Optimum Filtering for Time-Varying Systems with Plant Parameter Ignorance

Abstract

A procedure for designing feedback compensation for a time-varying linear plant is developed for the case of gross plant parameter ignorance. The given time-varying plant P is considered to be a member of a statistical ensemble of possible plants, because of ignorance of the plant parameters. There is a desired time-varying relationship T0 for the system as a whole, specified, for example, in terms of a time-varying impulse response function.An ensemble of input functions r, in general nonstationary, is applied to the system, and it is desired to minimize the output error. This error has three components:1)due to the statistical uncertainty associated with the plant parameters.2)due to an external non-stationary disturbance d acting at the output of the system.3)due to the noise n inevitably introduced by the sensor in the feedback loop. This noise may also be non-stationary.There are available to the designer two degrees of freedom, conveniently chosen as the operators L0 and P0, which are respectively the nominal loop transmission and nominal plant operators.The above problem can be made equivalent to the non-stationary filter problem by imposing the following requirements:1)zero expected output error.2)minimum expected output error squared at every instant of time.The resulting equations are solved using a matrix representation of the operators. The system is assumed to have a finite operating time and the signals are sampled N times during this interval. Thus the signals can be represented by N-dimensional column vectors and the causal operators by N × N lower triangular matrices. The matrix equations are solved iteratively on the digital computer to yield the optimum compensation.