Abstract

This paper focuses on estimation of a nonlinear function of state vector (NFS) in discrete-time linear systems with time-delays and model uncertainties. The NFS represents a multivariate nonlinear function of state variables, which can indicate useful information of a target system for control. The optimal nonlinear estimator of an NFS (in mean square sense) represents a function of the receding horizon estimate and its error covariance. The proposed receding horizon filter represents the standard Kalman filter with time-delays and special initial horizon conditions described by the Lyapunov-like equations. In general case to calculate an optimal estimator of an NFS we propose using the unscented transformation. Important class of polynomial NFS is considered in detail. In the case of polynomial NFS an optimal estimator has a closed-form computational procedure. The subsequent application of the proposed receding horizon filter and nonlinear estimator to a linear stochastic system with time-delays and uncertainties demonstrates their effectiveness.

1. Introduction

The Kalman filtering for dynamic systems with time-delays and its variations are well-known state estimation techniques in wide use in a variety of applications such as navigation, target tracking, vehicle state estimation, communications engineering, air traffic control, biomedical and chemical processing, and many other areas [1–6]. Ignorance of the effect of time- delays could cause unpredictable and unsatisfactory system performance with traditional Kalman filters [3, 6, 7]. These applications have motivated researchers to study the control and filtering problem for systems with time-delays.

However, some applications require the estimation of not only a state vector but also a nonlinear function of the state (NFS), which expresses practical and worthwhile information for control systems. For instance, in a mechanical application, such measurands include displacement (linear and angular), which can be interpreted as a quadratic form of a state vector, mass (weight, load, and density), and force (absolute, relative, static, torque, and pressure) [8]. In particular, prediction of future measurands depicted by an NFS can be helpful in several applications, such as system control and target tracking. It is well known that, for dynamical systems in which the stability issue depends explicitly on the time-delay and model uncertainty, in order to overcome this practical issue, it is necessary to study the estimation of NFS in dynamical systems with time-delay and model uncertainty. Most authors have not focused on estimation of NFS but have considered state estimation (filtering) only. To the best of our knowledge, there are no effective methods for estimation of an NFS in a dynamic system with time-delay and model uncertainty in the literature.

The Kalman filters are very difficult to be applied to dynamic system with uncertainty. The standard Kalman filter estimates the state of system if it fully fits the system model; that is, there are no model uncertainties. Otherwise, its algorithm may become unstable [9] and the estimate diverges [10] due to the limited knowledge of the system’s dynamic model and unestimated measurement noise. Thus, it is not robust against modeling uncertainty and disturbances [1]. For estimation problems where we attempt to achieve robustness against temporary uncertainty, numerous strategies have been suggested, rigorously investigated, and implemented over the past few decades. For this reason, in terms of structural adaptation, there are a number of methods currently available for such systems [11, 12]. The structure adaptation filters are based on the Bayesian approach in which the unknown parameter is assumed to be random with a prior known probability [13–15]. Next approach relative to the robust filtering problem with respect to various filtering performance criteria, such as -infinity filtering and energy-to-peak for linear continuous and discrete-time-delay systems [2, 6, 16, 17].

In this paper, we are concerned with usage of the receding horizon strategy for designing robust receding horizon (finite-memory) filter/estimator for time-delay stochastic systems with model uncertainties. In this way the information obtained over the most recent time interval is only used [15, 18, 19]. As a result, the receding horizon filters are more robust against model uncertainties and numerical errors than standard Kalman filters, which utilize all measurements [20, 21]. Based on aforementioned literature, and to the best of the authors’ knowledge, there are no existing results for the receding horizon estimation of NFS for linear systems with time-delays.

Motivated by the above problems, in this paper, we focus on estimating an arbitrary nonlinear function of the state in a linear system with time-delay and model uncertainty. The main contribution of the paper is derivation of receding horizon estimators based on the crucial Lyapunov-like equations for receding horizon initial conditions.

This paper is organized as follows. Section 2 presents a statement of the estimation problem for NFS within the linear system with time-delays. The optimal estimator for NFS is proposed. In Section 3, we present the receding horizon filter/estimator for discrete-time linear system with time-delays. Here, the exact recursive equations for determining receding horizon initial conditions (mean and covariance) are derived and discussed. In Section 4, we present several practical examples of NFS. In Section 5, we derive effective closed-form computational procedure for polynomial NFS. In Section 6, the unscented transformation is introduced, and its application for estimation of NFS is proposed. In Section 7, the efficiency of the proposed receding horizon estimators is studied on real dynamical model with time-delays. Finally, a brief conclusion is given in Section 8.

2. Problem Statement

We first consider a discrete-time linear system described by stochastic recursive equation with time-delays:where is an unknown state, , , are time-varying matrices, , , are initial conditions, is a zero-mean white Gaussian noise with covariance , and is the Kronecker function.

The discrete measurement is also described by recursive equation with time- delays:where is the measurement matrix, , and is a zero-mean white Gaussian noise with covariance .

We also assume that the initial states , , system noise , and measurement error are mutually uncorrelated; that is,A problem associated with such system (1), (2) is that of estimation of the nonlinear function of state (NFS) variablesfrom the overall noisy sensor measurementsTypical examples of such an NFS may be an arbitrary quadratic form , representing an energy-like function of an object or square distance between the current and nominal states, respectively.

We propose an optimal estimation algorithm for NFS and its performance in the subsequent sections.

The optimal mean square estimate of the NFS based on the overall sensor measurements (5) represents a conditional mean:where is a conditional probability density function (pdf).

To calculate a conditional pdf we need to solve the filtering problem consisting in calculation of good estimates of the state of a stochastic dynamical system based on noisy partial measurements (5).

