Examen de Sistemas Autom´aticos. Convocatoria de Julio. 19/07/2010

Examen de Sistemas Automáticos. Convocatoria de Julio. 19/07/2010
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Cuestión. Las bicicletas muestran un interesante comportamiento dinámico que ha intrigado a los
cientı́ficos desde su aparición en la mitad del siglo XIX. Por ejemplo, las bicicletas son estáticamente
inestables (se caen, como un péndulo invertido), pero bajo ciertas condiciones son estables durante la
marcha. Un modelo muy simplificado de la estabilidad vertical de la bicicleta es el siguiente:
the forces acting between ground and wheel. Since we do
2 tight turns, we
not consider extreme conditions and
assume that the bicycle tire rolls without longitudinal or
2
lateral slippage. Control of acceleration and braking is not
considered explicitly, but we often assume that the forward
velocity is constant. To summarize, we simply assume that
the bicycle moves on a horizontal plane and that the
wheels always maintain contact with the ground.
Geometry
2
TheaV
parameters
the geometry of a bicycle
d ϕ(t) g
dδ(t) that Vdescribe
− ϕ(t) =
+1. The δ(t)
are defined in Figure
key parameters are wheelbase b,
dt
h
bh
dt
bh
head angle λ, and trail c. The front fork is angled and
so that the contact point of the front wheel with
donde g es la aceleración de la gravedad, V es la shaped
velocidad
de marcha de la bicicleta, ϕ(t) es el ángulo
the road is behind the extension of the steer axis. Trail is
as the horizontal
distance
the contact
de inclinación de la bicicleta, δ(t) es el giro deldefined
manillar
y el resto
dec between
los parámetros
geométricos
point and the steer axis when the bicycle is upright with
pueden verse en las figuras adjuntas:
zero steer angle. The riding properties of the bicycle are
h
C1
P1
left with the roll angle ϕ as the only degree of
freedom.affected
All
strongly
by the trail. In particular, a large trail
angles are assumed to be small so that the equations can
improves stability but
y makes steering less agile. Typical z
be linearized. λ
values for c range 0.03–0.08 m.
Top and rear views of the bicycle are shown in Figure 3.
O
Geometrically,
it is convenient to view the bicycle as
The coordinate system xyz rotates around the vertical
axis
of two hinged planes, the frame plane and the
with the angular velocity ω = Vδ/b, where bcomposed
is the wheel
frontxyz
fork
plane. The frame and the rear wheel lie in the
base. An observer fixed to the coordinate system
expeframe
plane, while the front wheel lies in the front fork ϕ
riences forces due to the acceleration of the
coordinate
system relative to inertial space.
plane. The planes are joined at the steer axis. The points
Let m be the total mass C
of2 the system. PConsider
the
1 and P2 are the contact points of the wheels with the
rigid body obtained when the wheels, the rider,
and the
horizontal
plane, and the point P3 is the intersection of the
front fork assembly are fixed to the rear frame with δ = 0,
steer axis with the horizontal plane (Figure 1).
let J denote the moment of inertia of this body with
δ
respect to the x-axis, and
P2let D
P3= − Jxz denote the inertia
Coordinates
x
y
product with respect to the xz axes. Furthermore,
let the x
P1
P2
used ato analyze the system, which foland za coordinates of the center of mass be a The
and h,coordinates
respeclow
the
ISO
8855
standard,
are
defined
in
Figure
2.
There
tively. The angular momentum of the system with respect
b
c
is an inertial system with axes ξ ηζ and origin O. The
to the x axisb is [62]
(a)
(b)
coordinate system xyz has its origin
at the contact point
dϕ
dϕ
VD P of the rear wheel and the horizontal plane. The x axis
1
− Dω = J
Figure 3. Schematic (a) top and (b) rear views of a naive
Figure 1. Parameters defining the bicycle geometry.
Thedt − b δ.
dt
is aligned with (λ
the
line
of contact
of angle
the rear
plane
= 0)
bicycle.
The steer
is δ, and
the with
roll angle is ϕ.
points P1 and P2 are the contact points of the wheels with the
Lx = J
(20 %) 1. Obtener la función de transferencia que describe
la inclinación
de also
la bicicleta
the horizontal
plane. The x axis
goes throughϕ(t)
the en función del
of the
axis with
the
ground, the point P3 is the
Theintersection
torques acting
onsteer
the system
are
due to gravity and
P3 , which
point
is the intersection between the steer
ángulo
del
manillar
δ(t),
para
una
velocidad
dada
V
.
centrifugal
forces,
and
the
angular
momentum
balance
horizontal plane, a is the distance from a vertical line through
the center of mass to P1becomes
, b is the wheel base, c is the trail, h is
axis and the horizontal
plane.
