On the reconstruction of continuous-time models from estimated ...

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ON THE RECONSTRUCTION OF CONTINUOUS-TIME MODELS FROM ESTIMATED DISCRETE-TIME MODELS OF STOCHASTIC PROCESSES 1 Kaushik Mahata ∗ and Minyue Fu ∗ ∗ Centre

for Complex Dynamic Systems and Control, University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected], [email protected]

Abstract: In this paper, we study the problem of reconstructing a continuous-time model from an identified discrete-time model for a continuous-time stochastic process. We present a new necessary and sufficient condition for the existence of the solution. We also show that the solution is unique if it exists. These results are then used to develop a continuous-time ARMA process modelling algorithm with guaranteed solution. The performance of this algorithm is tested using numerical simulations. Our approach is applicable to multivariable processes. Keywords: Continuous-time stochastic processes, stochastic modelling, spectrum estimation, linear matrix inequalities.

1. INTRODUCTION Due to the fact that most real-world dynamical processes are in continuous-time (CT), model identification in continuous-time is an important research topic (Larsson and Mossberg, 2003). Although the signals are CT, in practice one works with sampled data. One popular approach is to identify an intermediate discrete-time (DT) model from uniformly sampled data (Larsson and Mossberg, 2003; Söderström, 1991). Subsequently, the estimated DT model is converted back to a CT model via a nonlinear transformation (Söderström, 1991). This approach will be referred to as the indirect approach. Apart from the obvious difficulty of solving nonlinear equations, this approach also suffers from several other setbacks: (i) At a fast sampling rate, the poles and the zeros of the associated discrete-time system cluster close to the point 1 + i0 in the complex plane, leading to a numerically ill-conditioned identification problem; (ii) There is no guarantee that a solution exists for DT to CT model conversion problem; (iii) It is not known how the reconstructed CT model depends on the realization 1

of the DT model. These issues have been discussed in (Fan et al., 1999; Söderström and Mossberg, 2000; Larsson and Mossberg, 2003; Söderström, 1991). A second approach is to identify the continuous-time parameters directly. This can be done by replacing the differentiation operator with the so-called delta operator (Feuer and Goodwin, 1996; Goodwin et al., 1992). Several methods have been developed using this approach for autoregressive models, see (Fan et al., 1999; Söderström and Mossberg, 2000; Larsson and Söderström, 2002). This approach is advantageous in many cases as it is computationally efficient and one avoids nonlinear transformations if the underlying model is autoregressive. This technique also benefits from non-uniform sampling (Larsson and Söderström, 2002). However, it is not well understood how we can extend this technique to continuoustime ARMA (CARMA) modelling. Therefore, for CARMA models, the only available approach seems to be the indirect method described earlier (Larsson and Mossberg, 2003; Söderström, 1991).

In this paper we focus on DT to CT model transformation. We present a necessary and sufficient condition for the solvability of the model transformaPreprint submitted 14th IFAC Symposium on System Identification. This work is supported by Australian Researchto Council Received September 1, 2005.


tion problem. Unlike the previous results (Söderström, 1990; El-Khoury and Crisalle, 1992), our result is valid for multivariable processes of any order. It is further shown that the solution to the model conversion problem is unique if it exists. The theoretical results are used to develop a CARMA estimation algorithm, where the CT model is estimated directly from the estimated covariance data by solving a semidefinite programming problem.

2. PRELIMINARIES Suppose that we observe a continuous-time stationary stochastic process y(t) ∈ Rm , t ≥ 0, given in terms of the linear stochastic differential equation dz(t) = Az(t)dt + Bde(t),

y(t) = Cz(t),

Rn ,

(1) Rm ,

where z(t) ∈ m ≤ n. The process e(t) ∈ is a Weiner process with unit incremental covariance matrix. The problem under consideration is to model y(t) from a sampled version y(kh), k ∈ {0, 1, . . . , N − 1}. The sampled signal admits a discrete-time state space ˚ om, representation (Söderström, 2002, Page 86), (Astr¨ 1970) zd {(k + 1)h} = eAh zd (kh) + w(kh) y(kh) = Czd (kh).

