30079_28a.pdf

CHAPTER 28

BASIC CONTROL SYSTEMS DESIGN

William J. Palm III

Mechanical Engineering Department

University of Rhode Island

Kingston, Rhode Island

28.1 INTRODUCTION

868

28.7.3 The Ziegler-Nichols

Rules

891

28.2 CONTROL SYSTEM

STRUCTURE

28.7.4 Nonlinearities and

Controller Performance 892

28.7.5 Reset Windup

869

28.2.1 A Standard Diagram

870

893

28.2.2 Transfer Functions

870

28.2.3 System-Type Number and

Error Coefficients

28.8 COMPENSATION AND

ALTERNATIVE CONTROL

STRUCTURES 893

28.8.1 Series Compensation 893

28.8.2 Feedback

Compensation

and Cascade Control 893

28.8.3 Feedforward

Compensation 894

28.8.4 State- Variable Feedback 895

28.8.5 Pseudoderivative

Feedback

871

28.3 TRANSDUCERS AND ERROR

DETECTORS

872

28.3.1 Displacement and Velocity

Transducers 872

28.3.2 Temperature Transducers 874

28.3.3 Flow Transducers

874

28.3.4 Error Detectors

874

28.3.5 Dynamic Response of

Sensors

875

896

28.4 ACTUATORS

875

28.9 GRAPHICAL DESIGN

METHODS

28.4.1 Electromechanical

Actuators 875

28.4.2 Hydraulic Actuators 876

28.4.3 Pneumatic Actuators 878

896

28.9.1 The Nyquist Stability

Theorem

896

28.9.2 Systems with Dead-Time

Elements

898

28.5 CONTROL LAWS

880

28.9.3 Open-Loop Design for

PID Control

28.5.1 Proportional Control

881

898

28.5.2 Integral Control

883

28.9.4 Design with the Root

Locus

28.5.3 Proportional-Plus-Integral

Control

899

884

28.5.4 Derivative Control

884

28.10 PRINCIPLES OF DIGITAL

CONTROL

28.5.5 PID Control

885

901

28.10.1 Digital Controller

Structure

28.6 CONTROLLER HARDWARE 886

28.6.1 Feedback Compensation

and Controller Design 886

28.6.2 Electronic Controllers 886

28.6.3 Pneumatic Controllers 887

28.6.4 Hydraulic Controllers 887

902

28.10.2 Digital Forms of PID

Control

902

28.11 UNIQUELY DIGITAL

ALGORITHMS

903

28. 1 1 . 1 Digital Feedforward

Compensation

28.7 FURTHER CRITERIA FOR

GAIN SELECTION 887

28.7.1 Performance Indices 889

28.7.2 Optimal Control Methods 891

904

28.11.2 Control Design in the

z-Plane

904

28. 1 1 .3 Direct Design of Digital

Algorithms

908

Revised from William J. Palm III, Modeling, Analysis and Control of Dynamic Systems, Wiley, 1983,

by permission of the publisher.

Mechanical Engineers'Handbook, 2nd ed., Edited by Myer Kutz.

867

28.12 HARDWARE AND

SOFTWARE FOR DIGITAL

CONTROL

28.13 FUTURE TRENDS IN

CONTROL SYSTEMS 912

28.13.1 Fuzzy Logic Control 913

28.13.2 Neural Networks

909

28.12.1 Digital Control

Hardware

914

909

28.13.3 Nonlinear Control

914

28.12.2 Software for Digital

Control

28.13.4 Adaptive Control

914

911

28.13.5 Optimal Control

914

28.1 INTRODUCTION

The purpose of a control system is to produce a desired output. This output is usually specified by

the command input, and is often a function of time. For simple applications in well-structured situ-

ations, sequencing devices like timers can be used as the control system. But most systems are not

that easy to control, and the controller must have the capability of reacting to disturbances, changes

in its environment, and new input commands. The key element that allows a control system to do

this is feedback, which is the process by which a system's output is used to influence its behavior.

Feedback in the form of the room-temperature measurement is used to control the furnace in a

thermostatically controlled heating system. Figure 28.1 shows the feedback loop in the system's block

diagram, which is a graphical representation of the system's control structure and logic. Another

commonly found control system is the pressure regulator shown in Fig. 28.2.

Feedback has several useful properties. A system whose individual elements are nonlinear can

often be modeled as a linear one over a wider range of its variables with the proper use of feedback.

This is because feedback tends to keep the system near its reference operation condition. Systems

that can maintain the output near its desired value despite changes in the environment are said to

have good disturbance rejection. Often we do not have accurate values for some system parameter,

or these values might change with age. Feedback can be used to minimize the effects of parameter

changes and uncertainties. A system that has both good disturbance rejection and low sensitivity to

parameter variation is robust. The application that resulted in the general understanding of the prop-

erties of feedback is shown in Fig. 28.3. The electronic amplifier gain A is large, but we are uncertain

of its exact value. We use the resistors Rl and R2 to create a feedback loop around the amplifier, and

pick Rl and R2 to create a feedback loop around the amplifier, and pick Rl and R2 so that AR2/Rl

» 1. Then the input-output relation becomes e0 « R^e^R^^ which is independent of A as long as

A remains large. If Rl and R2 are known accurately, then the system gain is now reliable.

Figure 28.4 shows the block diagram of a closed-loop system, which is a system with feedback.

