The University of Melbourne

Control Systems
Cheat Sheet
Contents
Linearization ......................................................................................................................... 3
Laplace Transforms ............................................................................................................... 4
Block Diagrams .................................................................................................................... 5
Bode, Steps and Responses ................................................................................................... 6
Frequency Response .......................................................................................................... 6
Bode Diagrams .................................................................................................................. 7
Pole-Zero Step Response ................................................................................................... 9
Feedback Control Structure ................................................................................................. 10
Steady State Error ........................................................................................................... 11
Unit Step Input ............................................................................................................ 11
Ramp input .................................................................................................................. 11
Parabolic input ............................................................................................................ 11
Sinusoidal Input .......................................................................................................... 11
Steady State Error for Disturbances ............................................................................. 11
Routh’s Algorithm .......................................................................................................... 12
Second-order polynomial............................................................................................. 12
Third-order polynomial ............................................................................................... 12
Fourth-order polynomial .............................................................................................. 12
Stability of K with restrictions ..................................................................................... 12
Impulse response modes .............................................................................................. 12
Root Locus ...................................................................................................................... 13
Rules of drawing a Root Locus .................................................................................... 13
Root Locus Plots (Asymptotes & Root locus) .............................................................. 15
Nyquist Stability Analysis ............................................................................................... 16
Nyquist Plot ................................................................................................................ 16
Phase Change in Nyquist Plots .................................................................................... 18
Relative Stability ......................................................................................................... 20
Robustness ...................................................................................................................... 20
Internal Model Principle .................................................................................................. 21
Controllers .......................................................................................................................... 22
PID ................................................................................................................................. 22
Ziegler-Nichols PID .................................................................................................... 22

1
Phase-Lag ....................................................................................................................... 23
Phase-Lead ...................................................................................................................... 23
Phase-Lead-Lag .............................................................................................................. 24
Formula ........................................................................................................................... 24
Classical Loop-Sharing ................................................................................................... 27
Advantages/Disadvantages .............................................................................................. 27
PID.............................................................................................................................. 27
Phase-Lag.................................................................................................................... 27
Phase-Lead .................................................................................................................. 27
Phase –Lead-Lag ......................................................................................................... 28
Pole Placement ................................................................................................................ 28
Sylvester Matrix .......................................................................................................... 28
Coefficient Matching ................................................................................................... 29

2
Linearization

Example-Consider the single pendulum with a mass of M attached at the end. Linearize the system

3
Laplace Transforms

4
Block Diagrams

Example-Consider the following block diagram of a linear voltage regulator. Find ( ) in terms
of ( ) ( ) ( ).

Solution:

( ) ( )( ( ) ( )) ( ) ( )

( )
( ) ( ( ) ( )) ( ) ( ) ( )
( )

( )
( ) ( ( )( ( ) ( )) ( )) ( )
( )

( ) ( ) ( ) ( )
( )
( )
( ( ) )

5
Bode, Steps and Responses

 Responses due to poles with +ve real part grows and these are said to be
‘UNSTABLE’
 Responses due to poles with –ve real part decays and these are said to be ‘STABLE’
 Responses due to imaginary axis poles is bounded if not repeated, else response grows
 A stable response is DOMINATED by ‘SLOWEST’ poles (i.e. closest to imaginary
axis)
 If only non-repeated imaginary axis pole are said to be ‘MARGINALLY STABLE’

Frequency Response
Unit Step Response: ( ) ( )

Final Value Theorem (Steady State Value):

6
Steady state value, the final value of the step response.

Rise time, tr: the time elapsed up to the instant at which the step response reaches for the first time.

Overshoot, Mp: the maximum instantaneous amount by which the step response exceeds its final
value. ( √ )

Undershoot, Mu: the absolute value of the maximum instantaneous amount by which the step
response falls below zero.

Settling time, ts: the time elapsed until the step response enters a specified deviation band of ,
around the final value.

Bode Diagrams
Cut-off frequency, ωc: This is a value of ω, such that | ( )| ̂ √ , where ̂ is respectively

 | ( )| for low pass filters and band reject filters

 | ( )| for high pass filters
 The maximum value of | ( )| in the pass band, for band pass filters

Bandwidth, Bw: This is a measure of frequency width of the pass band. It is define as
, where . In addition, and are cut-off frequencies on either side of the
pass band or reject band.