(2)

where w(kh) is a fictitious discrete-time zero-mean white noise with

E {w(k1 h)w′ (k2 h)} = Rd δk1 ,k2 . Furthermore, Rd is given by Rd =

Z h

′

eAt BB′ eA t dt.

(3)

Now integrating Π(t) by parts we get −

Z h 0

′

Π(t)dt = P − eAh PeA h +

Z h 0

Π′ (t)dt,

which gives (5) after a rearrangement. One approach to identify a CARMA model is to identify an equivalent DT model first (Larsson and Mossberg, 2003). One then seeks for a transformation which maps the identified DT to an equivalent CT model. ˆ Cˆ and Rˆ d be the estimates of eAh , C and Let F, Rd , respectively. The DT to CT model transformation algorithm (Söderström, 1991) is as follows: Algorithm 1. (1) Estimate A as 1 ˆ Aˆ = log(F) (6) h (2) Estimate P by solving the discrete-time Lyapunov equation [see (5)] Pˆ − Fˆ Pˆ Fˆ = Rˆ d . (3) The CT transfer function G(s) := C(sI − A)−1 B is estimated by solving a spectral factorization problem ˆ Gˆ ′ (−s) = Φ ˆ − A) ˆ −1 × ˆ c (s) := −C(sI G(s) (Aˆ Pˆ + Pˆ Aˆ ′ )(−sI − Aˆ ′ )−1Cˆ ′ . (7)

In the following we remark on the existence and ˆ uniqueness of the solution G(s) in (7). In the rest of the presentation we make the following standard assumption (Larsson and Mossberg, 2003; Söderström, 1991).

0

The model (2) is the DT equivalent to the underlying CT model (1) in the sense that the second order statistics of the discrete-time model is consistent with the continuous-time process at the sampling instants. The continuous-time state z(t) and the discrete-time state zd (kh) have the same covariance matrix. Indeed, if P is the covariance matrix of z(t) then it must satisfy the continuous-time Lyapunov equation AP + PA′ + BB′ = 0.

(4)

It can be shown that P also satisfies the discrete-time Lyapunov equation ′

Rd = P − eAh PeA h .

(5)

To give a quick proof, we substitute (4) in (3) to obtain Rd = −

Z h

eAt (AP + PA′ )eA t dt

=−

Z h

Π(t)dt −

′

0

Z h

Π′ (t)dt,

Assumption 1. The spectrum of Fˆ lies in the interior of the open unit disc. Moreover, none of the eigenvalues of Fˆ lies on the interval (−1, 0]. The spectral factorization problem in (7) may not ˆ Cˆ and Rˆ d . This always admit a solution for all F, ˆ problem is encountered when Φc (s) fails to be positive definite on the imaginary axis. Several authors have investigated the conditions on the solvability. An analysis for SISO second order process is given in (Söderström, 1990; El-Khoury and Crisalle, 1992). See also (Wahlberg, 1988) for a discussion on sampling zeros. Given the spectrum of the estimated discrete-time model, Rˆ d is not unique for a given choice of coorˆ This is explicit in the following dinates of Fˆ and C. general result.

Lemma 1. Given Mi ∈ R(m+n)×(m+n) , i ∈ {1, 2}, let us define where we define ¸ · −1 ¸ · ′ −1 ′ ′ −1 ′ ′ (z I − F ) C (zI − F ) C ′ ′ ′ d . Mi Φi (z) := Π(t) = eAt APeA t = AeAt PeA t = {eAt }PeA t . I I Identification. dt to 14th IFAC Symposium on System Preprint submitted Received September 1, 2005. 0

0


Then Φ1 (z) = Φ2 (z), ∀z if and only if there exists Q = Q′ ∈ Rn×n such that M1 = M2 + H (F, Q,C), where

H (F, Q,C) :=

·

¸ Q − FQF ′ −FQC′ ′ ′ . −CQF −CQC

on the imaginary axis. Using the Kalman-YakubovitzPopov lemma (Rantzer, 1996), this is true if and only if there exists S = S′ ∈ Rn×n such that ¸ · ˆ + SAˆ ′ (S + P) ˆ Cˆ ′ AS ≥ 0, ˆ + P) ˆ C(S 0 which is equivalent to ˆ + SAˆ ′ > 0, AS

