Theory of LTR for non-minimum phase systems, recoverable target loops, and recovery in a subspace Part 2. Design

Abstract

This Part focuses on the design of full order observer based controllers for the recovery of target loop transfer function or sensitivity and complimentary sensitivity functions when the given system is not necessarily left invertible and not necessarily of minimum phase. Four design tasks are considered. The first task concerns with arbitrarily specified target loop transfer functions and develops an observer design which has the capability to shape the inevitable recovery error according to the designer's needs whenever they are feasible. The second task considers observer design for exactly recoverable target loop transfer functions. The third task is similar to the second one in that it makes use of the specific properties of the target loop transfer function, but it considers observer design for asymptotically recoverable target loop transfer functions. The fourth task can be thought of as a generalization of the second and third tasks, and it considers observer design so that the achieved and target sensitivity and complimentary sensitivity functions match each other either exactly or asymptotically over a given subspace of the control space whenever it is possible. For all these tasks, observer design constraints and the available design freedom are reviewed. In view of the available freedom, possible specifications on the time-scale and/or eigenstructure of the observer dynamic matrix are formulated. In the case of first task, the conventional approach of designing observer based controllers by Kalman filter formalism which requires solving algebraic Riccati equations, is shown to have several fundamental limitations. A method of design based on asymptotic time-scale and eigenstructure assignment (ATEA) developed here overcomes these limitations. For the other tasks, no design methods other than the ones developed here are available in the literature. All the developed design methods are implemented in a ‘Matlab’ software package. A bank of examples illustrate that the proposed methods of design are capable of directly exploiting all the available freedom so as to achieve the desired results.