Gain Margin: additional gain that makes the system on the verge of instability.
(| ( )|
). Where is the phase cross over (i.e. when phase hits -180°)

Phase Margin: additional phase lag that makes the system on the verge of instability.
( ). Where is the gain cross over (i.e. when gains hits 0 dB)

Delay Margin: time delay, for the system to be on the verge of instability. In terms of transfer
functions the time delay function uses the Heaviside Function (Laplace table, 27):

7
 ( ) ( )↔ ( )

Note: ( ) , use correct units (radians/degrees)

Example (2012 Exam)

The plot below shows 3 instances of the controller using the

parameters ( ) (√ ) ( √ ) ( √ ). Using the Bode plots find the
open loop-gain cross-over frequency for each coupling networks specified. Which coupling exceeds
phase margin of 30°?

Loop gain cross over frequency is found such

that when the plant is coupled with the
controller its gain is zero. This is apparent at
ω =102 rad/s for 2 controllers and one at ω
=60 rad/s (solid line about +8dB and dashed
line -8dB =0dB.

Earlier rise occurs for larger τp values,

therefore from bode plot, from top to bottom
of the dashed lines represent pairs 1, 2 and 3
respectively. For the phase the graph gets
shifted to the left as τp increases.

The first pair has ωc=102 rad/s.

The second pair also has ωc=102 rad/s.

The third pair has ωc=60 rad/s.

First pair has cross over frequency of 100 rad/s, at this value the controller’s PM=180 + (-180) =0 and the plant
has PM=180+(-168) = 12°⇒ PM = 12 + 0 = 12° <30°

Second pair has cross over frequency of 100 rad/s, at this value the controller’s PM = 180 + (-168) = 12° and the
plant has PM=180+(-168)=12° ⇒ PM = 12+12=24°<30°

Third pair has cross over frequency of 60 rad/s, at this value the controller’s PM=180 + (-160)=20° and the plant
has PM = 180+(-160)=20° ⇒ PM = 20+20=40°>30°

Thus if the plant is coupled with the controller using the third pair of parameters, the network exceeds the
phase margin specification.

8
Pole-Zero Step Response

 Adding a LHP zero to the transfer function makes the step response faster (decreases
the rise time and the peak time) and increases overshoot, the smaller the zero the
larger the overshoot.
 Adding a RHP zero to the transfer function makes the step response slower, and can
make the response undershoot
 Adding a LHP pole to the transfer function makes the step response slower
 If the system has imaginary poles then the system should have oscillations

Examples 𝑠+𝑧
PZA: For 𝑠+𝑝, large p→ approach steady state faster,
small z, larger overshoot∴ SRD
𝑧 𝑠 𝑧 𝑧
PZB: For ⇒ 𝑠2 (𝑠𝑡𝑒𝑝) ⇒ 𝑦(𝑡)
𝑠 𝑠 𝑠
𝑧𝑡 ∴ 𝑆𝑅𝐴
𝑧+𝑠 𝑧 𝑧
PZC: For ⇒ 𝑠2 (𝑠𝑡𝑒𝑝) ⇒ 𝑦(𝑡)
𝑠 𝑠 𝑠
𝑧𝑡 ∴ 𝑆𝑅𝐶
𝑠+𝑧
PZD: For 𝑠+𝑝, small p→ long settling time, large z→
small overshoot∴ SRB

PZA: Zero in LHP, therefore overshoot must exist

→ SRC

PZB: Zero in RHP, therefore undershoot must

exist→ SRB

PZC: System has Imaginary components in poles,

therefore oscillations must exist→ SRA

9
Feedback Control Structure

For a simple controller, with gain K:

 Increasing the controller gain K reduces the ‘SENSITIVITY’ of the controlled

output to low frequency disturbances at the plant output
 Increasing the controller gain K increases the range of frequencies (i.e. oscillating
signals) that can be tracked by the output without attenuation or lag in steady-state.

10
In analysis and design it is often useful to think in terms of how the sensitivities depend on the open
( )
loop-gain across frequency: | ( ) ̇ ( ) ( )|. NOTE ( )
+ ( )
and ( )

+ ( )

Steady State Error

A common performance requirement is for the controlled output to follow a specific type of
reference without error in steady-state.