Proof: See (Genin et al., 1999b; Genin et al., 1999a). Corollary 1. Suppose we are given Rˆ d ∈ Rn×n and the associated spectrum estimate ˆ − F) ˆ d (z) := C(zI ˆ −1 Rˆ d (z−1 I − Fˆ ′ )−1Cˆ ′ . (8) Φ

ˆ Cˆ ′ = 0. (S + P)

(11)

The second condition in (11) is equivalent to S + Pˆ = C⊥ X for some X ∈ R(n−m)×n . Moreover, S + Pˆ is ′ = 0. Since C has symmetric. Hence we have CX ′C⊥ ⊥ a full column rank, this implies that ′ S + Pˆ = C⊥ ΩC⊥

(12)

Then ˆ − F) ˆ d (z) = Φ ¯ d (z) := C(zI ˆ −1 R¯ d (z−1 I − Fˆ ′ )−1Cˆ ′ , ∀z Φ

for some Ω = Ω′ ∈ R(n−m)×(n−m) . Combining the first condition in (11) and (12) we get (10). The following result addresses the uniqueness issue.

for some R¯ d 6= Rˆ d if and only if there exists Q = Q′ ∈ Rn×n satisfying QCˆ = 0 such that ˆ Fˆ ′ . R¯ d = Rˆ d + Q − FQ (9)

Theorem 2. Let R1 , R2 ∈ Rn×n , R1 6= R2 , be such that

ˆ d (z) = Φ ¯ d (z), ∀z if and only if Proof: By Lemma 1 Φ there Q = Q′ ∈ Rn×n such that · ¸ ¸ · R¯ d 0 Rˆ d 0 ˆ ˆ Q, C). + H (F, = 0 0 0 0 That is Q must be such that (9) holds, and ˆ Cˆ ′ = 0. ˆ ′=0 FQC CQ By Assumption 1, Fˆ is nonsingular. Hence QCˆ ′ = 0. From the above observations a question about the uniqueness arises naturally: If two different estimates of Rd give the same discrete-time spectrum, will they lead to the same continuous-time spectrum? 3. EXISTENCE AND UNIQUENESS In this section we give existence and uniqueness results for the DT to CT model transformation algorithm, Algorithm 1. These results will be used later for developing a new algorithm. Our first result gives a necessary and sufficient condition for the solvability ˆ of G(s) in (7). Theorem 1. Let C⊥ ∈ Rn×(n−m) be a full column rank ˆ ⊥ = 0. Then the spectral factorizamatrix such that CC tion problem (7) admits a solution if and only if ′ ′ ˆ′ ˆ ⊥ ΩC⊥ AC +C⊥ ΩC⊥ A > Aˆ Pˆ + Pˆ Aˆ ′ . (10) for some Ω = Ω′ ∈ R(n−m)×(n−m) . Proof: We can rewrite (7) as ˆ − A) ˆ −1 {Aˆ Pˆ + Pˆ Aˆ ′ }(−sI − Aˆ ′ )−1Cˆ ′ ˆ c (s) = −C(sI Φ ˆ − A) ˆ −1 {sPˆ − Aˆ Pˆ − sPˆ − Pˆ Aˆ ′ }(−sI − Aˆ ′ )−1Cˆ ′ = C(sI

ˆ − F) ˆ d (z) = C(zI ˆ −1 Ri (z−1 I − Fˆ ′ )−1Cˆ ′ Φ for i = 1, 2. Let P1 , P2 ∈ Rn×n satisfy ˆ i Fˆ ′ , Ri = Pi − FP

i = 1, 2.

