Gang of Four

Topics: Feedback Systems

Overview

The Gang of Four is a set of four fundamental closed-loop transfer functions that characterize the input-output behavior and robustness of a standard one-degree-of-freedom feedback control system. They describe how different inputs (reference, disturbance, and sensor noise) propagate to the system output and control effort.

Considering the above negative feedback loop with plant P and controller K, the Gang of Four consists of:

1. Sensitivity Function (S):

   • transfer function between the error term and the reference: $e=S \cdot r$
   • Formula: $S=\frac{1}{1+PK}$
   • Role: measures how much the system's output changes when its internal parameters or external inputs shift
   • Insight: a high loop gain is desired to move the sensitivity to 0

2. Complimentary Sensitivity (T):

   • transfer function between the output and the reference: $y=T \cdot r$
   • Formula: $T = \frac{PK}{1+PK}$
   • Role: determines the reference tracking performance of the system, as well as the robustness to model uncertainty
   • Insight: a high look gain $K$ is desired to move the complementary sensitivity to 1

3. P-Sensitivity (PS):

   • transfer function between the output and the plant input disturbance: $y=PS \cdot d_i$
   • Formula: $PS=\frac{P}{1+PK}$
   • Role: specifies the impact of input disturbance on the output
   • Insight: a high look gain $K$ is desired to move $PS$ towards 0.

4. K-Sensitivity (KS):

   • transfer function between the control input and the reference: $u=KS \cdot r$
   • Formula: $KS=\frac{K}{1+PK}$
   • Role: relates the control effort to the desired reference
   • Insight: a low loop gain is desired for $KS$ in order to have a reasonable control effort

Trade-Offs

1. The sensitivity, complementary sensitivity and P-sensitivity all benefit from a high loop gain. However, this is at the expense of control effort, which is often not desired.

2. The complementary sensitivity $T$ is desired to be 1 at low frequencies for good reference tracking, and to be 0 at high frequencies so that high-frequency noise is not tracked. The sensitivity is desired to be small at low frequencies for small stead-state error and disturbance rejection.
   The trade-off is that, following the equations, $S+T=1$ for a given frequency.

References