A New Input Design Framework for Asymptotic Active Fault Diagnosis with Application to Integrated Diagnosis and Control

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A New Input Design Framework for Asymptotic
Active Fault Diagnosis with Application to
Integrated Diagnosis and Control
Feng Xu∗
Tsinghua Shenzhen International Graduate School, Tsinghua University, 518055 Shenzhen, P.R.China.
Abstract
This paper proposes a new input design framework for set-based asymptotic active fault diagnosis and then applies it to
integrated design of active fault diagnosis and control. First, a new input design method is proposed for asymptotic active fault
diagnosis by maximizing a newly-defined excluding degree of the origin from all healthy and faulty residual zonotopes, which
has the advantage of low conservatism. Second, further using the excluding degree notion, an input set is constructed based
on the mathematical expression of the excluding degree at each time instant such that any input out of the input set can be
used for active fault diagnosis. Third, with the input set, an optimal input is designed out of the input set at each time instant
for simultaneous active fault diagnosis and control. Finally, detailed analysis and comparisons of the proposed methods with
other existing methods are made. Besides, an optimal input can also be designed out of the input set to achieve integrated
design of active fault diagnosis and other objectives (e.g., minimal input energy to reduce harm to the system, etc.). At the
end of this paper, several examples are used to illustrate the effectiveness of the proposed methods.
Key words: Input sets, active fault diagnosis, control, discrete linear systems, bounded uncertainties, zonotopes.
1 Introduction
There exist two classes of robust fault diagnosis methods:
stochastic and set-based [1, 2, 3]. The former describes
uncertainties by stochastic distributions while the later
describes uncertainties by their bounds. Moreover, both
of them include passive and active methods. This paper
focuses on set-based active fault diagnosis (AFD). Par-
ticularly, set-based passive fault diagnosis (PFD) can be
done by testing consistency between outputs and their
output sets [1, 4, 5]. Differently, set-based AFD further
designs inputs to excite the system to obtain more fault
information for diagnosis [6, 7, 8, 9, 10, 11]. In general,
the former is more conservative than the latter. In the lit-
erature, set-based AFD further includes two classes: the
classical set separation-based methods and the newly-
proposed set separation tendency-based methods.
On the first class, if an N-step input sequence can be
designed to separate the output sets of all healthy and
faulty modes, when the input sequence is injected into
the system, fault diagnosis can be guaranteed. The early
⋆This paper was not presented at any IFAC meeting. Cor-
responding author: Feng Xu (xu.feng@sz.tsinghua.edu.cn).
works on set-based AFD follow this logic to design AFD
inputs [12, 13, 14]. Within the author’s knowledge scope,
the earliest work on set-based AFD was proposed in [7],
where AFD inputs were designed by separating a healthy
output polytope and a faulty one. In [15], AFD inputs
were designed to separate more than two healthy and
faulty polytopes. In [16], an AFD input design method
was proposed based on the separation of a group of poly-
topes for linear parameter-varying (LPV) systems. Par-
ticularly, [16] considered the convex hull of the comple-
ment of the nonconvex set of separating inputs and then
used the convex hull to simplify AFD input design based
on its geometric features. When polytopes are used, de-
signing inputs to separate a group of polytopes has high
computational complexity. Moreover, for systems with
dimensions larger than 10, the AFD input design prob-
lem becomes computationally unstable [8]. To overcome
this issue, [8] proposed to use zonotopes instead of poly-
topes. However, the computional complexity issue was
still there. The reason is that [8] transformed the sep-
aration constraints of zonotopes into a group of mixed
integer constraints and solved a mixed integer quadratic
programming (MIQP) problem with exponentially in-
creasing complexity. Besides, the method in [8] was ex-
tended to closed-loop AFD in [9]. In [17], an AFD input
Preprint submitted to Automatica 28 January 2024

design method was propoed based on the separation of
a group of zonotopes for LPV systems. In [18], invariant
sets instead of time-varying sets were used for AFD. In
[19], AFD inputs were designed online, where the input
sequence length was set to 1. Particularly, a one-step in-
put was first designed based on one-step output set sepa-
ration but without considering the hard input constraint
set. Then the hard input constraint set was used to trun-
cate the input to obtain an AFD input. As a summary, it
should be mentioned that the computational complex-
ity issue heavily limits set separation-based AFD and its
application to faut-tolerant control (FTC).
The second class proposes a new notion named set sep-
aration tendency to replace set separation conditions to
implement AFD online [20, 21], which avoids set separa-
tion constraints and MIQP problems. Particularly, [20]
first proposed a qualitative notion of set separation ten-
dency and an observer-based AFD framework. In [22], a
maximin optimization problem was formulated to design
optimal gains for the observer-based AFD framework in
[20]. In [21], a quantitative definition of set separation
tendency was proposed, i.e., the dispersity of zonotopes
defined in [21]. For example, we use Yi
k+1 and Yj
k+1 to de-
note two output zonotopes at time instant k+ 1. Then,
the dispersity of Yi
k+1 and Yj
k+1 is defined in [21] as
D(Yi
k+1, Y j
k+1) = ∥yi,c
k+1 −yj,c
k+1∥2
2
∥Hi,y
k+1∥2
F+∥Hj,y
k+1∥2
F
,(1)
where yi,c
k+1 and yj,c
k+1 are the centers of Yi
k+1 and Yj
k+1,
Hi,y
k+1 and Hj,y
k+1 are the generator matrices of Yi
k+1 and
Yj
k+1, and ∥Hi,y
k+1∥Fand ∥Hj,y
k+1∥Fare the Frobenius ra-
diuses of Yi
k+1 and Yj
k+1, respectively. In [21], ∥Hi,y
k+1∥2
F
is used to measure the size of Yi
k+1.D(Yi
k+1, Y j
k+1) de-
notes the ratio of their centers distance and their sizes
sum, which can characterize the separation tendency of
Yi
k+1 and Yj
k+1 to some extent. For more faults, a disper-
sity for nf+ 1 (nf>1) output zonotopes is defined as
D(Y0
k+1,· · · , Y nf
k+1) = Pnf−1
i=0 Pnf
j=i+1 ∥yi,c
k+1 −yj,c
k+1∥2
2
Pnf
i=0 ∥Hi,y
k+1∥2
F
.
(2)
Finally, AFD inputs were designed in [21] by maximiz-
ing a weighted dispersity (2). As a summary, although
set separation tendency-based AFD has lower complex-
ity than set separation-based AFD, their AFD inputs are
designed based on maximizing the set separation ten-
dency. Thus, they can only design inputs with the max-
imal energy, which is harmful to the system [20, 21].
When AFD is used for FTC, the low conservatism advan-
tage of AFD over PFD can improve the whole FTC per-
formance. Within the author’s knowledge scope, there
only exist few works on integrated design of set-based
AFD and control. In [23], an FTC scheme was proposed
by combining model predictive control (MPC) and the
set separation-based AFD method in [8]. [24] proposed
an FTC scheme integrating set-based AFD with MPC,
where a constant input set was designed by trial and
error offline to guarantee separation of all healthy and
faulty output sets for AFD. In [25], an FTC scheme
based on interpolation optimization and AFD was pro-
posed to deal with component or actuator faults. Be-
sides, some other FTC schemes based on MPC but with-
out using AFD can be found in [26, 27, 28] for LPV sys-
tems and [29] for quasi-LPV (qLPV) systems. In [30], an
integrated design method of set-based AFD and control
was proposed, where the dipersity in (2) was weighted
with an output-tracking control objective to balance the
AFD and control performance. In [31], an auxiliary in-
put signal was designed to integrate conservative one-
step output set separation-based AFD and control. In
[32], preliminary results were proposed to design an in-
put set based on the dipersity (2) for integrated AFD
and control, where the performance was restricted by the
shortcomings of the dipersity. As a summary, although
the works above provide several ideas to integrate AFD
into FTC, they still have the disadvantages of high com-
plexity [23] or performance conservatism [24, 30, 31, 32].
Based on the analysis above, this paper proposes a new
and more effective online AFD input (set) design frame-
work to overcome the shortcomings of the existing meth-
ods based on some preliminary results in [32, 33]. The
main contributions of this paper are summarized:
•Compared with [20, 21], a new excluding degree notion
is proposed to characterize the set-based fault diagno-
sis performance. Since the excluding degree is defined
from the set-based fault diagnosis criterion and has an
analytical expression, it owns more explicit physical
meaning than the qualitative description in [20] and
the dispersity (1) in [21, 30], respectively.
•Compared with [30, 32], a new method is proposed to
design input sets for AFD by increasing the exclud-
ing degree and then an integrated design of AFD and
control is further proposed by selecting optimal in-
puts from AFD input sets. The AFD input sets can
also be used for integration of AFD with other objec-
tives (e.g., smallest energy). Moreover, the proposed
method has reasonable complexity and can achieve op-
timal control performance under the AFD objective.
The remainder of this paper has five sections. Section
2 includes some preliminary knowledge. Section 3 in-
troduces the system model and classical set separation-
based AFD. The main results are presented in Section
4. Section 5 illustrates the effectiveness of the proposed
methods. The paper is finally concluded in Section 6.
2

