6.01 lecture notes March 4 - Massachusetts Institute of...
Transcript of 6.01 lecture notes March 4 - Massachusetts Institute of...
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March 4, 2008 6.01 lecture notes 1
6.01 lecture notesMarch 4
Modeling and abstraction with signals and linear systems
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k= 0.5 k= 1.2
k= 2 k= 4 k= 6
Robot meets wall
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Big themes of 6.01
1. Controlling complexity – abstraction and modularity
2. Interacting with physical systems –models
3. Coping with error and incomplete information
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Signal and system abstraction
Describe a system by the way it transforms sequences of inputs in to sequences of outputs
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polymorphism,inheritance
higher-order fns
Patterns
ADTs, classesdefAbstraction
lists, objectsif, f(g(x))Combination
numbers, strings+, *, ==Primitives
DataProcedures
PCAP framework for Python
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Adder
][][][ 21 nxnxny +=
Gain
][][ nxkny ⋅=
21 xxy +=
xky ⋅=
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]1[][ −= nxny
Rxy =
R is called the delay operator
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Quiz
Find the operator equation and the difference equation corresponding to this delay-adder-gain block diagram
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Three representations
Block diagram: good for understanding signal flow
Difference equation: good for computing numerical outputs to given numerical inputs
][]1[002.1][ nxnyny +−=
Operator equation: good for doing algebraic manipulation and analysis
xRyy += 002.1
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Linear time-invariant systems (LTI)
• As block diagrams: Build from adders, (constant) gains, and delays
• As difference equations: Linear with constant coefficients
• As operator equations: Built from addition, multiplication by constants, and “multiplication” by R
][]1[][
][]1[][
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10
jnxbnxbnxb
knyanyanya
j
k
−+−+=−+−+
�
�
xRbRxbxb
yRaRyayaj
j
kk
�
�
++=
++
10
10
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Quiz
• Write the operator equation corresponding to the following difference equation
]3[8]2[6][10]2[7]1[4][3 −+−+=−−−+ nxnxnxnynyny
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Quiz
• Write the operator equation corresponding to the Fibonacci equation
][]2[]1[][ nxnynyny =−+−=
xyRRyy =−+ 2Answer:
Note: The Fibonacci numbers are generated by taking x=0, but starting with y[0]=0 and y[1]=1
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The big fact
• The operator algebra mirrors the behavior of the system, so we can reason about combining systems by doing algebra.
• This is captured by the idea of a system function
• This is, the system can be represented by the ratio of output y to input x as expressed by the operator equation.
xy
H =
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xnynyny
xyRRyy
RRRRxw
wy
xy
H
=−+−−=+−
+−=
−⋅
−=⋅==
]2[6]1[5][65
6511
311
211
2
2
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Patterns
system functionAbstraction
adder, gain, delayCombination
signalPrimitives
PCAP framework for signals and systems
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s( )( )][]1[][
]1[][]1[]1[][
nsxndknv
sndknv
nvndnd
−−=−−=
−−−= δRobot motion:
controller:
Put controller eqnin the form of a DE
where x[n]=1 for all n
xv
vd
xd
H ⋅==
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Rpxy
H01
1−
==
y gets multiplied by p0 each time around the cycle. So the result is an exponential: y[n]=pn
p0 is called the pole of the system
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p = -1.5 p = 1.5
p = 0.8
p = -0.8
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The general picture: explanation next week
• The system function can be written in the form
• The p’s are the poles• The poles are in general complex numbers• The positions of the poles in the complex plane
determine the stability and oscillation of the system’s response
)1()1)(1(in polynomial
21 RpRpRpR
k−−− �