A control system is an interconnection of components forming a system configuration that will provide a desired system response. The input-output relationship represents the cause-and-effect relationship of the process, which in turn represents a processing of the input signal to provide a desired output signal.
An open-loop control system utilizes an actuating device to control the process directly without using feedback.
A closed-loop control system uses a measurement of the output and feedback of this signal to compare it with the desired output (reference or command).
The system shown is a negative feedback control system, because the output is subtracted from the input and the difference is used as the input signal to the controller.
The feedback of the system is called negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.
A closed-loop control has many advantages over open-loop control, including the ability to reject external disturbances and improve measurement noise attenuation. External disturbances and measurement noise are inevitable in real-world applications and must be addressed in practical control system designs.
Standard Feedback Control System
The block diagram of a closed-loop control system. The role of the feedback element whose transfer function is \(H(s)\) is to modify the output \(C(s)\) before it is compared with the input \(R(s)\). In most cases, the feedback element is a sensor that measures the output of the plant (process) \(G(s)\). The output \(C(s)\) of the sensor is compared with the system input \(R(s)\), and the actuating error signal \(E(s)\) is generated.
By analyzing the block diagram, the following is obtained:
Feedback signal, \(\displaystyle B(s) = H(s)\,C(s)\)
Actuating error signal, \(\displaystyle E(s) = \frac{C(s)}{G(s)} = R(s) - B(s)\)
The ratio of the feedback signal \(B(s)\) to the actuating error signal \(E(s)\) is called the open-loop transfer function (OLTF).
\(\boxed{\displaystyle \mathrm{OLTF} = \frac{B(s)}{E(s)} = G(s)\,H(s)}\)
The ratio of the output \(C(s)\) to the actuating error signal \(E(s)\) is called the feedforward transfer function (FFTF).
\(\boxed{\displaystyle \mathrm{FFTF} = \frac{C(s)}{E(s)} = G(s)}\)
The transfer function relating \(C(s)\) to \(R(s)\) is called the closed-loop transfer function (CLTF). It relates the closed-loop system dynamics to the dynamics of the feedforward elements and feedback elements.
\(\displaystyle E(s) = R(s) - B(s) = R(s) - H(s)\,C(s)\)
\(\displaystyle C(s) = G(s)\,E(s) = G(s)\left[R(s) - H(s)\,C(s)\right]\)
\(\displaystyle C(s) + G(s)\,H(s)\,C(s) = G(s)\,R(s)\)
Then, the closed-loop transfer function is
\(\boxed{\displaystyle \mathrm{CLTF} = \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)\,H(s)}}\)
Unity Feedback Control System
If the feedback transfer function \(H(s)\) is unity, then the open-loop transfer function and the feedforward transfer function are the same.
For a unity feedback system, \(H(s) = 1\)
\(\boxed{\displaystyle \mathrm{OLTF} = \mathrm{FFTF} = G(s)}\)
\(\boxed{\displaystyle \mathrm{CLTF} = \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)} = \frac{\mathrm{num}\left[G(s)\right]}{\mathrm{den}\left[G(s)\right] + \mathrm{num}\left[G(s)\right]}}\)
where \(\mathrm{num}[G(s)]\) and \(\mathrm{den}[G(s)]\) are numerator and denominator of a transfer function \(G(s)\), respectively.
