min-plus linear systems theory
DESCRIPTION
Min-Plus Linear Systems Theory. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:. Linear:. Time invariant:. (Classical) System Theory. Linear Time Invariant (LTI) Systems. Linear Systems Theory. Consider an input signal: .. and its output at a system: - PowerPoint PPT PresentationTRANSCRIPT
Min-Plus Linear Systems TheoryMin-Plus Linear Systems Theory
System
(Classical) System Theory(Classical) System Theory
Linear Time Invariant (LTI) Systems
Linear:Time invariant:
Consider an input signal:
.. and its output at a system:
Note:
Linear Systems TheoryLinear Systems Theory
Linear Systems TheoryLinear Systems Theory
• Consider an arbitrary function
• Approximate by
Now we let
Linear Systems TheoryLinear Systems Theory
• The result of
“convolution”
System
(Classical) System Theory(Classical) System Theory
Linear Time Invariant (LTI) Systems
• If input is Dirac impulse, output is the system response
• Output can be calculated from input and system response:
“convolution”
Min-Plus Linear SystemMin-Plus Linear System
Networkmin-plus Linear:Time invariant:
data
time t
A(t)
D(t)
delay
backlog
Consider arrival function:
.. and departure function:
Note:
Min-Plus Linear SystemMin-Plus Linear System
Min-Plus Linear SystemMin-Plus Linear System
• Consider an arbitrary function
• Approximate by
Now we let
Min-Plus Linear SystemMin-Plus Linear System
• The result of
“min-plus convolution
”
Min-Plus Linear SystemsMin-Plus Linear Systems
Network
• If input is burst function , output is the service curve
Min-Plus Linear SystemsMin-Plus Linear Systems
Network
• Departures can be calculated from arrivals and service curve:
“min-plus convolution”
System
Back to (Classical) SystemsBack to (Classical) Systems
• Now:
Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant
system
Time Shift System
eigenfunction
eigenfunction eigenvalue
System
Back to (Classical) SystemsBack to (Classical) Systems
• Solving:
• Gives:
eigenvalue
Fourier Transform
System
Now Min-Plus Systems againNow Min-Plus Systems again
• Now:
Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant
system
Time Shift System
eigenfunction
eigenfunctioneigenvalue
System
Back to (Classical) SystemsBack to (Classical) Systems
• Solving:
• Gives:
eigenvalue
Legendre Transform
TransformsTransforms
• Classical LTI systems
Fourier transform
• Min-plus linear systems
Legendre transform
Time domain
Frequency domain
Time domain
Rate domain
Properties: (1) . If is convex:
(2) If convex, then
(3) Legendre transforms are always convex