SCE3113 - Controle
Systems 3
Nicky Magro
Lab Assignment – 3
Introduction
The Proportional Integral Derivative (PID) controller is a very popular type of controller. The
goal of this lab was to investigate the implementation of a PID controller, by implementing
a proportional and proportional-derivative controller schemes on the servo mechanism.
Theory
The controller output m(t) depends on the error signal e(t) according to the equation below.
Taking 𝐾𝑃 , 𝑇𝑑 and 𝑇𝑖 as the tuning parameters, respectively being proportional, derivative
and integral parameters.
𝑚(𝑡) = 𝐾𝑝 (𝑒(𝑡) + 𝑇𝑑
𝑑𝑒 1
+ ∫ 𝑒(𝑡)𝑑𝑡)
𝑑𝑡 𝑇𝑖
The above equation was represented by the block diagram in figure 1, the transfer function
represented by the plant and the controller represented by the PID controller block.
-figure 1: System Block diagram
The proportional term is a control action which is directly proportional to the error,
increasing the proportional gain reduces the systems response time. This can lead to
overshoot at larger values of gain which may lead to oscillations and instability which is not
ideal. The integral term is dependent on the accumulation of error over time, this results in
reduced steady state error by adjusting the degree of control. This leads to improved
accuracy, however it results in a slower response time and an increased value for
overshoot, potentially negatively impacting the system stability. The derivative term is
proportional to the rate of change of the error, this proves a predictive action that can
reduce future error resulting in larger values of damping. While this doesn’t inherently
effect the steady state error it may reduced to increased noise sensitivity.
The plant consists of a permanent magnet DC motor, this was placed in a system that was
configured in closed loop format to work as a servo motor. The they of control was adjusted
using the program, using this the systems PID terms were adjusted as needed. The DC
motor consists of an armature, with a coil providing the inductance 𝐿 and resistance 𝑅,
since the system made use of a small motor the inductive term was assumed to be
negligible due to the size of 𝑅. The voltage at the armature 𝑉 was responsible for controlling
the torque of the motor, by controlling the magnitude of the magnetic field. It is also
responsible for the back emf which was generated as the armature rotated through the
field, this is directly proportional to the angular velocity 𝜔. Assuming the constant of
proportionality of the torque and the coils current and the coils back emf and the angular
velocity are equal, they were denoted by the constant 𝑘𝑒 . Hence the transfer function of the
motor was derived according to the diagram below.
-figure 2: DC motor equivalent diagram
By taking Kirchoff’s equation of the circuit above the below equation was derived:
𝑣(𝑡) = 𝐿
•
𝑑𝑖
+ 𝑅𝑖 (𝑡) + 𝑘𝑒 𝜔(𝑡)
𝑑𝑡
The inductance was taken to be very low and the Laplacian was calculated:
𝑉(𝑠) = 𝑅𝐼(𝑠) + 𝑘𝑒 𝜔(𝑠)
𝐼(𝑠) =
𝑉(𝑠) − 𝑘𝑒 𝜔(𝑠)
𝑅
The motor torque can be denoted by the following equation:
𝑇𝑚 (𝑡) = 𝑘𝑖 𝑖(𝑡)
𝐽𝜔′ (𝑡) + 𝐹𝜔(𝑡) = 𝑇𝑚 (𝑡)
•
The Laplacian was calculated:
(𝐽𝑠 + 𝐹)𝜔(𝑠) = 𝑘𝑒 𝐼(𝑠)
•
Substituting the equation for I(s) into the above equation:
(𝐽𝑠 + 𝐹)𝜔(𝑠) =
𝑘𝑒 (𝑉(𝑠) − 𝑘𝑒 𝜔(𝑠))
𝑅
(𝑅𝐽𝑠 + 𝑅𝐹)𝜔(𝑠) = 𝑘𝑒 (𝑉(𝑠) − 𝑘𝑒 𝜔(𝑠))
(𝑅𝐽𝑠 + 𝑅𝐹 + 𝑘𝑒2 )𝜔(𝑠) = 𝑘𝑒 (𝑉(𝑠))
(𝑅𝐽𝑠 + 𝑅𝐹 + 𝑘𝑒2 )𝜔(𝑠)
= 𝑉(𝑠)
𝑘𝑒
(𝑅𝐽𝑠 + 𝑅𝐹 + 𝑘𝑒2 ) 𝑉(𝑠)
=
𝑘𝑒
𝜔(𝑠)
𝑘𝑒
𝜔(𝑠)
=
(𝑠𝐽𝑅 + 𝐹𝑅 + 𝑘𝑒2 ) 𝑉(𝑠)
𝑘𝑒
(𝐹𝑅 + 𝑘𝑒2 ) [
𝑠𝐽𝑅
+ 1]
𝐹𝑅 + 𝑘𝑒2
𝑘𝑒
(𝐹𝑅 + 𝑘𝑒2 ) [
𝑠𝐽𝑅
+ 1]
𝐹𝑅 + 𝑘𝑒2
=
𝜔(𝑠)
𝑉(𝑠)
=
𝜔(𝑠)
𝑉(𝑠)
𝑘𝑒
𝜔(𝑠)
𝐹𝑅 + 𝑘𝑒2
=
𝑠𝐽𝑅
[
+ 1] 𝑉(𝑠)
𝐹𝑅 + 𝑘𝑒2
Method: 1
The setup was connected to the computer, and the program was set to allow the
adjustment of the 𝜏 and 𝑘 terms according to the equation below. The system was initially
set to open loop operation, and the terms were adjusted so that the output of the system
Results and Discussion: 1
matches the transfer function wave in figure 3. This was accomplished by setting the 𝜏 term
to 0.1 and the 𝑘 term to 17.4, which was rounded to 17 in subsequent calculations.
