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Stable oscillations, arising from nonlinear non-equilibrium dynamics, are an important class of behavior in both nature and engineering. For engineering applications like robotic locomotion, it is usually desirable to induce oscillations in a controlled manner. While plenty of feedback control methods have been developed for equilibrium behaviors in control theory, we still lack a systematic design framework to excite non-oscillatory systems into oscillations via feedback control. As a step towards that, this thesis aims to provide theoretically guaranteed feedback controller design protocols for stable oscillations. Inspired by the oscillator models in biology and electrical engineering, we propose two architectures for oscillation control design: the mixed feedback oscillator and the resonant positive feedback oscillator. Our design methods take advantage of the dominant system theory to trigger and modulate stable oscillations in closed loop. For the mixed feedback oscillator, we show how to achieve oscillations by tuning the control parameters: the balance between the positive and negative feedback networks and the global feedback gain. For the resonant positive feedback oscillator, we show how to trigger oscillations by designing its linear and nonlinear control units. Then we show how to implement the two control architectures with basic linear and nonlinear circuit elements. Finally, we extend the control design for the mixed feedback oscillator. Based on linear matrix inequalities (LMIs) for dominance analysis, we develop control algorithms that automatically return state feedback gains that also take robustness and passivity-based interconnection into consideration.

The study of non-linear input-output maps can be summarized by three concepts: Gain, Positivity and Dissipativity. However, in order to make efficient use of these theorems it is necessary to use loop transformations and weightings, or so called ”multipliers”.The first problem this thesis studies is the feedback interconnection of a Linear Time Invariant system with a memoryless bounded and monotone non-linearity, or so called Absolute Stability problem, for which the test for stability is equivalent to show the existence of a Zames-Falb multiplier. The main advantage of this approach is that Zames–Falb multipliers can be specialized to recover important tools such as Circle criterion and the Popov criterion. Albeit Zames-Falb multipliers are an efficient way of describing non-linearities in frequency domain, the Fourier transform of the multiplier does not preserve the L1 norm. This problem has been addressed by two paradigms: mathematically complex multipliers with exact L1 norm and multipliers with mathematically tractable frequency domain properties but approximate L1 norm. However, this thesis exposes a third factor that leads to conservative results: causality of Zames-Falb multipliers. This thesis exposes the consequences of narrowing the search Zames-Falb multipliers to causal multipliers, and motivated by this argument, introduces an anticausal complementary method for the causal multiplier synthesis in [1].The second subject of this thesis is the feedback interconnection of two bounded systems. The interconnection of two arbitrary systems has been a well understood problem from the point of view of Dissipativity and Passivity. Nonetheless, frequency domain analysis is largely restricted for passive systems by the need of canonically factorizable multipliers, while Dissipativity mostly exploits constant multipliers. This thesis uses IQC to show the stability of the feedback interconnection of two non-linear systems by introducing an equivalent representation of the IQC Theorem, and then studies formally the conditions that the IQC multipliers need. The result of this analysis is then compared with Passivity and Dissipativity by a series of corollaries.

This report describes a method to control the density distribution of a large number of autonomous agents. The approach is based on the fact that there are a large number of agents in the system, and hence the time evolution of the probabilistic density distribution of agents can be described as a Markov chain. The main contribution of this paper is the synthesis of a Markov matrix which will guide the multi-agent system density to a desired steady-state density distribution, in a probabilistic sense, while satisfying some motion and safety constraints. Also, an adaptive density control method based on real time density feedback is introduced to synthesize a time-varying Markov ma- trix, which leads to better convergence to the desired density distribution. Finally, a decentralized density computation method is described. This method guarantees that all agents will have a best, and common, density estimate in a finite, with an explicit bound, number of communication updates.

This work details the development of a stability and control module for implementation into a Multidisciplinary Design Optimization (MDO) framework for the conceptual design of conventional and advanced aircraft. A novel approach, called the Variance Constrained Flying Qualities (VCFQ) approach, is developed to include closed-loop dynamic performance metrics in the design optimization process. The VCFQ approach overcomes the limitations of previous methods in the literature, which only functioned for fully decoupled systems with single inputs to the system. Translation of the modal parameter based flying qualities requirements into state variance upper bounds allows for multiple-input control laws which can guarantee upper bounds on closed-loop performance metrics of the aircraft states and actuators to be rapidly synthesized. A linear matrix inequality (LMI) problem formulation provides a general and scalable numerical technique for computing the feedback control laws using convex optimization tools. The VCFQ approach is exercised in a design optimization study of a relaxed static stability transonic transport aircraft, wherein the empennage assembly is optimized subject to both static constraints and closed-loop dynamic constraints. Under the relaxed static stability assumption, application of the VCFQ approach resulted in a 36% reduction in horizontal tail area and a 32% reduction in vertical tail area as compared to the baseline configuration, which netted a weight savings of approximately 5,200 lbs., a 12% reduction in cruise trimmed drag, and a static margin which was marginally stable or unstable throughout the flight envelope. State variance based dynamic performance constraints offer the ability to analyze large, highly coupled systems, and the linear matrix inequality problem formulation can be extended to include higher-order closed-loop design objectives within the MDO. Recommendations for further development and extensions of this approach are presented at the end.

