Signal Estimation for Second-Order Vector Difference Equations (Filtering, Prediction, Higher-Order Systems).

Stochastic processes viewed as the output signal of a system described by a linear second-order vector difference equations are considered. There are a number of important estimation problems associated with such stochastic processes including: prediction, which is estimation of future values of the process from past noisy measurements; and filtering, which is estimation of the present value of the stochastic process from past noisy measurements. An innovations approach is applied directly to develop recursive formulas for the one-stage prediction and the filtered estimates and associated error covariances for systems described by second-order vector difference equations driven by white or Markovian noise inputs. It is shown that for each of these estimation problems, the estimator can be expressed as a second-order recursion that preserves the form of the given signal model with innovations feedback loops. It is also shown that the innovations can be computed through a first-order recurrence relation in terms of the measurements. Although the results are derived using fundamental properties of innovations, it is also demonstrated that these results can be derived by converting the given model into an augmented state-variable form and applying Kalman filtering techniques. These ideas are generalized to systems described by higher-order vector difference equations. General formulas for the one-stage prediction estimator and associated gains are given for the case of a p'th-order vector difference equation model driven by a white-noise process. It is shown that the one-stage prediction estimator is expressed as a p'th-order recursion that preserves the form of the signal model, with innovations feedback loops. The innovations, in this case, are computed through a p-1'st-order recursion in terms of the measurements. It is shown that the well known Kalman filter and Autoregressive (AR) predictor are special cases of this general result. As an application, these ideas are applied to a particular class of models that arise in diverse engineering applications such as mechanics and elastic systems. These models are described by second-order vector differential equations. Discrete-time measurements are assumed. A first-order discrete-time signal model of twice the dimension of the given system results. Second-order recursions are obtained from the first-order discrete-time signal model. The one-stage prediction estimator is obtained by applying the developed theory to the second-order recursion. On the other h and , Kalman filtering techniques are applied to the first-order discrete-time model and a first-order discrete-time Kalman filter is obtained. The results are compared.PhDElectrical engineeringUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/160343/1/8502848.pd