Penalized Least-Squares Method for LQR Problem of Singular Systems

The linear quadratic regulator (LQR) algorithms devised using the Riccati equation possess two key attributes-they are recursive and have easily met conditions of existence. Nevertheless, these features only apply for the transformed structure of the regulated dynamics in singular systems, otherwise their optimal performance will be compromised under violation of constraints in non-singular versions. This technical note presents the LQR problem for a time-varying discrete linear singular system in a direct manner avoiding any transformations. This approach eliminates the requirement of making regularity assumptions for the system. To achieve this, first, we formulate a quadratic cost function for LQR derivation based on a penalized weighted least-squares method. Then, by using Bellman's principle of optimality and performing variable substitutions, we connect the formulation to a constrained and recursive minimization problem. We then proceed with investigating the existence conditions and using dynamic programming in a backward strategy at the finite horizon to derive a recursive regulator algorithm for the original system in a matrix array framework, without degrading its optimal performance. The achieved algorithm has more general features compared to the classical LQR problem for standard systems. This study concludes with numerical evaluation of the algorithm and confirmation of the results.