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    Assignment 5: Design using state space

    5.12 Let us consider a system shown in figure below
    a) Write the equation of motion of the system depicted in the figure for small $\theta_1$ and $\theta_2$.
    Hints: $M\dot{v}=-mg\theta_1-mg\theta_2+u$; $m(\dot{v}+l_i\ddot{\theta_i})=mg\theta_i; $for $i$=1,2. $v$ is the velocity of pendulum, $u$ is the external force applied to the cart.
    b) Write the state space equation of the system in $\dot{x}=Ax+Bu$ form.
    c) Is it always possible to control both the pendulums i.e. to keep them upright vertical position.
    d) Is the system is observable with the output $y=\theta_1$.

    Two inverted Pendulum

    Fig 5.2 Two inverted pendulum on cart

    5.13 Approximate equation of motion of a hot air balloon are $\dot{\theta}=-\theta/\tau_1 +u$; $\dot{v}=- v /\tau_2 +\sigma\theta + w/\tau_2$; $\dot{h}=v$. Where $\theta$=temperature change of air in balloon away from equilibrium temperature, $u$ is proportional to change in heat added to air in balloon (control parameter), $v$ is vertical velocity, $h$ is altitude from reference, $w$ is vertical wind velocity (disturbance).
    a) Can the $\theta$ and $w$ be observed by a contunious measurement of altitude change $h$ ? (assume $u$ is known).
    b) Is the system is completely controllable by $u$? Is it completely controllable by $w$?

    Hot air balloon

    Fig 5.3 Hot air balloon

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