In this thesis, LMI approach is employed to design a static output feedback controller so that all poles of the considered closed-loop continuous-time system are located within a prescribed LMI region, named D region. Based on the coordinate transformation, an analysis about the derived LMI-based sufficient condition is also established. The result is, moreover, extended to treat pole placement in the generalized LMI region, denoted by DR region. In addition to the requirement on pole location, two commonly exploited system performances in robust control, i.e. the H2 and Hinf designs, are also considered so that the multiobjective control by static output feedback is investigated in this thesis. To address robustness issue of the designed controllers, three different uncertainty descriptions, i.e. norm bounded uncertainty, positive real uncertainty, and polytopic uncertainty, are considered and LMI conditions for quadratic D stabilization by static output feedback have been derived. The bounded realness and positive realness with respect to an LMI D region are studied as well. Numerical examples are provided in the end of chapters 3, 4, and 5 to illustrate the obtained results there.

The work presented in this thesis was motivated by the desire to establish an alternative approach to the design of robust polynomial controllers. The procedure of pole-placement forms the basis of the design and for polynomial systems this generally involves the solution of a diophantine equation. This equation has many possible solutions which leads directly to the idea of determining the most appropriate solution for improved performance robustness. A thorough review of many of the aspects of the diophantine equation is presented, which helps to gain an understanding of this extremely important equation. A basic investigation into selecting a more robust solution is carried out but it is shown that, in the polynomial framework, it is difficult to relate decisions in the design procedure to the effect on performance robustness. This leads to the approach of using a state space based design and transforming the resulting output feedback controller to polynomial form. The state space design is centred around parametric output feedback which explicitly represents a set of possible feedback controllers in terms of arbitrary free parameters. The aim is then to select these free parameters such that the closed-loop system has improved performance robustness. Two parametric methods are considered and compared, one being well established and the other a recently proposed scheme. Although the well established method performs slightly better for general systems it is shown to fail when applied to this type of problem. For performance robustness, the shape of the transient response in the presence of model uncertainty is of interest. It is well known that the eigenvalues and eigenvectors play an important role in determining the transient behaviour and as such the sensitivities of these factors to model uncertainty forms the basis on which the free parameters are selected. Numerical optimisation is used to select the free parameters such that the sensitivities are at a minimum. It is shown both in a simple example and in a more realistic application that a significant improvement in the transient behaviour in the presence of model uncertainty can be achieved using the proposed design procedure.

This thesis is concerned with robust control of an offshore steel jacket platform subject to nonlinear wave-induced forces. Since time delay and uncertainty are inevitably encountered for an offshore structure and their existence may induce instability, oscillation and poor performance, it is very significant to study on how the delay and uncertainty affect the offshore structure. In this thesis, a memory robust control strategy is, for the first time, proposed to reduce the internal oscillations of the offshore structure under wave-induced forces, so as to ensure the safety and comfort of the offshore structure. Firstly, when the system's states are adopted as feedback, memory state feedback controllers are introduced for the offshore structure. By using Lyapunov-Krasovskii stability theory, some delay-dependent stability criteria have been established, based on which, and by combining with some linearization techniques, memory state feedback controllers are designed to control the offshore structure. The simulation results show that such controllers can effectively reduce the internal oscillations of the offshore structure subject to nonlinear wave-induced forces and uncertainties. On the other hand, a new Lyapunov-Krasovskii functional is introduced to derive a less conservative delay-dependent stability criterion. When this criterion is applied to the offshore structure, an improved memory state feedback controller with a small gain is obtained to control the system more effectively, which is sufficiently shown by the simulation. Secondly, when the system's outputs are adopted as feedback, memory dynamic output feedback controllers are considered for the offshore structure. By employing a projection theorem and a cone complementary linearization approach, memory dynamic output feedback controllers are derived by solving some nonlinear minimization problem subject to some linear matrix inequalities. The simulation results show that the internal oscillations of the offshore structure subject to nonlinear wave-induced forces are well attenuated. Finally, robust H control is fully investigated for the offshore structure. By employing Lyapunov-Krasovskii stability theory, some delay-dependent bounded real lemmas have been obtained, under which, via a memory state feedback controller or a dynamic output feedback controller, the resulting closed-loop system is not only asymptotically stable but also with a prescribed disturbance attenuation level. The simulation results illustrate the validity of the proposed method.

