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Robust control is an aspect of control theory which explicitly considers uncertainties and how they affect robust stability in the analysis and design of control decisions. A basic requirement for optimal robust guaranteed control in a real life scenario is the stabilization of systems in the presence of uncertainties or perturbations. In this thesis, the system uncertainties are embedded into a norm bounded uncertainty elements. The perturbation function is modelled as a class of nonlinear uncertainty influencing a neutral system with infinite delay. It is assumed to have delay in state and is input dependent; which implies the effect of control action can directly or indirectly influence the nonlinear perturbation function. In recognition of the fact that stability and controllability are fundamental in obtaining the optimal robust guaranteed cost control design for neutral functional integro-differential systems with infinite delays (NFDSID), total asymptotic stability results were developed using Razumikhin technique, unique properties of eigenvalues, and the uniform stability properties of the functional difference operator for neutral systems. The new results, obtained using Razumikhin’s technique, extend and complement basic stability results in neutral systems to NFDSID. Novel sufficient conditions were developed for the null controllability of nonlinear NFDSID when the controls are constrained. By exploring the knowledge gained through other controllability results; conditions are placed on the perturbation function. This guaranteed that, if the uncontrolled system is uniformly asymptotically stable, and the controlled system satisfies a full rank condition, then the control system is null controllable with constraint if it satisfies some algebraic conditions. The investigation of optimal robust guaranteed cost control method has resulted in a novel delay dependent stability criterion for a nonlinear NFDSID with a given quadratic cost function. The new design is based on a model transformation technique, Lyapunov matrix equation and Lyapunov-Razumikhin stability approach. The Lyapunov-Razumikhin technique is adopted for this investigation because it is considered more scalable for optimal robust guaranteed cost control design for NFDSID. It is demonstrated that a memory less feedback control can be synthesized appropriately to ensure: (i) the closed-loop systems robust stability, and (ii) guarantee that the closed-loop cost function value remains within a specified bound. The problem of designing the optimal guaranteed cost controller is also realized in terms of inequalities. The Lyapunov-Krasovskii method is used to obtain stability conditions in comparison to the Razumikhin method. This method leads to linear matrix inequality (LMI) for the delay-independent case which is known to be conservative. To illustrate the potential practical applicability of the theoretical results; a cascade connection of two fully filled chemical solution mixers, and an integrated lossless transmission line which has a capacitance, inductance, resistance and terminated by a nonlinear function are modelled. A neutral control system model for NFDSID is derived from each of these systems. Simulation studies on the transmission line system confirm the theoretical robust stability results. The new results and methods of analysis expounded in this thesis are explicit, computationally more effective than existing ones and will serve as a working document for the present and future generations in the comity of researchers and industries alike.

This thesis is concerned with the choice of the uncertainty structure in robust control problems. This choice affects the optimization carried out to obtain a robust feedback controller, and determines how robust a feedback loop will be to discrepancies in the parameters or dynamics of the plant model. Firstly, it presents readily applicable distance measures, robust stability margins and associated robust stability and robust performance theorems for several commonly used uncertainty structures for linear time-invariant systems (additive, multiplicative, inverse multiplicative, inverse additive, right coprime factor uncertainty).Secondly, the thesis discusses the robust stabilization problem for linear plants with a coprime factor uncertainty structure where the coprime factors of the plant are not necessarily normalized. The problem considered here is a generalization of the normalized coprime factor robust stabilization problem. It is shown that the minimum of the ratio of (non-normalized) coprime factor distance over (non-normalized) coprime factor robust stability margin, termed the robustness ratio, is an important bound in robust stability and performance results. A synthesis method is proposed which maintains a lower bound on the normalized coprimefactor robust stability margin (as a proxy for nominal performance) while also robustly stabilizing a particular perturbed plant, potentially far outside a normalized coprime factor neighbourhood of the nominal plant. The coprime factor synthesis problem is also considered in a state-space framework. It is shown that it admits a simple and intuitive controller implementation in observer form. Via the solution of one Riccati equation, an optimally robust observer gain L can be obtained for any state-feedback matrix F. One particular method for obtaining a suitable F is also proposed, ensuring that the feedback loop is particularly robust to uncertain lightly damped poles and zeros.

