Estimation of state and parameters of continuous non-linear systems

A new theory of estimating variables characterizing the state of a system and unknown parameters of a system for non-linear systems described by ordinary differential equations is given. The estimates given by this theory satisfy certain ordinary differential equation of a given order and specified form and minimize the mean square error. An exact solution of this problem is given based on the author's equation for the onedimensional characteristic function of a random process determined by a stochastic differential equation. The main feature of the theory presented is the computational simplicity of estimating processes since all the calculations using observations are reduced to integrating an ordinary differential equation. The algorithms and filters based upon our theory may thus be applied in on line identification processes using simple computing devices. All complicated and tedious computations involved are based only on prior data and may therefore be fulfilled during the processes of designing estimating filters or algorithms. The well known linear estimation theory of Kalman and Bucy follows from our general theory as a special case. Moreover our theory yields also an explicit expression for the distribution of the error in this case which may serve for determining confidence regions for estimated variables.