Two robust control strategies are proposed in this dissertation for a class of multi-input multi-output dynamic systems with matched and mismatched perturbations. First of all, a novel design methodology of switching variables is proposed for solving the regulation problems. A serial state transformations are needed in order to design pseudo feedback gains and adaptive mechanisms. By utilizing the pseudo control input gain embedded in each of the switching variable, the proposed controller can not only suppress the mismatched perturbations when the controlled systems are in the sliding mode, but also attain locally asymptotic stability. The design of a robust output tracking controller is presented next for solving the tracking problems. Without utilizing the information of state variable, the proposed output feedback tracking controllers are capable of driving the state tracking errors into a small bounded region whose size can be adjusted through the designed parameters, and guarantee the stability of controlled systems. These two robust control schemes are designed by means of the variable structure control technique with sliding mode and Lyapunov stability theorem. Each controller contains three parts. The first part is for eliminating measurable feedback signals. The second part is used for adjusting the convergent rate of state variables (or tracking errors) of the controlled system. The third part is an adaptive control mechanism, which is to adapt some unknown constants of the least upper bounds of perturbations, so that the knowledge of the least upper bounds of matched and mismatched perturbations are not required. Several numerical examples and an application of controlling aircraft's velocity are demonstrated for showing the feasibility of the proposed control methodologies.

In this thesis, digital controller design methods for sampled-data nonlinear systems are considered. Although sampled-data nonlinear control has attracted much attention in recent years, the controller design methods for sampled-data nonlinear systems are still limited. Therefore, a range of controller design methods for sampled-data nonlinear systems are developed such as backstepping, adaptive and robust backstepping, reduced-order observer-based output feedback controller design methods based on the Euler approximate model. These controllers are designed to compensate the effects of the discrepancy between the Euler approximate model and exact discrete time model, parameter estimation error in adaptive control and observer error in output feedback control which behave as disturbance. A dual-rate control scheme is presented for output-feedback stabilization of sampled-data nonlinear systems. It is shown that the designed controllers semiglobally practically asymptotically (SPA) stabilize the closed-loop sampled-data nonlinear system. Moreover, various applications of these methods are given and their performances are analyzed with simulations.

Based on the Lyapunov stability theorem, an adaptive output feedback controller is proposed in this thesis for a class of multi-input multi-output (MIMO) dynamic systems with time-varying delay and disturbances. With an adaptive mechanism embeded in the proposed control scheme, the controller will automatically adapt the unknown upper bound of perturbation, so that the information of upper bounded of perturbations is not required. Once the controlled system reaches the switching hyperplane, not only the dynamics of system can be stabilized, but also the state trajectories can be driven into a small bounded region whose size can be adjusted through the design parameter. Two numerical examples are given for demonstrating the feasibility of the proposed control scheme.

A networked control system (NCS) is a control system where sensors, actuators, and controllers are interconnected over a communication network. This dissertation presents a framework for modeling, stability analysis, optimal control, and bandwidth allocation of the NCS. A ball magnetic-levitation (maglev) system, four DC motor speed-control systems, and a wireless autonomous robotic wheelchair are employed as test beds to illustrate and verify the theoretical results of this dissertation. This dissertation first proposes an output feedback method to stabilize and control the NCSs. The random time delays in the controller-to-actuator and sensor-to-controller links are modeled with two time-homogeneous Markov chains while the packet losses are treated with Dirac delta functions. An asymptotic mean-square stability criterion is established to compensate for the network-induced random time delays and packet losses in the NCS. Then, an algorithm to implement the asymptotic mean-square stability criterion is presented. Experimental results illustrate effectiveness of the proposed output feedback method compared to conventional controllers. The proposed output feedback controller could reduce the errors of the NCS by 13% and 30–40% for the cases without and with data packet losses, respectively. The optimal bandwidth allocation and scheduling of the NCS with nonlinear-programming techniques is also presented in the dissertation. The bandwidth utilization (BU) of each client is defined in terms of its sampling frequency. Two nonlinear approximations, exponential and quadratic approximations, are formulated to describe the system performance governed by discrete-time integral absolute error (DIAE) versus sampling frequency. The optimal sampling frequencies are obtained by solving the approximations with Karush-Kuhn-Tucker (KKT) conditions. Simulation and experimental results are given to verify the effectiveness of the proposed approximations and the bandwidth allocation and scheduling algorithms. In simulations and experiments, the two approximations could maximize the total BU of the NCS up to about 98% of the total available network bandwidth.