𝑘
𝜔(𝑠)
=
𝑠𝜏 + 1 𝑉(𝑠)
-figure 3: System in open loop
Using the values obtained through experimentation for 𝜏 and 𝑘 the viscous damping
coefficient F was calculated. Comparing the above equation to the previously derived
equation the relationships below were derived.
•
Using the parameters from table 1, the viscous damping coefficient was calculated.
𝑘𝑒
𝐹𝑅 + 𝑘𝑒2
𝜏=
0.0502
𝐹 × 10.6 + 0.0502
0.1 =
𝑘=
17 =
𝐹 = 40.84𝜇𝑚Nms/rad𝑠−1
𝐽𝑅
𝐹𝑅 + 𝑘𝑒2
22.1𝜇 × 10.6
𝐹𝑅 + 𝑘𝑒2
𝐹 = −4.515𝑚Nms/rad𝑠−1
The difference in the calculated damping coefficients was due to finding the 𝜏 and 𝑘 values
using experimental methods, which resulted in slight differences between input and output
results. However the calculated values resulted in numbers close to zero and similar
values which aligned with what was expected.
DC MOTOR PARAMETERS
PARAMETE
R
Symbol
Value
Units
ROTOR
RESISTAN
CE
R
10.6 Ω
TORQUE
CONSTAN
T
BACK EMF
CONSTAN
T
𝑘𝑒
0.0502 Nm/A
𝑘𝑒
0.0502 Vs/rad
ARMATUR
E AND
LOAD
MOMENT
OF
INERTIA
J
22.1 𝜇kgm²
VISCOUS
DAMPING
COEFFICIE
NT
F
Unknown
Nms/rad
ROTOR
COIL
INDUCTAN
CE
L
Negligible
—
-table 1: DC motor parameters
Method: 2a
The system was then set to operate in closed loop mode with a PID controller with variable
proportional integral and differential terms. The equation used by the system varied when
compared to the standard system as seen in the equation below.
𝑚(𝑡) = 𝑘𝑝 𝑒(𝑡) + 𝑘𝑑
𝑑𝑒
+ 𝑘𝑖 ∫ 𝑒(𝑡)𝑑𝑡
𝑑𝑡
While the proportional gain 𝑘𝑝 is equivalent to the 𝑘𝑝 in the equation, however the
𝑘𝑝
derivative gain 𝑘𝑑 is equivalent to 𝑘𝑝 𝑇𝑑 and the integral gain 𝑘𝑖 is equivalent to 𝑇 . For a
𝑖
proportional gain controller, the integral and derivative terms were set to 0. The 𝜔(𝑠) term
was taken in terms of the shaft angle 𝜃(𝑠) below.
𝜔(𝑠) = 𝑠𝜃(𝑠)
The equation was rearranged to relate to give the motor transfer function as seen below.
