Joint state filtering and parameter estimation for linear stochastic time-delay systems

Signal Processing 91 (2011) 782–792

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Joint state filtering and parameter estimation for linear stochastic time-delay systems Michael Basin a,, Peng Shi b,1,2,#, Dario Calderon-Alvarez a a

Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Apdo postal 144-F, C.P. 66450, San Nicolas de los Garza, Nuevo Leon, Mexico Department of Computing and Mathematical Sciences, Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, United Kingdom

b

a r t i c l e i n f o

abstract

Article history: Received 12 February 2010 Received in revised form 2 June 2010 Accepted 26 August 2010 Available online 8 September 2010

This paper presents the joint state filtering and parameter estimation problem for linear stochastic time-delay systems with unknown parameters. The original problem is reduced to the mean-square filtering problem for incompletely measured bilinear timedelay system states over linear observations. The unknown parameters are considered standard Wiener processes and incorporated as additional states in the extended state vector. To deal with the new filtering problem, the paper designs the mean-square finite-dimensional filter for incompletely measured bilinear time-delay system states over linear observations. A closed system of the filtering equations is then derived for a bilinear time-delay state over linear observations. Finally, the paper solves the original joint estimation problem. The obtained solution is based on the designed mean-square filter for incompletely measured bilinear time-delay states over linear observations, taking into account that the filter for the extended state vector also serves as the identifier for the unknown parameters. In the example, performance of the designed state filter and parameter identifier is verified for a linear time-delay system with an unknown multiplicative parameter over linear observations. & 2010 Elsevier B.V. All rights reserved.

Keywords: Filtering Parameter identification Bilinear stochastic time-delay system

1. Introduction The optimal filtering problem for linear system states and observations without delays was solved in 1960s [1], and this closed-form solution is known as the Kalman– Bucy filter. However, the mean-square filtering problem for linear states with delay has not been solved in a closed form, regarding as a closed-form solution a closed system

 Corresponding author. Tel.: + 52 81 83294030; fax: + 52 81 83522954. E-mail addresses: [email protected], [email protected], [email protected] (M. Basin), [email protected] (P. Shi), [email protected] (D. Calderon-Alvarez). # EURASIP. 1 Tel.: +44 1443 482147; fax: + 44 1443 482711. 2 Also with School of Engineering and Science, Victoria University, Melbourne, Vic 8001, Australia.

0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.08.011

of a finite number of ordinary differential equations for any finite filtering horizon. Some particular cases, the meansquare filtering problems for linear systems with state delay and/or observation delays, have been solved in [2–4]. There exists a large bibliography related to filtering problems for time-delay systems, such as [5–31] and many others. Comprehensive reviews of theory and algorithms for time-delay systems can be found in [32–39]. Although the general solution to the mean-square filtering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the Kushner equation for the conditional density of an unobserved state with respect to observations [40], there are a very few known examples of nonlinear systems where the Kushner equation can be reduced to a finite-dimensional closed system of filtering equations for a certain number of lower conditional moments (see [1,41–43]). The last two papers actually deal with

M. Basin et al. / Signal Processing 91 (2011) 782–792

specific types of polynomial filtering systems. There exists a considerable bibliography on robust filtering for the ‘‘general situation’’ systems (see, for example, [5,8–11,14,15,17,18]). Apart form the ‘‘general situation,’’ the mean-square finite-dimensional filters have recently been designed [44–46] for certain classes of polynomial non-delayed system states with Gaussian initial conditions over linear observations. As follows from the given historical review, the mean-square filtering problems were solved only for linear time-delay systems. However, most practically used technical systems, such as typical mechanical, power, thermochemical, and biochemical ones, are intrinsically bilinear. Thus, designing a meansquare finite-dimensional filter for bilinear time-delay systems leads to a broad spectrum of possible applications. Moreover, the next natural step is to solve the joint state and parameter mean-square estimation problem for linear stochastic time-delay systems with unknown parameters, incorporating the unknown parameters as additional states in the extended state vector. Solution of these two indicated problems constitutes the main result of this paper. This paper presents the joint state filtering and parameter estimation problem for linear stochastic timedelay systems with unknown parameters over linear observations. The solution starts with reduction of the original problem to the mean-square filtering problem for incompletely measured bilinear time-delay system states over linear observations with an arbitrary, not necessarily invertible, observation matrix, upon considering the unknown parameters as additional system states satisfying linear stochastic Ito equations with zero drift and unit diffusion, i.e., standard Wiener processes. In doing so, the unknown parameters are incorporated as additional states in the extended state vector, which should be estimated solving the mean-square filtering problem for bilinear time-delay states. To deal with the new filtering problem for the extended state vector, this paper presents the mean-square finitedimensional filter for incompletely measured bilinear time-delay system states over linear observations with an arbitrary, not necessarily invertible, observation matrix, thus generalizing the results of [44–46]. The filtering problem is treated proceeding from the general expressions for the stochastic Ito differentials of the mean-square estimate and the error variance [47,48]. As the first result, the Ito differentials for the mean-square estimate, the error variance, and the error covariance between the currenttime and delay-shifted values corresponding to the stated filtering problem are derived. Next, a closed finite-dimensional system of the filtering equations with respect to a finite number of filtering variables can be obtained for a bilinear time-delay state and linear observations with an arbitrary observation matrix. Finally, the closed system of the filtering equations with respect to three variables, the mean-square estimate, the error variance, and the error covariance between the current-time and delay-shifted values is derived in the explicit form in a particular case of a bilinear equation with state delay. The paper then focuses on the original joint state filtering and parameter estimation problem for linear

783

stochastic time-delay systems with unknown parameters over linear observations, whose solution is based on the obtained mean-square filter for incompletely measured bilinear time-delay states. The designed filter for the extended state vector also serves as the identifier for the unknown parameters. This presents the mean-square algorithm for the joint state and parameter estimation in linear time-delay systems with unknown parameters over linear observations. Note that since the original estimation problem is reduced to the filtering problem for the extended system state including both state and parameters, the identifiability condition for the original system coincides with the observability condition for the extended system. As follows from the results of Theorem 7.4 and Section 7.7 in [47], the observability condition for the extended system implies the almost sure convergence of the state and parameter estimates to their real values. In the illustrative example, performance of the designed mean-square filter is verified for a linear timedelay system with an unknown multiplicative parameter over linear observations. The simulations are conducted for both, negative and positive, values of the parameter, thus considering stable and unstable linear systems. The simulation results demonstrate reliable performance of the filter: in both cases, the state estimate converges to the real state and the parameter estimate converges to the real parameter value rapidly, in less than 10 time units. The result is viewed even more promising, taking into account large deviations in the initial values for the real state and its estimate and large values of the initial error variances. The paper is organized as follows. Section 2 presents the filtering problem for incompletely measured bilinear time-delay system states over linear observations with an arbitrary observation matrix. The Ito differentials for the mean-square estimate, the error variance, and the error covariance between the current-time and delay-shifted values are derived in Section 3. Section 4 presents the joint state filtering and parameter estimation problem for an incompletely measured linear time-delay system state with unknown parameters over linear observations. In Section 5, the stated problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states. The resulting filtering system is bilinear in time-delay state, which is incompletely measured, and linear in observations. The meansquare filtering equations are given, based on the meansquare filtering equations from Section 3. In Section 6, performance of the designed filter is verified for a linear time-delay system with incompletely measured states and an unknown multiplicative parameter over linear observations. Both, stable and unstable, linear systems are examined. The simulation results show rapid convergence of the obtained estimates to the real state and parameter values.