2 Preliminaries
The Minkowski sum of two sets Xand Yis X⊕Y=
{x+y|x∈X, y ∈Y}. A zonotope is defined as
Z=g⊕HBr⊂Rn(abbreviated as Z=⟨g, H ⟩),
where g∈Rnand H∈Rn×rare its center and gen-
erator matrix, respectively, and Bris an r-dimensional
unitary box. Given Z1=⟨g1, H1⟩and Z2=⟨g2, H2⟩,
Z1⊕Z2=⟨g1+g2,[H1H2]⟩. Given Z=⟨g, H⟩and an
appropriate matrix K,KZ =⟨K g, KH⟩. The Frobenius
radius of a zonotope Zwith generator matrix H∈Rn×r
is the Frobenius norm of H, i.e., ∥Z∥F=∥H∥F=
pPr
i=1 ∥hi∥2, where ∥hi∥=p(hi)Thiand hiis the i-th
column of H[34]. Given a zonotope Z=g⊕HBr⊂Rn
and an integer t(with n≤t < r), denote by ˚
H=
[˚
H>˚
H<] the matrix resulting from the reordering of
the columns of the matrix Hin decreasing Euclidean
norm. Z⊆ ↓t(Z) =↓t(˚
H) = g⊕[˚
H>2(˚
H<)]Bt
where the symbol ↓t(·) denotes the reduction of the
number of generators of a zonotope to t,˚
H>is ob-
tained from the first t−ncolumns of the matrix ˚
Hand
2(˚
H<)∈Rn×nis a diagonal matrix whose elements sat-
isfy 2(˚
H<)ii =Pr
j=t−n+1 |˚
Hij |, i = 1, . . . , n. Note
that the subscript ii (or ij) of 2(˚
H<)ii (or ˚
Hij ) denotes
the i-th row and i-th (or j-th) column element of the
matrix 2(˚
H<) (or ˚
H) ([34]).
3 Problem Formulation
3.1 System Model
This paper considers the discrete linear time-invariant
(LTI) system under multiplicative actuator faults [35]:
xk+1 =Axk+BGuk+Eωk,(3a)
yk=Cxk+F ηk,(3b)
where A∈Rnx×nx,B∈Rnx×nu,C∈Rny×nx,E∈
Rnx×nωand F∈Rny×nηare parametric matrices, k
denotes the k-th discrete time instant, G∈Rnu×nuis a
diagonal matrix whose i-th diagonal element gimodels
a fault in the i-th actuator (gi= 1 means that the i-
th actuator is healthy while 0 ≤gi<1 means that the
i-th actuator is faulty), xk∈Rnxand yk∈Rnyare
the state and output vectors, respectively, uk∈Rnuis
the input vector, ωk∈Rnωis the disturbance vector,
and ηk∈Rnηis the measurement noise vector. Different
values of Gmodel different actuator modes. We consider
nf+ 1 modes (i.e., G=Gi(i∈If={0,1,2, . . . , nf}),
where G0is an identity matrix modeling the healthy
mode and Gi(i= 0) model the nffaulty modes.
Assumption 3.1 The matrix Ais a Schur matrix.
Assumption 3.2 The input, disturbance and noise
vectors uk,ωkand ηkare bounded by zonotopes
U=⟨uc, Hu⟩,W=⟨ωc, Hω⟩and V=⟨ηc, Hη⟩, respec-
tively, where uc,ωc,ηc,Hu,Hωand Hηare the corre-
sponding centers and generator matrices, respectively.
Note that Assumptions 3.1 and 3.2 are two classical as-
sumptions in set-based AFD and state estimation.
3.2 Set-Based Active Fault Diagnosis
Under Assumptions 3.1 and 3.2, corresponding to the
i-th mode, a set-based version of (3) is obtained as
Xi
k+1 =AXi
k⊕BGiuk⊕EW, (4a)
Yi
k+1 =CX i
k+1 ⊕F V, (4b)
where Xi
k+1 and Yi
k+1 are the state and output sets,
respectively. By using the properties of zonotopes, (4) is
transformed into its center and generator matrix form:
xi,c
k+1 =Axi,c
k+BGiuk+Eωc,(5a)
Hi,x
k+1 =[AHi,x
kEHω],(5b)
yi,c
k+1 =Cxi,c
k+1 +F ηc,(5c)
Hi,y
k+1 =[CH i,x
k+1 F Hη],(5d)
where xi,c
k,yi,c
k+1,Hi,x
kand Hi,y
k+1 are the centers and
generator matrices of Xi
kand Yi
k+1, respectively. In (5),
ukaffects the centers of output sets such that all output
sets are separatable under the AFD framework.
For convenience of expression, it is assumed that all con-
sidered faults are detectable and isolable in one step 1.
According to [7, 8], considered actuator faults are guar-
anteed to be detectable and isolable at time instant k+1
if the following set separation conditions are satisfied:
Yi
k+1 ∩Yj
k+1 =∅,∀i=j, i, j ∈If,(6)
where Yi
k+1 and Yj
k+1 are the output sets of the i-th and
j-th actuator modes, respectively. Under the conditions
(6), only the output set matching the current mode can
always include the output. Thus, fault diagnosis is done
by testing whether or not the following inclusions
yk+1 ∈Yj
k+1,∀j∈If(7)
hold online to locate a unique output set satisfying (7),
which can finally indicate the current mode. This means
1Here we make the assumption that faults are detectable
and isolable in one step just for brevity of introducing the
classical set-based AFD principle. In general, the AFD meth-
ods do not require to design one-step input to separate all
output sets. Instead, an N-step input sequence is usually
designed for set-based AFD (see [8] for details).
3

