The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations. With minor technical adjustments, the theory is broadened to include considerations of unstable subsystems as well as instability of composite systems.

Asymptotic stability and instability of large-scale systems

Survey paper: A survey of some recent results in linear multivariable feedback theory

The problem of finding the optimal, constant output feedback matrix for linear time-invariant multivariable systems with quadratic cost is reconsidered. Simple formulae for the gradient matrix are developed and used in a Fletcher-Powell-Davidon algorithm. Computational results are presented.

Solution of the optimal constant output feedback problem by conjugate gradients

A recursive algorithm is shown to solve the above equation accurately for large ( n \leq 146 ), lightly damped ( \zeta \geq 10^{-3} ) systems. About 2.5n^{2} storage locations are required, and about 2.5n^{3} multiplications are performed per recursion, ten recursions being typical.

Numerical solution of the matrix equation AX + XA T + B = 0

A numerical comparison is made of most published methods for solving the linear matrix equations which arise when a quadratic form Liapunov function is applied to a constant linear system (continuous or discrete, real or complex). Generally, for the real equations direct methods are satisfactory for systems of order ten or less, whereas for larger order systems iterative methods (based upon expressing the solution in terms of an infinite series) are to be preferred. For the complex equations the most convenient numerical method uses an explicit representation for the solution in terms of the eigenvalues and vectors of the system matrix. If the system matrix is in companion form then algorithms taking account of this structure offer minor improvements.

Comparison of numerical methods for solving Liapunov matrix equations

The problem of expanding the matrix equation PA + A'P = - Q into a system of algebraic equations is considered. Starting from the previously published results, it suggests an alternate approach which is rather simple and very effective in computer applications. This approach is particularly useful when the given equation has to be solved iteratively which is the case in solution of the algebraic matrix Riccati equation appearing in optimal control problems.

http://ieeexplore.ieee.org/iel5/9/24111/01099386.pdf

S. Bingulac

Papers

An alternate approach to expanding PA+A'P=-Q