured Gauss-Markov-proce ss-noise, which is described by the following dynamiC model of order n:

445

P. Blessing

446 ~(k+1)=~~ (k)+E:v~(k)

Because there are many equivalent systems which all describe the same input-output behavjour, it is necessary to represent Eq. (1) in a canonical form, where the number of parameters is reduced. With the equivalence transformation

asymptotically stable and {A*,Q*} is a completely controllable pair. Eqs. (7) and (9) describe the input-output- and noise-dynamics of the system completely. The matrices of these equations are unknown. The identification problem considered in this paper is the estimation of the parameters of a canonic innovation-model of the system which is equivalent to Eq. (7). The values of the orders m and n as well as the structural indices of the canonical form are assumed to be known a priori. The case of unknown order and indices is outlined in Blessing (1979). In this paper the identification problem is solved in following stages:

x(k)=T x*(k)

o

( 3)

v* (k) =D v(k) • -
It is assumed that A* and V* are asymptotically stable. In addition the pairs {A*,C*} and {~*,~*} are assumed to be complet;ly-observable and controllable respectively.

-

(4)

-0-

and by state-vector-augmentation Eqs. (3) can then be combined in the form

[ ~(k+1)1 v(k+1) -

=

[~

(1) and

~ll~(k)l + [~l -u(k)+ [E:.]--x w (k)

0 V -nxm -

v(k) -

0 -nxq

F

-
'---y----"

'--y----'

'-v--'

A*

8*

F*

X(k)] [

o (5)

[c 0 ] +w (k) ,,- ;pxn, v(k) --y

C*

-

where ~, B, C, D and F denote the transformed matrices ~,-~*~ ~*, ~v and E* respectively. The output y(.) is the sum of two effects:

Zu(·) due to the deterministic input ~(.) and ¥W(.) due to the stochastic sequences. The problem of separating the sequences ~(.) and ~y(') in identification procedures was found to be intractable (Tse and others, 1972, 1973). It is possible however to use another representatiop of the multivariable stochastic system. Using the innovation theory (Kailath, 19 70), the state model (5) with white noise sequences ~ ( .) and ~y ( .) can be replaced by an equivalent representation with a single white noise process. This innovation representation is given by

(7)

Identification of the input-outputdynamics

In order to identify the input-output-model {A, B, C} the inputs ~(k) are perturbed simulta;eously by pseudo-random-test-sequences, which show 'white noise character' . Using correlation techniques the unknown parameters may be estimated by a leastsquares fit.

o (6)

y'(k)

Selection of a suitable canonical form for the input-output-dynamics The input-output-dynamics are described by the realization {A, B, cl. To obtain a unique solution a-suitable canonic structure of this matrix triple should be selected which favors the computational treatment of the system equations.

Identification of the noise-dynamics

The identification of the noise-dynamics is based on the evaluation of the residual

~(k) = y(k)-iu(k)

where Yu(.) denotes the effect of the testinput ~(.) on the output, calculat~d ~i~ the estimated input-output-model { ~, ~, ~} . The residual is due to the noise processes acting on the output and to the errors in the estimated input- o utput-model. When the identification of the input-output-model converges, it can be shown that ~(k) converges to ¥W(k). In that case the residual can be described in the following innovation form of order m+n: X(k+1)

~~ (k) +Q~ (k )

z (k)

f~ (k)+~(k).

(11 )

=f*?f (k) +~(k).

X*(k) denotes the predicted state estimate of the state-vector in Eq. (5), i.e.: (8)

~ is the steady-state Kalman-gain of Eq. (5), which can be calculated by the recursive Kalman-filter algorithm. ~(k) describes the innovation sequence of the filter. For an o ptimal filter ~(k) is a white noise sequence with covariance

(9) It is assumed that the matrix { A*-Q*~* } is

(10)

This presentation is a canonical form of Eq. (7), omitting its deterministic part. It is related to Eq. (7) by

C = C*T- 1 -

--0

(12)

where ~o denotes an appropriate nonsingular transformation matrix. The parameters of the canonical noise-model { ~, f, Q, ~} are estimated using the results of time series analysis.

o

Formulation of a global model Because the input-output- and noise-model are identified separately using different coordi-

Identification of input-output- and noise-dynamics nate bases they finally have to be combined in a global model.

