ENME403 LAB REPORT
3-Cart System and Inverted Pendulum
Ben Buckley
96647561
1.
Introduction
An investigation into the aspects of controller design was conducted for an inverted
pendulum system and a triple cart system. In both cases, control was implemented with the
aim of meeting a set of design criteria when responding to a sequence of unit steps. The
inverted pendulum controller design was conducted using both pole placement and an LQR
approach with the results compared. A controller for the 3-cart system was designed using
LQR, with the focus centred around robustness to additional weight added to the system.
2.
Inverted Pendulum Control Design
The system comprised of a metal rod balanced on top of a motorised cart, creating an
inverted pendulum. A feedback control design was required to balance the inverted pendulum
as the system is inherently unstable. The system was also subject to step inputs of 0.1m in
both directions. The feedback system varied the voltage on the motor, moving the cart back
and forth to balance the rod. A free body diagram can be seen in Figure 1.
Figure 1: Free body diagram of the inverted pendulum system
There was a total of 4 possible pendulum configurations, therefore the control design needed
to be robust enough to successfully operate under all 4 possible configurations as well as
withstand as much sensor noise as possible as a further test of robustness. Successful
operation was defined by the design criteria in Table 1.
Table 1: Design criteria for the inverted pendulum
Settling time
Rise time
Overshoot
Steady-state error
Input voltage
Slew rate
2.1
5-7 s
1-3 s
10 degrees
< 2%
< 10 V
< 30 V/s
Method
Both pole placement and LQR control designs were implemented on the inverted pendulum
system. Using the gains generated from each design, the system was simulated in MATLAB,
with the results compared against design criteria. Further incrementation of gains was
performed to further refine the gains to achieve a fast response with minimal oscillation when
subjected to step inputs and increasing signal noise.
Pole placement was conducted by selecting a set of closed-loop poles for the system and
deriving the gains required to place the poles. Starting from a set of poles along the left-hand
side of the real axis suitably far from the origin. The system was then simulated, and the
resulting system behaviour compared to the design criteria. Adjustments to the pole locations
were then made, with increasing imaginary components tending to an oscillatory response
and real components increasing damping. The process was iterated until the design criteria
were met, with minimal controller effort.
The LQR approach to calculating controller gains required the tuning of a control weighting
matrix Q. The weightings in Q represent the cart position (x), pendulum angle (θ), change in
𝑑𝑥
𝑑θ
cart position ( 𝑑𝑡 ) and change in pendulum angle ( 𝑑𝑡 ) respectively. As the focus of the
controller was to balance the pendulum, a greater weighting was placed on pendulum angle,
with cart position also weighted. Change in position and angle were determined to be less
important and therefore were left at 1.
2.2
Results
Table 2 shows the controller parameters found by pole placement and LQR designs, and their
respective gains.
Table 2: Control designs and their gains
Parameters
Gains
Pole Placement
[-5-3j -5+3j -1.5-2j -1.5+2j]
K = [1.0 5.1 4.0 1.1]
LQR
Diag(Q) = [100 1000 1 1]
K = [10.0 42.0 13.0 9.0]
Figure 2 shows the system response to step inputs from 0.1 m to -0.1 m for the LQR designed
control system. The characteristics of the response are recorded in Table 3. The pole
placement control system was unable to stabilise the system, unsuccessfully balancing the
pendulum.
Figure 2: Cart position (left) and pendulum angle (right) for LQR control gains
Table 3: System response characteristics for LQR and pole placement control designs
Pole Placement
Characteristic Simulated Experimental
2.9 s
N/A
ts
0.5 s
N/A
tr
0.80°
N/A
Mp
0.8 V
N/A
Vmax
2.3
LQR
Simulated Experimental
2.2 s
N/A
1.2 s
1.9 s
2.86°
5.73°
1.60 V
4.58 V
Discussion
As expected, the LQR method of controller design was more successful in stabilising and
controlling the system. The system response was within the design criteria of rise time,
overshoot and motor voltage. Due to the oscillatory nature of the system at rest, settling time
and steady-state error could not be determined. The oscillations also have the added effect of
rise time and system behaviour during switching being largely dependent on the current
position of the cart.
