Ellipsoidal estimates of limit sets of discrete dissipative systems (Q1064301)

The authors consider the problem of constructing approximating ellipsoids for the discrete linear dissipative system \(x_{t+1}=Ax_ t+bu_ t+cv_ t\), where \(x_ t\in E^ n\), \(u_ t\in E^ 1\), \(v_ t\in E^ r\), A is a matrix with all its eigenvalues in magnitude less than unity and the input disturbance \(v_ t\) is subject to the constraint \(<v_ t,Mv_ t>\leq 1\), M is positive definite matrix. An approximating ellipsoid is one which contains the limit set of the system and the construction given utilises absorbing ellipsoids, ellipsoids such that, for any \(t_ 0\in T=\{0,1,2,...\}\) and any \(x_{t_ 0}\in {\mathcal T}\), \(x_ t\in {\mathcal T}\) for all \(t>t_ 0\). Approximating ellipsoids are constructed for the uncontrolled system, and in particular, a construction is given for the best such ellipsoid, the one of minimal volume. Two numerical examples are given. For the controlled system a correcting control law is defined as one for which the best approximating ellipsoid for the control law \(u=Kx\) is no worse than the corresponding ellipsoid for the uncontrolled system. A construction is given for correcting control laws and illustrated by a numerical example.