MIT OpenCourseWare
http://ocw.mit.edu
2.004 Dynamics and Control II
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.004 Dynamics and Control II
Spring Term 2008
Lecture 11
Reading:
• Nise: Chapter 1
1 Elements of the Course
(a) System Dynamics: System dynamics provides a unified approach to the modeling
and dynamic behavior of linear systems in many energy domains. For example:
a) An electrical network:
R
V
in
(t)
+
v (t)
C
-
U s in g K V L & K C L :
R C
d v
d t
+
v
= V
in
(t)
b) A simple mechanical system:
fo rc e
F (t)
v (t)
v e lo c ity
m a s s
m
m
F ro m
d v
B d t +
a fo r c e b a la n c e :
v
1
= B F (t)
B ( v i s c of u s f r i c t i o n )
c) A fluidic system:
p u m p p re s s u re
Ps
g
p
C
f
flu id c a p a c ita n c e
R
flu id r e s is ta n c e
f
p re s s u re a t
b o tto m o f ta n k
1
o u tp u t flo w
c D.Rowell 2008
copyright �
1–1
U s in g flu id ju n c tio n e q u a tio n s
R fC
f
d p
d t
+
p
= P (t)
We recognize a common form to the ODE describing each system and create analogs
in the various energy domains, for example
⎧
⎫
⎧
⎫
⎧
⎫
⎨ voltage
⎬
⎨ current
⎬
⎨ electrical
⎬
velocity
force
mechanical
and
in the
domains.
⎩
⎭
⎩
⎭
⎩
⎭
pressure
volume flow rate
fluidic
Then for each system, we can write a differential equation:
τ
dy
+ y = u(t)
dt
where u(t) is the system input
y(t) is the system output, and
τ is a system parameter (time-constant).
We will frequently use block diagrams to represent the input/output relationships of
systems
u (t)
y (t)
t
u (t)
in p u t
y (t)
o u tp u t
L in e a r s y s te m
t
S y s te m is u s u a lly r e p r e s e n te d b y a n O D E
w ith c o n s ta n t c o e ffic ie n ts .
usually represented by an ODE with constant coefficients.
In addition we will investigate energy transducers that convert power/energy from
one domain to another, for example a motor converts power P from the electrical do­
main to the rotational domain.
c u rre n t i
+
e le c tr ic a l
m e c h a n ic a l ( r o ta tio n a l)
a n g u la r v e lo c ity
W
v o lta g e
v
T
-
to rq u e
v = K v W
1
i = - _ _ T
K v
A transducer is bi-directional, that is it can transmit power P in both directional, so
that the motor can also act as a generator.
P = vi
electrical
�
transduction
⇒ motor ⇒
⇐ generator ⇐
5
1–2
�
P = T Ω
rotational
(b) Linear System Theory A generalized method for the description of the dynamic
response of systems described by linear ordinary differential equations (ODEs) with
constant coefficients without regard to the particular energy domain.
s y s te m
r e p r e s e n te d in a " s ta n d a r d " fo r m
d 2y
+
d t2
u (t)
in p u t
2 z w n d y +
d t
w
n
2
=
u
y (t)
o u tp u t
Typical questions we might ask about the system might be what is the response to:
(a) a sinusoidal input?
(b) a step input ?
(c) a short pulse ?
(c) Feedback Control Theory The use of feedback to modify the dynamic behavior
of a system
u (t)
in p u t
" O p e n - lo o p " s y s te m
y (t)
o u tp u t
Often the inherent system behavior is unsatisfactory, for example:
• The response might be too slow.
• The response might be unstable.
• The system might be susceptible to external influences.
• Components inside the system might change their values as they age, causing
the system response to change.
Often control involves monitoring the system response - comparing the response to the
desired behavior - and generating a system input so as to drive the system toward the
desired response.
d e s ir e d r e s p o n s e
C o n tr o lle r
A c tu a to r
(m o to r, e tc )
fe e d b a c k p a th
S y s te m
S e n s o r
m e a s u re m e n t
1–3
a c tu a l re s p o n s e
(1) monitor the response in real-time.
(2) compare the actual response with the desired response.
(3) decide how to modify the response by changing the input.
Example 1
Temperature control in a home:
h e a t lo s s fr o m
e x te rn a l te m p e ra tu re T o
h o u s e
in te r n a l te m p e r a tu r e T
th e rm o s ta t
h e a t fro m
fu rn a c e
fu rn a c e
• Desired behavior - maintain temperature at T
• Control law (algorithm) using a thermostat:
(a) if T > T0 + Δ - turn off furnace.
(b) if T < T0 − Δ - turn on furnace.
(c) if T0 − Δ ≤ T ≤ T0 + Δ - do nothing.
A typical response as the thermostat turns on and off might be:
T
T
T
d
d
+ D
T d
- D
t
o n
F u rn a c e h e a t
o ff
t
Notice that the temperature rises while the furnace is on, and falls while it is
off. The thermostat acts to keep the temperature centered about the desired
value Td (known as the set-point).
1–4
Example 2
Cruise Control for a car:
Goals - maintain the speed of a car at a prescribed value in the presence
of external disturbances (external forces such as wind gusts, gravita­
tional forces on a incline, etc).
Also - improve the dynamic response of the car as the driver ”steps on the
gas”.
(a) Form a dynamic model of the car. Assume a simplified model
v (t)
F d
Fp (t)
F p (t)
v (t)
v e lo c ity
m a s s
m
Fd (t)
B ( v is c o u s fr ic tio n )
• Model the car as a simple lumped mass element m, sliding on a
viscous friction element, FB = BVB - simplification
• Assume two external forces
Fp (t) - the propulsive force Fp (t) from the engine.
Fd (t) - a ”disturbance” force Fd (t) from the environment.
Also assume Fp (t) = Ke θ(t) where θ(t) is the gas-pedal depression
and Ke is a constant. Then from a simple force balance:
m
m
dv
+ Bv = Fp (t) + Fd (t)
dt
dv
+ Bv = Ke θ(t) + Fd (t)
dt
and draw a block diagram
F
e n g in e
q
g a s p e d a l
K
e
F
+
p
d
(t)
+
(t)
m
c a r m o d e l
d v
d t
+ B v = F
V
c a r s p e e d
(b) Closed-loop Control:
Now design the controller. Assume we will use error-based control. In
other words, given a desired speed vd , and the measured car speed v(t),
we define the error e(t) as
e(t) = vd (t) − v(t)
1–5
and choose a control law that tells us to depress the gas pedal by an
amount proportional to the error:
θ(t) = Kc e(t) = Kc (vd (t) − v(t)),
so that the propulsive force acting on the car is
Fp (t) = Ke Kp e(t) = Ke Kc (vd (t) − v(t)),
V d (t)
+
e rro r
e (t)
-
F
c o n tr o lle r
K c
e n g in e
K e
q
F
p
+
(t)
fe e d b a c k p a th
This is known as proportional control.
1–6
d
(t)
+
m
c a r m o d e l
d v
d t
+ B v = F
c a r s p e e d
V