Minimax Controllers

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Transcript Minimax Controllers

Continuous Control
MAE 443/543
Closed Loop Control system
Control
input
Reference
Input
+
Output
Process
(Plant)
Controller
-
Sensor
Positive vs Negative Feedback Story
“The distinction between the stabilizing and destabilizing character of negative
and positive feedback loops is neatly captured in the story of the misconnected
electric blanket. The newlyweds were given an electric blanket for their queensize double bed. The blanket had separate temperature settings for the two
sides of the bed, one for him and one for her. Properly connected, there should
have been two separate negative feedback systems, each attempting to
control the temperature of the blanket for the comfort of each individual. The
story goes that the newlyweds misconnected the blanket so that his setting
controlled her blanket temperature and hers controlled his. The result…was a
nasty positive feedback system. She felt cold, turned up her setting, making
his side too warm for him so he turned down his setting, making her even
colder, so she raised her setting even further, and so on. How such a scenario
would end is left up to the fertile imagination of the reader.
-Richardson, G. and Pugh, A. Introduction to System Dynamics Modeling with
Dynamo, MIT Press, Cambridge, MA 1981, pp 11-12
Centrifugal Governor
http://en.wikipedia.org/wiki/Centrifugal_governor
Centrifugal governor, Boulton & Watt, 1798
http://en.wikipedia.org/wiki/James_Watt
Controlled Flight
The key innovation, which had no clear predecessor, was the use of wing
warping to effect lateral control; that is, control for turning (Fig. 7). This
innovation provided a full complement of movable aerodynamic surfaces to
allow control over all three axes of rotational motion. This innovation was
critical, since it made controlled flight possible.
Feedback Control: An Invisible Thread in the History of Technology,
Dennis Bernstein
IEEE Control System Magazine, April 2002
MagLev Train
http://www.o-keating.com/hsr/mlx01.htm
The principal of a Magnet train is that
floats on a magnetic field and is
propelled by a linear induction motor.
They follow guidance tracks with
magnets. These trains are often
referred to as Magnetically Levitated
trains which is abbreviated to MagLev.
The MLX01
ML stands for maglev,
and X for experimental.
The Aerodynamic brakes on the MLX01
Control of Epidemics
• I(t): infective class
• S(t): susceptive class
• R(t): removed class
• V(t): vaccinated individuals,
is the vaccination rate
Control of Epidemics
0.2
Infective Class I(t)
0.18
0.16
0.14
0.12
0.1
Without Vaccination
With Vaccination
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
Time
Introduction to Optimal Control, (pp 17-18)
Jack Macki and Aaron Strauss
Springer-Verlag, 1982
35
40
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Tuned Mass Damped (TMD)
http://www.oiles.co.jp/en/menshin/building/control/amd.html
Hydrocephalus Therapy
http://www.medit.hia.rwthaachen.de/en/research/hydrocephalus_implant/index.html
The human brain is immersed in a fluid (cerebrospinal fluid, CSF),
which, among other things, protects the brain from mechanical
stress (e.g. concussion) and helps support its weight through
buoyancy. In normal situations, the production and reabsorption
of this fluid are equal. However, a constant overproduction,
blockage (i.e. tumor) or reabsorption difficulty can upset this
natural balance, resulting in a build-up of fluid in the skull
(Hydrocephalus). In adults, this excess fluid causes large
pressures to develop rapidly in the skull, and impairs brain
function. The most common solution today is the implantation
of a passive pressure-control valve and catheter system (shunt).
Once the pressure in the skull exceeds a certain critical value,
the excess fluid is released through the open valve and typically
drained into the stomach cavity (Fig. 1). Unfortunately, these
passive valves encounter many problems, including over- and
under- drainage, occlusions, and system failure. These
problems may be avoided through the use of a mechatronic
valve, which could monitor the patient's health and properly
regulate the amount of fluid in the skull through pressure,
flow and inclination sensors.
Stages in Control System Design
Modeling
Physics based model derivation
Non-Physics based modeling
System Identification
mode structure/class selection
parameter estimation
model validation
Analysis
Stability, Controllability, …
Control System Design
Performance specification
Synthesis of controller
Simulation and testing
System Models
Dynamic
Systems
Distributed
Parameter
Lumped
Parameter
Stochastic
Deterministic
Continuous
Time
Discrete Time
Nonlinear
Time Varying
Linear
Time
Invariant
Online examples
http://www.engin.umich.edu/group/ctm/
http://www.mathworks.com/applications/controldesign/
http://lorien.ncl.ac.uk/ming/Dept/Swot/connotes.htm
http://www-control.eng.cam.ac.uk/extras/Virtual_Library/Control_VL.html
http://www.jhu.edu/~signals/explore/index.html
Example
For the spring-mass dashpot system
u
m
k
y
Force balance leads to the free body diagram
m
At initial time (t=0), we need the values of and to solve the differential
equation which are referred to as initial conditions.
Overview of Control Approaches
• Classical Control: employs primarily frequency domain tools to achieve
control objectives
•MAE 443/543 (Continuous Control)
•MAE 444/544 (Digital Control)
• Modern Control: employs primarily time-domain tools
• MAE 571 (System Analysis)
• MAE 672 (Optimal Control)
• MAE 670 (Nonlinear Control)
•Post-Modern Control: integrate time-domain and frequency
domain tools.
Software Tools
• MATLAB (MATrix LABoratory)
powerful numerical software package with toolboxes for control,
optimization, system identification, etc.
http://www.engin.umich.edu/group/ctm/
•MAPLE (MAniPulator LanguagE)
symbolic manipulator for analytically solving algebraic, differential
equation and for linear algebra besides other functionality
http://www.mapleapps.com/categories/whatsnew/html/SCCCmapletutorial.shtml
•MATLAB is now bundled with MAPLE permitting the user to exploit the
strengths of both packages
Laplace Transform
Let f(t) be a function of time t, such that f(t) = 0, for t < 0, then
is the Laplace transform of f(t).
s is a complex variable
Example:
Laplace Transform
Example:
Example (Step function):
Laplace Transform of the Derivative of a Function
Consider
Let
Laplace Transform of the Derivative of a Function
Similarly
Final and Initial Value Theorem
Final Value Theorem:
If f(t) and
f(t), and if
Consider
Since
are Laplace transformable, if F(s) is the Laplace transform of
exists, then
Final and Initial Value Theorem
we have
Thus,
Initial Value Theorem:
If f(t) and
f(t), and if
are Laplace transformable, if F(s) is the Laplace transform of
exists, then
Inverse Laplace Transform
where
Not easy. We use table to find the Inverse Laplace Transforms
Partial Fraction method for Inverse Laplace Transform
Let
B(s), A(s) are polynomials in s with the order of A(s) > B(s). If F(s) can be
represented as
then
Inverse Laplace Transform
Example: (Distinct Poles)
We can write
where ak – residue at the pole s=pk.
Example:
Inverse Laplace Transform
therefore,
Example:
This has to be written as
Inverse Laplace Transform
therefore,
Example: (Multiple Poles)
Inverse Laplace Transform
Consider,
Similiarly
Matlab function
Matlab command for determining residue for the transfer function G(s) = B/A:
[R,P,K] = RESIDUE(B,A)
[R,P,K] = residue([1 2 3],[1 3 3 1])
R=
1.0000
0.0000
2.0000
P=
-1.0000
-1.0000
-1.0000
K=
[]