## Section: Scientific Foundations

### Failure detection in dynamical systems

#### Active failure detection

Failure detection has been the
subject of many studies in the past.
Most of these works are concerned with the problem of *passive failure
detection* . In the passive approach, for material or security
reasons, the detector has no way of acting upon the system; the detector
can only monitor the inputs and the outputs of the system and then decides
whether, and if possible what kind of, a failure has occurred. This is done by
comparing the measured input-output behavior of the system with the “normal”
behavior of the system. The passive
approach is often used to continuously monitor the system although
it can also be
used to make periodic checks.

In some situations however failures can be masked by the operation of the system. This often happens in controlled systems. The reason for this is that the purpose of controllers, in general, is to keep the system at some equilibrium point even if the behavior of the system changes. This robustness property, desired in control systems, tends to mask abnormal behaviors of the systems. This makes the task of failure detection difficult. An example of this effect is the well known fact that it is harder for a driver to detect an under-inflated or flat front tire in a car which is equipped with power steering. This tradeoff between detection performance and controller robustness has been noted in the literature and has lead to the study of the integrated design of controller and detector.

But the problem of failures being masked by system operation is not limited to controlled systems. Some failures may simply remain hidden under certain operating conditions and show up only under special circumstances. For example, a failure in the brake system of a truck is very difficult to detect as long as the truck is cruising down the road on level ground. It is for this reason that on many roads, just before steep downhill stretches, there are signs asking truck drivers to test their brakes. A driver who ignores these signs would find out about a brake failure only when he needs to brake going down hill, i.e., too late.

An alternative to passive detection which could avoid the problem of
failures being
masked by system operation is *active detection* . The active
approach to failure detection consists in acting upon the system on a
periodic basis or at critical
times using a test signal in order to detect abnormal behaviors which
would otherwise remain
undetected during normal operation.
The detector in an active approach can act either by taking over the
usual inputs of the system or through a special input channel. An
example of using
the existing input channels is testing the brakes by stepping on the
brake pedal.

The active detection problem has been less studied than the passive detection problem. The idea of injecting a signal into the system for identification purposes has been widely used. But the use of extra input signals in the context of failure detection has only been recently introduced.

The specificity of our approach for solving the problem of auxiliary
signal design is that we have adopted a deterministic point of view
in which we model uncertainty using newly developed techniques from
control theory. In doing so, we can deal efficiently with
the robustness issue which is in general not properly dealt with in
stochastic approaches to this problem. This has allowed us in
particular to introduce the notion of *guaranteed failure detection* .

In the active failure detection method considered an auxiliary signal v is injected into the system to facilitate detection; it can be part or all of the system inputs. The signal u denotes the remaining inputs measured on-line just as the outputs y are measured online. In some applications the time trajectory of u may be known in advance but in general the information regarding u is obtained through sensor data in the same way that it is done for the output y .

Suppose we have only one possible type of failure. Then we have two sets of input-output behaviors to consider and hence two models. The set is the set of normal input-outputs {u, y} from Model 0 and the set is the set of input-outputs when failure occurs. That is, is from Model 1. These sets represent possible/likely input-output trajectories for each model. Note that Model 0 and Model 1 can differ greatly in size and complexity but they have in common u and y .

The problem of auxiliary signal design for guaranteed failure detection is to find a “reasonable” v such that

That is, any observed pair {u, y} must come only from one of the two models. Here reasonable v means a v that does not perturb the normal operation of the system too much during the test period. This means, in general, a v of small energy applied over a short test period. However, depending on the application, “reasonable" can imply more complicated criteria.

Depending on how uncertainties are accounted for in the models, the mathematics needed to solve the problem can be very different. For example guaranteed failure detection has been first introduced in the case where unknown bounded parameters were used to model uncertainties. This lead to solution techniques based on linear programming algorithms. But in most of our works, we consider the types of uncertainties used in robust control theory. This has allowed us to develop a methodology based on established tools such as Riccati equations that allow us to handle very large multivariable systems. The methodology we develop for the construction of the optimal auxiliary signal and its associated test can be implemented easily in computational environments such as Scilab. Moreover, the online detection test that we obtain is similar to some existing tests based on Kalman filters and is easy to implement in real-time. The main results of our research can be found in a book published in 2004. We have developed many extension since, which have been published in various journals and presented at conferences.

#### Passive failure detection

##### Modal analysis and diagnosis

We consider mechanical systems with the corresponding stochastic state-space models of automatic control.

The mechanical system is assumed to be a time-invariant linear dynamical system:

where the variables are : : displacements of the degrees of freedom, M , C , K : mass, damping, stiffness matrices, t : continuous time; : vector of external (non measured) forces modeled as a non-stationary white noise; L : observation matrix giving the observation Y (corresponding to the locations of the sensors on the structure).

The modal characteristics are: the vibration modes or eigen-frequencies and the modal shapes or eigenvectors. They satisfy:

By stacking Z and and sampling at rate 1/ , i.e.,

we get the following equivalent state-space model:

with

The mechanical systems under consideration are vibrating structures and the numerical simulation is done by the finite element model.

The objectives are the analysis and the implementation of statistical model-based algorithms, for modal identification, monitoring and (modal and physical) diagnosis of such structures.

For modal analysis and monitoring, the approach is based on subspace methods using the covariances of the observations: that means that all the algorithms are designed for in-operation situation, i.e., without any measurement or control on the input (the situation where both input and output are measured is a simple special case).

The identification procedure is realized on the healthy structure.

The second part of the work is to determine, given new data after an operating period with the structure, if some changes have occurred on the modal characteristics.

In case there are changes, we want to find the most likely localization of the defaults on the structure. For this purpose we have to do the matching of the identified modal characteristics of the healthy structure with those of the finite element model. By use of the different Jacobian matrices and clustering algorithms we try to get clusters on the elements with the corresponding value of the "default criterion".

This work is done in collaboration with the INRIA-IRISA project-team SISTHEM (a spin-off of the project-team SIGMA2) (see the web-site of this project-team for a complete presentation and bibliography) and with the project-team MACS for the physical diagnosis (on civil structures).

##### Robust failure detection and control

Failure detection problems are formulated in such a way that mathematical techniques in robust control can be used to formulate and solve the problem of robust detection. Concepts developed for control can be used in particular to formulate the notion of robustness and provide numerically tractable solutions.

This system approach can also be used to formulate both the detection and the control in a single framework. The Simultaneous Fault Detection and Control problem is formulated as a mixed optimization problem and its solution is given in terms of Riccati equations. It is shown that controllers/detectors resulting from this approach have reasonable complexity and can be used for practical applications.