that if an input uk∈Uis designed such that all output
sets Yj
k+1 (j∈If) are separate, it is guaranteed that all
considered faults are detectable and isolable by AFD.
4 Main Results
4.1 Performance Metric of Active Fault Diagnosis
Based on (4), when the system is in the i-th mode, a
residual zonotope corresponding to the j-th set-based
dynamics (i.e., the j-th mode) is defined as
Rij
k+1 =Yj
k+1 ⊕(−yk+1),∀j∈If.(8)
The fault diagnosis criterion (7) is thus transformed into
0∈Rij
k+1,∀j∈If.(9)
Similarly, the center and generator matrix of Rij
k+1 are
rij,c
k+1 =yj,c
k+1 −yk+1,(10a)
Hij,r
k+1 =Hj,y
k+1,(10b)
where rij,c
k+1 and Hij,r
k+1 are the center and generator ma-
trix of Rij
k+1, respectively.
At each time instant, a group of residual zonotopes are
obtained by a group of set-based dynamics (4) and (8)
for all modes. In order to implement diagnosis, we expect
that all residual zonotopes not matching the real system
mode can be excluded as soon as possible by testing (9)
online. This implies that if the origin is excluded from
the residual zonotopes quickly, fault diagnosis can be
done quickly. Motivated by (9) and the analysis above,
a new notion named the excluding degree of the origin
from a zonotope is defined in Definition 4.1.
Definition 4.1 ([33]). The excluding degree of the ori-
gin from a zonotope Z=⟨g, H ⟩is defined as
E(Z) = ∥g∥2
2
∥H∥2
F
,(11)
where ∥g∥2
2measures the distance from the center of Z
to the origin and ∥H∥2
Fmeasures the size of Z.
Based on Definition 4.1, the excluding degree of the ori-
gin from Rij
k+1 can be obtained as
E(Rij
k+1) = ∥rij ,c
k+1∥2
2
∥Hij,r
k+1∥2
F
.(12)
Remark 4.1 Notice that the excluding degree E(Rij
k+1)
in (12) is different from the dipersity D(Yi
k+1, Y j
k+1)
in (1). On one hand, E(Rij
k+1)measures the degree of
the residual zonotope Rij
k+1 to exclude the origin while
D(Yi
k+1, Y j
k+1)characterizes the separation tendency of
two output zonotopes Yi
k+1 and Yj
k+1. On the other hand,
E(Rij
k+1)can employ the system output information
while D(Yi
k+1, Y j
k+1)cannot. Besides, since the excluding
degree E(Rij
k+1)is directly originated from the criterion
(9), it has a more explicit physical meaning than the
dipersity. At time instant k+ 1, the larger E(Rij
k+1)is,
the better the fault diagnosis performance is.
Under Remark 4.1, it is known that if we design inputs
at time instant kto increase the excluding degree of the
origin from residual zonotopes at time instant k+1, fault
diagnosis can be speeded up. However, it is observed in
(10) that rij,c
k+1 is related to yk+1, where yk+1 is unavail-
able at time instant k. In order to deal with this prob-
lem, when the system is in the i-th mode, the idea is
to use an appropriate signal to estimate yk+1. Following
this idea, we define a nominal system for the i-th mode:
¯xi
k+1 =A¯xi
k+BGiuk+Eωc,(13a)
¯yi
k+1 =C¯xi
k+1 +F ηc,∀i∈If.(13b)
where ¯xi
k+1 and ¯yi
k+1 are the nominal state and output,
respectively. Before successful AFD, we do not know
which mode the system is currently in. This means
that all modes and their corrsponding nomimal systems
should be considered when constructing an appropriate
signal to estimate yk+1 for set-based AFD.
However, different modes have different probabilities to
be the real system mode. This means that we should con-
sider different weights for different modes. Thus, we fur-
ther define a weighting coefficient to evaluate the match-
ing degree of the j-mode to the real mode, which is also
based on the excluding degree E(Rij
k). Since E(Rij
k) can
be computed at time instant k, the larger E(Rij
k) is, the
higher the possibility that the j-th mode does not match
the real mode is. Thus, we should give a smaller weight
to the j-th mode when constructing a signal to estimate
yk+1. Following this idea, a weighting coefficient 2for
the nominal model of the j-mode is defined as follows:
λij
k=
1
E(Rij
k)
Pnf
j=0 1
E(Rij
k)
,(14)
where we have Pnf
j=0 λij
k= 1 and 0 < λil
k≤1.
2For a more reliable performance, the N-step historical data
of the excluding degree before the time instant k+ 1 can be
considered in (14) to compute the weighting coefficients.
4

Using (13) and (14), an appropriate signal to estimate
yk+1 is constructed as
˚
¯yi
k+1 =
nf
X
j=0
λij
k¯yj
k+1.(15)
Theorem 4.1 The estimation error between yk+1 and
˚
¯yi
k+1 given by (13),(14) and (15) is bounded.
Proof 1 First, based on (15), an output estimation error
between yk+1 and ˚
¯yi
k+1 is defined as follows:
˚
¯ei,y
k+1 =yk+1 −˚
¯yi
k+1 =
nf
X
j=0
λij
k(yk+1 −¯yj
k+1).(16)
We assume that the real mode of the system is l∈Ifand
then (16) can be transformed into
˚
¯ei,y
k+1 =λil
k¯el,y
k+1 −
nf
X
j=0,j=l
λij
k(yk+1 −¯yj
k+1),(17)
where ¯el,y
k+1 =yk+1 −¯yl
k+1. By further defining ¯el,x
k+1 =
xk+1 −¯xl
k+1, we derive the dynamics of ¯el,x
k+1 and ¯el,y
k+1
by using (3) and (13) for j=las follows:
¯el,x
k+1 =A¯el,x
k+E(ωk−ωc),(18a)
¯el,y
k+1 =C¯el,x
k+1 +F(ηk−ηc).(18b)
Under Assumption 3.2, we have (ωk−ωc)∈¯
W=⟨0, Hω⟩
and (ηk−ηc)∈¯
V=⟨0, Hη⟩, and a set version of (18)
is derived as follows:
¯
El,x
k+1 =A¯
El,x
k⊕E¯
W , (19a)
¯
El,y
k+1 =C¯
El,x
k+1 ⊕F¯
V , (19b)
where ¯
El,x
k+1 and ¯
El,y
k+1 are the sets of ¯el,x
k+1 and ¯el,y
k+1, re-
spectively. Therefore, as long as ¯el,x
0∈¯
El,x
0is given,
¯el,x
k∈¯
El,x
kand ¯el,y
k∈¯
El,y
khold for all k > 0. This implies
that the following inclusion always holds:
˚
¯ei,y
k+1 ∈λil
k¯
El,y
k+1 ⊕(−
nf
X
j=0,j=l
λij
k(yk+1 −¯yj
k+1)).(20)
Furthermore, under Assumption 3.1 and according to
[36], as ktends to ∞,¯
El,x
ktends to the minimal robust
positively invariant set ¯
El,x
∞of ¯el,x
kand moreover, ¯
El,x
∞is
unique for (18). Thus, ¯
El,y
kalso tends to its minimal set
¯
El,y
∞=C¯
El,x
∞⊕F¯
V. Based on (20), as long as the consid-
ered faults are diagnosable by the proposed AFD method,
the term −Pnf
j=0,j=lλij
k(yk+1 −¯yj
k+1)in (20) will tend
to 0as all the unmatched modes are gradually excluded
by the proposed AFD method. This means that ˚
¯ei,y
k+1 con-
verges to the set λil
∞¯
El,y
∞. Furthermore, since 0< λil
∞≤1,
˚
¯ei,y
k+1 finally converges to a fixed set ¯
El,y
∞, which means that
˚
¯yi
k+1 converges to a set around yk+1, i.e., yk+1 ⊕¯
El,y
∞.
Moreover, it should be mentioned that for any l∈If,
¯el,x
k+1 and ¯el,y
k+1 are subject to the same dynamics (18) and
bounded by the same set-based dynamics (19). Therefore,
¯
El,x
k+1 and ¯
El,y
k+1, and ¯
El,x
∞and ¯
El,y
∞are the same for all the
considered modes. This means that the results obtained
above are general for all the considered modes.
Besides, if some considered faults are not diagnosable
(i.e., there remains more than one mode that cannot be
excluded finally by the proposed AFD method), a similar
result can be obtained. Here we assume that the indices
of all the remaining modes that cannot be distinguished
from the l-th mode form a new index set I′
fand have
˚
¯ei,y
k+1 ∈λil
k¯
El,y
k+1 ⊕(−X
j∈I′
f
λij
k(yk+1 −¯yj
k+1)),(21)
where the term −Pj∈I′
fλij
k(yk+1 −¯yj
k+1)does not tend
to 0in this case. Under Assumption 3.1,˚
¯ei,y
k+1 is bounded
by El,y
∞⊕(−Pj∈I′
fλij
k(yk+1 −¯yj
k+1)) at steady state and
˚
¯yi
k+1 is also bounded by a set around yk+1 at steady state,
i.e., yk+1 ⊕ El,y
∞⊕(−Pj∈I′
fλij
k(yk+1 −¯yj
k+1)).□
Remark 4.2 Instead of (13), we can also design a bank
of observers based on the nominal models of different
modes to make an estimation for yk+1 and the whole
procedure is similar. Moreover, when observers are used,
extra gains can be obtained to optimize the performance.
Before AFD, ˚
¯yi
k+1 is used to replace yk+1 and thus one
obtains a nominal residual zonotope:
¯
Rij
k+1 =Yj
k+1 ⊕(−˚
¯yk+1),(22)
where ¯
Rij
k+1 is represented as its zonotopic form:
¯
Rij
k+1 =⟨¯rij,c
k+1,¯
Hij,r
k+1⟩,(23)
where
¯rij,c
k+1 =CA(xj,c
k−
nf
X
l=0
λil
k¯xl
k) + CB (Gj−
nf
X
l=0
λil
kGl)uk,
(24a)
¯
Hij,r
k+1 =[CAH j,x
kCE HωF Hη].(24b)
5