CANONICAL STATE-SPACE- AND CORRESPONDING DIFFERENCE EQUATIONS OF THE INPUT-OUTPUT-DYNAMICS

447

Using !O, Eq. (13) can be transformed into the canonical representation

x

x(k+l)

A

l.(k)

£~

(k) +~~ (k)

( 19) (k)+Y.,..(k)

where the matrices are related by Observable Canonic Representation (20)

B = T B* In the following the realization {~*, ~*, £* } of the input-output-dynamics is transformed into a canonic form, which is based on the observable structure of the pair { ~*, £*}. For that purpose it is more convenient to represent the system (1) - (3) in the form x' (k+l)

-0-

This observable canonic form is considered in several publications, e.g. Mayne (1972), Tse, Weinert (1975). The transformed matrices show following structures:

•

o

-m -lxl

1 ... _.......

(13)

~~1

I

i

I

-m -ll . ... .. 1-. ...

,

I I

I

____ __ - - ---.1'

A= where it is assumed that all the stochastic processes affecting the system are combined into an additive (coloured) output vector Yw (k), see Eq. (6).

0--------, -m -lxm p 1 ·· · ·· ··T

1

P

~---------

I

I

I

I

I

~1

Consider the completely observable pair {~*, £*}. Using the partitioned matrix

~ xm

L _________ _

I

~ -lxl

.!.m-l p

p ' ~T -pp

(21 )

The vectors * * · · · c*]T c=[c - 1 -p

(14)

T ~;J' = [a . . ... a. .

•

1JO

1Jmj-

l

J

j

= 1. .. i

(22)

the following vector sequences can be calculated

may be c o mbined to the parameter vector

*T2 * * A*T * ~1 ~1' ~ ~1 ' -

-1

T

a.

(15) c* A*Tc* A* T2 c* -p' -p' -p

• B

and selected in the following manner to construct the transformation-matrix

(23)

... b JT b -p,I1lp -Pl

[b 11 · .. b 1 - ,ml JT

b. . -1J

[b ij 1· .. b

ijq

•

[1

-1]

(24 )

(25)

T

--.J

(16) * A*T * ~'- ~'

...

A*T(mp-l) *]T '~ .

which are additionally observable in the output component Yi(·) in comparison to the foregoing outputs. For simplicity it is assumed that > 0, i

=

1 ... p

(17)

i.e. all o utputs are necessary to ensure complete ob s ervability. The general case mi ~ 0, i = 1 .. . p, is outlined in Blessing (1979). In Eq. (16) it is easy to see that

2.1xm

p (26)

1

2.1xm

p

- 1]

Input-Output-Difference Equations Using the observable canonic representation the state-vecto r ~(k ) in Eq. ( 19 ) c an be eliminated by calculating each output Yi(l) f o r specified values o f 1, given by the set o f output indices. Rearranging the terms and substituting them into Eq. (18) l e ads to the f o llowing input-output-difference equations i

y . (k+m.) 1

(1 8 )

The set of indices {ml ... mp } completely defines the structure of the pair { ~*, C* } .

[1

0

-lxm

L

1

mj -l E: i L a .. y . (k+ v )+ L uT (k+j)t; .. +

j=l v=o i

m = m1 + ... + mp .

1

C

A vector ~*T(i)~j is included in this matrix if and only if it is independent fr o m all previously selected vectors. The interpendence of a further vector is tested, afte r the foregOing vectors in (15) have been selected. The index mi can be interpreted as the number of states, which are controllable by ~(. ) and

m i

2.1xm

-

m

j

-1

1J V J

j=O-

- 1J (27)

L L a. . y (k+ v )+y (k+m. ) Wj 1 j=l v=O 1J V Wj i = 1 .•• p

with

r

448

P. Blessing for m. < m. ~

max -1

E.

~

m. - 1

for m. ~ m. ~ ~

~

Max{m.

m.

~

j

)

max

~

1

. ..

