30079_28b.pdf

Then the system is switched into automatic mode. Digital computers are often used to replace the

manual adjustment process because they can be readily coded to produce complicated functions for

the start-up signals. Care must also be taken when switching from manual to automatic. For example,

the integrators in electronic controllers must be provided with the proper initial conditions.

28.7.5 Reset Windup

In practice, all actuators and final control elements have a limited operating range. For example, a

motor-amplifier combination can produce a torque proportional to the input voltage over only a

limited range. No amplifier can supply an infinite current; there is a maximum current and thus a

maximum torque that the system can produce. The final control elements are said to be overdriven

when they are commanded by the controller to do something they cannot do. Since the limitations

of the final control elements are ultimately due to the limited rate at which they can supply energy,

it is important that all system performance specifications and controller designs be consistent with

the energy-delivery capabilities of the elements to be used.

Controllers using integral action can exhibit the phenomenon called reset windup or integrator

buildup when overdriven, if they are not properly designed. For a step change in set point, the

proportional term responds instantly and saturates immediately if the set-point change is large enough.

On the other hand, the integral term does not respond as fast, It integrates the error signal and saturates

some time later if the error remains large for a long enough time. As the error decreases, the pro-

portional term no longer causes saturation. However, the integral term continues to increase as long

as the error has not changed sign, and thus the manipulated variable remains saturated. Even though

the output is very near its desired value, the manipulated variable remains saturated until after the

error has reversed sign. The result can be an undesirable overshoot in the response of the controlled

variable.

Limits on the controller prevent the voltages from exceeding the value required to saturate the

actuator, and thus protect the actuator, but they do not prevent the integral build-up that causes the

overshoot. One way to prevent integrator build-up is to select the gains so that saturation will never

occur. This requires knowledge of the maximum input magnitude that the system will encounter.

General algorithms for doing this are not available; some methods for low-order systems are presented

in Ref. 1, Chap. 7, and Ref. 2, Chap. 7. Integrator build-up is easier to prevent when using digital

control; this is discussed in Section 28.10.

28.8 COMPENSATION AND ALTERNATIVE CONTROL STRUCTURES

A common design technique is to insert a compensator into the system when the PID control algo-

rithm can be made to satisfy most but not all of the design specifications. A compensator is a device

that alters the response of the controller so that the overall system will have satisfactory performance.

The three categories of compensation techniques generally recognized are series compensation, par-

allel (or feedback) compensation, and feedforward compensation. The three structures are loosely

illustrated in Fig. 28.34, where we assume the final control elements have a unity transfer function.

The transfer function of the controller is G}(s). The feedback elements are represented by H(s), and

the compensator by Gc(s). We assume that the plant is unalterable, as is usually the case in control

system design. The choice of compensation structure depends on what type of specifications must

be satisfied. The physical devices used as compensators are similar to the pneumatic, hydraulic, and

electrical devices treated previously. Compensators can be implemented in software for digital control

applications.

28.8.1 Series Compensation

The most commonly used series compensators are the lead, the lag, and the lead-lag compensators.

Electrical implementations of these are shown in Fig. 28.35. Other physical implementations are

available. Generally, the lead compensator improves the speed of response; the lag compensator

decreases the steady-state error; and the lead-lag affects both. Graphical aids, such as the root locus

and frequency response plots, are usually needed to design these compensators (Ref. 1, Chap. 8; Ref.

2, Chap. 9).

28.8.2 Feedback Compensation and Cascade Control

The use of a tachometer to obtain velocity feedback, as in Fig. 28.24, is a case of feedback com-

pensation. The feedback-compensation principle of Fig. 28.3 is another. Another form is cascade

control, in which another controller is inserted within the loop of the original control system (Fig.

28.36). The new controller can be used to achieve better control of variables within the forward path

of the system. Its set point is manipulated by the first controller.

Cascade control is frequently used when the plant cannot be satisfactorily approximated with a

model of second order or lower. This is because the difficulty of analysis and control increases rapidly

with system order. The characteristic roots of a second-order system can easily be expressed in

analytical form. This is not so for third order or higher, and few general design rules are available.

Fig. 28.34 General structures of the three compensation types: (a) series; (b) parallel (or feed-

back); (c) feed-forward. The compensator transfer function is Gc(s).1

When faced with the problem of controlling a high-order system, the designer should first see if the

performance requirements can be relaxed so that the system can be approximated with a low-order

model. If this is not possible, the designer should attempt to divide the plant into subsystems, each

of which is second order or lower. A controller is then designed for each subsystem. An application

using cascade control is given in Section 28.11.

28.8.3 Feedforward Compensation

The control algorithms considered thus far have counteracted disturbances by using measurements

of the output. One difficulty with this approach is that the effects of the disturbance must show up

in the output of the plant before the controller can begin to take action. On the other hand, if we

can measure the disturbance, the response of the controller can be improved by using the measurement

to augment the control signal sent from the controller to the final control elements. This is the essence

of feedforward compensation of the disturbance, as shown in Fig. 28.34c.

Feedforward compensation modified the output of the main controller. Instead of doing this by

measuring the disturbance, another form of feedforward compensation utilizes the command input.

