Automatic control theoryA Course used for analyzing and designing a automatic control system

Chapter 1 Introduction1) A water-level control system 21 century information age, cybernetics(control theory), system approach and information theory , three science theory mainstay(supports) in 21 century.1.1 Automatic control A machine(or system) work by machine-self, not by manual operation.1.2 Automatic control systems1.2.1 examples

Chapter 1 Introduction2) A temperature Control system (shown in Fig.1.3) Another example of the water-levelcontrol is shown in figure 1.2.

Water exit

waterentrance

float

lever

Figure 1.2

+

e

ua=k(ur-uf)

ur

uf

amplifier

Gearassembly

thermometer

container

Figure 1.3

Chapter 1 Introduction3) A DC-Motor control system

+

-

+

regulator

trigger

rectifier

DC motor

techometer

load

e

Uf (Feedback)

ur

Fig. 1.4

ua

Uk=k(ur-uf)

Chapter 1 Introduction4) A servo (following) control system

Fig. 1.5

load

+

-

Inputr

outputc

servopotentiometer

servomechanism

servo motor

servomodulator

Chapter 1 IntroductionFig. 1.65) A feedback control system model of the family planning (similar to the social, economic, and political realm(sphere or field))

government(Family planning committee)

census

society

excessprocreate

Desire population

population

+

-

Policy orstatutes

Chapter 1 IntroductionFig. 1.7Example: 1.2.2 block diagram of control systems The block diagram description for a control system : Convenience

+

-

+

x1

x2

x3

e

Adders (comparison)e=x1+x3-x2

Signal(variable)

xxx

Components(devices)

x

Chapter 1 IntroductionActuatorProcesscontrollermeasurement (Sensor)ErrorFeedback signalFig. 1.8 For the Fig.1.1, The water level control system:

Chapter 1 Introduction For the Fig. 1.4, The DC-Motor control system

Desired rotate speed n

uk

ua

Regulator

Trigger

Rectifier

DC motor

uf

Techometer

Actuator

Process

controller

measurement (Sensor)

comparator

Actual rotate speed n

Error

Feedback signal

Reference input ur

Output n

Fig. 1.9

e

Chapter 1 Introduction1.2.3 Fundamental structure of control systems 1) Open loop control systems Features: Only there is a forward action from the input to the output.

Controller

Actuator

Process

Disturbance(Noise)

Input r(t)

Reference desired output

Output c(t)(actual output)

Controlsignal

Actuating signal

uk

uact

Fig. 1.10

Chapter 1 Introduction2) Closed loop (feedback) control systems

Controller

Actuator

Process

Disturbance(Noise)

Input r(t)

Reference desired output

Output c(t)(actual output)

Controlsignal

Actuating signal

uk

uact

Fig. 1.11

measurement

Feedback signal b(t)

+

-(+)

e(t)=r(t)-b(t)

Chapter 1 Introduction Notes: 1) Positive feedback; 2) Negative feedbackFeedback.1.3 types of control systems1) linear systems versus Nonlinear systems.2) Time-invariant systems vs. Time-varying systems.3) Continuous systems vs. Discrete (data) systems. 4) Constant input modulation vs. Servo control systems.1.4 Basic performance requirements of control systems 1) Stability. 2) Accuracy (steady state performance). 3) Rapidness (instantaneous characteristic).

Chapter 1 Introduction1.5 An outline of this text1) Three parts: mathematical modeling; performance analysis ;compensation (design). 2) Three types of systems: linear continuous; nonlinear continuous; linear discrete. 3) three performances: stability, accuracy, rapidness.in all: to discuss the theoretical approaches of the control system analysis and design. 1.6 Control system design process shown in Fig.1.12

Chapter 1 Introduction1. Establish control goals

Chapter 1 Introduction1.7 Sequential design example: disk drive read systemA disk drive read system Shown in Fig.1.13 Configuration Principle

Chapter 1 IntroductionSequential design: here we are concerned with the design steps 1,2,3, and 4 of Fig.1.12.Identify the control goal:(2) Identify the variables to control: Position the reader head to read the date stored on a track on the disk.the position of the read head.(3) Write the initial specification for the variables: The disk rotates at a speed of between 1800 and 7200 rpm and the read head flies above the disk at a distance of less than 100 nm. The initial specification for the position accuracy to be controlled: 1 m (leas than 1 m ) and to be able to move the head from track a to track b within 50 ms, if possible.