Similarly, the excluding degree of the origin from the
nominal residual zonotope ¯
Rij
k+1 is computed as
E(¯
Rij
k+1) = ∥¯rij,c
k+1∥2
2
∥¯
Hij,r
k+1∥2
F
=uT
k¯
Pij
1,kuk+¯
Pij
2,kuk+¯
Pij
3,k
¯
Pij
4,k
,
(25)
¯
Pij
1,k =(Gj−
nf
X
l=0
λil
kGl)TBTCTCB (Gj−
nf
X
l=0
λil
kGl),
(26a)
¯
Pij
2,k =2(xj,c
k−
nf
X
l=0
λil
k¯xl
k)TATCTCB (Gj−
nf
X
l=0
λil
kGl),
(26b)
¯
Pij
3,k =(xj,c
k−
nf
X
l=0
λil
k¯xl
k)TATCTCA(xj,c
k−
nf
X
l=0
λil
k¯xl
k),
(26c)
¯
Pij
4,k =trCAH j,x
k(Hj,x
k)TATCT+CE HωHT
ωETCT
+F HηHT
ηFT.(26d)
Remark 4.3 In (25) and (26), inputs only affect the
nomerator of E(¯
Rij
k+1)but do not affect its denomina-
tor, which means that the center plays a primary role in
the excluding degree. The reason is that in the set-based
dynamics (4) and (5), inputs only change the centers of
output sets but do not change their generator matrices.
This paper uses E(¯
Rij
k+1) to replace E(Rij
k+1) to describe
the excluding degree of the origin from the j-th residual
zonotope Rij
k+1. This replacement is a reasonable way to
handle the unknown yk+1 due to the following reasons:
•The output-tracking control mechanism in Section 4.3
designs inputs to minimize the differences between the
remaining candidate modes and the i-th mode;
•As the AFD mechanism works, the remaining modes
not matching the real mode are removed step by
step and the number of candidate modes gradually
decreases. The less the number of candidate modes
is, the stronger the impact of the replacement (15)
and output-tracking control mechanisms is, and the
smaller the error between E(¯
Rij
k+1) and E(Rij
k+1) is.
Similar to (14), the larger E(Rij
k) is, the higher the pos-
sibility that the j-th mode does not match the real mode
is. This implies that when designing inputs for AFD, it
is more necessary to exclude the j-th mode at time in-
stant k+1 and we should give a larger weight to the j-th
mode to speed up its exclusion. Thus, for AFD, a new
weighting coefficient for the j-mode is defined as
γij
k=E(Rij
k)
Pnf
j=0 E(Rij
k).(27)
With (25) and (27), a total excluding degree of the origin
for all candidate modes 3is defined as
¯
Ei
k+1 =
nf
X
j=0
γij
kE(¯
Rij
k+1),(28)
which describes the whole fault diagnosis performance
at time instant k+ 1. The proposed AFD logic here is
to design inputs at time instant kto increase ¯
Ei
k+1 such
that AFD is finally achieved at a certain time instant.
Remark 4.4 Although we do not know whether the sys-
tem has become faulty or which mode the system is in
before successful AFD, as the AFD mechanism works,
the remaining modes not matching the real system mode
will be excluded gradually. This means that the number
of sum terms in (15),(28) and (40) decreases gradually,
and ˚
¯yi
k+1,¯
Ei
k+1 and ˜yi
k+1 should be updated step by step
to remove the terms corresponding to unmatch modes.
Similarly, the index set Ifof all candidate modes should
also be updated to remove unmatched modes step by step
during AFD. Thus, corresponding to If,Pnf
j=0 does not
always mean nfsum terms, whose meaning should also
be updated online. However, for brevity, we do not em-
phasize the updation of Ifand Pnf
j=0 in the following. But
the readers should realize that an updating mechanism of
Ifand Pnf
j=0 is default included in the proposed methods.
4.2 Input Set for Active Fault Diagnosis
Substituting (25) and (26) into (28), the expression of
¯
Ei
k+1 with respect to ukis derived as
¯
Ei
k+1 =uT
kQi
1,kuk+Qi
2,kuk+Qi
3,k,(29)
where
Qi
1,k =
nf
X
j=0
γij
k¯
Pij
1,k
¯
Pij
4,k
, Qi
2,k =
nf
X
j=0
γij
k¯
Pij
2,k
¯
Pij
4,k
, Qi
3,k =
nf
X
j=0
γij
k¯
Pij
3,k
¯
Pij
4,k
.
At time instant k,¯
Ei
kis known. In order to achieve AFD,
we need to design an input at time instant kto increase
the excluding degree of all candidate modes, i.e.,
¯
Ei
k+1 ≥¯
Ei
k+δi
k,(30)
3When the system operates in the i-th mode before AFD, it
is also reasonable to remove the excluding degree component
of the i-th mode from the total excluding degree.
6

