# Optimal Robust Fault Detection for Linear Discrete Time Systems

Authors: Nike Liu; Kemin Zhou
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper considers robust fault-detection problems for linear discrete time systems. It is shown that the optimal robust detection filters for several well-recognized robust fault-detection problems, such asℋ−/ℋ∞, ℋ2/ℋ∞, and ℋ∞/ℋ∞ problems, are the same and can be obtained by solving a standard algebraic Riccati equatio
n. Optimal filters are also derived for many other optimization criteria and it is shown that some well-studied and seeming-sensible optimization criteria for fault-detection filter design could lead to (optimal) but useless fault-detection filters.


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## Body

## 1. Introduction

It is well recognized that many practical dynamical
systems are subject to various environmental changes,
unknown disturbances, and
changing operating conditions, thus sensors/actuators/components failure and
faults in those systems are inevitable.
Since any faults/failures in a
dynamical system may lead to significant performance
degradation, serious
system damages, and even loss of human life,
it is essential to be able to
detect and identify faults and failures in a timely
manner so that necessary
protective measures can be taken in advance.
To that end, fault diagnosis of
dynamic systems has received much attention
and significant progress has been
made in recent years in searching for both
data-driven and model-based
diagnosis techniques: see [1–5] and the references therein.Much attention has been devoted to the development of
robust fault-detection methods under external
disturbances for continuous time
systems. Since most (continuous) dynamical
systems are nowadays controlled
using digital devices, it is also important
to understand those theoretical
development in the digital (sampled-data) setting.
Furthermore, it has been
shown in [6] that
sample-data fault-detection problem
can be converted to equivalent
discrete time-detection problem using
certain discretization method and thus
discrete time fault-detection is of
great importance and most nature for modern
digital implementation.One of the particular interesting techniques among all
the model-based techniques is observer-based
fault-detection filter design [1].
It has been shown in many theoretical studies and
applications that suitably
designed observer-based fault-detection
filters are easy to implement in
discrete systems and can be very effective
in detecting sensors, actuators, and
system components faults. There are significant
amount of works addressing this
problem using Kalman filter related techniques [7–9]. Nevertheless, finding systemic design methods for
systems subject to unknown disturbances and model
uncertainties have been
proven to be difficult. Since known/unknown disturbances,
noise, and model
uncertainties are unavoidable for any practical systems, it is essential in the
design of any fault-detection filter to take these
effects into consideration
so that fault detection can be done reliably and
robustly. To that effect, many
robust filter design techniques, such as ℋ∞
optimization,
LMI, parity space, and eigen-structure assignment
techniques, have been applied
to fault detection filter design with
limited success [10–15]. The reason is
that a fault detection filter design
is really a multiple objective design task.
It needs not only rejecting
disturbance, noise and being insensitive to model
uncertainties, it also needs
to be as sensitive as possible to possible faults
so that early detection of
faults is possible. Unfortunately, these two design
objectives are almost
always conflicting with each other. Hence a
design tradeoff between these two
objectives is unavoidable and needs to be
addressed explicitly in the design
process. To do that, some suitable
design criteria for both objectives have to
be defined. It has been widely accepted in
the field that ℋ2 norm and ℋ∞ of the transfer
matrix from disturbances to fault detection
residuals are good candidates for
measuring up the disturbance rejection
capability of a fault detection system.
In some cases, ℋ2 norm of the
transfer function matrix from faults to
fault detection residual signals is
also suitable for evaluating the fault detection system's sensitivity to
faults. It has also been recognized that the ℋ−
index, first
introduced by Hou and Patton
[16] and further
extended by Liu et al. [17],
seems to be a very appropriate measure
of the fault-detection sensitivity [1–3]. Although this
concept was originally proposed for
continuous time system, it is quite straightforward
to extend this concept to
discrete time systems. With such defined performance
objectives, several
discrete time fault detection design problems
can be formulated as multiple
objective optimization problems by minimizing
the effects of disturbances and
maximizing the fault sensitivity, for example, ℋ−/ℋ∞ problem, ℋ∞/ℋ∞ problem, ℋ2/ℋ∞ problem, ℋ−/ℋ2 problem, and ℋ2/ℋ2 problem. These
problems have attracted a great deal of attention recently, [6, 18–25]. However,
most of the results obtained in the
existing literature are either conservative or
complicate to apply.
Furthermore, they are usually not guaranteed to
be optimal. A notable exception
is the work by Ding et al. in [26],
where optimal solutions to some formulations of
continuous ℋ−/ℋ∞ and H∞/ℋ∞ problems are given.
Zhang et al. in [27] also give an optimal solution to ℋ∞/ℋ∞ problem for
linear discrete time periodic systems.We have developed a new technique to solve the above
problems for continuous time systems in
[28]. In this paper,
we will carry out
the parallel development for discrete
time problems. Although there are
considerable similarities between the
continuous and the discrete time
solutions, there are also significant differences
in some cases where we can
give more explicit solutions in discrete time
cases that cannot be done in
continuous time cases. In addition, explicit
discrete time solutions have their
own merits in applications. It turns out that
our solutions are surprising
simple once the problems are suitably formulated.The rest of this paper is organized as follows:
Section2 introduces the notations and summarizes some key facts that will be
used in the later sections. Section 3 gives the mathematical formulations of
various fault-detection problems to be solved in this paper.
The analytic and
optimal solutions for ℋ−/ℋ∞ problem and ℋ∞/ℋ∞ problem are
given in Section 4.
The solution for ℋ2/ℋ∞ problem is
given in Section 5.
The solutions for various ℋ−/ℋ2 problems are
discussed in Sections 6–8.
Some numerical examples of our fault detection
designs are shown in Section 9. Some conclusions are
given in Section 10.


