Stabilization problems for a system with output feedback (review)

Abstract

Algorithms for solving the problem of design of static output feedback controllers for stationary linear systems with continuous
and discrete time are reviewed. The inverse problem is considered. The algorithms of synthesis of output feedback controllers
are generalized to the case of a periodic discrete-time system. To solve such problems, it might be more natural to use an
approach based on multi-criterion optimization. It is also shown that these algorithms can be used for the optimal stabilization
of unstable systems with delay. In this connection, the parameters of a controller with given structure for a controlled unstable
scalar system with delay are optimized. To this end, the system is first approximated by a system without delay, with the
exponent approximated by a fractionally rational function. Since the structure of the controller is given, the quality of
approximation is estimated as the difference (in the space of controller coefficients) between the stability domains of the
original and approximating systems. At the next stage, the gain coefficients of the controller for the reduced system are
optimized. The efficiency of the thus synthesized controller is assessed through mathematical modeling of a system with delay
whose feedback loop is defined by the gain coefficients found. The approach is illustrated by stabilizing an inverted simple
pendulum with a proportional–derivative controller with delay. The problem of synthesis of a robust controller for this example
is considered. Some examples of designing a robust controller, including for a third-order system in which the delay rather
than some parameter is uncertain are presented

KeywordsLQ-problem–inverse LQ-problem–static output feedback–periodic system–system with delay–proportional-derivative controller