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    Assignment 4: State space equation and it's solution

    4.1 Consider the states of the system (Fig 4.1) as $i_{L}(t)$ and $v_{c}(t)$. $x_{1}(t)=i_{L}(t)$ and $x_{2}(t)=v_{c}(t)$. Write state space model for the system. Consider two inputs $u_{1}(t)=e_{1}(t)$ and $u_{2}(t)=e_{2}(t)$. Now consider the initial condition$x_{1}(0)=i_{L,0}$ and $x_{2}(0)=v_{0}$ and solve the state space equation to findout $i_{L}(t)$ and $v_{c}(t)$. Does your result matches with the results obtained from circuit theory analysis?

    RLC parallel

    Fig 4.1

    4.2 Communication satellite: Let us consider a satellite of mass $m$ in earth orbit. The satellite position is expressed in spherical polar coordinate system ($r$,$\theta$,$\phi$. Assume state vector of the state space model of the satellite is $x(t) \triangleq [r(t)\hspace{.5cm} \dot{r}(t)\hspace{.5cm}\theta(t)\hspace{.5cm} \dot{\theta}(t)\hspace{.5cm} \phi(t)\hspace{.5cm}\dot{\phi}(t)]^T$. The input of the satellite is the thrust vector along $r$, $\theta$, $\phi$ direction. Therefore $u(t) \triangleq [u_{r}(t)\hspace{.5cm} u_{\theta}(t)\hspace{.5cm} u_{\phi}(t)]^T$. The equation of motion is obtained as \begin{equation*} \dot{x}(t)=f(x,u)= \begin{pmatrix} \dot{r} \\ r\dot{\theta}^2 cos^2 \phi +r\phi^2- k/r^2 + u_{r}/m\\ \dot{\theta}\\ -2\dot{r}\dot{\theta}/r +2\dot{\theta}\\dot{\phi} sin\phi /cos\phi+u_{0}/mrcos\phi\\ \dot{\phi}\\ -\dot{\theta}^2cos\phi sin\phi-2\dot{r}\\dot{\phi}/r+u_{\phi}/mr \end{pmatrix} \end{equation*} Assume at any instant of time satellite is in circular path $\therefore x_{0}(t) \triangleq\ \begin{pmatrix} r_{0} & 0 & w_{0}t & w_{0} & 0 & 0 \end{pmatrix}^T$ and $u_{0}(t)=0$. Linearise the nonlinear state space equation about the nominal circular path describe above.

    4.3 Consider a simple pendulum. Assume $\theta$ and $\dot{\theta}$ are two state of the system. Write the state space equation of the system. Note that the state space equation is nonlinear. Linearize the system about the nominal point $[0\hspace{.5cm} 0]^T$.

    Pendulum

    Fig 4.2 Pendulum


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