Question 10.19: Stability Analysis in State Space a. Compute the poles of th...
Stability Analysis in State Space
a. Compute the poles of the system described by
\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{cc} 0 & 1 \\ 0 & -16.883 \end{array}\right] \mathbf{x}+\left[\begin{array}{c} 0 \\ 3.778 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \mathbf{x} . \end{aligned}
b. Verify the results by converting the state-space representation to a transfer function and then identifying the poles of the transfer function.
a. The characteristic equation is
|s \mathbf{I}-\mathbf{A}|=\left|\begin{array}{cc} s & -1 \\ 0 & s+16.883 \end{array}\right|=s^{2}+16.883 s=0
which yields the poles s_{1}=0 and s_{2}=-16.883.
b. As presented in Section 4.4, state-space equations for a single-input/single-output system can be converted to a transfer function using
G(s)=\mathbf{C}(s \mathbf{I}-\mathbf{A})^{-1} \mathbf{B}+D .
Substituting the system matrices \mathbf{A}, \mathbf{B}, \mathbf{C}, and D, which is 0 in this example, gives
\begin{aligned} G(s) & =\left[\begin{array}{ll} 1 & 0 \end{array}\right]\left[\begin{array}{cc} s & -1 \\ 0 & s+16.883 \end{array}\right]^{-1}\left[\begin{array}{c} 0 \\ 3.778 \end{array}\right]+0=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \frac{\left[\begin{array}{cc} s+16.883 & 1 \\ 0 & s \end{array}\right]}{s^{2}+16.883 s}\left[\begin{array}{c} 0 \\ 3.778 \end{array}\right] \\ & =\frac{3.778}{s^{2}+16.883 s} . \end{aligned}
The characteristic equation is s^{2}+16.883 s=0, which yields the poles at 0 and -16.883 . The results agree with the poles obtained in Part (a).