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Continuous-Time Subspace Model Identification Method Using Laguerre Filtering
3. REFORMULATION OF THE STATE SPACE MODEL AND THE DATA EQUATION
transformation of the state space equations under the w operator and derives some properties of this transformation. In section 4 the data equation is introduced, and a noise-free identification algorithm is derived. Section 5 discusses the use of instrumental variables to identify the system under the presence of process and measurement noise. In section 6, treatment of the initial conditions is covered. Finally section 7 shows an example of the algorithm and discusses some properties.
We first introduce the following continuous-time all-pass filter: s-a w= - - , s+a
(1 - w)wj,j
(1)
L(S)=~(s-a)j
= xo Jll)77l
J Jll)l
The above description can be thought of as:
= Ax(t) + Bu(t) + n(t)
y(t) = i(t) = Cx(t) x(O) = Xo
s+a
s+a
Jll)nxn
h were x E J& ; u E J&. ; z E J&. • A E J&. , B E 1!])nX771 C Jll)lxn d D 1!])lX771 h d . . . J&. , E J&. an E J&. are t e etermmlstlc n system matrices. The noise n(t) E IR and m(t) E IRi are Wiener processes with incremental covariances Qdt and Sdt. xo , n(t) and m(t) are independent. The system is assumed to be stable and minimal.
x(t)
QC
Proof: This follows directly from the observation that the filter (1 - w)w j is in fact the j-th order Laguerre filter,
dx = Axdt
Jll)n
= 0, .. ,
is an orthogonal basis for the function space 11.2.
In this paper we will consider the following linear, finite dimensional, continuous-time model:
x(O)
(5)
Lemma 1 With this all-pass filter w we have the following property:
2. NOTATION AND PROBLEM FORMULATION
+ Budt + dn dz = Cxdt + Dudt + dm
a>O
+ Du(t) + m(t)
for which this property is derived e.g in (MakiHi 1990) .
•
Lemma 2 With the all-pass filter (5), the model description (1) can be transformed into the following state space model:
= Awx+Bw(1-w)u+(l-w)nw+Kl(l-w)xo (l-w)y = Cwx+Dw(1-w)u+(1-w)mw+K2(1-w)xo wx
(2)
(6)
(3)
with
(4)
+ aI)-ln(t) mw(t) = C(aJ + A)-ln(t) + m(t) Aw = (A + aI)-l (A - aI) Bw = (A + aI)-l B C w = 2aC(A + aI)-l Dw = D - C(A + aI)-l B Kl = (A + aI)-l K2 = C(A + aI)-l nw(t) = (A
Q and S can be seen as the spectral densities of n(t) and m(t). But n(t) and m(t) strictly speaking do not exist since their variances are infinity.
As we can see, also the output y(t) of the stochastic model strictly speaking does not exist. Therefore we have to apply a pre-filter to both u(t) and y(t) , before sampling takes place. The choice of this pre-filter is discussed later in this paper. Problem formulation: Given a finite set of sampled, pre-filtered input and output data, at time instances kT , with k = O... N - 1, in the presence of measurement noise m(t) and process noise n(t) , according to the model structure eqn. (1), our task is to find the order n and the matrices [A , B, C, DJ, of the FDLTI state space representation of the system, up to a similarity transform.
The proof is left out for the sake of brevity.
•
In the rest of this article, we will use the following notation: lj (t) denotes the time domain representation (impulse response) of the j-th order Laguerre filter and Lj(s) its Laplace-domain representation. We will use the notation [ljy](t) to denote the convolution of y(t) with lj(t), i.e. [ljy](t) = J~ lj(t - T)y(T)dT .
For the sake of brevity, we restrict our discussion to the approximation of the extended observability matrix , necessary to estimate the pair [A, Cl. In (Verhaegen and Dewilde 1992) it was shown for the discrete-time case, that once this quantity is known, the full state space quadruple and the initial conditions are known. The same holds for the continuous-time case.
Although y(t) does not really exist, because of the infinite variance of m(t) , [ljy](t) is a £2 signal, since the filter lj(t) is a strictly proper stable function. With the use of this pre-filter we can sample the signals [ljy](t) 1094
Sinha, N.K. and G .P. Rao (1991). Identification of continuous-time systems. Kluwer Academic Publishers. Unbehauen, H. and G.P. Rao (1990) . Continuous-time approaches to system identificaton - A survey. A utomatica 26(1) , 23-35. van Overschee, Peter and Bart de Moor (1994). N4SID: Subspace algorithms for the identification of combined deterministic stochastic systems. A utomatica 30(1) , 75-93. Verhaegen, Michel (1994). Identification of the deterministic part of MIMO state space models given in innovations form from input-output data. A utomatica 30(1) , 61-74. Verhaegen, Michel and Patrick Dewilde (1992) . Subspace model identification part 1. The output-error statespace model identification class of algorithms. Int. J. Control 56(5), 1187-1210.
