Methods for Fault Diagnosability Analysis of a Class of Affine Nonlinear Systems

The fault diagnosability analysis for a given model, before developing a diagnosis algorithm, can be used to answer questions like “can the fault f i be detected by observed states?” and “can it separate fault f i from fault f j by observed states?” If not, we should redesign the sensor placement. This paper deals with the problem of the evaluation of detectability and separability for the diagnosability analysis of affine nonlinear system. First, we used differential geometry theory to analyze the nonlinear system and proposed new detectability criterion and separability criterion. Second, the relatedmatrix between the faults and outputs of the system and the fault separablematrix are designed for quantitative fault diagnosability calculation and fault separability calculation, respectively. Finally, we illustrate our approach to exemplify how to analyze diagnosability by a certain nonlinear system example, and the experiment results indicate the effectiveness of the fault evaluation methods.


Introduction
Analyzing the fault diagnosability of system, which is very important for optimizing sensors placement, should be solved firstly before the fault diagnosis.In theory, the more sensors it has the better it is for fault diagnosis and monitoring.However, the sensors placement is constrained by the cost of sensitive component and the installation space, such as the satellite system.Diagnosability assessment is the key to evaluate the performance that one can expect from a diagnoser at run time and to define the appropriate set of sensors to be included in the design of a system [1,2].
The model-based approach to fault detection and isolation (FDI) has received a lot of attention.At first, the focus is on the linear system.Based on the transfer function method [3][4][5], the transfer function between residual and the fault is obtained.Through the transfer function, the diagnosability of faults is evaluated.In [6], the fault is detectable if the transfer function is not equal to zero.At the same time, the transfer function between residual and disturbance is used to determine whether the disturbance can be decoupled or not [7].In the case of the above conditions, the fault is regarded as separable, and then a set of unknown input observers are designed to realize fault isolation [8].The FDI of nonlinear system is researched in [9,10], and the FDI of uncertain linear systems is also concerned [11,12].However, compared with the rich results in FDI of linear and nonlinear systems, only a limited number of contributions about fault diagnosability analysis have been found, especially for the nonlinear system.In the field of linear continuous systems, diagnosability is formulated in terms of fault detectability and isolability which provides a survey of definitions from different points of view [13][14][15], and diagnosability which includes detectability and distinguishability are analyzed for discrete-event systems using labeled Petri net [16].
To the best of the authors' knowledge, up to now, almost no attention has been paid to the study of fault detectability criterion and fault separability criterion for nonlinear systems with faults occurring, though the diagnosability analysis and judgment of linear continuous system are focused [17].But we notice that the differential geometry based method is one of the most important methods in robust fault diagnosis of nonlinear system [18][19][20][21].The problem of fault diagnosis is converted into residual generator problem [18].A solution to the problem of fault detection and isolation is characterized in terms of properties of certain distributions, which can be considered as the nonlinear analog of the unobservability subspaces [19].The same problem is solved and the sufficient and necessary conditions are given by some distributions named unobservability codistribution [20].And the appropriate coordinate transformation based on the principle of differential geometry can be designed to make one subsystem only affected by one fault, and then the fault is detectable and separable.
In this paper, the methods, based on the principle of differential geometry, are proposed to analyze the fault detectability and separability (FDS) different to FDI.Usually, the attention is on the approach to FDI, and the conditions that make the faults be diagnosable are given based on the designed approaches.However, in this paper, the diagnosability of fault is regarded as the inherent attribute of the system and is not changed with the approach of diagnosis.
Affine nonlinear system is the most common system, such as induction/synchronous motor, rigid robot, space craft, and satellite.The main contributions of this paper are summarized as follows.For a class of affine nonlinear systems with faults occurring, (1) the definitions of detectability and separability are given and the criteria to judge detectability and separability are proposed and proved and (2) the detectability incidence matrix and separability incidence matrix are defined for quantitative fault diagnosability calculation and fault separability calculation, respectively.

System Model and Basic Properties
Consider the following affine nonlinear system: where  ∈ R  is the state vector,   is the input ( = 1, 2, . . ., ), and  ∈ R  is the output vector.
According to the principle of differential geometry, let us introduce the following lemmas which will be used in deriving our main results.

Fault Diagnosability Analysis
In the affine nonlinear system, its faults are regarded as the inputs of the system.The fault diagnosability evaluation includes two aspects: faults detection and faults separation.

Definition 3 (detectability).
Given the affine nonlinear model (3), if outputs y will be not influenced by   (i.e., y is invariant to   ), then fault   will be not detected by y; that is,   is undetectable; otherwise,   will be detected by y; that is,   is detectable.The fact that the ability of faults will be detected by output is named as detectability.
And then, we propose the following proposition as the criteria to judge detectability.

