Graphs, Goals and NPCs Advanced Programming for 3D Applications CE00383-3.
GOALS: To provide advanced students
Transcript of GOALS: To provide advanced students
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GOALS: To provide advanced students in mechanical engineering with a solid background in dynamic system modeling and analysis and to enable them to analyze and design linear control systems.
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FORMAT:
Lecture: 3 credits
Lab: 1 credit
You must enroll in both MEG 421 and MEG 421L
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MEG 421:
Prerequisites by Topic:
1. Electrical Circuits
2. Mathematics for Engineers.
3. Analysis of Dynamic Systems
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–Most General Definition:
To Produce a Desired Result
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“Trust is good, control is better.”
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Definition of Automation
• “Having the capability of starting, operating, moving, etc., independently.” 1
• “The use of machines to perform tasks that require decision making.” 2
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• Technical Control Systems
Open Loop Control Systems
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Open Loop Control SystemsExample: Batch Filling
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Block Diagram for Feedback
(or Closed Loop) Control
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England - Eighteenth Century ADWatt Steam Engine
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England - Eighteenth Century AD
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The accelerating technological change of the 19th century was reflected in literature and art.
In Jacques Offenbach’s opera ‘Les contesd’Hoffmann’ the hero falls in love with Olympia, a mechanical doll. Olympia can sing and dance. She needs rewinding every 5 minutes or so.
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Another famous example is Mary Shelley’s ever popular Frankenstein (1831).
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Automation Today
The world
around us
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Rapid Growth of Machine Intelligence
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Robots for Hazardous Areas
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Around the House
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Warm AndCuddly …
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…orCyborg
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Control SystemsClosed-loop control.Benefits: • System corrects “errors” (e.g. your fridgecorrects for temperature variations due todoor openings and other events.)•Labor savingDrawbacks:• More expensive and complex. • Need for sensors • System can become unstable
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Control System Example: Inverted Pendulum
The Problem: The cart with an inverted pendulum is "bumped" with an impulse force, F. Determine the dynamic equations of motion for the system, and find a controller to stabilize the system.
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Control System Example: Inverted Pendulum
Force analysis and system equations
At right are the two Free Body Diagrams of the system.
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Control System Example: Inverted Pendulum
Equation of motion for the cart:
Equation of motion for the pendulum:
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Control System Example: Inverted Pendulum
Without control we get the velocity response shown below, i.e. the pendulum falls to one side and the system is unstable.
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Control System Example: Inverted Pendulum
The equation at right describes the four “States” of the system:Cart Pos. xCart Vel x-dtAngle PhiAng. Vel. Phi-dt
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Control System Example: Inverted Pendulum
Control Loop Schematic:R = ‘reference’ = desired stateK = ‘state controller’
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Control System Example: Inverted Pendulum
After some mathematical analysis, the controller stabilizes the inverted pendulum:
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Control System Example: Inverted Pendulum
We can make the system respond faster, but it will oscillate more:
If we drive the controller gain too high, the system will become unstable.
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Control System Example: Inverted Pendulum
There is still a small problem:The cart has wandered off too far. So we add a requirement to return to (almost) where it started:
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The Airplane as Computer Peripheral
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The Future of Aviation
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The Future: More Automation. Manufacturing
“More Automation”
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Households and Service Industries:Repetitive Jobs will be automated.
Robotic mower forGolf courses.(Carnegie-Mellon)
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Medicine:Robodoc (Surgical Robot for hip
replacement)
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MedicineProsthetics
Mobility Assistance
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Artificial IntelligenceWith better brains and sensors, robots will interact better with
humans, and perform more functions.
Sony’s‘Aibo’
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…Remember the poor poet who fell in love with the robot doll?
‘Love’ is reality for many Aiboowners who seem to think that
their robot loves them.
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Quoted from: NY Times, May2, 2002
DIANE wasn't well. Her owner, Harry Brattin, placed a white muffler around her neck. She sat quietly on a metal desk in the meeting room while the others scampered around the floor playing."I get very sad when one of my dogs gets ill," said Mr. Brattin, 63, a motorcycle dealer from San Diego. "When Diane's head stopped moving I felt bad. I truly felt grief."Diane is an Aibo, a computer-controlled robot made by Sony, and D.H.S. is Droopy Head Syndrome, which is caused when a clutch wears out (it's repairable by replacing the head). Weird, perhaps, but not unusual.
