me 431 system dynamics dept of mechanical engineering
TRANSCRIPT
ME 431 System Dynamics
Dept of Mechanical Engineering
Lecture 1: Overview and Intro
• Introduction to the control system design process
• Control system example• open loop vs. closed loop
• Introduction to modeling
• Solving differential equations• Free response• Forced response
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Control System Design Process
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TranslatePlant
Design (andConstruction)
Model Analyze Controller Design
customerinput / gov’tregulations eng specs physical
systemdiagrams
math behavior controlsystem
purpose of models• analysis• design• verification
types of models• physical vs. empirical• mathematical• graphical
types of analysis• time domain• frequency domain• simulation• hardware in the loop (HIL)
types of control• supervisory logic control• on/off control• P, PI, PD, PID• advanced techniques
Control System Example
• Cruise Control Example
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ControlAlgorithm Engine Car
desiredspeed
throttleangle
(voltage)
forceactualspeed
Open-loop Control [feedforward]
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+-
disadvantages• sensitive to errors in model• sensitive to disturbances• needs periodic recalibration
advantages• simple to design• inexpensive• doesn’t affect stability• fast response wind force,
gravity force
ControlAlgorithm Engine Car
throttleangle
(voltage)
forceactualspeed
desiredspeed
Closed-loop Control [feedback]
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disadvantages• extra complexity• extra cost• can affect stability• can be slow to respond
advantages• robust to errors in model• robust to disturbances
+-
wind force,gravity force
ControlAlgorithm Engine Car
throttleangle
(voltage)
forceactualspeed
Speedometer
+-
measuredspeed
R E
D
U YCONTROLLER ACTUATOR PLANT
SENSOR
desiredspeed
Introduction to Modeling
• A model is an abstraction of the physical world
• Used for analysis and design, possibly before physical system exists
• Can be obtained from first principles or experimentally
• Purpose determines level of abstraction, form
• Complex enough, but no more
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Model Derivation
• From first principles• Use physical laws to derive models• Provides understanding• Can use empirical data to determine
parameters, validate model ME
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Model Derivation
• From empirical data• Feed a known input and observe output, fit model to
data
• Good for complicated systems (IC engine, battery) • Good for black-box systems (driver model)• Does not provide intuition, can’t be widely applied
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SYSTEM
Complexity Depends on Purpose
• Design/analysis model: simpler• Simple enough to generate closed-form solution• Less accurate, but provides intuition
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a a a b a
diR i L K e
dt
T b J
Complexity Depends on Purpose
• Simulation model: more accurate
Static vs. Dynamic SystemsStatic Systems
• Output is determined only by the current input, reacts instantaneously
• Relationship does not change (it is static!)
• Relationship is represented by an algebraic equation
Dynamic Systems• Output takes time to react• Relationship changes with
time, depends on past inputs and initial conditions (it is dynamic!)
• Relationship is represented by a differential equation M
E 4
31 L
ectu
re 1
12
SYSTEMinput output
Static vs. Dynamic SystemsMotor from a Dynamic ViewpointMotor from a Static Viewpoint
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0 0.5 1 1.5 2 2.5 30
50
100
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500
Mot
or S
peed
Time
0 0.5 1 1.5 2 2.5 30
1
2
3
4
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Mot
or T
orqu
e
Time
speed
torq
ue
T
Tstall
ea1
ea2
wwno-load
Solving Differential Equations
• Homogenous differential equations
• Righthand side of equation equals 0• Represents free response of system• Solution consists of exponentials
where exponents are roots of the characteristic eq.
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0mx bx kx
1 21 2( ) t tx t a e a e
Solving Differential Equations
• Homogenous differential equations
• For the above, the characteristic equation is
• Roots can be found from the quadratic formula ME
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0mx bx kx
2 0m b k
2
1,2
4
2 2
b b km
m m
Solving Differential Equations
• Recalling that• If the roots are completely real, then the solution
is exponential• If all negative, stable• If any positive, unstable
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1 21 2( ) t tx t a e a e
time
disp
lace
men
t, x
Solving Differential Equations
• If the roots are complex, then can rewrite in sines and cosines using Euler’s identity:
• Therefore, ME
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cos sinj te t j t
( ) ( )1 2
d dj t j ta e a e
( cos sin )td de A t B t
1 2d djt jtt ta e e a e e
Solving Differential Equations
• Above follows when have complex roots of char. eq.
real part = rate of decay (growth)imag part = freq of oscillation
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d j
( ) ( cos sin )td dx t e A t B t
Solving Differential Equations
• Forced differential equations
• Solution consists of two parts
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( )mx bx kx F t
( ) ( ) ( )h px t x t x t xh is the homogenous solution
- same form as before, natural response of systemxp is the particular solution
- generally same form as F(t), due to the input
Example
has a solution of the form
where
the homogenous portion dies out (transient) the particular portion remains (steady state)
8 25 2x x x t
4( ) ( cos3 sin 3 )tx t e A t B t at b
determined from characteristic equationhx
21,2where +8 +25=0 has roots 4 3 j
and has same form as ( )px F t
xh(t) xp(t)
Example
• Consider other types of forcing functions:
8 25 5x x x 4( ) ( cos3 sin 3 )tx t e A t B t a
28 25 3 tx x x e 4 2( ) ( cos3 sin 3 )t tx t e A t B t ae
8 25 5sin 2x x x t 4( ) ( cos3 sin 3 ) sin(2 )tx t e A t B t a t
cos 2 sin 2C t D t
Example
• Find the solution x(t) for 3 0, (0) 5x x x
Example
• Find the solution x(t) for23 , (0) 5tx x e x
Example (continued)