EE3060A-Control Engineering Lab
Exp 2 - State Space Modelling and State Feedback Controller
Design for a DC Motor
Objective:
To perform the state space modeling and pole placement through state feedback for a given DC
motor system.
Expectations:
• Before coming to the lab-session, read the material presented below and go through related
textbooks.
• Write a brief report about your design with the diagrams/plots, results obtained/noted and
inferences, if any. Indicate your answers in observation tables wherever necessary.
• Submit the report along with the simulation files. MATLAB© uploaded files should be error
free. There will be a penalty for erroneous code.
• Report should be written clearly and briefly.
• Ensure the following are presented in your report:
– Name and Roll Number.
– All relevant plots.
– Results and inferences.
• Avoid copying, it will attract zero marks. We encourage discussions, of course. In case you
need any clarifications, please contact:
Athira M (122204001@smail.iitpkd.ac.in) or Amala E G (122404004@smail.iitpkd.ac.in) or
Anna S Varghis (122404003@smail.iitpkd.ac.in) or Vidya A P (122404002@smail.iitpkd.ac.in)
or Dr.Sneha Gajbhiye (snehagajbhiye@ iitpkd.ac.in) .
Theory
Consider Figure 1. Torque generated by a DC motor is directly proportional to armature current
and strength of magnetic field. Assume that magnetic field is a constant and hence torque is
directly proportional to armature current given by:
T = Kt i
The back emf, e is proportional to angular velocity of the shaft given by:
e = Ke θ̇
In SI units, torque constant Kt and back emf constant Ke are equal. Take it as K. From Figure
1, the following two equations can be derived by applying Newton’s second law and Kirchhoff’s
voltage law:
J θ̈ + bθ̇ = Ki
di
L + Ri = v − K θ̇
dt
Figure 1: DC motor
Controller design using state feedback
Full state feedback, or pole placement, is a method employed in feedback control system theory to
place the closed-loop poles of a plant at predetermined locations in the s-plane. For this technique,
all state variables must be always known to the controller. The schematic diagram of this control
scheme is shown in Figure 2, where the reference pre-filter Kr is designed to eliminate the steady
state offset. It can be chosen as the reciprocal of dc gain of the system in closed loop. In this
experiment, the aim is to design the input v using state feedback techniques to stabilize the system
states.
Figure 2: Pole placement technique
Problem
1. Make a choice of state variables as x1 = ω = θ̇, x2 = i; input u = v and output y = ω = x1 .
Obtain the state-state space representation of the given system for the given choice of states,
inputs and output.
2. Using the physical parameters of the system provided in Table 1, simulate the system using
ode function in MATLAB© editor.
3. Find out eigenvalues of the system both manually and through MATLAB© . Also Comment
on the stability of the system based on it. [Hint: use eig() for finding the eigenvalues of the
system].
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Parameters
J
b
Ke
Kt
R
L
Description
Rotor moment of inertia
Motor viscous friction
constant
Back emf constant
Motor torque constant
Electric resistance
Electric inductance
Values
0.01 kg/m2
0.1 N.m.s
0.01 V/rad/sec
0.01 N.m/ Amp
1Ω
0.5 H
Table 1: Parameter description and values
4. Using an arbitrary initial condition, design a state feedback controller u = −K1 x1 − K2 x2 .
Make a proper choice of feedback gains K1 and K2 to asymptotically stabilize the system
states to origin i.e., (x1 , x2 ) = (0, 0).
5. Design a state feedback tracking controller to track unit step reference angular velocity (1
rad/s). [Note: Control law u takes the form, u = −Kx + Kr r (refer Figure 2).]
Try to automate the calculations as much as possible.
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