̇ ( ) ( ) ( ) ( ) ( )
( )

Unit Step Input

Consider ( ) the step input of the height A. The steady state error:

( ) ( )
( ) ( )

Ramp input
Consider the steady state error in response to a ramp input ( ) ( ) 2 :

( ) ( )
( ) ( )

Parabolic input
2
For a parabolic input ( ) , we take ( )

( ) ( )
( ) ( )

Sinusoidal Input
For a sinusoidal reference input ( ) ( ), we take ( ) 2+ 2

( ) ( )
( ) ( ( ))( )

Steady State Error for Disturbances

11
Routh’s Algorithm
Consider a polynomial p(s) of degree n, defined as: ( ) ∑ . The Routh’s algorithm is based
on the following numerical array:

Considering
a feedback control system with a characteristic polynomial, Routh’s Algorithm can be used to
determine K values so that the closed-loop is stable. NOTE: K>0

Second-order polynomial
Let ( ) for stability all coefficients, .

Third-order polynomial
Let ( ) for stability ( ) ( )

Fourth-order polynomial
Let ( ) for stability ( ) ( )
( )

Stability of K with restrictions

Finding values of K that have zeros of the characteristic all have a real part less than ‘n’.

Let ⇒ and note that ( ) ( ) . Now ( ) ( ) and

apply Routh’s Algorithm.

Impulse response modes

Faster Decay
Assume that the closed loop impulse response has a mode that decays FASTER than .
⇒ ( ) ( ), now apply Routh’s Algorithm. NOTE:

Slower Decay
Assume that the closed loop impulse response has a mode that decays SLOWER than .

⇒ ( ) ( ), now apply Routh’s Algorithm and take the INVERSE RELATION of the
solution as this method gives values of K for SLOWER decay, i.e. if Routh’s stability criterion says for
faster decay for slower decay. NOTE: .

12
 ( + )
Example (2013 Exam)-Consider a plant ( ) 2+ and a compensator ( ) in a
+ +
standard negative feedback configuration. Apply Routh’s Stability test to find the positive
compensator parameter values , such that the closed-loop impulse response has a mode that
decays SLOWER than .

Solution: Slower than implies at least 1 pole is between . To do this, we find values
of K for which the system decays FASTER than .

( )
( ) ( ) ( ) ( )( )( ) ( )

( ) ( )( ) ( ) ( )

( )

( )

All values must be positive:

⇒ ⇒

This is for a faster decay; therefore for a slower decay we take the inverse relationship

Root Locus

The goal is to find the locus of all points in the complex plane that satisfy these conditions as K is
varied over the positive real numbers. Below are the rules for drawing Root Locus:

Rules of drawing a Root Locus

Start from the form of ( )

1. There are n lines (loci) where n is the degree of the numerator/denominator of F(s)
2. As K increases from 0 to ∞ the roots move from the poles of F(s) to the zeros of F(s).
3. When roots are complex they are conjugate pairs
4. At no time will the same roots cross over path

13
 5. The portion of the real axis to the left of an odd number of open loop and zeros are
part of the loci.
6. Lines leave (break out) and enter (break in) the real axis at 90°
7. If there are not enough poles or zeros to make a pair then the extra lines go to or come
from ∞
8. Lines go to ∞ along asymptotes:

9. If there is at least 2 lines going to ∞ then the sum of all roots is constant
10. K going from 0 to ∞ can be drawn by reversing rule 5 and adding 180° to the
asymptote angle

Example- Consider the plant ( ) ( )( )

with a proportional gain controller ( ) . Show
the zeros and poles on the root locus plot and find where the asymptotes intersect the real axis as
well as the angle of asymptotes.

Solution:

For the open loop transfer function ( ) ( ), we have n=3 poles at and have m=0
finite zeros. We take the form of ( ) , and find .
( )( )

( ) ( )

All poles are in LHP and we have 3 poles, therefore 1 pole must tend to ∞, the plot below shows
that the system will be stable until K is increased to a point that the root locus enters the RHP:

14
Root Locus Plots (Asymptotes & Root locus)

15
Nyquist Stability Analysis
A graphical technique for determining the stability of a nominal feedback control system from the
frequency response of the nominal open loop transfer function: ( ) ( ) ( ).