2. State filtering problem statement Let ðO,F,PÞ be a complete probability space with an increasing right-continuous family of salgebras Ft ,t Z t0 ,

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M. Basin et al. / Signal Processing 91 (2011) 782–792

and let ðW1 ðtÞ,Ft ,t Zt0 Þ and ðW2 ðtÞ,Ft ,t Z t0 Þ be independent standard Wiener processes, EðW1 ðtÞW1T ðtÞÞ ¼ tI1 and EðW2 ðtÞW2T ðtÞÞ ¼ tI2 , where Ik, k ¼ 1,2, are the identity matrices of appropriate dimensions. The Ft-measurable random process (x(t),y(t)) is described by a nonlinear differential equation with a bilinear polynomial timedelay drift term for the system state dxðtÞ ¼ f ðx,tÞ dt þ bðtÞ dW 1 ðtÞ,

ð1Þ

where f ðx,tÞ ¼ a0 ðtÞ þa1 ðtÞxðtÞ þa2 ðtÞxðtÞxT ðthÞ, with the initial condition xðsÞ ¼ fðsÞ,s 2 ½t0 h,t0 , h is the state delay value, and a linear differential equation for the observation process dyðtÞ ¼ ðA0 ðtÞ þ AðtÞxðtÞÞ dt þBðtÞ dW 2 ðtÞ: n

ð2Þ m

Here, xðtÞ 2 R is the state vector and yðtÞ 2 R is the linear observation vector, mr n, a0(t) is an n-dimensional vector, a1(t) is an n  n-matrix, a2(t) is 3D tensor of dimension n  n  n. The initial condition x0 2 Rn is a Gaussian vector such that x0, W1 ðtÞ 2 Rp , and W2 ðtÞ 2 Rq are independent. The system state x(t) dynamics depends on the delayed state x(t h), which actually makes the system state space infinite-dimensional (see, for example, [34]). The observation matrix AðtÞ 2 Rmn is not supposed to be invertible or even square. It is assumed that B(t)BT(t) is a positive definite matrix, therefore, m r q. All coefficients in (1)–(2) are deterministic functions of appropriate dimensions. Note that the Eq. (1) is an Ito stochastic differential equation, since its right-hand side does not depend on time-advanced arguments. Therefore, its solution can be defined in the standard manner employing the Ito integral of its right-hand side. The rigorous definition of the unique weak solution of an Ito equation can be found, for example, in [49]. The same definition holds for solutions of other Ito stochastic differential equations throughout the paper, including the obtained filter equations. The function f ðx,tÞ 2 Rn , f ðx,tÞ ¼ a0 ðtÞ þ a1 ðtÞxðtÞ þ a2 ðtÞ xðtÞxT ðthÞ, which is a bilinear polynomial of n variables, components of the state vector xðtÞ 2 Rn , with timedependent coefficients, can also be expressed in the summation form (see [44] for more details) fk ðx,tÞ ¼ a0 k ðtÞþ

X X a1 ki ðtÞxi ðtÞ þ a2 i

kij ðtÞxi ðtÞxj ðthÞ,

k,i,j ¼ 1, . . . ,n:

ij

In general, a 3D tensor a2 2 Rnnn multiplied by a matrix P A 2 Rnn yields a vector a 2 Rn defined by ak ¼ ij a2 kij Aij , nnn and a 3D tensor a2 2 R multiplied by a vector b 2 Rn P yields a matrix B 2 Rnn defined by Bij ¼ k a2 ijk bk . The estimation problem is to find the mean-square ^ estimate xðtÞ of the system state x(t), based on the observation process YðtÞ ¼ fyðsÞ,t0 r s rtg, that minimizes the mean-square criterion T Y ^ ^ ðxðtÞxðtÞÞjF J ¼ E½ðxðtÞxðtÞÞ t 

of the system state x(t) with respect to the salgebra FYt generated by the observation process Y(t) in the interval [t0,t]. As usual, the matrix function PðtÞ ¼ E½ðxðtÞmðtÞÞðxðtÞmðtÞÞT jFtY  is the estimation error variance. The proposed solution to this optimal filtering problem is based on the formulas for the Ito differential of the conditional expectation EðxðtÞjFtY Þ and its variance P(t) (cited after [47,48]) and given in the following section. 3. Filter design The stated filtering problem is solved by the following theorem. Theorem 1. The mean-square filter for the polynomial bilinear time-delay state x(t) (1) over the incomplete linear observations y(t) (2) is given by the following equations for the mean-square estimate mðtÞ ¼ EðxðtÞjFtY Þ, the estimation error variance PðtÞ ¼ E½ðxðtÞmðtÞÞðxðtÞmðtÞÞT jFtY , and the estimation covariance Pðt,thÞ ¼ EððxðtÞmðtÞÞðxðthÞ mðthÞÞT jFtY Þ dmðtÞ ¼ a0 ðtÞ þ a1 ðtÞmðtÞ þ a2 ðtÞ½Pðt,thÞ þmðtÞmT ðthÞ þPðtÞAT ðtÞðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þ AðtÞmðtÞÞ dtÞ,

ð3Þ

dPðtÞ ¼ ð2a2 ðtÞmðthÞPðtÞ þð2a2 ðtÞmðthÞPðtÞÞT þa1 ðtÞPðtÞ þPðtÞaT1 ðtÞ þbðtÞbT ðtÞ PðtÞAT ðtÞðBðtÞBT ðtÞÞ1 AðtÞPðtÞÞ dt,

ð4Þ

dPðt,thÞ ¼ ð2a2 ðtÞmðthÞPðt,thÞ þ ð2a2 ðtÞmðt2hÞPðth,tÞÞT

þa1 ðtÞPðt,thÞ þP T ðt,thÞaT1 ðtÞ þ 1=2½bðtÞbT ðthÞ þbðthÞbT ðtÞð1=2Þ½PðtÞAT ðtÞðBðtÞBT ðtÞÞðBðtÞBT ðthÞÞ1 ðBðthÞBT ðthÞÞAðthÞPðthÞÞ

þPðthÞAT ðthÞðBðthÞBT ðthÞÞðBðthÞBT ðtÞÞ1 ðBðtÞBT ðtÞÞAðtÞPðtÞÞ dt:

ð5Þ

with the initial conditions mðsÞ ¼ EðfðsÞÞ, s 2 ½t0 h,t0 Þ, mðt0 Þ ¼ Eðfðt0 ÞjFtY0 Þ,Pðt0 Þ ¼ E½ðxðt0 Þmðt0 ÞÞðxðt0 Þmðt0 ÞÞT jFtY0 , and Pðs,shÞ ¼ E½ðxðsÞmðsÞÞðxðshÞmðshÞÞT jFsY  for s 2 ½t0 ,t0 þ hÞ. The system of filtering Eqs. (3)–(5) becomes a closed-form finite-dimensional system after expressing the superior conditional moments of the system state x(t) with respect to the observations y(t) as functions of only three lower conditional moments, m(t), P(t), and P(t,t  h). Proof. The mean-square filtering equations could be obtained using the formula for the Ito differential of the conditional expectation mðtÞ ¼ EðxðtÞjFtY Þ (see [47,48]) dmðtÞ ¼ Eðf ðx,tÞjFtY Þ dt þ Eðx½j1 ðxÞEðj1 ðxÞjFtY ÞT jFtY Þ ðBðtÞBT ðtÞÞ1 ðdyðtÞEðj1 ðxÞjFtY Þ dtÞ,

at every time moment t. Here, E½xðtÞjFtY  means the conditional expectation of a stochastic process T ^ ^ xðtÞ ¼ ðxðtÞxðtÞÞ ðxðtÞxðtÞÞ with respect to the salgebra FYt generated by the observation process Y(t) in the interval ½t0 ,t. As known [47,48], this mean-square estimate is given by the conditional expectation

where f ðx,tÞ ¼ a0 ðtÞ þa1 ðtÞxðtÞ þa2 ðtÞxðtÞxT ðthÞ is the bilinear drift term in the state equation, and j1 ðxÞ ¼ A0 ðtÞ þAðtÞxðtÞ is the linear drift term in the observation equation. Upon performing substitution, the estimate equation takes the form

^ ¼ mðtÞ ¼ EðxðtÞ j FtY Þ xðtÞ

dmðtÞ ¼ Eða0 ðtÞ þ a1 ðtÞxðtÞ þ a2 ðtÞxðtÞxT ðthÞjFtY Þ dt

M. Basin et al. / Signal Processing 91 (2011) 782–792

þEðxðtÞ½AðtÞðxðtÞmðtÞÞT jFtY Þ

yields

ðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þ AðtÞmðtÞÞ ¼ ða0 ðtÞ þ a1 ðtÞEðxðtÞjFtY Þ þa2 ðtÞEðxðtÞxT ðthÞjFtY ÞÞ dt

dPðt,thÞ ¼ ðEððxðtÞmðtÞÞðf ðxth ,thÞÞT j FtY Þ þ Eðf ðx,tÞðxðthÞmðthÞÞT Þ j FtY Þ

þEðxðtÞðxðtÞmðtÞÞT jFtY ÞAT ðtÞ

þ ð1=2Þ½bðtÞbT ðthÞ þ bðthÞbT ðtÞ ð1=2Þ½EðxðtÞ½j1 ðxÞEðj1 ðxÞjFtY ÞT jFtY ÞðBðtÞBT ðtÞÞ

ðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þ AðtÞmðtÞÞ dtÞ ¼ ða0 ðtÞ þ a1 ðtÞmðtÞ þa2 ðtÞEðxðtÞxT ðthÞjFtY ÞÞ dt þPðtÞAT ðtÞðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þAðtÞmðtÞÞ dtÞ ¼ ða0 ðtÞ þ a1 ðtÞmðtÞ þa2 ðtÞ½Pðt,thÞ þ mðtÞmT ðthÞÞ dt þPðtÞAT ðtÞðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þAðtÞmðtÞÞ dtÞ, ð6Þ where Pðt,thÞ ¼ EððxðtÞmðtÞÞðxðthÞmðthÞÞT jFtY Þ is the covariance of the estimation error values at the current time t and the delay-shifted moment t h. The Eq. (6) should be complemented with the initial condition mðsÞ ¼ EðfðsÞÞ, s 2 ½t0 h,t0 Þ, mðt0 Þ ¼ Eðfðt0 ÞjFtY0 Þ. Trying to compose a closed system of the filtering equations, the Eq. (6) should be complemented with the equations for the error variance P(t) and covariance P(t,t h). For this purpose, the formula for the Ito differential of the variance PðtÞ ¼ EððxðtÞmðtÞÞðxðtÞmðtÞÞT jFtY Þ is used (cited again after [47,48]): dPðtÞ ¼ ðEðððxðtÞmðtÞÞðf ðx,tÞÞT jFtY Þ þEðf ðx,tÞðxðtÞmðtÞÞT ÞjFtY Þ þbðtÞbT ðtÞEðxðtÞ½j1 ðx,tÞEðj1 ðx,tÞjFtY ÞT jFtY Þ ðBðtÞBT ðtÞÞ1 Eð½j1 ðx,tÞEðj1 ðxÞjFtY ÞxT ðtÞjFtY ÞÞ dt þEððxðtÞmðtÞÞðxðtÞmðtÞÞ½j1 ðx,tÞEðj1 ðx,tÞjFtY ÞT jFtY Þ ðBðtÞBT ðtÞÞ1 ðdyðtÞEðj1 ðx,tÞjFtY Þ dtÞ,

ðBðtÞBT ðthÞÞ1 ðBðthÞBT ðthÞÞEð½j1 ðxth ,thÞ Eðj1 ðxth ,thÞjFtY ÞxT ðthÞjFtY ÞÞ dt

þ EðxðthÞ½j1 ðxth ,thÞ Eðj1 ðxth ,thÞjFtY ÞT jFtY Þ ðBðthÞBT ðthÞÞðBðthÞBT ðtÞÞ1 ðBðtÞBT ðtÞÞEð½j1 ðxÞEðj1 ðxÞjFtY ÞxT ðtÞjFtY ÞÞ dt þ EððxðtÞmðtÞÞðxðthÞmðthÞÞ½j1 ðxth ,thÞ Eðj1 ðxth ,thÞjFtY ÞT jFtY Þ ðBðthÞBT ðthÞÞ1 ðdyðthÞEðj1 ðxth ,thÞjFtY Þ dtÞ

þ EððxðthÞmðthÞÞðxðtÞmðtÞÞ½j1 ðx,tÞ Eðj1 ðx,tÞjFtY ÞT jFtY Þ  ðBðtÞBT ðtÞÞ1 ðdyðtÞ Eðj1 ðx,tÞjFtY Þ dtÞ:

Upon substituting the expressions for f(x,t) and j1 ðx,tÞ and using the formulas for the variance P(t) and covariance Pðt,thÞ ¼ EððxðtÞmðtÞÞðxðthÞmðthÞÞT jFtY Þ, the last equation can finally be represented as dPðt,thÞ ¼ ða2 ðtÞEðððxðtÞxT ðthÞÞðxðthÞmðthÞÞT ÞjFtY Þ þ ða2 ðtÞEðððxðthÞxT ðt2hÞÞðxðtÞmðtÞÞT ÞjFtY ÞÞT þa1 ðtÞPðt,thÞ þ PT ðt,thÞaT1 ðtÞ þ 1=2½bðtÞbT ðthÞ

where the last term should be understood as a 3D tensor (under the expectation sign) multiplied by a vector, which yields a matrix. Upon substituting the expressions for f ðx,tÞ and j1 ðx,tÞ, the last formula takes the form dPðtÞ ¼ ðEðððxðtÞmðtÞÞða0 ðtÞ þ a1 ðtÞxðtÞ þ a2 ðtÞxðtÞxT ðthÞÞT jFtY Þ T

T

þ Eðða0 ðtÞ þ a1 ðtÞxðtÞ þ a2 ðtÞxðtÞx ðthÞÞðxðtÞmðtÞÞ T

785

ÞjFtY Þ

T

þ bðtÞb ðtÞðEðxðtÞðxðtÞmðtÞÞ jFtY ÞAT ðtÞ ðBðtÞBT ðtÞÞ1 AðtÞEððxðtÞmðtÞÞxT ðtÞÞjFtY ÞÞ dt T

þ bðthÞbT ðtÞð1=2Þ½PðtÞAT ðtÞðBðtÞBT ðtÞÞðBðtÞBT ðthÞÞ1 ðBðthÞBT ðthÞÞAðthÞPðthÞÞ þ PðthÞAT ðthÞðBðthÞBT ðthÞÞðBðthÞBT ðtÞÞ1 ðBðtÞBT ðtÞÞAðtÞPðtÞÞ dt þð1=2Þ½EððxðtÞ mðtÞÞðxðthÞmðthÞÞðxðthÞmðthÞÞT jFtY Þ AT ðthÞðBðthÞBT ðthÞÞ1 ðdyðthÞðA0 ðthÞ þAðthÞmðthÞÞ dtÞ þ EððxðthÞmðthÞÞðxðtÞ mðtÞÞðxðtÞmðtÞÞT jFtY Þ  AT ðtÞðBðtÞBT ðtÞÞ1 ðdyðtÞ

þ EððxðtÞmðtÞÞðxðtÞmðtÞÞðAðtÞðxðtÞmðtÞÞÞ jFtY Þ

ðA0 ðtÞ þAðtÞmðtÞÞ dtÞ:

ðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þ AðtÞmðtÞÞ dtÞ: T

Using the variance formula PðtÞ ¼ EððxðtÞmðtÞÞx the last equation can be represented as

ðtÞÞjFtY Þ,

dPðtÞ ¼ ða2 ðtÞEðððxðtÞxT ðthÞÞðxðtÞmðtÞÞT ÞjFtY Þ þða2 ðtÞEðððxðtÞxT ðthÞÞðxðtÞmðtÞÞT ÞjFtY ÞÞT þa1 ðtÞPðtÞ þPðtÞaT1 ðtÞ þbðtÞbT ðtÞ PðtÞAT ðtÞðBðtÞBT ðtÞÞ1 AðtÞPðtÞÞ dt þEðððxðtÞmðtÞÞðxðtÞmðtÞÞðxðtÞmðtÞÞT jFtY Þ AT ðtÞðBðtÞBT ðtÞÞ1 ðdyðtÞðA0 ðtÞ þ AðtÞmðtÞÞ dtÞ:

ð7Þ

The Eq. (7) should be complemented with the initial condition Pðt0 Þ ¼ E½ðxðt0 Þmðt0 ÞÞðxðt0 Þmðt0 ÞÞT jFtY0 . Applying now the Ito differential formula to the covariance Pðt,thÞ ¼ EððxðtÞmðtÞÞðxðthÞmðthÞÞT jFtY Þ

ð8Þ

The Eq (8) should be complemented with the initial condition Pðs,shÞ ¼ E½ðxðsÞmðsÞÞðxðshÞmðshÞÞT jFsY  for s 2 ½t0 ,t0 þhÞ. The Eqs. (6)–(8) for the mean-square estimate m(t), the error variance P(t), and the error covariance P(t,t  h) form a non-closed system of the filtering equations for the polynomial time-delay state (1) over linear observations (2). The non-closeness means that the system (6)–(8) includes terms depending on x, such as EðððxðtÞxT ðthÞÞ ðxðtÞmðtÞÞT ÞjFtY Þ and EðððxðtÞxT ðthÞÞðxðthÞmðthÞÞT ÞjFtY Þ, which are not expressed yet as functions of the system variables, m(t), P(t), and P(t,t h). The closed system composed of the filtering Eqs. (3)–(5) is eventually obtained performing the following steps substantiated previously in [44–46]. First, the original

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M. Basin et al. / Signal Processing 91 (2011) 782–792

filtering problem with a non-invertible observation matrix A(t) in (2) is transformed (see [46] for more details) into the filtering problem with an invertible one. Next, as shown in [44,45], a closed system of the filtering equations for a polynomial system state over linear observations can be obtained, if the observation matrix A(t) is invertible for any t Zt0 . Indeed, applying the technique of representing superior conditional moments of the random variable x(t) as functions of lower conditional moments, expectation m(t) and error variance P(t) (see [44,45] for more details), to both random variables x(t) and x(t  h), and taking into account the specific expressions for m(t) and P(t), obtained in [46] for polynomial third-order systems, including bilinear systems as a particular case, the filtering Eqs. (3)–(5) are finally derived. & Remark. Note that the general result of Theorem 1 allows one to design a mean-square finite-dimensional filter for any bilinear time-delay system state over linear observations. This leads to a broad spectrum of possible applications, given that typical mechanical, power, thermochemical, and biochemical nonlinear systems are intrinsically bilinear. The obtained result would be certainly useful for estimation and signal processing in those systems, since the mostly used approach, linearization at an operating point, frequently yields divergent estimates, especially for bilinear systems with time delays. The next challenging problem to be solved is the mean-square filtering problem for bilinear systems involving multiple time delays in the state equation. 4. Joint state filtering and parameter estimation problem

state x(t) dynamics depends on the delayed state x(t  h), which actually makes the system state space infinitedimensional (see, for example, [34]). The observation matrix AðtÞ 2 Rmn is not supposed to be invertible or even square. It is assumed that B(t)BT(t) is a positive definite matrix. All coefficients in (17)–(18) are deterministic functions of time of appropriate dimensions. It is considered that there is no useful information on values of the unknown parameters yk ðtÞ, k ¼ 1, . . . ,p, and this uncertainty even grows as time tends to infinity. In other words, the unknown parameters can be modeled as Ft-measurable Wiener processes dyðtÞ ¼ dW 3 ðtÞ,

ð19Þ

with unknown initial conditions yðt0 Þ ¼ y0 2 Rp , where ðW3 ðtÞ,Ft ,t Zt0 Þ is a Wiener process independent of x0, W1(t), and W2(t). The estimation problem is to find the mean-square ^ y^ ðtÞT of the combined vector of the estimate z^ ðtÞ ¼ ½xðtÞ, system states and unknown parameters zðtÞ ¼ ½xðtÞ, yðtÞT , based on the observation process YðtÞ ¼ fyðsÞ,0 r sr tg, that minimizes the the mean-square criterion J ¼ E½ðzðtÞz^ ðtÞÞT ðzðtÞz^ ðtÞÞjFtY  at every time moment t. Here, E½xðtÞjFtY  means the conditional expectation of a stochastic process xðtÞ ¼ ðzðtÞz^ ðtÞÞT ðzðtÞz^ ðtÞÞ with respect to the salgebra FYt generated by the observation process Y(t) in the interval [t0,t]. As known [47,48], this mean-square estimate is given by the conditional expectation z^ ðtÞ ¼ mðtÞ ¼ EðzðtÞjFtY Þ of the system state zðtÞ ¼ ½xðtÞ, yðtÞT with respect to the salgebra FYt generated by the observation process Y(t) in the interval [t0,t]. As usual, the matrix function PðtÞ ¼ E½ðzðtÞmðtÞÞðzðtÞmðtÞÞT jFtY 

Let ðO,F,PÞ be a complete probability space with an increasing right-continuous family of salgebras Ft ,t Z t0 , and let ðW1 ðtÞ,Ft ,t Zt0 Þ and ðW2 ðtÞ,Ft ,t Z t0 Þ be independent Wiener processes. The Ft-measurable random process (x(t),y(t)) is described by a linear delay-differential equation with unknown vector parameter yðtÞ for the system state

is the estimation error variance. The stated optimal filtering problem for the extended state is solved by the following theorem.

with the initial condition xðsÞ ¼ fðsÞ, s 2 ½t0 h,t0 , h is the state delay value, and a linear differential equation for the observation process

Theorem 2. The mean-square finite-dimensional filter for the extended state vector zðtÞ ¼ ½xðtÞ, yðtÞT , governed by the (17),(19) over the linear observations (18) is given by the following equations for the mean-square estimate ^ z^ ðtÞ ¼ mðtÞ ¼ ½xðtÞ, y^ ðtÞT ¼ Eð½xðtÞ, yðtÞjFtY Þ, the estimation error variance PðtÞ ¼ E½ðzðtÞmðtÞÞðzðtÞmðtÞÞT jFtY , and the estimation error covariance Pðt,thÞ ¼ E½ðzðtÞ mðtÞÞðzðthÞ mðthÞÞT jFtY 

dyðtÞ ¼ ðA0 ðtÞ þ AðtÞxðtÞÞ dt þBðtÞ dW 2 ðtÞ:

dmðtÞ ¼ ðc0 ðtÞ þa1 ðtÞmðtÞ þ a2 ðtÞ½Pðt,thÞ þ mðtÞmT ðthÞ

dxðtÞ ¼ ða0 ðy,tÞ þ aðyÞxðthÞÞ dt þbðtÞ dW 1 ðtÞ,

n

ð17Þ

ð18Þ m

Here, xðtÞ 2 R is the state vector, yðtÞ 2 R is the linear observation vector, m r n, and yðtÞ 2 Rp , p r n  n þn, is the vector of unknown entries of matrix aðyÞ and unknown components of vector a0 ðy,tÞ. The latter means that the vector a0 ðy,tÞ contains unknown components a0i ðtÞ ¼ yk ðtÞ, k ¼ 1, . . . ,p1 r n, as well as known components a0i ðtÞ whose values are known functions of time, and the matrix aðyÞ contains nonzero unknown entries aij ðtÞ ¼ yk ðtÞ, k ¼ p1 þ1, . . . ,p r n  n þ n, as well as zero entries. The initial condition x0 2 Rn is a Gaussian vector such that x0, W1(t), and W2(t) are independent. The system

þPðtÞ½AðtÞ,0mp T ðBðtÞBT ðtÞÞ1 ½dyðtÞAðtÞmðtÞ dt,

ð20Þ

^ ¼ EðfðsÞÞ,s 2 ½t0 h,t0 Þ, xðt ^ 0 Þ ¼ Eðfðt0 ÞjFtY Þ, Eðyðt0 ÞjFtY Þ, xðsÞ 0 dPðtÞ ¼ ða1 ðtÞPðtÞ þPðtÞaT1 ðtÞ þ2a2 ðtÞmðthÞPðtÞ þ ð2a2 ðtÞmðthÞPðtÞÞT þ ðdiag½bðtÞ,Ip Þðdiag½bðtÞ,Ip T ÞÞ dt

PðtÞ½AðtÞ,0mp T ðBðtÞBT ðtÞÞ1 ½AðtÞ,0mp PðtÞ dt,

ð21Þ

Pðt0 Þ ¼ Eððzðt0 Þmðt0 ÞÞðzðt0 Þmðt0 ÞÞT jFtY Þ, dPðt,thÞ ¼ ð2a2 ðtÞmðthÞPðt,thÞ þ ð2a2 ðtÞmðt2hÞPðth,tÞÞT

M. Basin et al. / Signal Processing 91 (2011) 782–792

þa1 ðtÞPðt,thÞ þP T ðt,thÞaT1 ðtÞ þð1=2Þ½ðdiag½bðtÞ,Ip Þðdiag½bðthÞ,Ip T Þ þðdiag½bðthÞ,Ip Þðdiag½bðtÞ,Ip T Þ ð1=2Þ½PðtÞ½AðtÞ,0mp T ðBðtÞBT ðtÞÞðBðtÞBT ðthÞÞ1 ðBðthÞBT ðthÞ½AðthÞ,0mp PðthÞ þ PðthÞ½AðthÞ,0mp T ðBðthÞBT ðthÞÞðBðthÞBT ðtÞÞ1

ðBðtÞBT ðtÞÞ½AðtÞ,0mp PðtÞ,

ð22Þ

Pðs,shÞ ¼ E½ðzðsÞmðsÞÞðzðshÞmðshÞÞT jFsY  for s 2 ½t0 ,t0 þ hÞ, where 0mp is the m  p—dimensional zero matrix. This filter, applied to the subvector yðtÞ, also serves as the identifier for the vector of unknown parameters yðtÞ in the Eq. (17), yielding the estimate subvector y^ ðtÞ as the meansquare parameter estimate. Proof. To solve this mean-square filtering problem, the following procedure is proposed for incorporating the unknown parameters as additional states in the extended state vector and writing the extended state vector equation in the polynomial bilinear form. For this purpose, a vector c0 ðtÞ 2 Rðn þ pÞ , a matrix a1 ðtÞ 2 Rðn þ pÞðn þ pÞ , and a cubic tensor a2 ðtÞ 2 Rðn þ pÞ ðn þ pÞ  ðn þ pÞ are introduced as follows. The equation for the i-th component of the state vector (17) is given by 0 1 n X @ aij ðtÞxj ðthÞAdt dxi ðtÞ ¼ a0i ðtÞ þ j¼1

þ

n X

bij ðtÞ dW 1j ðtÞ,

xi ðt0 Þ ¼ x0i :

j¼1

Then: 1. If the variable a0i ðtÞ is a known function, then the i-th component of the vector c0(t) is set to this function, c0i ðtÞ ¼ a0i ðtÞ; otherwise, if the variable a0i ðtÞ is an unknown function, then the (i,n+ i)-th entry of the matrix a1(t) is set to 1. 2. If the variable aij(t) is zero, then the (i,j)-th component of the matrix a1(t) is set to zero, a1ij ðtÞ ¼ 0; otherwise, if the variable aij(t) is an unknown function, then the ði,n þ p1 þk,jÞth entry of the cubic tensor a2(t) is set to 1, where k is the number of this current unknown entry in the matrix aij(t), counting the unknown entries consequently by rows from the first to n-th entry in each row. 3. All other unassigned entries of the matrix a1(t), cubic tensor a2(t), and vector c0(t) are set to 0. Using the introduced notation, the state (17) (19) for the vector zðtÞ ¼ ½xðtÞ, yðtÞT 2 Rn þ p can be rewritten as dzðtÞ ¼ ðc0 ðtÞ þa1 ðtÞzðtÞ þa2 ðtÞzðtÞzT ðthÞÞ dt þdiag½bðtÞ,Ipp d½W1T ðtÞ,W3T ðtÞT ,

ð23Þ

where the matrix a1(t), cubic tensor a2(t), and vector c0(t) have already been defined, and Ipp is the p  p—dimensional

787

identity matrix. The Eq. (23) is bilinear with respect to the extended state vector zðtÞ ¼ ½xðtÞ, yðtÞ. Thus, the estimation problem is now reformulated as to ^ y^ ðtÞT find the mean-square estimate z^ ðtÞ ¼ mðtÞ ¼ ½xðtÞ, T for the state vector zðtÞ ¼ ½xðtÞ, yðtÞ , governed by the bilinear Eq. (20), based on the observation process YðtÞ ¼ fyðsÞ,0 r s rtg, satisfying the Eq. (18). The solution of this problem is obtained using the filtering Eqs. (3)–(5) for incompletely measured bilinear time-delay states over linear observations. Indeed, directly applying the meansquare filter (3)–(5) for incompletely measured bilinear time-delay states over linear observations to the bilinear state zðtÞ ¼ ½xðtÞ, yðtÞT , governed by (23), and incomplete linear observations (18), the filtering Eqs. (20)–(22) are ^ y^ ðtÞT , PðtÞ ¼ E½ðzðtÞ obtained for mðtÞ ¼ z^ ðtÞ ¼ mðtÞ ¼ ½xðtÞ, mðtÞÞðzðtÞmðtÞÞT jFtY , and Pðt,thÞ ¼ E½ðzðtÞmðtÞÞðzðthÞ mðthÞÞT jFtY . & Thus, using the filtering Eqs. (3)–(5) for incompletely measured bilinear time-delay states over linear observations, the joint state filter and parameter identifier is obtained for the linear time-delay system state (17) with unknown parameters, based on the incomplete linear observations (18). Note that since the original estimation problem is reduced to the mean-square filtering problem for the extended system state including both state and parameters, the identifiability condition for the original system coincides with the observability condition for the extended system. As follows from the results of Theorem 7.4 and Section 7.7 in [47], the observability condition for the extended system (17)–(19) implies the almost sure convergence of the state and parameter estimates to their real values. In the next section, performance of the designed joint state and parameter estimator is verified in an illustrative example. 5. Example This section presents an example of designing the mean-square joint state and parameter estimator for an incompletely measured linear time-delay system state with an unknown multiplicative parameter, based on linear state measurements. Let the bi-dimensional system state xðtÞ ¼ ½x1 ðtÞ,x2 ðtÞ satisfy the linear time-delay equations with unknown parameter y x_ 1 ðtÞ ¼ x2 ðtÞ,

