Transient specifications

There are several key parameters in the time response (specifically the transient response) of a control system that we design for:¹ Important parameters we’re interested in:

• The yellow line denotes an arbitrary threshold. The time such that it never goes beyond the threshold is called the settling time $T_s$.
  • Usually we set this threshold as within 2%.
  • $T_s=4\tau =\frac{4}{\zeta {\omega }_n}$
• The time between 10% and 90% of the first rise to the final value is the rise time $T_r$. We can define $T_r$ such that $T_r\approx \frac{1.8}{{\omega }_n}$.
• The peak time $T_p$ is the time at which the system attains its maximum value.

• The percent overshoot is defined as: $\frac{y(T_p)-fv}{fv}$, where $fv$ is the final value of the response as ${\lim }_{t\to \infty }$. We can also define it as:

$$\%OS=\frac{y(T_p)-f_v}{f_v}=100\exp \left(-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}\right)$$

Design

Often we want to design for specific desired specifications. The exponent just denotes “desired” and isn’t a real exponent.

$$T_r\le T_r^d$$ $$T_s\le T_s^d$$ $$\%OS\le \%O{S}^d$$

For all to be satisfied, we must have a pole in the shaded area below.¹ We determine the bounds of the shaded area by finding ${\omega }_n$ and $\theta$. Note that our formulas above (involving ${\omega }_n,\zeta$) are only valid for second-order systems with complex conjugate poles. However, we can use them for higher-order systems if the additional poles and zeroes are sufficiently to the left of the dominant pair of poles (at least 5 times smaller).

Footnotes

1. From Modern Control Systems, by Richard Dorf and Robert Bishop. ↩ ↩²