---
title: "Controller to change transient properties"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:controller-transient"
source: https://www.bctengineer.com/q/semester_3/control_systems/controller-transient
site: BCT Engineering
---

# Controller to change transient properties

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2078 Kartik, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/controller-transient
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

What kind of controller would you recommend to bring changes in transient properties of a system and how?

## Short answer

To improve transient properties, a PD controller or lead compensator is used. Derivative action adds phase lead, increases effective damping, and reduces overshoot and settling time without affecting steady-state error.

## Answer

To change (improve) the **transient** properties of a system (overshoot, settling time, rise time), a controller with **derivative action** is recommended: a **PD controller** (or derivative/rate feedback), or in compensator terms a **lead compensator**.

#### How PD control changes the transient

$$
G_c(s)=K_p+K_ds=K_p(1+T_ds)
$$

For the standard plant $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ (with $K_p=1$):

$$
\frac{C(s)}{R(s)}=\frac{\omega_n^2(1+T_ds)}{s^2+(2\zeta\omega_n+T_d\omega_n^2)s+\omega_n^2},\qquad\zeta'=\zeta+\frac{T_d\omega_n}{2}
$$

1. **Damping increases** (ζ′ > ζ), so peak overshoot falls.
2. **Settling time falls** ($t_s=4/\zeta'\omega_n$).
3. **Anticipation:** D action responds to the rate of change of error, applying correction before the error becomes large, like braking a car before the stop line.
4. The added zero gives **phase lead**, which raises the phase margin and bandwidth, so the response is faster and more stable.
5. Steady-state error is not changed (the system type is the same); if needed, it is handled separately by I action.

| Transient spec | Effect of PD |
|---|---|
| Overshoot $M_p$ | decreases |
| Settling time $t_s$ | decreases |
| Rise time $t_r$ | decreases slightly |
| Stability | improves |

**Caution:** derivative action amplifies high-frequency noise, so $K_d$ is kept moderate or a filtered derivative is used.

**Example:** $\omega_n=4$ rad/s, $\zeta=0.25$ ($M_p=44\%$, $t_s=4$ s). With $T_d=0.15$ s: $\zeta'=0.25+0.3=0.55$, $t_s\approx4/(\zeta'\omega_n)=1.8$ s, and the overshoot drops from 44% to about 16% (a little above the 12.6% of a pure ζ = 0.55 system, because of the added zero).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which controller is recommended to improve transient properties?
   A. PD (derivative action), or a lead compensator
   B. PI controller
   C. P controller with a very low gain
   D. Lag compensator
   Answer: A. Derivative action adds damping and phase lead, improving overshoot and settling time. (Key point: Recommended controller)

2. For PD control Gc = 1 + Td s on wn^2/[s(s+2 zeta wn)], what is the new damping ratio?
   A. zeta' = zeta*Td*wn
   B. zeta' = zeta + Td*wn/2
   C. zeta' = zeta - Td*wn/2
   D. zeta' = zeta + Td*wn^2
   Answer: B. The s-coefficient becomes 2 zeta wn + Td wn^2, so zeta' = zeta + Td wn/2. (Key point: Effect on damping)

3. What does PD control do to steady-state error?
   A. It removes it completely
   B. It increases it a lot
   C. It does not change it; the system type is unchanged
   D. It removes it for step inputs only
   Answer: C. D action gives zero output for constant error, so I action handles steady-state error. (Key point: Steady state)

4. What is the main practical caution with derivative action?
   A. It reduces phase margin
   B. It makes the system type lower
   C. It reduces damping
   D. It amplifies high-frequency noise
   Answer: D. Gain K_d*omega grows with frequency, so noise is amplified; Kd is kept moderate or filtered. (Key point: Caution)

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