x1 ð0Þ ¼ x10 ,

x_2 ðtÞ ¼ yx2 ðthÞ þ c1 ðtÞ,

x2 ðsÞ ¼ fðsÞ,

s 2 ½t0 h,t0 ,

ð24Þ

and the observation process be given by the linear equation yðtÞ ¼ x1 ðtÞ þ c2 ðtÞ,

ð25Þ

where c1 ðtÞ and c2 ðtÞ are white Gaussian noises, which are the weak mean-square derivatives of standard Wiener processes (see [47,48]). The Eqs. (24)–(25) present the conventional form for the Eqs. (17)–(19), which is actually used in practice. The parameter y is modelled as a standard Wiener process, i.e., satisfies

788

M. Basin et al. / Signal Processing 91 (2011) 782–792 2 P_ 22 ðtÞ ¼ 1 þ4P23 ðtÞm2 ðthÞP12 ðtÞ,

the equation dyðtÞ ¼ dW 3 ðtÞ,

yð0Þ ¼ y0 ,

P_ 23 ðtÞ ¼ 2P33 ðtÞm2 ðthÞP12 ðtÞP13 ðtÞ,

which can also be written as

y_ ðtÞ ¼ c3 ðtÞ, yð0Þ ¼ y0 ,

2 ðtÞ, P_ 33 ðtÞ ¼ 1P13

ð26Þ

where c3 ðtÞ is a white Gaussian noise. The filtering problem is to find the mean-square estimate mðtÞ ¼ ½m1 ðtÞ,m2 ðtÞ,m3 ðtÞ for the bilinear–linear state (23),(25), ½x1 ðtÞ,x2 ðtÞ, y, using linear observations (24) confused with independent and identically distributed disturbances modelled as white Gaussian noises. The filtering horizon is set to T ¼ 9:5. The filtering Eqs. (20)–(22) take the following particular form for the system (24)–(26)

with the initial condition Pð0Þ ¼ Eðð½x10 ,x20 , y0 mð0ÞÞ ð½x10 ,x20 , y0 mð0ÞÞT jyð0ÞÞ ¼ P0 , P_ 11 ðt,thÞ ¼ ½P12 ðt,thÞ þP21 ðt,thÞP11 ðtÞP11 ðthÞ, P_ 21 ðt,thÞ ¼ P22 ðt,thÞ þ 2P31 ðt,thÞm2 ðthÞ ð1=2Þ½P11 ðtÞP21 ðthÞ þ P11 ðthÞP12 ðtÞ, P_ 31 ðt,thÞ ¼ P32 ðt,thÞð1=2Þ½P11 ðtÞP31 ðthÞ þ P11 ðthÞP13 ðtÞ,

_ 1 ðtÞ ¼ m2 ðtÞ þP11 ðtÞ½yðtÞm1 ðtÞ, m

P_ 12 ðt,thÞ ¼ P22 ðt,thÞ þ 2P13 ðt,thÞm2 ðt2hÞ ð1=2Þ½P11 ðthÞP12 ðtÞ þ P11 ðtÞP12 ðthÞ,

_ 2 ðtÞ ¼ m2 ðthÞm3 ðtÞ þ P32 ðt,thÞ þ P12 ðtÞ½yðtÞm1 ðtÞ, m _ 3 ðtÞ ¼ P13 ðtÞ½yðtÞm1 ðtÞ, m

ð28Þ

ð27Þ

P_ 22 ðt,thÞ ¼ 1 þ2P32 ðt,thÞm2 ðthÞ þ 2P23 ðt,thÞm2 ðt2hÞ

with the initial conditions m1 ð0Þ ¼ Eðx10 jyð0ÞÞ ¼ m10 , m2 ðsÞ ¼ EðfðsÞÞ, s 2 ½t0 h,t0 Þ, m2 ðt0 Þ ¼ Eðfðt0 ÞjFtY0 Þ, and m3 ð0Þ ¼ Eðy0 jyð0ÞÞ ¼ m30 ,

ð1=2Þ½P12 ðtÞP12 ðthÞ þ P12 ðtÞP21 ðthÞ, P_ 32 ðt,thÞ ¼ 2P33 ðt,thÞm2 ðt2hÞ ð1=2Þ½P12 ðthÞP13 ðtÞ þ P12 ðtÞP31 ðthÞ,

P_ 12 ðtÞ ¼ P22 ðtÞ þ 2P13 ðtÞm2 ðthÞP11 ðtÞP12 ðtÞ,

P_ 13 ðt,thÞ ¼ P23 ðt,thÞð1=2Þ½P11 ðthÞP13 ðtÞ þ P11 ðtÞP13 ðthÞ,

P_ 13 ðtÞ ¼ P23 ðtÞP11 ðtÞP13 ðtÞ,

P_ 23 ðt,thÞ ¼ 2P33 ðt,thÞm2 ðthÞ

Parameter esimate m3

Estimation error x2

Estimation error x1

2 P_ 11 ðtÞ ¼ 2P12 ðtÞP11 ðtÞ,

1000 500 0 −500

0

1

2

3

4

5 Time

6

7

8

9 9.5

0

1

2

3

4

5 Time

6

7

8

9 9.5

0

1

2

3

4

5 Time

6

7

8

9 9.5

1000 500 0 −500

0.5 0 −0.5 −1

Fig. 1. Graphs of the estimation error between the reference state variable x1(t) and the optimal state estimate m1(t) (above), the estimation error between the reference state variable x2(t) and the optimal state estimate m2(t) (middle), the optimal parameter estimate m3(t) (below) in the entire simulation interval [0,9.5] for the unstable system (24).

