Showing papers by "Emmanuel G. Collins published in 1993"

Efficient computation of the solutions to modified Lyapunov equations

Abstract: This paper develops a solution method for modified Lyapunov equations in which the modification term $\mathcal{F}( Q )$ is a linear function of the solution Q. Equations of this form arise in robustness analysis and in homotopy algorithms developed for solving the nonstandard Riccati and Lyapunov equations that arise in robust reduced-order design. The method relies on decomposing $\mathcal{F}( Q )$ as $\mathcal{F} ( Q ) = \mathcal{G} ( \phi ( Q ) )$, where $\phi ( Q )$ is an m-dimensional vector. It is shown that if m is small, the new solution procedure is much more efficient than are solution procedures based on a straightforward transformation of the modified Lyapunov equation to a linear vector equation in $n( n + 1 )/2$ unknowns. The results are extended to develop an efficient procedure for computing the solutions to an arbitrary number of coupled Lyapunov equations in which the coupling terms are linear operators.

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

Abstract: 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, H 2 optimal controllers. The algorithm has been implemented in MATLAB and the results are illustrated using a benchmark, non-colocated flexible structure control problem.

Robust stability analysis using the small gain, circle, positivity, and Popov theorems: a comparative study

Abstract: The stability robustness of a maximum-entropy controller designed for a benchmark problem is examined. Four robustness tests are used, i.e., small gain analysis, circle analysis, positive real analysis, and Popov analysis, each of which is guaranteed to give a less conservative result than the previous test. The analysis is performed graphically. The Popov test is seen, for this example, to yield highly nonconservative robust stability bounds. The results illuminate the conservatism of analysis based on traditional small-gain type tests and reveal the effectiveness of analysis tests based on Popov analysis and related parameter-dependent Lyapunov functions. >

A Homotopy Algorithm for Maximum Entropy Design

Abstract: Maximum entropy design is a generalization of LQG that was developed to enable the synthesis of robust control laws for flexible structures. The method was developed by Hyland and motivated by insights gained from Statistical Energy Analysis. Maximum entropy design has been used successfully in control design for ground-based structural testbeds and certain benchmark problems. The maximum entropy design equations consist of two Riccati equations coupled to two Lyapunov equations. When the uncertainty is zero the equations decouple and the Riccati equations become the standard LQG regulator and estimator equations. A previous homotopy algorithm to solve the coupled equations relies on an iterative scheme that exhibits slow convergence properties as the uncertainty level is increased. This paper develops a new homotopy algorithm that does not suffer from this defect and in fact has quadratic convergence rates along the homotopy curve.

Fixed Structure Computation of the Structured Singular Value

Abstract: This paper addresses the robest stability and performance analysis problem using a fixed structure approach of the structured singular value. Specifically, using recent results on H 2 / H ? , a Riccati equation approach is formulated for complex-? with constant D-scales along with an H 2 performance bound. An optimization procedure is proposed in computing optimal D-scales with respect to the H 2 performance bound. Finally, it is shown that this new approach compares favorably to the standard structured singular value involving frequency-dependent D-scales.

Optimality Conditions for Reduced-Order Modeling, Estimation and Control for Discrete-Time Linear Periodic Plants

Abstract: For linear time-invariant systems it has been shown that the solutions to the optimal reduced-order modeling, estimation, and control problems can be characterized using optimal projection equations, sets of Riecati and Lyapunov equations coupled by terms containing a projection matrix. These equations provide a strong theoretical connection between standard full-order results such as linear-quadratic Gaussian theory and have also proved useful in the comparison of suboptimal reduction methods with optimal reduced-order methods. In addition, the optimal projection equations have been used as the basis for novel homotopy algorithms for reduced-order design. This paper considers linear periodic plants and develops necessary conditions for the reduced-order modeing, estimation, and control problems. It is shown that the optimal reduced-order model, estimator, and compensator is characterized by means of periodically time-varying systems of equations consisting of coupled Lyapunov and Riccati equations.

Generalized Fixed-Structure Optimality Conditions for H 2 Optimal Control

Abstract: Over the last several years, researchers have shown that when it is assumed a priori that a fixed-order optimal compensator is minimal, the necessary conditions can be characterized in terms of coupled Riccati and Lyapunov equations, usually termed "optimal projection equations." When the optimal projection equations for H 2 optimal control are specialised to full-order control, the standard LQG Riccati equations are recovered. This paper relaxes the minimality assumption on the compensator and derives necessary conditions for fixed-structure H 2 optimal control that reduce to the standard optimal projection equations when the optimal compensators are assumed to be minimal. The results are then specialized to full-order control. The results show that the standard LQG Riccati equations can be derived using fixed-structure theory even without the minimality assumption. They also show for the first time that a reduced-order optimal projection controller is a projection of one of the extremals to the full-order H 2 optimal control problem. This latter result is used to discuss suboptimal projection methods that are able to produce minimal-order realizations of nonminimal LQG compensators. For this special case, the similarity transformation relating the projection matrix used by these suboptimal methods to the optimal projection matrix from the standard optimal projection theory is explicitly defined.

Computationally Efficient Homotopies for the H 2 Model order Reduction Problem

Abstract: The H 2 optimal model reduction problem, i.e., the problem of approximating a higher order dynamical system by a lower order one so that a model reduction criterion is minimized, is of significant importance and is under intense study. Several earlier attempts to apply homotopy methods to the H 2 optimal model order reduction problem were not entirely satisfactory. Richter devised a homotopy approach which only estimated certain crucial partial derivatives and employed relatively crude curve tracking techniques. Žigic, Bernstein, Collins, Richter, and Watson formulated the problem so that numerical linear algebra techniques could be used to explicitly calculate partial derivatives, and employed sophisticated homotopy curve tracking algorithms, but the number of variables made large problems intractable. We propose here several ways to reduce the dimension of the homotopy map so that large problems are computationally feasible.

A homotopy algorithm for digital optimal projection control GASD-HADOC

Abstract: The linear-quadratic-gaussian (LQG) compensator was developed to facilitate the design of control laws for multi-input, multi-output (MIMO) systems. The compensator is computed by solving two algebraic equations for which standard closed-loop solutions exist. Unfortunately, the minimal dimension of an LQG compensator is almost always equal to the dimension of the plant and can thus often violate practical implementation constraints on controller order. This deficiency is especially highlighted when considering control-design for high-order systems such as flexible space structures. This deficiency motivated the development of techniques that enable the design of optimal controllers whose dimension is less than that of the design plant. A homotopy approach based on the optimal projection equations that characterize the necessary conditions for optimal reduced-order control. Homotopy algorithms have global convergence properties and hence do not require that the initializing reduced-order controller be close to the optimal reduced-order controller to guarantee convergence. However, the homotopy algorithm previously developed for solving the optimal projection equations has sublinear convergence properties and the convergence slows at higher authority levels and may fail. A new homotopy algorithm for synthesizing optimal reduced-order controllers for discrete-time systems is described. Unlike the previous homotopy approach, the new algorithm is a gradient-based, parameter optimization formulation and was implemented in MATLAB. The results reported may offer the foundation for a reliable approach to optimal, reduced-order controller design.