Browsing by Author "Pearson, J. Boyd"
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Item Analysis and robust control of hybrid continuous/discrete time systems(1992) Bamieh, Bassam Abbas; Pearson, J. BoydHybrid Continuous/Discrete Time Systems are systems in which one part evolves in continuous time and another in discrete time. Important examples of such systems are digital (sampled-data) control systems, in which a continuous time plant is controlled by a digital controller using sample and hold devices. Traditionally, such systems have been discretized and treated as discrete-time systems thereby ignoring their continuous time (or inter-sample) behavior. The traditional approach is not applicable in modern theories of robust control such as ${\cal H}\sp\infty$ or $\ell\sp1,$ where for continuous time plants it is more natural that plant and signal uncertainties are given in continuous time. Therefore, to pose robust control problems with sampled-data controllers, one needs to consider the continuous time behavior of these systems. The continuous time behavior of sampled-data systems is in general periodically time varying. A new framework is developed for continuous-time periodic systems which has many of the useful qualities of the time invariant theory such as transforms and constant state space realizations. The framework is based on a lifting technique which establishes a strong correspondence between periodically time-varying systems in continuous time, and certain types of infinite dimensional time-invariant systems in discrete time. The utility of this framework is shown by posing and providing exact and constructive solutions to both the ${\cal H}\sp\infty$ and the $\ell\sp1$ sampled-data control problems, where the performance specifications are given in terms of continuous time uncertainties. A natural generalization is given for the ${\cal H}\sp2$ problem in the case of time-varying systems. This is used to pose the ${\cal H}\sp2$ problem for sampled-data systems, and a complete solution is also obtained.Item Design of control systems to meet l(infinity) specifications(1993) McDonald, James Stuart; Pearson, J. BoydThree control system design problems aimed at meeting specifications given in terms of the $l\sb\infty$ norm, or peak magnitude, of disturbances and errors are solved. The specification in the first problem requires simply that the error satisfy an $l\sb\infty$ norm constraint for all time provided that the disturbance satisfies a corresponding constraint. This is the standard $l\sb1$ problem, and the results here include generalizations of many known results on this problem: existence of optimal compensators, FIR sub-optimal approximation, and super-optimal approximation. The specifications in the remaining two problems are based on $l\sb\infty$ measures of weighted disturbances and errors; these weights can be chosen such that constraining a weighted signal constrains, for example, its peak rate and/or acceleration. One is an incremental weighted specification, requiring that the weighted error satisfy an $l\sb\infty$ norm constraint up until any given time provided that the weighted disturbance satisfies a corresponding constraint up until the same time. The other is a weighted specification which requires that the weighted error satisfy an $l\sb\infty$ norm constraint for all time provided that the weighted disturbance satisfies a corresponding constraint for all time. The associated design problems turn out to be distinct. For each of the two weighted specifications an appropriate system norm (or gain with respect to the given weights) is defined and it is shown that it can be computed by solving a standard $l\sb1$ problem (in the incremental weighted case) or a very similar problem (in the weighted case). Results for each weighted design problem parallel those for the unweighted, or standard $l\sb1,$ case: existence of optimal compensators, FIR sub-optimal approximation, and super-optimal approximation.Item EIGENVALUE ASSIGNMENT IN LINEAR CONTROL SYSTEMS USING ACCESSIBLE STATES(1970) BRASCH, FREDERICK MARTIN, JR.; Pearson, J. BoydItem Modelling, simulation, and control of robots(1989) Chatterjee, Trinanjan; Pearson, J. BoydThe continuous dynamic model of a robot is represented by nonlinear coupled differential equations. These can be discretized using first or second order Taylor's series approximations. One such model, the Greenspan Discrete Mechanics model appears to be the best for control applications. Global linearization of this model is straightforward and the linearized model is used to design $l\sp1$ optimal controllers for the robot. The Multi Input Multi Output optimization problem splits up into decoupled Single Input Single Output problems which are individually solved. Optimal systems thus designed are simulated and compared to traditional robot controllers. Simulations confirm the $l\sp1$ optimality of the controllers. This thesis thus provides a useful new tool for the robot control engineer.Item On the load-frequency control problem(1971) Bratton, Timothy Lee; Pearson, J. BoydThis thesis reports an investigation of tie-line bias load-frequency control. This type of control is currently used on many electric energy supply systems, but has