M. Basin et al. / Signal Processing 91 (2011) 782–792

ð1=2Þ½P12 ðtÞP13 ðthÞ þ P21 ðthÞP13 ðtÞ, P_ 33 ðt,thÞ ¼ 1ð1=2Þ½P13 ðtÞP13 ðthÞ þ P13 ðtÞP31 ðthÞ,

ð29Þ

Parameter esimate m3

Estimation error x2

Estimation error x1

with the initial condition Pðs,shÞ ¼ Eðð½x1 ðsÞ, x2 ðsÞ, yðsÞ mðsÞÞð½x1 ðshÞ,x2 ðshÞ, yðshÞmðshÞÞT jFsY Þ ¼ RðsÞ for s 2 ½t0 ,t0 þhÞ. Numerical simulation results are obtained solving the system of filtering Eqs. (27)–(29). The numerical solver is based on the Monte-Carlo algorithm averaging over realizations of the stochastic processes, optimal estimate (conditional mean) and estimation error variance and covariance. It is realized in MatLab 7.0 as a Simulink scheme, including Subroutine ODE 4.5, which is combined with an external cycle to run Monte Carlo realizations. The obtained values of the estimates [m1(t),m2(t)] for [x1(t),x2(t)], and m3(t), estimate for y, are compared to the real values of the state variable xðtÞ ¼ ½x1 ðtÞ,x2 ðtÞ and parameter y in (24)–(26). For the filter (27)–(29) and the reference system (24)– (26) involved in simulation, the following initial values are assigned: x10 ¼ x20 ¼ 1000, m10 ¼ 0:1, m2 ð0Þ ¼ 0:1 and m2 ðsÞ ¼ 0 for any s 2 ½h,0Þ, m30 ¼ 0, P110 ¼ P220 ¼ P330 ¼ 100, P120 ¼ 10, P130 ¼ P230 ¼ 0, R11 ðhÞ ¼ R22 ðhÞ ¼ R33 ðhÞ ¼ 100, R12 ðhÞ ¼ R21 ðhÞ ¼ 10 and R11 ðsÞ ¼ R22 ðsÞ ¼ R33 ðsÞ ¼ R12 ðsÞ ¼ R21 ðsÞ ¼ 0, for any s 2 ½0,hÞ, and the other entries of R(s) are equal to zero for any s 2 ½0,h. The delay value is set to h ¼ 5. The unknown parameter y

789

is assigned as y ¼ 0:1 in the first simulation and as y ¼ 0:1 in the second one, thus considering the system (24) unstable and stable, respectively. Gaussian disturbances c1 ðtÞ, c2 ðtÞ, and c3 ðtÞ in (24)–(26) are realized using the built-in MatLab white noise functions with the equal discretization step and amplitude values, 0.01, which correspond to discrete MatLab representations of the standard Gaussian white noise with unit intensity (noise power). The following graphs are obtained: graphs of the estimation errors between the reference state variable x1(t) and the optimal state estimate m1(t) and between the reference state variable x2(t) and the optimal state estimate m2(t), graph of the optimal parameter estimate m3(t) in the entire simulation interval [0,10] for the unstable system (24) (y ¼ 0:1); graphs of the estimation errors between the reference state variable x1(t) and the optimal state estimate m1(t) and between the reference state variable x2(t) and the optimal state estimate m2(t), graph of the optimal parameter estimate m3(t) in the entire simulation interval [0,10] for the stable system (24) (y ¼ 0:1) The graphs of all those variables are shown in the entire simulation interval from t0 ¼ 0 to T ¼ 9:5 in Figs. 1 and 3 for the unstable and stable cases, respectively. Figs. 2 and 4 show the graphs of those three filtering variables with more visualization details in the simulation interval [6.5,9.5] for the unstable and stable cases, respectively.

0.4 0.2 0 −0.2 −0.4

6.5

7

7.5

8 Time

8.5

9

9.5

6.5

7

7.5

8 Time

8.5

9

9.5

6.5

7

7.5

8 Time

8.5

9

9.5

5

0

−5

0.11 0.1 0.09 0.08

Fig. 2. Graphs of the estimation error between the reference state variable x1(t) and the optimal state estimate m1(t) (above), the estimation error between the reference state variable x2(t) and the optimal state estimate m2(t) (middle), the optimal parameter estimate m3(t) (below) in the simulation interval [6.5,9.5] for the unstable system (24).

M. Basin et al. / Signal Processing 91 (2011) 782–792

Parameter esimate m3

Estimation error x2

Estimation error x1

790

1000 500 0 −500

0

1

2

3

4

5 Time

6

7

8

9 9.5

0

1

2

3

4

5 Time

6

7

8

9 9.5

0

1

2

3

4

5 Time

6

7

8

9 9.5

1000 500 0 −500

0 −0.5 −1 −1.5 −2

Fig. 3. Graphs of the estimation error between the reference state variable x1(t) and the optimal state estimate m1(t) (above), the estimation error between the reference state variable x2(t) and the optimal state estimate m2(t) (middle), the optimal parameter estimate m3(t) (below) in the entire simulation interval [0,9.5] for the stable system (24).

It can be observed that, in both cases, the state estimates [m1(t),m2(t)] converge to the real state [x1(t),x2(t)] and the parameter estimate m3(t) converges to the real value (0.1 or 0.1) of the unknown parameter yðtÞ. Note that the square roots of the gain matrix entries Pii(t), i ¼ 1,2,3, converge to finite values in the range from 0 to 10 as time tends to infinity, which are expected. Thus, it can be concluded that, in both cases, the designed optimal state filter and parameter identifier (20)–(22) yields reliable estimates of the unobserved system state and the unknown parameter value. The simulation results show that the state and parameter estimates calculated using the designed joint state filter and parameter identifier for linear systems with unknown parameters converge to the real state and parameter values rapidly, in less than 10 time units. This behavior can be classified as very reliable, especially taking into account large deviations in the initial values for the real state and its estimate and large values of the initial error variances. Another advantage to be mentioned is that the designed joint state and parameter estimator works equally well for stable and unstable systems, which correspond to operation of linear systems in nominal conditions and under persistent external disturbances, respectively. Although this conclusion fol-

lows from the developed theory, the numerical simulation serves as a convincing illustration.

6. Conclusions This paper designs the mean-square joint state filter and parameter estimator for linear stochastic time-delay systems with unknown multiplicative and additive parameters, which has not yet obtained even for linear timedelay systems. The original problem is reduced to the mean-square filtering problem for incompletely measured bilinear time-delay system states over linear observations, where the unknown parameters are considered standard Wiener processes and incorporated as additional states in the extended state vector. To deal with the new filtering problem, the paper designs the mean-square finitedimensional filter for incompletely measured bilinear time-delay system states over linear observations, which itself makes a noticeable advance with respect to the previously known results in the mean-square filtering for polynomial time-delay systems. The developed technique of reducing the joint state filtering and parameter estimation problem to the mean-square state filtering one for an extended polynomial system provides a

Parameter esimate m3

Estimation error x2

Estimation error x1

M. Basin et al. / Signal Processing 91 (2011) 782–792

791

0.5

0

−0.5

6.5

7

7.5

8 Time

8.5

9

9.5

6.5

7

7.5

8 Time

8.5

9

9.5

6.5

7

7.5

8 Time

8.5

9

9.5

5

0

−5

−0.08 −0.1 −0.12 −0.14

Fig. 4. Graphs of the estimation error between the reference state variable x1(t) and the optimal state estimate m1(t) (above), the estimation error between the reference state variable x2(t) and the optimal state estimate m2(t) (middle), the optimal parameter estimate m3(t) (below) in the simulation interval [6.5,9.5] for the stable system (24).

consistent procedure for obtaining further results in nonlinear polynomial filtering for time-delay systems.

Acknowledgments The authors thank the UK Engineering and Physical Sciences Research Council (EPSRC) and the Mexican National Science and Technology Council (CONACyT) for financial support under EPSRC Grant EP/F029195/1 and CONACyT Grants 55584 and 52953. References [1] R.E. Kalman, R.S. Bucy, New results in linear filtering and prediction theory, ASME J. Basic Eng., Ser. D 83 (1961) 95–108. [2] M.V. Basin, R. Martinez-Zuniga, Optimal linear filtering over observations with multiple delays, Int. J. Robust and Nonlinear Control 14 (2004) 685–696. [3] M.V. Basin, J.G. Rodriguez-Gonzalez, R. Martinez-Zuniga, Optimal filtering for linear state delay systems, IEEE Trans. Autom. Control 50 (2005) 684–690. [4] M.V. Basin, M.A. Alcorta-Garcia, J.G. Rodriguez-Gonzalez, Optimal filtering for linear systems with state and observation delays, Int. J. Robust and Nonlinear Control 15 (2005) 685–696. [5] P. Shi, Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control 43 (1998) 1022–1027. [6] M.S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, Marcel Dekker, 2000. [7] C.E. de Souza, P.M. Palhares, P.L.D. Peres, Robust H1 filtering design for uncertain linear systems with multiple time-varying state delays, IEEE Trans. Signal Process. 49 (2001) 569–576.

