Abstract
The paper is intended to provide algorithmic and computational support for solving the frequently encountered linear-quadratic regulator (LQR) problems based on receding-horizon control methodology which is most applicable for adaptive and predictive control where Riccati iterations rather than solution of Algebraic Riccati Equations are needed. By extending the most efficient computational methods of LQG estimation to the LQR problems, some new algorithms are formulated and rigorously substantiated to prevent Riccati iterations divergence when cycled in computer implementation. Specifically developed for robust LQR implementation are the two-stage Riccati scalarized iteration algorithms belonging to one of three classes: 1) Potter style (square-root), 2) Bierman style (LDLT), and 3) Kailath style (array) algorithms. They are based on scalarization, factorization and orthogonalization techniques, which allow more reliable LQR computations. Algorithmic templates offer customization flexibility, together with the utmost brevity, to both users and application programmers, and to ensure the independence of a specific computer language.
Highlights
A thorough insight into the history of Automatic Control Systems theory gained by reading volumes such as the Systems and Control Encyclopedia [1] convinces us that ACS theory as a model-based science has passed through the three epochs of its development (Figure 1)
The paper is intended to provide algorithmic and computational support for solving the frequently encountered linear-quadratic regulator (LQR) problems based on receding-horizon control methodology which is most applicable for adaptive and predictive control where Riccati iterations rather than solution of Algebraic Riccati Equations are needed
The emphasis in this paper has been on the robust linear quadratic regulator computations where the single Riccati iteration algorithm is an integral part and where seeking a steady-state Riccati solution (Algebraic Riccari Equation) does not apply
Summary
A thorough insight into the history of Automatic Control Systems theory gained by reading volumes such as the Systems and Control Encyclopedia [1] convinces us that ACS theory as a model-based science has passed through the three epochs of its development (Figure 1). (C3) Analytical Relations based Adaptive Model approach uses the analytical relations between the optimal system equations and the Data Source (DS) statistics They are used with the current estimates of unknown DS statistics (parameters) substituted for the exact (unknown) values. In finite RHC, the attainment of the steady-state Riccati solution is not the case due to the very sense of words “finite horizon” and “system adaptation” as can be seen from the generalized adaptive stochastic control system structure (in Figure 2, reproduced from [4]) This explains why we do not consider the above surveyed methods of solving ARE advisable for regulator modification (re-design) in the adaptive control structure of Figure 2. The paper closes with the concluding remarks about the novelty of the new algorithmic insights
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