control//fault diagnosis//fault model
A fault model is the description of how a fault enters the equations of a system, written in the same state-space model the estimator and the controller use, and it is what lets a diagnosis designer predict which residuals a fault will move and under what conditions it becomes visible. Two shapes cover most cases.
A fault model is the description of how a fault enters the equations of a system, written in the same state-space model the estimator and the controller use, and it is what lets a diagnosis designer predict which residuals a fault will move and under what conditions it becomes visible. Two shapes cover most cases.
An additive fault adds a signal to a command or a measurement. With xkx_kxk the state, uku_kuk the command, yky_kyk the measurement and wkw_kwk, vkv_kvk the usual noises,
xk+1=Axk+B (uk+fka)+wk,yk=Cxk+fks+vk,x_{k+1}=Ax_k+B\,(u_k+f^{a}_k)+w_k,\qquad y_k=Cx_k+f^{s}_k+v_k ,xk+1=Axk+B(uk+fka)+wk,yk=Cxk+fks+vk,
where fkaf^{a}_kfka is an actuator fault added to the command (a valve that opens 5 % less than ordered) and fksf^{s}_kfks a sensor fault added to the reading (a pressure transmitter with a 0.3 bar offset, a temperature probe drifting with age). Additive faults behave like unknown inputs: they shift the residual whatever the system is doing, so a detection threshold or a CUSUM can see them at rest.
A multiplicative fault changes the matrices themselves. A cracked propeller produces 80 % of the thrust it should, which multiplies one column of BBB by 0.8; a fouled heat exchanger lowers a coefficient of AAA. Its effect on the output is proportional to the signal it multiplies, so it only shows when that signal moves: a gain fault on an actuator that is barely used produces almost no residual.
Feedback hides faults until the system is asked to move.
A degraded motor goes unnoticed while a multirotor hovers, because the controller's integral action raises that motor's command until thrust balances weight; the residual stays quiet. The first sharp manoeuvre asks for more than the motor can give, and the fault appears at the worst moment.
The masking is a general property of good control: the loop removes from the output exactly the deviations a diagnosis would look for. Two remedies follow. Watch the control effort as well as the output (the integrator's level is itself a residual), and excite the system on purpose during a check, a short test manoeuvre or a step on the setpoint.
Seeing a multiplicative fault means estimating a parameter, so it needs inputs rich enough to make that parameter identifiable (persistent excitation); an online estimate of the parameter compared with its nominal value is one form of analytical redundancy.
Faults in a model can also be classified by where they enter (sensor, actuator, process) and by their shape in time, which are the categories of fault.