Optimal Estimation and Control

Estimation and control are the two fundamental problems for dynamical systems. Many engineering design tasks can be formulated into either an estimation or control problem associated ith some appropriate performance index. State-space representations facilitate the development of the computational tool for optimal design and enabling design of optimal estimators and controllers. Without exaggeration optimal estimation and control are the two most celebrated results in engineering system design. They have brought in not only the design algorithms but also the new design methodology that has had far reaching impacts as evidenced by the wide use of Kalman filtering and feedback control in almost every aspect of the system engineering. Nevertheless it is the conceptual notions from linear system theory that empower the state of the art design algorithms and allow applications of optimal estimation and control in engineering practice. This chapter covers the well-known results in Kalman filtering and quadratic regulators that have enriched engineering system design. It also covers optimal output estimators and full information control that are developed more recently. Estimation aims at design of state estimators that reconstruct the state vector based on measurements of the past and present input and output data. Due to the unknown and random nature of the possible disturbance at the input and the corrupting noise at the output, it is impossible to reconstruct the true state vector in real-time. Therefore the design objective for state estimators will be minimization of the estimation error variance by assuming white noises for input disturbances and output measurement errors. The focus will be on design of optimal linear estimators. Disturbance rejection has been the primary objective in feedback control system design in which white noises are the main concern. The emphasis will be placed on the design of state-feedback controllers to not only stabilize the feedback control system but also minimize the adverse effect due to white noise disturbances. With the variance as the performance measure, optimal control leads to linear feedback controllers that are dual to optimal linear estimators.