-1

where it is assumed that

max

i-I}.

i m1jJ-j-1 ~ .. = b.

.- E Ea. ,I, AbI . \ -~,mi-J 1jJ=1 A=l ~~,m1jJ~,m1jJ-J-A

with b .. = 0

-qx

-~J

(29)

for j < o.

1

From Eqs. (27) it is obvious that the output sequences are only coupled in one direction. This property makes it possible to estimate the parameters of the difference equations which imply the unknown parameters of the matrices A and B in a successively operating identification procedure starting with a small number of unknown parameters.

IDENTIFICATION OF THE INPUT-OUTPUT-DYNAMICS

If operating signals were used as inputsequences, the evaluation of Eq. (31) for different time shifts T would be very difficult in general, because a matrix of rather high order has to be inverted. But if it is assumed that the Markov-parameter-sequence has decayed to zero approximately for k>kmax the evaluation of Eq. (31) is simplified by using input sequences with white noise character as follows:

JP

(T)=E{s(k)rT(k-T)}=

-'-rS

-

-

[f~'Sl

for ITI < k

!.r,s

(30)

(T)

q

with

~ Ijl

r 1s i

(T) ••• Ijl

rpsi

(T)

J.

Ijl

~Yi

(35) unknown parameters, see Eq.

which is calculated by (36)

Formulating Eq. (34) for shift-intervals T=l··.Qi' with Qi > Ni, an equation-error vector ~ can be introduced by

5. (m. )-S.a.

=

-~

~

(37)

-~-~

where ~i is the unknown parameter-vector corresponding to the i-th subsystem of A, see Eq. (23). The terms ~(mi) and ~ co~tain input-output-correlation-functions. They are completely described by ~

~Yi

-T (m.+1) ... 1jl ~

~Yi

8.=[5 (0) ••• 5 (m -1): - 1 - 1 I 1

(m.+Q.) JT ~

(38)

~

5. (0) •.. 5. (m . -1)

-~

-~

~

J. (39)

Defining a loss function

T > 0

i

(21).

Now each element of the correlation vectors ~Yk(V) is replaced by its estimate iuyk(V)

-1.

(m.+T)= E E a .. Ijl (V+T)+ ~ j=l v=o ~J~Yj ~

i=1. .. p (34) T=1,2 ...

and

)

E.

m .-1 )

In the following only the i-th equation of (34) is considered, which contains

[_T s.(m.)= Ijl

m.-1

+ E !pT (j+T)~. j=O -uu -~J

~

(m. +T) = E E a .. Ijl (V+T) Yi ~ j=l v=o ~J~Yj

-~

Multiplying Eq. (27) by U(k-T) and taking the expectation leads to-the following equations: i

.'fu

e.

(T)j

max

Input sequences with these properties based on pseudo-random-binary-sequences are described in Tsafestas (1977) and Blessing (1977). If Eq. (33) is fulfilled Eq. (31) can be expressed for T > 0 by

-~

T

(33)

for T = 0

.

In this section a method for the identification of the unknown parameters in the input-output-model is described. This approach first identifies a nonparametric model (Markov-parameters) by correlation analysis and then estimates the state-spaceparameters with the aid of the least squares fit. As shown in Isermann, Baur (1974) this two-step identification method has some nice properties for the identification of singleinput single-output processes. The proposed method for multivariable systems makes use of the pseudo-statistical properties of testsequences. These test-sequences excite the inputs of the system simultaneously. The structural indices mi' i = 1 ••• p, of the input-output-aynamics are assumed to be known a priori. In the following a matrix of crosscorrelation functions of a p-vector ~(k) and a q-vector s(k) is defined by

(32)

1jJ > O.

(28)

max

Using Eqs. (21), (22) and (24) the vectors C. in Eq. (27) can be calculated by -~J

-~J

1. .. i

j

-1

1 .•. p

(31)

V.

~

=

T

e.e.