## 2. Notations and Preliminary Results

The notations
used in this paper are quite standard. The set ofm by n real (complex)
matrices is denoted as ℛm×n (𝒞m×n). For a matrix A∈𝒞m×n we use AT to denote its
transpose and A′ for its
complex conjugate transpose. For a Hermitian matrix A=A′∈𝒞m×n, λ¯(A) represents the
largest eigenvalue of A and λ̲(A) represents the
smallest eigenvalue value of A. For A∈𝒞m×n, σ¯(A)=λ¯(AA′)=λ¯(A′A) denotes the
largest singular value of A and σ̲(A)=λ̲(AA′) (λ̲(A′A)) denotes the smallest singular value of A if m≤n (m≥n). The n×n identity matrix
is denoted as In and the m×n zero matrix is
denoted as 0m,n, with the subscripts dropped if they can be inferred
from context.Discrete transfer matrices and𝒵-transforms of
signals are represented using bold characters and sometimes in dependence of
the variable z. A state-space realization of a transfer matrix G(z) is denoted as(1)G(z)=[ABCD]
such that G(z)=D+C(zI−A)−1B. We define G~(z):=GT(z−1) and denote G−1(z) as the inverse
of G(z) if G(z) is square and
invertible. Now suppose
(2)G(z)=[ABCC]
is square and D
is nonsingular,
then we
have from [29](3)G−1=[A−BD−1C−BD−1D−1CD−1].We useℛℒ2m×n to denote the set of m×n real rational
transfer matrices with no poles on the unit circle.
The superscripts for
dimensions will usually be dropped when
they are either not important or clear
from context. ℛℋ2 (=ℛℋ∞) is the set of all stable proper transfer matrices.ForG(z)∈ℛℋ2 we define the ℋ2 norm of G as
(4)∥G∥2=12π∫02πTrace[G~(ejθ)G(ejθ)]dθ.
For G∈ℛℋ∞ we define the ℋ∞ norm of G as(5)∥G∥∞=supθ∈[0,2π]σ¯[G(ejθ)].
Similar to the ℋ− definitions of
continuous system in [16, 17], we define the ℋ− index of a
discrete transfer matrix G on the whole
unit circle as(6)∥G∥−[0,2π]=infθ∈[0,2π]σ̲(G(ejθ)).
The ℋ− index of G over a finite
frequency range [θ1,θ2] is defined as
(7)∥G∥−[θ1,θ2]=infθ∈[θ1,θ2]σ̲(G(ejθ)).
In particular the ℋ− index defined
at θ=0 is(8)∥G∥−[0]=σ̲(G(1)).
If no superscript is added to the ℋ− symbol, such as ∥G∥−, then it represents all possible ℋ− definitions. In
many literatures ℋ− index is also
called ℋ− norm, although
it is actually not a norm.It is easy to show from the definition of singular
value of a matrix that we have the following result [30].Lemma 1.
LetA∈𝒞m×n and B∈𝒞n×p be two matrices
with appropriate dimensions, then σ̲(AB)≤σ¯(A)σ̲(B).The following transfer matrix factorizations will be
frequently used in this paper and can be found from [29].Lemma 2 (Left Coprime Factorization).
LetG(z) be a proper
real rational transfer matrix. A left coprime
factorization (LCF) of G is a
factorization
(9)G=M−1N,
where N and M are
left-coprime over ℛℋ∞. Let
(10)G=[ABCD]
be a detectable state-space realization of  G and let L be a matrix
with appropriate dimensions such that A+LC is stable, then
a left coprime factorization of G is given by
(11)[MN]=[A+LCLB+LDCID].Lemma 3 (Spectral Factorization).
LetG(z) be a proper
real rational transfer matrix and let
(12)G=[ABCD]
be a detectable realization of G. Suppose D has full row rank and [A−ejθIBCD] has full row
rank for all θ∈[0,2π]. Let P≥0 be the
stabilizing solution to the following algebraic Riccati equation:
(13)APA′−P−(APC′+BD′)(DD′+CPC′)−1×(DB′+CPA′)+BB′=0
such that A−(APC′+BD′)(DD′+CPC′)−1C is stable and
let R:=DD′+CPC′. Then the following spectral factorization holds
(14)WW~=GG~,
where W−1∈ℛℋ∞ and
(15)W=[A(APC′+BD′)R−1/2CR1/2].