For closed-loop stability we need ( ) ( ) ( ) to have no zeros in the RHP or on the

imaginary axis (+ no unstable pole-zero cancellation). No zeros on the imaginary axis iff ( )
.

Nyquist Plot
Let in transfer function then sweep ω from 0 to ∞, draw reflection about real axis to have
the plot.

For a simple Transfer function:

Find , and Real/Imaginary intercepts. Nyquist plot starts at the value found for
and ends at . If plot does not have open loop poles in RHP and no encirclements of -1,
the closed loop system is stable.

16
Example-Consider the transfer function ( ) , draw the Nyquist plot.
+

Solution: Substitute

( ) ( ) ( ) ( )

As ( ) ∴ ( ) . Now we find the Real and Imaginary intercepts:

( ( )) ⇒ ( ( )) ⇒

Now we calculate what the intercepts are. Since we already know what give, we pretty
much have the values . We now know that the Nyquist starts at (1,0) goes to the
origin without crossing the Imaginary axis, this is because the phase is at high frequency. Phase
calculations:

( )

⇒ ⇒

The plot is shown below:

17
Strictly Proper
Where degree of DEN > degree of NUM. Picking a point at ∞ in RHP maps the point to the origin in
the Nyquist contour. | ( )| ∴what happens at ∞ doesn’t matter.

Proper
Where degree of DEN = degree of NUM. Picking a point at ∞ in RHP maps the point somewhere on a
circle in the Nyquist contour. | ( )| . At ∞ constant magnitude, angles to all
poles and zeros are the same ∴ ( ) ∑ ∑ .

Phase Change in Nyquist Plots

Assume the transfer function G has the following phases and gains for 2 cases. The Nyquist plot is
shown.

18
Example (Special12 July Exam)

As we have 1 open loop RHP pole, no encirclements of -1. P=1, N=0. Thus we need 1
anticlockwise encirclement to have closed loop stability, N=-1. As the Nyquist stability criterion
mentions, the closed-loop is stable iff N=-P. This is only exists for real axis values between -0.5 and -
0.136:

| | | |

These K values will give stability of the closed-loop system.

19
Relative Stability

Robustness

20
Internal Model Principle

21
Controllers

PID
Proportional action provides a contribution which depends on the instantaneous value of the
control error. A proportional controller can control any stable plant, but it provides limited
performance and nonzero steady state errors.

Integral action, on the other hand, gives a controller output that is proportional to the accumulated
error, which implies that it is a slow reaction control mode. This forces the steady state error to zero
in the presence of a step reference and disturbance.

Derivative action, acts on the rate of change of the control error. Consequently, it is a fast mode
which ultimately disappears in the presence of constant errors. The main limitation of the derivative
mode is its tendency to yield large control signals in response to a high frequency control errors.

Ziegler-Nichols PID
First method of Ziegler and Nichols applies when the plant dynamics is well approximated by the
first-order system with transport delay. ( )
+

The second method of Ziegler and Nichols involves the critical gain and critical frequency. Assume
plant poles all stable except one at origin and zeros are all in LHP. The critical gain is the gain of a
proportional controller that just causes the closed-loop to oscillate, critical frequency, . This is
best understood by the root locus method where the critical gain is where the root locus crosses the
imaginary axis and the critical frequency is how far along it crosses.

22
Phase-Lag

Phase-Lead

23
Phase-Lead-Lag

Formula
The phase margin and cross-over frequency specifications correspond to the following constraints
on the open loop transfer function:

(|| ( ) ( )|| ) ( ( ( ) ( )))

PID

24
Phase-Lag

Phase-Lead

25
Phase-Lead-Lag

( + )
Example (2012 Exam)-Consider the controller ( ) ( + )
, using appropriate phase and gain
conditions, find values of the controller parameters that achieve closed-loop stability with
a phase margin of 30° and a loop-gain cross-over frequency of √ rad/s. What is the steady-state
value of the signal in response to a unit step disturbance d? The plant is ( ) ( 2 )
.

