Inverted Pendulum: Designing a Controller for ROTPEN Kit

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*** Semester
Inverted Pendulum
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Table of Contents
Question-1.................................................................................................................................................................... 1
Question-2.................................................................................................................................................................... 1
Question-3.................................................................................................................................................................... 2
Question-4.................................................................................................................................................................... 3
Question-5.................................................................................................................................................................... 5
Question-6.................................................................................................................................................................... 6
Question-7.................................................................................................................................................................... 8
Question-8.................................................................................................................................................................. 10
Question-9.................................................................................................................................................................. 12
Output Graphs............................................................................................................................................................... 14
Step Input to the System............................................................................................................................................ 14
Response of pendulum position to an Impulse Disturbance.........................................................................................15
Pole – zero map of the System..................................................................................................................................... 15
Step Response with estimator...................................................................................................................................... 16
Real Pole plot.............................................................................................................................................................. 16
Complex pole plot....................................................................................................................................................... 17
Real zero plot.............................................................................................................................................................. 17
Complex zero plot....................................................................................................................................................... 18
Integrator plot............................................................................................................................................................ 18
Differentiator Plot....................................................................................................................................................... 19
Notch filter with zero and pole plot............................................................................................................................. 19
Open loop of Bode plot............................................................................................................................................... 20
Open Loop of Nyquist Diagram.................................................................................................................................. 20
Closed Loop response of Bode and Nyquist plot..........................................................................................................21
Discrete -time system response of Inverted Pendulum.................................................................................................21
Input – Output Plot of Pendulum position...................................................................................................................22
Simulation of Simulink Model of the Inverted Pendulum............................................................................................22
Step response comparison........................................................................................................................................... 23
Change in position of pendulum at a point..................................................................................................................23
Impulse disturbance rejection in a pendulum..............................................................................................................24
Summary....................................................................................................................................................................... 24
References...................................................................................................................................................................... 25

Question-1
ANSWER:
Introduction
The Rotary Inverted Pendulum is an exemplary control issue that is investigated
frequently as an undertaking in control courses because of its effectively created elements that
area mix of its multifaceted nature of control design 1. It is a system which is built using the
pendulum that is attached to the rotary arm’s end, which the motor controls. In general, the motor
includes the servomotor coupled, by using the gear-chain. The principle objective includes
keeping the pendulum in the upright position of unsteady equilibrium. The next objective
includes keeping the motor at the specifically mentioned angular position, when the first task is
being performed. Whereas, the last task includes destabilizing the motor starting from the
hanging position of the equilibrium which is not stable, with the goal to achieve the stability
range (i.e., here the mode controller could easily start the stabilization.) 2.
Aim
The aimincludesdesigning the controller for the ROTPEN kit, with the help of an
effective linearised pendulum model.
Question-2
ANSWER:
The below mentioned steps help to build the inverted pendulum model in Simulink, they
are 3,
1 P S A, "THE STABILIZATION OF FORCED INVERTED PENDULUM VIA FUZZY CONTROLLER",
in International Journal of Research in Engineering and Technology, vol. 05, 2016, 152-155.
2 J Babu & E Vargheese, "Stabilization of Rotary Arm Inverted Pendulum using State Feedback
Techniques", in International Journal of Engineering Research and, vol. V4, 2015.
3 H Ali, "Robust Stabilizing Controller Design for Inverted Pendulum System", in Jurnal Teknologi, vol. 71,
2014.
1

1) In the MATLAB command window type Simulink and it opens the Simulink
environment. Next,in Simulink open a new model window by selecting New >
Simulink > Blank Model of the open Simulink Start Page window or by
pressing Ctrl-N.
2) From the Simulink/User-Defined Functions library, 4 Fcn Blocks are inserted. The
following equations for , , , and are built by employing the blocks.
3) Every single Fcn block must be changed so that itmatches with its linked function.
4) From the Simulink/Continuous library4 Integrator blocks must be inserted. Every
single Integrator block’s output will be the system’s state variable, , , , and .
5) Every single Integrator block must be double-clicked for adding the “State Name:” of
the linked state variable. Next, the “Initial condition:”must be changedfor
(pendulum angle) to "pi", for representing that the pendulum starts to point straight up.
6) From the Simulink/Signal Routing library, 4 Multiplexer (Mux) blocks must be
inserted, for every single Fcn block.
7) From the Simulink/Sinks and Simulink/Sources libraries, 2 Out1 blocks and one In1
block must be inserted, respectively. Next, the labels must be double-clicked, as it
helps to change the names of the blocks. For "Position" of the cart and the "Angle" of
the pendulum, two outputs are provided when one input is for "Force" that is applied
on the cart.
8) Mux blocks’ each output is connected to the corresponding input of the Fcn block.
9) In the function blocks the below equations are filled 4.
( J eq+ M p r2 ) ̈θ + M p Lp rsinα ( ̇α ) 2−M p Lp rcosα ̈α=τoutput− β1 ̇θ
4
3 M p Lp
2 ̈α−M p Lp rcosα ̈θ−M p g Lp sinα =−β2 ̇α
τ output= Kt [ V m− Km ̇θ (t) ]
Rm
4 L Wan, J Lei & H Wu, "Design of LQR Controller for the Inverted Pendulum", in Advanced Materials
Research, vol. 1037, 2014, 221-224.
2

Question-3
ANSWER:
The MATLAB command window is plotted and represented as follows 5,
Figure: MATLAB command window
Question-4
ANSWER:
The pendulum swings frequent through the full revolutions where the point moves over at
radians. Besides, the position of the cart becomes unbounded, however it oscillates affected
by the swinging pendulum. Such outcomes vary a lot from the aftereffects of the open-loop
5 B Zhang & L Gu, "Robust Control Design and Simulation for Cantilever-Typed Inverted Pendulum",
in Advanced Materials Research, vol. 187, 2011, 548-553.
3

simulation as mentioned at the beginning. This is expected obviously to the way that this
simulation utilized a completely nonlinear model 6.
Based on the pendulum’s reaction to the impulse of 1-Nsec implemented to the cart, the
pendulum’s design prerequisites are as follows:
1) For and the settling time is lower than 5 seconds
2) Pendulum angle will never be greater than 200 (0.35 radians) vertically.
In addition, the system’s response requirement for the 0.2-meter step command in the cart
position includes:
a) For and the settling time is lower than 5 seconds.
b) For the rise time is less than 0.5 seconds.
c) Pendulum angle will never be greater than 200 (0.35 radians) vertically.
Impulse response of the Open-loop
In the below equations, ‘v’ helps to return the poles and zeros as the column vectors.
6 I Boussaada, I Morărescu & S Niculescu, "Inverted Pendulum Stabilization Via a Pyragas-Type
Controller: Revisiting the Triple Zero Singularity", in IFAC Proceedings Volumes, vol. 47, 2014, 6806-
6811.
4

The system’s zeros and poles where the pendulum position denotes the output are observed as
the following,
Time response can be retrieved from the poles of the system. Here, the system uses two transfer
functions, where the system contains a couple of outputs and a single input. Basically, all the
transfer functions work based on the input and output of the multi-input, multi-output (MIMO)
system which contains same poles, until there occurs a pole-zero cancellation.
Question-5
ANSWER:
The process of linearization comprises of producing the nonlinear system’s linear
5