# State space representation of coupled nonlinear ordinary differential equation

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.

• You cannot state the linear state space representation without linearizing the system. Commented Nov 6, 2017 at 9:21
• I am not having the main problem in linearizing it. Let's assume this question is linearized about some point. How to decouple the terms where the two states are multiplied? Commented Nov 6, 2017 at 9:23
• The linearization automatically eliminates all terms that are nonlinear e.g. $\dot{\theta}_1\dot{\theta}_2$. In your question, you asked how to state the state space representation and this is what @LutzL has answered. What you want is to get the linear time-invariant form of the system Commented Nov 6, 2017 at 9:33

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.
• @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 $\dot{x}=A\,x+B\,u$. Commented Nov 6, 2017 at 15:06