PD Controller

A PD controller combines proportional action with derivative action to improve transient response by anticipating future error trends. While the P term reacts to the current error, the D term reacts to the rate of change of error, providing a predictive correction that reduces overshoot and improves damping. GATE frequently tests the effect of derivative action on system poles and stability margins.

Core Concept Explanation

The derivative term in a PD controller computes de(t)/dt, the rate at which the error is changing. If the error is decreasing rapidly, the derivative term produces a braking action that prevents overshoot. If the error is increasing rapidly, the derivative term provides an additional corrective boost ahead of the proportional term. This anticipatory behavior is why derivative action is often called predictive control.

In the frequency domain, the derivative term Kd*s adds a zero to the open-loop transfer function at s = -Kp/Kd. This zero adds positive phase (phase lead) to the system at frequencies above Kp/Kd. The increased phase margin results in better damping and reduced overshoot. From a root locus perspective, the zero attracts poles toward the left half-plane, making closed-loop poles more stable and better damped.

The main drawback of pure derivative action is noise amplification. Because differentiation amplifies high-frequency components, even small measurement noise in the feedback signal gets magnified, causing rapid fluctuations in the control output. In practice, the derivative term is always implemented with a first-order filter in series, limiting the derivative action to a finite bandwidth. The ideal PD controller is therefore a theoretical construct.

Mathematical Expression

The transfer function of a PD controller is Gc(s) = Kp + Kd*s, which can also be written as Kp(1 + Td*s) where Td = Kd/Kp is the derivative time constant. In the Bode plot, this controller has a magnitude that increases at +20 dB/decade above the frequency 1/Td, and contributes a phase lead that peaks at +90 degrees asymptotically. The zero is at s = -1/Td = -Kp/Kd.

The characteristic equation of the closed-loop system with a PD controller has higher damping because the zero shifts the closed-loop poles. If the original system without the controller has damping ratio zeta, adding derivative action with appropriate Td increases zeta, reducing overshoot. For a second-order plant, the overshoot percentage can be precisely controlled by choosing Td to place the zero at the desired location.

Practical Understanding

PD controllers are used in servo systems, robotic joint control, and positioning systems where fast and accurate response is needed. A PD controller does not eliminate steady-state error since it has no integral action. It is therefore used when the plant naturally has zero or acceptable steady-state error (for example, a type-1 plant), or when it is combined with other elements to handle steady-state behavior. The derivative action is most effective for second-order and higher plants where oscillatory transients are the primary concern.

Given:
Plant G(s) = 1 / (s^2 + 2s + 1)
PD Controller: Kp = 5, Kd = 2
Find closed-loop characteristic equation.

Why this formula applies:
PD controller adds zero at s = -Kp/Kd = -5/2 = -2.5
Closed-loop characteristic equation = 1 + Gc(s)*G(s) = 0

Formula:
Gc(s) = 5 + 2s
Open-loop = (5 + 2s) / (s^2 + 2s + 1)

Substitution:
Characteristic equation:
1 + (5 + 2s)/(s^2 + 2s + 1) = 0
s^2 + 2s + 1 + 5 + 2s = 0

Calculation:
s^2 + 4s + 6 = 0
Roots: s = (-4 +/- sqrt(16-24)) / 2 = -2 +/- j*sqrt(2)

Final Answer:
Closed-loop poles at s = -2 +/- j1.41
Damping ratio zeta = 2/(2*sqrt(2)) = 0.707 (well damped, ~4% overshoot)

Exam Tip: PD controller adds a zero at s = -Kp/Kd in the open-loop transfer function. On root locus, this zero pulls poles toward the left half-plane, improving damping. PD never eliminates steady-state error; it only improves transient response. Ideal PD is not physically realizable due to noise amplification.

Quick Revision

• PD controller: Gc(s) = Kp + Kd*s = Kp(1 + Td*s). Adds a zero at s = -Kp/Kd.
• Derivative action senses rate of change of error and provides anticipatory correction.
• Zero in open-loop TF adds phase lead, increases phase margin, improves damping and reduces overshoot.
• Does NOT eliminate steady-state error since there is no integrator. Offset may still remain for type-0 plants.
• Major drawback: amplifies high-frequency noise in feedback signal. Practical derivative term uses filter.
• Root locus perspective: zero at s = -Kp/Kd attracts closed-loop poles toward left half-plane.
• Exam trap: PD controller is a phase-lead type. Do not confuse adding a zero with adding an integrator.