Proposition 4 (detectability judgment). Given the affine nonlinear model (3), the necessary and sufficient condition for 𝑤 𝑖 being detectable is
where ⟨,   , . . .,   , According to the distribution algorithm where Δ 1 and Δ 2 are any distribution), we get the conclusion that that is, This is contradictory with the supposed equation ( 5).So, the necessary condition is true. ( So, according to Lemma 1, for all  = 1, . . ., ,   = ℎ  () (where ℎ  ∈ {ℎ 1 , . . ., ℎ  }) will be not influenced by   ; that is,   is not detectable, which is contradictory with the supposing that   is detectable.So, the sufficient condition is true.This completes the proof of Proposition 4.

Evaluation Indicator of the Detectability.
Here, we design the incidence matrix and quantitative fault diagnosability calculation method as the following definitions.
Definition 6 (diagnosability calculation).DFD (Degree of Fault Detectability) is as follows: where  is the number of the nonzero column vectors in Π  *  and  is the sum of the faults.

Fault Separability Judgment.
First, we give the definition of faults separability.And then, we propose the following proposition as the criteria to judge separability.
Definition 7 (separability).For the detectable faults   and   of nonlinear system (3), there is the ability to separate   from   when the influence of outputs y affected by   is different from   , and this ability is named as separability.
For the detectable faults, we can judge their separability by the distribution (space) spanned by the fault vector field (i.e., the coupled mode of fault).
(1) If the spanned distributions span(  ) and span(  ) are equivalent, then the influence of   on the outputs y is same as the influence of   ; that is,   and   are inseparable.Otherwise,   and   are separable.
(2) When the distributions (space) span(  ) and span(  ) are not equivalent, the following proposition will be used to judge the separability of   and   .
Proposition 8 (separability judgment).Given the affine nonlinear system (3)  It is contradictory with (13).So, the fact that the faults   and   are separable is true.This completes the proof of Proposition 8.

Evaluation Indicator of Separability.
Similarly, we design the incidence matrix and quantitative fault separability calculation method as the following definitions.

Mathematical Problems in Engineering
Definition 9 (separability incidence matrix).For separability incidence matrix Γ  *  , its element is denoted by Γ  (1 ≤  ≤ , 1 ≤  ≤ ).Γ  ( ̸ = ) is to express the separability of   and   .If   and   are separable, then set the value of Γ  as 1 or else the value as 0; and set Γ  = 0 particularly.Definition 10 (separability calculation).DFS (Degree of Fault Separation) is as follows: where Γ  is the element of Γ  *  and  is the sum of the detectable faults.

Illustration Example
The following affine nonlinear system is given as an example to demonstrate the fault diagnosability analysis methods outlined above: where  1 ,  2 , and  3 are the faults of the system and So, according to Proposition 4,  1 is detectable,  2 is not detectable, and  3 is detectable.Then, the detectability incidence matrix where So,  1 and  3 are separable according to Proposition 8.The separability incidence matrix So, the DFS of  1 and  3 is Next, we design the state observer and get the residual error to verify the detectability and separability of the faults.Let and then subsystem (26) changed as 2 where Design the state observer as follows: where ( + ) is Hurwitz; here and then where The residual error output is shown in Figure 1 when giving step fault input  1 at time 10∼15 s into the system; and when giving step fault input  3 at time 10∼15 s, the residual error output is shown in Figure 2. In Figures 1 and 2, we can see that the residual error obviously changed at time 10∼15 s.Therefore, the faults  1 and  3 are detectable, same as the above analyses of the detectability.
The residual error output is shown in Figure 3 when giving step fault input  2 at time 10∼15 s into the system; from Figure 3, we can see that fault  2 is not detectable, same as the above analyses of the detectability.
When giving step fault input  1 at time 10∼15 s and step fault input  3 at time 20∼30 s, the residual error output is shown in Figure 4.In Figure 4, we can see that the residual error obviously changed at time 10∼15 s and time 20∼30 s separately.Therefore, the faults  1 and  3 are separable, same as the above analyses of the separability.

Conclusion
In this paper, we use the principle of differential geometry to analyze the problem of diagnosability of a class of affine nonlinear systems with faults.We first give the necessary and sufficient conditions for (1) detectability judgment, that is, the conditions which the system should satisfy whether the faults are detectable or not, and (2) separability judgment, that is, the conditions which the system should satisfy whether the faults are separable or not.And the measurement, that is, DFD (Degree of Fault Detectability) and DFS (Degree of Fault Separation), based on the defined DIM (detectability incidence matrix) and SIM (separability incidence matrix) is established to quantify diagnosability performance.Then, we have provided an example to analyze and test the faults diagnosability.However, the simple quantitative measure indexes, DFD and DFS, have not taken full advantage of the full information of DIM and SIM.More comprehensive

1 Figure 1 :
Figure 1: The residual error when giving step fault  1 at time 10∼15 s.