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Exploration
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Control Systems in Entertainment
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Control Systems in
Entertainment
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Control Systems in Entertainment
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Control Systems in Entertainment
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Control
Open-Loop,Benefit:
• Simple, always stable• Widely used in well-
defined situations, e.g. Batch filling
Closed-Loop• Maintains desired
output in the presence of disturbances
• Can become unstable
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Feedback Control
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Feedback Control
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Feedback Control
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Each element or ‘Block’ has one input and one output variable
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For instance, the plant in the preceding block diagram can be modeled as:
Gplant(s) = Y(s) /U(s)
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Transfer Function
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Transfer Function
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Chapter 2: Chapter 2: Dynamic ModelsDynamic Models
Differential Equations in State-Variable Form
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State-Variables: Example
• x is the variable that describes any arbitrary position of the system (also called system variable)
• are the state-variables of the system.
• Since , the state-variables can be defined as
⎩⎨⎧
==++
vxukxbvvm
&
&
vx =&
xandx &
andx v
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State-Variable FormDeriving differential equations in state-
variable form consists of writing them as a vector equation as follows:
uJXHyuGXFX
+=+=&
where is the output
and u is the input
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Definitions
• is called state of the system.
• X is the state vector.It contains n elements for an nth-order system, which are the n state-variables of the system.
X&
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
nx
xx
X
&
M
&
&
& 2
1
• The constant J is called direct transmission term
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
nx
xx
XM2
1
ImportantYou can
forget that
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Deriving the State Variable Form requires to specify F, G, H, J for a
given X and u
{ {
[ ]
inputtheisuand
outputtheisuJ
x
xx
hhhywhere
u
g
gg
x
xx
ff
ffff
x
xx
n
H
n
G
n
X
n
F
nnn
n
n
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
===
M44 344 21L
MM
444 3444 21LL
MOM
MO
K
&
M
&
&
2
1
21
2
1
2
1
1
21
11211
2
1
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Satellite Altitude Control Example
Assumptions:• ω is the angular velocity• The desired system
output is θQ: Dynamic model
in state-variable form?
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Strategy (recommended but not required)
1. Derive the dynamic model.2. Identify the input control variable, denoted by u.
3. Identify the output variable, denoted by y.
4. Define a state vector, X , having for elements the system variables and their first derivative.
5. Determine 6. Determine F and G, in manner that
7. Determine H and J, in manner that
X&GuFXX +=&
JuHXy +=
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Ex 1: Dynamic Model
• Applying Newton’s law for 1-D rotational motion leads to:
=> (1)
θ&&IMdF Dc =+
IMdF DC +
=θ&&
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Example 1 (cont’d)Given:• The control input,
denoted by u, is given by:
• The output, denoted by y, is the displacement angle: y = θ
Assumption:• The state vector,
denoted by X, is defined as:
Known: The dynamic model
(1)
Required: Rewrite (1) as:
DC MdFu +=
IMdF DC +
=θ&&
{ { {
[ ] {
=
×+⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
==
====
uand
uywhere
u
JH
GXFX
ωθ
ωθ
ωθ
321
321&
&
&
θωωθ &=⎥⎦
⎤⎢⎣
⎡= withX
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Example 1 (cont’d)
• By definition: . Thus,
• Expressing the dynamic model:
as a function of ω and u (with ) yields:
ωθ =&
Iu
=ω&
IMdF DC +
=θ&&
ωθ &&& =
Dc MdFu +=
(1)
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Example 1 (cont’d) Equivalent form of
available eq:
where
Available equations:
• From the dynamic model:
• The output y is defined as: y = θ
• The input u is given by:
⎩⎨⎧
×+×+×=×+×+×=
u)I(u
100010
ωθωωθθ
&
&
uy ×+×+×= 001 ωθ
DC MdFu +=
{ { {
[ ] {
DC
JH
GXFX
MdFuand
uywhere
uI
+=
×+⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
==
====
001
10
0010
ωθ
ωθ
ωθ
321
321&
&
&
DC MdFu +=
⎩⎨⎧
==
Iuωωθ
&
&
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Example 1: Dynamic Model in State-Variable Form
By defining X and u as: The state-variable form is given by:
[ ] 001
10
0010
==
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
JH
IGF
DC MdFu
X
+=
⎥⎦
⎤⎢⎣
⎡=
ωθ
inputstate-variables
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Ex 1½: Recall the Satellite Altitude Control Example
Assumptions:• ω is the angular velocity• The system output is
Q: Dynamic model in state-variable
form?
