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Open-Loop vs Closed-Loop Control: Disturbances and Stability

Open-loop control decides the manipulated variable from the setpoint and applies it to the plant, without measuring the resulting controlled variable to check it. Closed-loop control measures the controlled variable with a sensor, compares it with the setpoint and keeps correcting the manipulated variable to make the difference (the error) smaller. Because the signal returns from the output to the input side and goes around a loop, it is also called feedback control. Open-loop control is simple and cheap, and with a stable plant the controller itself never causes oscillation, but it cannot correct deviations caused by disturbances or changes in the plant. Closed-loop control corrects them automatically, but it needs a sensor and oscillates or becomes unstable if the gain is raised too far.

AspectOpen-loop controlClosed-loop control (feedback control)
Signal flowSetpoint → controller → plant → controlled variable; the output is not fed backThe controlled variable is measured by a sensor and the difference from the setpoint (the error) is fed back to the controller; the signal goes around a loop
Sensor (measuring the controlled variable)Not neededNeeded; the sensor's accuracy and noise limit the accuracy of control
Response to disturbancesCannot correct them; if the load or ambient temperature changes, the output stays off targetCorrects any deviation that appears as an error, whatever its cause
Changes in the plantAppear directly in the result: if the plant gain drops by 50%, the output drops by 50%Their effect is reduced: with negative feedback, G = 100 and H = 0.1, halving G lowers the closed-loop gain by only about 8.3%
Speed of responseOnly the plant's own lag; it acts without waiting for an error to appearThere is a delay, because it corrects after an error appears; with a higher gain, however, it can respond faster than the plant alone
StabilityWith a stable plant, the controller never causes oscillation or divergenceOscillates and diverges if the gain is too high or lags such as dead time are large; checked with gain margin and phase margin
Steady-state errorCan be 0 if the model is accurate, but errors from model mismatch and disturbances remain as they areProportional action alone leaves an error (1/(1 + G(0)) for a step in a type-0 system); adding integral action makes it 0
Structure and costSimple and cheap; accuracy depends on how accurate the model and parts areNeeds a sensor and a circuit or software that compares and corrects; the loop must be designed and tuned with stability in mind
ExamplesA toaster that browns bread by time alone, positioning a stepper motor by counting pulses, traffic lights that switch at fixed timesRoom temperature in an air conditioner, cruise control in a car, the output voltage of a DC-DC converter, motor speed and position

Figures are typical orders of magnitude and vary by process generation and product.

When to use which

Open-loop control
When disturbances are small and the plant is well known and does not change much. It is also used when a sensor cannot be fitted or would be too expensive. With a stable plant there is no risk of oscillation and tuning is simple.
Closed-loop control (feedback control)
When the result must match the setpoint accurately despite disturbances, load changes, part-to-part variation and aging. Most control of temperature, speed, position and voltage is of this kind, with PID control as the typical example.

FAQ

Is feedforward control a kind of open-loop control?

Looking at the feedforward path alone, yes: it decides the manipulated variable from a model without measuring the controlled variable, so it has an open-loop structure. Because it measures disturbances or setpoint changes and adds a canceling action before an error appears, it responds quickly, but it cannot correct unmeasured disturbances or model errors. It is therefore usually combined with feedback control, which corrects what remains. The feedforward path lies outside the loop, so it does not change the loop's stability.

Why is closed-loop control robust to changes in the plant?

With negative feedback, forward gain G and feedback gain H, the closed-loop gain is G/(1 + GH). When GH is much larger than 1, this is about 1/H and hardly depends on G. With G = 100 and H = 0.1 the closed-loop gain is about 9.09, and even if G halves to 50 it is about 8.33, a drop of only about 8.3%. In open-loop control, halving G halves the result.

Why does closed-loop control become unstable?

Because of lags in the plant and sensor, corrective actions take effect late. With a high gain, the controller corrects a deviation so strongly that it overshoots in the opposite direction, and repeating this causes oscillation. If the loop gain is 1 (0 dB) or more at the frequency where the phase of the loop transfer function reaches −180°, the oscillation keeps growing and diverges. Designers check how far the loop is from instability with the gain margin and phase margin.

Check your understanding

Which of the following correctly describes feedback control (closed-loop control)?

  1. AIt runs only on a predetermined manipulated variable or procedure without measuring the controlled variable, so a deviation caused by a disturbance is not corrected automatically
  2. BIt measures the controlled variable, compares it with the setpoint and changes the manipulated variable to reduce the difference (the error), so it can reduce the effect of disturbances and of changes in the plant
  3. CIt measures a disturbance and applies a manipulated variable that cancels it before it reaches the controlled variable, so it always drives the error to zero, even for unmeasured disturbances and model errors
Show answer and explanation

Answer: B. It measures the controlled variable, compares it with the setpoint and changes the manipulated variable to reduce the difference (the error), so it can reduce the effect of disturbances and of changes in the plant

Feedback control compares the controlled variable measured by a sensor with the setpoint and decides the manipulated variable according to the error. Because the signal returns from the output to the input side and goes once around a loop, it is also called closed-loop control. Its strength is robustness to disturbances and to changes in the plant, because it corrects the deviation of the result whatever the cause. On the other hand, it corrects only after an error has appeared, so there is a delay, and too much gain makes the loop oscillate or become unstable.

Which of the following is the correct reason for combining feedforward control with feedback control?

