Robust controller design : a bounded-input bounded-output worst-case approach

Caption title.Includes bibliographical references (leaves 38-41).Research supported by the NSF. 9157306-ECS Research supported by Wright Patterson AFB. F33615-90-C-3608 Research supported by C.S. Draper Laboratory. DL-H-441636Munther A. Dahleh

Analysis and synthesis of SISO H[subscript infinity] controllers

Classical feedback control theories are traditionally concerned with issues like stability and performance, however, they typically fail to address issues such as robustness and plant perturbation. This thesis is concerned with the robust stability and the robust performance of single-input single-output plants. The basic issue under analysis is how to realize the benefits of the usual feedback control structure in the presence of model uncertainty. This is accomplished by seeking feedback controllers providing robust stability and performance by minimizing weighted sensitivity functions of a linear system represented by its transfer function. A characterization of models for plants with unstructured uncertainty is introduced. Specifications and measures of stability and performance for robust controllers and the necessary and sufficient conditions to test the robust stability and the robust performance conditions of a control design are explored. A parametrization of feedback controllers that guarantee closed loop stability for both stable and unstable plants is shown and a systematic procedure for synthesizing robust controllers, known in the literature as HK controllers, is presented. These systematic algorithms are based on the theory of interpolation by analytic functions and the solution to the model matching problem. A case study of the inverted pendulum positioning system is developed to illustrate the concepts of robust analysis and the design algorithms. The controller is compared to a classic state variable feedback solution

Optimal Disturbance Rejection and Robustness for Infinite Dimensional LTV Systems

In this paper, we consider the optimal disturbance rejection problem for possibly infinite dimensional linear time-varying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. This approach does not assume any state space representation and views LTV systems as causal operators. After reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, which satisfy a "time-varying" allpass or flatness condition. Under mild assumptions the optimal TV controller is shown to be essentially unique. Next, the concept of M-ideals of operators is used to show that the computation of time-varying (TV) controllers reduces to a search over compact TV Youla parameters. This involves the norm of a TV compact Hankel operator defined on the space of causal trace-class 2 operators and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is provided. These results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz. The final outcome of the approach developed here is that it leads to two tractable finite dimensional convex optimizations producing estimates to the optimum within desired tolerances, and a method to compute optimal time-varying controllers.Comment: 30 pages, 1 figur

Optimal Control with Information Pattern Constraints

Despite the abundance of available literature that starts with the seminal paper of Wang and Davison almost forty years ago, when dealing with the problem of decentralized control for linear dynamical systems, one faces a surprising lack of general design methods, implementable via computationally tractable algorithms. This is mainly due to the fact that for decentralized control configurations, the classical control theoretical framework falls short in providing a systematic analysis of the stabilization problem, let alone cope with additional optimality criteria. Recently, a significant leap occurred through the theoretical machinery developed in Rotkowitz and Lall, IEEE-TAC, vol. 51, 2006, pp. 274-286 which unifies and consolidates many previous results, pinpoints certain tractable decentralized control structures, and outlines the most general known class of convex problems in decentralized control. The decentralized setting is modeled via the structured sparsity constraints paradigm, which proves to be a simple and effective way to formalize many decentralized configurations where the controller feature a given sparsity pattern. Rotkowitz and Lall propose a computationally tractable algorithm for the design of H2 optimal, decentralized controllers for linear and time invariant systems, provided that the plant is strongly stabilizable. The method is built on the assumption that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that some decentralized, strongly stablizable, stabilizing controller is available beforehand. For this class of decentralized feedback configurations modeled via sparsity constraints, so called quadratically invariant, we provided complete solutions to several open problems. Firstly, the strong stabilizability assumption was removed via the so called coordinate free parametrization of all, sparsity constrained controllers. Next we have addressed the unsolved problem of stabilizability/stabilization via sparse controllers, using a particular form of the celebrated Youla parametrization. Finally, a new result related to the optimal disturbance attenuation problem in the presence of stable plant perturbations is presented. This result is also valid for quadratically invariant, decentralized feedback configurations. Each result provides a computational, numerically tractable algorithm which is meaningful in the synthesis of sparsity constrained optimal controllers

