On solving periodic Riccati equations.

*(English)*Zbl 1212.65254Summary: Numerically reliable algorithms to compute periodic non-negative definite stabilizing solutions of the periodic Riccati differential equation (PRDE) and the discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represent extensions of the periodic QZ algorithm to non-square periodic pairs, whereas the third method represents an extension of a quotient-product swapping and collapsing “fast” algorithm. All proposed approaches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting

##### MSC:

65K10 | Numerical optimization and variational techniques |

93B40 | Computational methods in systems theory (MSC2010) |

93C55 | Discrete-time control/observation systems |

34A34 | Nonlinear ordinary differential equations and systems |

34C25 | Periodic solutions to ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

##### Keywords:

periodic system; periodic Riccati equation; periodic Lyapunov equation; periodic deadbeat control; periodic Riccati differential equation; discrete-time; multiple shooting-type algorithm; periodic solutions; periodic QZ algorithm; quotient-product swapping and collapsing “fast” algorithm##### Software:

Periodic Systems Toolbox
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\textit{A. Varga}, Numer. Linear Algebra Appl. 15, No. 9, 809--835 (2008; Zbl 1212.65254)

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