EE C128 / ME C134 Fall 2010 Lab 5a

University of California, Berkeley Rev. 1.5

Lab 5a: Pole Placement for the Inverted Pendulum

I. Purpose
The objective of this lab is to achieve simultaneous control of both the angular position of the
pendulum and horizontal position of the cart on the track using full-state feedback. We will be
considering small angle perturbations and sine wave reference tracking.

II. Theory
The setup consists of a pendulum attached to a movable cart as shown in figure below.

Figure 1: Cart-inverted pendulum setup with free body diagram

We ignore friction and assume that the entire mass of the pendulum is concentrated at its
center of mass, which is half-way up the length of the pendulum (𝐿𝑝 = 𝐿/2). N and P are the horizontal
and vertical components, respectively, of the reaction force between the cart and the pendulum.
Here are the values of the physical system:

Symbol Description Value

Resolution of cart position encoder 439.6 counts/cm
Resolution of angle encoder 666.7 counts/rad
𝑴 Mass of cart and motor 0.94 kg
𝒎 Mass of pendulum 0.230 kg
𝑳𝒑 Pendulum distance from pivot to center of mass 0.3302 m
2
𝑰𝒄 Moment of inertia of pendulum about its center mLP /3
2
𝑰𝒆 Moment of inertia of pendulum about its end 4*mLp /3
𝑲𝒕 Motor torque constant 0.00767 N*m/A
𝑲𝒎 Motor back emf constant 0.00767 V*s/rad
𝑲𝒈 Motor gearbox ratio 3.71
𝑹𝒎 Motor winding resistance 2.6 Ω
𝒓 Radius of motor gear 0.00635 m
-7 2
𝑱𝒎 Motor moment of inertia 3.9 × 10 kg*m

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EE C128 / ME C134 Fall 2010 Lab 5a
University of California, Berkeley Rev. 1.5

III. Pre-lab
a) Derive the equations of motion of the inverted pendulum-cart system. One way of doing this is by
considering the free-body diagrams of the cart and the pendulum separately and writing their
equations of motion.
(Hint: While deriving use the small-signals approximation 𝐬𝐢𝐧 (𝜽) ≈ 𝜽 and 𝐜𝐨𝐬 (𝜽) ≈ 𝟏. This will
simplify the math a little.)
The equations should turn out to be the following:
(𝑀 + 𝑚)𝑥̈ + 𝑚𝐿𝑃 𝜃̈ = 𝐹𝑎 (1)
4𝑚𝐿 2
𝑚𝐿𝑃 𝑥̈ + 3 𝑃 𝜃̈ − 𝑚𝑔𝐿𝑃 𝜃 = 0 (2)

Here, 𝑭𝒂 equals the force exerted on the cart by the attached motor.

b) Use the motor dynamics derived in the Lab 2 (in the form 𝑭𝒂 = 𝑓(𝑽, 𝑥, 𝑥̇ )) and substitute this into
the cart-pendulum dynamics from part a to obtain the complete system dynamics.
The outputs of our interest are the position of the cart (𝒙) and the pendulum angle (𝜽) and the
available control input is the voltage applied to the motor (𝑽). Thus, our system is a 1-input, 2-
output system (SIMO).
Substitute the model parameters and obtain the state-space model for the complete system. You
may use 𝑿 = [𝒙 𝒙̇ 𝜽 𝜽̇]𝑻 as your state vector. Make sure you do this derivation symbolically.
Once you have the expressions for your state space representation, be sure to use a MATLAB script
file to plug in your values. We actually have two different sets of pendulums in the lab, and this will
save you a lot of headaches down the road if you need to alter your parameters.
To help you verify your solution, you may check your numerical answers against what is shown
below. Derived formula errors tend to propagate across entire rows.
* * * *  * 
* − 6.8312 * *  * 
A= B=  C =? D=?
* * * *  * 
   
* * * * − 3.4625

c) Determine the eigenvalues of the state matrix 𝐴 and the poles of the state-space representation. Is
the open-loop system internally stable? BIBO stable? Also, check if the system is controllable and
observable. (For this part you may use the MATLAB commands eig, ctrb, obsv, and rank)

d) Simulate the output response of the system for a step input in the applied voltage. What would you
expect to happen to 𝑥 and 𝜃 in the physical system (assume infinite track length)? What are the
discrepancies between the simulation and the physical system? Why might these discrepancies be
there?

e) We will use a state-feedback controller to achieve the desired performance specifications. For the
purpose of design, we assume that all the state variables are available for measurement and can use
them for feedback (i.e. the entire state vector 𝑋 is known).