Robust stabilisation of multivariable systems: A super-optimisation approach

The work aims to derive extended robust stability results for the case of unstructured uncertainty models of multivariable systems. More specifically, throughout the thesis, additive and coprime unstructured perturbation models are considered. Refined robust stabilisation problems of MIMO systems are defined and maximally robust controllers are synthesised in a state-space form. Unstructured perturbations which destabilise the feedback system for every optimal (maximally robust) controller are identified on the boundary of the optimal ball, i.e. the set of all admissible perturbations with norm equal to the maximum robust stability radius. Boundary perturbations are termed "uniformly destabilising" if they destabilise the closed-loop system for every optimal controller and it is shown that they all share a common characteristic, i.e. a projection of magnitude equal to the maximal robust stability radius, along a fixed direction defined by a pair of maximising vectors (scaled Schmidt pair) of a Hankel operator related to the problem. By imposing a directionality constraint it is shown that it is possible to increase the robust stability radius in every other direction by a subset of all optimal controllers. In order to solve this problem, super-optimisation techniques are developed. Independently a natural extension of Hankel norm approximations, the so-called super­ optimisation problem is posed and solved explicitly for the case of one-block problems in a state-space setting. It is thus shown that a subset of all maximally robust controllers, namely the class of super-optimal controllers, stabilises all perturbed plants within an extended stability radius 11,*(b), subject to a directionality constraint. In addition, the work is related to robust stabilisation subject to structured perturbations. The notions of structured robust stabilisation problem, and structured set approximation are defined in connection with the maximised set of permissible perturbations. It is further shown that µ*(J) can serve as an upper bound the structured robust stabilisation problem. The effect of µ*(J) as an upper bound depends on the compatibility between the two structures, the true structure and the artificial structure of the extended permissible set. [Look inside the thesis' abstract for an exact version of formulas and equations

Model-based control for high-tech mechatronic systems

Motion systems are mechanical systems with actuators with the primary function to position a load. The actuator can be either hydraulic, pneumatic, or electric. The feedback controller is typically designed using frequency domain techniques, in particular via manual loop-shaping. In addition to the feedback controller, a feedforward controller is often implemented with acceleration, velocity, and friction feedforward for the reference signal. This chapter provides an overview of a systematic control design procedure for motion systems that has proven its use in industrial motion control practise. A step-by-step procedure is presented that gradually extends single-input single-output (SISO) loop-shaping to the multi-input multi-output (MIMO) situation. This step-by-step procedure consists of interaction analysis, decoupling, independent SISO design, sequential SISO design, and finally, norm-based MIMO design. Extreme ultraviolet is a key technology for next-generation lithography

Optimization and analysis of the current control loop of VSCs connected to uncertain grids through LCL filters

Premio Extraordinario de Doctorado 2011This thesis focuses on the design and analysis of the control of voltage source converters connected to the grid through LCL filters. Particularly it is centered on grids presenting uncertainty in their intrinsic dynamic parameters and their influence over the inner control loop of a grid converter: the current control. To that end, the thesis follows a three-fold discussion. Firstly, the thesis studies the grid model, its uncertain parameters and presents a proposal to recursively estimate them. The estimation is based on a recursive least-squares optimization procedure applied to the current and voltage measurements, performed in the point of common coupling, expressed in a synchronous reference frame. The synchronization and the reference frame transformation process is specially designed for the proposed system. The optimization process is complemented with an estimation evaluation block that gives a real-time measure of the estimation quality. The influence of those uncertain parameters over the stability of the current control loop of grid converters is the second topic of this thesis. For the case of linear controllers, the analysis is performed by applying the structured singular value mu theory to a parametric uncertainty model that is described in the document. The proposed method extracts safe grid parameters ranges from a previously defined controller and plant model. Special attention is payed to important practical considerations as pure real uncertainty and sampled-data systems analysis. To test the method performance and illustrate its behavior, this dissertation discusses the robustness of three particular examples: a SISO control approach, a MIMO servo-controller approach and a robust H_inf design. For the case of non-linear controllers, the thesis focuses on hysteresis controllers and presents some practical conclusions. After that analysis, the thesis deals with the complementary problem: the design of a robust controller for grid converters connected through LCL filters to grids whose parameters range between known values. As a prior stage, the thesis presents an LQ servo-controller design procedure that may be complemented with the use of state estimators. The control is faced in a synchronous reference frame and directly controls the grid injected current. Once the framework is settled, the thesis proposes a design technique based on a robust Loop-shaping H_inf design procedure complemented with the nu-gap analysis tool. The final part of this dissertation describes the experimental set-up used for testing the presented proposals. After this, a summary of experimental results and waveforms is presented