Kalman Filtering When the Large Bandwidth Control is Not Known

In the standard Kalman filtering (KF) paradigm it is assumed that the control signal is known, or, alternatively, it is assumed that the dynamical system is in a "free fall." This is problematic when maneuvering targets must be tracked, in which case the input signal is not known to the observer. The KF paradigm for discrete-time control systems is revisited and it is not assumed that the control signal is known. Moreover, a large bandwidth input signal is allowed for. It is shown that under the assumption that, e.g., the measurement and control matrix product CB is full (column) rank - an assumption used in direct adaptive control - it is possible to jointly estimate the input signal and the control system\´s state. It is not necessary to assume that the control signal is constant and therefore large bandwidth input signals are accommodated. A recursive algorithm for the calculation of the minimum variance estimates of the state and control signal is developed. Similar to classical KF, a linear estimation problem is solved; therefore, the minimum variance estimates of the state and input signal are obtained, and thus, the Cramer-Rao lower bound (CRLB) is attained. The filter\´s gain is constant and whereas in conventional KF the calculation of the covariance of the state estimation error entails the solution of a Riccati equation, the covariances of the state and input estimation errors are determined here by the solution of a Lyapunov equation, and explicit formulae are obtained.