Research Doctorate - Doctor of Philosophy (PhD) / This thesis addresses the problem of robustness in model predictive control (MPC) of discrete-time systems. In contrast with most previous work on robust MPC, our main focus is on robustness in the face of both imperfect state information and dynamic model uncertainty. For linear discrete-time systems with model uncertainty described by sum quadratic constraints, we propose output-feedback MPC policies that: (i) treat soft constraints using quadratic penalty functions; (ii) respect hard constraints using 'tighter' constraints; and (iii) achieve robust closed-loop stability and non-zero setpoint tracking. Our two main tools are: (1) a new linear matrix inequality condition which parameterizes a class of quadratic MPC cost functions that all lead to robust closed-loop stability; and (2) a new parameterization of soft constraints which has the advantage of leading to optimization problems of prescribable size. The stability test we use for MPC design builds on well-known results from dissipativity theory which we tailor to the case of constrained discrete-time systems. The proposed robust MPC designs are shown to converge to well-known nominal MPC designs as the model uncertainty (description) goes to zero. Furthermore, the present approach to cost function selection is independently motivated by a novel result linking MPC and minimax optimal control theory. Specifically, we show that the considered class of MPC policies are the closed-loop optimal solutions of a particular class of minimax optimal control problems. In addition, for a class of nonlinear discrete-time systems with constraints and bounded disturbance inputs, we propose state-feedback MPC policies that input-to-state stabilize the system. Our two main tools in this last part of the thesis are: (1) a class of N-step affine state-feedback policies; and (2) a result that establishes equivalence between the latter class and an associated class of N-step affine disturbance-feedback policies. Our equivalence result generalizes a recent result in the literature for linear systems to the case when N is chosen to be less than the nonlinear system's 'input-state linear horizon'.

Circadian rhythms exist in almost all of living species, and they occupy an important role in daily biological activities of these species. This thesis deals with reduction of measurements in circadian models, and recovery of circadian phases. Two mathematical models of circadian rhythms are considered, with a 3rd order model for Neurospora, and a 7th order model for Mammals. The reduction of measurements of circadian models is shown by the proposals of observer designs to the two mathematical models of circadian rhythms. Both mathematical models contain strong nonlinearities, which make the observer design challenging. Two observer designs, reduced-order and one-sided Lipschitz, are applied to the circadian models to tackle the nonlinearities. Reduced-order observer design is based on a state transformation to make certain nonlinearities have no impact on the observer errors, and the design of one-sided Lipschitz observer is based on systems with one-sided Lipschitz nonlinearities. Both observer designs are based on the existing methods in literature. The existing method of reduced-order observer has been applied to a class of multi-output nonlinear systems. A new reduced-order observer design which extends the existing one in literature is presented in this thesis. In this new reduced-order observer method, the observer error dynamics can be designed by choosing the observer gain, unlike the existing one, of which the observer error dynamics depend on the invariant zeros under certain input-output map. The recovery of circadian phases is carried out to provide a solution to phase shifts occurred in circadian disorders. The restoration of circadian phases is performed by the synchronizations of trajectories of a controlled model with trajectories of a reference model. The reference model and the controlled model have phase differences, and both these models are based on a given 3rd order model of Neurospora circadian rhythms. The phase differences are reflected by different initial conditions, and by parameter uncertainty. The synchronizations of the two models are performed by using back-stepping method for the case of different initial conditions, and by using adaptive back-stepping method for the remaining case. Several simulation studies of the proposed observer designs and the proposed schemes of synchronizations are carried out with the results shown in this thesis.

A compact model representing the dynamics between piezoelectric voltage inputs and cantilever probe positioning, including nonlinear surface interaction forces, for atomic force microscopes (AFM) is considered. By considering a relatively large cantilever stiffness, singular perturbation methods reduce complexity in the model and allows for faster responses to Van der Waals interaction forces experienced by the cantilever's tip and measurement sample. In this study, we outline a nonlinear near-optimal feedback control approach for non-contact mode imaging designed to move the cantilever tip laterally about a desired trajectory and maintain the tip vertically about the equilibrium point of the attraction and repulsion forces. We also consider the universal instance when the tip-sample interaction force is unknown, and we construct cascaded high-gain observers to estimate these forces and multiple AFM dynamics for the purpose of output feedback control. Our proposed output feedback controller is used to accomplish the outlined control objective with only the piezotube position available for state feedback. / Master of Science / In this thesis, the idea of an atomic force microscope (AFM), specifically the applications of the non-contact mode, will be discussed. An atomic force microscope (AFM) is a tool that measures the surface height of nanometer sized samples. To improve the speed and precision of the machine under a non-contact mode objective, a controller is designed based on optimality and is applied to the system. The system contains a series of equations designed to steer the system towards a desired trajectory and minimal vibrations. Given the complexity of the system, resulting from nonlinearities, we will apply singular perturbation principles on the system's stiffness property to separate the larger problem into two smaller ones. These two problems are inserted into a near-optimal controller and a series of simulations are conducted to demonstrate performance. Alongside this, we will outline an observer to estimate the unknown dynamics of the system. These estimates are then applied to our controller to demonstrate that only the AFM's piezotube position is to be known in order to estimate and control the remaining dynamics of the system.