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EE C128 / ME C134 Fall 2010 Lab 5a
University of California, Berkeley Rev. 1.5

The full-state feedback controller is 𝒖 = −𝑲𝑿. The gain matrix 𝐾 is chosen such that the closed-
loop eigenvalues lie at some desired values. These desired values of the eigenvalues are found
based on the performance specifications desired to be achieved.
We would like our closed-loop eigenvalues to lie at −𝟏. 𝟕 ± 𝟗𝒋 and −𝟐 ± 𝟏. 𝟔𝒋. Using MATLAB,
find the feedback gain matrix 𝐾 that will help us achieve this. You may use the acker or place
commands for pole placement.
f) Now your system dynamics are 𝑿̇ = 𝑨𝑿 + 𝑩𝒖, 𝒖 = 𝑲(𝒓 − 𝑿), where 𝒓 is your reference input.
Using the calculated gain matrix 𝑲 and MATLAB, calculate the closed loop transfer function from
the first component in the reference 𝒓 (position reference input) to the output 𝒙 (only the position
of the cart). You may use the ss and tf commands. Plot the Bode plot of this transfer function. You
should notice that the DC gain for this transfer function is unity.

IV. Lab
Based on which station you use, you will either have a long or short pendulum. The different
pendulum physical properties are listed below:
Symbol Description Value
𝒎𝒍 Mass of pendulum (long) 0.230 kg
𝑳𝒑,𝒍 Pendulum distance from pivot to center of mass (long) 0.3302 m

𝒎𝒔 Mass of the pendulum (short) 0.127 kg

𝑳𝒑,𝒔 Pendulum distance from pivot to center of mass (short) 0.1778 m

a) Implement the designed state-feedback controller on the actual setup (with saturation limit 8V and
-8V for the voltage input). However, since we have measurements only for the linear and angular
position of the cart (𝒙) and pendulum (𝜽) and not their velocities, 𝒙̇ and 𝜽̇, we can numerically
differentiate 𝒙 and 𝜽 to approximate 𝒙̇ and 𝜽̇, respectively. This helps obtain the entire state vector
𝑿 for feedback. Don’t forget to include your QuaRC diagram and subsystem (if applicable) in your
lab report.

Note 1: The direction we have defined 𝜽 to increase and the direction the angular
encoder on the hardware increases are opposite of each other. Therefore, make
sure your angular unit conversion is negative.

Note 2: As always, watch your units! Everything in this system should match the
units that you used in your state-space derivation.

b) Run the controller on the hardware (with 𝒓 set to 0) and make sure it balances. You will need to
hold the pendulum vertical before starting the program. Give the pendulum small perturbations
manually and check the response.

c) Now we will introduce a sine wave reference position input into the system and analyze the results.
Your reference would be like 𝒓 = [𝑴𝒔𝒊𝒏(𝝎𝒕) 𝟎 𝟎 𝟎]′ . Start with a reasonable amplitude

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EE C128 / ME C134 Fall 2010 Lab 5a
University of California, Berkeley Rev. 1.5

(𝑴 =0.10 m), use frequencies (𝝎) of 1, 5, and 10 rad/sec. Make sure to start at the center of the
track! You will also need to hold the pendulum vertical before starting the program. Check your
response.

d) Be sure to save all your data (especially reference input and position output) from the workspace
for your post lab. Demonstrate to your GSI your working stage for check off.

V. Post-lab Section

For Part b) in the lab section, plot the variation of the cart and pendulum position with time for
small perturbations (manually-induced) about the equilibrium value. Comment on the controller’s
general performance. Why does the hardware continue to oscillate about the equilibrium point?
For Part c) in the lab section, calculate the gain and phase for each of the frequencies in your
frequency response (ignoring the offset from the hardware response). Locate these frequencies on the
Bode plot from Part f) of your prelab and compare the results. Do your values match for each
frequency? If not, explain possible causes for the difference.

VI. Revision History

Semester and Revision Author(s) Comments
Fall 2010 Wenjie Chen and Jansen Added a post lab section and modified other sections
Rev 1.5 Sheng
Winter 2009 Justin Hsia Corrections and additions after Fall 2009 run-through.
Rev. 1.4
Fall 2009 Justin Hsia Updated table of constants for new equations including
Rev. 1.3 moment of inertia.
Winter 2008 Justin Hsia Corrections and additions after Fall 2008 run-through
Rev. 1.2 and feedback.
Fall 2008 Justin Hsia Converted lab to Word, fixed free body diagram and
Rev. 1.1 other errors.
Fall 2008 Pranav Shah Initial lab write-up.
Rev. 1.0