  1. AFeedforward can respond to measurable disturbances and setpoint changes before an error appears, but it cannot correct unmeasured disturbances or model errors, so feedback takes care of the rest
  2. BAdding feedforward changes the gain and phase of the loop transfer function and increases the phase margin, so the loop stays stable even with a larger feedback gain
  3. CWith feedback alone a steady-state error always remains even with integral action, so feedforward is essential to bring the steady-state error to a constant setpoint or disturbance to zero
Show answer and explanation

Answer: A. Feedforward can respond to measurable disturbances and setpoint changes before an error appears, but it cannot correct unmeasured disturbances or model errors, so feedback takes care of the rest

Feedforward control measures disturbances or setpoint changes and, before they affect the controlled variable, applies a manipulated variable calculated from a model. It does not wait for an error to appear, so it responds quickly, but it assumes an accurate model and measurable disturbances. A common combination is therefore to let feedforward handle the main disturbances and setpoint tracking and to let feedback correct the remaining deviation. The feedforward path lies outside the loop, so it does not change the stability of the loop (the phase margin and so on).

In a negative-feedback block diagram, the forward path has the transfer function G(s) and the feedback path has the transfer function H(s). Which of the following is the closed-loop transfer function Y(s)/R(s) from the setpoint R(s) to the output Y(s)?

  1. AG(s)/(1 − G(s)H(s))
  2. BG(s)H(s)/(1 + G(s)H(s))
  3. CG(s)/(1 + G(s)H(s))
Show answer and explanation

Answer: C. G(s)/(1 + G(s)H(s))

The error is E(s) = R(s) − H(s)Y(s) and the output is Y(s) = G(s)E(s). Eliminating E(s) from the two equations gives Y(s) = G(s)R(s) − G(s)H(s)Y(s), and so Y(s)/R(s) = G(s)/(1 + G(s)H(s)). The term G(s)H(s) in the denominator is the transfer function that a signal sees going once around the loop, called the loop transfer function (open-loop transfer function). With unity feedback, H(s) = 1, it becomes G(s)/(1 + G(s)).

  1. A negative-feedback system has a forward gain G = 100 and a feedback gain H = 0.1. When G halves to 50 because a component degrades, which of the following is closest to the fractional decrease in the closed-loop gain?
  2. In a unity-feedback system the forward transfer function is G(s) = 9/(1 + 2s). Which of the following is the steady-state error for a step setpoint of size 1?
  3. Which of the following correctly describes the gain margin and phase margin read from the Bode plot of the loop transfer function?

Practice Control Engineering Basics questions

Key terms explained

Feedback control
Control that measures the controlled variable with a sensor, compares it with the setpoint and changes the manipulated variable to reduce the difference (the error). Because the signal goes around a loop, it is also called closed-loop control. It corrects deviations in the result whatever their cause, so it is robust to disturbances and plant changes, but it corrects only after an error appears, so there is a delay, and too high a gain leads to oscillation or instability. PID control is the typical example, widely used to control temperature, speed, voltage, position and more.
Feedforward control
Control that measures disturbances or setpoint changes and, before they affect the controlled variable, applies a manipulated variable calculated from a model of the plant. It does not wait for an error to appear, so it responds quickly, but it cannot correct model errors or unmeasured disturbances, so it is usually combined with feedback control. Its path lies outside the loop, so it does not change the stability of the feedback loop. In semiconductor fabs, adjusting the conditions of a process step based on measurements from an earlier step is also called feedforward control.
Block diagram
A diagram of a control system that shows each element as a box (block) containing its transfer function, the flow of signals as arrows, the addition and subtraction of signals as summing points and the branching of signals as pickoff points. Blocks in series combine into the product of their transfer functions and blocks in parallel into the sum, and a negative-feedback loop can be replaced by a single block G/(1 + GH). Repeating such equivalent transformations gives the transfer function from the setpoint or a disturbance to the output.
Transfer function
The input–output relation of a linear time-invariant system expressed as the ratio G(s) = Y(s)/X(s) of the Laplace transforms of output and input with all initial values zero. It equals the Laplace transform of the impulse response. Values of s that make the denominator zero are poles and those that make the numerator zero are zeros; the pole positions show stability and speed of response. Setting s = jω gives the frequency response. For discrete-time systems it is written H(z) using the z-transform.
Steady-state error
The difference between the setpoint and the controlled variable that remains after the response of a control system has settled; it is also called offset. In a unity-feedback system it is 1/(1 + Kp) for a step setpoint, where Kp = G(0) is the position error constant, and it does not become zero in a system with no integrator in the forward path (type 0). In a type 1 system, with one integrator, the steady-state error to a step is zero, and an error of 1/Kv (Kv is the velocity error constant) remains for a ramp setpoint. The integral action of PI control is used to remove this error.
Gain margin
The amount, in decibels, by which the gain of the loop transfer function is below 0 dB at the frequency where its phase reaches −180° (the phase crossover frequency). It shows by what factor the loop gain could be raised before the loop becomes unstable; for example, with a gain margin of about 9.5 dB, tripling the gain reaches the stability limit. Together with the phase margin it expresses the stability margin, and about 10 to 20 dB is a common guideline for servo systems.
PID control
Feedback control that sets the manipulated variable by adding a term proportional to the error e (P), a term proportional to the integral of the error (I) and a term proportional to the derivative of the error (D). It is written u = Kp(e + (1/Ti)∫e dt + Td·de/dt), where Kp is the proportional gain, Ti the integral time and Td the derivative time. The proportional term provides speed, the integral term removes the steady-state error and the derivative term suppresses oscillation. Its structure is simple, and it is used in most industrial control loops, for temperature, motors, power supplies, chemical processes and more.

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