What is a Closed-Loop Control System?

How to Design a Closed-Loop Control System?

The basic objective of design is the same in both open-loop and closed-loop control systems. Depending on our output requirement we set input to the controller, which generates the control signal, and this control signal is fed to the processing unit to achieve the output. So far, the process and design are the same for open-loop and closed-loop control systems, but the basic difference is the achieved output is measured with the help of a measuring instrument to compare with a set value of the output. Hence the measuring instrument acts as feedback to the output. The error detector will compare the measured value with the pre-set value of the output and an appropriate actuating signal is sent to the controller to correct the output. The basic block diagram representation of a closed-loop control system is shown below.

Based on the need of the user and the system complexity, there may be any number of feedback loops.

Examples of a Closed Loop Control System

Human beings are the most complex and sophisticated feedback control systems in existence. Hence consider an example of picking an apple from the desk. After you are reaching the desk, the brain sends a control signal to your hand to pick the apple, your eyes are acting as the feedback, and the distance between your hand and the apple is the error. The brain and eyes are coordinated to minimize the error and to achieve the objective that is picking the apple. If you are blindfolded and asked to pick the same apple, then the feedback that is your eyes are absent, you are reaching towards the object with a blind estimation, hence there is a chance that you might miss the apple by a significant margin. So, in the first case, your body is acting as a closed-loop control system with feedback in the form of eyes, and in the second case, your body acts as an open-loop control system.

In a similar way, if the system can correct the error by using the feedback provided in the form of measuring instruments and various controllers such systems are the best examples of closed-loop control systems such as automatic lighting systems, automatic voltage controllers, pilotless locomotives, etc.

Canonical Form of Closed Loop Control System

Let,

$X_{i} (s) \to$ Input signal

$X_{0} (s) \to$ Output signal

G(s) $\to$ Transfer function of the open-loop control system

H(s) $\to$ Gain of the elements in the feedback path

B(s) $\to$ The output of feedback elements

E(s) $\to$ Error signal

Then the closed-loop control system can be represented as shown below.

From the above diagram

$G (s) = \frac{X_{0} (s)}{E (s)}$

$\Rightarrow E (s) = \frac{X_{0} (s)}{G (s)}$

E(s) can also be written as

$E (s) = X_{i} (s) - B (s)$

$\Rightarrow \frac{X_{0} (s)}{G (s)} = X_{i} (s) - B (s)$

From the diagram,

$H (s) = \frac{B (s)}{X_{0} (s)}$

$\Rightarrow \frac{X_{0} (s)}{G (s)} = X_{i} (s) - H (s) X_{0} (s)$

$\Rightarrow X_{0} (s) = G (s) X_{i} (s) - G (s) H (s) X_{0} (s)$

$\Rightarrow X_{0} (s) [1 + G (s) H (s)] = G (s) X_{i} (s)$

$\Rightarrow \frac{X_{0} (s)}{X_{i} (s)} = \frac{G (s)}{1 + G (s) H (s)}$

This is the standard form of representing the closed loop control system with negative feedback. However, the summing point (⊗) may have more than one feedback input in the form of negative feedback (feedback will be subtracted from the input) or positive feedback (feedback will be added to the input), but its output will be the algebraic sum of all inputs.

Effect of Feedback on the System Performance

The feedback construction in the closed loop control system is responsible for the accurate and effective output of the system, but while gaining this advantage we have to compromise on some system parameters. Here we will discuss how the feedback is affecting the key parameters of the system.

Effect on System Gain

If G(s) be the gain of the open-loop control system, then from the standard form of the closed-loop control system we can observe that the feedback affects the gain by a factor 1+G(s)H(s). There is no rule that the magnitude GH be always greater than 1, as it will be a negative or positive quantity so based on this value and the type of feedback the gain may increase or decrease.

Practically, G(s) and H(s) are frequency parameters so, the value of 1+G(s)H(s) may be less than one in one frequency range and may be greater than one in another range of frequencies. Hence feedback will certainly affect the system gain, but how the gain changes will depend on the value of 1+G(s)H(s).

Effect on Sensitivity

Sensitivity is one of the most significant considerations in the design of a control system. As the properties of elements involved in the design vary with age and environmental conditions. For example, the electrical properties of the transformer change with the temperature and age of the machine.

A control system is said to be highly sensitive if its control objective is affected due to any internal or external disturbances. In any control system, the desirable feature is it should be less sensitive to external disturbances. Here we will discuss how sensitive the closed loop control system is to the disturbances in its forward path as well as feedback path.

