User:Egm6321.f12.team7.parikh.a/HW16.3

=R*3.12 Linear time-variant system, Rocket Roll Control in matrix notation=

Given

 * $$ \displaystyle \dot{\phi}=\omega $$ $$\displaystyle (1) p. 16-2

$$


 * $$ \displaystyle \dot{\omega}=-\frac{1}{\tau }\omega+\frac{Q}{\tau }\delta $$$$\displaystyle (2) p. 16-2

$$


 * $$ \displaystyle \dot{\delta}=u $$ $$\displaystyle (3) p. 16-2

$$

See lecture notes [[media:pea1.f11.mtg16.djvu|Mtg 16 (b)]] for problem description.


 * $$ \displaystyle \mathbf{\dot{x}}(t)=\mathbf{A}(t)\, \mathbf{x}(t)+\mathbf{B}(t)\, \mathbf{u}(t) $$$$\displaystyle (1) p. 14-4

$$

Reference lecture notes[[media:pea1.f11.mtg14.djvu|Mtg 14]] for form of equation.

Find
Put (1-3)p.16-2 in the form of (1)p.14-4

Solve

 * $$ \displaystyle \begin{bmatrix}

\dot{\phi }\\ \dot{\omega }\\ \dot{\delta } \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0\\ 0 & -\frac{1}{\tau } & \frac{Q}{\tau }\\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \phi \\ \omega \\ \delta \end{bmatrix}

+ \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \mathbf {u}(t) $$