Transfer Function of Closed Loop System — Derivation & Examples
- What is a Transfer Function?
- Closed Loop Control System
- Block Diagram of Closed Loop System
- Derivation for Negative Feedback
- Derivation for Positive Feedback
- Unity Feedback System
- Comparison Table — Positive vs Negative Feedback
- Characteristics of Closed Loop Transfer Function
- Solved Example
- Applications
- FAQs
- Related Articles
In control systems engineering, the transfer function of a closed loop system is one of the most fundamental concepts. It mathematically relates the output of a system to its input using Laplace transforms. Understanding this derivation is essential for analyzing stability, transient response, and steady-state error of any feedback control system.
What is a Transfer Function?
A transfer function is defined as the ratio of the Laplace transform of the output signal to the Laplace transform of the input signal, with all initial conditions assumed to be zero.
The transfer function is a property of the system itself — it does not depend on the input signal. It is expressed as a ratio of polynomials in the complex variable s, making it convenient for frequency-domain analysis.
Closed Loop Control System
A control system in which the control action depends upon the output is called a closed loop or feedback control system. The output is measured, fed back through a sensor (feedback element), and compared with the reference input. The difference (error signal) drives the controller to reduce the error.
Key advantages of closed loop systems over open loop systems include:
- Reduced sensitivity to parameter variations
- Improved disturbance rejection
- Better accuracy and repeatability
- Ability to stabilize inherently unstable systems
Block Diagram of Closed Loop System
The block diagram above shows the standard closed loop control system with the following components:
Derivation for Negative Feedback
Negative feedback is the most common configuration in control systems. The error signal is the difference between the reference input and the feedback signal.
From the block diagram, we can write three fundamental equations:
B(s) = H(s) · Y(s) ...(2)
E(s) = X(s) − B(s) ...(3) [negative feedback]
Step 1: Substitute E(s) from equation (3) into equation (1):
Y(s) = G(s)·X(s) − G(s)·B(s) ...(4)
Step 2: Substitute B(s) from equation (2) into equation (4):
Step 3: Collect Y(s) terms on the left side:
Y(s) [1 + G(s)·H(s)] = G(s)·X(s)
Step 4: The closed loop transfer function (CLTF) for negative feedback:
The denominator 1 + G(s)·H(s) is called the characteristic polynomial. Setting it to zero gives the characteristic equation, whose roots determine system stability.
Derivation for Positive Feedback
In positive feedback, the feedback signal adds to the reference input instead of subtracting:
Following the same algebraic steps as above, the transfer function becomes:
Positive feedback systems are generally unstable and are rarely used in control applications. However, they find use in oscillators, Schmitt triggers, and latching circuits.
Unity Feedback System
When the feedback element is simply a direct connection (H(s) = 1), the system is called a unity feedback system. The output is directly compared with the input without any sensor scaling.
Negative feedback: Y(s)/X(s) = G(s) / [1 + G(s)]
Positive feedback: Y(s)/X(s) = G(s) / [1 − G(s)]
Unity feedback systems are widely used because they simplify analysis and the steady-state error can be directly determined from the system type number.
Comparison — Positive vs Negative Feedback
Characteristics of Closed Loop Transfer Function
- Characteristic Equation: 1 + G(s)·H(s) = 0 determines all closed loop poles
- Open Loop Transfer Function: G(s)·H(s) is used for Bode plots and Nyquist analysis
- Sensitivity: Sensitivity to plant parameter changes is reduced by factor (1 + GH)
- Disturbance Rejection: Effect of disturbances is attenuated by the loop gain
- Steady-State Error: Depends on system type (number of integrators in G(s))
Solved Example
Problem: A unity feedback system has G(s) = 10/(s+2). Find the closed loop transfer function and poles.
= [10/(s+2)] / [1 + 10/(s+2)]
= [10/(s+2)] / [(s+2+10)/(s+2)]
= 10 / (s + 12)
Closed loop pole: s = −12 (stable, in left half plane)
Notice that the open loop pole was at s = −2, but negative feedback moved it to s = −12, making the system respond faster (smaller time constant: τ = 1/12 s instead of 1/2 s).
Applications
- PID Controllers: Temperature, speed, and position control in industrial automation
- Cruise Control: Automotive speed regulation using negative feedback
- Voltage Regulators: Maintaining constant output voltage despite load changes
- Robotic Arms: Precise position and force control using encoder feedback
- Power Systems: Automatic generation control (AGC) for frequency regulation
Frequently Asked Questions
1. What is the transfer function of a closed loop system?
The transfer function of a closed loop system with negative feedback is Y(s)/X(s) = G(s)/[1 + G(s)·H(s)], where G(s) is the forward path transfer function and H(s) is the feedback transfer function.
2. Why is negative feedback preferred over positive feedback in control systems?
Negative feedback improves stability, reduces sensitivity to parameter variations, increases bandwidth, and provides better disturbance rejection. Positive feedback tends to make systems unstable and is only used in special applications like oscillators.
3. What is the characteristic equation of a closed loop system?
The characteristic equation is 1 + G(s)·H(s) = 0. Its roots are the closed loop poles that determine system stability — if all roots have negative real parts, the system is stable.
4. What happens to gain in a negative feedback system?
The overall gain is reduced by a factor of (1 + GH) compared to the open loop gain. This is called "gain sacrifice" — we trade gain for improved stability, reduced sensitivity, and increased bandwidth.
5. What is the difference between open loop and closed loop transfer function?
The open loop transfer function is G(s)·H(s) — the product of forward and feedback path gains without closing the loop. The closed loop transfer function is G(s)/[1 ± G(s)·H(s)] — the actual input-output relationship when feedback is connected.