Controlled and conditioned invariant subspaces in linear system theory

A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system

“…(1) the closed-loop matrix A − B K is stable; (2) for all x 0 ∈ L, the state trajectory generated by (1) with initial state x(0) = x 0 and with the input u(t) = −K x(t) minimizes the quadratic cost function…”

“…The keystone is the interpretation of the LQ regulator as an output nulling problem referred to the Hamiltonian system. This new approach enables to solve the cheap LQ regulator by only using the basic and computationally efficient tools of the geometric approach, namely the algorithms for controlled and conditioned invariance and invariant zeros, [2,4,18,20]. By writing the conditions for optimality in the form of the Hamiltonian system, whose output has to be maintained identically equal to zero, the cheap LQ problem reduces to finding a state feedback such that the statecostate trajectory entirely lies on an internally stabilizable output nulling subspace of the Hamiltonian system.…”

A geometric approach to the singular filtering problem

“…(cf. [20], [2], [3], [21]). The sequence { Vk(k)} is decreasing and will, after a finite number of steps, reach a limit which is denoted by V*(:E).…”

Disturbance and input–output decoupling of singular systems

“…For the class of state space systems with proper or strictly proper transfer functions numerous papers were published over the last decades and several aspects of disturbance decoupled systems have been investigated (see for example [4] , [22] where the problem was tackled by using geometric control theory, [6] where a structural approach was followed for the solution of diagonal and disturbance decoupling of a state space system). In [17] the problem of fixed poles of disturbance decoupling was solved by using the geometric approach.…”