State space representation of coupled nonlinear ordinary differential equation


You have an equation of the form $$ r=M(θ)\ddot θ+b(θ,\dot θ) $$ Introduce $\omega = \dot θ$ to find the first order system $$ \begin{bmatrix}\dot θ\\\dot ω\end{bmatrix} = \begin{bmatrix}ω\\M(θ)^{-1}(r-b(θ,ω))\end{bmatrix} $$ provided the matrix $M(θ)$ is invertible.


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Aniket Sharma
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Aniket Sharma

Undergraduate Electrical engineer at SVNIT, Surat and a maker at Drishti - technical club of the college. I know things about control systems, embedded systems and microcontrollers. Familiar with C, MATLAB, C++ and Python. Other than that, huge anime fan!

Updated on August 01, 2022


  • Aniket Sharma
    Aniket Sharma over 1 year

    I have a DH matrix (Denavit-Hartenberg) of a two link manipulator having differential equation of the form:

    $$ \begin{bmatrix} \tau_1 \\ \tau_2 \end{bmatrix}= \begin{bmatrix} k_1+k_2\cos\theta_2 & k_3+k_4\cos\theta_2 \\ k_5+k_6\cos\theta_2 & k_7 \end{bmatrix} \begin{bmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{bmatrix}+ \begin{bmatrix} k_8\dot{\theta}_2^2 - k_9\dot{\theta}_1\dot{\theta}_2 \\k_{10}\dot{\theta}_1^2 \end{bmatrix}+f(\theta_1,\theta_2) $$

    where $f$ is a nonlinear function. $k_i$ are constants

    How can such a nonlinear coupled differential equation be expressed in state space form? How to decouple terms which contain product of both states?

    Edit : I want to design a state observer for this, so wanted it to be represented in state space form.

  • Aniket Sharma
    Aniket Sharma about 6 years
    But the new matrix that you formed still has theta and omega coupled in the function b. How to express this in dX/dt = AX+BU form?
  • Lutz Lehmann
    Lutz Lehmann about 6 years
    You can't as the system is deeply non-linear. If you have the goal to construct a controller you should post it as relevant information in your question.
  • Aniket Sharma
    Aniket Sharma about 6 years
    I want to design a state observer and controller. So I want to represent it in state space form.
  • Kwin van der Veen
    Kwin van der Veen about 6 years
    @AniketSharma The first order differential equation in this answer is a state space form. Only it is nonlinear, so you would have to linearize it in order to obtain the form $\dot{x}=A\,x+B\,u$.