where δi
kis a given positive scalar. As said in Remark 4.4,
all modes not matching the real system mode should be
first removed at time instant k and then we compute ¯
Ei
k
and formulate ¯
Ei
k+1 of the remaining modes for (30).
Besides, another key problem is the range of δi
k, which
should be known when a value of δi
kis given for (30). As
said in (30), δi
k>0 should be given from the perspective
of AFD. However, δi
k≤0 also has an explict meaning
for passive FTC. Therefore, when analyzing the range
of δi
kin the following, we first consider a more general
case and omit the limitation δi
k>0. Following this idea,
we could formulate two optimization problems to obtain
the lower and upper bounds of δi
k. First, based on (29),
the lower bound of ¯
Ei
k+1 can be obtained by solving the
following optimization problem:
min
uk∈U
¯
Ei
k+1.(31)
and we use the notation ¯
Ei
k+1 to denote the optimal value
of (31) (i.e., the lower bound of ¯
Ei
k+1). The upper bound
of ¯
Ei
k+1 can be obtained by solving the problem:
max
uk∈U
¯
Ei
k+1 (32)
and the notation ¯
Ei
k+1 is used to denote the optimal
value of (32) (i.e., the upper bound of ¯
Ei
k+1).
For brevity, Uis considered as a zonotope in this paper.
According to (26), Qi
1,k is a positive semi-definite ma-
trix. Without loss of generality, we consider that Qi
1,k
is positive definite here. Thus, (31) is a convex problem
that can be solved easily to obtain ¯
Ei
k+1, while (32) can
be solved by enumerating the vertices of Uand then
evaluating the function values of ¯
Ei
k+1 on the vertices of
Uto obtain ¯
Ei
k+1. Thus, one has
¯
Ei
k+1 ≤¯
Ei
k+1 ≤¯
Ei
k+1,(33)
which means that the given value of δi
ksatisfies
δi
k≤δi
k≤δi
k,(34)
with δi
k=¯
Ei
k+1 −¯
Ei
kand δi
k=¯
Ei
k+1 −¯
Ei
k.
Remark 4.5 Both δi
kand δi
khave explicit meanings.
δi
kmeans the maximal energy to force all output sets
to get close to each other at time instant k+ 1 . The
input ukdesigned for δi
kreflects the maximal passive FTC
potential at time instant k.δi
kmeans the maximal energy
to force all output sets at time instant k+ 1 to leave
from each other. The input ukdesigned for δi
kreflects
the maximal AFD potential at time instant k. Besides,
when δi
k≤0, it means that it is impossible to design
an input ukto further increase the excluding degree at
time instant k+ 1. If at time instant k,δi
k≤0holds
and there still exist some candidate modes that cannot
be distinguished, we should terminate the AFD process
to give up to distinguish them at this time. Instead, once
δi
k≤0is found at a time instant, we can turn to the idea
of passive FTC to further design inputs to minimize the
excluding degree to tolerate their effect. When δi
k≥0, a
similar analysis can be done. Besides, the dispersity (2)
in [21, 30] cannot provide a specific condition like δi
k≤0
to terminate or switch the AFD process during operation.
Based on (30), an AFD input set is further defined below
and it should be emphysized that δi
k>0 needs to be
given to construct an AFD input set in Definition 4.2.
Definition 4.2 An AFD input set of ukis defined as
Uf
k={uk∈Rnu:uT
kQi
1,kuk+Qi
2,kuk+Qi
3,k
−¯
Ei
k−δi
k≥0}.(35)
Based on the analysis, at time instant k, any input uk∈
Uf
kcan be used to increase the excluding degree such that
AFD is speeded up. Theoretically, as long as we select
an input from its AFD input set at each time instant,
AFD will be achieved at a certian time instant when all
candidate modes not matching the real system mode are
excluded by testing (7) or (9) online. Since we consider
that Qi
1,k is positive definite here, it means that (35)
represents the complementary set of an ellipsoid ¯
Uf
k=
{uk∈Rnu:uT
kQi
1,kuk+Qi
2,kuk+Qi
3,k −¯
Ei
k−δi
k<0}
(we omit the boundary here for brevity). Moreover, δi
kis
a key parameter that determines the AFD input set Uf
k.
Actually, δi
khas an explict meaning in the adjustment
of the AFD performance. In particular, the larger δi
k
is, the faster the AFD process is, and consequently the
smaller Uf
kis. When a control objective is considered
together with AFD, the smaller Uf
kis, the lower the
control performance is. This means that δi
kplays a key
role in balancing the AFD and control performances.
Based on the analysis above, two ways are proposed in
the following to design AFD inputs, which depend on
the requirements on inputs. If our objective is to achieve
the fastest AFD, the corresponding AFD input at time
instant kcan be designed by solving the optimization
problem (32) to achieve the maximal excluding degree at
time instant k+ 1. This is what the existing asymptotic
AFD methods have done such as [20, 21, 22]. However,
the AFD inputs for the fastest AFD always have large
energy, which are harmful to the system. Therefore, in
7

a more general case, on one hand, we want to achieve
AFD. On the other hand, we hope that the AFD inputs
have the minimal energy such that the harm of AFD to
the system can be reduced to the smallest. In this case,
the AFD inputs can be designed by solving
min
uk∈Uf
k,uk∈U
uT
kQuk,(36)
where Qis a given positive definite matrix.
Remark 4.6 Based on the AFD input set above, any
input inside Uf
kcan facilitate the AFD objective. The
optimization problem (36) means to select the AFD input
with the smallest energy out of the AFD input set and
input-constraint set, which can simultaneously achieve
both the AFD and minimal input energy objectives.
4.3 Integrated Design of AFD and Control
As analyzed above, any input inside Uf
kcan be used for
AFD at time instant k. If selecting an input out of Uf
kto
achieve the best control performance, integrated AFD
and control can be achieved. Without loss of generality,
we consider a simple output-tracking problem to intro-
duce the idea in the following. Particularly, when the
system is in the i-th mode before AFD, a reference sys-
tem for the i-th mode is designed as
xi,ref
k+1 =Axi,ref
k+BGiui,ref
k+Eωc,(37a)
yi,ref
k+1 =Cxi,r ef
k+1 +F ηc,∀i∈If,(37b)
where ui,ref
kis the given reference input, and xi,ref
kand
yi,ref
k+1 are the reference state and output, respectively.
In this case, the control objective is to track the reference
output yi,ref
k+1 and thus a tracking error of the j-th mode
to the i-th reference output is defined as
˜yij
k+1 =yj,c
k+1 −yi,ref
k+1 ,∀j∈If.(38)
At a time instant, if a fault occurs, the system mode
changes from the mode ito another mode j(j=i). How-
ever, before successful AFD, we do not know whether a
mode change induced by faults has occurred or which
mode the system newly enters into. Thus, during AFD
(including both active fault detection and active fault
isolation), all modes should be considered in the output
tracking problem. In order to measure the tracking per-
formance during AFD, a tracking error of all modes to
the i-th reference output is defined as
˜yi
k+1 =
nf
X
j=0
˜yij
k+1.(39)
However, we do not use (38) and (39) for the track-
ing control problem here. Instead, we further propose
a weighted tracking control performance specification
here. Similar to AFD, since we do not know the real
mode before AFD, when considering the tracking con-
trol problem, it is also necessary to give a weighting co-
efficient to each mode to evaluate its importance. Simi-
lar to (14), the larger E(Rij
k) is, the higher the possibil-
ity that the j-th mode does not match the real mode is.
However, when designing inputs for control, we should
give a smaller weight to the j-th mode to less suppress
its tracking error such that it is easier to be excluded at
time instant k+ 1. Based on this idea, a weighted track-
ing error is defined based on (14) as
˜yi
w,k+1 =
nf
X
j=0
λij
k˜yij
k+1.(40)
During AFD, an optimal input u∗
kfor integrated design
of AFD and control is obtained to suppress the effect
of faults on the control performance and improve the
system safety by solving the optimization problem:
min ˜yi
w,k+1
s.t. uk∈Uf
k, uk∈U, (41)
where u∗
kis the optimal solution of (41).
Since Uf
kis not convex, (41) is not a convex problem.
Thus, solving (41) depends on handing the constraint
uk∈Uf
k(similar to (36)). In [32] and [37], two dif-
ferent methods were proposed to handle the constraint
uk∈Uf
kand solve the optimization problems with the
constraint uk∈Uf
klike (41). In particular, the first
method uses a nonlinear programming solver fmincon in
MATLAB, which has low computational complexity and
obtains a locally optimal solution. However, the main
drawbacks are that the method is not an analytical one
and that the obtained locally optimal solution depends
on a given initial point, which result in that the qual-
ity of the locally optimal solution cannot be evaluated
accurately. The second method is a branch and bound
method, which obtains a globally optimal solution with
an arbitrarily given precision. However, the main draw-
back is its high computational complexity. In this paper,
we aim to achieve a trade-off performance between the
locally optimal method in [32] and the globally optimal
method in [37] and thus propose a new method to solve
(41) such that both the quality of the obtained solution
and the computational complexity are satisfactory.
First, one makes a detailed analysis on the two con-
straints uk∈Uf
kand uk∈U, where the former is a time-
varying set and the complement of an ellipsoid while the
8

