Your room air-conditioner, an alternator’s automatic voltage regulator (AVR), a water-tank float valve and even your body temperature are all control systems. The first chapter of every control-engineering course — and a regular question in BCS (EEE), PGCB, BPDB, power-plant and instrumentation job exams — is the difference between open-loop and closed-loop (feedback) control. This post explains it with block diagrams, an animated simulation, transfer-function examples and MCQs.
- Open-loop: control action does not depend on the output. Simple and cheap, but cannot correct errors or disturbances.
- Closed-loop: output is measured and fed back; the controller acts on the error = reference − feedback. Accurate and self-correcting, but more complex and can become unstable.
- Closed-loop transfer function (negative feedback): C/R = G / (1 + GH).
- Feedback reduces sensitivity to parameter changes and disturbances by the factor (1 + GH), increases bandwidth, but reduces overall gain.
1. Basic Terms
- System: a combination of components acting together to perform a function.
- Control system: a system that regulates its output to a desired value by manipulating its input.
- Reference input r(t) / set point: the desired value (e.g. 25 °C).
- Controlled variable / output c(t): the actual value (room temperature).
- Plant (process): the part being controlled (room, motor, boiler).
- Controller: decides the control action (thermostat, PID controller, PLC).
- Actuator: applies the action (heater, valve, motor drive).
- Sensor / feedback element H: measures the output (thermistor, tachogenerator, CT/PT).
- Error signal e(t): r(t) − b(t), the difference between reference and feedback.
- Disturbance: an unwanted input that affects the output (open window, load change).
2. Open-Loop Control System
In an open-loop system, the control action is independent of the output. The controller sends a predetermined command; nobody checks whether the result is correct.
Examples: a washing machine running on a fixed timer, an electric toaster, a fixed-time traffic signal, a hand dryer, an electric immersion heater, a stepper motor driven by a fixed pulse train.
Advantages: simple, cheap, easy to maintain, generally stable. Disadvantages: inaccurate, cannot correct for disturbances or parameter changes, needs frequent recalibration.
3. Closed-Loop (Feedback) Control System
In a closed-loop system, the output is measured by a sensor and fed back to be compared with the reference. The controller acts on the error, continuously driving it toward zero.
Examples: air-conditioner/refrigerator thermostat, automatic voltage regulator (AVR) of an alternator, governor speed control of turbines, automatic power-factor correction panel, cruise control, water-level control with float, servo motor position control, auto-pilot, human body temperature regulation.
4. Seeing the Difference (Animation)
Both systems heat a room to 25 °C. The open-loop heater is set to a fixed power that gives 25 °C under normal conditions. At 20 minutes a window is opened (a disturbance). The open-loop room settles 6 °C too cold because the heater never “knows”. The closed-loop thermostat sees the error and increases heating until the temperature returns to 25 °C.
5. Open-Loop vs Closed-Loop – Comparison Table
| Basis | Open-loop | Closed-loop |
|---|---|---|
| Feedback | Absent | Present |
| Control action depends on output? | No | Yes (on error) |
| Accuracy | Low; depends on calibration | High |
| Effect of disturbances / parameter change | Not corrected | Automatically corrected |
| Construction & cost | Simple, cheap | Complex, costly (sensors, comparator) |
| Stability | Generally stable | Can become unstable / oscillate if gain too high |
| Bandwidth | Smaller | Larger (faster response) |
| Overall gain | G | G/(1 + GH) — reduced |
| Also called | Non-feedback system | Feedback / automatic control system |
| Examples | Toaster, timer washing machine, traffic light | AC thermostat, AVR, governor, servo |
6. Transfer Function of a Feedback System
The transfer function is the ratio of the Laplace transform of the output to that of the input, with all initial conditions zero. For the loop in Fig. 2:
E(s) = R(s) − H(s)C(s), C(s) = G(s)E(s) ⇒ C(s)/R(s) = G(s) / [1 + G(s)H(s)]- Positive feedback: C/R = G/(1 − GH) — used in oscillators, rarely in control.
- Unity feedback (H = 1): C/R = G/(1 + G).
- Characteristic equation: 1 + G(s)H(s) = 0. Its roots (closed-loop poles) decide stability — all must lie in the left half of the s-plane.
- G(s)H(s) is called the open-loop (loop) transfer function.
7. Effects of Feedback
- Reduces sensitivity to parameter variations: S = (∂T/T)/(∂G/G) = 1/(1 + GH).
- Reduces the effect of disturbances and noise entering the forward path.