-~-~

(40)

and minimizing it with regard to the parameter-

449

Identification of input-output- and noise-dynamics vector ~ leads to a least-squares-estimate based on correlation functions :

follows from the assumption (17):

m. (41 )

~

i = 1 ••• p

> 0

(43)

If input-sequences satisfying Eq. (33) are used then the parameters of ~, see Eq. (24), can simply be calculated by

For identification of the noise model { ~, C, G, W} the correlation-functions of the ;esidual z(k), Eq. (10), are evaluated using results-of time series analysis. It is assumed that the structural indices mi' i = 1 ... p, are known a priori. This assumption and Eq. (43) imply that all the parameters of matrix C are given, see Eq. (26). Formulating the correlation-matrix fzZ(T) and using Eqs. (9) - (11) leads to following set of equations (Tse and Weinert, 1975):

biJ' k = ~

P = A P AT+Q~QT

This procedure is applied to Eq. (34) for i = 1 ... p. Because of the one-directional coupling in the observable canonic representation of the system, it is easy to see that the order of the matrix to be inverted increases from m1 (for the first subsystem) to m.

(42) (j)/a~ ~Yi where ~ (j) denotes the crosscorrelation ukYi between the input- and and Yi. Because all structural assumed to be nonzero, describe the matrix £,

Gutput-components


-'-'Zz

(0)

(44)

ff.e+~

(45)

~

indices have been they completely see Eq. ( 26 ).

In the general case mi = 0, iE{l ... p}, the unknown parameters in C can be estimated analogously to the estimation of ~, Blessing (1979).

(46) where P denotes the covariance of the statevector-X(k). With the aid of these equations the noise-model is estimated in two steps. The matrix A is identified first through a least squares-fit based on_correlation functions. Using the estimated A the parameters of G and Ware determined in the second step.

Estimation of A IDENTIFICATION OF THE NOISE DYNAMICS In the following an approach for the identification of the noise-dynamics is described using the same input-output-data as in the foregoing section. It is assumed that the identification of the input-output-model C} has converged. The noise-dynamics a;e described in an innovation-model, see Eq. (12). The transformation-matrix is constructed analogously to !o by using the pair {A*, C*} instead of {A*, C*}. Then the struct~re of the canonic pair T~, f } of this model corresponds to that of the input-outputpair {A, cl. The correspondences between the notations-of the input-output- and noisemodel are shown in table 1.

{A, B,

la

Corresponding to Eq. (23), the unknown parameters of the matrix A are combined in parameter-vectors~, i =-l ... p. Because of the properties of the observable canonic form of the noise model, these vectors can be estimated separately. In the following the estimation of ~ is considered, which contains

unknown parameters. Using Eqs. (46) , (21), (23) and the correspo ndenses in table 1 this vector is related to the crosscorrelations of the residual (Tse and Weinert, 1975; Blessing, 1979) by ~

z .z. J

TABLE

Correspondences between the InputOutput- and Noise-Dynamic-Model Input-OutputModel m

m

Structural indices

m.

m.

Outputmatrix

~

T

(T+m.) 1

= 1,2, ...

~

with

•••
Z . Z.

J

(T+m.) 1

J.

1

(a. ,a .. ,a. 'k) _A, -1 -1J

C

(49)

1

If each correlation function

_A,

(48)

jEt 1. .. p}

Noise-Model

Order

Systemmatrix

(47)

Ni = m + ... +m i 1

1J

(a . ,a . . ,a . 'k) -1 -1J

1J

c

~z . z. (v)

is subJ 1 stituted by its estimate and Eq. (48) is formulated for T = 1 ... Qi' with ~ > Ni, an errorvector ~i can be introduced by

e. =.6-J1 .. (m.)-S .. a. 1 -J1-1

(50)

-1

An output index mi of the noise-model can be interpreted as the number of states which are additionally observable by the output Yi(k) in comparison to the foregoing outputs. It is easy to show that mi ~ mi. Then it

with (51)

450

P. Blessing

I ~

~

••• :.6 .. (0) ••• .6 . . (m.-1)J. ,-:J~

-J~

~

(52)

Minimizing the scalar product of the errorvector yields the parameter estimate (53)

It should be noted that the estimation of the parameter vectors~, i = 1 •.• p, is nearly independent of the estimated input-outputmodel {A, B, C} which is only necessary to calculate the-residual z(k). An approach which considers the inp;t-output-dynamics as an auxiliary condition in estimating the vectors~, i = 1 ••. p, is described in a following section.

