Inverted Pendulum Laboratory Experiment

Inverted Pendulum Laboratory Experiment
Part I - Introduction
This experiment aims to demonstrate the design of a stabilizing controller for an inverted pendulum
by means of a linear control law. In implementing such a control system the following
topics will be covered.
Modeling the dynamics of an inverted pendulum using the Euler-Lagrange equation.
Obtaining a linear state-space representation of the system.
Creating a Simulink model to simulate the non-linear behaviour of the pendulum.
Designing a state-feedback control law that improves damping for the pendulum in the
downward position and balances it at its vertical upward position.
Implementing the control law on the Quanser SRV02 + SIP (Single Inverted Pendulum)
plant and evaluating its performance.
The non-linear model is obtained using classical Lagrangian mechanics. The Lagrangian function,
L(·, ·), is defined as
L(q(t), q˙(t)) = T(q(t), q˙(t)) V (q(t)) , (1)
where T and V are the kinetic and potential energies of the mechanical system, respectively and
(q, q˙) are the generalized coordinates.
The dynamics of the system can, then, be obtained through the Euler-Lagrange equation
dtLq˙(t, q(t), q˙(t)) Lq(t, q(t), q˙(t)) = τ , (2)
where Ly denotes the partial derivative of the function L with respect to the variable y, and
τ describes the non-conservative forces applied to the system with respect to the generalized
coordinates, q.
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Part II - Derivation of the model
For the considered pendulum, the generalized coordinates, q(t), are taken to be the vector
(θ(t), α(t)) representing the angles of the rotary arm and of the pendulum, respectively, see
Figure 1.
The kinetic energy of the pendulum is given by (all parameters are defined in Table 2)
where τm is the motor torque applied at the load gear, Barm is the viscous friction force, Bp the
damping coefficient of the pendulum and kθ and kα are the constant coefficients of the static
friction terms acting on the shaft and the pendulum respectively. Since the pendulum is not
actuated, the only force acting on the link is the damping. Considering the dynamics of the