Chapter 1 Introduction(4) Establish an initial system configuration: It is obvious : we should propose a closed loop system , not a open loop system. An initial system configuration can be shown as in Fig.1.13. We will consider the design of the disk drive further in the after-mentioned chapters.

Chapter 1 IntroductionExercise: E1.6, P1.3, P1.13

Chapter 2 mathematical models of systems2.1 Introduction 1) Easy to discuss the full possible types of the control systemsin terms of the systems mathematical characteristics. 2) The basis analyzing or designing the control systems. For example, we design a temperature Control system : The key designing the controller how produce uk.2.1.1 Why

Chapter 2 mathematical models of systems2.1.3 How get 1) theoretical approaches 2) experimental approaches 3) discrimination learning2.1.2 What is Mathematical models of the control systems the mathematical relationships between the systems variables.Different characteristic of the process different uk:

Chapter 2 mathematical models of systems2.2.1 Examples2.2 Input-output description of the physical systems differential equations 2.1.4 types 1) Differential equations 2) Transfer function 3) Block diagramsignal flow graph 4) State variables(modern control theory) The input-output descriptiondescription of the mathematical relationship between the output variable and the input variable of the physical systems.

Chapter 2 mathematical models of systemsExample 2.1 : A passive circuit define: input ur output uc we have

ur

uc

R

L

C

i

Chapter 2 mathematical models of systemsExample 2.2 : A mechanismDefine: input F output y. We have:Compare with example 2.1: ucy; urF analogous systems

y

k

f

F

m

Chapter 2 mathematical models of systems Example 2.3 : An operational amplifier (Op-amp) circuitInput ur output uc(2)(3); (2)(1); (3)(1)

ur

uc

R1

C

R2

R4

R1

R3

i3

i1

i2

+

-

Chapter 2 mathematical models of systems Example 2.4 : A DC motorInput ua output 1(4)(2)(1) and (3)(1):

ua

w1

Ra

La

ia

w3

w2

(J3,f3)

(J1,f1)

(J2,f2)

Mf

i1

i2

Chapter 2 mathematical models of systemsMake:

Chapter 2 mathematical models of systems Assume the motor idle: Mf = 0, and neglect the friction: f = 0, we have:The differential equation description of the DC motor is:

Chapter 2 mathematical models of systemsExample 2.5 : A DC-Motor control systemInput urOutput ; neglect the friction:

+

-

+

trigger

rectifier

DC motor

techometer

load

Uf

ur

ua

w

-

-

uk

R3

R1

R1

R2

R3

Chapter 2 mathematical models of systems2134we have2.2.2 steps to obtain the input-output description (differential equation) of control systems1) Determine the output and input variables of the control systems.2) Write the differential equations of each systems components in terms of the physical laws of the components. * necessary assumption and neglect. * proper approximation.

Chapter 2 mathematical models of systems2.2.3 General form of the input-output equation of the linear control systemsA nth-order differential equation:3) dispel the intermediate(across) variables to get the input-output description which only contains the output and input variables.4) Formalize the input-output equation to be the standard form: Input variable on the right of the input-output equation . Output variable on the left of the input-output equation. Writing polynomial according to the falling-power order.Suppose: input r output y

Chapter 2 mathematical models of systems2.3 Linearization of the nonlinear components2.3.1 what is nonlinearity The output is not linearly vary with the linear variation of the systems (or components) input nonlinear systems (or components).2.3.2 How do the linearization Suppose: y = f(r) The Taylor series expansion about the operating point r0 is:

Chapter 2 mathematical models of systemsExamples:Example 2.6 : Elasticity equation Example 2.7 : Fluxograph equationQ Flux; p pressure difference

Chapter 2 mathematical models of systems

2.4 Transfer function Another form of the input-output(external) description of control systems, different from the differential equations.2.4.1 definition Transfer function: The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable,with all initial condition assumed to be zero and for the linear systems, that is:

Chapter 2 mathematical models of systems

C(s) Laplace transform of the output variable R(s) Laplace transform of the input variable G(s) transfer function * Only for the linear and stationary(constant parameter) systems.* Zero initial conditions.* Dependent on the configuration and the coefficients of the systems, independent on the input and output variables.2.4.2 How to obtain the transfer function of a system1) If the impulse response g(t) is knownNotes:

Chapter 2 mathematical models of systems Because:We have:Then:Example 2.8 :2) If the output response c(t) and the input r(t) are knownWe have:

Chapter 2 mathematical models of systems Example 2.9:Then:3) If the input-output differential equation is known Assume: zero initial conditions;Make: Laplace transform of the differential equation;Deduce: G(s)=C(s)/R(s).

