Polynomial Wiener LQG Controllers Based on Toeplitz Matrices

Abstract

This paper re-examines the discrete-time Linear Quadratic Gaussian (LQG) regulator problem. The normal approach to this problem is to use a Kalman filter state estimator and Kalman control state feedback. Though quite successful, an alternative approach in the frequency domain was employed later. That method used z-transfer functions or polynomials in the z-domain. The transfer function approach is similar to the method used in Wiener filtering and requires the use of Diophantine equations (sometimes bilateral) to find the optimal controller. The contribution here uses a similar approach but uses lower triangular Toeplitz matrices instead of polynomials to gain advantage of eliminating the use of Diophantine equations. This is because the single Diophantine equation approach fails when the system has non-relative prime polynomials and the need for bilateral Diophantine equations is computationally far more complex.