Diffusive Realization of a Lyapunov Equation Solution, and its FPGA Implementation

Abstract

International audienceIn Yakoubi [2010] and Lenczner et al. [2010] we developed a theoretical framework of diffusive realization for state-realizations of some linear operators. Those are solutions to certain operator linear differential equations inone-dimensional bounded domains. We also illustrated the theory and developed a numerical method for a Lyapunov equation arising from optimal control theory of the heat equation. However, the principles of our numerical methods were only sketched, and now we provide more details. Then, we do not only provide validation results of the method, but we also report our experience in its implementation on a Field Programmable Gate Arrays (FPGA), for the purpose of promoting embedded real-time computation

Diffusive Realization of a Lyapunov Equation Solution, and Parallel Algorithms Implementation

“…When it is analytic in its second variable then see Montseny [8]. The case where P is the solution of an operator equation, so p is neither explicitly given nor analytic, is reported in Yakoubi et al [12]. Their method was announced in Lenczner and Montseny [8] and fully developed in Lenczner et al [5].…”

“…In Yakoubi et al [12] the calculations for both causal and anti-causal parts have been detailed, we simply recall them without repeating their justification. Defining the parameters α ± (ξ) = e −θ ± (ξ)h , and β ± (ξ) = α ± (ξ)−1 −θ ± (ξ) , the recurrence relations yield…”