Dampening Controllers via a Riccati Equation Approach

An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed–loop system dynamics. In other words, a feedback gain matrix is computed such that the eigenvalues of the closed–loop state matrix are within the region of the left half– plane where the magnitude of the real part of each eigenvalue is greater than that of the imaginary part. This may be accomplished by solving a damped algebraic Riccati equation and a degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms. Finally, the formulation of damped Riccati equations is unusual in that it may be viewed as an invariant subspace problem for a periodic Hamiltonian system. This periodic Hamiltonian system induces two damped Riccati equations, one which is associated with a symmetric solution and the other with a skew symmetric solution. These two solutions result in two different state feedbacks, both of which dampen the system dynamics, but produce different closed–loop eigenvalues, thus giving the controller designer greater freedom in choosing a desired feedback.