( + )
Solution: ( ) ( )
( )( + )
( )
(
⇒ ( )
( + 2
( + ) + )( ) )+ ( )

|| ( )|| || || ⇒ √( ) ( )
( ) ( )

( ( )) ⇒ ( ( )) ( ) ( ) ( )

√ ( ) ( )
⇒ ( ) ( )⇒ ( )
( ) ( )

√ ⇒

The steady state error due to disturbances is:

26
 ( )
( ) ( ) ( ( ))
( ) ( ) ( )
( )

Classical Loop-Sharing
The challenge is to shape the loop gain to fit the constraints while ensuring sufficient phase margin
at the cross-over frequency so as to limit peaking of sensitivity and complementary sensitivity
functions, which otherwise leads to oscillatory transients and poor robustness. Bode’s gain-phase
relation highlights a potential trade-off between phase margin and the roll-off at the cross-over
frequency. When the plant has no RHP or imaginary axis poles or zeros and approach to
compensator design is to select a desirable loop transfer function Λ(s) then let C(s)=Λ(s)/G(s). If the
plant has unstable poles or zeros, then these must be retained in the desired open loop transfer
function in order for ALL sensitivities to be stable.

Advantages/Disadvantages

PID
 The P controller shows a relatively high maximum overshoot, a long settling time as
well as a steady-state error.
 The I controller has a higher maximum overshoot than the P controller due to slowly
starting I behaviour, but no stead-state error.
 The PI controller fuses the properties of the P and I controllers. It shows a maximum
overshoot and settling time similar to the P controller but no steady-state error.
 The real PD controller has a smaller maximum overshoot due to the 'faster' D action.
Also in this case a steady-state error is visible, which is smaller than in the case of the
P controller. This is because the PD controller generally is tuned to have a larger gain
due to the positive phase shift of the D action.
 The PID controller fuses the properties of a PI and PD controller. It shows a smaller
maximum overshoot than the PD controller and has no steady state error due to the I
action.

Phase-Lag
 Gain cross-over frequency increases
 Bandwidth decreases
 Phase Margin will be increased
 Response will be slower before due to decreasing bandwidth, the rise time and the
settling time will increase
 Phase lag network allows low frequencies and high frequencies are attenuated
 Due to the presence of phase lag compensation, the steady state accuracy increases
 Due to the presence of phase lag compensation, the speed of the system decreases

Phase-Lead
 The velocity constant Kv increases

27
  The slope of the magnitude plot reduces at the gain crossover frequency so that
relative stability improves and error decreases since error is directly proportional to
the slope
 Phase margin increases and response becomes faster
 Due to the presence of phase lead network, the speed of the system increase because it
shifts gain crossover frequency to a higher value
 Due to the presence of phase lead compensation, maximum overshoot of the system
decreases
 Steady State error is not improved
 Improves stability

Phase –Lead-Lag
 Due to the presence of phase lead-lag network, the speed of the system increases
because it shifts gain crossover frequency to a higher value
 Accuracy is improved

Pole Placement

Sylvester Matrix

28
When controller is given it might not take the form of : ( )

+ + +
If the given controller has more components to it such as: ( ) +
, we must take
and multiply it the plant ( ), then the modified ( ) will be used in the Sylvester Matrix. Generally
in exams, they give the controller in a way so that when you factorize ‘a’ from the controller above,
the controller’s component will cancel out with a component of ( )when multiplied.

The inverse of a 4x4 matrix that takes the form below:

Coefficient Matching
Express the closed loop function by using Diophantine’s equation: ( ) ( ) ( ) ( ) ( ).
+
As mentioned earlier, be sure that the controller takes the form of : ( ) +
. This method is
generally asked when we have 5th degree polynomial. This method is simpler, basically match
coefficients! Take the above ( ) polynomial and expand it out, then match the coefficients to the
given polynomial in the question. NOTE: ( ) must have the parameters that the controller has.

( + )( + )
Example (2011 Exam)-Consider a controller of the form ( ) ( + )
.Use the pole
placement procedure to determine the values of that make the thee free closed-
poles coincide with the roots of the polynomial . The plant is ( )

( 2 )
.

Solution: We must find ( ) ( ) and make sure the controller has the main form.

( )( ) ( )
( ) ( )
( )( ) ( ) ( )

Note: ( ) ( ) ( )( ), ( ) ( )
and ( ) ( ) ( ) ( ). Then:

( ) ( )( ) ( )

Therefore we have:

[ ] [ ] [ ] [ ]

Example-Do the same as above by matching coefficients.

( ) ( )( ) ( )

29
( ) ( ) ( )

( )

⇒ ⇒

[ ] [ ]

30