Generalized eigenproblem algorithms and software for algebraic Riccati equations

An Output Stabilization Problem of Distributed Linear Systems Approaches and Simulations

“…In this section we presen lculatio adient of the system (11). By the previous result th Equation (20).…”

On the numerical solution of large-scale sparse discrete-time Riccati equations

“…Inspired by Kleinman's formulation of a Newton method for continuous-time algebraic Riccati equations [37], Hewer [32] proposed a Newton method for solving DAREs. The algorithm was extended to the generalized equation as given in (1) by Arnold and Laub [3]. A discussion of its convergence properties can be found in [39,44].…”

“…We will follow in particular the approach taken in [13]. That is, we will make use of the fact that R(X) = 0 defines a system of nonlinear equations and can hence be solved by an appropriate Newton method as proposed in [32,3]. Newton's method is reviewed in Section 2.…”

Control System Radii and Robustness Under Approximation

“…On the one hand the finite dimensional systems converge to the distributed parameter system for which (approximate) controllability is not robust with respect to bounded perturbations ( [37; 10]). On the other hand, the dimension of the finite dimensional approximating systems grows as the discretization is refined making it more likely that small perturbations will destroy controllability ( [1]). In addition, the measures considered here do not take into account the typical structure of the finite dimensional approximating systems.…”

“…As noted in [1] numerical algorithms which assume a specific system property such as controllability or stabilizability can be expected to be numerically ill-conditioned if the system model is nearly uncontrollable (or nearly unstabilizable). The following simple example illustrates the type of difficulties that one can encounter.…”