Building fault detection and isolation related objects

FDIModel <: AbstractFDDObject

Type for synthesis models employed to solve fault detection and isolation problems.

If sysf::FDIModel is the synthesis model object, the underlying descriptor system model can be obtained via sysf.sys and the dimensions of the control, disturbance, fault, noise and auxiliary vectors are contained in the integers sysf.mu, sysf.md, sysf.mf, sysf.mw and sysf.ma, respectively. The ranges of indices of control, disturbance, fault, noise and auxiliary inputs can be accessed as sysf.controls, sysf.disturbances, sysf.faults, sysf.noise and sysf.aux, respectively.

fdimodset(sys; controls, c, disturbances, d, faults, f, fa, faults_sen, fs, noise, n, aux) -> sysf::FDIModel

Build for a given linear time-invariant system model sys = (A-λE,B,C,D), a synthesis model object sysf::FDIModel to be used in conjunction with the analysis and synthesis functions of FDI filters. If sys is a vector of system models, then sysf results as a vector of synthesis model objects.

The information on the partition of the input components in control, disturbance, fault, noise and auxiliary inputs can be specified using the following keyword arguments:

controls = inpu or c = inpu specifies the indices inpu of the control inputs (default: void)

disturbances = inpd or d = inpd specifies the indices inpd of the disturbance inputs (default: void)

faults = inpf or f = inpf specifies the indices inpf of the fault inputs (default: void)

fa = inpfa specifies the indices inpfa of control inputs subject to actuator fault (default: void)

faults_sen = inpfs or fs = inpfs specifies the indices inpfs of the system outputs subject to sensor fault inputs (default: void)

noise = inpn or noise = inpn specifies the indices inpn of the noise inputs (default: void)

aux = inpa specifies the indices inpa of the auxiliary inputs (default: void)

The indices of inputs or outputs can be specified as integer vectors, integer scalars or integer UnitRanges.

The resulting sysf contains the partitioned system sysf.sys = (A-λE,[Bu Bd Bf Bw Bv],C,[Du Dd Df Dw Dv]), where Bu, Bd, Bf, Bw and Bv are the input matrices from the control inputs u, disturbance inputs d, fault inputs f, noise inputs w and auxiliary inputs v, respectively, and Du, Dd, Df, Dw and Dv are the feedthrough matrices from those inputs. The dimensions of control, disturbance, fault, noise and auxiliary input vectors are contained in sysm.mu, sysm.md, sysm.mf, sysm.mw and sysm.ma, respectively.

Method: If G(λ) is the p x m transfer function matrix of sys, then the resulting system sysf has an equivalent input output form [Gu(λ) Gd(λ) Gf(λ) Gw(λ) Gv(λ)], where the following relations define the component matrices: Gu(λ) = G(λ)*Su, Gd(λ) = G(λ)*Sd, Gf(λ) = [G(λ)*Sf Ss], Gw(λ) = G(λ)*Sw, Gv(λ) = G(λ)*Sv, with the selection matrices Su, Sd, Sf, Sw and Sv formed from the columns of the m-th order identity matrix and Ss is formed from the columns of the p-th order identity matrix.

FDFilter <: AbstractFDDObject

Type for fault detection filters resulted as solutions of fault detection problems.

If filter::FDFilter is the fault detection filter object, the underlying descriptor system model can be obtained via filter.sys and the dimensions of the partitioned filter input vectors as measured outputs and control inputs, can be accessed as the integers contained in filter.ny and filter.mu, respectively. The ranges of the indices of output and control inputs can be accessed as filter.outputs and filter.controls, respectively.

FDFilter(sys, ny, mu) -> Q::FDFilter

Build for a given linear time-invariant descriptor system model sys = (A-λE,B,C,D), a fault detection filter object Q, as determined with the synthesis functions of FDI filters. ny and mu are the number of measured outputs and the number of control inputs, respectively. It is assumed that B = [By Bu Bv] and D = [Dy Du Dv] are partitioned matrices such that By and Dy have ny columns, and Bu and Du have mu columns, where By and Bu are the input matrices from the measured outputs y and control inputs u, Dy and Du are the feedthrough matrices from the measured outputs y and control inputs u.

The resulting Q contains the partitioned system Q.sys = (A-λE,[By Bd],C,[Dy Du]) and the dimensions of the partitioned filter input vectors as measured outputs and control inputs, can be accessed as the integers contained in Q.ny and Q.mu, respectively.

