Accounting I Final Question and Answers: Which of the following is not a step in providing accounting information to stakeholders? prepare accounting surveys Equipment with an estimated market value of $45,000 is offered for sale at $65,000. The equipment
e
z0ty(t)dt
(4.8.22)
Clearly for z0 = 3,we have
I(3) =
_
0
e
3te2tdt
0
_
=
e
tdt
= 1 (4.8.23)
Notice that this is correctly predicted by Lemma 4.1 since
Y (3) =
1
32
= 1 (4.8.24)
However,if we take z0 = 1,th en Y (1) = 1. Yet clearly I(z0) is . This is in
21.2 Completely Decentralized Control 631
21.3 Pairing of Inputs and Outputs 635
21.4 Robustness Issues in Decentralized Control 638
21.5 Feedforward Action in Decentralized Control 641
21.6 Converting MIMO Problems to SISO Problems 643
21.7 Industrial Ca
1
2t 1
4
(4.4.6)
Laplace transforms are useful in the study of linear differential systems since
they convert differential equations to algebraic equations.
4.5 Transfer Functions
4.5.1 High Order Differential Equation Models
Consider again the linear hig
fashion if the systems were to operate safely and efficiently. A major development
at this time was Watts fly ball governor. This device regulated the speed of a
steam engine by throttling the flow of steam, see Figure 1.1. These devices remain
in service
feedback. Nevertheless, the feedback interaction is seen to have a profound impact
on the behavior of the engaged systems.
This behavior altering effect of feedback is a key mechanism that control engineers
exploit deliberately to achieve the objective of
do not affect the steady-state
response
dt
de t
E s K s U s u t Kd d
()
( ) ( )_ ( )
MCEN 467 Control Systems
Effect of change for gain
PD controller
Increase in gain:
Upgrade transient
response
Decrease the peak and
rise time
Increase overshoot
and set
Ex (contd): Open-loop
step response
1/20=0.05 is the final value
of the output to an unit step
input.
This corresponds to a
steady-state error of 95%,
quite large!
The settling time is about
1.5 sec.
MCEN 467 Control Systems
Ex (contd): Proportional
Co
u=uQ
xQ f
u
_
x=xQ
u=uQ
uQ (3.10.12)
F = g(xQ, uQ) g
x
_
x=xQ
u=uQ
xQ g
u
_
x=xQ
u=uQ
uQ (3.10.13)
In general, A, B, C, D, E and F, will depend on time. However, in the case
when we linearize about an equilibrium point, then they will be time-invariant.
I
Closed-loop Response
Small
change
Small Decrease Decrease
change
D
I Decrease Increase Increase
Eliminate
Small Decrease
change
P Decrease Increase
Steadystate
error
Settling
time
Maximum
overshoot
Rise time
Note that these correlations may not
be exactl
The transfer function of a pure delay is of the form (see Table 4.2):
Section 4.5. Transfer Functions 73
H(s) = e
sTd (4.5.16)
where Td is the delay (in seconds). Td will typically vary depending on the transportation
speed.
Example 4.4 (Heating system).
The remainder of the book is devoted to developing systematic answers to these
and other related questions.
2.4 Definition of the Problem
The example presented in section 2.3 motivates the following more formal statement
of the nature of the control probl
possible. This approach is connected to our assumption that readers will have ready
access to modern computational facilities including the software package MATLABSIMULINK.
This assumption allows us to put the emphasis on fundamental ideas
rather than on
taking a holistic viewpoint. Some of the issues that are embodied in a typical control
design include:
plant, i.e. the process to be controlled
objectives
sensors
actuators
communications
12 The Excitement of Control Engineering Chapter 1
computing
Multiplicative modelling error
Figure 3.3. AME and MME due to saturation
Of course, the exact model errors are rarely known since the true plant is not
precisely known. However, certain information about the size of errors may still
be available. This wil
B.5 Poles and Zeros
B.6 Matrix Fraction Descriptions (MFD)
C RESULTS FROM ANALYTIC FUNCTION THEORY
C.1 Introduction
C.2 Independence of path
C.3 Simply connected domains
C.4 Functions of a Complex Variable
C.5 Derivatives and Differentials
C.6 Analytic Fu
all signals to a common point. Obvious objections to this include complexity, cost,
time constraints in computation, maintainability, reliability, etc.
Thus one usually partitions the control problem into manageable sub-systems.
How one does this is part
One of the things that makes control science interesting is that all real life systems
are acted on by noise and external disturbances. These factors can have a significant
impact on the performance of the system. As a simple example, aircraft are subject
Control design aims to achieve a desired level of performance in the face of
disturbances and uncertainty
Section 1.7. Further Reading 19
Examples of disturbances and uncertainty include
System Actuators Sensors Disturbances Uncertainties
Aircraft Throt
computing a corrective action to bring the actual system to the desired
state
applying the corrective action to the system via actuators
repeating the above steps.
The principal components in a feedback loop are shown in Figure 2.12.
Referenceof
outpu
settling time, and had
small effect on the rise
time and the steady-state
error.
300, 10 p d K K
MCEN 467 Control Systems
Ex (contd): PI Controller
The closed loop transfer function is
given by:
p
p
p
p
i
i
i
i
ssKsK
KsK
ss
KKs
ss
KKs
Fs
Xs
10 (20 )
10 2
This idea is illustrated in Figure 2.6
r
+
f1_
u
f_
+
+
y
d
Conceptual controller Plant
Figure 2.6. Conceptual controller
This is a conceptual solution to the problem. However a little thought indicates
that the answer given in (2.5.3) presupposes certain
Upgrade both steadystate
and transient
responses
Reduce steady-state
error
Reduce stability!
MCEN 467 Control Systems
P Controller with high
gain
MCEN 467 Control Systems
Integral Controller
Integral of error with a
constant gain
increase the system type
the proportional controller
reduced both the rise time
and the steady-state error,
increased the overshoot, and
decreased the settling time
by small amount.
300 p K
MCEN 467 Control Systems
Ex (contd): PD Controller
The closed loop transfer function is
g
presented, we will use state space models frequently since it simplifies much of the
presentation.
3.8 High Order Differential and Difference Equation Models
An alternative model format that is frequently used is that of a high order differential
equation
1 + 2 (4.8.5)
This system has two complex conjugate poles, s1 and s2, which are given by
s1,2 = n jd = ne
j() (4.8.6)
where is the angle such that cos = .
For this system, the Laplace transform of its unit step response is given by
Y (s) = 2n
(s2 + 2ns +
u = h_r z
Thus
= h_r f_u
(2.6.1)
Section 2.6. High Gain Feedback and Inversion
33
z
+
uy
Plant
r
f_
h_
Figure 2.7. Realization of conceptual controller
h
1_u
= r f_u
(2.6.2)
from which we finally obtain
u=f
1_r h
1_u
(2.6.3)
Equation (2.6.3) suggests that
(double effect).
The above response shows that the
integral controller eliminated the
steady-state error.
30, 70 p i K K
MCEN 467 Control Systems
Ex (contd): PID
Controller
The closed loop transfer function is
given by:
d
d
p
p
p
p
d
d
i
i
i
i
sKsKsK
Ks
until the dynamic behavior of model and plant match sufficiently well.
An alternative approach for dealing with the modeling problem is to use physical
laws (such as conservation of mass, energy and momentum) to construct the
model. In this approach one u