Mean Square Tracking Error Research Articles

Impact of Doppler shift error on least‐squares MIMO channel estimation for high‐speed railway

Owing to the availability of Doppler shift measurement and the existence of dominant line-of-sight (LOS) path in high-speed railway (HSR) communications, Doppler shift is often pre-compensated on the received signal before performing further processing. In this study, the authors address the impact of erroneous Doppler shift measurement on the tracking performance of least-squares multiple-input multiple-output (MIMO) channel estimation for HSR. The closed-form tracking mean-square error (MSE) is derived over time-varying Rician fading channel assuming Gaussian distributed Doppler shift measurement. By transforming the Gaussian distributed measurement into its characteristic function and performing approximations with practical considerations, the tracking MSE of the approach with Doppler shift pre-compensation is written as a function of that with no pre-compensation, contributed from the previous research. Accordingly, they are able to develop a simple relation between MSE gain and MSE loss for channels with dominant LOS path. It is revealed that erroneous Doppler shift measurement impacts stronger LOS or faster time-varying channel more severely. However, if the measurement is perfect, the presence of a dominant LOS path provides tremendous gain approximately equal to the Rician factor in high signal-to-noise ratio (SNR) regime.

Hydrodynamic Control of Wave Energy Devices [Bookshelf

A control system can be optimized for various fundamentally distinct performance objectives, leading to various canonical design problems such as disturbance rejection and tracking. Most of these can be recast as design problems, where it is desired to make some aspect of the closed-loop system (such as the norm of an input/output channel or mean-square tracking error) as close to zero as possible. However, not all control design objectives fit neatly into this paradigm, and one of the notable exceptions is the problem of harvesting maximal energy from dynamic phenomena. For certain classes of problems (characterized by the passivity of the mapping u v and the boundedness of the disturbance), such objectives can be shown to have a finite lower bound, resulting in a finite maximum available energy. This fact has been known to control and system theorists for decades and is of particular importance in robustness analysis. However, beyond the obvious connections to the classical concept of “impedance matching,” the idea that this objective is useful as the primary measure of a control system’s performance—a thing to be optimized in closed loop and balanced against other objectives—is a concept that has received considerably less attention. It may, therefore, be surprising to learn that this problem has a rich and decades-long history in the area of ocean wave energy conversion. This book begins with a brief but fascinating overview of this still-evolving history, from its early beginnings in Japan in the 1950s, through its surge of activity in Europe and America during the oil crisis of the 1970s, the rally of global industrial activity in the 1990s, and into the present. The text succeeds at illustrating these various aspects, while also giving the reader a strong appreciation for the history and evolution of the field. One is left with an accurate and compelling snapshot of where the research community is headed and an appreciation for the central role that control theory will play in its future. Although this is not the first text written on wave energy technology, it does focuses primarily on advanced control techniques.

Adaptive Low Rank Matrix Completion

The low-rank matrix completion problem is fundamental to a number of tasks in data mining, machine learning, and signal processing. This paper considers the problem of adaptive matrix completion in time-varying scenarios. Given a sequence of incomplete and noise-corrupted matrices, the goal is to recover and track the underlying low rank matrices. Motivated from the classical least-mean square (LMS) algorithms for adaptive filtering, three LMS-like algorithms are proposed for estimating and tracking low-rank matrices. Performance of the proposed algorithms is provided in form of nonasymptotic bounds on the tracking mean-square error. Tracking performance of the algorithms is also studied via detailed simulations over real-world datasets.

Robust adaptive tracking control of an underactuated ship with guaranteed transient performance

ABSTRACTIn this paper, a robust adaptive controller for a underactuated ship is proposed to make the ship follow a predefined path with guaranteed transient performance and arbitrarily small steady-state position error and orientation error, despite the presence of environmental disturbances induced by wave, wind and ocean current and unknown system parameters. We design a new adaptive state feedback controller and parameter estimators to guarantee converging exponentially by assuming all states are available. It is shown that the mean-square tracking errors are of order of an arbitrarily small design parameter, and the position error and the orientation error also decrease at a pre-specified exponential rate. Simulations with a commercial monohull ship using Matlab also demonstrate the effectiveness of the controllers.

