Browsing by Subject "Robust Control"
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Item Control and optimization with dimensionality constraints(2008-10) Takyar, Mir ShahrouzThe purpose of this thesis is to develop synthesis tools for control design with dimensionality constraints. In particular, given a model for a physical process, the goal is to characterize all possible controllers of a certain dimension which satisfy given performance criteria. In classical feedback design, the complexity of controller adversely affects robustness of the regulatory mechanisms of the feedback and adds to the fragility of the system. The complexity is often due to the difficulty in imposing performance specifications in a natural mathematical context. Typically, this is done using "weight functions" which encapsulate the specifications, and then introducing those in a suitable optimization problem. A contribution of this work is to address a certain type of optimization problem and the choice of weight functions. More precisely, we develop a new design approach which leads to a controller achieving both requirements, the performance specifications and low complexity, at the same time. Further, this thesis generalizes the previous methods for multivariable systems by developing analogous theory and techniques. The main contribution in multivariable analytic interpolation is to characterize a family of minimal McMillan degree solutions by a choice of spectral-zero dynamics. In addition to application of this theory for model-matching in control design, we show how to use the same techniques for maximum power transfer in circuit theory, and for spectral estimation in signal analysis. Also in this thesis we give new results on implementation of controllers with some very specific elements. One such, which is hard to simulate on a digital computer, is what could be described as "half capacitor". It implements a "fractional integration" and can be used to a great advantage in classical feedback design, providing gain but without introducing time-lag.Item Optical Tweezers: Characterization and systems approach to high bandwidth force estimation.(2010-04) Sehgal, HullasIn recent times, the hard boundaries between classical fields of sciences have almost disappeared. There is a cross-pollination of ideas between sciences, engineering and mathematics. This work investigates a modern tool of micro-manipulation of microscopic particles that is used primarily by bio-physicists and bio-chemists for single cell, single molecule studies. This tool called the Optical Tweezers can trap microscopic dielectric particles using radiation pressure of light. Optical tweezers is increasingly being used in bio-assays as it provides a means to observe bio-molecules non invasively and offers a spatial resolution in nanometers and force resolution in femto-Newtons at millisecond timescales. In this work, physics governing the operating principle behind optical tweezers is presented, followed by a step by step procedure to build an optical tweezers system having measurement and actuation capability along with a controller logic for feedback implementation. The working of optical tweezers system is presented using a spring mass damper model and the traditional methods of optical tweezers characterization are discussed. A comprehensive view of Optical tweezers is then presented from a system theoretic perspective, underlying the limitations of traditional methods of tweezers characterization that are based on the first principle. The role of feedback in Optical tweezers is presented along with the fundamental limitations that the plant model imposes on optical tweezers performance to be used as a force sensor for fast dynamics input force. The purpose of optical tweezers as a pico-newton force probe is emphasized and a classical controls based method to improve the bandwidth of force estimation using an ad-hoc approach of system inversion is presented. The efficacy of system inversion based method in improving the force probe capability of feedback enhanced optical tweezers is validated by experimental results. It is shown experimentally that the system inversion method results in an order of magnitude improvement in the bandwidth of external force estimation. Finally, a robust control strategy is presented, where the problem of estimation of high bandwidth force is casted as an H-infinity optimization problem along with other performance objectives. This strategy is then compared with the traditional method using PI-controllers and experimental results presented. The robust control strategy is found to further improve the ability of optical tweezers as a force sensor for fast changing force profile by approximately three times over the system inversion approach.Item A Robust Control Perspective on Optimization of Strongly-Convex Functions(2016-07) Hu, BinLarge-scale optimization is a central topic in big data science. First-order black-box optimization methods have been widely applied in machine learning problems, since the oracle complexity of these methods can be independent of the parameter dimension. In this dissertation, we formulate linear matrix inequality (LMI) conditions to analyze the convergence rates of various deterministic and stochastic optimization methods. We derive these LMIs using integral quadratic constraints (IQCs) and dissipation inequalities. The first part of this dissertation analyzes deterministic first-order methods (gradient descent, Nesterov's method, etc) as generalized eigenvalue problems (GEVPs). A standard dissipation inequality requires a non-negative definite storage function and ``hard'' IQCs which must hold over all finite time horizons. We develop a modified dissipation inequality that requires neither non-negative definite storage functions nor hard IQCs. Then we show that linear rate analysis of a given deterministic first-order method is equivalent to uniform stability analysis of a related scaled system. This enables derivation of linear rate analysis conditions using standard IQCs for a scaled operator. A soft Zames-Falb IQC is derived and used in the modified dissipation inequality, leading to a GEVP formulation for linear rate analysis of first-order optimization methods. In the second part of this dissertation, we extend the IQC framework to analyze stochastic optimization methods which have been widely applied in empirical risk minimization and machine learning problems. We first combine jump system theory with IQCs to derive LMI conditions for rate analysis of the stochastic average gradient (SAG) method and its variants (SAGA, etc). The resultant LMI conditions can be used to analyze the convergence rates of SAG, SAGA, and other related variants with uniform or non-uniform sampling strategies. Then we develop LMI conditions to analyze the stochastic gradient (SG) method and its variants. The SG method with a constant stepsize typically achieves a linear convergence rate only up to some fixed tolerance. We develop stochastically averaged quadratic constraints with disturbance terms quantifying the inaccuracy of the SG method. Several known results about the SG method have been recovered using our proposed LMI conditions. We also obtain new results regarding the convergence of the SG method under different conditions.Item Synthesis and validation of flight control for UAV.(2009-11) Paw, Yew ChaiUnmanned Aerial Vehicles (UAVs) are widely used worldwide for a board range of civil and military applications. There continues to be a growing demand for reliable and low cost UAV systems. This is especially true for small-size mini UAV systems where majority of systems are still deployed as prototypes due to their lack of reliability. Improvement in the modeling, testing and flight control for the small UAVs would increase their reliability during autonomous flight. The traditional approach used in manned aircraft and large UAV system synthesizing, implementing and validating the flight control system to achieve desired objectives is time consuming and resource intensive. This thesis aims to provide an integrated framework with systematic procedures to synthesize and validate flight controllers. This will help in the certification of UAV system and provide rapid development cycle from simulation to real system flight testing. The effectiveness of the approach is demonstrated by applying the developed framework on a small UAV system that was developed at the University of Minnesota. The thesis is divided into four main parts. The first part is mathematical modeling of the UAV nonlinear simulation model using first principle theory. Flight test system identification technique is used to extract model and model uncertainty parameters to update the nonlinear simulation model. The nonlinear simulation model developed must be able to simulate the actual UAV flight dynamics accurately for real-time simulation and robust control design purposes. Therefore it is important to include model uncertainties into the nonlinear simulation model developed, especially in small UAV system where its dynamics is less well understood than the full-size aircraft. The second part of the work provides the approach and procedures for uncertainty modeling into the nonlinear simulation model such that realization of linear uncertain model is possible. The third part of work describes the flight control design and architecture used in the UAV autopilot system. Classical and model-based control synthesis approaches are presented for roll angle tracking controller to demonstrate the controller synthesis approaches and practical controller implementation issues on the embedded flight computer system. The last part of work blends in all the previous works into the integrated framework for testing and validation of the synthesized controllers. This involves software-in-the-loop, processor-in-the-loop and flight testing of the synthesized controllers using the integrated framework developed.