Robert Shumar
December 9, 2008
ENME 403
Automatic Controls Project
Section 9.5 – Feedback Compensation
Cascading a compensator with a plant is not the only way to improve transient response
and stead-state response independently. Another way would be through transfer functions.
Figure 1: Feedback control system with minor loop
In the feedback control, displayed in Figure 1 [Feedback systems], there is a minor loop.
The minor loop, H(s), gives an output at a much fast rate of speed. If an engineer wanted
to get a value in the middle of a piece of code, he can insert a minor loop into a feedback
control system. When then system runs, the minor loop path will be taken and there will
be an output of the desired value. Signals that leave the compensator are at a higher level
of amplification than the receiving path, which has a low level. When dealing with a
single-input-single-output system, no theoretical advantages are obtained when using a
minor loop. They would only be inserted for obtaining specific information at any point
in time within a code. When multiple inputs and outputs are introduced, minor loops
become very important. There may be certain minor aspects in a system that are very
important and need specific attention to make sure that they are running properly and
contributing to the system as a whole [Dorf].
Another type of feedback compensator is one that deals with the design of feedback
compensation. There are two approaches that can be made. The first approach involves
rearranging the different functions so that there is one G(s) and one H(s). The approach
replaces poles and zeros by H(s) to reshape the root to pass through the design point. The
zeros dealing with H(s) are not closed-loop zeros. An example of this would start in
Figure 2.
Figure 2: Minor loop feedback system
To apply the first approach to this system, the functions need to be joined together to
1
form a G(s) and an H(s). To do this, you would multiply K to
to get your
s( s  7)( s  10)
new G(s) and you would add K to Kfs to get your new H(s). After completing these two
steps, the necessary information may be acquired. The second approach is basically the
opposite of the first. Instead of making the feedback system into two functions, the two
functions are split up to make a feedback-compensated system. For example, Figure two
would be the end result of the second approach. The difference is that the poles change
through a gain adjustment.
Section 9.6 – Physical Realization of Compensation
The equation
Vo ( s)
Z ( s)
 2
Vi ( s)
Z1 ( s)
(1)
derived back in Chapter 2 is used again dealing with compensators and controllers.
Active realization functions can be found using this equation such as gain, integration,
differentiation, PI, PD, and PID controllers, and lag and lead compensation. There also
exists passive realizations of compensators such as lag compensation, lead compensation,
and lag-lead compensation. These passive realizations can function as either passive or
active. In order to create a lag-lead compensation, the lag compensation and the lead
compensation are combined together. In order for the transfer function to be the product
of the individual transfer function, there needs to be a point of isolation between the lag
and the lead compensation. To create the isolation, operational amplifiers are put in
between the two. An example of a lag-lead compensator for an active and passive
realization can be seen in Figure 3.
Figure 3: (Top) Active lag-lead compensator where left half is lag and the right half
is lead
(Bottom) Passive lag-lead compensator where left section is lag,
right section is lead and the middle is the isolation section.
Section 10.1 – Introduction
This chapter will be talking about another way of looking at feedback control systems
through the use of frequency response. Four key advantages of the frequency response
over the other methods of feedback are as follows:
1. When modeling transfer functions from physical data
2. When designing lead compensators to meet a stead-state error requirement and a
transient response requirement
3. When finding the stability or nonlinear systems
4. In settling ambiguities when sketching a root locus
Before further discussion about frequency response, sinusoids must be understood.
Sinusoid may also be called phasors. The notation for a sinusoid looks like M1∟Φ1,
which is also known as the frequency response. The M is the amplitude of the sinusoid
and the Φ is the phase angle of the sinusoid. When a sinusoid is input into a system, the
output will result in a sinusoid. In order to calculate the output of a system where
sinusoids are present, the input, Mi, is multiplied by the system magnitude, M, and the
phase angles are added together.
When given a Laplace transform such as r(t) = Acos(ωt) + Bsin(ωt) =
B
A2  B 2 cos(t  tan 1  ) , there are three ways that it can be represented. First is in
 A
polar form, Mi∟Φi, then rectangular form, A-jB, and finally in Euler’s formula, MiejΦi.
After displaying the equation in one of the three forms, the forced response can be found.
To simplify the equation partial fractions can be formed. Within the partial fractions, the
value that comes from the poles of the waveform is the stead-response. When the transfer
function of a frequency response is G(s) the following expression may be used.
G(jω) = G(s)|s→jω
(2)
The next step is learning how to plot G(jω) = MG (ω) < φG (ω). There are two ways in
which this expression can be plotted. The first way is as a function of frequency with
separate magnitude and phase plots where its x and y axis are labeled as frequency (rad/s)
and 20logM (decibels) respectively. The second way to plot the expression is as a polar
plot where the phasor length is the magnitude and the phasor angle is the phase. The x
and y axis for this plot are frequency (rad/s) and phase (degrees). Also, jω1 can be
plugged into G(s) for s. In these situations, the magnitude can be calculated from
Product of vector lengths from zeros of G(s)
Magnitude = Product of the vector lengths form the poles of G(s) (3)
whereas the phase response can be calculated from
Phase reponse = Sum of angles from zeros of G(s) − Sum of angles from poles of G(s)
Examples of both plots are shown below in Figure 4 [Controls Tutorial].
