A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Derivatives of eigenvalues and eigenvectors of a general complex matrix

Analysis of a general recursive prediction error identification algorithm

A robust H ∞ observer-based controller design for uncertain T-S fuzzy systems with unknown premise variables via LMI

Linear systems of ordinary differential equations

The paper considers the problem of eigenvalue assignment of linear multivariable two-dimensional systems using dynamic state or output feedback. The particular compensator applied is of the PID form, i.e. it involves a proportional (P), an integral (I) and a derivative (D) term. The algorithm proposed reduces the problem to that of solving a linear system of equations. Furthermore, conditions are established for the problem to have a solution.

Eigenvalue assignment of two-dimensional systems using PID controllers

In this paper, a design procedure for the synthesis of dead-beat control policies for a class of sampled-data systems with bounded input is presented The design algorithm is illustrated by a numerical example for a second-order system

Dead-beat control of sampled-data systems with bounded input

In this paper a number of important properties of the mode-controllability matrix of a linear continuous-time system are presented. It is shown that examination of the form of this matrix leads simply and directly to complete knowledge of the controllability characteristics of such a system.

Properties of the mode-controllability matrix†

In a previous paper (Crossley and Porter 1969) simple and explicit derivations were given for the first-order eigenvalue and eigenvector sensitivity coefficients for the Fundamental eigenproblem associated with the behaviour of linear dynamical systems governed by equations of the form [xdot] = Ax. In the present paper, similar techniques are used to derive the corresponding second-order eigenvalue and eigenvector sensitivity coefficients. The second-order theory is then applied to two examples in order to compare the results obtainable from this theory with the corresponding results obtainable from the first-order theory and from exact calculations.

High-order eigenproblem sensitivity methods: theory and application to the design of linear dynamical systems†

A single-input modal control theory is developed whereby the loop gains of a single-input time-invariant multi-variable system may be calculated using simple formulae for the cases when both the open-loop plant matrix and the closed-loop plant matrix have a number of sets of distinct and confluent eigenvalues. The results may be applied in a sequential manner for multi-input systems with repeated sets of confluent eigenvalues provided that the appropriate mode-controllability conditions are satisfied. A number of illustrative examples are included.

Generalized single-input modal control theory

In this paper a procedure for the design of feedback loops for a class of linear time-invariant systems is presented. These feedback loops contain multivariable three-term controllers whose parameters can be selected such that the matrix of the resulting closed-loop system can be assigned any prescribed set of eigenvalues.

T. R. Crossley

Papers

Dead-beat control of sampled-data systems with bounded input

Properties of the mode-controllability matrix†

High-order eigenproblem sensitivity methods: theory and application to the design of linear dynamical systems†

Generalized single-input modal control theory

Eigenvalue assignment in linear time-invariant closed-loop systems incorporating multi-variable three-term controllers†