Abstract
We use a second-order learning algorithm for numerically solving a class of the algebraic Riccati equations. Specifically, the extended Hamiltonian algorithm based on manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm, and the new subspace iteration method. Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms.
Highlights
The algebraic Riccati equations (AREs) have been widely used in control system syntheses [1, 2], especially in optimal control [3], robust control [4], signal processing [5], and the LMI-based design [6]
The structure-preserving doubling algorithm (SDA) was given under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to continuous-time AREs [12]
The result shows that the extended Hamiltonian algorithm (EHA) has the fastest convergence speed among four algorithms and needs 28 iterations to obtain the numerical solution of the ARE as follows: (11..40104020103060 02..98929894929791)
Summary
The algebraic Riccati equations (AREs) have been widely used in control system syntheses [1, 2], especially in optimal control [3], robust control [4], signal processing [5], and the LMI-based design [6]. The structure-preserving doubling algorithm (SDA) was given under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to continuous-time AREs [12]. It is the purpose of this paper to investigate the unique (under suitable hypotheses) symmetric positive definite solution of an algebraic Riccati equation.
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