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Problem 1. Consider the following block diagram of a coupled two-input two-output control system:
(a) Find the Transfer Function T(s) = Y2(s)
R1(s)
R2(s)=0
(b) Find G5(s) in terms of the other transfer functions in order to make T(s) = 0 and thus decouple
Y2(s) from R1(s).
Problem 2. Operational Amplifiers (OpAmps): OpAmps are key active building blocks in electrical circuits, and are commonly used in many applications such as amplifying sensor signals, or
implementing various filters and compensators in electronic controllers. Due to very high open-loop
gain, OpAmps are typically used with negative feedback, and with proper characteristics, the closedloop gain of the amplifier can be controlled primarily by the relatively stable and accurate passive
elements in the feedback path.
(a) Consider the following configuration. We typically assume an ideal OpAmp. This means that the
input impedance is assumed to be infinity and the output can essentially act like a voltage source
with zero impedance.
Using the superposition rule, and the voltage division, show that:
Va =
Z2
Z1 + Z2
Vi +
Z1
Z1 + Z2
Vo
These homework problems are compiled using the different textbooks listed on the course syllabus
1
ECE141 – Principles of Feedback Control Homework 2 2
(b) Given the polarity of the input connection in the above configuration, we have Vo = −aVa where a
is the open-loop gain of the OpAmp. Substitude in the above equation and show that the transfer
function Vo
Vi
can be obtained as:
Vo
Vi
=
−aZ2/(Z1 + Z2)
1 + aZ1/(Z1 + Z2)
Remember that both the open loop gain a, as well as the impedances Zi
, can be frequencydependent complex quantites. The quantity L(jω) shown below is called the Loop Transfer
Function and, as we shall see in future lectures, it is generally a very critical quanitity in any
feedback system and can virtually impact all the performance aspects of the closed-loop system.
L(jω) ,
a(jω)Z1(jω)
Z1(jω) + Z2(jω)
(c) With OpAmps, we typically assume ideal closed-loop gain by assuming very large open loop gain
leading to |L(jω)| 1, which will reduce our closed-loop transfer function for the above configuration to the following:
Vo(s)
Vi(s)
= −
Z2(s)
Z1(s)
Notice how our ideal closed-loop gain no longer depends on the open-loop gain of the OpAmp and
can be controlled by the passive Zi elements. This is a key advantage of using feedback with large
loop gain. Show how you could have obtained this transfer function by simply writing a KCL
node equation at the negative input terminal while assuming that negligible current flows into the
OpAmp terminals and also that the differential voltage at the input to the OpAmp is negligible
(i.e., in the above configuration, the negative terminal can be considered as a virtual ground with
zero voltage).
(d) The configuration we studied above is called an inverting amplifier due to the opposite sign relationship between input and output voltages. Now, consider the following configuration:
Using similar analysis and under the assumption of an ideal OpAmp along with an ideal closed-loop
gain, show how the transfer function Vo(s)
Vi(s)
can be obtained as:
Vo(s)
Vi(s)
=
Z1(s) + Z2(s)
Z1(s)
This configuration is called a non-inverting amplifier.
ECE141 – Principles of Feedback Control Homework 2 3
(e) Now that we have seen the basic OpAmp configurations, let’s place a parallel RC for Z1 and a
series RC for Z2, as shown below:
Again assume ideal OpAmp with very high open loop gain so that both the current flowing into the
OpAmp terminals as well as the differential voltage at the input to the OpAmp may be neglected.
Show that the Transfer Function Vo(s)
Vi(s) may be written in the following form, and find the constants
KP , KD, and KI in terms of R’s and C’s:
Vo(s)
Vi(s)
= KP + KDs +
KI
s
As we will see in future lectures, this is a simple basic ciruit realization of a so-called ProportionalIntegral-Derivative (PID) controller.
(f) Finally, let’s look at another configuration where a bit more complicated passive network forms
the feedback around our OpAmp, and see how the dynamics of the network may be represented
in State-Space form.
Under the same ideal assumptions as before, show that the dynamics of this circuit network may
be written in the State-Space form as:
dx
dt =
”
−
1
R1C1
−
1
RaC1
0
−
Rb
Ra
1
R2C2
−
1
R2C2
#
x +
1
R1C1
0
u, y =
0 1
x, (1)
where u = v1 is the input voltage, and y = v3 is the output voltage. (Hint: Use v2, i.e., the voltage
across the capacitor C1, and the output voltage v3 as your two state variables).
Problem 3. Mechanical System Modeling: The hanging crane structure supporting the Space
Shuttle Atlantis, along with its simple schematic representation are shown below, where M is the mass
of the cart, m is the mass of the payload, L is the length of the massless rigid connector, x(t) is the
cart displacement, Fb(t) = −bx˙(t) is the friction force, φ(t) is the connector angle with respect to the
vertical, and u(t) is the force applied to the cart.
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