Reduced cost of sensors and increased computing power is enabling the development and implementation of control systems that can simultaneously regulate multiple variables and handle conflicting objectives while maintaining stringent performance objectives. To make this a reality, practical analysis and design tools must be developed that allow the designer to trade-off conflicting objectives and guarantee performance in the presence of uncertain system dynamics, an uncertain environment, and over a wide range of operating conditions. As a first step towards this goal, we organize and streamline a promising robust control approach, Robust Linear Parameter Varying control, which integrates three fields of control theory: Integral Quadratic Constraints (IQC) to characterize uncertainty and nonlinearities, Linear Parameter Varying systems (LPV) that formalizes gain-scheduling, and convex optimization to solve the resulting robust control Linear Matrix Inequalities (LMI). To demonstrate the potential of this approach, it was applied to the design of a robust linear parametrically varying controller for an ecosystem with nonlinear predator-prey-hunter dynamics.

Abstract: Sampled-data control theory is the mathematical foundation required for the analysis and design of sampled-data systems. An important assumption in the development of conventional sampled-data control theory is that measurement sampling periods are uniform. However, with the widespread use of networked and embedded control systems it is not possible or practically feasible to have constant measurement sampling periods. Consequently, the conventional sampled-data theory needs to be re-evaluated before designing this class of control systems. Motivated by this, this thesis develops mathematical approaches for the analysis and synthesis of sampled-data systems with nonuniform sampling periods. Two types of variations in sampling period are considered. For the first type, we assume that the measurement is sampled irregularly but the input is updated regularly. For the second type, we assume that both measurement sampling and input updating occur synchronously but with nonuniform intervals. These timing models are general enough to capture many different types of variations in sampling periods. Both state estimation and control problems are considered. For state estimation, two types of filters are developed: a sampling period dependent for the first type of variations and a robust one for the second type of variations. A novel Markov model of the irregular sampling process together with the theory of Markovian jump systems lead to the design of the first type of filter. The second filter is designed by modelling the nonuniform sampling system as uncertain feedback system with linear fractional transformation uncertainty. For the control problem, a dynamic robustly stabilizing output feedback controller and a robust H-infinity controller are developed. Both controllers are designed for the second type of variations in sampling period and analysis is based on modelling the nonuniform sampling system as uncertain feedback system. All theoretical development in this thesis is in discrete time and design conditions are formulated as linear matrix inequalities (LMI's). Many solvers, such as the LMI toolbox of MATLAB, exist and can solve these convex optimization problems very efficiently. Most constant-parameter filters or controllers designed are easily implemented. The effectiveness of the proposed filters or controllers is demonstrated using simulation results.

In this thesis, we consider the distributed parameter-dependent modeling and control of non-uniform flexible structures, which are classified under spatially varying distributed systems. A distributed state space model of a non-uniform flexible cantilever beam is developed, in which the spatial variation of the beam parameters is treated as parametric uncertainty, assuming that the system depends on the spatially varying parameters in linear fractional manner. We are particularly interested in the systems discretized in spatial dimension for practical reasons. Spatial discretization is obtained via the central finite difference scheme. We assume that the displacements at each discretized node are measurable in real time for controller use. Based on the proposed distributed model, sufficient conditions for analysis and synthesis of a distributed output-feedback controller are presented using the induced L2 norm as the performance criterion. The controller synthesis condition is characterized in terms of linear matrix inequalities, which are convex optimization problems and can be solved efficiently using available software. The distributed controller inherits the same structure as the plant, which results in a localized control architecture and a simple implementation. Each local controller unit processes the available local displacement measurement while sharing information with its adjacent units. We present the main advantage of distributed control; its reliability in the case of malfunctioning actuators or sensors, where many other control techniques would probably fail.

Uncertainties in a plant's dynamics and in its input and output channels make it difficult for a control system to obtain good performance characteristics and stability of the closed-loop system. These difficulties arise both in continuous-time and discrete-time, but are exemplified in the sampled-data control problem. In the present work, the effects of uncertainties on the robust stability and performance of continuous-time, discrete-time and sampled-data systems is quantified using an H2 performance measure. For continuous-time and discrete-time systems, robust stability and performance bounds are formulated in terms of guaranteed cost inequalities. New guaranteed cost bounds are derived for plants with real structured uncertainty and are given in their equivalent LMI form. For continuous-time systems, a shifted linear bound and a shifted inverse bound are obtained. LMI forms are given for these bounds and for the shifted bounded real bound and shifted Popov bound. For discrete-time systems, a linear bound, an inverse bound, a shifted bounded-real bound, and a shifted Popov bound are obtained. LMI forms are given for these bounds. For both systems, comparisons are made between the shifted bounds, their unshifted counterparts and bounds based on standard LMI techniques. Finally, the sampled-data robust stability and performance problem is considered. The closed-loop performance of a sampled-data control system under plant uncertainty is investigated by utilizing exact discretization techniques. In particular, for an H2 performance measure, exact expressions for the closed-loop, state, cross-weighting, and control costs are obtained for a given sample interval h. LMI techniques are used to bound the worst-case performance given input and output uncertainties, and additional consideration is given to the case in which controllability is lost due to sampling. After applying discrete-time LQG synthesis to the sampled-data system, the achievable performance is evaluated for fast sampling h → 0 and for slow sampling h → infinity.