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 $$ \begin{bmatrix}\dot θ\\\dot ω\end{bmatrix} = \begin{bmatrix}ω\\M(θ)^{1}(rb(θ,ω))\end{bmatrix} $$ provided the matrix $M(θ)$ is invertible.
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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, 2022Comments

Aniket Sharma over 1 year
I have a DH matrix (DenavitHartenberg) 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 about 6 yearsBut 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 about 6 yearsYou can't as the system is deeply nonlinear. If you have the goal to construct a controller you should post it as relevant information in your question.

Aniket Sharma about 6 yearsI want to design a state observer and controller. So I want to represent it in state space form.

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 $\dot{x}=A\,x+B\,u$.