PhD Course on Model Reduction

1
PhD Course onModel Reduction
2
Model Reduction strategy for automatic
simplification of very complicated systems
3
Signal Wire in an Integrated Circuit
Application Example
Signal Wire
Wire has resistance
Wire and ground plane form a capacitor
Logic Gate
Logic Gate
Ground Plane
Reduced-Order Model

• Assess wiring impact on IC performance
• Wire Model must preserve terminal behavior

4
Jet Engine Design
Application Example

• Generate Low-order models directly from
  Navier-Stokes Equation based physical simulators.

Operating Conditions
Sensing
Reduced-Order Inlet Model
Actuator Model
Flow Disturbances
Actuation
Flow features of Interest
5
Realistic Simulation of Heart
Application Example
6
CD Player Controller Design
Application Example

• The control task is to achieve track following,
  which basically amounts to pointing the laser
  spot to the track of pits on the CD that is
  rotating.
• The challenge is to find a low-cost controller
  that can make the servo system faster and less
  sensitive to external shocks.

7
(No Transcript)
8
Common features

• Devices have Well-defined Inputs and Outputs
• Signal Transmission on a Wire
• Left and right end Voltages and Currents
• Jet Engine Nozzle Design
• Nozzle in and out flow, Nozzle size
• Dynamics is important
• Internal state must be somehow represented

9
Traditional Approach to Generating Models
Application Example
Interconnect Expert
MEMS Expert
10
Wire
Wire and ground plane form a capacitor
Ground Plane
11
Advantages of replacing the person who does the
reduction

• Faster results
• The simulation wont make mistakes

12
The Numerical Macromodeling or Model Reduction
Paradigm

• To extract input/output behavior
• Automatic extraction of reduced order models

13
Heat Conducting Bar
Demonstration Example
Lamp
Input of Interest
Output of Interest
14
Heat Conducting Bar
Demonstration Example
Basic Equations

• Temperature Differential Equation
• Spatial Discretization (except at end)

15
Heat Conducting Bar
Demonstration Example
Input-Output Discrete Equations
16
Heat Conducting Bar
Demonstration Example
State-Space Description
Given the right scaling
17
Heat Conducting Bar
Demonstration Example
Nonlinear State-Space Description
18
Model Order Reduction
Why???
19
Some reasons

• A high order system is expensive to simulate and
  store in computers memory
• Simulation and Analysis is time consuming
• Design procedure and control is complicated

20
Objective reducing the order of dynamic systems
21

• Stability preservation
• Global Accuracy
• Error bound

22

• Computational issues