Sensors always have some systematic and random error. Hopefully they are both small enough to deal with (or ignore). Sometimes their error will not be small, such as when they fail or when they are somehow fooled into giving the wrong reading. It may be worth designing a contingency for these cases, depending on the value of the sensor reading in question.

Error propagation in systems such as closed-loop controllers is a well studied but too complex for a single homework question. Generally, systems with positive gain tend to magnify errors, and those with negative gain tend to minimize them. Typical feedback systems have large negative feedback gains, which minimizes the effects of sensor error. System stability should still be investigated.

Right off the bat, a great way to address this issue is to sanitize and average all sensor readings. Throw out those readings that don’t make sense, and do some form of averaging. These efforts alone will go a long way towards protecting against open or short circuit failures, random errors, and other extreme effects.

Smaller, systematic errors need to be treated with more subtlety. Consider a sensor with a steady-state error and a proportional (gain) error characterized by a polynomial, and a time delay or phase error. Integral control (I) can track and correct the accumulation of steady state errors, but it won’t reduce it to less than the error itself. Gain errors, those that differ by constant multiples (and higher order multiples) of the actual value, change the way proportional (P) and derivative (D) control act, as well as higher order control systems. The generic closed-loop control system in the figure above can be used to show how sensor errors propagate.

The textbook Modern Control Systems under the heading Sensitivity of Control Systems to Parameter Variations (Dorf & Bishop, 2011, pp. 239-242) holds the following valuable information:

In the open-loop system, all these errors and changes result in a changing and inaccurate output. However, a closed-loop system senses the change in the output due to the process changes and attempts to correct the output. …A primary advantage of a closed-loop feedback control system is its ability to reduce the system’s sensitivity.

System sensitivity is the ratio of the change in the system transfer function T(s) to the change of a process transfer function G(s) (or parameter) for a small incremental change:

Often, we seek to determine [the sensitivity with respect to] a parameter within the [process] transfer function G(s). Using the chain rule, we find that

Very often, the transfer function of the system T(s) is a fraction of the form

The sensitivity to [the parameter is then]:

An important advantage of feedback control systems is the ability to reduce the effect of the variation of parameters of a control system by adding a feedback loop. …A closed-loop systems allows G(s) to be less accurately specified, because the sensitivity to changes or errors in G(s) is reduced by the loop gain.

Other sections expand on this theme, especially Disturbance Signals in a Feedback Control System.

Referenced textbook: Dorf, R. C., & Bishop, R. H. (2011). Modern Control Systems, 12th ed. Upper Saddle River, NJ: Pearson Education, Inc. [Amazon.ca link],[Pearson link (13th Ed.)]