Q2. What are the main functions of a residual generator?
A residual generator has at least two basic functions: (1) generating zero residuals in the fault-free case; (2) generating nonzero residuals when any fault occurs in the system.
Q3. What are the other tools of the DESCRIPTOR TOOLBOX?
Besides snull and smcover, fd also calls other tools of the DESCRIPTOR TOOLBOX as the minimal realization of generalized systems, finite-infinite/stable-unstable spectral separations, coprime factorization, etc.
Q4. What is the main computational problem in (Frisk and Nyberg, 2001)?
The main computational problem in (Frisk and Nyberg, 2001) is the numerical computation of a minimal polynomial basis for the left nullspace of a certain rational matrix.
Q5. What is the first limitation of the proposed approach?
The first limitation is the intrinsic ill-conditioning of polynomial representations because of possible extremely wide range of polynomial coefficients.
Q6. What is the second limitation of the proposed approach?
The second limitation, pointed out by Van Dooren (1981), is that many algorithms based on polynomial manipulations are numerically unstable.
Q7. How can the authors obtain least order nullspaces?
Least order fault detectors can be obtained by selecting an appropriate linear combination of the basis vectors by eliminating non-essential dynamics (see Section 4).
Q8. What is the definition of a residual generator?
From a system theoretic point of view, the residual generators are physically realizable systems having as inputs the measured outputs and the control inputs of the monitored system, and as output the generated residual.
Q9. What is the simplest way to compute a nullspace basis?
Using the staircase form (8), it is shown in (Beelen, 1987) that a minimal polynomial basis can be computed by selecting νi−1 − νi polynomial basis vectors of degree i − 1, for i = 1, . . . , ` + 1. This basis can be used to construct a minimal rational basis by making each column proper with appropriate order denominators.
Q10. What are the main computational ingredients of the least order detector?
The involved main computational ingredients are: (1) the computation of a rational nullspace basis of a rational matrix; (2) the reduction of the dynamical order of the detector.
Q11. What is the main idea behind the proposed approach to design least order detectors?
The authors believe that the overall approach to design least order detectors is a viable alternative to polynomial bases based approaches.
Q12. What is the main limitation of the proposed approach to design least order detectors?
Their approach to design least order detectors (see Section 2) is based on a new numerically stable algorithm to compute least order rational nullspace bases of rational matrices (see Section 3 ).
Q13. What is the resulting system of least McMillan order?
The resulting system of least McMillan order is(Â11 − λÊ11, B̂12, Ĉ1 + Dr,2F̂11, Dr,1 + Dr,2Lr)The proposed method to compute rational nullspace bases has been implemented in a MATLAB m-function snull, based on the computation of orthogonal Kronecker-like forms available in the DESCRIPTOR TOOLBOX (Varga, 2000).
Q14. How many uncontrollable eigenvalues can be obtained?
by taking Lr such that B21Lr + B22 = 0 andFr,2 = [F̂11 0 0 0] Z−1with F11 satisfying B̂21F̂11 + Â21 = 0, the authors achieve the cancellation of the maximum number of uncontrollable eigenvalues.
Q15. What is the simplest way to determine the TFM of a linear residual generator?
A linear residual generator (or detector) of least dynamical order is sought having the general formr(λ) = R(λ) [ y(λ) u(λ) ] (1)such that: (i) r(t) = 0 when f(t) = 0; and (ii) r(t) 6= 0 when fi(t) 6= 0, for i = 1, . . . , q. Besides the requirement that the TFM of the detector R(λ) has least possible McMillan degree, it is also necessary, for physical realizability, that R(λ) is a proper and stable TFM.