An Algebraic Method for Analyzing OpenLoop Dynamic Systems

1
An Algebraic Method for Analyzing Open-Loop
Dynamic Systems

• Wenqin Zhou, D. J. Jeffrey and G. J. Reid
• University of Western Ontario, Canada
• CASA 2005

2
Outline

A
C
B
DynaFlex
RifSimp
Numerical solver
3
Traditional Subsystem Models
Parameters
Subsystem
Inputs
Outputs
Subsystem

• Inputs, outputs, and parameters are numeric
• Subsystems combined at simulation time

4
Symbolic Subsystem Models
Parameters
Relationships defined for Internal Variables
Output Expressions
Input Expressions

• Inputs, parameters and outputs are symbolic
  expressions
• Subsystems combined at formulation time, as
  opposed to simulation time.

5
Introduction on DynaFlex

• Symbolic manipulation applied to mechanical
  systems to get the governing dynamical equations.
  
• Using graph theory to describe the mechanical
  system as input file, then automatically generate
  the dynamical equations.4

6
DynaFlex Output

• General forms for a multibody dynamical system
• Here is a DAE system of second order but usually
  of high differential index.
• Now we only consider the open-loop systems which
  means the system without algebraic constraints.

7
3D Spinning Top
The model equations for a top from DynaFlex
are without algebraic
constraints. They are
Fig1 The three-dimensional top. The centre of
mass is at C and , Gravity
acts in the -Z direction.
8
Advantages for Symbolic Modelling

• The Advantages for using DynaFlex are

Easy to get physical insight into the system
Good for communication
Facilitates subsequent real-time simulations.
9
Difficulties for Symbolic Modelling

• The Difficulties with DynaFlex are
• The generation of Large Expressions
• It is hard to analyse the equations when they are
  very big.

Large Expression Management
RifSimp or diffalg
10
How to solve these symbolic models?

• Three ways for solving these symbolic models

• A?B Just symbolic solve these equations. Too
  complicated for Maple to get analytical solutions.

2. A?C Directly numeric solve these models.
Hard to get the consistent initialization of
numerical solution procedures.
3. A?B?C Symbolic simplify then numeric solve.
Such as using differentiation and elimination
methods to get a simplified canonical form and
including all constraints for the system. Then
the initial value problem of the original DAE
system has a unique solution. Like using Maple
package RifSimp and diffalg.
11
RifSimp Package

• RifSimp Symbolic simplification of ODE and PDE
  systems and reduction to canonical differential
  forms.
• Main features
• Computation with nonlinear systems including ODE
  and PDE
• Advance case splitting capabilities for discovery
  of all cases with their own special properties
• A visualization tool for examination of the
  binary tree
• Algorithms for working with formal power series
  solutions
• of the systems.

12
How does RifSimp work?
ODE/PDE System
Define a ranking, classify the whole system
L0
N0
Solves L0 for their highest derivatives
Differentiate N0
Eliminate w.r.t. L0 and N0
0
0
END
13
Application to 3D Top

• First need to convert trig functions to
  polynomials with and
• get rational
  polynomial differential equations.
• Getting 24 cases and 9 cases after recording
  physical facts, like and . (Maple
  worksheet).
• Pay price the total degree increased and the
  complexity is higher.

14
A Generic Case Case 1 An Oscillating System
15
Example of Special Case

• From the 9 case tree, case 5 (Altgt0) and case 6
  (A0) have the same equations
• with a constraint
• by the relation
• We have which means the top is
  moving horizontally in the x-y plane, i.e. it is
  precessing without nutation.

16
Some Difficulties

• Matrix inverse for solving . For
  example if we try to invert a 6 by 6 matrix, the
  output from Maple is huge. So it is hard to get
  symbolic canonical forms. We present a new idea
  for this problem in paper 1. Also we organize a
  special session about large expression in ACA05
  meeting.
• Membership test. If the leading linear part is
  too lengthy, it is harder to do reductions
  symbolically with respect to such lengthy
  equations.

17
Reference

• 1 W. Zhou, D.J. Jeffrey, G.J. Reid, C. Schmitke
  and J. McPhee. Implicit Reduced Involutive Forms
  and Their Application to Engineering Multibody
  Systems. IWMM 2004, LNCS 3519, pp. 31-43, 2005.
• 2 A.D. Wittkopf, G.J. Reid. The Reduced
  Involutive Form Package. Maple Software Package.
  First distributed as part of Maple 7 (2001).
• 3 A.D. Wittkopf. Algorthims and Implementation
  for Differential Elimination. Ph.D. Thesis. Simon
  Fraser University, Burnaby (2004).
• 4 P. Shi, J. McPhee. DynaFlex Users Guide,
  Systems Design Engineering, University of
  Waterloo (2002).

18
Thank you for your attention!

19
ACA 2005 Special Session

• Title Handling Large Expressions in Symbolic
  Computation
• Abstract Large Expressions appear in many
  symbolic contexts. In this session, we invite
  papers expressing different viewpoints concerning
  large expressions. Papers describing applications
  in which large expressions arise are also welcome.

20

21
(No Transcript)