- Reduces overall gain by the factor (1 + GH) — the price paid for accuracy.
- Increases bandwidth and speeds up response (smaller time constant).
- Reduces steady-state error: for a unity-feedback type-0 system with step input, ess = 1/(1 + Kp).
- Affects stability — feedback can stabilise an unstable plant, or make a stable one oscillate if the loop gain is too high.
8. Solved Problems
Problem 1. A system has forward gain G = 100 and feedback H = 0.1. Find the closed-loop gain. If G falls by 10 %, what is the percentage change in closed-loop gain?
- T = 100/(1 + 100 × 0.1) = 100/11 = 9.09
- With G = 90: T = 90/(1 + 9) = 9.00 ⇒ change = (9.09 − 9.00)/9.09 ≈ 1 %
- A 10 % change in G causes only ≈ 1 % change in T — sensitivity reduced by (1 + GH) = 11.
Problem 2. Find the closed-loop transfer function for G(s) = 10/(s + 2) with unity feedback, and its time constant.
- T(s) = [10/(s + 2)] / [1 + 10/(s + 2)] = 10/(s + 12)
- Open-loop time constant = 1/2 = 0.5 s; closed-loop time constant = 1/12 ≈ 0.083 s — feedback made the system six times faster.
- DC gain: open loop 5, closed loop 10/12 = 0.83 ⇒ steady-state error for unit step = 1 − 0.83 = 1/(1 + 5) = 0.167.
9. Practice MCQs (click to see answer)
1. In an open-loop control system: (a) Output is fed back (b) Control action is independent of output (c) Error is measured (d) Sensor is essential
2. Closed-loop transfer function with negative feedback is: (a) G/(1 − GH) (b) G/(1 + GH) (c) GH/(1 + G) (d) 1 + GH
3. Which is an example of closed-loop control? (a) Toaster (b) Fixed-time traffic signal (c) Air-conditioner with thermostat (d) Electric hand dryer
4. Feedback in a control system reduces: (a) Bandwidth (b) Sensitivity to parameter variation (c) Accuracy (d) Stability always
5. The characteristic equation of a feedback system is: (a) G = 0 (b) H = 0 (c) 1 + GH = 0 (d) GH = 1
6. A major disadvantage of closed-loop systems is: (a) Inaccuracy (b) Possibility of instability (c) No disturbance rejection (d) Low bandwidth
7. The AVR of an alternator is a: (a) Open-loop system (b) Closed-loop system (c) Manual system (d) Non-linear open system
8. If G = 50 and H = 1, the closed-loop gain is: (a) 50 (b) 51 (c) 0.98 (d) 1
9. Transfer function is defined assuming: (a) Non-zero initial conditions (b) Zero initial conditions (c) Non-linear system (d) Time-varying system
10. Steady-state error of a unity-feedback type-0 system for unit step input is: (a) 0 (b) 1/(1 + Kp) (c) 1/Kv (d) ∞
10. বাংলায় দ্রুত রিভিশন
- ওপেন-লুপ: আউটপুট পরিমাপ করে ফেরত দেওয়া হয় না; সহজ ও সস্তা, কিন্তু ত্রুটি বা বিঘ্ন সংশোধন করতে পারে না। উদাহরণ: টোস্টার, টাইমার ওয়াশিং মেশিন, নির্দিষ্ট-সময়ের ট্রাফিক সিগন্যাল।
- ক্লোজড-লুপ: সেন্সর দিয়ে আউটপুট মেপে ফিডব্যাক দেওয়া হয়; কন্ট্রোলার ত্রুটি (রেফারেন্স − ফিডব্যাক) অনুযায়ী কাজ করে। উদাহরণ: থার্মোস্ট্যাট, অল্টারনেটরের AVR, টারবাইনের গভর্নর।
- ক্লোজড-লুপ ট্রান্সফার ফাংশন: C/R = G/(1 + GH); বৈশিষ্ট্য সমীকরণ: 1 + GH = 0।
- ফিডব্যাকের প্রভাব: সংবেদনশীলতা ও বিঘ্নের প্রভাব কমে, ব্যান্ডউইথ বাড়ে, সামগ্রিক গেইন কমে, তবে অস্থিরতার ঝুঁকি থাকে।
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References: Katsuhiko Ogata, Modern Control Engineering; I. J. Nagrath & M. Gopal, Control Systems Engineering; Norman Nise, Control Systems Engineering. Images and animations: original work by Totalecer.
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