Stephen Richter

A Homotopy Algorithm for Solving the Optimal Projection Equations for Fixed-Order Dynamic Compensation: Existence, Convergence and Global Optimality

The purpose of this paper is to present a homotopy algorithm for solving the Optimal Projection Equations. Questions of existence and the number of solutions will also be examined. It will be shown that the number of stabilizing solutions to the given Optimal Projection Equations can be determined and that all solutions can be computed via a homotopic continuation from a simple problem. For an important special case, where the number of inputs or the number of outputs to the system is less than or equal to the dimension of the compensator, there is only one solution to the OPE, thus guaranteeing that globally optimum reduced order controller can be computed.

On direct versus indirect methods for reduced-order controller design

Comparisons are obtained between direct (parameter optimization) and indirect (controller reduction) methods in reduced-order controller design. In particular, the authors compare the controller canonical correlation coefficients method of C. De Villemagne and R.E. Skelton (1988) to the optimal projection theory of D.C. Hyland and D.S. Bernstein (1984). It is noted that suboptimal methods may, in certain cases, be simpler to implement than optimal methods. Thus, the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control-system performance. >

Design of Reduced-Order, H2 Optimal Controllers Using a Homotopy Algorithm

The minimal dimension of a linear-quadratic-gaussian (LQG) compensator is usually equal to the dimension of the design plant. This deficiency can lead to implementation problems when considering control-design for high-order systems such as flexible structures and has led to the development of methodologies for the design of optimal (or near optimal) controllers whose dimension is less than that of the design plant. This paper develops a new homotopy algorithm for the design of reduced-order, H2 optimal controllers. The algorithm has been implemented in MATLAB and the results are illustrated using a benchmark, non-colocated flexible structure control problem.

A continuation algorithm for eigenvalue assignment by decentralized constant-output feedback

This paper describes a continuation approach to eigenvalue assignment by decentralized constant-output feedback for interconnected systems. The method is a homotopy technique which embeds the decentralized control problem into a parametrized family of control problems. The parametrization is based on the system interconnection structure, and represents a continuous deformation of a system consisting of the uncoupled subsystems into the original system with full interconnections. Based on such a deformation, a differential equation is constructed whose solution trajectory has an endpoint which is a decentralized constant-output feedback matrix assigning the desired spectrum to the interconnected system. The derivation of the differential equation and considerations which guarantee the existence of a solution are given. Two examples are presented.

Feedback gain optimization in decentralized eigenvalue assignment

Abstract This paper develops a new design procedure for minimizing the norm of a decentralized output feedback matrix which assigns a user specified set of eigenvalues. Two distinct approaches which can be meshed together as a unified design tool are derived. Assuming one has computed a decentralized feedback matrix which assigns a desired spectrum, the first approach describes an iterative algorithm which reduces an algebraic cost function (e.g. the Frobenius norm) of the feedback gains while maintaining the desired spectrum. This algorithm allows for small movements in the eigenvalues. The iteration step is based on the first order variational behaviour of the eigenvalue-eigenvector equations. The second algorithm modifies a continuation method for decentralized eigenvalue assignment to include an optimizing factor. Numerical considerations for the design procedures are discussed. An example showing the improvement possible by the application of the procedure is also given.

Homotopy Methods in Control System Design and Analysis

Recent technologies have led to stringent control system requirements. This has increased the importance and complexity of the analysis and design of control systems, which often require the solution of systems of nonlinear equations of high order. Some challenging computational problems in control design include model order reduction, high dimensional Riccati equations, fixed-structure optimization, robust analysis and feedback synthesis, sensor/actuator placement, and simultaneous controller/structure design. This paper describes these problems, and the directions in which globally convergent homotopy methods must be extented in order to be applicable to computational problems in control. By way of illustration, a computationally effective probability-one homotopy algorithm is presented for the optimal projection formulation of the reduced order model problem, together with some numerical results.

Reliable design of H-2 optimal reduced-order controllers via a homotopy algorithm

Due to control processor limitations, the design of reduced-order controllers is an active area of research. Suboptimal methods based on truncating the order of the corresponding linear- quadratic-Gaussian (LQG) compensator tend to fail if the requested controller dimension is sufficiently small and/or the requested controller authority is sufficiently high. Also, traditional parameter optimization approaches have only local convergence properties. This paper discusses a homotopy algorithm for optimal reduced-order control that has global convergence properties. The exposition is for discrete-time systems. The algorithm has been implemented in MATLAB and is applied to a benchmark problem.