recently been the subject of much controversy in the literature. Claims have been made that typical industry values of the load-frequency controller parameters result in smaller "stability margins" than other gains which minimize integral quadratic performance indices which weight only tie-line power and frequency deviations. Through analog simulation studies of the dynamic model of two interconnected steam-electric areas, it was found in this research that the "optimum" LFC gains resulted in rates of change of generated power which were beyond the capabilities of typical industry generating units. A digital-computer parameter optimization routine was written which demonstrated that typical industry LFC gains also satisfy an integralquadratic performance index -- one which weights speed-changer rate more heavily than frequency or tie-line power. A case is then made against the use of parameter optimization as a tool for designing control systems. The eigenvalues of the Two-Area Model were studied and it was found that the LFC gains have little effect on any interconnection poles other than those which are characteristic of the motion of the speedchanger. In particular, the synchronizing oscillations become less damped as the LFC integrator gain increases from zero. When the two areas of the Model were identical and a decentralized control strategy was implemented, it was found that half the poles of the interconnection were fixed as the interconnection coefficient increased from zero. These fixed-pole modes were not observable in the transfer function between an area's load change and its angle deviation. Classical root-locus techniques were used to study the observable modes. It was found that as the interconnection coefficient went to infinity, the synchronizing oscillation poles approached a vertical asymptote whose position depends only on the inertia and damping of each area, not on any feedback of an area's state variables to its governor input. It is indicated that further improvement in interconnection response may be difficult to achieve via a decentralized control strategy which acts through the governor.Item Optimal compensation and the circle condition(1971) Randall, Robert Louis; Pearson, J. BoydA desirable property of an optimal system with full state feedback was shown by Kalman to be satisfaction of a circle condition in the A(jw) plane where A(s) is the loop gain function (return ratio) of the optimal system. This thesis demonstrates that this same property is satisfied by systems which are designed to be optimal using output feedback. The implications of this result are applied to the problem of system stability in the presence of a time-varying, nonlinear gain at the plant input. This thesis also investigates the optimality of dual mode control for systems using output feedback.Item Optimal control with incomplete state measurements(1969) Bhattacharyya, S. P. (Shankar P.), 1946-; Pearson, J. BoydThis thesis considers the problem of designing a controller for a linear plant, which, using output measurements only, will make the plant state variables follow optimally any input signal belonging to a known class of signals. The dynamic order of the controller is determined by the observability properties of the plant and by the class of input signals. The structure of the controller is determined from the requirement that the closed loop system consisting of plant and controller be an optimum system. Two approaches to the optimization procedure are indicated: (1) Optimizing the dynamics of an augmented system consisting of plant, controller and a dynamic system generating the input signals, to minimize appropriate errors. (2) Regarding the general problem as one of optimum regulation of errors desired to be minimized. The results allow the linear servomechanism problem to be treated, for the first time, in a realistic manner.Item Robust steady-state tracking(1996) Sadeghi, Behnam; Pearson, J. BoydThe thesis solves the problem of finding an LTI controller that minimizes the steady-state tracking error of uncertain discrete-time systems. If a system is LTI, use of the "internal model principle" will assure that the error signal converges to zero. But this principle no longer applies when the physical plant is time-varying. This leads to the problem of how the steady-state value of the tracking error can be minimized in the time-varying case. The solution is provided in the following mathematical setting. The nominal model of the plant is SISO and LTI, and plant uncertainty is modeled by an arbitrary fading memory operator that is SISO, linear time-varying, and norm-bounded. A special case of the more general n-perturbation problem is also solved.Item Specific optimal control using stochastic approximation(1967) Tarver, Loyd Estes; Pearson, J. BoydThis thesis is concerned with the solution of a specific optimal control problem. Because of the effect of the initial state of a system on the solution, it is desired to obtain a specific optimal controller which will be optimum in a well-defined sense for a region of initial states. The problem is formulated by first assuming the initial state to be a random variable. Then the "stochastic optimal controller" is defined as the set of constant