𝑠𝜃(𝑠)
𝑘
=
𝑉(𝑠) 𝜏𝑠 + 1
𝑠𝜃(𝑠)
17
=
𝑉(𝑠) 0.1𝑠 + 1
•
Taking into consideration the proportional gain 𝑘𝑝
𝑠𝜃(𝑠)
17
=
× 𝑘𝑝
𝑉(𝑠) 0.1𝑠 + 1
170𝑘𝑝
𝜃(𝑠)
=
𝑉(𝑠) 𝑠(𝑠 + 10)
•
Substituting 170𝑘𝑝 as 𝑘
𝜃(𝑠)
𝑘
=
𝑉(𝑠) 𝑠(𝑠 + 10)
The root locus was then plotted in figure 4:
•
•
Taking the poles at s = 0, -10
Taking the breakaway points at:
o
1
𝜃
1
+ 𝜃+10 = 0
𝜃 + 10 + 𝜃
=0
𝜃(𝜃 + 10)
2𝜃 + 10 = 0
𝜃 = −5
•
The damping ratio was taken to be 0.5, and the angle was computed as follows
o cos(𝜁) = 0.5
𝜁 = 60°
•
The asymptotes were computed below
o
±𝑙𝜋
𝑛−𝑚
±𝜋
2
-figure 4: Proportional gain root locus
Using this plot the blue line, which is 𝜔𝑛 was calculated as seen below:
cos(60) =
𝜔𝑛
5
𝜔𝑛 = 10
From the diagram it is evident that the blue line and the green line are equal, hence the
calibration rule was applied to compute a value for the system gain 𝑘.
𝑘=
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑝𝑜𝑙𝑒𝑠
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑧𝑒𝑟𝑜𝑠
𝑘=
10 × 10
1
𝑘 = 100
The value for 𝑘𝑝 was then computed from the previous substitution.
𝑘 = 170𝑘𝑝
100 = 170𝑘𝑝
0.588 = 𝑘𝑝
Results and Discussion: 2a
Using the previous value for damping ratio the theoretical percentage peak overshoot was
computed.
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑒
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑒
(−
(−
𝜁𝜋
)
√1−𝜁 2 × 100
0.5𝜋
)
√1−0.52 × 100
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 16.3%
Using experimental results the experimental percentage peak overshoot was computed.
%𝑀𝑝−𝑒𝑥𝑝 =
%𝑀𝑝−𝑒𝑥𝑝 =
𝑃
× 100
𝑐𝑠𝑠
(4.78 − 0.04) − 4
× 100
4
%𝑀𝑝−𝑒𝑥𝑝 = 18.5%
The resulting values were very similar meaning the constructed system behaved as expected from
the theory.
%𝑒𝑠𝑠 =
𝑒𝑟𝑟𝑜𝑟
× 100
𝑅
%𝑒𝑠𝑠 =
0.04
× 100
4
%𝑒𝑠𝑠 = 1%
The percentage steady state error in theory was assumed to be zero or very close to zero, this
aligned with the experimental calculations as the value obtained was very small.
Using the value of 𝜔𝑛 computed previously the theoretical rise time was computed as follows.
𝑡𝑟−𝑡ℎ𝑒𝑜𝑟𝑦 =
1.8
𝜔𝑛
𝑡𝑟−𝑡ℎ𝑒𝑜𝑟𝑦 =
1.8
10
𝑡𝑟−𝑡ℎ𝑒𝑜𝑟𝑦 = 0.18𝑠
The experimental rise time was computed by taking 10% and 90% of the input voltage and taking
the time at each point and finding the difference between the two points.
𝑡𝑟−𝑒𝑥𝑝 = 11.28 − 11.12
𝑡𝑟−𝑒𝑥𝑝 = 0.156𝑠
The experimental rise time was very similar to the theoretical rise time which means the system
aligned with theory.
The discrepancies between the experimental and theoretical values were due to several modelling
assumptions taken, such as assuming a perfectly linear motor, ignoring the play in the motor gears
and friction of the system. Taking also into consideration the parameter estimations that were rough
values rather than calculated ones, all these factors and more were responsible for the
discrepancies observed.
Method: 2b
The system’s response time was improved by adding a derivative component and using a
PD controller instead of a P controller. This meant the transfer function from before needed
to be computed again now considering the derivative term.
𝜃(𝑠) 170 × 𝑘𝑝 (1 + 𝑠𝑇𝑑 )
=
𝑉(𝑠)
𝑠(𝑠 + 10)
𝜃(𝑠) 170 × 0.588(1 + 𝑠𝑇𝑑 )
=
𝑉(𝑠)
𝑠(𝑠 + 10)
𝜃(𝑠) 99.96(1 + 𝑠𝑇𝑑 )
=
𝑉(𝑠)
𝑠(𝑠 + 10)
1
𝜃(𝑠) 𝑇𝑑 × 99.96(𝑇𝑑 + 𝑠)
=
𝑉(𝑠)
𝑠(𝑠 + 10)
The previous root locus diagram was modified by including the additional zero in the plot
which resulted in a shift.
-figure 5: Proportional Derivative gain root locus
Using the diagram above the value for s’ was computed.