(4)
Figure 4: Two different examples of plotting G(jw) where the top graph is a function
of frequency and the bottom graph is expressed in a polar plot.
Section 10.2 – Asymptotic Approximations: Bode Plots
Bode plot are sketched for simplicity because of straight-line approximations. Some
important definitions are as follows:




Straight-line approximations – Asymptotes
Low-frequency approximations – Low-frequency asymptote
High-frequency approximations – High-frequency asymptote
Frequency “a” – Break frequency
The frequency “a” is called the breaking frequency because it separates the high and low
frequencies. In order to draw the phase response, it is known that at low frequencies, the
phase is at 0 degrees and at higher frequencies, the phase angle is 90 degrees. Zero phase
can be drawn to a magnitude of 10 below “a” in the first graph then start to increase in a
straight line with an angle of 45 degrees up until a magnitude of 10 above “a” in the first
1
graph. For bode plots consisting of G(s) = (s+a) , the function has a low-frequency
asymptote, where s is allowed to approach zero. Because the dBs equal −20 log ω, the
graph will decrease by a factor of 20 db/decade after the break frequency. This forces the
phase graph to end at -90 degrees at its high-frequency. For Bode plots that have a
function of G(s) = s, the magnitude of 20logω will stay constant making the phase graph
stay constant at +90 degrees. For Bode plots with functions of G(s) = 1/s, the magnitude
is a constant -20logω making the phase graph a constant -90 degrees. Bode plots can also
take form in second-order systems. An example would be G(s) = s2 + 2ζωn s + ω2n . The
function is found to have twice the slope of the first-order magnitude. Since ω=ωn, it is
proven that ωn is the breaking frequency for second-order systems. To get a relationship
between the actual values and the values calculated, equations for the magnitude and
phase angles are used. These equations are shown below.
M = √(ω2n − ω2 )2 + (2ζωn ω)2
2ζω ω
n
Phase = tan−1 ω2 −ω
2
(5)
(6)
n
The last type of second-order Bode plot described in the book is
1
G(s) = (s2 +2ζω
2
n s+ωn )
(7)
where the values for the magnitude and slope are the same as the first second-order
system with the exception that they are negative.
Section 10.3 – Introduction to the Nyquist Criterion
This criterion relates the stability of a closed-loop system to the open-loop frequency
response and open-loop pole location. Throughout the derivation of this criterion, there
are four concepts that should be looked over, they include:
1. The relationship between the poles of 1+G(s)H(s) and the poles of G(s)H(s)
2. The relationship between the zeros of 1+G(s)H(s) and the poles of the closed-loop
transfer function
3. The concept of mapping points
4. The concept of mapping contours
Mapping is when there is a substitution of the complex number, which is on the s-plane,
into the function F(s). An example of this is when a pre-determined value is plugged in
for a variable, for example, s. The contours have their own set of rules that need to be
followed, they include [Contours]:
1.
2.
3.
4.
The contour must close
The contour may not cross directly through a pole of the system
Contours must have a direction, either clockwise or counter clockwise
A contour is called "simple" if it has no self-intersections
An example of a contour is shown below in Figure 5 [Kashani].
Figure 5: A contour in the s-plane where its roots are in the clockwise direction and its poles are in the
counterclockwise direction.
References
"Contours." Control Systems/Nyquist Stability Criteria. 28 Nov. 2008. Wiki Books. 1
Dec. 2008
<http://en.wikibooks.org/wiki/control_systems/nyquist_stability_criteria>.
Dorf, Richard C., ed. The Electrical Engineering Handbook. New York: CRC P, 1997.
"6.302 Feedback Systems." Department of Electrical Engineering and Computer Science.
Fall 2002. MASSACHUSETTS INSTITUTE OF TECHNOLOGY. 30 Nov. 2008
<http://mit.nelc.edu.eg/nr/rdonlyres/electrical-engineering-and-computerscience/6-302feedback-systemsfall2002/7f84a8bb-9449-402c-a03025608c4e3710/0/ps8.pdf>.
Kashani, Reza. "THE NYQUIST STABILITY CRITERION." 1 Dec. 2008
<http://www.engr.udayton.edu/faculty/rkashani/mee527/nyq_stab/nyq_stability.ht
m>.
Nise, Norman S. Control Systems Engineering. New York: John Wiley & Sons,
Incorporated, 2007.
O., L. J. "Sample Bode Plots." Control Tutorials for Matlab. 27 Aug. 1996. Carnegie
Mellon. 1 Dec. 2008 <http://www.engin.umich.edu/group/ctm/freq/dfr.html>.