On solving periodic differential matrix equations with applications to periodic system norms computation

Varga, Andreas

doi:10.1109/cdc.2005.1583212

Cited by 35 publications

(39 citation statements)

References 21 publications

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“…Similar discretizations of appropriate Hamiltonian matrices are employed to solve periodic Lyapunov, Sylvester or Riccati differential equations (Varga, 2005b). The applications of these multishot methods to evaluate Hankel, H 2 -, or H ∞ -norms (Varga, 2005b) or to solve continuoustime output feedback linear-quadratic problems (Viganò et al, 2007) are relatively straightforward.…”

Section: Algorithms For Continuous-time Periodic Systemsmentioning

confidence: 99%

“…The applications of these multishot methods to evaluate Hankel, H 2 -, or H ∞ -norms (Varga, 2005b) or to solve continuoustime output feedback linear-quadratic problems (Viganò et al, 2007) are relatively straightforward. Recently, an evaluation of the multi-shot method to solve periodic Riccati equations in conjunction with special structure preserving (symplectic) integrators has shown the high effectiveness of these techniques (Johansson et al, 2007).…”

Section: Algorithms For Continuous-time Periodic Systemsmentioning

confidence: 99%

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An Overview of Recent Developments in Computational Methods for Periodic Systems

Varga

2007

IFAC Proceedings Volumes

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Section: Algorithms For Continuous-time Periodic Systemsmentioning

confidence: 99%

Section: Algorithms For Continuous-time Periodic Systemsmentioning

confidence: 99%

An Overview of Recent Developments in Computational Methods for Periodic Systems

Varga

2007

IFAC Proceedings Volumes

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“…As an alternative to the one-shot method, we consider the multi-shot method proposed in (Varga, 2005). The main idea is to turn the continuoustime problem into an equivalent discrete-time problem.…”

Section: Multi-shot Methodsmentioning

confidence: 99%

“…The one-shot periodic generator method solves an ODE with unstable dynamics in both steps 1 and 3, and therefore this method is unreliable for systems with large periods (Varga, 2005).…”

Section: One-shot Methodsmentioning

confidence: 99%

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COMPARING ONE-SHOT AND MULTI-SHOT METHODS FOR SOLVING PERIODIC RICCATI DIFFERENTIAL EQUATIONS1 1Financial support has partially been provided by the Swedish Foundation for Strategic Research under the frame program grant A3 02:128.

Johansson

Kågström

Shiriaev

et al. 2007

IFAC Proceedings Volumes

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One-shot methods and recently proposed multi-shot methods for computing stabilizing solutions of continuous-time periodic Riccati differential equations are examined and evaluated on two test problems: (i) a stabilization problem for an artificially constructed time-varying linear system for which the exact solution is known; (ii) a nonlinear stabilization problem for a devil stick juggling model along a periodic trajectory. The numerical comparisons are performed using both general purpose and symplectic integration methods for solving the associated Hamiltonian differential systems. In the multi-shot method a stable subspace is determined using recent algorithms for computing a reordered periodic real Schur form. The results show the increased accuracy achievable by combining multi-shot methods with structure preserving (symplectic) integration techniques.

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On solving periodic Riccati equations

Varga

2008

Numerical Linear Algebra App

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Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and 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, the 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 matrices.Provided a non-negative N -periodic stabilizing solution X k of the DPRE (5) is known, the periodic state-feedback matrix F k in the optimal control law u * k = F k x k , which minimizes the performance index (7), results asAlternatively, it is possible to determine the periodic generator X 0 := X (0) by solving the algebraic Riccati equation [3]: 21 + 22 X 0 − X 0 11 − X 0 12 X 0 = 0

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On solving periodic differential matrix equations with applications to periodic system norms computation

Cited by 35 publications

References 21 publications

An Overview of Recent Developments in Computational Methods for Periodic Systems

An Overview of Recent Developments in Computational Methods for Periodic Systems

COMPARING ONE-SHOT AND MULTI-SHOT METHODS FOR SOLVING PERIODIC RICCATI DIFFERENTIAL EQUATIONS1 1Financial support has partially been provided by the Swedish Foundation for Strategic Research under the frame program grant A3 02:128.

On solving periodic Riccati equations

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