Time-Delay Systems and Its Applications in Engineering 2014View this Special Issue
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Estimation of Nonlinear Functions of State Vector for Linear Systems with Time-Delays and Uncertainties
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.
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) . 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  and the estimate diverges  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 . 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).
3. Receding Horizon Filter and Nonlinear Estimator
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
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 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 . 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 .
Application of the obtained results is illustrated by example of wind tunnel system with model uncertainties.
7. Numerical Example: The Wind Tunnel System
A comparative experimental analysis of the proposed RHF (11)–(19b) and KFTD (7)–(10) is considered on example of nonlinear estimation of a kinetic energy of the wind tunnel system with two model time-invariant uncertainties and . In this case, we numerically evaluate and compare three mean square errors (MSEs) of the RHF, KFTD, and nonlinear estimator for kinetic energy.
A discretization of the state differential equations and the design based on a discrete-time model has been applied to the high-speed closed-air unit wind tunnel model by Armstrong and Tripp . The state vector consists of the state variables , , and , representing derivatives from a chosen equilibrium point of the following quantities: = Mach number, = actuator position guide vane angle in a driving fan, and = actuator rate. Then the discretized system model with a sampling period of (sec) is given bywhere and are unknown time-invariant model parameters and a wind tunnel system with the initial mean and covariance is and . The covariance of the system white Gaussian noise is subjected to .
The first coordinate , related to Mach number of the wind tunnel system, is observable through a measurement model. We havewhere is zero-mean white Gaussian measurement noise with covariance . We assume that the true values of the model uncertainties and on the given uncertainty interval (UI) areand without it, that is,The horizon length of the RHF (13)–(19b) and RHE (20) takes three values as , and .
The total kinetic energy of an actuator representing the NFS, , can be expressed as the sum of the translational kinetic energy of the center of mass and the rotational kinetic energy about the center of mass . The kinetic energies can be expressed in the following quadratic form (see Section 4.2):where is rotational inertia, is angular velocity, is mass, and is linear velocity. Concrete numerical values for , , , and .
7.1. Simulation Results with Parametric Uncertainty
We assume that time-invariant uncertain model parameters take the form (40). We specially focus on comparing the MSEs of the Mach number of the wind tunnel system that directly contain the uncertainties in (38) and the total kinetic energy in (42).
Our point of interest is the behavior of the nonlinear RHE using proposed RHF around the uncertainty interval . Here we describe the results of simulations of two filters (KFTD, RHF) and RHE with three different receding horizon lengths in terms of MSEs such thatIn Figures 1 and 2, we observe that, within the uncertainty interval , the time histories of the MSEs for the KFTD are notably larger than the other RHFs. However, the KFTD performs slightly worse than the RHF with the biggest horizon length of . Also, the RHF/RHE with the small horizon length is more accurate than the RHF/RHE with the big horizon lengths and . Therefore, The reason for the presence of such a robustness property (44) is to compensate for the given uncertainties , as the horizon length for sensor (memory of RHF) should be minimal. In this case, it is equal, as .
The comparison shows us that performance of the RHF/RHE is quite good and it can be widely used in different areas of applications: industrial, military, space, inertial navigation, and others .
In this paper we propose a new RHF for discrete-time linear system with time-delays and model uncertainties with application to estimation of nonlinear functions of state vector. We have derived the Lyapunov-like recursive equations for horizon initial mean and covariance of a system state with an arbitrary number of time-delays and verified the effectiveness of the proposed filter RHF through numerical simulation and comparison with KFTD.
In order to estimate an arbitrary NCF in uncertain environment, a new RHE is proposed. Two important classes of an NCF (quadratic, polynomial) are considered in detail, and effective computational procedures based on formulas for high-order moments of a Gaussian distribution and UT transformation are proposed.
In verifying we implemented example for real dynamic system with time-delays and parametric model uncertainty . Through the implementations, the robustness of the RHF/RHE in terms of MSEs is demonstrated and the proposed receding horizon estimators have a potential to be used in a wide range of practical applications.
During the last decades, there has been extensive interest in the study of a class of physical systems modeled by hybrid system dynamics known as Markovian jump systems including multiple delays , mode-dependent delays , singular time-delays , and time-variant delays . As a generalization of the obtained results for linear stochastic systems with time-delays and uncertainties we would like to point out that it is possible to extend the main results to Markovian jump systems with time-delays and complex dynamical systems with prevalent network induced phenomena such as missing measurements, sensor delays, multiple fading measurements, and signal quantization.
Derivation of Equation For Horizon Initial Covariance (19a). Subtracting (18) from (1) we obtain time propagation of the centered state:Next we haveTaking expectation on both sides of (A.2) and using the fact that current noise does not depend on current and past states , we obtain recursive equation for covariances (19a) and (19b),Note that (A.3) contains autocovariance:
Derivation of Equation for Autocovariance (19b). Using “symmetric” property of autocovariance and without loss of generality we can assume that . Substituting in (A.1) we obtainMultiplying both sides of (A.6) by and using (A.5) we obtain It remains to calculate expectation in (A.8); that is,Calculating product using (A.6) and after that taking expectation we getAccording to assumption , “future” noise does not depend on current and past states ; therefore . Next using property of white noise we obtainFinally using (A.8), (A.10), and (A.11) we get equation for autocovariance (19b).
This completes the derivation Lyapunov-like equations for receding horizon mean and covariances.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A2000679).
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