Control Lab Assignment 2 ECSE 403

Control Lab Assignment 2
ECSE 403
1 Objective
The main goal of this assignment is to review some concepts from the linear control course(ECSE
307), and to become familiar with some useful tools in MATLAB which helps us in designing
and implementing controllers. Following concepts are covered in this assignment: response of a
second order system, root-locus diagram, state space representation, pole-zero plots, and stability
of uncontolled dynamics.
2 Your duty
Your duty is to answer all questions which have been asked throughout this assignment and submit
all your answers in addition to MATLAB codes in MyCourses website. Please submit your work
both in the PDF format and MATLAB executable notebook files.
3 Model Description
The final goal of this lab is to model and control an inverted pendulum system. In that system
the input is applied to a DC motor connected to a cart-pole. The first step to model the whole
system is to model the DC motor. We saw the (approximate) model of the DC motor in lab 1 as:
Jm
¨θ + (b +
KtKe
Ra
)
˙θ =
Kt
Ra
va
where θ is the shaft angle (in radians) of the motor and va is the applied voltage.
System’s parameters are as following:
• Jm = 0.01kgm2 be the inertia of the rotor and the shaft.
• b = 0.001Nmsec be the viscous friction coefficient
• Ke = 0.02V sec be the back emf constant
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• Kt = 0.02Nm/A be the motor torque constant.
• Ra = 10Ω be the armature resistance
Note that using SI units Ke = Kt
.
4 Questions
4.1 Proportional Controller
1. Consider the transfer function θ(s)
va(s)
in lab assignment 1 as open loop system. Suppose we
are using a proportional gain with a unity feedback loop. Using the standard form of second
order systems:
ω
2
n
s
2 + 2ζωns + ω2
n
,
find the parameters ωn, ζ of the closed loop system as a function of proportional controller(gain) K. [5 marks]
2. For the closed loop system in Question 1, plot the the roots of the closed loop system for
proportional gains K ∈ [0.01, 0.1, 1, 10, 100, 103
] in one plot to see the movement of poles as
a function of proportional gain K.[5 marks]
Hint: You can use command pzplot(sys) to plot the poles and zeros on the complex plane.
3. Explain what root locus diagram describes and plot the root locus diagram for θ(s)
va(s)
as open
loop function.[5 marks]
Hint: You can use command rlocus(sys) to plot the root locus diagram. Notice that this
command receives the open loop system as input.
4. Using the time-domain step response of second order systems find a value of proportional
gain K such that closed loop step response has approximately 20% overshoot. Verify your
proportional controller using step response of closed loop system.[10 marks]
Hint: You can use the formulas required in Question 4 and 5 from any reference.
5. Ignoring the over shoot constraint in Question 4, find the proportional gain K such that closed
loop step response has peak time of approximately 4s. Verify your proportional controller
using step response of closed loop system.[10 marks]
6. Suppose a controller with transfer function s+0.9
s+0.75 is added to the system. Is there a proportional gain for this system which makes it unstable?[5 marks]
Hint: You can find the gain by visual inspection of root locus diagram(theoretical derivation
is not required.)
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4.2 state space representation
7. The electormechanical equations of the DC motor can be described as following:
Jm
¨θ + b
˙θ = Kti
La
˙i + Rai = va − Ke
˙θ,
where i and La = 0.5H are the current and inductance of the armature. The transfer function
in Lab 1, is an approximation of the model which is described here. We can find the exact
model of a DC motor by state space representation. Describe the state space representation
of the DC motor in the standard from of:
x˙(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
D = 0
with two different setups:
(a) Variables ( ˙θ, ia) are assumed to be states of the system and variable ˙θ is assumed to be
the output.[5 marks]
(Please denote matrices corresponding to state space representation by A1, B1, C1.)
(b) Variables (θ, ˙θ, ia) are assumed to be states of the system and variable θ is assumed to
be the output.[5 marks]
(Please denote matrices corresponding to state space representation by A2, B2, C2.)
8. Consider the state space model (A1, B1, C1). Find the transfer function(denote it by H1(s))
and plot the step response.[5 marks]
Consider transfer function of ˙θ(s)/va(s) in Lab 1(denote it by G1(s)). Are transfer functions
H1(S) and G1(s) exactly the same? Compare the step responses of H1(s) and G1(s), are
they exactly the same?[5 marks]
Hint : You can use the command [num,den] = ss2tf(A,B,C,D) to find the numerator and denumerator of the transfer function corresponding to state space representation of (A, B, C, D).
9. Graph the pole-zero plot for both H1(s) and G1(s). Based on these Graph explain why the
approximation in Lab 1 was a ”good” approximation. [5 marks]
10. Consider the state space model (A2, B2, C2). Find the transfer function(denote it by H2(s))
and plot the step response.[5 marks]
Consider transfer function of θ(s)/va(s) in Lab 1(denote it by G2(s)). Are transfer functions
H2(S) and G2(s) exactly the same? Compare the step responses of H2(s) and G2(s), are
they exactly the same? [5 marks]
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11. Graph the pole-zero plot for both H2(s) and G2(s). Based on these Graph explain why the
approximation in Lab 1 was a ”good” approximation. [5 marks]
12. Consider the uncontrolled dynamics(or zero input dynamics):
x˙ = A1x(t)
Find the eigen-values and eigen-vectors of A1. Find the response of the uncontrolled system
to the eigen-values as initial conditions. Is there an initial condition which results in non
exponentially stable response?[5 marks]
Hint: Command [V1,D1] = eig(A1) returns eigen-values and eigen-vectors in matrix format.
Hint: Command initial(sys,x0) plots the response of the system to the initial condition x0.
Hint: Any exponentially stable response goes to 0 as time goes to infinity.
13. Plot the system’s response to initial conditions [1; 0] and [0; 1]. Explain intuitively what is
the physical interpretation of these initial conditions. Explain physical interpretation of the
final value of the system response to each of these initial conditions. (for example if system
starts with an initial speed and zero initial current, intuitively why this speed approaches
0?) [10 marks]
14. Consider uncontrolled dynamics
x˙ = A2x(t)
Find the eigen-values and eigen-vectors of A2. Find the response of the uncontrolled system
to the eigen-values as initial conditions. Is there an initial condition which results in non
exponentially stable response?[5 marks]
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