Thethe
orientation
of thefrom steer
It follows from
(1) that
transfer function
rear wheel plane is defined by the angle ψ , which is the
angle δ to tilt angle ϕ is
the height of the
centersi
of mass,
and λ is the head
angle.
(20 %) 2. Demostrar
que,
la bicicleta
circula
amVla
velocidad
conandelthemanillar
(valor de
2hangle
x-axis. The zbloqueado
between theVξ -axis
axis is
d2 ϕ
DV dδ
J 2 − mghϕ =
+
δ.
(1)
b vertical, and y is perpendicular to x and positive on the
dt
δ(t) = cte) la bicicleta es inestable
yb dtse cae.
V(Ds + mVh)
ζ
= a right-hand
left side of the bicycleGso
that
system is
ϕδ (s)
2
The term mghϕ is the torque generated by gravity. The
b( Js − mgh)
obtained. The roll angle ϕ of the rear frame is positive
ϕf
mVh
V
(50 %) 3. Si se libera el manillar
ythe
seright-hand
hace circular
dethenuevo
la
a la velocidad
Vfront
,s +puede
comprobarse
terms on
side of (1) are
torques
gen-bicicleta
s + of theaV
when
leaning
to the right. The roll
fork
VD angle
D ≈
a .
η with the first term due to inertial
erated by steering,
(4)
g
ϕf . The steergira
δ=
plane
ismanillar
angle ligeramente,
is bthe
of
intersection
mgh
que cuando la bicicleta
se
inclina
ligeramente,
el
por
gravedad,
hacia el
z
J angle
bh
2
s −
forces and the second term due to centrifugal forces.
s2 −
ϕ
h
between the rear and front planes, positive
when steerJ
The model la
is called
the inverted
pendulum
modelsu inclinación estabilizándola. La bicicleta se
lado en el que se inclinó
bicicleta,
lo que
corrige
left. The
because of the similarity with the linearizeding
equation
for effective steer angle δf is the angle between
lines
of intersection
of
thethe
rear
and
planes
with
mantiene sin caerse
variospendulum.
metros hasta quethesu
velocidad
disminuye
yfront
finalmente
cae.
the inverted
Notice that
both
gain
and
the
zero
of this
transfer
func-De forma
J ≈horizontal
mh2 and plane.
Approximating the moment of inertia asthe
tion depend on the velocity V.
aproximada, este the
comportamiento
del
manillar
puede
describirse
como
un
giro
proporcional
a la
C2 D ≈ mah,δfthe model becomes
inertia product as
The model (4) is unstable and thus cannot explain why
x
C1
it is possible to rideModels
with no hands. The system (4), howevSimple Second-Order
inclinación
er, can will
be stabilized
by activebased
controlonusing
the proporSecond-order models
now be derived
addid2Pϕ2 P3g
aV dδ
V2
= −k
δ. tional
tionalassumptions.
feedback law It is assumed that the
2 ϕ(t)
P1
simplifying
ψ dt 2 − h ϕ = bh dt + bhδ(t)
bicycle rolls on the horizontal plane, that the rider has
Esta relación convierte a la bicicleta enξ un fixed
sistema
δ =to
−kthe
positionrealimentado.
and orientation relative
2 ϕ, frame, and
The model (1), used in [37] and [21], is a linear dynamical
system of second order with two real poles that the forward velocity at the rear wheel V is constant.
For simplicity, which
we assume
that
the steersystem
axis is vertical,
yields the
closed-loop
Figure 2. Coordinate systems. The orthogonal system ξ ηζ is
! orthogo!
which implies that the head angle λ is 90◦ and that the
fixed to inertial space, and the ζ -axis is vertical. The
mgh
g
g
V
= ±of the rear
p1,2
≈±
(2) We also assume that the steer angle δ is the
trail c is zero.
nal system xyz has its origin at the contact
point
"
#
J
h
d2aϕ
DVk2 dδ h mV 2hk2
control variable. The rotational
associwheel with the ξ η plane. The x axis passes through the points
J 2 + degree+of freedom
− mgh
ϕ = 0.
b dt
dt disappears,
P1 and P3 , while the z axis is vertical and passes through P1 .
ated with the front fork then
andbthe system is
2
and one zero
(5)
a) Obtener el diagrama de bloques, indicando las variables y sistemas.
�
b) Trazar el lugar de las raı́ces del sistema, suponiendo | | > |
|.
