Genral state space model - runge kutte

Click For Summary

Discussion Overview

The discussion revolves around solving general n-order state space models, particularly in the context of numerical methods like the Runge-Kutta method (RK4). Participants explore the application of these methods to state space representations of systems, addressing both theoretical and practical aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the concept of a "general n order state space model" and the Butcher tableau, indicating a lack of familiarity with these terms.
  • Another participant clarifies that a state space model can be represented as a matrix ODE and suggests using MATLAB/Simulink for practical implementation, arguing that Butcher tableau is not necessary.
  • A different perspective is offered regarding the conversion of higher-order ODEs into first-order systems, which can then be solved using various methods, including Runge-Kutta.
  • There is a mention of RK4 being a common choice due to its ease of implementation, but alternatives like Heun's method are also noted as potentially effective.
  • One participant questions the original poster about the specific problems they are facing and what methods they have already tried.
  • A post was edited to remove content that introduced a new problem unrelated to the original query, indicating a focus on maintaining topic relevance.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concepts discussed, with some confusion evident about the terminology. There is no clear consensus on the necessity of the Butcher tableau, and multiple approaches to solving the problem are presented without agreement on a single method.

Contextual Notes

Participants have not fully defined their assumptions or the specific context of their problems, leading to potential gaps in understanding. The discussion includes unresolved mathematical steps and varying interpretations of the state space model.

Who May Find This Useful

This discussion may be useful for individuals interested in numerical methods for solving differential equations, particularly in the context of state space models, as well as those seeking clarification on related concepts like the Butcher tableau.

ash4sigh
Messages
5
Reaction score
0
Hi,

Really glad to have found this site... I was hoping someone could help me - have been trying to decipher how to find a way to solve general n order state space model - I started off with RK4 and now I want to expand that but I can't get my head wrapped around the butcher tableau to generate code for n order state space model..

any help would be appreciated..

sorry *general
 
Last edited:
Physics news on Phys.org
hell i don't even know if i need butcher tableau - lost!
 
You've lost me too! I thought I knew differential equations but I don't recognize "n order state space" nor "Butcher tableau"!
 
sniff
 
What he means is just a matrix ODE given as in a compact fashion
[tex] \dot{x} = Ax+Bu, y = Cx + Du[/tex]
where A,B,C,D are matrices. This is usually referred as a State-Space model of a system.


You can just use MATLAB/simulink for your system. For that you don't need anything else, it does it for you. Or you can just pick up a sample time and update your states at each sample time e.g. Newton's method. If you insist on RK4 it is similar to Runge Kutta 4 for scalar ODEs. just make sure that what you are doing is the same with why you are doing.

Of course if you have the input function known in time domain, you can solve the convolution integral

[tex] y=Ce^{At}x_0 + C\int^t_0{e^{A(t-\tau)}Bu(\tau)d\tau}[/tex]

or the Laplace equivalent of this (in terms of a transfer function), (if you know your input function's Laplace domain representation!)
[tex] y(s) = (C(sI-A)^{-1}B + D)u(s)[/tex]

Butcher Tableau is not useful here. Just forget about it.
 
ash4sigh said:
[ I have ] been trying to decipher how to find a way to solve general n order state space model - I started off with RK4 ...
What do you mean by "general n order state space model"? I assume you mean that you have a functional representation for the nth derivative of the state with respect to time,

[tex]\frac {d^n}{dt^n} x(t) = x^{(n)}(t) = g(t,x(t),x^{(1)}(t),\cdots,x^{(n-1)}(t))[/tex]

Such systems can be converted to a first-order ODE by making the derivatives up to order n-1 a part of the state. You then have a plethora of first-order ODE techniques from which to choose, Runge-Kutta methods being just one class.

RK4 is the gold standard, not so much because it is particularly good but more because it is often adequate and particular easy to implement. Lower order methods such as Heun's method (aka velocity verlet) can yield good results at a lower cost. What problems are you confronting, and what have you tried?

You probably don't need to use the Butcher tableau. Most of the work on Runge-Kutta methods was done a long time ago.
 
help on state space HW

Edited by HallsofIvy
I have removed this because

1) It introduced a totally new problem, not that of the original poster.

2) It has also been posted under "homework".
 
Last edited by a moderator:

Similar threads

  • · Replies 65 ·
3
Replies
65
Views
9K
  • · Replies 4 ·
Replies
4
Views
3K
Replies
0
Views
2K
  • · Replies 0 ·
Replies
0
Views
2K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 1 ·
Replies
1
Views
4K
  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K