Min-Plus Linear Systems Theory

1. Min-Plus Linear Systems Theory TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

2. Linear: Time invariant: (Classical) System Theory Linear Time Invariant (LTI) Systems

3. Linear Systems Theory Consider an input signal: .. and its output at a system: Note:

4. Linear Systems Theory • Consider an arbitrary function • Approximate by Now we let

5. Linear Systems Theory • The result of Impulse Response “convolution”

6. If input is Dirac impulse, output is the system response • Output can be calculated from input and system response: “convolution” (Classical) System Theory Linear Time Invariant (LTI) Systems

7. Min-Plus Linear System min-plus Linear: Time invariant:

8. Min-Plus Linear System Consider arrival function: .. and departure function: Note:

9. Min-Plus Linear System • Consider an arbitrary function • Approximate by Now we let

10. Min-Plus Linear System • The result of “min-plus convolution” Service Curve

11. Min-Plus Linear Systems • If input is burst function , output is the service curve

12. Min-Plus Linear Systems • Departures can be calculated from arrivals and service curve: “min-plus convolution”

13. Back to (Classical) Systems Time Shift System eigenfunction eigenvalue eigenfunction • Now: • Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system

14. Back to (Classical) Systems eigenvalue • Solving: • Gives: Fourier Transform

15. Now Min-Plus Systems again Time Shift System eigenfunction eigenvalue eigenfunction • Now: • Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system

16. Back to (Classical) Systems eigenvalue • Solving: • Gives: Legendre Transform

17. Transforms • Classical LTI systems Frequency domain Time domain Fourier transform • Min-plus linear systems Rate domain Time domain Legendre transform Properties: (1) . If is convex: (2) If convex, then (3) Legendre transforms are always convex