(57)

with the auxiliary condition (54). This optimization-problem can be solved by extending Eq. (57) by a symmetric Lagrange multiplier matrix L and the auxiliary condition (54). Then the modified loss function T =I+tr[r (P-A P A-G w G ) ] ~

I

m

-

-

~

(58)

--- ---

has to be minimized. The necessary conditions for the solution of this problem are given by:
(G,w,P,r)

m -

-

--

o -mxm

2>r

G and

Estimation of

(G,W,P,r)

C>I

W

m -

In the following the estimation of the matrices G and W of the noise-model is outlined, assumi;g that ~, f and lzz(v) are known exactly. This estimation-problem is often discussed in the literature as an adaptive filter-problem. The estimation is based on Eqs. (44) - (46). From our point of view however, the matrices A and ~zz(v), v = 1,2 ... , are not available. Theref~re they are replaced by their estimates! and ~z(v). Eqs. (44) and (45) are then given by:

-

Cl P

-

-

= r-2CT{~

-

~

-'-Zz

(0)

=

T

C P C +w .

-

-~

8I (G,W, P, r) m---o~

-'-Z,z

--

-

-2·{~

-'-Zz

-

--

= ~xm

(60)

(O)-CPC T -W}+ ----

- PCT-G W} -GT r G + -G T {~-* -A -'-Z - - -

m----

--

-

= 0

-pxp

(61 )

T

~

-A PC -GwL w+

-'-Z - - -

-2rGw

---

T

PC -Gw} G

~*

= -2.{~

--

T

-'-Z - - -

cG

-

=

--

-

0

(62)

-mxp

To obtain the solution of the optimizationproblem, these equations may be calculated iteratively as described in Blessing (1979). The resulting matrices are the estimates and ~ of the noise-dynamic-matrices Q and~. -

G

It is important to notice that~th~ identification of the noise-dynamics {A, G, depends for its accuracy on how well the inp;t-outputmatrices have been identified.

W}

(m ) JT

P

--

-* - T {~-~ff -Q~}f

-{~*-A

(55)

••• cp

---

-'-Z - - -

~T

----

If in addition the components of izz(v) are arranged in the following manner for different time-shifts T:

(O)-C PCT_W}C+

-'-Zz

_CT{~*_A P CT-G W}TA_A?r A +

OI (G,w,P,r) (54)

-

(59)

.

P Then Eq. (46) can be simplified using the properties of the observable canonic structure: (56) First one should note that in Eqs. (54) (56) in addition to G and W the covariance P is also unknown. These equations are ;atisfied only by one set {Q,~, E}. Tse and Weinert (1975) estimate these matrices by an iterative procedure, in which at each step a matrix inversion is necessary. In the following an iterative gradient-algorithm is described which is closely relaced to an algorithm proposed by Mehra (1970).

FORMULATION OF A GLOBAL MODEL construction of a Transformation Matrix In the preceeding sections the m-dimensional input-output-model x(k+1) (63)

C x (k) and the m-dimensional noise model ~(k+l)

~~ (k) +Q~ (k)

z (k)

f~ (k) +~ (k)

(64) To solve the nonlinear equations (54) - (56) an equivalent optimization problem is defined by:

with the innovation-covariance Whave been estimated using different coordinate bases.

Identification of input-output- and noise-dynamics These models are now combined to a global model by a similar transformation of the statevector ~(k). It should be pointed out that the identified noise-model is of less quality compared with the input-output-model, which is identified first. Therefore it should be ensured that the errors in the noise-model do not affect the accuracy of the input-outputmodel. Thus it follows that the transformation of ~(k) should be applied such that the first m components of the resulting state-vector ~(k) coincide with the state x(k) of the inputoutput-model. In.th~t case each of the inputoutput-matrices A, Band C is a sub-matrix of the corresponding matrices in the global model which may then be formulated as:

=f'~(k)+~(k)

(65)

c'

~

D']

-A

-

~'J .