ω
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Example 1½: Dynamic Model in
State-Variable FormBy defining X and u as: The state-variable form is given by:
DC MdFu
X
+=
⎥⎦
⎤⎢⎣
⎡=
ωθ
[ ] 010
10
0010
==
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
JH
IGF
Only change
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Analysis in Control Systems
• Step 1: Derive a dynamic model
• Step 2: Specify the dynamic model for software by writing it
in STATE-VARIABLE form
in terms of its TRANSFER FUNCTION (see chapter 3)
either
or
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Example 2: Cruise Control Step Response
• Q1: Rewrite the equation of motion in state-variable formwhere the output is the car velocity v ?
• Q2: Use MATLAB to find the step response of the velocity of the car ?Assume that the input jumps from being u(t) = 0 N at time t = 0 sec to a constant u(t) = 500 N thereafter.
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Reminder: Strategy
1. Derive the dynamic model.2. Identify the input control variable, denoted by u.
3. Identify the output variable, denoted by y.
4. Define a state vector, X , having for elements the system variables and their first derivative.
5. Determine 6. Determine F and G, in manner that
7. Determine H and J, in manner that
X&
GuFXX +=&JuHXy +=
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Ex 2, Q1:Dynamic Model
• Applying Newton’s law for translational motion yields:
(2)mux
mbx +−= &&&
xmuxb &&& =+−
=>
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Example 2, Question 1 (cont’d)Given:• The input ( = external
force applied to the system) is denoted by u
• The output, denoted by y, is the car’s velocity:
y = vAssumption:• The state-vector is
defined as:
Known: The dynamic model
(2)
Required: Rewrite (2) as:mux
mbx +−= &&&
{ { {
[ ] {
=
×+⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
==
====
uand
uvx
ywhere
uvx
vx
JH
GXFX
321
321&
&
&
xvwithvx
X &=⎥⎦
⎤⎢⎣
⎡=
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Example 1 (cont’d)
• By definition: . As a result,
• Expressing the dynamic model:
as a function of v and u leads to:
vx =& yx &&& =
(2)
mmuvbv +−=&
mux
mbx +−= &&&
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Ex 2, Q1 (cont’d)
Equivalent form of available eq:
where
Therefore:
⎪⎩
⎪⎨
⎧
×⎟⎠⎞
⎜⎝⎛+×⎟
⎠⎞
⎜⎝⎛ −+×=
×+×+×=
um
vmbxv
uvxx10
010
&
&
uvxy ×+×+×= 010
{ {
[ ] { uvx
ywhere
umv
xmbv
x
JH
GXFX
×+⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡
==
====
010
10
010
321
32143421&
&
&
Available equations:
• From the dynamic model:
• The output y is defined as:
y = v
• The input is the step function u
⎪⎩
⎪⎨⎧
+−=
=
muv
mbv
vx
&
&
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Example 2, Question 1: Dynamic Model
in State-Variable FormBy defining X as: The state-variable form results as:
⎥⎦
⎤⎢⎣
⎡=
vx
X
[ ] 010
10
010
==
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
=
JH
mG
mbF
![Page 76: GOALS: To provide advanced students](https://reader034.fdocuments.us/reader034/viewer/2022051903/628452e993adbe4c55227357/html5/thumbnails/76.jpg)
Example 2, Question2: Step Response using MATLAB?
Assumptions: m = 1000 kg and b = 50 N.sec/m.
[ ] 010
00100
050010
==
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
=
JH
.G
.F
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Ex 2, Q2: Step Response with MATLAB?
• The step function in MATLAB calculates the time response of a linear system to a unit step input.
• In the problem at hand, the input u is a step function of amplitude 500 N:
u = 500 * unity step function.
• Because the system is linear ( ):G * u = (500 * G) * unity step function
GuFXX +=&
G * Step 0 to 500 N 500*G * Step 0 to 1 N
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MATLAB StatementsF = [0 1;0 -0.05];G = [0;0.001];H = [0 1];J = 0;sys = ss(F, 500*G, H, J);
t = 0:0.2:100;y = step(sys,t);plot (t,y)
% defines state variable matrices
% defines system by its state-space matrices
% setup time vector ( dt = 0.2 sec)
% computes the response to a unity step response
% plots output (i.e., step response)
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Response of the car velocity to a step input u of amplitude 500 N
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