A method of fault diagnosis in dynamic systems based on a fuzzy approach is proposed. The new method possesses two basic specific features which distinguish it from the other known fuzzy methods based on the application of fuzzy logic and a bank of state observers. First, this method uses a bank of interacting observers instead of traditional independent observers. The second specific feature of the proposed method is the assumption that there is no strict boundary between the serviceable and disabled technical states of the system, which makes it possible to specify a decision making rule for fault diagnosis.

A dynamic system model is widely used to describe technical systems in solving various problems of analysis and synthesis, including diagnosis, as applied to these systems. Although the literature on problems of diagnosis is abundant, the interest to them still persists and the investigations are continuing. The problems of how to increase accuracy, or diagnostic depth, and how to take into account different uncertainties that are inherent in the solution of diagnosis problems are conventionally central to the studies of fault diagnosis. It is to these problems that this paper is devoted, wherein the technical states of the system are assumed uncertain.

In the literature, the diagnosis problem is considered in different formulations, depending primarily on the models used to describe a system: deterministic [

Each of the afore mentioned approaches fits different lines of investigations. Among the most efficient of them is the one that relies on the models of the diagnosed system for synthesis of diagnostic tools. This line is developed in the framework of all approaches and is realized by application of either single state observers [

Structure of the diagnostic system.

In this study, we propose a method for diagnosing dynamic systems in the framework of a fuzzy approach. The new method possesses two basic specific features which distinguish it from the other known fuzzy methods based on the application of fuzzy logic and a bank of state observers

First of all, let us discuss the notion of a “fuzzy” technical state and, as a consequence, “fuzzy” fault, which is understood as a transition from the serviceable technical state to the disabled one. The notion of a fuzzy technical state used later seems to be quite adequate to describe the existing engineering approach. Indeed, judging from the value of the parameter which indicates the technical state of an object, an engineer can conclude that the object is either serviceable or disabled. Depending on a particular value of this parameter, the engineer can conclude that the object is serviceable or, correspondingly, disabled to a certain extent.

We define a

Figure

Illustration of the ideas of “crisp” and “fuzzy” technical states.

We should also make two more remarks concerning the problems touched upon in this paper. First, for simplicity, we assume that there are no perturbations in the considered model of the system. Second, we do not discuss the already known procedures of synthesis of stable observers [

The rule is based on the notion of the confidence coefficient

Let us clarify the procedure of calculation of confidence coefficient. It is based on two groups of parameters that define the technical state of a dynamic system: the residuals

As for the variables

In order to obtain the confidence coefficients

The explanation of this expression is obvious. Indeed, the observer adequate to the technical state of the system forms a small residual, whereas the others form large residuals. This being so, we may say that a fault does exist if

Then, the confidence coefficient

In this paper, we consider successively three different models of fault: structural changes, faults in the signal space, and faults in the parameter space. In so doing, we study various structures of diagnostic tools different in the organization of the banks of the state observers and decision making rules. The proposed structures are compared with the known variant of diagnostic tools, which uses a bank of independent observers (Figure

Structure of an attitude control system of a space launch vehicle (a) and an example of structural changes (b).

From here on, the diagnosed dynamical system is described in the time domain (by differential or difference equations); that is, for a linear system,

while for a nonlinear system,

The problem of synthesis of tools for fault diagnosis involves two main questions: formation of a rule for decision making and synthesis of a bank of observers. Assuming that the answer to the first question is given in the previous section, we proceed to the discussion of the second question.

Let us next study some variants of fault diagnosis that employ the banks of both independent and interacting observers. By this we mean the method used to form the system state vector estimate in each of the observers. If it is formed independently, we have independent observers and treat the obtained estimates as conditional with respect to a certain technical state. If the formation of the estimate also takes into account the estimates obtained in other observers, we have interacting observers. This chapter is referring to the independent observers.