latter is a fixed zonotope. Moreover, they together are
equivalent to the following constraint:
uk∈Uf
k∩U, (42)
which is the intersection of the complementary set of an
ellipsoid with a zonotope. In general, this intersection
has four possible relations in Figure 1, where the first
case is an empty set (i.e., ¯
Uf
kcontains U), the second
case is a convex set (i.e., ¯
Uf
kand Uhave no intersection),
the third case is a concave set (i.e., Ucontains ¯
Uf
k),
and the fourth case is also a concave set ( ¯
Uf
kand U
have an intersection but do not contain mutually). In
particular, in the first case, (41) has no solution. In the
second case, (41) reduces to a simple convex problem:
min ˜yi
w,k+1 s.t. uk∈U. Thus, we need to focus on the
third and fourth cases (i.e., the original form (41)).
In these two cases, the idea is to use a zonotope Zkto
overapproximate the ellipsoid i.e., Zk⊇¯
Uf
k. Since an
ellipsoid can be overapproximated by a zonotope with
an arbitrarily given precision, as long as computational
complexity is acceptable, it is reasonable to use Zkwith
a satisfactory overapproximate precision to replace ¯
Uf
k
to handle (41). An effective method to construct an over-
approximate zonotope of an ellipsoid with an arbitrarily
given precision can be found in [38]. Based on this idea,
uk∈Uf
kis approximated by
uk∈∁RnuZk.(43)
With (43), the problem (41) is further approximated by
min ˜yi
w,k+1,
s.t. uk∈ Uk=∁RnuZk∩U, (44)
where Ukis a concave set. In order to solve the problem
(44), we further use the faces of Zkto partition Ukinto
a group of convex sets Ui,k (i∈IU={1,2,· · · , nU}):
Uk=U1,k ∪ U2,k ∪ · ·· ∪ UnU,k ,(45)
where nUis the number of the obtained convex set. The
partitioning of Ukby a red dashed line is illustrated in
Figure 2. Based on (45), (44) is transformed into a group
of convex problems, where each convex problem has a
convex constraint uk∈ Ui,k (i∈IU). Finally, the opti-
mal input u∗
kis designed by solving nUconvex problems
and then comparing their optimal values to select the
smallest one and the corresponding optimal input.
Remark 4.7 (45) represents a polyhedral partition of Uk
(see Definition 4.6 of [39] for details on polyhedral par-
tition). Moreover, an algorithm to implement polyhedral
partition (45) is also given in Theorem 4.2 of [39].
(a) ¯
Uf
k⊃U
(b) ¯
Uf
k∩U=∅
(c) ¯
Uf
k⊂U
(d) ¯
Uf
k⊃ U,¯
Uf
k∩U=∅and ¯
Uf
k⊂ U
Fig. 1. Relations between Uf
kand U
After injecting u∗
kinto the system, yk+1 and a group
of output sets Yj
k+1 are obtained and then we employ
them to accelerate AFD by testing (7) or (9) at each
time instant. If some output sets violate (7), then they
are removed at this time instant. After testing (7) or (9)
and removing unmatched modes step by step, AFD will
9

Fig. 2. Partitioning of Ukinto a group of Ui,k
be achieved at a certain time instant. The procedure of
integrated AFD and control is summarized as follows:
•For the i-th mode, design an AFD input set Uf
kat
time instant kbased on (29), (30) and (35);
•Using (43), u∗
kfor integrated AFD and control is de-
signed at time instant kwith (44) and (45);
•By injecting u∗
kinto (3), (4) and (13), the output yk+1
and output sets Yj
k+1 are obtained to test (7) or (9) to
exclude some modes not matching the output;
•Repeat the above steps at the following time instants.
Remark 4.8 Computational complexity of the proposed
method is mainly originated from non-convexity of Uf
k.
In order to handle Uf
k, an overapproximate zonotope
Zk⊇¯
Uf
kis constructed and the feasible region Ukof (44)
is partitioned into a group of polytopes in (45), which
leads to a group of solvable convex optimization problems.
Since the overapproximating precision of Zkto ¯
Uf
kcan be
arbitrarily given, the number of vertices and faces of Zk
can be adjusted by setting an appropriate precision such
that the scale of the final subsets in (45) is acceptable.
Additionally, the number of candidate residual zonotopes
also decreases gradually by testing (7) online, which can
further reduce computational complexity.
Remark 4.9 This paper only considers multiplicative
actuator faults for brevity. However, the idea and princi-
ple of the proposed methods are extendable to system and
sensor faults. Due to the limitation of length, instead of
detailed extensions, a brief introduction on the extensions
is given for multiplicative system and sensor faults as ex-
amples. Since multiplicative system and sensor faults af-
fect the system matrix Aand output matrix C, we could
model them as A+ ∆Aand C+ ∆C, respectively, where
∆Aand ∆Cdenote the faults, respectively. Based on the
modeling, the system model (3) and set-based dynamics
(4) can be modified accordingly to consider the effect of
faults. Consequently, following the definition of the ex-
cluding degree in Section 4.1, the remaining procedure
and derivations can be modified in similar ways to handle
multiplicative system and sensor faults.
5 Illustrative Examples
A numerical example with three inputs, three states and
three outputs is used to illustrate the proposed method
and the parameters of this example are given as fol-
lows: A=h0.6 0.1 0.2
0.1 0.4 0.1
0.3 0.1 0.5i,B=h0.05 0.08 0.04
0.06 0.07 0.05
0.03 0.04 0.06i,C=h1 0 0
0 1 0
0 0 1i,
E=h0.05 0.03 0.02
0.04 0.05 0.03
0.02 0.01 0.05i, and F=h0.1 0 0
0 0.1 0
0 0 0.1i. The uncer-
tainties are bounded by W=⟨h0
0
0i,h0.5 0 0
0 0.5 0
0 0 0.5i⟩and
V=⟨h0
0
0i,h0.1 0 0
0 0.1 0
0 0 0.1i⟩, respectively. The input con-
straint set is given as U=⟨h0
0
0i,h2 0 0
0 2 0
0 0 2i⟩.Besides, in all
simulations, once the number of generators of zonotopes
becomes larger than 12, it will be reduced to 12.
First, we analyze the single AFD performance of the
proposed method, where AFD inputs are designed by
(32). The healthy and four faulty modes G0=h1 0 0
0 1 0
0 0 1i,
G1=h0.2 0 0
0 0.9 0
0 0 0.8i,G2=h0.8 0 0
0 0.2 0
0 0 0.9i,G3=h0.9 0 0
0 0.8 0
0 0 0.2i
and G4=h0.3 0 0
0 0.3 0
0 0 0.3iare considered. It is set that the
system operates in the mode G1. In this simulation, the
initial state is given as x0= [0 0 0]T. The initial state
sets and initial nominal states for all the actuator modes
are given as ˆ
Xi
0=⟨x0,diag([0.01 0.01 0.01])⟩and ¯xi
0=
x0, i ∈If={0,1,2,3,4}, respectively. During AFD, a
specific input can be designed for AFD by solving (32)
and the upper bound of δi
kcan be computed by (34)
at each time instant. If the upper bound satisfies ¯
δi
k>
0, it implies that the designed AFD input from Ucan
increase the excluding degree at time instant k+ 1. In
this simulation, the AFD inputs are designed as u0=
[2 2 2]Tand u1= [−2 2 −2]T. Moreover, the
entire AFD process is illustrated in Figure 3, where yk,1,
yk,2and yk,3are the first, second and third components
of yk, respectively. In particular, at k= 1, y1∈ Y0
1,y1∈
Y3
1and y1∈ Y4
1are tested, which mean that the faulty
modes G0,G3and G4are excluded. At k= 2, y2∈ Y2
2
is tested, while only y2∈Y1
2holds. This means that the
faulty mode G2can be excluded, and the faulty mode G1
is diagnosed at k= 2. The results verify the effectiveness
of the proposed AFD method. In this case, the average
computing time for one-step zonotopic computation is
0.000371s. Note that all simulations in this paper are
done using Matlab 2019a on a desktop with Intel(R)
Core(TM) i7-6700 CPU@3.40GHz and 16GB RAM. The
optimization problem is solved by the CVX toolbox.
Second, we consider another situation where some faulty
modes are not isolable. In this situation, the upper bound
of δi
kmay not satisfy ¯
δi
k>0, which means that it is im-
possible to design an input ukfrom Uto further increase
the excluding degree at next time instant. Instead, the
proposed method turns to passive FTC to design inputs
to decrease the excluding degree as much as possible
to tolerate the unisolable modes. In order to simulate
10

(a) k= 1
(b) k= 2
Fig. 3. Output sets of five actuator modes for single AFD
this situation, we only reset the fourth faulty mode as
G4=h0.9 0 0
0 0.9 0
0 0 0.9iand the rest of the parameters remain
the same with the fist simulation. Besides, we set the to-
tal simulation period to 15 steps and consider that the
system operates in the faulty mode G4. The inputs are
shown in Figure 4, the output and the centers of all out-
put sets are shown in Figure 5, and the AFD and pas-
sive FTC processes are shown in Figure 6. According
to these results above, the first, second and third faulty
modes are excluded at k= 1. However, the healthy and
the fourth faulty modes cannot be distinguished and the
excluding degree cannot be increased at k= 5. Then, as
stated in Remark 4.5, we use the passive FTC idea to
tolerate the two unisolable modes after k= 5, which can
be realized by minimizing the excluding degree. From
Figure 5, the centers of the two output sets gradually
move away from each other before k= 5, and gradually
move closer after k= 5. At k= 15, the two output sets
coincide with each other. These results verify that the
proposed method can passively tolerate faults that are
not isolated by the proposed AFD strategy. The super-
scripts and subscripts in Figures 4, 5 and 6 have similar
meanings with those in Figure 3.
Third, in order to further illustrate the AFD perfor-
mance of the proposed method, we make a comparison
of the proposed method here with the method in [21]. In
this simulation, we maximize the dispersity of all output
sets proposed in [21] to design inputs for AFD. With-
out loss of generality, we only consider the healthy mode
G0=h1 0 0
0 1 0
0 0 1iand a faulty mode as G1=h0.85 0 0
0 0.85 0
0 0 0.85i
Fig. 4. Inputs for AFD and passive FTC
Fig. 5. Outputs and output set centers for AFD and FTC
for brevity, and set the total simulation period as 20
steps. Besides, the other parameters are set as the same
with the first simulation, and the system operates in the
faulty mode. The results are shown in Figure 7. The pro-
posed method (i.e., (32)) can implement AFD at k= 5,
but the method in [21] cannot distinguish the two modes
within 20 steps. Actually, we have done lots of simula-
tions to compare the AFD performances of the two meth-
ods. A general result is that the proposed AFD method
has better performance in most cases. Besides, there also
exist some cases that the two methods have similar per-
formance and a few cases that the method in [21] has
better performance. As a summary, since the excluding
degree is defined directly based on the set-based fault di-
agnosis criterion which includes the information of both
the output and output sets, the proposed method has
higher AFD performance potential.
Fourth, the proposed method can achieve integrated
design of AFD and control by selecting an input
from Uf
k∩Uto achieve an optimal control perfor-
mance. In this simulation, the system outputs are re-
quired to track the reference outputs of the healthy
mode, and the reference system of the healthy mode
is given in (37). The initial reference state is given
as x0,ref
0= [0 0 0]T, and the reference input is given
11

(a) k= 1
(b) k= 5
(c) k= 15
Fig. 6. Entire process of AFD and passive FTC
as u0,ref
k=h0.8cos(2πk
60 ) 0.8sin(2πk
60 ) 0.8cos(2πk
60 )iT
.
Here, the different values of δi
kare obtained by changing
β(0 ≤β≤1) in δi
k= max{δi
k,0}+β(δi
k−max{δi
k,0}),
where the operation max{·} takes the larger one be-
tween two elements. The value of βis given as β= 0.5
in this case and other parameters are the same with the
first simulation. As said in Section 4.3, we use an enclos-
ing zonotope Zkto overapproximate the ellipsoid ¯
Uf
k.
The enclosing zonotope is constructed by the method
in [38] and the number of generators of the enclosing
zonotope is set to 10 (see Figure 8). The AFD inputs
and input sets are shown in Figure 8, and the AFD
process is shown in Figure 9. The output-tracking per-
formance and inputs are shown in Figures 10 and 11,
respectively. The subscript and superscript here have
similar meanings with the previous figures.
Fifth, the overapproximating precision of the enclos-
(a) AFD results of the proposed method
(b) AFD results of the method in [21]
Fig. 7. Comparison of the two AFD methods
(a) k= 0
(b) k= 1
Fig. 8. AFD input sets and corresponding inputs designed
for integrated AFD and control
12

