Actuator fault diagnosis is often studied under strong assumptions on available sensors.
Typically, it is assumed that the sensors are either fault free or sufficiently redundant. The
purpose of this paper is to present a new method for

In the design of modern control systems, fault
diagnosis is often considered for component faults, sensor faults, and actuator
faults. Various methods for fault diagnosis are generally based on the
processing of sensor signals [

The purpose of this paper is to present a method for
actuator fault diagnosis which is robust to sensor distortion. It is assumed
that each sensor can be affected by an

Robust methods for fault diagnosis has been studied in
different contexts by many authors. In [

The preliminary results presented in [

The paper is organized as follows. The problem
considered in this paper is formulated in Section

The considered linear state-space system with sensor distortion is formulated as

For notation simplicity, the parenthesis

Sensor distortion is modeled by the component-wisely defined nonlinear function

Each sensor distortion

The system dynamics matrix

No additive uncertainty is assumed in the sensor distortion equation (

With the above formulation, the problem considered in this paper is the detection and
isolation of multiplicative actuator faults, modeled as changes in the coefficient vector

The main difficulty of the problem formulated in the previous section is caused by the

Let

Notice that

Zero initial condition of the state

Let

Let

If

Let us first consider the case

Remind that

Proposition

Assume that the system matrix pair

This proof applies to the residual

Define the vector

Notice that the equality

Similarly, it is also shown that

Now, let us consider

Notice that the existence of such input signals is ensured by the controllability of the matrix
pair

Because

If

Assume that

For notation simplicity, the sampling period is assumed to be 1 here.

at discrete time instantsAfter the detection of an actuator fault, the purpose of fault isolation is to figure out which
actuators are faulty. In terms of (

It should be first remarked that, because of the arbitrary unknown function

After having clarified the limitation related to unknown sensor distortion, let us look for an
algorithm for fault isolation. The basic idea is to design residuals similar to (

Let

The notations

For the purpose of fault isolation, different partitions of

For two time instants

Because

For each chosen partition of

Remark that the inequality (

The constrained minimization (

If the true parameter vector

The proof of this result is quite straightforward. Let us first derive from (

It is assumed that

Property (

For fault isolation, various matrices

Keep in mind that fault isolation cannot distinguish the cases such that

There exists a permutation matrix

In this section, the presented fault diagnosis method will be illustrated with a simulated distillation column.

In [

Numerical simulation is first made in continuous time. The three input variables are randomly
drawn with uniform distributions ranged within the intervals [20, 40] (mol/min), [10, 30] (mol/min) and [10, 20] (

The residual for fault detection, computed with the averaging window length

Fault detection residual. The same residual is plotted twice at different scales. The time unit is the minute.

For fault isolation, three residuals are computed with the signals from the 1801th minute to
the 2000th minute. Each of the three residuals is designed to reject a fault affecting one of the three
actuators. The residuals rejecting the faults of actuator 1, 2, and 3 are, respectively, plotted in
the top, middle, and bottom pictures of Figure

Fault isolation residuals. The time unit is the minute. Top: residual rejecting the fault of actuator 1. Middle: residual rejecting the fault of actuator 2. Bottom: residual rejecting the fault of actuator 3.

Despite unknown nonlinear distortions of sensors, the information provided by such sensors is still useful for fault diagnosis, even when there is no redundant sensors, if the distortions are strictly monotonous. The monotonousness is a weak assumption since nonmonotonous distortion would make the sensor information useless. The main idea of the method presented in this paper is about how to use the information provided by such sensors. Because of the unknown nature of nonlinear distortion, neither the absolute value of the measured physical variable nor its sign can be determined from the sensor signal. The strict monotonousness of the nonlinear distortion is not helpful in this aspect. However, for any two different time instants, the relative sign of the measured variable is preserved by the monotonous nonlinear distortion. By using the information residing in the relative sign of sensor signals, the method for actuator fault diagnosis presented in this paper is conceptually robust to sensor distortions, as illustrated by the numerical example presented in this paper.