Figure 28.37 is an example of this approach. The closed-loop transfer function is

nw = Kf + K

flr(s) Is + c + K

Fig. 28.35 Passive electrical compensators: (a) lead; (b) lag; (c) lead-lag.

For a unit-step input, the steady-state output is a>ss = (Kf + K)/(c + K). Thus, if we choose the

feedforward gain Kf to be Kf = c, then a)ss = 1 as desired, and the error is zero. Note that this form

of feed forward compensation does not affect the disturbance response. Its effectiveness depends on

how accurately we know the value of c. A digital application of feedforward compensation is pre-

sented in Section 28.11.

28.8.4 State-Variable Feedback

There are techniques for improving system performance that do not fall entirely into one of the three

compensation categories considered previously. In some forms these techniques can be viewed as a

type of feedback compensation, while in other forms they constitute a modification of the control

law. State-variable feedback (SVFB) is a technique that uses information about all the system's state

variables to modify either the control signal or the actuating signal. These two forms are illustrated

in Fig. 28.38. Both forms require that the state vector x be measurable or at least derivable from

other information. Devices or algorithms used to obtain state variable information other than directly

Fig. 28.36 Cascade control structure.

Fig. 28.37 Feedforward compensation of the command input to augment proportional control.2

from measurements are variously termed state reconstructors, estimators, observers, or filters in the

literature.

28.8.5 Pseudoderivative Feedback

Pseudoderivative feedback (PDF) is an extension of the velocity feedback compensation concept of

Fig. 28.24.1>2 It uses integral action in the forward path, plus an internal feedback loop whose operator

H(s) depends on the plant (Fig. 28.39). For G(s} = 11 (Is + c), H(s) = K^ For G(s) = 1 /Is2, H(s)

= Kl + K2s. The primary advantage of PDF is that it does not need derivative action in the forward

path to achieve the desired stability and damping characteristics.

28.9 GRAPHICAL DESIGN METHODS

Higher-order models commonly arise in control systems design. For example, integral action is often

used with a second-order plant, and this produces a third-order system to be designed. Although

algebraic solutions are available for third- and fourth-order polynomials, these solutions are cumber-

some for design purposes. Fortunately, there exist graphical techniques to aid the designer. Frequency

response plots of both the open- and closed-loop transfer functions are useful. The Bode plot and

the Nyquist plot all present the frequency response information in different forms. Each form has its

own advantages. The root locus plot shows the location of the characteristic roots for a range of

values of some parameters, such as a controller gain. A tabulation of these plots for typical transfer

functions is given in the previous chapter (Fig. 27.8). The design of two-position and other nonlinear

control systems is facilitated by the describing function, which is a linearized approximation based

on the frequency response of the controller (see Section 27.8.4). Graphical design methods are dis-

cussed in more detail in Refs. 1, 2, and 3.

28.9.1 The Nyquist Stability Theorem

The Nyquist stability theorem is a powerful tool for linear system analysis. If the open-loop system

has no poles with positive real parts, we can concentrate our attention on the region around the point

-1 + /O on the polar plot of the open-loop transfer function. Figure 28.40 shows the polar plot of

the open-loop transfer function of an arbitrary system that is assumed to be open-loop stable. The

Nyquist stability theorem is stated as follows:

Fig. 28.38 Two forms of state-variable feedback: (a) internal compensation of the control sig-

nal; (b) modification of the actuating signal.1

Fig. 28.39 Structure of pseudoderivative feedback (PDF).

A system is closed-loop stable if and only if the point —1 + iO lies to the left of the open-

loop Nyquist plot relative to an observer traveling along the plot in the direction of increasing

frequency a>.

Therefore, the system described by Fig. 28.39 is closed-loop stable.

The Nyquist theorem provides a convenient measure of the relative stability of a system. A

measure of the proximity of the plot to the -1 + /O point is given by the angle between the negative

real axis and a line from the origin to the point where the plot crosses the unit circle (see Fig. 28.39).

The frequency corresponding to this intersection is denoted a>g. This angle is the phase margin (PM)

and is positive when measured down from the negative real axis. The phase margin is the phase at

the frequency a)g where the magnitude ratio or "gain" of G(ia))H(ia)) is unity, or 0 decibels (db).

The frequency a>p, the phase crossover frequency, is the frequency at which the phase angle is -180°.

The gain margin (GM) is the difference in decibels between the unity gain condition (0 db) and the

value of \G(a)p)H((op)\ db at the phase crossover frequency a>p. Thus,

gain margin = -\G((op)H(a)p)\ (db)

(28.34)

A system is stable only if the phase and gain margins are both positive.

The phase and gain margins can be illustrated on the Bode plots shown in Fig. 28.41. The phase

and gain margins can be stated as safety margins in the design specifications. A typical set of such

specifications is as follows:

gain margin > 8 db and phase margin > 30°

(28.35)

In common design situations, only one of these equalities can be met, and the other margin is allowed

to be greater than its minimum value. It is not desirable to make the margins too large, because this

results in a low gain, which might produce sluggish response and a large steady-state error. Another

commonly used set of specifications is

Fig. 28.40 Nyquist plot for a stable system.1

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