= [C

~', ~'

=

(63),

(64),

(66) and

i-1G

(68)

a global model can be formulated in which the submatrices ~', ~' and~' are described by a relative small number of parameters.

B

[o v' -m-mxm -

p

~+

where ~ denotes the quadratic Ni-dimensional left upper submatrix of the input-outputmatrix ~ and ~i = mi - mi. It should be noticed that T is only dependent on the parameters of ~, ;hich have been identified in the first phase of the proposed method. It is easy to show that f is nonsingular. The inversion of may si;ply be realized analytically.

G'

.,

-m . xm . 1 + .. • +m

Using Eqs.

with

A

o

H. =

-~

i

~(k+l)=~'X(k)+~'~(k)+Q'~(k) X(k)

451

Identification of

and~'

are appropriate dimensioned matrices with arbitrary structures. The models (63) and (64) are combined to the model (65) by choosing a nonsingular transformation matrix which satisfies:

i,

A Using

Auxiliary Conditions

To fulfill the assumptions stated in the foregoing section the parameter-vectors~, i=l ... p, should be estimated with regard to the inputoutput pair {~, ~}. It can be shown that each vector ~ can be described by a linear transformation as follows:

(66)

H -p

~11 : L -p

(67)

fi

is of

lIN. = N. -N .. ~

~

(70)

~

The matrix ~ and the vector ~ are of appropriate dimensions. Their elements are known, because they only depend upon the matrix A, Blessing (1979), which has already been estimated. Eq. (69) and (70) imply that only lINi degrees of freedom of ~ are described independently .• The remaining degrees are fixed by the pair {A, C}. This fact is used in estimating the-parameter-vector ~. Substituting Eq. (69) into Eq. (50) yields

6 ..

-J~

(71 )

(m . )-S .. {K. 8 .+k.} ~

-J~

-~

~

-~

Then the scalar product of ~ is minimized with respect to the vector ~. The resulting estimate of fi is given by: -~

(72)

Using ~, the parameter-vector ~ is then calculated by

1

-m.xm-m ~

- - - -

[

-~.

. T·T· -1 T·T· . 8.=[K .S .. S .. K.J K.S .. CL.(m.)-S. k} -~-J~-J~-~ -~-J~ -J~ ~ -Ji-i·

J

with

~i =

-~-~

While ~ is a Ni-vector, the vector dimension

-i

l.!!l I. =:

(69)

= K. 8 . +k .

Q.

-~

These conditions are fulfilled if it is assumed that the eigenvalues of A occur definitely in A and that the inter~al couplings in the pair TA, C} are also described by {A, C}. It must be stated that these assumptions do not hold if A is estimated as shown in the foregoing chapter. However they can be accomplished by estimating A as described in the following section. Using these assumptions and generalizing the results of Defalque and others (1976) to multivariable systems a general form of f is found which satisfies Eq. (66), Blessi~g (1979). Because only m columns of T are fixed by Eq. (66) the remaining n columns may be used to obtain an appropriate parametrization of the submatrices D', v' and C'. A particular simple structure-of is obtained, if it is arranged as follows:

o

- -

- - - - 0T - - , - - -

-n i xn 1+ ... +n. 1 ~-

I I

I

I

-no

I

I ~ I

0

-n . xn ~

- - - - - --

1+1

+ ... +n

p_

( 73) This approach is used for the estimation of

~, i = 1 ... p. The noise-dynamic pair is then matching to the pair {~, C}.

