Extension of Kalman filter theory to nonlinear systems with application to wing rock motion

The formulations for nonlinear optimal control and nonlinear Kalman filter theory are developed in this paper. It is shown that the solution of the nonlinear Kalman filter problem is governed by a Hamilton-Jacobi-Bellman inequality (HJBI). Choosing the closed loop Lyapunov function in a symmetric matrix form of the state vector results in the reduction of the HJBI to an algebraic Riccati inequality along with several other algebraic inequalities. These inequalities are formulated into a series of closed loop Lyapunov inequalities which are turned into equalities by adding a positive state vector function. Closed loop Lyapunov functions are then obtained successively by solving these equations. This procedure guarantees a nonlinear filter system with a large stable region. Control of a nonlinear wing rock motion is employed as an example to illustrate the theory.