(13)

Denote the reconstructed CT spectrums by ˆ − A) ˆ −1 {−AP ˆ i − Pi Aˆ ′ }(−sI − Aˆ ′ )−1 . Φci (s) = C(sI Then Φc1 (s) = Φc2 (s), ∀s. Proof: By Corollary 1 there exists Q = Q′ ∈ Rn×n such that ˆ Fˆ ′ . QCˆ = 0, R1 − R2 = Q − FQ Combining with (13) we have ˆ 1 − P2 − Q)Fˆ ′ . P1 − P2 − Q = F(P Since the spectrum of Fˆ lies in the interior of the open unit disc, we must have P1 − P2 = Q. Consequently, Φc1 (s) − Φc2 (s) = C(sI − A)−1 {−AQ − QA′ }(−sI − A′ )−1C′ = C(sI − A)−1 {sQ − AQ − sQ + −QA′ }(−sI − A′ )−1C′ = CQ(−sI − A′ )−1C′ +C(sI − A)−1 QC′ = 0 for all s, and the theorem follows.

4. PARAMETERIZATION VIA HALF-SPECTRUM In DT modelling, one often identifies the innovations model. We seek a rank-m positive definite Rˆ d = JJ ′ , where J ∈ Rn×m has a full column rank. The estimated ARMA transfer function

ˆ −1 PˆCˆ ′ . ˆ = Cˆ P(−sI − Aˆ ′ )−1Cˆ ′ +C(sI − A)

ˆ − F) ˆ −1 J Gˆ d (z) = C(zI

is minimum phase provided Rˆ d satisfy some additional ˆ c (s) admits a stable minimum-phase constraints (Anderson, 1973). However, our ultimate The function Φ aim is toonidentify theIdentification. underlying CT model, and we spectral factor if Preprint and onlysubmitted if it is positive to 14thdefinite IFAC Symposium System

Received September 1, 2005.


may choose not to impose such strict constraints on Rˆ d . This additional degree of freedom may lead to a simple and elegant estimation algorithm. In this section we explore such an approach. This framework is similar to (Mari et al., 2000; Stoica et al., 2000; Byrnes et al., 2000; Mahata and Fu, 2005). Decompose the spectrum of the DT sequence y(kh) in (2) in the causal and the anti-causal parts: ′

Φd (z) := C(zI − eAh )−1 Rd (z−1 I − eA h )−1C′ = L(z) + L′ (z−1 ), L(z) = D +C(zI − eAh )−1 H, eAh PC′

D′

(14) CPC′

where H = and D + = for some unique positive definite P satisfying (4) (Söderström, ˆ however, must 2002, page 96). The estimates Dˆ and H, satisfy some additional constraints in order to ensure that the associated DT spectrum and the reconstructed CT spectrum are positive definite on the unit circle and imaginary axis, respectively.

(15) n×(n−m) ˆ Let A be defined in (6), and C⊥ ∈ R be a ˆ ⊥ = 0. Then full column rank matrix such that CC the solution to the CT spectral density reconstruction problem for the DT function ˆ d (z) = L(z) ˆ + Lˆ ′ (z−1 ) Φ (16) exists if and only if ˆ Cˆ ′ , Hˆ = FQ

ˆ Cˆ ′ , Dˆ + Dˆ ′ = CQ ˆ Fˆ ′ > 0. Q − FQ ˆ′

ˆ + QA > AQ

(17) (18) (19)

for some Q = Q′ ∈ Rn×n and Ω = Ω′ ∈ R(n−m)×(n−m) . When the above holds then the reconstructed CT spectrum is given by ˆ − A) ˆ −1 {−AQ ˆ − QAˆ ′ }(−sI − Aˆ ′ )−1Cˆ ′ . ˆ c (s) = C(sI Φ (20) Proof: The positive real lemma (Anderson, 1967) implies that (17) and (18) are the necessary and suffiˆ d (z) to be strictly proper and cient conditions for Φ ˆ Φ(z) > 0, ∀|z| = 1. Using (16) and (17) we have ¸ ¸′ · −1 · (z I − F ′ )−1C′ (zI − F ′ )−1C′ ˆ M Φd (z) = I I where M=

