3
$\begingroup$

I have a question about state space representation. How can I represent an equation in which I have only the second and first derivatives? For example

equation

where $u$ is the control input.

If I put $x_1=x$ and $x_2=\dot{x}$ I will not have $x_1$ in my state space representation, and when finding equilibrium points or Jacobian to check controllability, I will obtain zero in the partial derivatives corresponding to $x_1$. Is there a way to overcome this with a more suitable state representation? I'm not sure how to solve this problem as I have to linearize the system around an operating point using SS. Thanks in advance.

$\endgroup$
12
  • $\begingroup$ What is the problem with the zeros? $\endgroup$
    – copper.hat
    Oct 30 '14 at 2:43
  • $\begingroup$ Well it's just I don't know what to say about the stability of the system in this case or how to find out its equilibrium points, nor how to design a feedback controller for this system, control is not my background and is a little bit difficult to me to get this kind of tricky questions, as I can solve "normal" nonlinear systems. So I'm thinking I could be wrong in my SS representation, or there is a way I can solve the rest of my exercise coming from this. $\endgroup$
    – Wobbler28
    Oct 30 '14 at 3:13
  • $\begingroup$ You need to provide more info. about what you are trying to do and what you are trying to control/design. Is your state $x,\dot{x}$ or just $\dot{x}$, does $u$ have some nominal value, etc, etc... $\endgroup$
    – copper.hat
    Oct 30 '14 at 3:19
  • $\begingroup$ Pitifully I do not have much more info than this. The equation is intended to rule the motion of a plane, what I call x is its attitude angle, and I have to put the equation in state space form, use linearization to analyze the stability of the system and develop feedback control for the system to track a reference angle, so x is what I want to control. $\endgroup$
    – Wobbler28
    Oct 30 '14 at 3:38
  • $\begingroup$ Well, if you need to control $x$ then you have no choice but to include it in the state space representation. However, I am still not clear about why you think this is a problem with the representation. $\endgroup$
    – copper.hat
    Oct 30 '14 at 3:41
2
$\begingroup$

By selecting $x_1=x$ and $x_2=\dot{x}$ you can write

$$\begin{align*} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= k_1 x_1 (1 - |x_1|) + k_2 u \end{align*}$$

Now the fixed points are $(0, 0), (1, 0), (-1, 0)$. The Jacobian around $(1, 0)$ (as an example) is

$$ \begin{bmatrix} 0 & 1 \\ k_1(1 - 2 x_1) & 0 \end{bmatrix} |_{x_1=1} = \begin{bmatrix} 0 & 1 \\ -k_1 & 0 \end{bmatrix} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.