If, $𝛼$ is the variable that changes its value,

$𝛽$ is a parameter that changes the value of $𝛼$ , then the sensitivity of $𝛼$ concerning $𝛽$ is given by

$S_{𝛽}^{𝛼} = \frac{% c h a n g e i n 𝛼}{% c h a n g e i n 𝛽} = \frac{\frac{𝜕 𝛼}{𝛼}}{\frac{𝜕 𝛽}{𝛽}}$

$S_{𝛽}^{𝛼} = \frac{𝛽}{𝛼} \frac{𝜕 𝛼}{𝜕 𝛽}$

Case (i): Sensitivity of closed-loop control system for disturbances occur in the forward path elements.

Let 𝛼 is closed-loop control system M(s),

M(s)= G(s)/[1+G(s)H(s)]

And β be the disturbance in the forward path elements, that is G(s), then

S^M(s)_G(s)=G(s)∂M(s)/M(s)∂G(s)

From the equation of M(s), we can write,

$\frac{G (s)}{M (s)} = 1 + G (s) H (s)$

$\frac{𝜕 M (s)}{𝜕 G (s)} = \frac{[1 + G (s) H (s)] (1) - G (s) H (s)}{{(1 + G (s) H (s))}^{2}}$

$\Rightarrow S_{G (s)}^{M (s)} = \frac{1 + G (s) H (s)}{{(1 + G (s) H (s))}^{2}}$

$\Rightarrow S_{G (s)}^{M (s)} = \frac{1}{1 + G (s) H (s)}$

So, in this case, the sensitivity function can be made small by increasing the value G(s)H(s).

Case (i): Sensitivity of closed loop control system for disturbances occur in the feedback path elements.

Let $𝛼 = M (s)$ , $𝛽 = H (s)$

$\Rightarrow S_{H (s)}^{M (s)} = \frac{H (s)}{M (s)} \frac{𝜕 M (s)}{𝜕 H (s)}$

From the equation of M(s), we can write,

$\frac{H (s)}{M (s)} = \frac{[1 + G (s) H (s)] H (s)}{G (s)}$

$\Rightarrow S_{G (s)}^{M (s)} = \frac{G (s) H (s)}{1 + G (s) H (s)}$

In this case, it is difficult to minimize the sensitivity, so the closed-loop control system or the feedback control system is more sensitive to the disturbances in the feedback elements.

Effect on Stability

If the system is able to respond to the input command effectively, then the system is said to be stable. If the output is increasing without any bounds, then the system is said to be out of control or unstable. In the standard representation of a closed-loop control system, if the value of G(s)H(s)=-1, then the system output will be infinitely large for a finite value of the input. It is to be noted that G(s)H(s)=-1 may not be the only condition for the instability. So, even the feedback is key in obtaining the controlled output, at the same time it is harmful when we use it inappropriately.

In such cases when G(s)H(s)=-1, the system can be stabilized by introducing another feedback loop in the system. Let us assume that the gain of the new feedback loop is L(s), then the closed loop system will be represented as

$\frac{X_{0} (s)}{X_{i} (s)} = \frac{G (s)}{1 + G (s) H (s) + G (s) L (s)}$

So, these are the key parametric variations due to the introduction of feedback in the system, we will have a deeper analysis of the closed loop control system in our lectures.

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FAQs

What is the closed-loop control system?

Any device or system that regulates its output to maintain or achieve the desired state of operation with the help of the feedback mechanism is known as the closed-loop control system. The construction of both open-loop and closed-loop control systems are similar in the forward path of system representation, but the closed-loop system will have one or more feedback paths based on the complexity of the system.

What are the advantages of the closed-loop control system?

The main advantage of the closed-loop control system is its automatic error correction feature that enables the system independent or less dependent on human monitoring. Along with that, it can stabilize the unstable system, can increase, or decrease the system sensitivity based on the user requirement, and can respond better to external disturbances as compared to the open-loop control system.

What are the applications of closed-loop control systems?

The closed-loop or feedback control systems found numerous applications in the form of machine tool control, quality control of manufacturing industries, transportation systems, computer science, automatic control of heating and air conditioning, advanced weapon systems and space technologies, etc.

What are the disadvantages of closed-loop control systems?

Proper care should be taken in the design of the feedback path, otherwise, it can affect the system stability. In addition, it can affect the system’s gain. They need a complex design, the feedback elements will incur significant cost spending, and their initial, as well as maintenance costs, are high.

What is the effect of feedback on the gain of the system?

The gain of the systems either can increase or decrease depending on factors like the connection of feedback, the magnitude of loop gain, and the operating frequency of the system. If the feedback is negative then the gain will reduce, for the positive feedback gain of the system may increase. If the magnitude of G(s)H(s) is negative in the negative feedback then there is a possibility of increasing the gain in the negative feedback, and the value of 1+G(s)H(s) may be greater in one frequency range or less than one in some other frequency. So, the feedback certainly affects the gain, but the connection and certain operating conditions will decide its effect.