Order reduction approaches for the algebraic Riccati equation and the LQR problem

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices

Algorithms for Computing Nash Equilibria in Deterministic LQ Games

In this paper we review a number of algorithms to compute Nash equilibria in deterministic linear quadratic differential games.We will review the open-loop and feedback information case.In both cases we address both the finite and the infinite-planning horizon.Algebraic Riccati equations;linear quadratic differential games;Nash equilibria

Linear Quadratic Games: An Overview

In this paper we review some basic results on linear quadratic differential games.We consider both the cooperative and non-cooperative case.For the non-cooperative game we consider the open-loop and (linear) feedback information structure.Furthermore the effect of adding uncertainty is considered.The overview is based on [9].Readers interested in detailed proofs and additional results are referred to this book.linear-quadratic games;Nash equilibrium;affine systems;solvability conditions;Riccati equations

Coupled Riccati equations for complex plane constraint

A new Linear Quadratic Gaussian design method is presented which provides prescribed imaginary axis pole placement for optimal control and estimation systems. This procedure contributes another degree of design freedom to flexible spacecraft control. Current design methods which interject modal damping into the system tend to have little affect on modal frequencies, i.e., they predictably shift open plant poles horizontally in the complex plane to form the closed loop controller or estimator pole constellation, but make little provision for vertical (imaginary axis) pole shifts. Imaginary axis shifts which reduce the closed loop model frequencies (the bandwidths) are desirable since they reduce the sensitivity of the system to noise disturbances. The new method drives the closed loop modal frequencies to predictable (specified) levels, frequencies as low as zero rad/sec (real axis pole placement) can be achieved. The design procedure works through rotational and translational destabilizations of the plant, and a coupling of two independently solved algebraic Riccati equations through a structured state weighting matrix. Two new concepts, gain transference and Q equivalency, are introduced and their use shown