1 Introduction
A key reason for using feedback is to reduce the effects of uncertainty which may appear in different forms as disturbance or as other imperfection in the models used to design the feedback law. Model uncertainty and robustness have been a central theme in the development of the field of automatic control. This paper gives an elementary presentation of the key results.
A central problem in the early development of automatic control was to construct feedback amplifiers whose properties remain constant in spite of variations in supply that emerged in the 1920s. The problem was solved by [9]. We quote from his paper:
�..by building an amplifier whose gain is deliberately made say 40 decibels higher than necessary (10 000 fold excess on energy basis) and then feeding the output back on the input in such a way as to throw away excess gain it has been found possible to effect extraordinarily improvement in constancy of amplification freedom from nonlinearity.�
Black invention had a tremendous impact and it inspired much theoretical work. This was required both for
understanding and for development of design method. A novel approach to stability was developed in [36], fundamental limitations were explored by [10] who also developed methods for designing feedback amplifiers, see [11]. A systematic approach to design controllers that were robust to gain variations were also developed by Bode.
The work on feedback amplifiers became a central part of the theory of servomechanisms that appeared in the 1940, see[22],[27]. Systems were then described using transfer functions or frequency responses. It was very natural to capture uncertainty in terms of deviations of the frequency responses. A number of measures such as amplifiers also found good use in design of servomechanisms. Bode�s work on robust design was generalized to deal with arbitrary variations in the process by Horowitz [24]. The design techniques used were largely graphical.
The state-space theory that appeared in the 1960s represented significant paradigm shift. Systems were mow described using differential equations. There was a very vigorous development that gave new insight, new concepts [32], [30] new design methods. Control design problems were formulated as optimization problems which gave effective design methods ,see [7],[8].and[40].control of system with Gaussian disturbances and quadratic criteria, the LQG problem ,was particularly attractive potations were also improved because it was possible to capitalize on advances in numerical linear algebra and efficient software. The controller obtained from LQC theory also had a very interesting structure. It was a composition of a kalman filter and a state feedback.
The state-space theory became the predominant approach, see [5]. Safonov and A thanes [42] showed that the phase margin is at least and the amplitude margin is infinite for an LQC problem where all state variables are measured. This result does unfortunately not hold for output feedback as was demonstrated in [15]. There were attempts to recover the robustness of state feedback using special design techniques called loop transfer recovery. The central issue is however that it is not straightforward to capture model uncertainty in a state variable setting. there. Was also criticism of the state-space theory, see [25].
The paper [49] represented a paradigm shift which brought robustness to the forefront. It started a new development that led to the so called H theory. the idea was to develop systematic design methods that were guaranteed to give stable closed loop systems for systems with model uncertainty. The original work was based on frequency responses and interpolation theory which led to compensators of high order. The seminal paper [17] showed how the problem could be solved using state-space methods. Game theory is another approach to H theory. The game is to find a controller in the presence of an adversary that changes the process, see[6]. The H theory is now well described in books, see[16][21]and [52].
Major advances in robust design was made in the book [35] where the H control problem was regarded as a loop shaping problem. This gave effective design methods and it also reestablished the links with classical control. This line of research has been continued by [48] who has obtained definite results relating modeling errors and robust control. To do this he also had to invent a novel metric for system, see[46]. This work brings H even closer to the classical results.
In this chapter we will try to present the essence of the development in the simple setting of single-input single-output system . we start with a presentation of some aspects of classical control theory in section 2. Robustness issues for state-space theory
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