In principal component analysis (PCA), the principal components (PC) are linear combinations of the variables that minimize some objective function. In the classical setup the objective function is the variance of the PC's. The variance of the PC's can be easily upset by outlying observations; hence, Chen and Li (1985) proposed a robust alternative for the PC's obtained by replacing the variance with an M-estimate of scale. This approach cannot achieve a high breakdown point (BP) and efficiency at the same time. To obtain both high BP and efficiency, we propose to use MM- and τ-estimates in place of the M-estimate. Although outliers may cause bias in both the direction and the size of the PC's, Chen and Li looked at the scale bias only, whereas we consider both. All proposed robust methods are based on the minimization of a non-convex objective function; hence, a good initial starting point is required. With this in mind, we propose an orthogonal version of the least median of squares (Rousseeuw and Leroy, 1987) and a new method that is orthogonal equivariant, robust and easy to compute. Extensive Monte Carlo study shows promising results for the proposed method. Orthogonal regression and detection of multivariate outliers are discussed as possible applications of PCA. / Science, Faculty of / Statistics, Department of / Graduate

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Since the launch of the first rocket by the scientists during the World War II , mankind continues their exploration of space. Those space explorations bring the benefits to human, such as high technology products like GPS, cell phone, etc. and in-depth insight of outside of the earth. However, they produce millions of debris with a total estimated mass of more than 3,000,000 kg in the space around the earth, which has and will continue to threat the safety of manned or unmanned space exploration. According to the research, at least tens of spacecraft were considered been damaged or destroyed by the debris left in the space. Thus, the increasingly cluttered environment in space is placing a premium on techniques capable of tracking and estimating the trajectory of space debris. Among debris, the pieces smaller than 1cm are unable to damage spacecraft because of the crafts’ shields, while the pieces larger than 10cm can be tracked by ground-based radars or a radar network. However, unlike the debris within these size ranges, the debris larger than 1 cm and smaller than 10 cm are able to hurt the shield of space craft and are hard to be detected by the exiting technical equipments because of their small size and cross-section area. Accordingly it is always a challenge for spacecraft or satellite mission designers to consider explicitly the ones ranged from 1 cm to 10 cm a priori. To tackle this challenge, a vision based debris’ trajectory tracking method is presented in the thesis. Unlike radar tracking, vision based tracking doesn’t require knowledge of a debris’ cross-section, regardless of its size. In this work, two cameras onboard of satellites in a formation are used to track the debris in iv close proximity. Also to differentiate the target debris from other clutters (i.e. the debris that are not tracked intentionally), a data association technique is investigated. A two-stage nonlinear robust controller is developed to adjust the attitude of the satellites such that the target debris is always inside of the field of view of the cameras. Capabilities of the proposed integrated estimation and control methods are validated in the simulations.

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The development of navigation filters that make use of robust estimation techniques is important due to the sensitivity of the typical minimum L2 norm techniques, such as the Kalman filter, to deviations in the assumed underlying probability distribution. In particular, those distributions with thicker tails than the Gaussian distribution can give rise to erratic filter performance and inconsistency of results. This dissertation discusses the development of an adaptive discrete-time robust nonlinear filtering technique based on a recursive form of Huber's mixed minimum L1/L2 norm approach to estimation, which is robust with respect to deviations from the assumed Gaussian error probability distributions inherent to the Kalman filter. This mixed norm approach is applied to a type of Sigma-Point Kalman filter, known as the Divided Difference Filter, which can capture second-order effects of nonlinearities in the system and measurement dynamics. Additionally, if these assumed parameters of the distribution differ greatly from the true parameters, then the filter can exhibit large errors and possibly divergence in nonlinear problems. This behavior is possible even if the true error distributions are Gaussian. To remedy these problems, adaptive filtering techniques have been introduced in order to automatically tune the Kalman filter by estimating the measurement and process noise covariances, however these techniques can also be highly sensitive to the nature of the underlying error distributions. The Huber-based formulations of the filtering problem also make some assumptions regarding the distribution, namely the approach considers a class of contaminated densities in the neighborhood of the Gaussian density. Essentially the method assumes that the statistics of the main Gaussian density are known, as well as the ratio or percentage of the contamination. The technique can be improved upon by the introduction of a method to adaptively estimate the noise statistics along with the state and state error covariance matrix. One technique in common use for adaptively estimating the noise statistics in real-time filtering applications is known as covariance matching. The covariance matching technique is an intuitively appealing approach in which the measurement noise and process noise covariances are determined in such a way that the true residual covariance matches the theoretically predicted covariance. The true residual covariance is approximated in real time using the sample covariance, over some finite buffer of stored residuals. The drawback to this approach is that the presence of outliers and non-Gaussianity can create problems of robustness with the use of the covariance matching technique. Therefore some additional steps must be taken to identify the outliers before forming the covariance estimates. In this dissertation, an adaptive scheme is proposed whereby the filter can estimate the process noise and measurement noise covariance matrices along with the state estimate and state estimate error covariance matrix. The adaptation technique adopts a robust approach to estimating these covariances that can resist the effects of outliers. The particular outlier identification method employed in this paper is based on quantities known as projection statistics, which utilize the sample median and median absolute deviation, and as a result are highly effective technique for multivariate outlier identification. These projection statistics are then employed as weights in the covariance matching procedure in order to reduce the influence of the outliers. The hybrid robust/adaptive nonlinear filtering methods introduced in this dissertation are applied to the problem of 6-DOF rendezvous navigation in elliptical orbit. The full nonlinear equations of relative motion are formulated in spherical coordinates centered on the target orbit. A relatively simple control law based on feedback linearization is used to track a desired rendezvous trajectory. The attitude dynamics are parameterized using Modified Rodrigues Parameters, which are advantageous for both control law development and estimation since they constitute a minimal 3-parameter attitude description. A switching technique which exploits the stereographic projection properties of the MRP coordinate is utilized to avoid singularities which inevitably arise in minimal attitude descriptions. This dissertation also introduces the proper covariance transformations associated with the singularity avoidance switching technique. An attitude control law based on backstepping is employed to track the target vehicle. A sensor suite consisting of a generic lidar or optical sensor, an Inertial Measurement Unit, consisting of accelerometers and gyroscopes, a star tracker, and a horizon sensor are utilized to provide measurement data to the navigation filters so that the chaser vehicle can estimate its relative state during the rendezvous maneuver. Several filters are implemented for comparison, including the Extended Kalman Filter, First and Second-Order Divided Difference Filters and Huber-based generalizations of these filters that include adaptive techniques for estimating the noise covariances. Monte-Carlo simulations are presented which include both Gaussian and non-Gaussian errors, including mismatches in the assumed noise covariances in the navigation filters in order to illustrate the benefits of the robust/adaptive nonlinear filters. Additionally, computational burdens of the various filters is compared. / Ph. D.

