6.2 Solution for complete fault-output decoupling
6.2.3 Solving the complete fault-output decoupling problem . 132
This section proposes a solution for the complete fault-output decoupling prob-lem (Probprob-lem 6.2). The solution is closely related to the solution for the input-output decoupling problem using a regular static state feedback (6.3) for a square analytic system (6.4) and (6.5), see e.g. Falb and Wolovich (1967); Gras and Ni-jmeijer (1989). Therefore, the solution for the input-output decoupling will be briefly introduced first and then a solution for the fault-output decoupling with its conditions is presented.
Input-output decoupling
One solution for input-output decoupling by a regular static state feedback for the system (6.1) and (6.2) can be obtained by following the solution given in Gras and Nijmeijer (1989). Taking the system (6.1) and (6.2), omitting the faults
, and considering it to be square, i.e. it has the same number of inputs as outputs
m=l, the following system remains:
_
x = Ax + Bu (6.4)
y = Cx + Du (6.5)
wherex 2 Rn describes the states,u 2 Rm the inputs, y 2 Rl the outputs,A,
B, C andDare matrices of appropriate size, and m = l. For this system the
6.2 Solution for complete fault-output decoupling 133
following derivatives can be calculated:
0
B
B
@ y
( u
1 )
1
...
y (
u
l )
l 1
C
C
A
= 0
B
B
@ C
1 A
u
1
...
C
p A
u
l 1
C
C
A
x + M u
dec
u (6.6)
Looking at the derivatives (6.6) it can be seen that it is possible to establish input-output decoupling from the new inputs wj to the outputs yi by the following regular static state feedback which has the same form as (6.3):
u= 0
B
B
@ u
1
...
u
m 1
C
C
A
= ( M u
dec )
1 0
B
B
@ C
1 A
u
1
...
C
p A
u
l 1
C
C
A
x + ( M u
dec )
1 0
B
B
@ w
1
...
w
m 1
C
C
A
(6.7)
as it leads to:
0
B
B
@ y
( u
1 )
1
...
y (
u
l )
l 1
C
C
A
= 0
B
B
@ w
1
...
w
m 1
C
C
A
=w (6.8)
The solution given by equation (6.7) obviously only holds ifMu
dec
is invertible and that means it is nonsingular or equivalently has full rank:
rankM u
dec
= m (6.9)
In Falb and Wolovich (1967); Gras and Nijmeijer (1989) the proof is given that the regular static state feedback input-output problem (as considered here) is solvable if and only if equation (6.9) holds true.
Complete fault-output decoupling
A solution for the complete fault-output decoupling problem (Problem 6.2), is derived corresponding to the solution for the input-output decoupling problem presented above. Also here the system (6.1) and (6.2) is considered to be square, i.e.l=m. First the case is considered that the characteristic numbers fulfill the following condition:
u
i
=
i
8i2l (6.10)
Later the other cases where condition (6.10) does not hold will be discussed in detail. Using the condition (6.10) the following derivatives similar to equation (6.6) can be obtained:
0
B
B
@ y
(
1 )
1
...
y (
l )
l 1
C
C
A
= 0
B
B
@ C
1 A
1
...
C
p A
l 1
C
C
A
x + M u
dec
u + M
dec
(6.11)
When looking at the derivatives (6.11) it can be seen that to achieve fault-output decoupling for the considered system two different aspects have to be consid-ered: Is there any cross-coupling of states in the CiAi x terms that causes different faults to affect the same output? and Which structure does the decou-pling matrixM
dec
have? If the later has more than one nonzero element in one row there is no possibility to obtain complete fault-output decoupling, as it is not possible to prevent certain faults from effecting the output signals yi by direct compensation. This is due to the important difference that fault signals are un-known, hence, they cannot be compensated by using them in a feedback.
To avoid the mentioned cross-coupling of states in the Ci A
i
x terms the in-putsui can be used for a regular static state feedback in the same way as in the input-output decoupling problem (see (6.7) and (6.8)):
u= 0
B
B
@ u
1
...
u
m 1
C
C
A
= (M u
dec )
1 0
B
B
@ C
1 A
1
...
