Question

{PD controller:

• A. Decreases steady state error and improves stability
• B. Rise time decreases
• C. Transient response becomes poorer
• D. Increases steady state error

Hint:

Remember these core controller effects for quick reference: - PD Controller (Adds a Zero): Acts like a high-pass filter. It reduces rise time, decreases overshoot, and improves transient stability, but does not change the steady-state error. - PI Controller (Adds a Pole at Origin): Acts like a low-pass filter. It eliminates steady-state error but can increase overshoot and degrade transient stability.

Answer 1

The Correct Option is B

Concept: A Proportional-Derivative (PD) controller determines its control output based on a combination of the current tracking error and its rate of change over time. The mathematical transfer function of a PD controller is given by: \[ G_c(s) = K_p + K_d s = K_p (1 + T_d s) \] Adding a PD controller to a system loop introduces a feedforward zero into the overall open-loop transfer function, which significantly alters both the transient performance and stability characteristics of the system.

Step 1: Analyzing the Effect of the Added Zero
Introducing a zero into the forward path adds a predictive component to the control loop. It anticipates changes in the error signal based on its current trend or velocity. This action adds a dampening effect that suppresses large oscillations, reducing the system's peak overshoot and improving overall relative stability.

Step 2: Effect on System Speed (Rise Time)
Because the derivative action responds to how fast the error is changing, it allows the system to react more aggressively at the start of a transient change when the error slope is steep. This quick initial response drives the system toward its target value much faster, which significantly reduces the rise time ($t_r$). A shorter rise time means the system responds faster to inputs.

Step 3: Effect on Steady-State Error
Let us look at the steady-state error characteristics. The derivative term $K_d s$ becomes zero under steady-state conditions because the error stops changing ($s \rightarrow 0$). This means a PD controller does not change the type number of the system, so it cannot reduce the steady-state tracking error the way an Integral (I) controller does.

Step 4: Evaluating the Options

• Option A and D: PD controllers do not change the system type, so they do not inherently decrease or increase the steady-state error.
• Option C: The transient response improves (less overshoot, faster settling), rather than becoming poorer.
• Option B: The rise time decreases, which correctly describes the faster response provided by PD control.