·

Theorem 3 can be seen to provide a necessary and ˆ H, ˆ d (z) ˆ Cˆ and Dˆ so that Φ sufficient condition on A, defined in (16) is positive-definite on |z| = 1 and that it has an associated continuous-time counterpart. This condition must be incorporated in the estimation algorithm. In fact, the condition in Theorem 3 can be incorporated in an estimation algorithm for Φc (s) using various statistics extracted from the data. Here we show how this can be done when we use estimated covariance function of y(t). It is also possible to adapt to the framework where the half-spectrum is estimated (Byrnes et al., 2000; Mahata and Fu, 2005). In the following we consider SISO CARMA processes. The procedure can be generalized easily for the multivariable situation. Let us denote the covariances of the observed process as rτ = E {y(t + τh)y′ (t)}, which can be estimated from the data in the standard way. Then it is well-known that

Theorem 3. Define ˆ − L] ˆ := Dˆ + C[zI ˆ −1 Hˆ L(z)

′ ′ ˆ′ ˆ ⊥ ΩC⊥ AC +C⊥ ΩC⊥ A

5. A NEW ESTIMATION ALGORITHM

¸ ˆ Fˆ ′ 0 Q − FQ ˆ ˆ −Q, C). + H (F, 0 0

Therefore, by Lemma 1 we have ˆ d (z) = C(zI − F)−1 {Q − FQ ˆ Fˆ ′ }(z−1 I − F ′ )−1C′ . Φ ˆ Fˆ ′ is an estimate of Rd . In particular, Rˆ d = Q − FQ Consequently, by Theorem 2, the reconstructed CT spectral density function is given by (20). Now by ˆ c (s) > 0, ∀ Re(s) = 0 if and only if (19) Theorem 1, Φ holds for some Ω = Ω′ ∈ R(n−m)×(n−m) .

r0 = D + D′ = CPC′ rτ = CeAh(τ−1) H = CeAh PC′ ,

(21)

where P satisfies the Lyapunov equation (4). Here we choose to work in the observer canonical form (Kailath, 1980, Page 107), where we choose eAh and C in the following form i.e.:  ′   1 0 1 ··· 0 0  .. .  .. ..    . ..  . C= .  , eAh =  . ,  ..   0 0 ··· 1  0 −γn −γn−1 · · · −γ1 and {γk }dk=1 are the coefficients of the characteristic polynomial of eAh . Now from (21) we can verify using the Caley-Hamilton theorem that γd rτ + γd rτ+1 + · · · + γ1 rτ+d−1 + rτ+d = 0,

(22)

for all τ ≥ 0. Provided q ≥ 2n we can set up a linear regression problem by considering (22) for τ = q 0, 1, . . . , q − d, and replacing the covariances {rτ }τ=0 q by their estimates {ˆrτ }τ=0 . We denote the resulting estimates by {ˆγk }dk=1 , and hence   0 1 ··· 0  .. .. .  ..  . ..  . Cˆ = C, Fˆ =  . (23) .  0 0 ··· 1  −ˆγn −ˆγn−1 · · · −ˆγ1 In the simplified SISO case there is no need to estimate C. For the mutivariable case, however, we have to estimate C for which there are several standard procedures (Mari et al., 2000; Van Overschee and De Moor, 1993).

It remains to estimate H and D. Note that D is not uniquely identifiable in the multivariable case. We identify D + D′ instead. The approach is similar to (Mari et on al.,System 2000). We solve (21) in the least-squares Preprint submitted to 14th IFAC Symposium Identification.



sense subject to the constraints (17), (18) and (19). Define rq = [ rˆ0 rˆ1 · · · rˆq ]′ (24) Oq = [ Cˆ ′ Fˆ ′Cˆ ′ · · · (Fˆ ′ )qC′ ]′ ,

for some user defined positive definite weighting matrix W . The problem of minimizing (25) subject to the constraints (18) and (19) reduces to the following equivalent semidefinite programming problem (Vandenberghe and Boyd, 1996):  minimize ℓ    ˆ Fˆ ′ > 0  subject to Q − FQ  ′ ′ ˆ′ ′ ˆ ˆ ˆ AC ΩC +C ΩC A > AQ + Q A ⊥ ⊥ ⊥ ⊥ · ¸   ˆ ′q ℓ r′q − CQO   > 0.  ′ rq − Oq QCˆ W (26) The resulting solution for Q can then be used to conˆ ˆ c (s), see (20). Subsequently, G(s) struct Φ is obtained via spectral factorization. The summary of the proposed algorithm is given below.

