Sensitivity (control systems)
Encyclopedia
The controller parameters are typically matched to the process
Process (engineering)
In engineering a process is a set of interrelated tasks that, together, transform inputs into outputs. These tasks may be carried out by people, nature, or machines using resources; so an engineering process must be considered in the context of the agents carrying out the tasks, and the resource...

characteristics and since the process may change it is important that the controller parameters are chosen in such a way that the closed loop system is not sensitive to variations in process dynamics. One way to characterize sensitivity is through the nominal sensitivity peak :

where and denote the plant and controller's transfer function in a basic closed loop control System, using unity negative feedback.

The sensitivity function , which appears in the above formula also describes the transfer function from measurement noise to process output, where measurement noise is fed into the system through the feedback and the process output is noisy. Hence, lower values of suggest further attenuation of the measurement noise. The sensitivity function also tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that is less than one are reduced by an amount equal to the distance to the critical point and disturbances with frequencies such that is larger than one are amplified by the feedback.
It is important that the largest value of the sensitivity function be limited for a control system and it is common to require that the maximum value of the sensitivity function, , be in a range of 1.3 to 2.

## Sensitivity Circle

The quantity is the inverse of the shortest distance from the Nyquist curve
Nyquist plot
A Nyquist plot is a parametric plot of a transfer function used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The...

of the loop transfer function to the critical point . A sensitivity guarantees that the distance from the critical point to the Nyquist curve is always greater than and the Nyquist curve of the loop transfer function is always outside a circle around the critical point with the radius , known as the sensitivity circle.

• Control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

• Control engineering
Control engineering
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...

• Robust control
Robust control
Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...

• PID controller
PID controller
A proportional–integral–derivative controller is a generic control loop feedback mechanism widely used in industrial control systems – a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a...