FDFilterIF <: AbstractFDDObject

Type for the internal form of fault detection filters resulted as solutions of fault detection problems.

If filter::FDFilterIF is the fault detection filter internal form object, the underlying descriptor system model can be obtained via filter.sys and the dimensions of the control, disturbance, fault, noise and auxiliary vectors are contained in the integers filter.mu, filter.md, filter.mf, filter.mw and filter.ma, respectively. The ranges of indices of control, disturbance, fault, noise and auxiliary inputs can be accessed as filter.controls, filter.disturbances, filter.faults, filter.noise and filter.aux, respectively.

FDFilterIF(sys; mu = 0, md = 0, mf = 0, mw = 0, ma = 0, moff = 0) -> R::FDFilterIF

Build for a given linear time-invariant descriptor system model sys = (A-λE,B,C,D), a fault detection filter internal form object R, as determined with the synthesis functions of FDI filters. mu, md, mf, mw and maux are the dimensions of control, disturbance, fault, noise and auxiliary input vectors, respectively. It is assumed that B = [Boff Bu Bd Bf Bw Bv] and D = [Doff Du Dd Df Dw Dv] are partitioned matrices such that Boff and Doff have moff columns, Bu and Du have mu columns, Bd and Dd have md columns, Bf and Df have mf columns, Bw and Dw have mw columns, and Bv and Dv have maux columns.

The resulting R contains the partitioned system R.sys = (A-λE,[Bu Bd Bf Bw Bv],C,[Du Dd Df Dw Dv]) and the dimensions of control, disturbance, fault, noise and auxiliary input vectors are contained in R.mu, R.md, R.mf, R.mw and R.ma, respectively.

FDIFilter <: AbstractFDDObject

Type for fault detection and isolation filters resulted as solutions of fault detection and isolation problems.

If filter::FDIFilter is the fault detection and isolation filter object, the underlying i-th descriptor system model can be obtained via filter.sys[i] and the dimensions of the partitioned filter input vectors as measured outputs and control inputs, can be accessed as the integers contained in filter.ny and filter.mu, respectively. The ranges of the indices of output and control inputs can be accessed as filter.outputs and filter.controls, respectively.

FDIFilter(sys, ny, mu) -> Q::FDIFilter

Build for a vector of linear time-invariant descriptor system models sys[i] = (Ai-λEi,Bi,Ci,Di) with the same number of inputs, a fault detection and isolation filter object Q, as determined with the synthesis functions of FDI filters. ny and mu are the number of measured outputs and the number of control inputs, respectively. It is assumed that each Bi = [Byi Bui Bvi] and Di = [Dyi Dui Dvi] are partitioned matrices such that Byi and Dyi have ny columns, and Bui and Dui have mu columns, where Byi and Bui are the input matrices from the measured outputs y and control inputs u, Dyi and Dui are the feedthrough matrices from the measured outputs y and control inputs u.

The resulting Q contains the vector of partitioned systems Q.sys[i] = (Ai-λEi,[Byi Bdi],Ci,[Dyi Dui]) and the dimensions of the partitioned filter input vectors as measured outputs and control inputs, can be accessed as the integers contained in Q.ny and Q.mu, respectively.

FDIFilterIF <: AbstractFDDObject

Type for the internal form of fault detection and isolation filters resulted as solutions of fault detection and isolation problems.

If filter::FDIFilterIF is the fault detection and isolation filter internal form object, the underlying i-th descriptor system model can be obtained via filter.sys[i] and the dimensions of the control, disturbance, fault, noise and auxiliary vectors are contained in the integers filter.mu, filter.md, filter.mf, filter.mw and filter.ma, respectively. The ranges of indices of control, disturbance, fault, noise and auxiliary inputs can be accessed as filter.controls, filter.disturbances, filter.faults, filter.noise and filter.aux, respectively. .

FDIFilterIF(sys; mu = 0, md = 0, mf = 0, mw = 0, ma = 0, moff = 0 ) -> R::FDIFilterIF

Build for a vector of linear time-invariant descriptor system models sys[i] = (Ai-λEi,Bi,Ci,Di), a fault detection and isolation filter internal form object R, as determined with the synthesis functions of FDI filters. mu, md, mf, mw and ma are the dimensions of control, disturbance, fault, noise and auxiliary input vectors, respectively. It is assumed that each Bi = [Boffi Bui Bdi Bfi Bwi Bvi] and Di = [Doffi Dui Ddi Dfi Dwi Dvi] are partitioned matrices such that Boffi and Doffi have moff columns, Bui and Dui have mu columns, Bdi and Ddi have md columns, Bfi and Dfi have mf columns, Bwi and Dwi have mw columns, and Bvi and Dvi have ma columns.