Low-complexity adaptive iteration algorithm for frequency tracking in OFDM systems

The carrier frequency offset (CFO), destroying the orthogonality between subcarriers, greatly degrades the performance of an orthogonal frequency-division multiplexing system. One of the most frequently used ways for a CFO estimator design is to adopt maximum-likelihood (ML) estimation, achieving high accuracy tightly close to the Cramer-Rao lower bounds (CRLBs). One of the ML-based algorithms, called linearly-combined CFO (LC-CFO), evaluates all the single-time-slot CFO estimates first and then linearly combined these CFO estimates in the minimum mean-square-error sense. Its tracking range is quite wide up to half the carrier spacing, and convergence speed is very fast, costing only several tens of iterations; moreover, its mean-square error (MSE) performances are very much close to CRLBs at medium-to-high signal-to-noise ratio (SNR) values. However, a set of arctangent functions is needed to be evaluated for each iteration, which increases the computational complexity. In this article, a low complexity, called simplified LC-CFO (SLC-CFO), is proposed that the set of arctangent functions are replaced by low-complexity limiters, resulting in simplifying the receiver design and reducing the computational load while keeping nearly the same tracking range and MSE performances. With proper choice of a parameter, SLC-CFO even shows faster convergence speed and lower MSE value at low SNR, compared with LC-CFO. Simulation results demonstrate all these aforementioned properties.

Global stable tracking control of underactuated ships with input saturation

In this paper, tracking control of underactuated ship in the presence of input saturation is addressed. By dividing the tracking error dynamic system into a cascade of two subsystems, the torques in surge and yaw axes are designed separately using the backstepping technique. More specifically, we design the yaw axis torque in such a way that its corresponding subsystem is finite time stable, which makes it to be de-coupled from the second subsystem after a finite time. This enables us to design the torque in the surge axis independently. It is shown that the closed-loop system is stable and the mean-square tracking errors can be made arbitrarily small by choosing design parameters. Simulation results also verify the effectiveness of the proposed scheme.

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

An observer-based controller is proposed for a class of uncertain MIMO nonlinear systems preceded by actuators with unknown asymmetric saturations. The gains of actuators are unknown nonlinear functions that are generally functions of states. The unknown nonlinearities are approximated by interval type-2 fuzzy systems where the consequent parameters are tuned adaptively. Meanwhile, the interval type-2 fuzzy systems due to their structure can cope with the uncertainties in antecedent membership functions. The states of the system do not need to be known. To tackle the unavailability of states an observer based on strictly positive real (SPR) theory is designed to estimate the states. To cope with the fuzzy approximation error and external disturbances an adaptive continuous robust structure is proposed. The continuous robust structure due to adaptiveness does not need the bound of uncertainties and due to continuity does not have the chattering problem. All the adaptive laws are derived via Lyapunov synthesis approach. Meanwhile, all the signals in the closed-loop system are guaranteed to be semi-globally uniformly ultimately bounded (SGUUB) and the mean-square tracking error can be made arbitrarily small. Simulation results reveal the effectiveness of the proposed method in terms of coping well with the uncertainties and saturation actuators while maintaining satisfactory performance.

An adaptive learning controller for MIMO uncertain feedback linearizable nonlinear systems

Most of available results in adaptive learning controllers (ALCs) with input learning technique have considered the single-input single-output nonlinear systems. This paper presents an ALC for MIMO uncertain feedback linearizable systems whose uncertainty is in their linear parameters. Since only an output signal is available for measurement, a high gain observer is used to estimate the unmeasurable state. The estimated state is then utilized to implement the ALC. The proposed ALC learns the input gain parameters of the state equation as well as the internal parameters. In addition, the desired input is also learned using an input learning rule to track the whole command history. In the proposed ALC, the tracking errors are bounded and the mean-square tracking error is \(O(\epsilon )\) as the task is repeated. Single-link and two-link manipulators are presented as simulation examples to confirm the feasibility and the performance of the proposed ALC.

Switched Kalman filter‐interacting multiple model algorithm based on optimal autoregressive model for manoeuvring target tracking

A manoeuvring target tracking algorithm based on the autoregressive (AR) model is proposed. First, the AR model is incorporated into the Kalman filter (KF) for target tracking. The closed-form solution of the AR model coefficients is obtained by minimising the mean-square tracking error, and subject to the polynomial constraint of target motion. Then, based on the AR model, the proposed algorithm is constructed by combining the KF with the interacting multiple model (IMM) filter, coupled with the proposed detection schemes for manoeuvre occurrence and termination, as well as for switching initialisation. Simulations are performed to demonstrate the effectiveness of the AR model, and the proposed algorithm is compared with the IMM filter and variable-dimension filter in the manoeuvring scenario.

