SESS3011
M 9:00 11:00, 11:00 13:00
Autumn 2015
Birkbeck Malet Street 254, 253
Homework #2 Answers
Due Thursday 9th October
1. Consider a marriage market where men come in two types: cads and lads. Lad
EC 101.03
Exercises for Chapter 9
FALL 2010
1. Using the graph, assume that the government imposes a $1 tariff on hammers. Answer the following
questions given this information.
a.
b.
c.
d.
e.
f.
g.
h
M11(s)
1 +M11(s)
0
0 M22(s)
1+M22(s)
(26.5.12)
from where it follows that the loop will not be internally stable unless both M11(s)
and M22(s) vanish at s = zo.
We then see that the NMP zero must app
simultaneously satisfied.
It is evident that had we chosen a lower triangular coupling structure,t hen the
NMP zero would have appeared in channel 2,bu t not necessarily in channel 1.
This example con
1.0 INTRODUCTION
1.1 Background of study
Land-use and land-cover change (LULCC) can be major threat to biodiversity as a result of the
destruction of the natural vegetation and the fragmentation or is
Autonomous
Non-Autonomous states
input
output
Linear Control
Output/State Feedback
Estimation: How to estimate states from input/output?
Keep the nonlinearities
6
ME 433 - State Space Control 11
Linea
(t) = Ax(t) + Bu(t) (26.4.10)
where
A=
_
Ao 0
Co 0
_
;B=
_
Bo
0
_
(26.4.11)
We next consider the composite system and design a state controller via LQR
theory.
We choose
=
Co
TC o 0 0
0 0.005 0
0 0 0.
transform
Time domain (t domain)
Complex frequency domain
(s domain)
Laplace Transform
ME 433 - State Space Control
20
Laplace Transform
Characteristic Equation
Transfer Function
Transfer Function
ME
dynamic decoupling for stable minimum phase systems
dynamic decoupling for stable non-minimum phase systems
dynamic decoupling for open loop unstable systems.
As might be expected, full dynamic dec
2.3
Land use impacts on water
Water quality and quantity in many regions of the world has been severely degraded and
endangered by increasing human demands Ojma et al., 1991.The water degradation is d
26.9 MIMO Anti Wind-Up Mechanism
In Chapter 11 the wind-up problems were dealt with using a particular implementation
of the controller. This idea can be easily extended to the MIMO case as
follows.
A
1.1
LITERATURE REVIEW
1.2
Land-Use and Land Cover Change
The terms "land cover" and "land use" are often used exchangeably. Land cover is "the observed
physical and biological cover of the earth's lan
_
s + 2 3
2 1
_
(26.2.33)
This model has a NMP zero at s = 4. Then,t o synthesize a controller following
the ideas presented above,we first compute a right interactor matrix which turns out
to have th
the water table, which determines where saturated land-surface areas develop and have the
potential to produce saturation overland flow.
TOPMODEL uses topographic information in the form of an index t
_
+
_
01
_
r (45)
y = [3 1]
_
x1
x2
_
(46)
Reverse the order of the state variables appeared previously in PVF to obtain,
_
x 2
x 1
_
=
_
01
7 2
_
x2
x1
_
+
_
01
_
r (47)
y = [3 1]
_
x2
x1
_
(48)
Ex
(26.4.16)
An interactor for this closed loop system is
L(s) =
_
(s + )2 0
0 (s + )2
_
; = 0.03
This leads to an augmented system having state space model (A
e,B
e,C
e,D
e)
with
A
e = Ae
B
e = Be
C
e =
Food shortage is a challenge in Kenya and more so in light of the high rate of population growth.
This can be greatly reduced by among other strategies increasing food production through
irrigated agr
ME 433 - State Space Control 3
State Space Control
State-space methods of feedback control system design and design
optimization for invariant and time-varying deterministic, continuous
systems; pole
where
840 Decoupling Chapter 26
GoN(s) =
_
5 s2
1 0.0023
_
; GoD(s) =
_
25s+ 1 0
0 s(s + 1)2
_
(26.4.3)
(i) Convert to state space form and evaluate the zeros.
(ii) Design a pre-stabilizing controller
(26.7.5)
856 Decoupling Chapter 26
from where it can be seen that the dimension of the null space is z = 1 and
the (only) associated (left) direction is hT = [5 6]. Clearly this vector has
two nonzero
Discrete Random Variables
1
1. Random Variables
2. Expectations
1. Linearity of expectations
1. Additivity of means
2. Additivity of independent variances
3. Binomial Distribution
1. Bernoulli Trials
EC 101.03
Exercises for Chapter 9
FALL 2010
1. Using the graph, assume that the government imposes a $1 tariff on hammers. Answer the following
questions given this information.
a.
b.
c.
d.
e.
f.
g.
h
u(t). Thus (Ai,Bi,Ci,Di) is a minimal realization of (A,Bei,C,Dei), where
ei is the ith column of the m m identity matrix.
1Note
that the NMP zeros of Go(s) are eigenvalues of A , i.e. A is assumed un
CD(s)1CN(s)
GoD(s)(s)
+
+
r(t) +
GoN(s)(s)
+
Y (s)
GoD(s)
r(t)
+
Figure 26.9. Q parameterization with two d.o.f for unstable MIMO plants
It can readily be shown that the nominal transfer function from
2(s + 1) 1
(s + 1)2 (s + 1)(s + 2)
(26.2.10)
Choose a suitable matrix Q(s) to control this plant,u sing the affine parameterization,
in such a way that the MIMO control loop is able to track referenc
_
s + 10 120
s + 10 40(s + 2)
_
(26.3.5)
If we repeat the procedure in Example 26.2,bu t this time for GTo
(s),we have that
[R(s)]1 = [GTo
(s)R(s)]1 =
1
s4
_
s + 1 2(s + 1)
3s+2
_
(26.3.6)
and,i n thi
2
Time [s]
Plant outputs and ref.
y
1
(t)
y
2
(t)
Figure 26.15. Decoupled design in the absence of saturation
(b)
We run a second simulation including saturation for the controller output in the
first