Abstract
Many control problems require solving the algebraic Riccati equation (ARE). Previous studies have focused more on solving the time-invariant ARE than on solving the time-varying ARE (TVARE). This paper proposes a typical recurrent neural network called zeroing neural network (ZNN) to determine the solution of TVARE. Specifically, the ZNN model, which is formulated as an implicit dynamic equation, is developed by defining an indefinite error function and using an exponential decay formula. Then, such a model is theoretically analyzed and proven to be effective in solving the TVARE. Computer simulation results with two examples also validate the efficacy of the proposed ZNN model.
Highlights
In control areas, a typical nonlinear matrix equation termed algebraic Riccati equation (ARE) is frequently encountered and needs to be solved accurately [1]–[7]
The negative definite solution of ARE has been utilized in the Lyapunov equation approach to solve the matrix differential Riccati equation involved in the linear quadratic optimal control [13]
In this paper, motivated by the inspiring work [28] and considering the advances in neural network [29]–[31], we provide a new and efficient model to determine the solution of time-varying ARE (TVARE)
Summary
A typical nonlinear matrix equation termed algebraic Riccati equation (ARE) is frequently encountered and needs to be solved accurately [1]–[7]. AREs with the continuous- and discrete-time forms are playing a remarkable role in many engineering applications of control problems [7]–[11], such as the linear-quadratic-Gaussian and H∞ control problems. The solution of ARE is generally utilized as an essential part of the solution to the aforementioned control problems. The positive definite solution of ARE has been used in each iteration of the homotopy algorithms for the fixed-architecture control [12]. The negative definite solution of ARE has been utilized in the Lyapunov equation approach to solve the matrix differential Riccati equation involved in the linear quadratic optimal control [13]. An effective algorithm/model for solving the ARE is worth designing and investigating due to its important role
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