Robust Adaptive Control of Nonlinear Systems with Uncertainties

This paper proposes a robust adaptive robust controller for nonlinear systems represented by input-output models with unmodeled dynamics. Under the circumstances that the output of the system is bounded, the proposed controller can guarantee that all the variables of the system are bounded in the presence of unmodeled dynamics and time-varying disturbances. The scheme does not need to generate an additional dynamic signal to dominate the effects of the unmodeled dynamics. It is shown that the mean-square tracking error can be made arbitrarily small by choosing some design parameters appropriately.

Dynamic waveform selection for manoeuvering target tracking in clutter

In recent years, cognitive radar (CR) with waveform diversity has exhibited significant performance improvements over the traditional fixed waveform radar and become an area of vigorous research and development. This study presents a dynamic waveform selection algorithm to strive for tracking error minimisation for CR manoeuvering target tracking in clutter. Based on the concepts of resolution cell and measurement extraction cell, the statistical characteristics of radar measurements are discussed without dependence upon the Cramer-Rao lower bound of the measurement errors and the high signal-to-noise ratio assumption. A particle filter combined with probabilistic data association is used as a tracker. To quantify the utility of available waveforms, the predicted tracking mean-square error, because of its dependence on actual future measurements, is approximated efficiently via Gaussian fitting of the prior density of the target state and statistical linearisation of the measurement equation. Monte Carlo simulation results show that the proposed dynamic waveform selection algorithm can improve tracking performance considerably, especially in terms of track loss probability.

Multiple Model Adaptive Mixing Control: The Discrete-Time Case

Most of the work in multiple model adaptive control with various forms of switching focused on continuous-time systems. The purpose of this technical note is to extend the results of one approach, the adaptive mixing control (AMC), to discrete-time systems. Further, the technical note solves the tracking problem which has not been addressed in most schemes of this class. Stability and robustness properties of the AMC scheme for discrete-time systems are analyzed. It is shown that in the ideal case, when no disturbances or unmodeled dynamics are present, the tracking error converges to zero. In the non ideal case, the mean-square tracking error is of the order of magnitude of the modeling error provided the unmodeled dynamics satisfy a norm-bound condition. While these robustness results are conceptually similar to those of traditional robust adaptive control, the proposed scheme does not suffer from the drawback of stabilizability of the estimated plant and in addition performs much better in simulation studies. Furthermore, it allows well developed results from robust control to be incorporated in the design.

Data‐driven tuning of linear parameter‐varying precompensators

AbstractMethods for direct data‐driven tuning of the parameters of precompensators for linear parameter‐varying (LPV) systems are developed. Since the commutativity property is not always satisfied for LPV systems, previously proposed methods for LTI systems that use this property cannot be directly adapted. When the ideal precompensator giving perfect mean tracking exists in the proposed precompensator parameterization, the LPV transfer operators do commute and an algorithm using only two experiments on the real system is proposed. It is shown that this algorithm gives consistent estimates of the ideal parameters despite the presence of stochastic disturbances. For the more general case, when the ideal precompensator does not belong to the set of parameterized precompensators, another technique is developed. This technique requires a number of experiments equal to twice the number of precompensator parameters and it is shown that the calculated parameters minimize the mean‐squared tracking error. The theoretical results are demonstrated in simulation. Copyright © 2009 John Wiley & Sons, Ltd.

QPSK DS-CDMA System over Rayleigh Channel with a Randomly-Varying Frequency Narrow-Band Interference: Frequency Tracking Analysis

This paper analyses frequency tracking characteristics of a complex-coefficient adaptive infinite-impulse response (IIR) notch filter used for suppression of narrow-band interference (NBI) with a randomly-varying frequency in a quadriphase shift keying (QPSK) modulated direct-sequence code-division multiple-access (DS-CDMA) communication system. The QPSK DS-CDMA signals are transmitted over a frequency non-selective Rayleigh fading channel. The analysis is based on a first-order real-coefficient difference equation with respect to steady-state instantaneous frequency tracking error from which a closed-form expression that relates frequency tracking mean square error (MSE) with number of DS-CDMA active users and NBI power is obtained. Closed-form expressions for optimum notch bandwidth coefficient and step size constant that minimize the frequency tracking MSE are also derived. Computer simulations are included to substantiate the accuracy of the analyses.