The continuous oscillations stem from the natural instability of the system. The oscillations
are not seen in the simulated displacement due to experimental factors that were not modelled
such as friction, gear backlash and limited slew rate. The simulations unlimited slew rate
allows for the simulation to instantly react to small changes in the system. Despite the slew
rate limitations in the experiment, maximum voltage was significantly higher than in the
simulation, but still within the design criteria limit. This is caused by factors such as friction
within the system and gear backlash.
3.
3-Cart Control Design
The system investigated consisted of three carts connected by springs. A full-state feedback
controller was designed to damper the system so that the position of cart 3 can be
successfully tracked through step inputs of 250mm and 500mm applied to cart 1 as fast and
accurately as possible. The system is shown in Figure 3.
Figure 3: 3-cart system
Accounting for the possibility of 500g masses placed on either cart 2 or 3 or both, and the
three possible springs used, there is a total of 12 system configurations. As such the designed
controller had to be robust enough to successfully operate in each configuration. A successful
controller satisfies the design criteria shown in Table 4.
Table 4: Design criteria for the 3-cart system
Steady-State Error
Motor Voltage
Motor Voltage Slew Rate
3.1
<10 mm
< +/-12 V
< +/-30 V/s
Method
The control design was conducted using LQR. Through iteratively changing the weighting
matrix Q and simulating the system on MATLAB, a set of gains were found that satisfy the
design criteria. As cart 3 position was the observed output, emphasis was placed on its
weighting. As cart 3’s position is highly dependant on carts 1 and 2, weighting was also
placed on their positions.
As there were restrictions on controller effort, motor voltage was also simulated to ensure
criteria were met. The Q matrix weightings were increased proportionally until motor voltage
approached its limits, making full use of the system for the fastest response possible.
3.2
Results
Table 5 shows the controller parameters found by pole placement and LQR designs, and their
respective gains. Figure 4 shows the system response to step inputs from 0.125 m to -0.125 m
for the LQR designed control system. The characteristics of the response are recorded in
Table 6. The experimental data uses a 3-cart system with a 0.5 kg mass added to cart 3, with
step inputs every 2 seconds.
Table 5: LQR parameters and gains for the 3-cart system:
Parameters
Gains
LQR
Diag(Q) = [100 100 1000 1 1 1]
K = [22.3 8.6 3.7 3.9 2.0 1.8]
Figure 4: Cart position (left) and Controller effort (right) for LQR control gains
Table 6: System response characteristics for LQR control design
LQR
Characteristic Simulated Experimental
1.8 s
1.2 s
ts
0.20 s
0.33 s
tr
5.9%
10.1%
Mp
N/A
12.5 mm
ess
5.40 V
9.05 V
Vmax
3.3
Discussion
While the controller was able to successfully respond to the step inputs and settle within the
required time frame of 2 seconds, its steady state error exceeded the design criteria. This error
is likely to be caused by friction forces in the system that are not modelled in MATLAB. The
experimental data appears as less oscillatory than the simulations, possibly caused by
unmodelled damping in the system. As a result, the experimental response surprisingly has a
faster settling time. The delay seen between the commanded step input and the simulated and
experimental data is caused by the system driver being cart 1.
The maximum voltages in both simulation and experiment were within the criteria with the
experimental having a significantly higher peak voltage due to the extra force required to
overcome static and dynamic friction and gear backlash. The limited slew rate in the
experiment does not appear to have a noticeable effect on the resulting response.
4.
Conclusion
An experiment into control design was conducted on two systems, an inverted pendulum and
a 3-cart system. In both cases the LQR method of controller design provided the better
response, with pole placement requiring further iteration to stabilise the inverted pendulum.
Methods of iteration were required for both designs, with focuses on meeting the design
criteria while minimising controller effort. Throughout the experiment, simulations were
compared to experimental responses and the differences in responses were apparent. This is
due to unmodelled factors present in the system such as friction, gear backlash and limited
slew rates.
The inverted pendulum LQR controller was able to stabilise the system and respond to 200
mm step input commands, while having a controller cost of 4.58 V. The LQR controller for
the 3-cart system was able to respond with a settling time of 1.2 s but had a steady-state error
outside the design criteria.