Välkomna till TSRT15 ReglerteknikFöreläsning 5

Summary of lecture 4Frequency responseBode plot

2Summary of last lecture

Given a pole polynomial with a varying parameter

P(s)+KQ(s)=0

We draw the location of the poles with respect to K in a root-locus

An approximate root-locus can be sketched using some simple drawing rules

3Summary of last lecture

Solution time is roughly 3/(distance to the origon for the pole closest to the origin) for systems with real poles and oscillating systems with a reasonable relative damping ξ (between 0.5 and 1)

A relative damping ξ of 0.7 gives an overshoot of 5%, which typically is what we want

Zeros are fun (will be added to the lecture notes)

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4Frequency response

5Frequency response

6Frequency response

Speaker test:

A test signal (sinusoidal voltage) is sent to the speaker

A microphone measures the sound and registers the amplification from voltage to sound level.

Typical behavior: The measurement signal (the sound) has the same frequency (it would sound terrible otherwise) but the amplification depends on the frequency

7Frequency response

A similar experiment can be performed on any system with an input

Car slalom: We perform a test with a sinusoidal movement of the steering whell, and measure the lateral position of the car (distance from center line)

The car dynamics (from steering wheel angle u(t) in radians to lateral position y(t) in meter) can approximately be modeled with the following linear system

8Frequency response

Input:

Output: : Sinusoidal with an amplitude of roughly 3 meters

9Frequency response

Input:

Output: : Sinusoidal with an amplitude of roughly 80 centimeters

10Frequency response

Input:

Output: : Sinusoidal with an amplitude of roughly 6 meters

11Frequency response

Experimental hypothesis: A sinusoidal input leads to a sinusoidal output (asymptotically after transients from initial conditions have faded away)

12Frequency response

Linear systems are described by linear differential equations whose solutions are constructed from the homogenuous part (which depends on the initial conditions) and the particular part (which depends on the input signal)

Now assume that a sinusoidal input has been used since t=-∞. The homogeneous part will then have disappeared at t=0 and we can use the convolution theorems for Laplace transforms

13Frequency response

A sinusoidal input with frequency ω passed through a linear system systemG(s) is amplified by a factor |G(iω)| and the phase changes arg(G(iω))radians

14Frequency response

We use these formulas for our car

15Frequency response

Phase change

16Frequency response

The amplification in the car from steering wheel angle to lateral position

17Bode plot

Amplitude-gain plots are typically hard to interpret in standard linear-linear scale, and log-log scale is used instead. Additionally, the gain is multiplied with 20 to obtain a decibel scale

Hence, we plot 20 log|G(iω)| with a logarithmically growing frequency

This is called a Bode plot. We will now learn how to sketch these

18Bode plot

Assume that the system is given in the following factorized form (i.e. n poles, p integrators and m zeros easily seen)

Amplitude and phase:

19Bode plotSketch method: Start with very small frequencies

This function is linear in log-log scale, with a slope of –p (i.e., it falls with p*20dB per decade, if we convert to dB)

When we increase the frequency, the other terms will increase, and we must finally take them into account too

When ω approaches a zero or a pole, the corresponding term must be added, since terms of the type starts to become significant

When such a terms is added, the amplitude curve changes direction, and either increases the slope with one unit (when passing a pole, it bends downwards) or decreases the slope with one unit (when passing a zero, it bends upwards)

20Bode plot

For small frequencies

When we increase the frequency, the other terms will start to become significant, just as in the amplitude curve.

A stable zero gives a asymptotic phase addition of while a stable pole gives a negative phase loss of

The phase plot is typically harder to sketch manually.

21Bode plot

22Bode plot

23Bode plot

Complex roots are harder to draw manually

They give rise to resonance frequencies where input signals have an extra large amplification

It occurs at ω0, and its size depends on the relative damping ξ

24Periodic signals

”Why are we interested in frequency responses? In practice, we will probably use other, more general, signals”

The reason is that all perioduic signals with period T can be written as a sum of sine and cosine signals with frequencies nω0where ω0=(2π/T) and n=0,1,2,…

This is called a Fourier expansion

25Periodiska signaler

Exemple: Given the following linear system

The input signal is a square-wave signal with frequency ω0. The Fourier expansion is given by

The output is (asymptotically)

If the relative damping ξ is low, we will have a significant resonance frequency at ω0 leading to a large value for |G(jω0)|. Hence, the first term will dominate the output

26Summary

Summary of todays lecture

A system can be analysed and evaluated based on how it responds to sinusoidal inputs

All linear systems driven by a sinusoidal input give a sinusoidal output with the same frequency, but another amplitude and phase

A plot showing this amplitude gain and phase change is called a Bode plot

Bode plots are drawn in log-log scale since this highlights some of the features better

A pole bends the amplitude curve in the Bode plot downwards while a zero bends the curve upwards

27Summary

Frequency response: Output from a system when the input is sinusoidal.

Bode plot: Plot showing the amplitude gain and phase change of a sinusoidal input for a linear system. Most often drawn in log-log and log-linear scale

Resonance frequency: A frequency for which input signals are amplified significantly more than at other frequencies

Resonance peak: The peak in the bode plot at a resonance frequency

Important words