Abstract
When system parameters vary at a fast rate, identification schemes based on model-free local estimation approaches do not yield satisfactory results. In cases like this, more sophisticated parameter tracking procedures must be used, based on explicit models of parameter variation (often referred to as hypermodels), either deterministic or stochastic. Kalman filter trackers, which belong to the second category, are seldom used in practice due to difficulties in adjusting their internal parameters such as the smoothness coefficient and the order of the hypermodel. The paper presents a new solution to this problem, based on the concept of preestimation of system parameters. The resulting identification algorithms, which can be characterized as decoupled Kalman trackers, are computationally attractive, easy to tune and can be optimized in an adaptive fashion using the parallel estimation approach. The decoupled KF algorithms can be regarded as an attractive alternative to the state-of-the-art algorithms which are much more computationally demanding.
Highlights
Consider the problem of identification/tracking of a time-varying finite impulse response (FIR) system governed by y(t) = φT (t)θ (t) + e(t) (1)where t = . . . , −1, 0, 1, . . . denotes discrete time, y(t) denotes system output, φ(t) = [u(t − 1), . . . , u(t − n)]T is the regression vector made up of past measurements of the observable wide sense stationary input signal {u(t)}, {e(t)} denotes white measurement noise, and θ (t) = [θ1(t), . . . , θn(t)]T is the parameter vector made up of unknown time-varying system coefficients, independent of {u(t)} and {e(t)}.Linear time-varying FIR models are used, among others, to describe rapidly fading mobile communication channels
Multi-path effect: due to scattering the transmitted signal reaches the receiver along different paths, i.e., with different time delays; the values of FIR coefficients depend on the strength of ‘‘natural reflectors’’ and their time variation is caused by the receiver motion [1], [5]
When system parameters vary slowly with time, their estimation can be successfully carried out using the local estimation approach, such as the method of weighted least squares (WLS) [6], [7]
Summary
M. Ciołek et al.: Decoupled Kalman Filter Based Identification of Time-Varying FIR Systems. The problem of estimation of θ(t) can be expressed and solved as the problem of estimation of a state vector of an associated dynamical system In such a setup parameter estimation can be carried out using appropriately designed Kalman filtering (KF) algorithms [24]–[31]. Decoupling means that identification is carried out independently for each coefficient of the analyzed system Such a solution is possible owing to a new estimation paradigm for identification of nonstationary stochastic systems, based on the concept of preestimation. It will be shown that optimization of the tracking performance of the decoupled KF algorithm, i.e., tuning of its design parameters to the unknown and possibly time-varying form and rate of parameter changes, can be achieved by means of parallel estimation and cross-validation. We will derive the simplified, steady state version of the proposed identification algorithm with reduced computational load (depending linearly on the number of estimated parameters), which is a computationally attractive alternative to the current state-of-the-art
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