21-05-2012, 10:22 AM
Transfer Functions
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Frequency Domain Description of SystemsThe idea of studying systems in the frequency domain is to characterize alinear time-invariant system by its response to sinusoidal signals. The ideagoes back to Fourier, who introduced the method to investigate propagationof heat in metals. Frequency response gives an alternative way of viewingdynamics. One advantage is that it is possible to deal with systems of veryhigh order, even in¯nite. This is essential when discussing sensitivity toprocess variations. This will be discussed in detail in Chapter ?.Frequency response also gives a di®erent way to investigate stability. InSection 2.3 it was shown that a linear system is stable if the characteristic
polynomial has all its roots in the left half plane. To investigate stability
of a the system we have to derive the characteristic equation of the closedloop system and determine if all its roots are in the left half plane. Even ifit easy to determine the roots of the equation numerically it is not easy todetermine how the roots are in°uenced by the properties of the controller.It is for example not easy to see how to modify the controller if the closedloop system is stable. The way stability has been de¯ned it is also a binaryproperty, a system is either stable or unstable. In practice it is highlydesirable to have a notion of the degrees of stability. All of these issues canbe related to frequency response. The key is Nyquist's stability criterion(the subject of the next chapter) which is a frequency response concept.
Poles and Zeros
The poles of the linear time-invariant system (?) are simply the eigenvaluesof the matrix A. To determine the zeros we use the fact that zeros are suchthat the pure response to input est is zero. The state and the output thatcorresponds to the input u0est are then x0est and y0est.
Bode Plots
A useful representation of the frequency response was proposed by Bodewho represented it by two curves, the gain curve and the phase curve. Thegain curve gives gain jG(i!)j as a function of ! and the phase curve phaseargG(i!) as a function of !. The curves are plotted as shown below withlogarithmic scales for frequency and magnitude and linear scale for phase.
Block Diagrams
Feedback systems are often large and complex. It is therefore a major challengeto understand, analyze and design them. This is illustrated by thefact that the idea of feedback was developed independently in many differentapplication areas. It took a long time before it was found that thesystems were based on the same idea. The similarities became apparentwhen proper abstractions were made. In this section we will developsomeideas that are used to describe feedback systems. The descriptions we arelooking for should capture the essential features of the systems and hideunnecessary details. They should be applicable to many di®erent systems.Schematic DiagramsIn all branches of engineering, it is common practice to use some graphicaldescription of systems. They can range from stylistic pictures to drastically
simpli¯ed standard symbols. These pictures make it possible to get an overallview of the system and to identify the physical components. Examplesof such diagrams are shown in Figure 6.8
Block DiagramsThe schematic diagrams are useful because they give an overall picture of asystem. They show the di®erent physical processes and their interconnection,and they indicate variables that can be manipulated and signals thatcan be measured.
A special graphical representation called block diagrams has been developedin control engineering. The purpose of block diagrams is to emphasizethe information °ow and to hide technological details of the system. It isnatural to look for such representations in control because of its multidisciplinarynature. In a block diagram, di®erent process elements are shown asboxes. Each box has inputs denoted by lines with arrows pointing toward
A Generic Control System with Error Feedback
Although the centrifugal governor, the autopilot and the feedback ampli¯erin Examples 22, 23 and 24 represent very di®erent physical systems, theirblock diagrams are identical apart from the labeling of blocks and signals,compare Figures 6.12, 6.13 and 6.15. This illustrates the universality ofcontrol. A generic representation of the systems is shown in Figure 6.16.The system has two blocks. One block P represents the process and theother C represents the controller. Notice negative sign of the feedback. Thesignal r is the reference signal which represents the desired behavior of theprocess variable x.Disturbances are an important aspect of control systems. In fact if therewere no disturbances there is no reason to use feedback. In Figure 6.16 there
are two types of disturbances, labeled d and n. The disturbance labeled dis called a load disturbance and the disturbance labeled n is calledmeasurementnoise. Load disturbances drive the system away from its desiredbehavior. In Figure 6.16 it is assumed that there is only one disturbancethat enters at the system input. This is called an input disturbance. InPractice there may be many di®erent disturbances that enter the system inmany di®erent ways.