Formula & Calculator
Closed-Loop Transfer Function
Relates system output to input for a feedback control loop.
Interpretation
T(s) = G(s) / (1 + G(s)H(s)). Describes the input‑output relationship of a feedback control system. Used to analyse stability and performance in the Laplace domain.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T(s) | Closed-loop transfer function | |
| G(s) | Forward path transfer function | |
| H(s) | Feedback transfer function |
What it means
The closed‑loop transfer function T(s) relates the output to the reference input in a unity feedback system. G(s) is the forward path transfer function (plant + controller) and H(s) is the feedback path transfer function. The denominator 1 + G(s)H(s) = 0 gives the characteristic equation, whose roots determine system stability. This formula is fundamental in control theory for analysing steady‑state error, transient response, and frequency response. It is used to design controllers (e.g., PID, lead‑lag) and to evaluate system robustness. Understanding T(s) is essential for control engineers to predict and shape system behaviour in the frequency domain.
Worked example
Closed‑Loop Transfer Function – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| G(s) | 1/s |
| H(s) | 1 |
| Parameter | Value |
|---|---|
| G(s) | 1/(s+1) |
| H(s) | 1 |
Common mistakes
- Closed‑loop transfer function: T(s) = G(s) / (1 + G(s)H(s)) – for unity feedback (H=1), it simplifies to G/(1+G).
- Sign convention: Negative feedback uses the denominator (1 + GH) – do not use (1 − GH) unless positive feedback.
- G(s) and H(s): Must be proper rational functions – check degrees of numerator/denominator.
- Stability: Poles of T(s) must be in the left half‑plane for stability – check via Routh‑Hurwitz.
- Laplace domain: Assumes zero initial conditions – for non‑zero, add initial condition terms.
Applications
The closed‑loop transfer function, T(s) = G(s)/(1 + G(s)H(s)), is the cornerstone of classical control theory. It relates the output of a feedback control system to its input, accounting for the plant dynamics G(s) and the feedback path H(s). Engineers use this formula to analyse system stability (via poles of the characteristic equation), to evaluate steady‑state accuracy, and to design compensators. By examining the frequency response and root locus, they can tune controllers to meet performance specifications such as overshoot, settling time, and bandwidth. This formula is essential for designing controllers for motors, robotic arms, aircraft, and industrial processes. Understanding closed‑loop transfer functions enables the systematic design of robust and stable control systems in aerospace, automotive, and manufacturing domains.
- Stability analysis of feedback control systems
- Controller design using root locus and Bode plots
- Performance evaluation of servo systems
- Aerospace flight control and autopilot design
- Process control in chemical and power plants
Frequently Asked Questions
For a unity feedback system, the closed‑loop transfer function relating output C(s) to input R(s) is T(s) = G(s) / (1 + G(s)·H(s)), where G(s) is the forward path transfer function and H(s) is the feedback path transfer function. It characterises the overall system dynamics.
The denominator is the characteristic equation. Its roots (poles) determine the stability and transient response of the closed‑loop system. For stability, all poles must lie in the left half of the s‑plane.
A unity feedback system has H(s) = 1. The closed‑loop transfer function then becomes T(s) = G(s) / (1 + G(s)). This is a common configuration for controller design.
The steady‑state error e_ss = lim_{s→0} s·E(s), where E(s) = R(s) – C(s). From the closed‑loop transfer function, the error transfer function can be derived. The error depends on the system type (number of integrators).
Open‑loop transfer function is G(s) (or G(s)H(s) for the loop gain). Closed‑loop includes feedback and is given by T(s). Feedback alters the system dynamics (poles move) and improves disturbance rejection and tracking.
The system type is the number of poles at s=0 in the forward path G(s). Type 0 has no integrator, Type 1 has one, etc. This determines the steady‑state error constants (position, velocity, acceleration).
By choosing the controller parameters, we place the poles of the closed‑loop system to achieve desired performance (e.g., settling time, overshoot). Methods like root locus, pole placement, and frequency response design are used.
Feedback can stabilise an unstable open‑loop system by shifting the poles into the left half‑plane, provided the controller is properly designed. This is the essence of feedback control.
The final value theorem states that lim_{t→∞} f(t) = lim_{s→0} s·F(s). It is used to find the steady‑state output or error of a system from its Laplace transform.
Controller design (PID, lead‑lag), stability analysis (Routh‑Hurwitz, Nyquist), performance analysis (overshoot, settling time), and simulation of control systems.