A system can be modeled by $(z + 3)(z + 2)(z + 1) + C = 0$, where $C > 0$, and $z = x + iy$. When it is marginally stable $Re(z) = 0$.
What are the values of the roots in marginally stable condition? At what values of $C$ do they occur?
Please give me help, as I don't know how to go about this question.
Thanks in advance, Adithya
If $(z + 3)(z + 2)(z + 1) + C = 0$ is the characteristic equation for a transfer function (usually denoted in powers of $s$), we can apply the Routh-Hurwitz criterion. Unless there is more to the question, I will assume this is the case. We have that $$ (z + 3)(z + 2)(z + 1) + C = z^3 + 6z^2 + 11z + 6 + C $$ Then \begin{array}{ccc} z^3 & 1 & 11\\ z^2 & 6 & 6 + C\\ z^1 & \frac{6 + C - 66}{6} & 0\\ z^0 & 6 + C \end{array} For the system to be stable, all the coefficients in the second column must have the same sign. Therefore, $\frac{6 + C - 66}{6} > 0$ and $6 + C > 0$. $$ C > -6 $$ and $$ C > 60 $$ Thus, $C > 60$. Now, if $C = 60$, $(z + 3)(z + 2)(z + 1) + C = z^3 + 6z^2 + 11z + 6 + C$, will have two roots on the imaginary axis. Using the second row of the Routh-Hurwitz table, we see that $z = \pm i\sqrt{11}$ are the two roots on the imaginary axis.