Next we propose the new receding horizon filtering algorithm for discrete-time linear systems with time-delays and uncertainties (1), (2) determining the conditional pdf in (6).

3. Receding Horizon Filter and Nonlinear Estimator

At first assume that the system model (1), (2) does not contain any uncertainties.

3.1. Kalman Filtering with Time-Delays for Model without Uncertainties

According to [22, 23] the optimal mean square estimate of state based on all current measurements (5) represents conditional mean which is described by the Kalman filter equations with time-delays (KFTD):where Kalman gains , , and error covariancesare given by To run KFTD equations (7)–(9) we need to know the initial conditions for estimates and error covariances, which represent a priori mean and covariance; that is,In practice any system model contains uncertainties; therefore, to achieve robustness against temporary uncertainty, numerous strategies have been proposed. In the paper we use the effective receding horizon (finite-memory) strategy using only the measurements obtained over the most recent time interval (receding horizon) [10, 20, 21].

3.2. Receding Horizon Filtering with Time-Delays for Model with Uncertainties

According to the receding horizon strategy the optimal (in mean square sense) estimate of the unknown state based on the overall receding horizon sensor measurements,with horizon time interval , represents the conditional mean,Using KFTD’s equations (7)–(9) we propose their receding horizon version for estimation of the state using receding horizon measurements on the interval . We obtainwhere the receding horizon filter gains , , and error autocovariancesare described byIn contrast to KFTD (7)–(9) the receding horizon filter (RHF) (13)–(15) needs to initialize horizon initial conditions at which represent unconditional means and covariances of the state ; that is,We have the following.

Theorem 1. The horizon initial means (16) are described by

Theorem 2. The horizon initial covariances (17) satisfy Lyapunov-like recursive equations

Derivation of Lyapunov-like equations for mean and covariances (18), (19a), and (19b) is given in the appendix.

Remark 3. Original initial conditions (10) for the KFTD (7)–(9) at different time instants are identical in contrast to new horizon initial conditions (18), (19a), and (19b) which are more realistic for practice.

Remark 4 (real-time implementation of RHF). The RHF equations (13)–(19b) can be divided into two parts as follows.
Part 1 (offline equations). We may note that the RHF gains and error autocovariances can be precomputed, since they do not depend on the receding horizon measurements , but only on the system matrices , , noise statistics , , and horizon initial conditions (18), (19a), and (19b) which are the part of system model (1) and (2).
Part 2 (online equations). Thus, once the measurement schedule has been settled, the real-time implementation of the RHF requires only the computation of the RHF estimates , , , using only current measurements .

3.3. Conditional PDF and Optimal Receding Horizon Estimator

From (6) the optimal mean square estimate of the NCF based on the receding horizon sensor measurements (11) also represents a conditional mean:where is a Gaussian probability density function with conditional mean and covariance determined by RHF equations (13) and (15).

Thus, the estimate in (20) represents the optimal receding horizon estimator (RHE):which depends on the receding horizon estimate and its error covariance .

Further, we consider several examples of application of the general nonlinear estimator (20).

4. Examples of Optimal Nonlinear Estimator

4.1. Estimation of Distance

Consider an NFS representing a distance between an unknown scalar position and given point . In this case an NFS becomes . Using (20) the best estimate of the distance between an unknown position and moving point , , iswhere and are determined by (13) and (15) (or (7) and (9)), respectively, and is the error function.

In the particular case with , the estimate of an unknown modulus takes the form

4.2. Quadratic Function of State Vector

Consider an arbitrary quadratic function of a state vector ,Then optimal nonlinear estimate (20) can be explicitly calculated in terms of a state estimate and its error covariance . Using the formula [24]we obtain an optimal estimate for the quadratic functionIn (25) and (26), is the trace of a matrix .

In the particular case a quadratic function represents a power of a signal . Using (26) at , , we obtain

4.3. Estimation of Sine and Cosine Functions

In this case an NCF becomeswhere is an unknown angle.

Then the best estimates of the trigonometrical functions of an unknown angle are

5. Multivariate Polynomial Nonlinear Function

Here we consider a special NFS representing an arbitrary multivariate polynomial function of the formIn this case, the algorithm for calculation of the optimal estimator has a closed form because conditional expectationdepends on high-order moments of a multivariate Gaussian distribution, which can be explicitly calculated in terms of first- and second-order conditional moments [24–26]. For example,whereHere and are determined by the RHF equations.

For general NFS the unscented transformation serves as a useful tool for their calculations.

Remark 5. For typical NFS the integral (20) for normal distribution is known; that is, it can be explicitly expressed in terms of a receding horizon state estimate and its error covariance (see examples in Sections 4 and 5). But for general NFS the integral (20) can be calculated only approximately, for example, using the unscented transformation.

6. General Nonlinear Function and Unscented Transformation

The unscented transformation (UT) was first proposed by Julier and Uhlmann in light of the intuition that it is significantly easier to approximate the statistics of a transformed random variable, for example, the mean and covariance [27, 28]. During the last decade, the UT has become a powerful approach for designing new filtering and control algorithms for nonlinear dynamic models [29, 30]. Following this approach, the procedure to calculate the best estimate of an NFS (conditional mean)using the UT can be summarized as follows.

Generate the sigma points with corresponding weights :where is the th column of the matrix square root of and is the scaling parameter influencing the spreading of the points in the state space and thus the accuracy of the approximation [31]. Propagate each of these sigma points through a nonlinear function asand the resulting best estimate of the NFS is given asThus, the estimate (37) is approximately represented by known function of the receding horizon estimate and error covariance .