(6)
c) Asumiendo un valor fijo de k , demostrar que la bicicleta es estable mientras k2 > bg/V 2 ,
lo que implica
cae cuando la velocidad llega a ser menor a una velocidad
�que la bicicleta
mVh Control
V
This closed-loop system is asymptotically
28
Auguststable
2005 if and only
z = − IEEE ≈
− . Systems Magazine
(3)
bg
if k2 > bg/V 2 , which is the case when V is sufficiently large.
D
a
crı́tica Vcr = k2 .
Authorized licensed use limited to: UNIVERSIDAD DE OVIEDO. Downloaded on July 15,2010 at 15:41:28 UTC from IEEE Xplore. Restrictions apply.
IEEE Control
Systemsque
Magazine
August 2005 geometrı́as de bicicleta
29
(10 %) 4. Se han sugerido otras
en las
el manillar está detrás (con
un giro
Authorized
licensed
use
limited
to:
UNIVERSIDAD
DE
OVIEDO.
Downloaded
on
July
15,2010
at
15:41:28
UTC
from
IEEE
Xplore.
Restrictions
apply.
similar al de las carretillas elevadoras). El mismo modelo anterior permite describir este tipo de
bicicleta cambiando el signo de la velocidad V < 0 (es decir, suponiendo que va hacia atrás). El
departamento de Transporte de los EE.UU. propuso la construcción de una motocicleta “segura”
con esta configuración. Resultó ser muy segura, pero de una forma insospechada: era tan inestable
que nadie podı́a montarla. Dar una justificación a este hecho.
Fuente: K.J. Aström, R.E. Klein y A. Lennartson. ”Bicycle Dynamics and Control”. IEEE Control Systems Magazine,
Ago. 2005, pp. 26–47. El documento completo puede descargarse de
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1499389 accediendo desde el dominio uniovi.es.
1
Nombre, Apellidos: ____________________________________________ Julio 2010
D.N.I.: _____________________
Sistemas Automáticos
La figura representa el diagrama de Bode de la función de transferencia en cadena
abierta G(s), de un sistema de realimentación unitaria. Se pide:
Ki
de forma que el sistema
s
en cadena cerrada presente un error ev!0.05 ante una rampa unitaria y un
MF!45º. (6 puntos)
2. Dibujar el diagrama de Bode del controlador diseñado. (2 puntos)
3. Obtener el algoritmo de control (ecuación en diferencias) utilizando la
discretización del controlador diseñado, tomando como transformación
1 ! z !1
s=
, siendo T el periodo de muestreo. (2 puntos)
T
1. Diseñar el regulador PI definido por D( s ) = K p +
Julio
Sistemas Automáticos
19 Julio 2010
4 Ingenierı́a Industrial
Nombre y apellidos:
Responda razonadamente y con letra clara a las cuestiones propuestas, utilizando el
espacio proporcionado.
No está permitido el uso de calculadora.
1. (2.5 pt.) Dado el sistema:
U (s)
U ∗ (s)
= e−0,0314s
∠U(s)/U*(s) (°)
|U(s)/U*(s)| (dB)
a. Dibuje su diagrama de Bode en el recuadro adjunto.
10
10
10
10
10
Frecuencia (rad/s)
b. Si a este sistema se le suministra como entrada la señal de la figura u∗ (t), represente de
forma aproximada, en la misma gráfica, cómo serı́a la respuesta u(t) en régimen permanente.
2
1.5
1
u*(t)
0.5
0
−0.5
−1
−1.5
−2
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
tiempo (s)
c. Exponga al menos tres ejemplos de sistemas que incorporen una función de transferencia
similar a la enunciada.
El ejercicio continúa en la página siguiente.
Página 1 de 2
Sistemas Automáticos
Julio (cont.)
4 Ingenierı́a Industrial
2. (2.5 pt.) (a) Explique en qué consiste el problema del windup.
(b) Esquematice en un diagrama de bloques una posible solución.
3. (2.5 pt.) Explique, de forma razonada, qué ocurre si el ancho de banda del sensor realmente
utilizado para medir la variable a controlar no es mucho mayor que el ancho de banda del
sistema controlado, para cuyo diseño se utilizó un sensor ideal.
4. (2.5 pt.) Una técnica para sintonizar reguladores es el modelado de la función de lazo.
Explique las relaciones de la función de lazo con el sistema realimentado que permiten
dicho procedimiento.
Fin del ejercicio. No olvide escribir su nombre y apellidos en todas las hojas antes de entregar.
Página 2 de 2