Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

In this paper, a ZNN model is introduced to solve the time-varying algebraic Riccati equation (TVE-ARE) problem in the context of infinite-horizon optimal control problems. The ZNNTVE-ARE model is designed to produce the unique nonnegative definite solution and can also solve the continuous-time Lyapunov equation. The paper also presents a hybrid ZNN model for stabilizing linear time-varying (LTV) systems, which utilizes the ZNNTVE-ARE model to solve the continuous-time algebraic Riccati equation (CARE). Experimental results demonstrate the effectiveness of both the ZNNTVE-ARE model and the HFTZNN-LTVSS model, with the latter showing slightly better asymptotic stability.