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Modern Control Theory

(Digital Control)

Lecture 1

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Course Overview

Analog and digital control systems MM 1 – introduction, discrete systems, sampling. MM 2 – discrete systems, specifications, frequency

response methods. MM 3 – discrete equivalents, design by emulation. MM 4 – root locus design. MM 5 – root locus design.

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Outline Short repetition of analog control methods Introduction to digital control

Digitization Effect of sampling

Sampling Spectrum of a sampled signals Sampling theorem

Discrete Systems Z-transform Transfer function Pulse response Stability

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Digitization Analog Control System

ctrl. filterD(s)

plantG(s)

sensorH(s)

r(t) u(t) y(t)e(t)+

-

continuous controllerFor example, PID control

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System caracteristics Transfer function

Characteristic equation 1+D(s)G(s)H(s) = 0

Poles are the roots of the characteristic equation

)()()(1

)()(

)(

)(

sHsGsD

sGsD

sR

sY

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Time functions associated with poles

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Second-order system Transfer function

is the damping ratio

is the undamped natural frequency

n

22

2

2)(

nn

n

sssH

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Rise time, overshoot and settling time

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Response og second-order system versus

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Bode-plot design Determin the open loop gain end phase as function

of

Evaluate the phase margin and gain margin

Adjust the margins by use of poles, zeros and gain scheduling.

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Bode plot )22.3(

07.13

22.3

07.13)(

223

sssssG

12

1054.0

123.3051.0

22.3

07.13)()(

23

s

s

sssDsG

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Digitization Analog Control System

ctrl. filterD(s)

plantG(s)

sensor1

r(t) u(t) y(t)e(t)+

-

continuous controllerFor example, PID control

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Digitization Digital Control System

T is the sample time (s) Sampled signal : x(kT) = x(k)

control:differenceequations

D/A andhold

sensor1

r(t) u(kT) u(t)e(kT)

+-

r(kT) plantG(s)

y(t)

clock

A/DT

T

y(kT)

digital controller

voltage → bit

bit → voltage

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Digitization Continuous control vs. digital control

Basically, we want to simulate the cont. filter D(s) D(s) contains differential equations (time domain) –

must be translated into difference equations.

Derivatives are approximated (Euler’s method)

T

kxkxkx

)()1()(

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DigitizationExample (3.1)Using Euler’s method, find the difference equations.

bs

asK

sE

sUsD

0)(

)()(

Differential equation

)()()()()( 00

1

aeeKbuusEasKsUbs L

Using Euler’s method

)1()()1()()1()1(

)()1()()1(

00

0

keKkeaTKkubTku

aeT

kekeKbu

T

kuku

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Digitization

sssssG

s

ssD

2

1

)1(

1)(,

10

270)(

Compare – investigate using Matlab

1) Closed loop step response with continuous controller.

2) Closed loop step response with discrete controller. Sample rate = 20 Hz

3) Closed loop step response with discrete controller.Sample rate = 40 Hz

Significance of sampling time TExample controller D(s) and plant G(s)

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DigitizationMatlab - continuous controllernumD = 70*[1 2]; denD = [1 10];numG = 1; denG = [1 1 0];sysOL = tf(numD,denD) * tf(numG,denG);sysCL = feedback(sysOL,1);step(sysCL);

Matlab - discrete controllernumD = 70*[1 2]; denD = [1 10];sysDd = c2d(tf(numD,denD),T);numG = 1; denG = [1 1 0];sysOL = sysDd * tf(numG,denG);sysCL = feedback(sysOL,1);step(sysCL);

sssG

s

ssD

2

1)(

10

270)(

Controller D(s)and plant G(s)

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Digitization

Notice, high sample frequency (small sample time T )gives a good approximation to the continuous controller

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Effect of samplingD/A in output from controller

The single most important impact of implementing a control digitally is the delay associated with the hold.

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Effect of sampling Analysis

Approximately 1/2 sample time delay

Can be approx. by Padè

(and cont. analysis as usual)

ctrl. filterD(s)

PadéP(s)

sensor1

r(t) u(t) y(t)e(t)+

-

plantG(s)

Ts

TsP

/2

/2)(

2

TTd

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Effect of sampling

Example of phase lag by sampling

Example from before with sample rate = 10 Hz

Notice PM reduction

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Spectrum of a Sampled Signal Spectrum

Consider a cont. signal r(t) with sampled signal r*(t) Laplace transform R*(s) can

be calculated

ks

k

jnsRT

sR

kTttrtr

)(1

)(

)()()(

r(t) r*(t)

T

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Spectrum of a Sampled Signal

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Spectrum of a Sampled Signal High frequency signal and low frequency signal

– same digital representation.

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Spectrum of a Sampled Signal Removing (unnecessary) high frequencies –

anti-aliasing filter

control:differenceequations

D/A andhold

sensor1

r(t) u(kT) u(t)e(kT)

+-

r(kT) plantG(s)

y(t)

clock

A/DT

T

y(kT)

digital controller

anti-aliasing

filter

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Spectrum of a Sampled Signal

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Sampling Theorem Nyquist sampling theorem

One can recover a signal from its samples if the sampling frequency fs=1/T (s=2/T) is at least twice the highest frequency in the signal, i.e.

s > 2 b (closed loop band-width)

In practice, we need 20 b < s < 40 b

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Discrete Systems

Discrete Systems Z-transform Transfer function Pulse response Stability