Understanding Closed-Loop Control Systems

A closed-loop control system is a system where the output is measured and fed back to compare it with the desired input. This comparison generates an error signal that is then used to adjust the system's input. The most common type of feedback used in these systems is negative feedback, where the feedback signal is subtracted from the input signal.

Negative feedback is crucial because it helps to reduce the error between the desired output and the actual output. Beyond just error reduction, it significantly impacts various system characteristics, including stability, bandwidth, and importantly, system sensitivity to parameter variations.

System Sensitivity Explained

System sensitivity is a measure of how much a system's output or a specific characteristic (like its transfer function) changes in response to a change in one of its parameters. In control systems, parameters can include gains of amplifiers, resistances, capacitances, or any component value within the system. High sensitivity means even small changes in a parameter can lead to large, undesirable changes in the system's performance, making the system less robust.

For a closed-loop control system, the sensitivity of the overall transfer function \(T(s)\) with respect to a change in the forward path gain \(G(s)\) (which represents a parameter variation) can be mathematically expressed. Let's consider a basic unity-feedback closed-loop system where:

• \(R(s)\) is the reference input.
• \(C(s)\) is the output.
• \(G(s)\) is the forward path transfer function.
• \(H(s)\) is the feedback path transfer function. (For unity feedback, \(H(s) = 1\)).

The closed-loop transfer function \(T(s)\) is given by:

The sensitivity of \(T(s)\) with respect to \(G(s)\), denoted as \(S_{G}^{T}\), is defined as:

First, let's find the partial derivative of \(T(s)\) with respect to \(G(s)\):

Now, substitute this into the sensitivity formula:

Impact of Negative Feedback on Variations

From the derived sensitivity function \(S_{G}^{T} = \frac{1}{1 + G(s)H(s)}\), we can clearly see the effect of negative feedback on system sensitivity to parameter variations. The term \(G(s)H(s)\) is known as the loop gain. In most practical closed-loop control systems, the magnitude of the loop gain \(|G(s)H(s)|\) is designed to be significantly greater than 1 over the system's operating frequency range, especially at low frequencies. This large loop gain ensures that the error signal is effectively reduced.

Since \(|G(s)H(s)| \gg 1\), the denominator \(|1 + G(s)H(s)|\) will also be much greater than 1. Consequently, the value of \(S_{G}^{T}\) will be much less than 1 (i.e., \(|S_{G}^{T}| \ll 1\)).

What this means is:

• If \(S_{G}^{T} = 0.01\), a 1% change in \(G(s)\) will cause only a 0.01% change in the overall closed-loop transfer function \(T(s)\).
• Without feedback, the transfer function is simply \(G(s)\), and its sensitivity to \(G(s)\) would be 1 (a 1% change in \(G(s)\) would cause a 1% change in the output).

Therefore, negative feedback effectively attenuates the effect of parameter variations within the forward path on the overall system performance. It makes the system more robust and less susceptible to changes in its internal components. This is one of the primary advantages of using negative feedback in control systems.

In summary, with negative feedback in a closed-loop control system, the system sensitivity to parameter variations decreases significantly, making the system more stable and reliable.