A Variational Method for Computing Static Output Feedback LQR Gains

A variational method for computing static output feedback LQR gain is presented in this paper. As a major drawback when attempting to find the LQR gain, many existing algorithms suffer from the well-known initialization problem - a stabilizing output feedback gain must be provided to start the search. Unfortunately, finding such a gain itself is very challenging, if not equally difficult. This initialization problem has been overcome by indirect methods which obtain the optimal output feedback gain through a state feedback gain. The second problem faced by typical methods is the sensitivity to the initial guess - a badly chosen guess inevitably results in algorithm failure. Following an indirect projection method, this research aims to solve the second problem by developing a variational method that is completely insensitive to the initial guess and is guaranteed to converge to the optimal gain. Technical tools utilized for the algorithm development include Kronecker algebra and LaSalle's Invariance Principle. Also proposed in the paper is a hybrid algorithm that enhances the convergence rate when a switching criterion is met amid solution iterations. Finally, a numerical example is given to validate the approach.