An open-loop system, such as a timer, has no feedback. Figure 28.4 serves as a focus for outlining

the prerequisites for this chapter. The reader should be familiar with the transfer-function concept

based on the Laplace transform, the pulse-transfer function based on the z-transform, for digital

control, and the differential equation modeling techniques needed to obtain them. It is also necessary

to understand block-diagram algebra, characteristic roots, the final-value theorem, and their use in

evaluating system response for common inputs like the step function. Also required are stability

analysis techniques such as the Routh criterion, and transient performance specifications, such as the

damping ratio £, natural frequency a)n, dominant time constant r, maximum overshoot, settling time,

and bandwidth. The above material is reviewed in the previous chapter. Treatment in depth is given

in Refs. 1, 2, and 3.

Fig. 28.1 Block diagram of the thermostat system for temperature control.1

Fig. 28.2 Pressure regulator: (a) cutaway view; (b) block diagram.1

28.2 CONTROL SYSTEM STRUCTURE

The electromechanical position control system shown in Fig. 28.5 illustrates the structure of a typical

control system. A load with an inertia / is to be positioned at some desired angle 6r. A dc motor is

provided for this purpose. The system contains viscous damping, and a disturbance torque Td acts

on the load, in addition to the motor torque T. Because of the disturbance, the angular position 6 of

the load will not necessarily equal the desired value 6r. For this reason, a potentiometer, or some

other sensor such as an encoder, is used to measure the displacement 6. The potentiometer voltage

representing the controlled position 0 is compared to the voltage generated by the command poten-

tiometer. This device enables the operator to dial in the desired angle dr. The amplifier sees the

difference e between the two potentiometer voltages. The basic function of the amplifier is to increase

the small error voltage e up to the voltage level required by the motor and to supply enough current

required by the motor to drive the load. In addition, the amplifier may shape the voltage signal in

certain ways to improve the performance of the system.

The control system is seen to provide two basic functions: (1) to respond to a command input

that specifies a new desired value for the controlled variable, and (2) to keep the controlled variable

near the desired value in spite of disturbances. The presence of the feedback loop is vital to both

Fig. 28.3 A closed-loop system.

Fig. 28.4 Feedback compensation of an amplifier.

functions. A block diagram of this system is shown in Fig. 28.6. The power supplies required for

the potentiometers and the amplifier are not shown in block diagrams of control system logic because

they do not contribute to the control logic.

28.2.1 A Standard Diagram

The electromechanical positioning system fits the general structure of a control system (Fig. 28.7).

This figure also gives some standard terminology. Not all systems can be forced into this format, but

it serves as a reference for discussion.

The controller is generally thought of as a logic element that compares the command with the

measurement of the output, and decides what should be done. The input and feedback elements are

transducers for converting one type of signal into another type. This allows the error detector directly

to compare two signals of the same type (e.g., two voltages). Not all functions show up as separate

physical elements. The error detector in Fig. 28.5 is simply the input terminals of the amplifier.

The control logic elements produce the control signal, which is sent to the final control elements.

These are the devices that develop enough torque, pressure, heat, and so on to influence the elements

under control. Thus, the final control elements are the "muscle" of the system, while the control

logic elements are the "brain." Here we are primarily concerned with the design of the logic to be

used by this brain.

The object to be controlled is the plant. The manipulated variable is generated by the final control

elements for this purpose. The disturbance input also acts on the plant. This is an input over which

the designer has no influence, and perhaps for which little information is available as to the magnitude,

functional form, or time of occurrence. The disturbance can be a random input, such as wind gust

on a radar antenna, or deterministic, such as Coulomb friction effects. In the latter case, we can

include the friction force in the system model by using a nominal value for the coefficient of friction.

The disturbance input would then be the deviation of the friction force from this estimated value and

would represent the uncertainty in our estimate.

Several control system classifications can be made with reference to Fig. 28.7. A regulator is a

control system in which the controlled variable is to be kept constant in spite of disturbances. The

command input for a regulator is its set point. A follow-up system is supposed to keep the control

variable near a command value that is changing with time. An example of a follow-up system is a

machine tool in which a cutting head must trace a specific path in order to shape the product properly.

This is also an example of a servomechanism, which is a control system whose controlled variable

is a mechanical position, velocity, or acceleration. A thermostat system is not a servomechanism, but

a process-control system, where the controlled variable describes a thermodynamic process. Typically,

such variables are temperature, pressure, flow rate, liquid level, chemical concentration, and so on.

28.2.2 Transfer Functions

A transfer function is defined for each input-output pair of the system. A specific transfer function

is found by setting all other inputs to zero and reducing the block diagram. The primary or command

transfer function for Fig. 28.7 is

Fig. 28.5 Position-control system using a dc motor.1

Fig. 28.6 Block diagram of the position-control system shown in Fig. 28.5.1

0£) = A(s)Ga(s)Gm(s)Gp(S)

V(s) 1 + Ga(s)Gm(s)Gp(s)H(S)

' }

The disturbance transfer function is

C(s) = ~Q(s)Gp(s)

D(s) 1 + Ga(s)Gm(s)Gp(s)H(s)

V ' ;

The transfer functions of a given system all have the same denominator.

28.2.3 System-Type Number and Error Coefficients

The error signal in Fig. 28.4 is related to the input as

E(s) = * R(s)

(28.3)

1 + G(s)H(s)

If the final value theorem can be applied, the steady-state error is

Elements

Signals

A(s) Input elements B(s) Feedback signal

Ga(s) Control logic elements C(s) Controlled variable or output

Gm(s) Final control elements D(s) Disturbance input

Gp(s) Plant elements E(s) Error or actuating signal

H(s) Feedback elements F(s) Control signal

Q(s) Disturbance elements M(s) Manipulated variable

R(s) Reference input

V(s) Command input

Fig. 28.7 Terminology and basic structure of a feedback-control system.1

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