Let us consider briefly the question of synthesis of a single observer. Observers may be synthesized by different rules, resulting from differences in the formulation of the problem. The procedure for the synthesis of a state observer for the linear system is known; however, for the sake of completeness, we will recall its main issues. As previously stated, each

As a result, an

To summarize this section, we formulate the algorithm for diagnosis of arbitrary faults in the formulation under consideration.

Formation of a list of faults.

Synthesis of the independent observer for each of the faults.

Assignment of membership functions for the considered fuzzy residuals based on the developer’s empirical knowledge of the system operability.

Decision making about faults by forming a confidence coefficient.

To illustrate the proposed algorithm, once again, we turn to the example in Figure

Let us synthesize the diagnosis tools for break of the velocity feedback. The nominal behavior of the system is described by the equation

We analyzed an intermittent fault, when a device recovers after the failure and then fails again, in the Simulink environment using a sinusoidal input signal. In practice, faults of this type usually are a severe problem for diagnosis. Simulation results of a problem of diagnosis are shown in Figure

Time diagrams for confidence coefficients of technical states of an attitude control system of a space launch vehicle for independent observers.

In the case of diagnosis in the signal space, the fault is simulated as an additional term

In this case of diagnosis in the signal space, the state vector

As a result, in the presence of the

Let us discuss the proposed method for obtaining an estimate of the state vector of the system in each of the observers. Further, consider two diagnostic algorithms with the application of a bank of interacting observers. In so doing, we use the decision making rule on the fault occurrence described in the previous section. It will be shown later that in the general case, the efficiency of the considered algorithms is higher than that in the case of independent observers.

An important specific feature of the first algorithm is that on each successive step of the calculation, each of the observers is based on the estimate of state

The result of it is a nonlinear state feedback. Indeed, the residual (estimation error) formed by an adequate observer tends to zero, and the corresponding confidence coefficient increases with the decrease of confidence coefficients for other technical states. Thus, in expression (

The second algorithm, though being similar to the previous one, differs from it, first of all, in the fact that the observers are matched not with technical states but with the transitions between them. In this case, it is assumed that the observer is matched with the transition

In order to make a decision according to rule (

It is evident that the analysis of the behavior of the diagnosed system used in the second algorithm is more detailed, which is why we can expect this method to be more efficient. In the general case, this is proved by the simulation results given later.

Let us illustrate the described algorithms by a particular example.

Let us consider the linear system characterized by the matrices

This system is a reduced model of an aircraft control loop at an altitude obtained by linearization of the aircraft motion equations in the neighborhood of the nominal trajectory. This description covers the controlled object, the rudder control servo drive, the altitude sensor, and the controller. For this example, the problem of diagnosis in the signal space was simulated in Simulink. An intermittent fault

Time diagrams for alternating fault and confidence coefficients of technical states of the control system for (a) independent and (b) interacting observers.

In the case of diagnosis in the parameter space, the fault is simulated as the deviation of the value of some system parameter from the nominal value. Thus, for example, for the linear system, it is simulated as the deviation of elements of the system matrices from the nominal, where

In this case, the number of types

Let us use the algorithms considered in the previous section in the diagnosis in the parameter space. Assume that the parameter

Let us analyze the efficiency of the diagnostic algorithms using the following example.

Let us consider the model of a water torpedo described by the nonlinear equation

Time diagrams for alternating fault and confidence coefficients of technical states of a torpedo for (a) independent and (b) interacting observers.

In this paper, we proposed the method of diagnosis of dynamic systems based on the application of the bank of fuzzy interacting state observers. We considered successively three different models of fault: structural changes, faults in the signal space, and faults in the parameter space. For these models of fault, we considered various structures of diagnostic tools different in the organization of the bank of the state observers and decision making rules. The proposed method was compared with the known method with independent observers by simulation in Simulink. As a result, it was demonstrated that the proposed method makes it possible to achieve higher quality of diagnosis.

This work was supported by the Russian Foundation for Basic Research, Project no. 10-08-00035a.