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.
i.
What is the domestic price and quantity demanded o
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. Lads are 60% of the
population while Cads make up the rema
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 confirms that a single NMP zero may be transferred to more
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 isolation of nature areas (Verburg,
2006).Land-use change
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
Linearity
Particular type of nonlinearities: Constraints
Ant
(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 0.1
; = 2I22 (26.4.12)
2For
simplicity we consider the re
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 433 - State Space Control
11
21
What is the solutions y
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 positioning, observability, controllability, modal cont
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 decoupling is a strong requirement and is
generally not co
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 due to the
changes in the land cover patterns within the
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.
Assume that the controller transfer function matrix, C(s
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 land, as vegetation or man-made features." In
contrast, la
_
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 the general form R(s) = diagcfw_s + , (s + )2. For numeri
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 that describes the tendency of
water to accumulate and 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)
Express them as such,
x 2 = x1 (49)
x 1 = 7x2 2x1 + r (50
(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 = 2Ce + 2CeAe + CeAe
2
D
e = CeAeBe
The exact inverse th
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 agriculture (Hillel, 1997). This requires adequate quality
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 to give static decoupling for reference signals.
(iii)
(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 elements,s o we could expect that there will be additi
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
2. Permutations
3. The binomial coefficient
4. Cumulati
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.
i.
What is the domestic price and quantity demanded o
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 unstable here
_
_
834 Decoupling Chapter 26
We next apply
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 r(t) to y(t) is
846 Decoupling Chapter 26
given by
Hcl
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 references
of bandwidths less than,or equal to, 2[rad/s] and 4[
_
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 this case, a state space realization for [R(s)]1 is
A =
_
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 channel,at symmetrical levels 2.5. The results are sho
1
_
ln |[To(j)]rr|d(o, ) ln(|gir|) (26.6.7)
Then,t he result follows on using(24.7.7).
26.7 The Cost of Decoupling
We can now investigate the cost of dynamic decoupling,by comparing the results
in Chapter 24 (namely Lemma 24.4 on page 760) with those in c