Overshoot of higher order systems? Due to several textbooks on control, the overshoot of a second order system can be calculated by the formula
$$
\text{Overshoot} = \exp \left[\frac{-\zeta\pi}{\sqrt{1 - \zeta^2}}\right] \,,
$$
where $\zeta$ is the damping ratio of the system.
Question: I wonder whether there also exist explicit formulas for higher order systems (or even arbitrary linear time-invariant systems)? If yes: where can I find them? If no: why not?
 A: The peak overshoot $\text{PO}$ (in percent) of a step response for stable systems can be calculated as the difference between the supremum of the time response $x_{\sup}$ (similar to the maximal value) and the terminal value $x_{\infty}$ of the step response for $t \to \infty$ divided by the terminal value. Hence
$$\text{PO}=100 \%\cdot \dfrac{x_{\sup}-x_{\infty}}{x_{\infty}}.$$
Let us consider the $n^{\text{th}}$ order system with a Heaviside step function $H(t)$ as input and zero initial conditions.
$$a_nx^{(n)}(t)+\ldots+a_1x^{(1)}(t)+a_0x(t)=H(t) \qquad x(0)=x^{(1)}(0)=\ldots=x^{(n-1)}(0)=0$$
In order to determine the solution of the previous ODE we need to determine the homogeneous $x_{\text{h}}(t)$ and the particular solution $x_{\text{p}}(t)$. The general solution is then given as the linear superposition of the homogeneous and particular solution:
$$x(t)=x_{\text{h}}(t)+x_{\text{p}}(t).$$
While the particular solution is easy to determine as $x_{\text{p}}(t)=1/a_0$ (note that $a_0>0$ is necessary for stability by the Stodola criterion) it is much more difficult to determine the homogeneous solution. In general, it is only possible to determine the exact solution for systems with order $n<5$, because we need to determine the roots of the characteristic polynomial of degree $n$ when determining the solution of the system (see Euler's exponential ansatz). It is a well-known fact that in general, it is not possible to determine the roots (using only radicals) for polynomials with degree higher than $4$.
Hence, if we cannot determine the roots of a general polynomial of degree $n\geq 5$ than we can not expect a closed form solution for the $\text{PO}$ for systems of order $n\geq 5$, because we cannot determine the exact form of $x(t)$.      
For systems with an order $n\leq 4$ you might be able to determine the $\text{PO}$ with a closed formula but I expect it to be quite messy (if possible). You will need to solve $\dot{x}(t)=0$ (Fermat criterion for an extremal value of function aka first derivative equal to zero).   
A: I don't know an explicit formula, but maybe the following could be useful. Let $$V(x)=x^T P x,\quad P^T=P$$ be a Lyapunov function for the stable system
$$
\dot x=Ax,\quad x\in\mathbb R^n;
$$
then the overshoot
$$
O\le \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}},
$$
i.e.
$$
\forall t\ge t_0\quad \|x(t)\|\le \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}\|x(t_0)\|,
$$
$\|x\|=\sqrt{x_1^2+\ldots+x_n^2}$.
