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  • EE 350 Control Systems Assignments

    Assignment 5: Design using state space

    5.1 Determine the response of a system $\dot{x}=Ax+Bu$, for unit step input, where $A=\begin{pmatrix}0 & 1 \\ -6 & -5 \end{pmatrix}$, $u=\begin{pmatrix}0 \\ 1 \end{pmatrix}$. Assume zero initial condition.

    5.2 Check the stability and controllability of the following pairs (A,B). Also determine the controllability Gramian.
    a) $A=\begin{pmatrix}0 & 1 \\ -2 & -3 \end{pmatrix}$, $B=\begin{pmatrix}0 \\ 2 \end{pmatrix}$.
    b) $A=\begin{pmatrix}1 & 1 \\ 1 & 0 \end{pmatrix}$, $B=\begin{pmatrix}0 \\ 1 \end{pmatrix}$.
    c) $A=\begin{pmatrix}-3 & 1 \\ -2 & 1.5 \end{pmatrix}$, $B=\begin{pmatrix}0 \\ 1 \end{pmatrix}$.
    d) $A=\begin{pmatrix}-3 & 1 \\ -2 & 1.5 \end{pmatrix}$, $B=\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}$.
    e) $A=\begin{pmatrix}0 & 1 & 0 \\ 0 & 1 & 1\\ 0 &-1 & -2\end{pmatrix}$, $B=\begin{pmatrix}0 \\ 1 \\1 \end{pmatrix}$.
    f) $A=\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1\\ a & b & c \end{pmatrix}$, $B=\begin{pmatrix}0 \\ 0 \\1 \end{pmatrix}$.

    5.3 Determine observability Gramian and comments on observability of the following pairs (A,C).
    a) $A=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -6 & -11 & -6 \end{pmatrix}$, $C=\begin{pmatrix}3 & 2 & 1 \end{pmatrix}$.
    b) $A=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -6 & -11 & -6 \end{pmatrix}$, $C=\begin{pmatrix}4 & 5 & 1 \end{pmatrix}$.

    5.4 Consider the state equation which is represented by $\dot{x}=Ax+Bu$, where $A=\begin{pmatrix}0 & 1 & 0 & 0\\ 3w^2 & 0 & 0 & 2w\\ 0 & 0 & 0 & 1\\ 0 & -2w & 0 & 0 \end{pmatrix}$ and $B=\begin{pmatrix}0 & 0\\ 1 & 0 \\ 0 & 0\\ 0 & 1 \end{pmatrix}$. You may have noticed that the given $A$ and $B$ matrices could be reached after linearization of nonlinear state space equation of a satellite (problem 4.2, 4.4) about a steady state solution. The states are radius ($x_1$), angle ($x_3$) and their derivatives ($x_2=\dot{x_1}; x_4=\dot{x_3}$). The input vectors $u_1$ and $u_2$ are radial and tangential thrust respectively.
    a) Is this system controllable from $u$? If $C = \begin{pmatrix}1 & 0\\ 0 & 1 \end{pmatrix}$, is the system observable?
    b) Can the system be controlled if the radial thruster fails? What if the tangential thruster fails?
    c) Is the system observable from $x_1$ only measurement? From $x_2$ only?


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