parameters which minimizes the expected value with respect to the initial state of a suitably chosen performance index. The method of stochastic approximation is proposed as a means of solution; however, it is shown that the theory of stochastic approximation as it now exists cannot be directly applied to the problem. Modifications of the theory which will allow solutions to be obtained are then proposed. The validity of these modifications has been verified by numerical experimentation, although an analytical verification has not been made. Both linear and nonlinear systems may be treated in this formulation. In cases where the expected value of the performance index may not be a good criterion function, the specific optimal controller for the worst-case initial state or input signal can be used as an alternative design. This min-max problem is discussed, and numerical examples are presented.Item Stability and performance robustness of discrete-time systems with structured uncertainty(1990) Khammash, Mustafa Hani; Pearson, J. BoydThe robustness of discrete-time systems in the presence of norm-bounded linear time-varying and/or nonlinear perturbations is addressed. A general framework for dealing with stability as well as performance robustness of such systems is set up whereby an arbitrary number of perturbations can be treated simultaneously. It is shown that a performance robustness problem can in fact be converted to a stability robustness problem. In addition, it is demonstrated how a stability robustness problem can be reduced to a simple algebraic problem, one which can be easily solved providing necessary and sufficient conditions for stability/performance robustness. These conditions are simple to compute and thus provide an effective tool for robustness analysis which can handle a large number of perturbations with relative ease. Furthermore, it is shown how in certain important cases controllers with optimal robustness properties can be designed. Finally, some results for computing the ${\cal A}$ norm of stable rational functions are provided.Item Studies in system representation and control(1995) Kishore, Ananda Papineni; Pearson, J. BoydRepresentation of an abstract system by a mathematical object such as a matrix or a differential equation is important because much of the analysis is done via representations. For discrete-time input-output systems, infinite-matrix kernel representation is the most general representation. We use the notion of finite-horizon systems to characterize all operators defined on $\ell\sb{p}$ spaces, $p\in\lbrack1, \infty$) that can be represented by an infinite-matrix kernel. We obtain a characterization of complete and closed behaviors in terms of finite-horizon systems. We give a new definition of stabilizability and obtain a characterization of stabilizability of systems on Hilbert spaces in terms of closable operators. A robustly stabilizing controller stabilizes every member of the model class to which the given plant belongs. We show that if the model class is an open set in $H\sb\infty$, there is a necessary and sufficient condition for robust stability. But if the model class is a closed set, the expected condition is only sufficient but not necessary. Same results hold for robust performance as well. Also, there is an equivalence between robust stability and robust performance problems if the model class is open, but none if it is closed. However, even with open model class, the expected conditions may be only sufficient but not necessary for uniform stability and performance. We present necessary and sufficient conditions for uniform stability in certain cases. There is sometimes a need to eliminate sinusoidal vibrations in the body of a mechanical system such as a cryogenic cooler. These vibrations are modeled as output disturbances. We assume that we know the frequencies of these sinusoids, but not their phases. There are infinitely many controllers resulting in notch filters that eliminate the sinusoids asymptotically. A weighted combination of the energies of the error signal and the control signal is a good criterion of optimality. We reduce this optimization problem to a weighted $H\sb2$-minimization problem. Due to some technical reasons, this problem is more difficult to solve than the regular Linear Quadratic Gaussian problem. We present a solution that involves solving two Riccati equations and a stable projection.Item Turbo-alternator modeling and control(1971) Kinra, Madan Mohan; Pearson, J. BoydThis thesis is concerned with the problem of designing a constant coefficient feedback controller that will allow arbitrary pole placement of a single synchronous generator connected through a reactance to an infinite bus. The nonlinear system model is derived and linearized. The controller regulates voltage, frequency and real power outputs of the synchronous machine. It is in cascade with the plant and uses voltage, frequency and real power deviations to generate the control signals. It has the following features, which others do not possess: (1) It is designed for arbitrary pole placement of the closed loop system; (2) It is a multi-input, multi-output controller and (3) It permits ah algorithmic approach which can be completely computerized.