{A,

~}

P. Blessing

452

CONCLUSIONS A method has been presented for the identification of linear multi variable systems perturbed by coloured noise. The described method separately identifies the parameters of an observable-canonic input-output-model and a noise-model in an innovation representation. Both stages of the identification are based on correlation techniques. The crosscorrelation-functions between the inputs and outputs - as a nonparametric model of the input-output-dynamics - allow a preliminary examination before estimating the parameters of the state-space model. As described in previous publications these correlationfunctions furtheron may be used to estimate the structural indices of the input-outputmodel before estimating the parameters. Hence an iteration on these indices by estimating the parameters of different models and then choosing an appropriate structure is avoided. The crosscorrelation-functions of the residuals between the outputs and its estimate - calculated with the identified input-output-model - are evaluated to obtain the parameters of a noise model by using a least squares approach and an iterative gradient algorithm. Finally these models are combined to a global model. This approach however has an inherent disadvantage that the accuracy of the noise model is dependent upon the quality of the identified input-outputdynamics.Nevertheless simulation-studies showed good results. The proposed method has also been applied successfully to the identification of a drying-process and an airconditioning system. ACKNOWLEDGEMENT This report publishes results of the researchproject DFG Is 14/13, supported by the Deutsche Forschungsgemeinschaft. REFERENCES Ackermann, J. and R.S. Bucy (1971). Canonical minimal realization of a matrix of impulse response sequences. Inf. Contr., 19, 224-231. Bingulac, S.P. and M.A.C. Farias (1977). Identification and minimal realization of multivariable systems. Prepr. IFACSymp. on Mult. Techn. Syst., Pergamon Press, 373-377. Blessing, P. (1977). Parameter estimation of state-space models for multivariable systems with correlation analysis and method of least squares. Prepr. IFACSymp. on Mult. Techn. Syst., Pergamon Press, 385-394. Blessing, P. (1979). Identifikation von linearen stocha~tisch gestor ten MehrgroBensystemen. PDV-Bericht, GfK Karlsruhe, to be published.

Budin, M.A. (1971). Minimal realization of discrete linear systems from input-output-measurements. IEEE-Trans. on AC, Vol. AC-16, 395-401. Defalque, B., M. Gevers and M. Installe (1976) Combined identification of the inputoutput ar.d noise dynamics of a closed loop controlled linear system. Int. J. Contr., Vol. 24, No. 3, 345-360. Glover, K. and J.C. Willems (1974). Parameterization of linear dynamical systems: Canonical forms and identifiability. IEEE Trans. on AC, Vol. AC-19, 640-646. Gopinath, B. (1960). On the identification of linear time invariant systems from input-output data. Bell Syst. Tech. J., Vol. 48, No. 5, 1101-1113. Guidorzi, R.P. (1973). Canonical structures in the identification of multi variable systems. IFAC-Symp. on Ident., paper TS.4, The Hague. Ho, B.L. and R.E. Kalman (1966). Effective construction of linear state-variable models from input-output functions. Regelungstechnik, 14, 545-548. Isermann, R. and U. Baur (1974). Two-step process identification with correlation analYSis and least squares parameter estimation. J. Dyn. Syst. Meas. and Contr., Paper 75-Aut-B. Kailath, T. (1970). The innovations approach to detection and estimation. Proc. IEEE, Vol. 58, 680-695. Mayne, D.Q. (1972). A canonical model for identification of multivariable systems. IEEE Trans. on AC, Vol. AC-17, 728-729. Mehra, R.K. (1970). An algorithm to solve matrix equations pHT=G and p=~p~T+rrT. IEEE Trans. on AC, Vol. AC-15, 600. Mehra, R.K. (1971). On-line identifications of linear dynamic systems with applications to Kalman filtering. IEEE-Trans. on AC, Vol. AC-16, 12-21. Rozsa, P. and N.K. Sinha (1975). Minimal realization of a transfer function matrix in canonical forms. Int. J. Contr., Vol. 21, No. 2, 273-284. Sinha, N.K. and Y.H. Kwong (1977). Recursive identification of the parameters of a multivariable system. Prepr. IFAC-Symp. on Mult. Techn. Syst., Pergamon Press, 323-327. Tzafestas, S.G. (1977). Multivariable control system identification using pseudorandom test inputs. Aut. Contr. Theo. and Appl., Vol. 5, No. 3. Tse, E. and J. Anton (1972). On the identifiability of parameters. IEEE Trans. Aut. Contr., Vol. AC-17, 637-646. Tse, E. and H. Weinert (1973). Correction and extension of 'On the identifiability of parameters'. IEEE Trans. on AC, Vol. AC-18, 687-688. Tse, E. and H. Weinert (1975). Structure determination and parameter identification for multivariable stochastic systems. IEEE Trans. on AC, Vol. AC-20, 603-613.