Chapter 2 mathematical models of systemsExample 2.10:4) For a circuit* Transform a circuit into a operator circuit.* Deduce the C(s)/R(s) in terms of the circuits theory.

Chapter 2 mathematical models of systems Example 2.11: For a electric circuit:

ur

uc

C1

C2

R1

R2

uc(s)

1/C1s

1/C2s

R1

R2

ur(s)

Chapter 2 mathematical models of systemsExample 2.12: For a op-amp circuit

C

ur

uc

R1

R2

R1

+

-

ur

uc

R1

R2

R1

+

-

1/Cs

Chapter 2 mathematical models of systems5) For a control system Write the differential equations of the control system, and Assume zero initial conditions; Make Laplace transformation, transform the differential equations into the relevant algebraic equations; Deduce: G(s)=C(s)/R(s). Example 2.13the DC-Motor control system in Example 2.5

+

-

+

trigger

rectifier

DC motor

techometer

load

Uf

ur

ua

w

-

-

uk

R3

R1

R1

R2

R3

Chapter 2 mathematical models of systems In Example 2.5, we have written down the differential equations as:Make Laplace transformation, we have:

Chapter 2 mathematical models of systems(2)(1)(3)(4), we have:

Chapter 2 mathematical models of systems2.5 Transfer function of the typical elements of linear systems A linear system can be regarded as the composing of several typical elements, which are:2.5.1 Proportioning elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples:amplifier, gear train, tachometer

R(s)

C(s)

k

1

k

t

r(t)

C(t)

t

Chapter 2 mathematical models of systems2.5.2 Integrating elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples: Integrating circuit, integrating motor, integrating wheel

t

r(t)

C(t)

t

1

R(s)

C(s)

1

TI

Chapter 2 mathematical models of systems2.5.3 Differentiating elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples: differentiating amplifier, differential valve, differential condenser

R(s)

C(s)

TDs

1

TD

t

r(t)

C(t)

t

Chapter 2 mathematical models of systems2.5.4 Inertial elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples: inertia wheel, inertial load (such as temperature system)

1

R(s)

C(s)

k

t

r(t)

C(t)

t

T

Chapter 2 mathematical models of systems2.5.5 Oscillating elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples: oscillator, oscillating table, oscillating circuit

1

R(s)

C(s)

k

t

r(t)

C(t)

t

Chapter 2 mathematical models of systems2.5.6 Delay elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples: gap effect of gear mechanism, threshold voltage of transistors

R(s)

C(s)

1

k

t

r(t)

C(t)

t

Chapter 2 mathematical models of systems2.6 block diagram models (dynamic) Portray the control systems by the block diagram models more intuitively than the transfer function or differential equation models.2.6.1 Block diagram representation of the control systems Examples:

+

-

+

X1(s)

X2(s)

X3(s)

E(s)

Adder (comparison)E(s)=x1(s)+x3(s)-x2(s)

Signal(variable)

G(s)

Component(device)

X(s)

Chapter 2 mathematical models of systemsExample 2.14 For the DC motor in Example 2.4In Example 2.4, we have written down the differential equations as:Make Laplace transformation, we have:

Chapter 2 mathematical models of systemsDraw block diagram in terms of the equations (5)(8):Consider the Motor as a whole:

Ua(s)

Cm

Ia(s)

M(s)

Ea(s)

Ce

-

-

Ua(s)

-

Chapter 2 mathematical models of systemsExample 2.15The water level control system in Fig 1.8:

amplifier

Motor

Gearing

Valve

Watercontainer

Float

-

e

ua

Q

Desired water level

Actualwater level

Feedback signal hf

Input hi

Output h

Chapter 2 mathematical models of systemsThe block diagram model is: