Eigenvalue placement for variable structure control systems
Variable Structure Control is a well-known solution to the problem of deterministic control of uncertain systems, since it is invariant to a class of parameter variations. A central feature of vsc is that of sliding motion, which occurs when the system state repeatedly crosses certain subspaces in the state space. These subspaces are known as sliding hyperplanes, and it is the design of these hyperplanes which is considered in this thesis. A popular method of hyperplane design is to specify eigenvalues in the left-hand half-plane for the reduced order equivalent system, and to design the control matrix to yield these eigenvalues. A more general design approach is to specify some region in the left-hand half-plane within which these eigenvalues must lie. Four regions are considered in this thesis, namely a disc, an infinite vertical strip, a sector and a region bounded by two intersecting sectors. The methods for placing the closed-loop eigenvalues within these regions all require the solution of a matrix Riccati equation : discrete or continuous, real or complex. The choice of the positive definite symmetric matrices in these Riccati equations affects the positioning of the eigenvalues within the region. suitable selection of these matrices will therefore lead to real or complex eigenvalues, as required, and will influence their position within the chosen region. The solution of the hyperplane design problem by a more general choice of the closed-loop eigenvalues lends itself to the minimization of the linear part of the control. A suitable choice of the position of the eigenvalues within the required region enables either the 2-norm of the linear part of the control, or the condition number of the linear feedback to be minimized. The choice of the range space eigenvalues may also be used, more effectively, in this minimization.