Robust control of a general servomechanism problem: The servo compensator

doi:10.1016/0005-1098(75)90022-9

Automatica

Volume 11, Issue 5, September 1975, Pages 461-471

https://doi.org/10.1016/0005-1098(75)90022-9 Get rights and content

Abstract

The robust control of a general servomechanism problem, which is an extension to the results of [1], is considered in this paper. Necessary and sufficient conditions, together with a characterization of all robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system, are obtained. A new type of compensator, introduced in [1], called a servo-compensator which is quite distinct from an observer is shown to play an essential role in the robust servomechanism problem. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a robust controller of a general servomechanism problem must consist of two devices (i) a servo-compensator and (ii) a stabilizing compensator. A study of the stabilizing compensator is made; in particular, it is shown that a new type of stabilizing compensator called a complementary controller, may be used together with the servo-compensator to form a robust controller for the servo-mechanism problem.

A study of the case when perturbations in the robust controller are also allowed is then made; this leads to the Strong robust servo limitation theorem which imposes a fundamental limitation on the ability of practical servomechanisms to regulate a system.

References (12)

E.J. Davison et al.
Properties and calculation of transmission zeros of linear multivariable systems
Automatica
(1974)
E.J. Davison
The systematic design of control systems for large multivariable linear time-invariant systems
Automatica
(1973)
E.J. Davison
The robust control of a general servomechanism for linear time-invariant multivariable systems
M. Heymann
On the input and output reducibility of multivariable linear systems
IEEE Trans. Aut. Control
(1970)
E.J. Davison et al.
Properties of linear multivariable systems subject to arbitrary output and state feedback
IEEE Trans. Aut. Control
(1973)
E.J. Davison
The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances
IEEE Trans. Aut. Control
(1972)

There are more references available in the full text version of this article.

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^☆: The original version of this paper was presented at the 6th IFAC Congress which was held in Boston, Cambridge, Massachusetts, during August 1975. It was recommended for publication in revised form by associate editor D. Tabak.

^☆☆: This work has been supported by the National Research Council of Canada under Grant No. A4396.

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Robust control of a general servomechanism problem: The servo compensatorContrôle robuste d'un problème général de servo-mécanisme: Le compensateur servoUnempfindlicher regler für ein allgemeines folgereglerproblem: der servokompensator☆,☆☆

Abstract

Automatica

Automatica

The robust control of a general servomechanism for linear time-invariant multivariable systems

On the input and output reducibility of multivariable linear systems

IEEE Trans. Aut. Control

Properties of linear multivariable systems subject to arbitrary output and state feedback

IEEE Trans. Aut. Control

The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

IEEE Trans. Aut. Control