Estimation methods such as the Kalman filter identify best state estimates based on certain optimality criteria using a model of the system and the observations. A common assumption underlying the estimation is that the noise is Gaussian. In practical systems though, one quite frequently encounters thick-tailed, non-Gaussian noise. Statistically, contamination by this type of noise can be seen as inducing outliers among the data and leads to significant degradation in the KF. While many nonlinear methods to cope with non-Gaussian noise exist, a filter that is robust in the presence of outliers and maintains high statistical efficiency is desired. To solve this problem, a new robust Kalman filter framework is proposed that bounds the influence of observation, innovation, and structural outliers in a discrete linear system. This filter is designed to process the observations and predictions together, making it very effective in suppressing multiple outliers. In addition, it consists of a new prewhitening method that incorporates a robust multivariate estimator of location and covariance. Furthermore, the filter provides state estimates that are robust to outliers while maintaining a high statistical efficiency at the Gaussian distribution by applying a generalized maximum likelihood-type (GM) estimator. Finally, the filter incorporates the correct error covariance matrix that is derived using the GM-estimator's influence function. This dissertation also addresses robust state estimation for systems that follow a broad class of nonlinear models that possess two or more equilibrium points. Tracking state transitions from one equilibrium point to another rapidly and accurately in such models can be a difficult task, and a computationally simple solution is desirable. To that effect, a new robust extended Kalman filter is developed that exploits observational redundancy and the nonlinear weights of the GM-estimator to track the state transitions rapidly and accurately. Through simulations, the performances of the new filters are analyzed in terms of robustness to multiple outliers and estimation capabilities for the following applications: tracking autonomous systems, enhancing actual speech from cellular phones, and tracking climate transitions. Furthermore, the filters are compared with the state-of-the-art, i.e. the <i>H_â</i>-filter for tracking an autonomous vehicle and the extended Kalman filter for sensing climate transitions. / Ph. D.

The autocorrelation function is a commonly used tool in statistical time series analysis. Under the assumption of Gaussianity, the sample autocorrelation function is the standard method used to estimate this function given a finite number of observations. Non-Gaussian, impulsive observation noise following probability density functions with thick tails, which often occurs in practice, can bias this estimator, rendering classical time series analysis methods ineffective. This work examines the robustness of two estimators of correlation based on memoryless nonlinear functions of observations, the Phase-Phase Correlator (PPC) and the Median- of-Ratios Estimator (MRE), which are applicable to complex-valued Gaussian random pro- cesses. These estimators are very fast and easy to implement in current processors. We show that these estimators are robust from a bias perspective when complex-valued Gaussian pro- cesses are contaminated with impulsive noise at the expense of statistical efficiency at the assumed Gaussian distribution. Additionally, iterative versions of these estimators named the IMRE and IPPC are developed, realizing an improved bias performance over their non- iterative counterparts and the well-known robust Schweppe-type Generalized M-estimator utilizing a Huber cost function (SHGM). An impulsive noise suppression technique is developed using basis pursuit and a priori atom weighting derived from the newly developed iterative estimators. This new technique is proposed as an alternative to the robust filter cleaner, a Kalman filter-like approach that relies on linear prediction residuals to identity and replace corrupted observations. It does not have the same initialization issues as the robust filter cleaner. Robust spectral estimation methods are developed using these new estimators and impulsive noise suppression techniques. Results are obtained for synthetic complex-valued Guassian processes and real-world digital television signals collected using a software defined radio. / Ph. D.