5 Transform Methods

5.2.2 Design Parameters in the z PlaneAll Activities

Assume that we are given the following transient response specifications:

Percent overshoot: $PO \le 15\% \rarr \zeta \ge 0.5$.
Rise time: $t_r \le 6 \text{sec} \rarr \omega_n \le \frac{1.8}{6} = 0.3$.
Settling time: $1\% t_s \le 20 \text{sec} \rarr \zeta\omega_n \ge 0.23$.

Regions in the s-plane corresponding to these specifications are shown in Figure 1.

Allowable regions in the z-plane.
The s-plane regions shown in the figure below can be mapped to the z plane using the pole mapping $z = e^{sT}$ . They are shown in the z-plane drawn in Figure 2. Recall the discussion in Section 2.5 of the textbook... lines of constant damping ratio $\zeta$ are logarithmic spirals; lines of constant natural frequency $\omega_n$ are nearly circular near $z = 1$, but are increasingly distorted as $\omega_n$ increases and the contours move to the left; and lines of constant "real part" or $\zeta\omega_n$ are contours of constant radius. (NOTE: to draw the $\omega_n$ and $t_s$ contours one must select a sampling period $T$ )

2nd order response 2nd order response
Figures 1 (left) and 2 (right): Regions in the s and z-plane corresponding to restrictions on PO, $t_r$, and $t_s$. z plane contours use a sample time $T$=1 sec.

When designing a control system, to achieve the specified transient response we must have the dominant poles within the allowable regions of Figure 1. Because clear dominance may not occur, the final design should always be checked with a simulation to verify acceptable response.


Use the controls below (or control "popout" ) to change lines of constant $\zeta$ (PO), $\omega_n$ ($t_r$) and $t_s$in Figures 1 and 2




    Assume T=1 for the following questions
  1. What area of the s plane corresponds to a percent overshoot of less than 10%? Use Eqn 5.24 to relate PO and $\zeta$.
  2. What region on the z plane corresponds to a percent overshoot of less than 10%.?
  3. What area of the s plane corresponds to a settling time of less than 2 sec?
  4. What area of the z plane corresponds to a settling time of less than 2 sec? How is this affected by the sampling time?
  5. What point on the s plane corresponds to $\omega_n = 2$rad/s and a damping ratio $\zeta = 0.8$? What is the settling time for that system?
  6. Identify the corresponding point, from the previous question, on the z-plane.
  7. Repeat the questions assuming $T$= 2 sec.
Controls
Constant $\zeta$: