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Question: Obtain a state-space representation of nonlinear multiple-input multiple-output (MIMO) system:

$$\dddot{y}_1 + 2\dot{y_1} + 3y_2 + 2 = u_1 y_2 \tag{1}$$ $$\ddot{y}_2 - 2 \dot{y}_2 + \dot{y}_1^3 + y_2 + y_1 = (u_2 - u_3)y_1 \tag{2}$$

I find it difficult solving the above equations. I have the following queries:

  1. What do I do with $(dy_1/dt)^3$? How do I represent it in state space model?
  2. Are $u_1, u_2$ and $u_3$ control inputs or just constants (coefficients of $y_1$ and $y_2$)?
  3. Does the constant $2$ in equation come in $\mathbf B$ (i.e., $\mathbf A x + \mathbf B u)$?
  4. Do I have to convert these equations into linear equations?
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    $\begingroup$ (1) Since you will set a new variable $v=y_1'$, that term becomes $v^3$ (2) Impossible to tell with further context. (3) I don't know what you're asking (4) No, there is no way to do so. $\endgroup$
    – Paul
    Oct 8 '18 at 16:02
  • $\begingroup$ @Paul 1. Yeah, I've taken v=y1' but then, how do I represent it in state-space(Matrix form i.e f(x,u)) 3. What do I do with the number '2'(scalar quantity) in equation 1. In which matrix does it come in State-space rep? $\endgroup$ Oct 8 '18 at 16:48
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Let $y_3=y_1'$, $y_4=y_2'$, $y_5=y_3'$. Then the state space equations look like

$$y_1'=y_3$$ $$y_2'=y_4$$ $$y_3'=y_5$$ $$y_4'-2y_4+y_3^3+y_2+y_1=(u_2-u_3)y_1$$ $$y_5'+2y_3+3y_2+2=u_1y_2$$

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  • $\begingroup$ Shouldn't the 3rd equation be y3''? $\endgroup$ Oct 8 '18 at 16:58
  • $\begingroup$ No, if $y_1'=y_3$, then $y_1''=y'_3$ $\endgroup$
    – Paul
    Oct 8 '18 at 17:09
  • $\begingroup$ Yeah, you are right. But then, in the question it is y1''' so it should be y3'' right? $\endgroup$ Oct 8 '18 at 17:18
  • $\begingroup$ Oh! I missed the third derivative. So will need a fifth variable. I will edit accordingly. $\endgroup$
    – Paul
    Oct 8 '18 at 17:36
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    $\begingroup$ No, only linear equations can be represented in matrix form. $\endgroup$
    – Paul
    Oct 8 '18 at 21:24

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