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Modern Control Theory (Digital Control)

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Modern Control Theory (Digital Control) Lecture 1 Based on notes from Jesper Sandberg Thomsen Course Overview Analog and digital control systems MM 1 introduction ... – PowerPoint PPT presentation

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Title: Modern Control Theory (Digital Control)


1
Modern Control Theory(Digital Control)
  • Lecture 1
  • Based on notes from Jesper Sandberg Thomsen

2
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.

3
Outline
  • Introduction to digital control
  • Digitization (3.1)
  • Effect of sampling (3.2)
  • Sampling
  • Spectrum of a sampled signals (5.2)
  • Sampling theorem (11.1, 11.2)
  • Discrete Systems
  • Z-transform (4.2.1)
  • Transfer function (4.2.2)
  • Pulse response (4.2.4)
  • Stability (4.2.5)

4
Digitization
  • Analog Control System

For example, PID control
continuous controller
ctrl. filter D(s)
plant G(s)
r(t)
u(t)
y(t)
e(t)

-
sensor 1
5
Digitization
  • Digital Control System
  • T is the sample time (s)
  • Sampled signal x(kT) x(k)

digital controller
bit ? voltage
control difference equations
D/A and hold
r(t)
u(kT)
u(t)
e(kT)
r(kT)
y(t)
plant G(s)

T
-
clock
y(kT)
sensor 1
A/D
T
voltage ? bit
6
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 (Eulers method)

7
Digitization
Example (3.1) Using Eulers method, find the
difference equations.
Differential equation
Using Eulers method
8
Digitization
Significance of sampling time T Example
controller D(s) and plant G(s)
  • 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

9
Digitization
Matlab - continuous controller numD 701 2
denD 1 10 numG 1 denG 1 1
0 sysOL tf(numD,denD) tf(numG,denG) sysCL
feedback(sysOL,1) step(sysCL)
Controller D(s) and plant G(s)
Matlab - discrete controller numD 701 2
denD 1 10 sysDd c2d(tf(numD,denD),T) num
G 1 denG 1 1 0 sysOL sysDd
tf(numG,denG) sysCL feedback(sysOL,1) step(sys
CL)
10
Digitization
Notice, high sample frequency (small sample time
T ) gives a good approximation to the continuous
controller
11
Effect of sampling
D/A in output from controller
The single most important impact of implementing
a control digitally is the delay associated with
the hold.
12
Effect of sampling
  • Analysis
  • Approximately 1/2 sample time delay
  • Can be approx. by Padè
  • (and cont. analysis as usual)

ctrl. filter D(s)
Padé P(s)
r(t)
u(t)
y(t)
e(t)
plant G(s)

-
sensor 1
13
Effect of sampling
Example of phase lag by sampling Example from
before with sample rate 10 Hz Notice PM
reduction
14
Spectrum of a Sampled Signal
  • Spectrum
  • Consider a cont. signal r(t)
  • with sampled signal r(t)
  • Laplace transform R(s) can be calculated

r(t)
r(t)
T
15
Spectrum of a Sampled Signal
16
Spectrum of a Sampled Signal
  • High frequency signal and low frequency signal
    same digital representation.

17
Spectrum of a Sampled Signal
  • Removing (unnecessary) high frequencies
    anti-aliasing filter

digital controller
control difference equations
D/A and hold
r(t)
u(kT)
u(t)
e(kT)
r(kT)
y(t)
plant G(s)

T
-
clock
anti-aliasing filter
sensor 1
y(kT)
A/D
T
18
Spectrum of a Sampled Signal
19
Sampling Theorem
  • Nyquist sampling theorem
  • One can recover a signal from its samples if the
    sampling frequency fs1/T (ws2p /T) is at least
    twice the highest frequency in the signal, i.e.
  • ws gt 2 wb (closed loop band-width)
  • In practice, we need
  • 20 wb lt ws lt 40 wb

20
Discrete Systems
  • Discrete Systems
  • Z-transform (4.2.1)
  • Transfer function (4.2.2)
  • Pulse response (4.2.4)
  • Stability (4.2.5)
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