The resulting R contains the vector of partitioned systems R.sys[i] = (A-λE,[Bui Bdi Bfi Bwi Bvi],C,[Dui Ddi Dfi Dwi Dvi]) and the dimensions of control, disturbance, fault, noise and auxiliary input vectors are contained in R.mu, R.md, R.mf, R.mw and R.ma, respectively.

fdIFeval(sysQ::FDFilter, sysf::FDIModel; minimal = false, atol, atol1 = atol, atol2 = atol, rtol, fast = true) -> sysR::FDFilterIF

Compute the internal form sysR of the fault detection filter sysQ applied to the synthesis model sysf. If sysf has the partitioned transfer function matrix G(λ) = [ Gu(λ) Gd(λ) Gf(λ) Gw(λ) Gv(λ) ] in accordance with the partitioned system inputs as controls, disturbances, faults, noise and auxiliary inputs, respectively, and Q(λ) = [ Qy(λ) Qu(λ) ] is the partitioned transfer function matrix of the fault detection filter sysQ in accordance with the partitioned filter inputs as measurable outputs and control inputs, then the transfer function matrix R(λ) of the resulting internal form sysR is given by

 R(λ) = | Qy(λ)  Qu(λ) | * | Gu(λ)  Gd(λ) Gf(λ) Gw(λ) Gv(λ) |
                           |  I       0     0     0     0   |

A minimal descriptor realization is computed if minimal = true and a possibly non-minimal realization is determined if minimal = false (default).

The minimal realization computation relies on pencil manipulation algorithms which employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

If (Ar-λEr,Br,Cr,Dr) is the full order descriptor realization of sysR.sys, then the keyword arguments atol1, atol2, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of matrices Ar, Br, Cr, Dr, the absolute tolerance for the nonzero elements of Er, and the relative tolerance for the nonzero elements of Ar, Br, Cr, Dr and Er. The default relative tolerance is nϵ, where ϵ is the working machine epsilon and n is the order of the system sysR. The keyword argument atol can be used to simultaneously set atol1 = atol and atol2 = atol.

fdIFeval(sysQ::FDIFilter, sysf::FDIModel; minimal = false, atol, atol1 = atol, atol2 = atol, rtol, fast = true) -> sysR::FDIFilterIF

Compute the internal form sysR of the fault detection and isolation filter sysQ applied to the synthesis model sysf. If sysf has the partitioned transfer function matrix G(λ) = [ Gu(λ) Gd(λ) Gf(λ) Gw(λ) Gv(λ) ] in accordance with the partitioned system inputs as controls, disturbances, faults, noise and auxiliary inputs, respectively, and Qi(λ) = [ Qyi(λ) Qui(λ) ] is the partitioned transfer function matrix of the i-th filter sysQ.sys[i] in accordance with the partitioned filter inputs as measurable outputs and control inputs, then the transfer function matrix Ri(λ) of the i-th filter in the resulting internal form sysR.sys[i] is given by

 Ri(λ) = | Qyi(λ)  Qui(λ) | * | Gu(λ)  Gd(λ) Gf(λ) Gw(λ) Gv(λ) |
                              |  I       0     0     0     0   |

Minimal descriptor realizations are computed if minimal = true and a possibly non-minimal realization is determined if minimal = false (default).

The minimal realization computation relies on pencil manipulation algorithms which employ rank determinations based on either the use of rank revealing QR-decomposition with column pivoting, if fast = true, or the SVD-decomposition. The rank decision based on the SVD-decomposition is generally more reliable, but the involved computational effort is higher.

If (Ari-λEri,Bri,Cri,Dri) is the full order descriptor realization of sysR.sys[i], then the keyword arguments atol1, atol2, and rtol, specify, respectively, the absolute tolerance for the nonzero elements of matrices Ari, Bri, Cri, Dri, the absolute tolerance for the nonzero elements of Eri, and the relative tolerance for the nonzero elements of Ari, Bri, Cri, Dri and Eir. The default relative tolerance is ni*ϵ, where ϵ is the working machine epsilon and ni is the order of the system sysR.sys[i]. The keyword argument atol can be used to simultaneously set atol1 = atol and atol2 = atol.