C
m A
l 1
C
C
A
x + ( M u
dec )
1 0
B
B
@ w
1
...
w
m 1
C
C
A
(6.12)
By applying the feedback (6.12) the equation (6.11) turns into
0
B
B
@ y
(
1 )
1
...
y (
l )
l 1
C
C
A
= 0
B
B
@ w
1
...
w
m 1
C
C
A + M
dec
(6.13)
Looking at equation (6.13) and keeping the result for the input-output decoupling problem in mind the following theorem can be given :
6.2 Solution for complete fault-output decoupling 135
Theorem 6.7 Complete fault-output decoupling can be achieved for a square analytic system of the form (6.1) and (6.2), which fulfills condition (6.10) by the regular static state feedback (6.12) if the following two conditions are fulfilled:
(i) The decoupling matrixMu
dec
is invertible.
(ii) The decoupling matrixM
dec
can be written in a diagonal form after pos-sible relabeling of the faults.
Remark 2: When looking at the feedback (6.12) it can be seen that the inputs are used to cancel out the cross-couplings Ci
A
i
x, withi2 l. This is done to achieve the wanted fault-output decoupling as described by (6.13). As already mentioned in Remark 1 a complete loss of theith actuator can be modeled by setting theithcolumn of the matrix Lxequal to theith column of the matrixB and theith component of x equal to ui. This would mean that the control action of theithinput (ui) is destroyed (compensated) by the fault. Hence, the compensation of the cross-coupling terms achieved by this input is lost. This will, however, only effect one predefined output. This is due to the fault-output decoupling. The actuator fault was modeled and taken care of during the fault-output decoupling design. Hence, the decoupling effect is only lost for exactly that output that by design is only affected by the actuator fault. So when this lost compensation affects the predefined input it has exactly the desired affect. If it would also affect other outputs it would proof that the fault-output decoupling was not designed correctly.
In the following the case will be considered that condition (6.10) is not fulfilled.
This is done by discussing the possibilities to achieve fault-output decoupling for the situationsui
<
i andui
>
i.
u
i
<
i
: If the characteristic number with respect to the inputs is smaller than the characteristic number with respect to the faults it means that there exists at least one derivative of the outputy(
u
i )
i
where the inputs enter explicitly before the faults do. This gives an additional freedom, because the inputsui,i 2 m could then be used for an additional state feedback to influence the outputyiin such a way, by compensating certain states, that some faults or even all faults do not enter/affect the output yi. The additional state feedback could be of the following structure
u = M
x + N
v
where M 2 R( m;n) andN 2 R( m;m) is a nonsingular matrix. That would give the possibility to influence the decoupling matrixM
dec
. The exact possi-bilities of the freedom available have been well studied in connection with the disturbance-decoupling problem (DDP), see e.g. Wonham and Morse (1970);
Wonham (1985); Isidori (1985); Nijmeijer and van der Schaft (1990). Another useful option might be in this case to apply a dynamic state feedback:
_ z = A
z z + B
zx x + B
zv
v (6.14)
u = C
z
z + D
zx
x + D
zv
v (6.15)
wherez2Rs describes the states of the dynamic feedback,x 2Rn the system states, u 2 Rm the inputs to the system,v 2 Rm the new inputs to the overall system, y 2 Rl the system outputs, and Az, Bzx, Bzv, Cz, Dzx, and Dzv are matrices of appropriate size. Applying a specific number of integrations on the input could for example lead to the situation that condition (6.10) is fulfilled for the new input signalsvi,i2m. However, this idea requires further study, which is not in the scope of this thesis.
u
i
>
i
: If the characteristic number with respect to the inputs is bigger than the characteristic number with respect to the faults it means that there exists at least one derivative of the outputy(
u
i )
i
where the faults enter explicitly before the inputs do. Hence, the fault-output decoupling problem cannot be solved in this case as there is no possibility to compensate possible cross-coupling of states in theCiAi xterms or to influence the decoupling matrixM
dec
.
6.3 Efficient fault-output decoupling
In the previous sections the concept of complete fault-output decoupling, see Definition 6.1 and Problem 6.2, has been introduced. A solution and the con-dition when it can be obtained was derived and stated in Theorem 6.7. In this section we take a look at what to do when complete fault-output decoupling can-not be obtained due to the structure of the considered system. Would that mean that fault-output decoupling is not applicable at all? The answer is ’no’. By dis-cussing problems with a non-diagonal decoupling matrixM
dec
and with sensor faults it will be illustrated why. As a result an additional definition of fault-output decoupling, next to the complete fault-fault-output decoupling, will be given.
It is called efficient fault-output decoupling for FDI.
6.3 Efficient fault-output decoupling 137