•
Taking the rise time the value for 𝜔𝑛 was computed
1.8
o 𝑡𝑟 = 𝜔
𝑛
90𝑚 =
1.8
𝜔𝑛
𝜔𝑛 = 20
•
Finding the value for 𝜔𝑑
𝜔
o sin(60) = 20𝑑
𝜔𝑑 = 17.32
•
Finding the value for 𝜎
𝜎
o cos(60) = 20
𝜎 = 10
The value of s’ was found to be: 𝑠 ′ = −10 + 𝑗17.32
To compute the angle of the additional pole, the phase criterion was used, taking the value
of ±𝑙 as positive:
∠𝐺(𝑠)𝐻(𝑠) = ±𝑙180°
∠𝐺(𝑠) = ±𝑙180° − ∠𝐻(𝑠)
∠𝐺(𝑠) = ±𝑙180° − (−90 − ∅)
∠𝐺(𝑠) = ±𝑙180° − (−90 − (180 − 60))
∠𝐺(𝑠) = ±𝑙180° + 210
∠𝐺(𝑠) = 30
Using the computed angle above, the value of 𝑇𝑑 was computed.
•
1
Computing the distance of 𝑇 from -10.
𝑑
o tan(30) =
17.32
𝑥
𝑥 = 30
•
1
Computing the value of 𝑇
o
1
𝑇𝑑
𝑑
= 10+30
𝑇𝑑 = 0.025
The value for the grey line was computed to find the value of 𝑘 using the calibration rule.
sin(30) =
17.32
𝑘
𝑘 = 34.64
Using the calibration rule the value of 𝑘 was computed.
𝑘=
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑝𝑜𝑙𝑒𝑠
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑧𝑒𝑟𝑜𝑠
𝑘=
17.32 × 20
34.64
𝑘 = 10
The value of 𝑘𝑝 was computed below by first computing the value of 𝑘𝑑 :
𝑘𝑑 = 𝑘𝑝 𝑇𝑑
𝑘𝑑 = 2.3 × 0.025
𝑘𝑑 = 0.0575
•
Using the previous substitution the value for 𝑘𝑝 was computed.
𝑘𝑝 𝑘𝑇𝑑
= 10
𝜏
𝑘𝑝 × 17.4 × 0.025
= 10
0.1
𝑘𝑝 = 2.3
Results and Discussion: 2b
Using the previous value for damping ratio the theoretical percentage peak overshoot was
computed.
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑒
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑒
(−
(−
𝜁𝜋
)
√1−𝜁 2 × 100
0.5𝜋
)
√1−0.52 × 100
%𝑀𝑝−𝑡ℎ𝑒𝑜𝑟𝑦 = 16.3%
Using experimental results the experimental percentage peak overshoot was computed.
%𝑀𝑝−𝑒𝑥𝑝 =
𝑃
× 100
𝑐𝑠𝑠
(4.7) − 4
× 100
4
%𝑀𝑝−𝑒𝑥𝑝 =
%𝑀𝑝−𝑒𝑥𝑝 = 17.5%
The resulting values were very similar meaning the constructed system behaved as expected from
the theory.
%𝑒𝑠𝑠 =
𝑒𝑟𝑟𝑜𝑟
× 100
𝑅
%𝑒𝑠𝑠 =
0.05
× 100
4
%𝑒𝑠𝑠 = 1.25%
The percentage steady state error in theory was assumed to be zero or very close to zero, this
aligned with the experimental calculations as the value obtained was very small.
Using the value of 𝜔𝑛 computed previously the theoretical rise time was computed as follows.
𝑡𝑟−𝑡ℎ𝑒𝑜𝑟𝑦 = 90𝑚𝑠
The experimental rise time was computed by taking 10% and 90% of the input voltage and taking
the time at each point and finding the difference between the two points.
𝑡𝑟−𝑒𝑥𝑝 = 2.014 − 2.069
𝑡𝑟−𝑒𝑥𝑝 = 55𝑚𝑠
The experimental rise time was very similar to the theoretical rise time which means the system
aligned with theory.
The experimental peak overshoot as observed was slightly lower than the previous P controller
result, this is due to the non-idealities in the damping factor. While the steady state error remained
the same, which was as expected since no integral term was used. The largest change was the
heavily reduced rise time while maintaining an acceptable level of peak overshoot which aligns with
the theoretical expectations for a system such as the one used.
Conclusion
The DC servomotor was modeled using P and PD controller schemes. The experimental
results aligned with the theoretical analysis, certain discrepancies were observed however.
These discrepancies are attributed to the assumptions and simplifications taken when
modelling the system, such as neglecting the friction, the play in the motor and the errors
due to quantization incase of a digital system. The PD controller consistently provided an
improved transient performance compared to the P controller, with a faster rise time and
better damping while maintaining an acceptable steady state error close to zero. Overall,
the results obtained align with the theory of controller design while highlighting the
advantages and disadvantages of the different controllers.