digital control - lecture 2 - aalborg universitethomes.et.aau.dk/yang/de5/dc/mm2.pdflecture 2...

Outline

Digital Control

Lecture 2

Lecture 2 Digital Control

Outline

Outline

1 Sampling Rate Selection

2 Equivalents Between Continuous and Digital Systems


What have we talked about in MM1?MM1?

•Discrete transfer function•Discretization (ZOH)

Sampling Rate SelectionEquivalents Between Continuous and Digital Systems

Exercises

Sampling TheoremSmoothnessEffect of Noise

Outline

1 Sampling Rate SelectionSampling TheoremSmoothnessEffect of Noise

2 Equivalents Between Continuous and Digital SystemsSampling EffectSystem Specifications



Exercises


Sampling of a continuous signal



Exercises


Representations of sampled signals - time domain

Sum of impulses

r∗(t) =∞

∑

k=−∞

r(t)δ(t − kT )

Fourier series

r∗(t) = r(t)1

T

∞∑

n=−∞

e jn2πt/T



Exercises


Representations of sampled signals - Laplace domain

Laplace representation

R∗(s) =

∫

∞

−∞

r(t)

{

1

T

∞∑

n=−∞

e jntωs

}

e−stdt ωs =2π

T

=1

T

∞∑

n=−∞

∫

∞

−∞

r(t)e−(s−jnωs)tdt

=1

T

∞∑

n=−∞

R(s − jnωs)



Exercises


Spectrum of sampled signal



Exercises


Aliasing



Exercises


Anti-aliasing

Use anti-aliasing filter before sampling a signal!



Exercises


Sampling theorem

Theoretical lower limit

ωsωb

> 2

ωs : sampling frequencyωb: required closed-loop bandwidth

Not practical!



Exercises


Smooth response

Practical limits

20 <ωsωb

< 40

ωs : sampling frequencyωb: required closed-loop bandwidth

External requirements to minimizing sampling time

Bandwidth of reference

Human interaction

Sensitivity to delays



Exercises


Sampling of sinusoid

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Step Response

Time (sec)

Am

plitu

de

4x8x20x40xcont



Exercises


Sampling of damped system

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Step Response

Time (sec)

Am

plitu

de

4x8x20x40xcont

Rule-of-thumb: 6-10 samples in closed-loop rise-time



Exercises


Effect of sampling on frequency response

−150

−100

−50

0

50

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

−270

−180

−90

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

4x8x20x40xcont



Exercises


Sampling of damped system with 1 sample delay

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Step Response

Time (sec)

Am

plitu

de

4x8x20x40xcont



Exercises


Effect of noise on sampling

Prefiltering is important!



Exercises


Requirements to prefilter

Breakpoint of prefilter, ωc

ωb < ωc <ωs2

ωs : sampling frequencyωb: required closed-loop bandwidthωc : breakpoint of filter

Prefilter should filter lowest noise frequencies while not

disturbing highest system frequencies!



Exercises

Sampling EffectSystem Specifications

Outline

1 Sampling Rate SelectionSampling TheoremSmoothnessEffect of Noise

2 Equivalents Between Continuous and Digital SystemsSampling EffectSystem Specifications



Exercises


Reconstruction of signal from ZOH

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Step Response

Time (sec)

Am

plitu

de

8xcontcont+16x delay

Reconstructed signal corresponds to continuous system with

half-sample delay



Exercises


Accommodating for sampling effect delay

Approximation of delay

e−sT/2 ≈2/T

s + 2/T

Accommodation

Investigate effect of half-sample delay on continuous system beforedigitizing:

Root locus wrt. T

Analyze using frequency based methods

Effect on phase margin/damping



Exercises


Digital equivalents of continuous-time specifications

Transient response

Overshoot/damping, Mp, ζ

Rise time, ωn

Settling time, ωn, ζ, σ

Steady-state response

Steady-state: z → 1, (limk→∞ f (k) = limz→1(z − 1)F (z))

System type: number of pure integrators in open-loop,(z = 1)

System input

Step, zz−1

Ramp, Tz(z−1)2

Parabola, T2

2z(z+1)(z−1)3



Exercises


Time-domain specifications in z-domain

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.9

0.80.70.60.50.40.3

0.2

0.1

π/T

0.9π/T

0.8π/T

0.7π/T

0.6π/T0.5π/T

0.4π/T

0.3π/T

0.2π/T

0.1π/T

π/T

0.9π/T

0.8π/T

0.7π/T

0.6π/T0.5π/T

0.4π/T

0.3π/T

0.2π/T

0.1π/T


OutlineSampling Rate SelectionSampling TheoremSmoothnessEffect of Noise

Equivalents Between Continuous and Digital SystemsSampling EffectSystem Specifications

Exercises

digital control - lecture 2 - aalborg universitethomes.et.aau.dk/yang/de5/dc/mm2.pdflecture 2...

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