1

10 Spectrum

where we consider covariances up to lag q − 1. Then imposing the equality constraints (17) we eliminate Hˆ and Dˆ (which are not necessary to estimate). The resulting least-squares objective function is ˆ ′W −1 (rq − Oq QC), ˆ (rq − Oq QC) (25)

2

10

0

10

−1

10

−2

10

0

2

4 6 Frequency [Hz]

8

10

Fig. 1. Comparison of the mean of estimated spectrum (dashed line) and the true spectrum (solid line). The mean ± standard deviation of the estimated spectrum is shown in dotted lines. Parameter a1 a2 a3 c1 c2 c3

True vale 0.3 9.0 0.9 1.0 0.5 6.0

Mean 0.3103 9.0179 0.9823 0.9855 0.5310 5.9908

Std Deviation 0.0492 0.1283 0.2255 0.0443 0.1972 0.4065

Table 1. Parameter estimation performance. q {ˆrτ }τ=0 ,

Algorithm 2. (1) Estimate the covariances for some q ≥ 2n. (2) Replace rτ in (22) by rˆτ and compute {ˆγi }ni=1 by solving a least-squares problem. Subsequently form Fˆ and Cˆ as in (23). Compute Aˆ as in (6). (3) Form rq and Oq as in (24). Choose a suitable W and compute Q by solving (26). ˆ c (s) using (20). Compute the unique (4) Compute Φ minimum-phase minimal degree spectral factor ˆ c (s) = Gˆ c (s)Gˆ ′c (−s). Gˆ c (s) satisfying Φ 6. NUMERICAL SIMULATION RESULTS

The algorithm proposed in the previous section is tested in a numerical simulation study. We consider a scalar continuous-time stochastic process with a spectrum c(s)c(−s) , Φc (s) = a(s)a(−s) where a(s) = s3 + 0.3s2 + 9s + 0.9,

c(s) = s2 + 0.5s + 6.

The correlation function of the chosen process has an oscillatory behaviour and a large time constant. To obtain a reliable estimate of such a process it is generally required to have a large observation time window. Here we work with a data-length of 500 seconds, sampled at a frequency of 20 Hz. We estimate the correlation function up to 5 seconds from the data, and use it in our estimation routine.

spectrum is compared with estimated mean value ± standard deviation. As can be seen in Figure 1, the estimated spectrum is unbiased and is accurate. We point out here that the weighting matrix W in this case is chosen as an identity matrix. In estimating the matrix logarithm (6) there are possible numerical problems if Fˆ has eigenvalues close to each other. In order to reduce the numerical problems the estimated correlation function (sampled at 20 Hz) is decimated and down sampling is done at 2Hz. The estimation of {γk }dk=1 in (22) is done with the decimated correlation function data at 2 Hz in 10 parallel channels. This approach is similar to (Haldar and Kailath, 1997) in principle. The estimation of Q uses the correlation data at a sampling interval of 0.5 seconds up to 10 seconds 1 . In fact it is natural to expect significant improvement in the accuracy if we use the whole correlation data and an optimized weighting matrix W , which is a part of the ongoing research. Finally we give the parametric estimation results in Table 1. Here ak and ck denote the coefficients of s3−k of a(s) and c(s), respectively. 7. CONCLUSIONS In this paper we have addressed the problem of reconstructing a continuous-time model from a given discrete-time model. A necessary and sufficient condition for the existence of the solution is given. It

The estimation results obtained from 100 Monte-Carlo 1 This is done to reduce the computational burden. simulations are shown in submitted Figure 1, where trueSymposium Preprint to 14ththeIFAC on System Identification.



is also shown that the solution is unique if it exists. Subsequently, we develop an algorithm to identify the continuous-time model directly from covariance data. In our approach it is not required to compute the underlying discrete-time model. Using the proposed algorithm it is guaranteed to get a solution and the amount of computation involved is mild. The numerical simulation results confirm that the proposed method is capable to delivering accurate and reliable estimates. The statistical properties of the proposed method is not considered in this paper, and is a part of ongoing research.

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