P Controller

A proportional controller is the simplest form of feedback controller, where the control action is directly proportional to the error signal. It is the foundation for understanding all PID-type controllers and is a mandatory topic for both university exams and GATE control systems problems.

Core Concept Explanation

In a proportional controller, the control signal u(t) = Kp * e(t) where Kp is the proportional gain and e(t) is the error between the desired output (setpoint) and the actual output. When the error is large, the controller applies a large correction. When the error is small, the correction is small. This intuitive behavior makes the P controller easy to understand and the simplest starting point for controller design.

As Kp increases, the loop gain increases, which makes the system respond faster and reduces the steady-state error in a type-0 plant. However, increasing Kp also reduces the phase margin of the system, eventually causing oscillations and instability for very large values. There is always a compromise between speed of response and stability when tuning Kp.

A fundamental limitation of the P controller is that it cannot eliminate steady-state error for a type-0 plant with a step input. The error that remains after all transients die out is called offset or proportional offset. For the error to be zero, the controller output must still be nonzero, which is impossible unless e(t) is nonzero. This is why integrators are added in PI and PID controllers.

Mathematical Expression

The transfer function of a P controller is simply Gc(s) = Kp, a constant gain in the s-domain. The closed-loop transfer function with a plant G(s) and unity feedback becomes T(s) = KpG(s) / (1 + KpG(s)). The steady-state error for a unit step input with a type-0 plant is ess = 1 / (1 + Kp*Kdc) where Kdc is the DC gain of the plant. Increasing Kp reduces ess but never makes it exactly zero.

Practical Understanding

Proportional control is used in applications where a small residual error is acceptable and simplicity of implementation is valued. A thermostat with proportional action, a simple motor speed controller, or a hydraulic valve actuator can all use pure P control if the offset is within tolerance. In more precise systems such as CNC machines or robot arms, pure P control is insufficient and is augmented with integral and derivative terms.

The gain margin and phase margin of the closed-loop system decrease as Kp increases. Root locus analysis shows that poles move toward open-loop zeros as gain increases. If all open-loop zeros are in the left half-plane, stability is maintained up to a critical gain, beyond which poles enter the right half-plane and the system becomes unstable.

Given:
Plant G(s) = 5 / (s + 5)
Controller: P with Kp = 3
Input: unit step R(s) = 1/s

Why this formula applies:
Steady-state error formula for type-0 plant with step input.

Formula:
ess = 1 / (1 + Kp * Kdc)
where Kdc = lim s->0 G(s) = 5/5 = 1

Substitution:
ess = 1 / (1 + 3 * 1) = 1 / 4

Calculation:
ess = 0.25

Final Answer:
Steady-state error = 0.25 (25% offset remains with Kp = 3)

Exam Tip: For a type-0 plant, P controller always leaves a nonzero steady-state error for step input. The error = 1/(1 + Kp*Kdc). For a type-1 plant with step input, even a P controller gives zero steady-state error because the plant already has an integrator.

Quick Revision

• P controller transfer function: Gc(s) = Kp (pure gain, no dynamics added).
• Control output: u(t) = Kp * e(t). Larger error produces proportionally larger correction.
• Increasing Kp reduces steady-state error but reduces stability margins (phase margin decreases).
• Steady-state error for type-0 plant with step: ess = 1 / (1 + Kp * Kdc). Cannot be made zero with P alone.
• For type-1 plant (plant already has integrator), P controller gives zero steady-state error for step input.
• Key trap: P controller does NOT eliminate offset for type-0 plants. An integrator is needed for that.