(a) k= 1
(b) k= 2
Fig. 9. Output sets of modes for integrated AFD and control
Fig. 10. Output-tracking control of the proposed method
Fig. 11. Inputs for integrated AFD and control
Table 1
Influence of overapproximating precisions on computational
complexity and performance of the proposed method
The number of generators 6 10 14
Average computing time 0.78s 5.85s 26.82s
Optimal value of (41) 0.0143 0.0136 0.0098
ing zonotope can affect the computational complexity
and performance of the proposed method. In order
to better display this influence of different precisions,
we do simulations using the same parameters used in
the fourth simulation and the optimization problem
formulated at k= 0. In this simulation, we set three
different overapproximating precisions by setting the
number of generators of the enclosing zonotope to 6,
10 and 14. The designed AFD input sets and inputs
are shown in Figure 12. The corresponding values
of u0are h2 1.93 −1.1973iT
,h2 1.86 −1.1313iT
and
h−1.7599 1.8106 2iT
, respectively. The computational
complexity and performance of this method are illus-
trated by the average computing time and the optimal
value of (41). The results are shown in Table 1. We
also compare the inputs generated from the proposed
method and the original nonconvex problem (41) solved
by the toolbox fmincon in MATLAB. The solution of
(41), solved by fmincon, depends on the selection of
initial points. The inputs and optimal values obtained
with different initial points are shown in Table 2
Sixth, we make a comparison of the proposed method
here with the method in [30]. Since the value of the
weighting coefficient αproposed in [30] can affect the
balance between the AFD and control objectives, we
consider two extreme cases to compare the proposed
method and the method in [30]. First, we give the val-
ues of αand βas close to zero as possible to make
the two methods focus more on control performance.
Therefore, the value of βis given as β= 0 + ϵ, and
the value of αis given as α= 0 + ϵ, where ϵis a
very small positive constant and set as ϵ= 0.01 in
this simulation. In this simulation, we set the simula-
tion period as 40 steps and give the reference input as
u0,ref
k=h0.5cos(2πk
60 ) 0.5sin(2πk
60 ) 0.5cos(2πk
60 )iT
. It is
set that the system operates in the faulty mode G4.
Other parameters are the same as those in the first sim-
ulation. The AFD performances of the two methods are
shown in Figure 13, and the tracking control perfor-
mances of the two methods are shown in Figure 14 by
the tracking errors between outputs and reference out-
puts, where ri
k(i= 1,2) = yk−y0,ref
kare the tracking er-
rors of the proposed method (r1
k) and the method in [30]
(r2
k), respectively. According to these results, the pro-
posed method can diagnose the faulty mode at k= 24,
but the method in [30] only excludes the healthy system
13

Table 2
Solutions of (41) by the toolbox fmincon in MATLAB with different initial points
Initial point [2 2 2]T[2 2 −2]T[−2 2 −2]T[−2 2 2]T[2 −2 2]T
Input [−1.48 1.92 1.55]T[1.47 1.20 −1.26]T[−1.31 2 −2]T[−1.52 1.95 1.56]T[2 −1.41 2]T
Optimal value 0.0085 0.0117 0.0439 0.0087 0.0095
(a) The number of generators is 6
(b) The number of generators is 10
(c) The number of generators is 14
Fig. 12. AFD input sets and corresponding inputs with dif-
ferent overapproximating precisions at k= 0
mode and cannot distinguish the remaining four faulty
modes within 40 steps. The reason is that the AFD in-
put sets proposed in this method guarantee that the in-
puts chosen from Uf
k∩Ucan make the excluding degree
at k+ 1 bigger than the excluding degree at k. There-
fore, although the proposed method pays more attention
to tracking control, the AFD task can still be realized.
When the objective in [30] is designed to focus more
on control, the effectiveness of AFD will be weakened.
Moreover, due to the lack of a mechanism to consistently
facilitate AFD, the AFD task cannot be realized in this
simulation. For the control performance, since the in-
puts designed by the proposed method are chosen from
Uf
k∩U, which is a smaller feasible domain for inputs
that can be used for control. Therefore, during AFD, the
control performance of the proposed method may have
lower potential than the method in [30]. However, since
the method in [30] just provides a way to obtain feasible
solutions to the optimization problem, the method does
not obtain the globally optimal inputs used for tracking
control. Additionally, when the two methods focus more
on control performance, the proposed method can real-
ize AFD while the method [30] cannot. Once the faulty
mode has been diagnosed, the control performance of the
proposed method has higher potential than the method
in [30], which is also illustrated in Figure 14.
The second case is that the two methods focus more on
the AFD performance. We give the values of αand βas
close to 1 as possible. When both the two methods focus
more on the AFD tasks, the proposed excluding degree
and the dispersity in [30] are required to be maximized.
This situation is similar to those results that have been
illlustrated in the third simulation. Therefore, the com-
parative analysis of the proposed integrated AFD and
control method and the method in [30] on the AFD per-
formance can refer to the results of the third simulation.
Besides, the computational complexity of the two meth-
ods is also presented in the seventh simulation for
complete comparison. In this simulation, the number of
generators of the enclosing zonotope used in the pro-
posed method is set to 10. Since the method in [30] only
searches a feasible point of the optimization problem,
and proposes to search a group of feasible points and
select the best one from them as the final input. In this
simulation, we search 50 feasible points and then select
the best one out of them as the final input. During the
AFD stage, the average computing time of the proposed
method and the method in [30] is 4.2876s and 10.2435s,
respectively. During the tracking-control stage, the av-
erage computing time of the proposed method and the
method in [30] is 0.4025s and 0.4241s, respectively.
6 Conclusions
This paper has proposed a new input design framework
for set-based asymptotic AFD based on a newly-defined
excluding degree of the origin from a zonotope. The core
idea of this framework is to design an AFD input set
14

(a) AFD results of the proposed method
(b) AFD results of the method in [30]
Fig. 13. AFD performances of the two methods
Fig. 14. Output-tracking performances of the two methods
based on the excluding degree such that any input out of
the AFD input set can be used for AFD. With this AFD
input set, the proposed method can be further extended
to achieve simultaneous AFD and other objectives such
as output-tracking control, minimal input energy and so
on. In order to show the value of this new framework, a
detailed analysis has also been done in this paper and
a plenty of simulations are made to show the effective-
ness of the proposed framework and compare its perfor-
mances with other existing methods in the literature.
In the future, the proposed method will be extended to
more complex systems and more types of faults.
Acknowledgements
Yushuai Wang is acknowledged for the examples and
simulations. This work is supported by the National
Natural Science Foundation of China (62003186), the
Guangdong Provincial Natural Science Foundation
(2021A1515012628) and the Shenzhen Science and Tech-
nology Innovation Committee (JCYJ20210324132606015).
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Feng Xu received his Bachelor’s degree
in Measurement and Control Technol-
ogy and Instrumentation from North-
western Polytechnical University, Xi’an,
P.R.China, in July 2010. In December
2014, he obtained his Ph.D. degree with
honor in Autom´atica, Rob´otica y Visi´on from Univer-
sitat Polit`ecnica de Catalunya, Barcelona, Spain. In
2014, he was a jointly-trained Ph.D. student in Centrale-
Sup´elec, Paris, France. From March 2015 to February
2018, he was a Postdoctoral Fellow in Control Science
and Engineering with Tsinghua University, P.R.China.
Currently, he is an Assistant Professor with Tsinghua
Shenzhen International Graduate School, Tsinghua Uni-
versity. His research interests include robust state esti-
mation, fault diagnosis and fault-tolerant control.
17