An algorithm to solve HJI equation arising in discrete nonlinear H ∞ control

Synthesis of the discrete nonlinear H_∞ control law leads 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 the nonlinear nature, it is rarely possible to obtain the closed form solution for DHJI equation. This paper proposes an approximation approach to solving 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 nonlinear H_∞ control law