Overview

After studying this chapter you will be able to

• use an observer to estimate the state vector of a linear time-invariant system;

• use a Kalman filter to estimate the state vector of a linear system using knowledge of the system matrices, the system input and output measurements, and the covariance matrices of the disturbances in these measurements;

• describe the difference among the predicted, filtered, and smoothed state estimates;

• formulate the Kalman-filter problem as a stochastic and a weighted least-squares problem;

• solve the stochastic least-squares problem by application of the completion-of-squares argument;

• solve the weighted least-squares problem in a recursive manner using elementary insights of linear algebra and the mean and covariance of a stochastic process;

• derive the square-root covariance filter (SRCF) as the recursive solution to the Kalman-filter problem;

• verify the optimality of the Kalman filter via the white-noise property of the innovation process; and

• use the Kalman-filter theory to estimate unknown inputs of a linear dynamical system in the presence of noise perturbations on the model (process noise) and the observations (measurement noise).

5.1 Introduction

Imagine that you are measuring a scalar quantity x(k), say a temperature. Your sensor measuring this quantity produces y(k). Since the measurement is not perfect, some (stochastic) measurement errors are introduced. If we let ?(k) be a zero-mean white-noise sequence with variance R, then a plausible model for the observed data is

A relevant question is that of whether...