Analysis of FDI synthesis models

fdigenspec Generation of achievable FDI specifications.
fdichkspec Feasibility analysis of a set of FDI specifications.

FaultDetectionTools.fdigenspec — Function

S = fdigenspec(sysf::FDIModel; sdeg, FDtol, FDGainTol, FDfreq, atol, atol1, atol2, atol3, rtol, fast = true)

Generate all achievable specifications S for a given synthesis model sysf with additive faults. Each row of the resulting binary matrix S contains a nonzero specification (or fault signature) which can be achieved using a linear fault detection filter (e.g., as obtainable with the help of function efdisyn).

FDFreq = freq specifies a vector of real frequency values or a scalar real frquency value for strong detectability checks (default: FDFreq = missing).

FDtol = tol1 specifies the threshold tol1 for assessing weak specifications (see also function fditspec) (default: tol1 = 0.0001).

FDGainTol = tol2 specifies the threshold tol2 for assessing strong specifications, i.e., the threshold for nonzero frequency responce gains for all frequency values specified in freq (see also function fdisspec) (default: tol2 = 0.01).

The keyword argument sdeg = β specifies a prescribed stability degree β for the poles of the internally generated candidate filters, such that the real parts of filters poles must be less than or equal to β, in the continuous-time case, and the magnitudes of filter poles must be less than or equal to β, in the discrete-time case. If sdeg = missing then no stabilization is performed if FDFreq = missing. If sdeg = missing and FDFreq = freq, then the following default values are employed : β = -0.05, in continuous-time case, and β = 0.95, in discrete-time case.

The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

The keyword arguments atol1, atol2, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of A, B, C, D, the absolute tolerance for the nonzero elements of E, and the relative tolerance for the nonzero elements of A, B, C, D and E. The default relative tolerance is n*ϵ, where ϵ is the working machine epsilon and n is the order of the system sysf.sys. The keyword argument atol3 is an absolute tolerance for observability tests (default: internally determined value). The keyword argument atol can be used to simultaneously set atol1 = atol, atol2 = atol and atol3 = atol.

Method: The Procedure GENSPEC from [1] is implemented. The nullspace method [2] is recursively employed to generate candidate fault detection and isolation filters, whose internal forms provide the structure matrices corresponding to the achievable weak specifications, if freq = missing, or strong specifications for the frequencies conatined in freq. The generation method is also described in [3].

References:

[1] A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques. Springer Verlag, 2017; sec. 5.4.

[2] A. Varga, On computing nullspace bases – a fault detection perspective. Proc. IFAC 2008 World Congress, Seoul, Korea, pages 6295–6300, 2008.

[3] A. Varga, On computing achievable fault signatures. Proc. SAFEPROCESS'2009, Barcelona, Spain.

FaultDetectionTools.fdichkspec — Function

fdichkspec(sysf::FDIModel, SFDI::BitMatrix; sdeg, FDtol, FDGainTol, FDfreq, 
             atol, atol1, atol2, atol3, rtol, fast = true, minimal = false) -> (rdims, orders, leastorders)

Check for a given synthesis model sysf::FDIModel the feasibility of a set of fault detection and isolation specifications SFDI. If SFDI has N rows (i.e., contains N specifications), then the N-dimensional integer vectors rdims, orders, leastorders are returned and contain information related to the synthesis of FDI filters to achieve the feasible specifications. For the i-th specification contained in SFDI[i,:], rdims[i] contains the number of residual outputs of a minimal nullspace basis based FDI filter which can be used to achieve this specification. If rdims[i] = 0, then the i-th specification is not feasible. For a feasible i-th specification, orders[i] contains the order of the minimal nullspace basis based FDI filter which can be used to achieve this specification. If the i-th specification is not feasible, then orders[i] is set to -1. If minimal = true, leastorders[i] contains the least achievable order for a scalar output FDI filter which can be used to achieve the i-th specification. If minimal = false or if the i-th specification is not feasible, then leastorders[i] is set to -1.

FDFreq = freq specifies a vector of real frequency values or a scalar real frquency value for strong detectability checks (default: FDFreq = missing).

FDtol = tol1 specifies the threshold tol1 for assessing weak specifications (see also function fditspec) (default: tol1 = 0.0001).

The keyword argument sdeg = β specifies a prescribed stability degree β for the poles of the internally generated candidate filters, such that the real parts of filters poles must be less than or equal to β, in the continuous-time case, and the magnitudes of filter poles must be less than or equal to β, in the discrete-time case. If sdeg = missing then no then no stabilization is performed if and FDFreq = missing. If sdeg = missing and FDFreq = freq, then the fllowing default values are employed : β = -0.05, in continuous-time case, and β = 0.95, in discrete-time case.

The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if fast = true or the more reliable SVD-decompositions if fast = false.

Method: The nullspace method of [1] is successively employed to determine FDI filters as minimal left nullspace bases which solve suitably formulated fault detection problems.

References:

[1] A. Varga, On computing nullspace bases – a fault detection perspective. Proc. IFAC 2008 World Congress, Seoul, Korea, pages 6295–6300, 2008.