SECTION 6: THE ROUTHHURWITZ STABILITY TEST
We showed in Section 5 that determining the ptability of a general interconnected system
boils down to finding the roots of the characteristic polynomial. (Recall that the
characteristic roots are found by looki
27
3.s MASON'S GAII\ RULE (MGR)
Motivation
Block diagram manipulation methods work fine for simple block diagrams, but it can get
confusing for complicated models. MGR allows us to systematically find a transfer
function without much effort. The steps to
11
3.3 LINEARIZATION
Motivation
We
will now talk about the impo rtantprocess of linearization. Ltnearrzation involves
finding a linear system whose behaviour approximates that of a nonlinear system. Now,
one cannot expect a linear system and a nonlinear s
SECTION 7: ROOT LOCUS METHODS
The closedloop poles tell us much about the behaviour of a control system. If both the
plant and controller are known, it is an easy matter of finding the closedloop poles
numerically fiust find the roots of the characteris
SECTION 8: NYQUIST PLOTS
Hopefully you can appreciate that root locus methods are quite powerful for control
systems analysis and design. We can use root locus plots to see the effect of changing
controller gain, or to see the effect of incorporating an e
ECE 4BBTutorial 1:
Reviewof Root LocusMethods
o Root locus techniquescan be used to study the motion of the closedlooppoles of a
control system as the value of a free parameter is varied over a specified range.
. It is useful to understand how the closed
21
8.3 QUANTIFYING STABILITY ROBUSTNESS
A major advarftage of Nyquist plot methods over RouthHurwitz methods is that we
can
determine not only whether or not a closedloop system is stable, but also how close the
system is to being unstable. Basically, t
SECTION 1: INTRODUCTION
In this brief section, we look at some basic idpas from control engineering and show in
what ways the new material in this course goes beyond what you already have learned in
ECE380 (or the equivalent, i.e., MTE360 or SYDE352).
Thi
t1
Erample
Find the zeros of the following systems (recall that
(a) G(s)= i16.
fbund the poles on page l0):
t2
L
G('r)=lt*'
(b)
s(s + l)
r,r,e
l6
L
(c)
G1s;
(s+1)
sl
sl
 (s+1)(s +2)l 6
I
t 2)
l
)l
4l
^'*1l
0
(s1)
(.s+lX,s+2)
(d) G(s) =
I
l
(s1)
t2
Manipulation
;l: Converting
from transfer function to statespace
Since statespace realizations are not unique, there is more than one way to get a statespace
rnodel tiom a given transfer function. Here we'll give a systematic method of deriving one
37
General Case
Let's summarize: we've shown that, if the statespace realization of a SISO plant is in CCF,
then we can place the closedloop poles anywhere we want in the complex plane. What do we
do if the statespace realization is not in CCF? One app
50
2.9 IIcfw_TRODUCTION TO LOOPSHAPING
2.9.1 Basic Loopshaping Idea
Consider the usual unity feedback control loop with an output disturbance and sensor noise:
d(t)
v(r)
n(r)
,
4r
\ k*J*'
vt^a I
ra
.
.(3)
Define the loop gain for this system to be
z(s)

58
2: Find
Step
a suitable Z(s)
The goal of Step 2 is to find a I(s) that will fit within the bounds shown on the plot at the
bottom of page 56; in addition to satis$ring the bounds, we must make sure that the unity
gain feedback loop (with loop gain equa
F"!*
f r ccfw_Ar Le*ur cp e'A I
nz (>* 0,t)" Sry cfw_r^r (Nflacfw_
7cfw_

1.3 EXTENDIcfw_G SISO TRANSFER FUNICTION METHODS:
35
MIMO NYQUIST
Sottretirnes the approach taken in Section 4.2. i.e., trying to break up the MIMO problem
into a collection
6
Weli, it appears that we've solved the control problem: simply use highgain control in
unity feedback loop, and you're guaranteed good tracking, good disturbance rejection,
and good sensitivity! Naturally, things are not as simple as thisthere arc atl
'
.
sk;rg
23
On the next page we give a convenient test for controllability, in terms of the socalled
controllability matrix, P. The proof requires the CayleyHamilton Theorem from linear
algrebra, which (you may recall from first year) states the follo
30
Theorem (observability tests)
There are two standard tests for observability:
'
Tlre system (C , A) is observable if and only if the socalled observability mutrix,
C
,
cfw_.r
v\
CA
Q_ CA2 I.
LA'
n
has rank
trtl
)\Jj l.r
\
1'
lcl> 
f.
'
* *h"s n
T4
2.4. TRANSIENT RESPOIcfw_SE CALCULATIONS
Steadystate calculations are easy and are always the same: simply apply the FVT. In
contrast, transient calculations are (in general) complicated. For example, suppose the
closedloop poles and zeros are as fol
20
1.2.2 Approach 2:
Decoupling control (Try to reduce crosschannel interactions)
The second approach applies to cases rvhere crosschannel interaction should not be
totally ignored. The idea is to design the controller in two stages:
Stage
l:
The first
ll
First. note that the theorem assumptions are satisfied:
o ft
+
L0
ol.
1_l
'
R:1
(A, B) is controllable since rank
(Q, A) is observable since rank
[f
ABf
f 0I  )
I oAl

2
.
L+J
Second. set up und
Substitute
inro
n,
r?*jEj\RE.'ro this end, ler P l:"
SECTION
3:
DESIGN FOR SIso SYSTEMS
PART 2: PERFORMANCE LIMITATIONS IN CONTROL
In this section we continue the study of SISO systems. and rve address
sonle very basic
"is it possible?" types of questions. There are
three particular obiectives:
r
flre
Dete
31
2.7 CLASSICAL CONTROLLER DESIGN: LEAD, LAG, PID
A. MultiStage Compensator
Design
Before reviewing lead, Iag, and PID compensation, it is useful to stand back and look at
compensation in general. (The term compensation is simply another word for contro
SECTION 2: DESIGN FOR SISO SYSTEMS
PART 1: CLASSICAL TECHNIQUES
In this section we restrict ourselves to SISO systems, just like in ECE3B0. Much of this
material is a review of key concepts and tools from ECE380, but there is also some new
material. Overa
7
What about more complicated feedback systemshow do we compute the characteristic
polynomial? Instead of going through the laborious process of considering a/ possible
closedloop transfer functions in order to find the characteristic polynomial, there
19
J.4.1 TimeDomain Performance Limitations
An tlndershoot Bound
lJsing the intuition developed on pages 4548, it's reasonable to expect that there is
alwaYs an undershoot if the plant has an ORHP zero. (The calculations on page 46
.iustify this claim
23
A Performance Limitation Due to IDOF Topology
Now that u,e have defined S and 7, we can easily state our first true (and very simple)
performance limitation. Specifically, note that
This equation implies that at any particular frequency t
and lr11ar)l
3.1.2. Youla Parameterization: General Case
We nou, relax the assumption that p(.s) is stable. The Youla parameterization result is the
conceptuall) exactlv the same as before. but the math details are significantly messier
now. Befbre we can even staGlEe
Finally. we turn to the topic of multiinput multioutput (MIMO) systems. The following
diagram summarizes the different design strategies we will consider:
MIMO Design Strategies
Break a MIMO problem into a
series of SISO problems and
Extend SISO transfe