Prelab 2: Motor System Identification
In Lab 2, we will experiment with several single-input single-output (SISO) control
strategies for the Lego motors, to gain hands-on intuition for both transient and steady state
behaviors.
Before designing any controllers, however, we will first characterize important model
elements that capture the dynamics of the plant; ideally, these should capture both linear and
nonlinear effects.
First, let us hypothesize a model.
Each model assumption is presented first in words and
then in equation form.
You should need only the equations to complete the prelab; the text is
included to justify the model.
During lab, you may find evidence our assumed model is not quite
right, and so understanding and later refining these assumptions could be useful if you wish to
improve performance in future labs.
We will assume the Power Level command to the motor,
p
m
, (which is a value from -100
to 100 for the Lego NXT system) can be modeled simply as a voltage (of as-yet unknown
scaling). In reality, there is a PWM (pulse-width modulated) signal involved, but the inductance
of the motor will (we hope) provide enough of a low-pass filtering effect that we can treat the
voltage as an average value, proportional (i.e., linear) to the
p
m
.
ܸ
ൌܭ
We also assume the electrical time constant of the motor is so fast (i.e.,
߬
ൌܮ
/ܴ
is
very small) that the electrical impedance relating voltage and current in the motor can be
approximated simply as a resistance,
R
m
:
݅
ൌܸ
/ܴ
And that motor torque is linearly proportional to current:
ܶ
ൌܭ
்
݅
Also, the back EMF (electromotive force) reduces the net voltage across the motor by an
amount linearly proportional to the angular velocity of the motor.
Since power is related as
ܸ
݅
ൌܶ
߱
, the same motor torque constant relating torque to current also relates back EMF
and motor speed:
ܸ
ൌെܭ
்
߱
This linear loss can be lumped with any other linear damping due to the mechanism
(gears, etc.) itself, collectively represented as a viscous damping term in the dynamics,
b
eff
, that
is proportional to the angular velocity
of the output shaft
,
ω
s
:
ܶ
௩௦௨௦
ൌെܭ
்