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motor, the motor torque is expressed as
The control variable is taken to be the motor voltage Vm.
The model is derived considering (0, 0) as the downward position and (0, π) as the upward position.
The non-linear model is given by
Figure 1: Kinematics of a single inverted pendulum system.
Q1) Show that the mechanical model can be expressed as in (7) and (8) using the EulerLagrange
equation (2) and considering equations (3)-(6).
Q2) Rewrite the system of differential equations derived in Q1) in the matrix form (9).
Q3) Design a simulink model describing the non-linear model given by (7) and (8) assuming
τm ≡ 0, in order to simulate the unforced response of the pendulum. This model will be
used to estimate the values of the parameters kθ and kα.
a) To estimate the value of the parameter kα let the real pendulum oscilate freely from a
small initial angle (between 3o and +3o) and adjust the value of kα in your simulink
model to match the observed oscilation of the pendulum to the theoretical one of
your model.
b) In the control block, add a constant term as shown in Figure 2. We can estimate the
value of the parameter kθ, by finding the greatest constant torque for which the shaft
remains stationary. (Ensure that the initial value is very small.)
Part III - Linear System Analysis
Q1) (a) Linearize the model (9) in the downward position, (θ, α) = (0, 0).
(b) Repeat the procedure for the upward position, (θ, α) = (0, π).
Q2) Obtain the state space representation of the linearized models derived in Q1), defining the
Figure 2: Modified Control Block.
[x1(t), x2(t), x3(t), x4(t)]T = [θ(t), α(t),
˙θ(t), α˙(t)]T and [x1(t), x2(t), x3(t), x4(t)]T = [θ(t), α(t)?
π, ˙θ(t), α˙(t)]T
for the downward and upward positions, respectively.
(HINTS: (i) Get an expression for q¨(t) and transform the two second-order differential
equations into a system of four first-order differential equations. (ii) Since static friction
is small, it can be ignored in the derivation of the linearised model.)
Part IV - System Analysis
For this section consider the linearized state space models ˙x(t) = Ax(t)+Bu(t) obtained in Part
III.
Q1) Evaluate the expression for the matrices A, B (see Table 2 for the parameter values). Assess
the stability of the linearized systems associated to both the upward and downward
configurations.
Q2) Check controllability of the linearized systems.
Part V - Controller design and implementation
The aim of the controller is to stabilize the equilibrium point of the pendulum in the upward
configuration and improve the damping of the pendulum in the downward. Assuming that all
the states are measurable, the stabilization of the upward and downward equilibria of the pendulum
is achieved through the implementation of a linear static state feedback law, K, of the
form u = K(xd x) where xd is the desired reference input and x the actual state vector.
Q1) If the pair (A, B) is controllable assign the poles to a location of your choice that improves
damping for the downward position and stabilises the upward. Scientifically support your
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answer. By using pole placement, obtain a pair of control laws corresponding to the poles
you have chosen and comment on their success.
Q2) Compare and contrast your ‘guess’ with the pair of LQR control laws resulting from the
and Rdown = 10, Rup = 100.
Part V - Matlab and Simulink
The Simulink model, Figure 3, is used to control the SRV02 pendulum using the QuaRC software.
The closed-loop system comprises the plant (System) and the controller.
SRV02 Inverted Pendulum: Balance Control
Figure 3: Closed-loop diagram.
The System block, Figure 4, contains the QuaRC sub-blocks required to communicate with the
DC motor and encoders of the actual system (SRV02-ET+ROTPEN-E), the swing-up controller
and a filter block (High-Gain Observer) that calculates the unmeasured states of the system.
As far as this experiment is concerned, all the blocks together make up the plant and, therefore,
the block is not to be tampered with.
The Set point block can be used to adjust the reference input to the system; it is initially set
to a square wave of frequency 0.2Hz. The amplitude of the reference signal can be entered in
degrees in the Amplitude block.
The Swing-up switch can be used to alternate between manual and automatic swing up. In
the OFF position, the user must manually swing the pendulum to the vertical position, where
the controller takes over. In the ON position, a separate energy-based controller swings the
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pendulum up.
The Stabilize switch allows selection between upward and downward stabilization. Because of
the way the angle of the pendulum is read, it is necessary to set it to the correct position
otherwise the controller will not function correctly.
Figure 4: Description of the System block.
In Lab Procedure
The MATLAB control toolbox can be used to create a state space representation of the system
and to design the static controller K. The state space representation can be obtained by
executing sys = ss(A,B,C,D). The matrices A and B are as calculated in Part IV, Q1) and.
Verify your answer to Part IV, Q2) by generating the controllability matrix using the ctrb command
and checking its rank.
Verify the control laws, obtained earlier, using the place command (for both upward and downward
setups). Note that MATLAB uses ?K for the control law.
The first task is to implement the controller for the downward position. Ensure that you have
the correct value for K before attempting to run the model. The variable K is passed into the
Simulink model. Open and run the configuration file ‘setup.m’. This sets all the pendulum
parameters to the correct values; then open the model file ‘q rotpen bal.mdl’.
Set the Stabilize down/up switch to the down position and the Swing-Up ON/OFF to
OFF.
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Click on QuaRC → Build.
Press the ‘Connect to target’ button on the Simulink toolbar. The experiment is ready
to run, by clicking the ‘Play’ button. You can always interrupt the simulation by clicking
‘Stop’ on the toolbar.
NOTE: KEEP YOUR DISTANCE. THE SWING CAN BE VIOLENT!
Q1) Set K = 0. This will simulate the open loop response of the system. Connect and run the
experiment. Using the scopes (in Scopes block) plot a graph of α(t) showing the undamped
oscillations. Now, set k to its downward position value. Repeat, plotting a graph for α(t).
Compare the two results.
Q2) Stop the simulation and change the amplitude of the reference signal in the Set point
block to 40 degrees. What do you expect to observe? Comment on the response of the
pendulum.
Q3) Repeat Q2 using K = [0.3, ?1.36, 0.28, 0.17]. Plot graphs of α(t) and θ(t) showing the
ideal and actual responses. Can you explain the shape of the wave form for θ(t) Repeat
the experiment using K = [0.6, 1.36, 0.28, 0.17], [1, 1.36, 0.28, 0.17] and experiment
changing the square wave reference to a sine. Does the controller follow the reference?
Comment on your results.
NOTE: You need to gently perturb manually the pendulum to observe the damped response.
Q4) Set K to the corresponding controller for the up position and the amplitude in the Set
pointblock to zero. Ensure that the Stabilize switch is set to the up position. Run the
model and manually rotate the pendulum counter-clockwise to the up position. At the top
of its trajectory, the controller should take over keeping the pendulum vertical. If your
controller works (stabilizes the pendulum), stop and enable the automatic swing. Recompile
and run your design. The pendulum should now swing to its vertical position before
your controller takes over.
Q5) While the controller is running, change the reference signal to a 30 degree sinewave with
frequency 0.2Hz. Plot graphs of α(t) and θ(t) showing the ideal and actual responses.
Q6) Try increasing the frequency to 2Hz in steps and comment on the robustness of your
controller. Repeat using a square wave.
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Appendix
Table 1 lists and describes the various files supplied for the experiment.
Table 1: List of MATLAB files
File Name Description
config srv02.m Returns the configuration-based SRV02 model specification
Rm, Kt
, Km, Kg, ηg and ηm and the sensor calibration
constants.
setup.m The main MATLAB scripts that sets the SRV02 motor and
sensor parameters, the SRV02 configuration-dependent
model parameters, and the SIP sensor parameters. Run this
file only to setup the laboratory.
Table 2: Parameter description and values
Symbol Description Value
Rm Motor armature resistance 2.6 ?
Kt Motor torque constant 7.68e-3 N · m/A
ηm Motor efficiency 0.69
Km Back-emf constant 7.68e-3 V /(rad/s)
Kg High-gear total gearbox ratio 70
ηg Gearbox efficiency 0.9
mp Mass of the pendulum 0.127 Kg
lp Distance from pivot to center of gravity 0.1556 m
Jp Pendulum moment of inertia 0.0012 Kg · m2
Jarm Rotary arm moment of inertia 0.002 Kg · m2
Bp Damping coefficient of the pendulum 0.0024 N · m · s/rad
Barm Viscous friction force 0.0024 N · m · s/rad
r Length of the rotary arm 0.2159 m
g Gravitational constant 9.81 m/s2

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