i.e. Q(s) is a multi-dimensional transfer-matrix with known signals y(t) and u(t) as inputs and a scalar residual as output. The requirement on the residual generator, i.e. Q(s), is that it is sensitive to monitored faults and not sensitive to disturbances (including non-monitored faults). This section introduces linear residual generation problem and also briefly describes the minimal polynomial basis solution. All derivations are performed in the continuous case but the corresponding results for the time-discrete case can be obtained by substituting s by z and improper by non-causal.

2.1 Linear Residual Generation
The systems studied in this work are assumed to be on the form

where y(t) is measurements, u(t) is known inputs to the system, d(t) is unknown disturbances including non-monitored faults, and f(t) is the monitored faults. The filter Q(s) in (1) is a residual generator if and only if the transfer function from u and d to r is zero, i.e. limt !1 r(t) = 0 for all d(t) and u(t) when f(t) = 0. To be able to detect faults, it is also required that r(t) <> 0 when f(t) <> 0.

Inserting (2) into (1) gives

To make r(t) = 0 when f(t) = 0, it is required that disturbances and the control signal are decoupled, i.e. for Q(s) to be a residual generator, it must hold that

Also, it is required that

with suitable properties, e.g. adequate DC-gain from faults. Thus, the linear residual generation problem is to find a suitable Q(s) that fulfills (3) and (4).

2.2 A Solution based on Minimal Polynomial Bases
Equation (3) implies that Q(s) must belong to the left null-space of

This null-space is denoted N_L(M(s)). The matrix Q(s) thus need to fulfill two requirements: belong to the left null-space of M(s) and give good fault sensitivity properties in the residual. If, in a first step of the design, all Q(s) that fulfills the first requirement is found, then a Q(s) with good fault sensitivity properties can be selected. Thus, in a first step of the design of the residual generator Q(s) we need not consider f or Gf (s). The problem is then to find all rational Q(s) that belong to N_L(M(s)). Of special interest are the residual generators with least McMillan degree, i.e. the number of states in a minimal realization. Finding all Q(s) in N_L(M(s)) can be done by finding a minimal basis for the rational vector-space N_L(M(s)). A minimal basis for a rational vector-space is a polynomial basis (Forney 1975). For now, assume that such a basis can be found and let the base vectors be the rows of a matrix denoted N_M(s). How to extract such a basis will be dealt with in Section 2.3. By inspection of (5), it can be realized that the dimension of N_L(M(s)) (i.e. the number of rows of N_M(s) is

where m is the number of outputs, i.e. the dimension of y(t), and k_d is the number of disturbances, i.e. the dimension of d(t). The last equality holds only if rank G_d(s) = k_d, but this should be the normal case. When a polynomial basis N_M(s) has been obtained, the second and final step in the residual generator design is to shape fault-to-residual responses as described next. The minimal polynomial basis N_M(s) is irreducible and because of this (Kailath 1980), all decoupling residual generators Q(s) can be parameterized as

where (s) is an arbitrary rational vector of suitable dimensions. This parameterization vector (s) can e.g. be used to shape the fault-to-residual response or simply to select one row in . Since N_M(s) is a basis, the parameterization vector (s) have minimal number of elements. The only constraint on (s) if the residual generator Q(s) is to be realizable, it must be chosen such that (6) is proper. This means that the dynamics, i.e. poles, of the residual generator Q(s) can be chosen freely. This also means that the minimal order of a realization of a decoupling filter is determined by the row-degrees of the minimal polynomial basis N_M(s).

%% Define state-space matrices
>> A =  [0, 0, 1.1320, 0,-1
            0,-0.0538,-0.1712, 0,0.0705
            0, 0, 0, 1,0
            0, 0.0485, 0,-0.8556,-1.0130
            0,-0.2909, 0, 1.0532,-0.6859];

>> Gd = Gu(:,3);
>> Gyf = [eye(3,3) Gu];
>> ndist = size(Gd,2);
>> nfault = size(Gyf,2);

>> B_d = B(:,3);
>> D_d = D(:,3);
>> rank(ctrb(A,[B B_d]))
ans =
        5

>> M = minreal([Gu Gd;eye(nctrl) zeros(nctrl,ndist)]);
>> [N,D] = ss2rmf(M.a, M.b, M.c, M.d);
>> Nm2 = null(N.’).’
Nm2 =
       0.071s 0.054 + s 0.091 0.12 -1 0
       15s + 23sˆ2 -6.7 -17 - 0.94s + sˆ2 31 0 0

where ss2rmf is a macro of the Polynomial Toolbox that converts a state-space model to a right MFD. As can be seen, the bases are identical. A minimal polynomial basis is not unique, but the algorithm implemented in the toolbox null command is based on a canonical echelon form resulting in identical bases. The above basis has a natural interpretation. It means that the following equations hold in the fault-free case and also for all values of the actuator fault f6 = d.

Figure 2: Transfer functions from the 6 faults to the residual. Note that fault 6 is decoupled, i.e. gain 0 from f6 to r.

Figure 3: Transfer functions from the 6 faults to the residual. Note that fault 6 is decoupled, i.e. gain 0 from f6 to r. The dashed lines corresponds to the results of the first design with '(s) given by (11). The solid lines corresponds to the second design where '(s) given by (12).

References