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. – MrYouMath Nov 6 '17 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? – Aniket Sharma Nov 6 '17 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 – MrYouMath Nov 6 '17 at 9:33

1 Answer

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.

• 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? – Aniket Sharma Nov 6 '17 at 9:03
• 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. – LutzL Nov 6 '17 at 10:06
• I want to design a state observer and controller. So I want to represent it in state space form. – Aniket Sharma Nov 6 '17 at 10:11
• @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 $\dot{x}=A\,x+B\,u$. – Kwin van der Veen Nov 6 '17 at 15:06