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.

  • $\begingroup$ You cannot state the linear state space representation without linearizing the system. $\endgroup$
    – MrYouMath
    Commented Nov 6, 2017 at 9:21
  • $\begingroup$ 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? $\endgroup$ Commented Nov 6, 2017 at 9:23
  • $\begingroup$ 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 $\endgroup$
    – MrYouMath
    Commented Nov 6, 2017 at 9:33

1 Answer 1


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.

  • $\begingroup$ 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? $\endgroup$ Commented Nov 6, 2017 at 9:03
  • $\begingroup$ 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. $\endgroup$ Commented Nov 6, 2017 at 10:06
  • $\begingroup$ I want to design a state observer and controller. So I want to represent it in state space form. $\endgroup$ Commented Nov 6, 2017 at 10:11
  • $\begingroup$ @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 $\dot{x}=A\,x+B\,u$. $\endgroup$ Commented Nov 6, 2017 at 15:06

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