Exercise 1

1.Nichols diagrams

Sketch the Nichols diagrams for the following systems

$\frac{1}{s(s+1)(s+10)}, \quad \frac{1}{s-1}, \quad \frac{\exp{(-s)}}{1+s}, \quad \frac{1-s}{s(1+s)}, \quad \frac{1}{s^2+2 \zeta s + 1}, \; (\zeta \textrm{ small})$

For what feedback gains is the closed loop system stable?

 

 

2. Analysis of design

Consider the design in Handin 1B:

Output error feedback control of $G(s)=(s+1)/s^2$ with closed loop poles given by the design polynomials $s^2+2\zeta\omega s+\omega^2$ with $\omega=10, \zeta=0.71$ for both the state feedback and the observer pole placement. 

 

Construct code, in language of your choice, to plot a) Gang of Four, b) Nyquist diagram c) Nichols diagram d) Root locus for the design.

You can use the matlab code ex01hint.m Download ex01hint.m to get started.

 

3 Cancelling slow process poles

Consider the following system with a slow process pole in $s = -0.1$
$G(s) =\frac{1}{10s+1}$
controlled with the controller
$C(s) =\frac{10(10s + 1)}{s}$

which cancels the slow pole and puts the closed loop pole in $s = −10$. Is fast
control achieved? Plot the Gang of Four and study the time response of an
input load disturbance.

 

4 Taking care of an oscillating load disturbance

Consider a system with transfer function

$G(s) = \frac{1}{(s+1)^3}.$

Assume a disturbance $l(t)=\sin \omega t$ with $\omega = 0.05$ acts on the process input.

 

a)  Investigate the error obtained with a PI

controller with $k=1$ and $T_i=2$. Try to trim the PI controller to reduce the impact of the disturbance. What is the trade-off ?

 

b) Construct an LTI controller $C\left(s\right)$ so that the error due to the sinusoidal disturbance, after initial transients, is less than 0.001,  i.e. $|y(t)|<0.001$ for large $t$. What is the main trade-off ?

Hint: Should the controller have low or high gain at $\omega=0.05$?

 

Plot GoF, Bode,  Nichols and Nyquist plots, root loci. Also show time domain  responses to reference step change and impact of the input disturbance $l(t)=\sin \omega t$.

 

5 A non-fractional transfer function related to heat propagation

The transfer function for one-dimensional heat propagation in a semi-infinite metal rod, without dissipation, is given by (see Example 9.6 in Åström-Murray 2020)

$G(s) = e^{-\sqrt{sT}}$.

a) Determine the critical gain under i) proportional control and ii) integral control.

b) The transfer function $G(s)$ relates temperatures at points within a given distance $x$. The time constant $T=\frac{C_V}{\lambda} x^2$ is related to material properties of the metal. Measurements of $G(i\omega)$ using sinusoidal temperature variations were used by Ångström in 1861 to determine thermal conductivity $\lambda$. One problem that Ångström had was that the experimental transfer function turned out to be closer to

$G_\mu(i\omega) = e^{-\sqrt{(i\omega+\mu)T}}$

where the parameter  $\mu>0$ is related to heat losses due to radiation and convection during the experiments, which might be hard to determine. Show that if

$A(\omega) = \log |G_\mu(i\omega)| \quad \textrm{and} \quad \varphi(\omega) = \mathrm{arg}\,G_\mu(i\omega)$

then the following remarkable relation holds

$A(\omega)\cdot \varphi(\omega) = \frac{\omega T}{2}$.

Note that this relation is independent of $\mu$, making it easy to estimate $T$ from experimental data.

c)  Suggest a way to simulate such a system (for example the step response of G(s) under PI control). Hint: Use FFT and IFFT.

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On the exercise we will also discuss the results of Handin 1 and familiarize ourselves with Handin 2: The Flexible Servo Benchmark problem