classical control - lecture 1homes.et.aau.dk/yang/de5/cc/lecture1.beamer.pdf · economic...

Outline

Classical Control

Lecture 1

Lecture 1 Classical Control

Outline

Outline

1 Introduction

2 Control Specifications


IntroductionControl Specifications

Prerequisites

Block diagram for system modeling

Modeling

MechanicalElectrical



BackgroundBasic SystemsModels/Transfers functions

Outline

1 IntroductionBackgroundBasic SystemsModels/Transfers functions

2 Control SpecificationsClosed Loop StabilitySystem PerformanceEffect of PolesEffect of Additional Poles/Zeros




Goal of Control Engineering

Graham C. Goodwin:

The fundamental goal of CE is to find technically, environmentally,and economically feasible ways of acting on systems to controltheir outputs to a desired level of performance in the face ofuncertainty of the process and in the presence of uncontrollableexternal disturbances acting on the process.




The Fly-Ball Governor

Historical Periods

The Industrial Revolution (1860s)

World War II (1940-1945)

The Space Race (1960s,1970s)

Economic Globalization (1980s)




Evolution of Control

1940 Classical ControlFrequency DomainMainly useful for SISO systemsA fundamental tool for many practicing engineers

1960 Modern ControlState-space approach to linear control theorySuitable for both SISO and MIMO systemsNo explicit definition of performance and robustness

1970 Optimal ControlMinimize a given objective function (fuel, time)Suitable for both open-loop and closed-loop control

1980 Robust ControlGeneralization of classical control ideas into MIMO contextBased on operator theory which can be interpreted intofrequency domain

Nonlinear control, Adaptive Control, Hybrid Control




Basic Continuous Control System

Ref. +

−

Feedforward Actuator Plant

Disturbances

Output

Sensor+

+

Measurement noise

Feedback




Basic Digital Control System

Ref. +

−

Feedforward D/A Act. Plant

Disturbances

Output

SensorA/DFeedback

Clock




Continuous System Models

Numerator and denominator

General formula

G (s) =Y (s)

U(s)=

b0sm + b1s

m−1 + · · · + bm

a0sn + a1sn−1 + · · · + an

Second order system (special case)

G (s) =K

s2 + 2ζωns + ω2n

Matlab

sys=tf(num,den)




Continuous System Models

Zeros and poles

General formula

G (s) =Y (s)

U(s)= K

∏mk=1

(s − zk)∏n

i=1(s − pi )

Second order system (special case)

G (s) =K

(s − p1)(s − p2)p1,2 = −ζωn ± jωn

√

1 − ζ2

Matlab

sys=zpk(z,p,k)



Closed Loop StabilitySystem PerformanceEffect of PolesEffect of Additional Poles/Zeros

Outline

1 IntroductionBackgroundBasic SystemsModels/Transfers functions

2 Control SpecificationsClosed Loop StabilitySystem PerformanceEffect of PolesEffect of Additional Poles/Zeros




Bounded Input Bounded Output

Impulse response

Roots of characteristic equation

Gain and phase margins

Nyquist stability criterion




Stability - Impulse Response

Continuous systems

The system is BIBO stable if and only if the impulse response h(t)is absolutely integrable

Discrete systems

The system is BIBO stable if and only if the impulse response h[n]is absolutely summable




Stability - Characteristic Roots

Asymptotic internal stability

Continuous systems

All poles of the system are strictly in the LHP of the s-plane

Discrete systems

All poles of the system are strictly inside the unit circle of thez-plane




Analyzing System Performance

System types

Continuous system

Discrete system

Analysis domain

Time-domain specifications

Frequency-domain specifications

Different periods

Dynamic (transient) responses

Steady-state responses




Continuous Time - Dynamic Response

0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Impulse Response

Time (sec)

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Step Response

Time (sec)A

mpl

itude

num1=[1];

den1=[1 2 1];

num2=[1 2];

den2=[1 2 3];

impulse(tf(num1,den1),’b’,tf(num2,den2),’r’)

step(tf(num1,den1),’b’,tf(num2,den2),’r’)Lecture 1 Classical Control



Continuous Time - Dynamic Response

0 2 4 6 8 10 120

0.1

0.4

0.6

0.9

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

tr

Mp

1%

ts

tp

Overshoot(Mp)

Rise time (tr )

Settling time(ts)

Peak time (tp)




Natural Frequency and Damping




Poles in the Continuous s-plane

−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0.985

0.94

0.86

0.76

0.640.5

0.340.16

0.985

0.94

0.86

0.76

0.640.5

0.340.16




Poles in the Continuous s-plane




Systems with Different Poles

−25 −20 −15 −10 −5 0−6

−4

−2

0

2

4

60.870.9250.965

0.984

0.996

0.350.60.760.870.9250.965

0.984

0.996

510152025

0.350.60.76

Pole−Zero Map

Real Axis

Imag

inar

y A

xis

syslhp1syslhp2syslhp3sys




Step Response of Different Systems

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

Step Response

Time (sec)

Am

plitu

de

origsyslhp1syslhp2syslhp3




Bode Plot of Different Systems

−150

−100

−50

0

50

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

−270

−180

−90

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

origsyslhp1syslhp2syslhp3




Systems with Different Zeros

−9 −8 −7 −6 −5 −4 −3 −2 −1 0−6

−4

−2

0

2

4

60.8

0.9

0.97

0.120.260.40.520.660.8

0.9

0.97

123456789

0.120.260.40.520.66

Pole−Zero Map

Real Axis

Imag

inar

y A

xis

syslhz1syslhz2syslhz3sys




Step Response of Different Systems

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

Step Response

Time (sec)

Am

plitu

de

origsyslhz1syslhz2syslhz3




Bode Plot of Different Systems

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

−180

−90

0

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

origsyslhp1syslhp2syslhz3




System with Zero in RHP

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−6

−4

−2

0

2

4

60.060.120.190.270.36

0.5

0.66

0.88

0.060.120.190.270.36

0.5

0.66

0.88

1

2

3

4

5

6

1

2

3

4

5

6

Pole−Zero Map

Real Axis

Imag

inar

y A

xis

sysrhzsys




Step Response

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Step Response

Time (sec)

Am

plitu

de

origrhz




Bode Plot of System with RHP

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

−180

0

180

360

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

origrhz


classical control - lecture 1homes.et.aau.dk/yang/de5/cc/lecture1.beamer.pdf · economic...

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