Is reviewed briefly in section 3 where we also present an example that illustrates that a blind use of state-space theory can lead closed loop systems with very poor robustness properties. This can be overcome by analyzing the robustness and modifying the design criteria. In section 4 we discuss fundamental limitations on performance due to time delays and right half plane poles and zeros. This does not appear naturally in state-space theory where the major requirements are absorbability and controllability.
In section 5 we present some key results from H -loop shaping. To do this we have to discuss the important problem of determining if two systems are close from the point of view of feedback. We also have to introduce better stability concepts and ways to characterize model uncertainty. We end the section with a discussion of vinnicombe�s theory which gives much insight, necessary conditions and very nice ties to classical control theory.
2. Classical Control Theory
The fields of automatic control emerged in the mid 1940s when it was realized that it was a common framework for problems associated with feedback control from a wide range of fields such as telecommunications where a key problem had been to design accurate reliable amplifiers from components with variable properties.
3. The Feedback Amplifier A schematic diagram of the amplifier is show in figure 1. let the raw gain of the amplifier be A the feedback amplifier has very remarkable properties can be seen from its input-output relation. It follows from figure 1 that.
Notice that the gain V2 /V1 is essentially given by the ration R2/R1. if raw amplifier gain is A is large is virtually independent of A assume for example that R2/R1=100 that A=10. A 10% change of a the variation gives only a gain variation of 0.1% feedback thus has the amazing property of reducing the effects of uncertainty drastically. The linearity is also increased significantly.
The risk for instability is a drawback of feedback. Nyquist developed a theory for analyzing stability of feedback amplifiers, see [36]. Systematic methods to design feedback systems were developed in [10] and further elaborated in [11]. These ideas formed one of the foundations of automatic control.
In today�s terminology, we could say that black used feedback to design an amplifier that was robust to variations in the gain of the process. As a side-effect, he also obtained a closed-loop system that was extremely linear.
4. Generalization
The ideas of feedback are applicable to a wide range of system. This is illustrated in figure 2 which shows a basic feedback loop consisting of a process and a controller. The
Purpose of the system is make the process variable follow the set point in spite the disturbances l and n that act on the system. The properties of the closed loop system should also be insensitive to variations in the process. There are two types of disturbances. The load disturbance l drives the system away from its desired state and the measurement noise n which corrupts the information about the system obtained from the sensors.
The system in figure 2 has three inputs T,l and n and four interesting signal x y, e and u there are thus 12 relations that are of potential interest. Assume that the process and the controller are linear time-invariant systems that are characterized by their transfer functions P and C respectively. The relations between the signals are
|