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transformation of the state space equations under the w operator and derives some properties of this transformation. In section 4 the data equation is introduced, and a noise-free identification algorithm is derived. Section 5 discusses the use of instrumental variables to identify the system under the presence of process and measurement noise. In section 6, treatment of the initial conditions is covered. Finally section 7 shows an example of the algorithm and discusses some properties.
We first introduce the following continuous-time all-pass filter: s-a w= - - , s+a
(1 - w)wj,j
(1)
L(S)=~(s-a)j
= xo Jll)77l
J Jll)l
The above description can be thought of as:
= Ax(t) + Bu(t) + n(t)
y(t) = i(t) = Cx(t) x(O) = Xo
s+a
s+a
Jll)nxn
h were x E J& ; u E J&. ; z E J&. • A E J&. , B E 1!])nX771 C Jll)lxn d D 1!])lX771 h d . . . J&. , E J&. an E J&. are t e etermmlstlc n system matrices. The noise n(t) E IR and m(t) E IRi are Wiener processes with incremental covariances Qdt and Sdt. xo , n(t) and m(t) are independent. The system is assumed to be stable and minimal.
x(t)
QC
Proof: This follows directly from the observation that the filter (1 - w)w j is in fact the j-th order Laguerre filter,
dx = Axdt
Jll)n
= 0, .. ,
is an orthogonal basis for the function space 11.2.
In this paper we will consider the following linear, finite dimensional, continuous-time model:
x(O)
(5)
Lemma 1 With this all-pass filter w we have the following property:
2. NOTATION AND PROBLEM FORMULATION
+ Budt + dn dz = Cxdt + Dudt + dm
a>O
+ Du(t) + m(t)
for which this property is derived e.g in (MakiHi 1990) .
•
Lemma 2 With the all-pass filter (5), the model description (1) can be transformed into the following state space model:
= Awx+Bw(1-w)u+(l-w)nw+Kl(l-w)xo (l-w)y = Cwx+Dw(1-w)u+(1-w)mw+K2(1-w)xo wx
(2)
(6)
(3)
with
(4)
+ aI)-ln(t) mw(t) = C(aJ + A)-ln(t) + m(t) Aw = (A + aI)-l (A - aI) Bw = (A + aI)-l B C w = 2aC(A + aI)-l Dw = D - C(A + aI)-l B Kl = (A + aI)-l K2 = C(A + aI)-l nw(t) = (A
Q and S can be seen as the spectral densities of n(t) and m(t). But n(t) and m(t) strictly speaking do not exist since their variances are infinity.
As we can see, also the output y(t) of the stochastic model strictly speaking does not exist. Therefore we have to apply a pre-filter to both u(t) and y(t) , before sampling takes place. The choice of this pre-filter is discussed later in this paper. Problem formulation: Given a finite set of sampled, pre-filtered input and output data, at time instances kT , with k = O... N - 1, in the presence of measurement noise m(t) and process noise n(t) , according to the model structure eqn. (1), our task is to find the order n and the matrices [A , B, C, DJ, of the FDLTI state space representation of the system, up to a similarity transform.
The proof is left out for the sake of brevity.
•
In the rest of this article, we will use the following notation: lj (t) denotes the time domain representation (impulse response) of the j-th order Laguerre filter and Lj(s) its Laplace-domain representation. We will use the notation [ljy](t) to denote the convolution of y(t) with lj(t), i.e. [ljy](t) = J~ lj(t - T)y(T)dT .
For the sake of brevity, we restrict our discussion to the approximation of the extended observability matrix , necessary to estimate the pair [A, Cl. In (Verhaegen and Dewilde 1992) it was shown for the discrete-time case, that once this quantity is known, the full state space quadruple and the initial conditions are known. The same holds for the continuous-time case.
Although y(t) does not really exist, because of the infinite variance of m(t) , [ljy](t) is a £2 signal, since the filter lj(t) is a strictly proper stable function. With the use of this pre-filter we can sample the signals [ljy](t) 1094
Sinha, N.K. and G .P. Rao (1991). Identification of continuous-time systems. Kluwer Academic Publishers. Unbehauen, H. and G.P. Rao (1990) . Continuous-time approaches to system identificaton - A survey. A utomatica 26(1) , 23-35. van Overschee, Peter and Bart de Moor (1994). N4SID: Subspace algorithms for the identification of combined deterministic stochastic systems. A utomatica 30(1) , 75-93. Verhaegen, Michel (1994). Identification of the deterministic part of MIMO state space models given in innovations form from input-output data. A utomatica 30(1) , 61-74. Verhaegen, Michel and Patrick Dewilde (1992) . Subspace model identification part 1. The output-error statespace model identification class of algorithms. Int. J. Control 56(5), 1187-1210.
1098