Estimation of unknown time-varying driving matrix and it's application to learning control

The problem of estimating state variables and unknown parameters is considered for discrete-time control system of random disturbances and measurement noise. Where, elements of a time-varying driving matrix are considered as unknown parameters. Making use of the fact that a time-varying function is able to be expanded into a polynomial for time, the author expanded unknown elements of a driving matrix into a polynomial for time. Then, estimating constants of a polynomial for time was intended, instead of estimating driving matrix. Estimating them, we can calculate an estimated value of the driving matrix. To estimate state variables and constants of a polynomial with respect to a driving matrix, the author constituted a new joint vector with them. After transforming the state transition equation and observation equation into a new transition equation and an observation equation for a joint vector, Kalman filter was adapted to estimate an unknown joint vector. As an application, a learning control was discussed to be optimized with respect to a quadratic criterion function with the aid of dynamic programming method.