Tracking Analysis of Complex Adaptive IIR Notch Filter for a Linear Chirp Signal

This paper analyzes frequency tracking characteristics of a complex-coefficient adaptive infinite impulse response (IIR) notch filter with a simplified gradient-based algorithm. The input signal to the complex notch filter is a complex linear chirp embedded in a complex zero-mean white Gaussian noise. The analysis starts with derivation of a first-order real-coefficient difference equation with respect to steady-state instantaneous frequency tracking error. Closed-form expression for frequency tracking mean square error (MSE) is then derived from the difference equation. Lastly, closed-form expressions for optimum notch bandwidth coefficient and step size constant that minimize the frequency tracking MSE are derived. Computer simulations are presented to validate the analysis.

Adaptive Fuzzy Tracking Control of a Class of Stochastic Nonlinear Systems with Unknown Dead-Zone Input

In this paper, a direct adaptive fuzzy tracking control scheme is presented for a class of stochastic uncertain nonlinear systems with unknown dead-zone input. A direct adaptive fuzzy tracking controller is developed by using the backstepping approach. It is proved that the design scheme ensures that all the error variables are bounded in probability while the mean square tracking error becomes semiglobally uniformly ultimately bounded (SGUUB) in an arbitrarily small area around the origin. Simulation results show the effectiveness of the control scheme.

Robust adaptive output feedback control of nonlinearly parameterized systems

The ideas of adaptive nonlinear damping and changing supply functions were used to counteract the effects of parameter and nonlinear uncertainties, unmodeled dynamics and unknown bounded disturbances. The high-gain observer was used to estimate the state of the system. A robust adaptive output feedback control scheme was proposed for nonlinearly parameterized systems represented by input-output models. The scheme does not need to estimate the unknown parameters nor add a dynamical signal to dominate the effects of unmodeled dynamics. It is proven that the proposed control scheme guarantees that all the variables in the closed-loop system are bounded and the mean-square tracking error can be made arbitrarily small by choosing some design parameters appropriately. Simulation results have illustrated the effectiveness of the proposed robust adaptive control scheme.

Performance Analysis of Noncoherent Digital Delay Locked Loops for Direct Sequence Spread Spectrum Systems With Doppler Shift and Quantized Adaptation

The noncoherent second-order digital delay locked loop (DLL) with the presence of Doppler shift is investigated. The loop performance such as lock-in range, transient response, mean time-to-lose lock (MTLL), and mean square tracking error (MSE) are analyzed. The analysis is unique in two respects. First, MTLL and MSE are evaluated more accurately than by previous methods. Second, the impact of quantized adaptation on the loop performance is examined for digital DLL that was neglected previously. Numerical results show that: 1) the quantized adaptation may significantly alter the loop behaviors, including the lock-in range, transient responses, MTLL, and MSE; and 2) the traditional analysis based on Gaussian approximation may result in a large error in performance evaluation when quantized adaptation is taken into account.

A Robust PN Code Tracking Algorithm for Frequency Selective Rayleigh-Fading Channels

In this letter, we propose a new tracking scheme that is robust against multipath fading for pseudonoise (PN) code tracking in direct-sequence-spread spectrum systems. The proposed scheme employs an adaptive filter whose taps are adapted using a block least-mean square algorithm and it results in minimizing the effect of multipath interference on the tracking performance. We show that the mean-squared tracking error performance of the proposed scheme is not affected by the presence of closely spaced paths (e.g., one to three chips), unlike that of conventional delay locked loops. We also show that the tap-weight distribution of the filter provides accurate estimates of the multipath delays. For example, at E/sub b//N/sub 0/=5 dB, 98% of the time the path estimates lie within one sample (1/5 of a chip) from the actual delays. Furthermore, simulation results suggest that multipath delays over a wide range of terminal speeds can be tracked successfully. The proposed scheme is well suited for wideband code-division multiple-access systems where a large number of closely-spaced multipath components need to be tracked and used in RAKE combining.

Robust adaptive control of nonlinear systems with unmodelled dynamics

A robust adaptive output feedback controller is proposed for a class of nonlinear systems represented by input–output models. In the design of the adaptive controller, a signal is used to characterise the unmodelled dynamics and a nonlinear damping term is introduced to counteract the effects of the unmodelled dynamics and bounded disturbances. With the proposed controller, all the variables in the closed-loop system are bounded in the presence of unmodelled dynamics and bounded disturbances. Moreover, the mean-square tracking error can be made arbitrarily small by choosing appropriate design parameters.