An algorithm to solve the discrete HJI equation arising in the L2 gain optimization problem

A synthesis of the discrete non-linear H control law boils down to the solution of a set of algebraic and partial differential equations known as the discrete Hamilton-Jacobi-Isaacs (DHJI) equation, an extension of the discrete algebraic Ricatti equation arising in the linear discrete H control problem. Due to its non-linear nature, it is rarely possible to obtain the closed form solution for DHJI equation. This paper proposes an approximation approach to solving the DHJI equation in terms of the Taylor series. It is shown that the coefficients of the Taylor series solution for DHJI equation are governed by one discrete algebraic Ricatti equation and a sequence of linear algebraic equations, respectively. This result lends itself to a systematic algorithm to approximately synthesize a discrete non-linear H control law.