[8] Z.D. Wang, H. Qiao, Robust filtering for bilinear uncertain stochastic discrete-time systems, IEEE Trans. Signal Process. 50 (2002) 560–567. [9] S. Xu, P.V. van Dooren, Robust H1 filtering for a class of nonlinear systems with state delay and parameter uncertainty, Int. J. Control 75 (2002) 766–774. [10] Z.D. Wang, J. Lam, X. Liu, Nonlinear filtering for state delayed systems with markovian switching, IEEE Trans. Signal Process. 51 (2003) 2321–2328. [11] M.S. Mahmoud, P. Shi, Robust Kalman filtering for continuous timelag systems with Markovian jump parameters, IEEE Trans. Circuits Syst. I 50 (2003) 98–105. [12] E. Fridman, U. Shaked, An improved delay-dependent H1 filtering, IEEE Trans. Signal Process. 52 (2004) 668–673. [13] H. Gao, C. Wang, A delay-dependent approach to robust H1 filtering for uncertain discrete-time state-delayed systems, IEEE Trans. Signal Process. 52 (2004) 1631–1640. [14] J. Sheng, T. Chen, S.L. Shah, Optimal filtering for multirate systems, IEEE Trans. Circuits Syst. II 52 (2005) 228–232. [15] H. Gao, J. Lam, L. Xie, C. Wang, New approach to mixed H2 =H1 filtering for polytopic discrete-time systems, IEEE Trans. Signal Process. 53 (2005) 3183–3192. [16] S. Xu, J. Lam, T. Chen, Y. Zou, A delay-dependent approach to robust H1 filtering for uncertain distributed delay systems, IEEE Trans. Signal Process. 53 (2005) 3764–3772. [17] H. Gao, J. Lam, C. Wang, Induced l2 and generalized H2 filtering for systems with repeated scalar nonlinearities, IEEE Trans. Signal Process. 53 (2005) 4215–4226. [18] P. Shi, M.S. Mahmoud, S. Nguang, A. Ismail, Robust filtering for jumping systems with mode-dependent delays, Signal Process. 86 (2006) 140–152. [19] H. Zhang, X. Lu, D. Cheng, Optimal estimation for continuous-time systems with delayed measurements, IEEE Trans. Autom. Control 51 (2006) 823–827. [20] H. Zhang, G. Feng, G. Duan, X. Lu, H1 filtering for multiple time-delay measurements, IEEE Trans. Signal Process. 54 (2006) 1681–1688.

792

M. Basin et al. / Signal Processing 91 (2011) 782–792

[21] Z.D. Wang, F. Yang, D.W.C. Ho, X. Liu, Robust H1 filtering for stochastic time-delay systems with missing measurements, IEEE Trans. Signal Process. 54 (2006) 2579–2587. [22] L. Wu, Z.D. Wang, H. Gao, C. Wang, Filtering for uncertain 2-D discrete systems with state delays, Signal Process. 87 (2007) 2213–2230. [23] M.V. Basin, D. Calderon-Alvarez, Alternative optimal filter for linear systems with multiple state and observation delays, Int. J. Innovative Comput., Inf. Control 4 (2008) 2889–2898. [24] Z.D. Wang, Y. Liu, X. Liu, H1 filtering for uncertain stochastic timedelay systems with sector-bounded nonlinearities, Automatica 44 (2008) 1268–1277. [25] L. Wu, D.W.C. Ho, Fuzzy filter design for Ito stochastic systems with application to sensor detection, IEEE Trans. Fuzzy Syst. 17 (2009) 233–242. [26] C.W. Liao, C.Y. Lu, K.Y. Zheng, C.C. Ting, A delay-dependent approach to design state estimator for discrete stochastic recurrent neural network with interval time-varying delays, ICIC Express Lett. 3 (2009) 465–470. [27] G. Wei, Z. Wang, H. Shu, Robust filtering with stochastic nonlinearities and multiple missing measurements, Automatica 45 (2009) 836–841. [28] B. Shen, Z. Wang, H. Shu, G. Wei, H1 filtering for nonlinear discretetime stochastic systems with randomly varying sensor delays, Automatica 45 (2009) 1032–1037. [29] B. Song, S. Xu, Y. Zou, Non-fragile H1 filtering for uncertain stochastic time-delay systems, Int. J. Innovative Comput., Inf. Control 5 (2009) 2257–2265. [30] L. Wu, D.W.C. Ho, Sliding mode control of singular stochastic hybrid systems, Automatica 46 (2010) 779–783. [31] L. Wu, Z.D. Wang, Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay, Signal Process. 89 (2009) 1739–1753. [32] V.B. Kolmanovskii, L.E. Shaikhet, Control of Systems with After effect, American Mathematical Society, Providence, 1996. [33] V.B. Kolmanovskii, A.D. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, New York, 1999. [34] M. Malek-Zavarei, M. Jashmidi, Time-Delay Systems: Analysis, Optimization and Applications, North-Holland, Amsterdam, 1987.

[35] S. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, Heidelberg, 2001. [36] E.-K. Boukas, Z.-K. Liu, Deterministic and Stochastic Time-Delay Systems, Birkhauser, 2002. [37] K. Gu, S.-I. Niculescu, Survey on recent results in the stability and control of time-delay systems, ASME Trans. J. Dyn. Syst., Meas. Control 125 (2003) 158–165. [38] J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003) 1667–1694. [39] H. Zhang, L. Xie, Control and Estimation of Systems with Input/ Output Delays, in: Series Lecture Notes in Control and Information Sciences, vol. 355, Springer-Verlag, Berlin, 2007. [40] H.J. Kushner, On differential equations satisfied by conditional probability densities of Markov processes, SIAM J. Control 2 (1964) 106–119. [41] W. Wonham, Some applications of stochastic differential equations to nonlinear filtering, SIAM J. Control 2 (1965) 347–369. [42] V.E. Benes, Exact finite-dimensional filters for certain diffusions with nonlinear drift, Stochastics 5 (1981) 65–92. [43] S.S.-T. Yau, Finite-dimensional filters with nonlinear drift I: a class of filters including both Kalman–Bucy and Benes filters, J. Math. Syst., Estimation, Control 4 (1994) 181–203. [44] M.V. Basin, J. Perez, M. Skliar, Optimal filtering for polynomial system states with polynomial multiplicative noise, Int. J. Robust Nonlinear Control 16 (2006) 287–298. [45] M.V. Basin, New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems, Springer-Verlag, Berlin, Heidelberg, 2008. [46] M.V. Basin, D. Calderon-Alvarez, Optimal filtering for incompletely measured polynomial states over linear observations, Int. J. Adaptive Control Signal Process. 22 (2008) 482–494. [47] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, 1970. [48] V.S. Pugachev, I.N. Sinitsyn, Stochastic Systems: Theory and Applications, World Scientific, Singapore, 2001. [49] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2006.