## 3. Problem Formulation

Consider a discrete time invariant system with
disturbance and possible faults as:(16)x(k+1)=Ax(k)+Bu(k)+Bdd(k)+Bff(k),y(k)=Cx(k)+Du(k)+Ddd(k)+Dff(k),
where x(k)∈ℛn is the state
vector, y(k)∈ℛny is the output
measurement, d(k)∈ℛnd represents the
unknown/uncertain disturbance and measurement noise, and f(k)∈ℛnf denotes the process,
sensor or actuator fault vector. f(k) and d(k) can be modeled
as different types of signals, depending on specific situations under
consideration. See Chapters 4 and 8 of [29] and
[1] for some detailed discussions. Two frequently used
assumptions on d(k) and f(k) are:(i)
unknown signal
with bounded energy or bounded power;(ii)
white noise.
Different assumptions on d(k) and f(k) will lead to
different fault detection problem formulations and the solutions for all these
problems will be discussed in this paper.All coefficient matrices in (16)
are assumed to be known constant matrices.
Furthermore, the following assumptions are made.Assumption 1.
(A,C) is detectable.This is a standard assumption for all fault-detection
problems.Assumption 2.
Dd has full row
rank.This means thatny≤nd and every
measurement of the output signals is either affected by some disturbance or
corrupted with some measurement noise. We argue that this assumption can be
made without loss of any generality since it is impossible to take perfect
measurement in any practical system and furthermore it is reasonable to assume
that the measurement noise is independent of each other. So it is reasonable to
assume that Dd has full row
rank. In the case of some simplified model where Dd does not have
full row rank, we can simply add some columns to make it full row rank. For
example, suppose that Dd does not have
full row rank, then let
(17)d˜=[ddϵ],B˜d=[Bd0n×ny],D˜d=[DdϵIny]
for a small ϵ>0. Then D˜d has full row
rank.Assumption 3.
[A−ejθIBdCDd] has full row
rank for all θ∈[0,2π] or, equivalently, the transfer function matrix
(18)Gd:=[ABdCDd]
has no transmission zero on the unit circle.This assumption can be relaxed in the same way as in
the continuous time case [28].Remark 1.
We want to point out that in several recent work
on continuous time fault detection problems [17, 19, 21, 22], it is assumed that Df has full column
rank. We believe that this assumption is extremely restrictive. The assumption implies that measurement y contains
directly the independent information on every faulty component of f. In particular, this implies that Df cannot be zero
which is usually not the case when there is only actuator/system component
fault and no sensor fault. Furthermore, we believe that the fault detection for
sensor fault is relatively easier than that for actuator/system fault.
By taking𝒵-transform of
(16) we have the system input/output equation
(19)y=Guu+Gdd+Gff,
where Gu, Gd, and Gf are ny×nu, ny×nd and ny×nf transfer
matrices, respectively and their state-space realizations are
(20)[GuGdGf]=[ABBdBfCDDdDf].
Since the state-space realization of Gu, Gd, and Gf share the same A and C matrices,
applying Lemma 2 we can find an
LCF for the system (20)
(21)[GuGdGf]=M−1[NuNdNf],
where
(22)[MNuNdNf]=[A+LCLB+LDBd+LDdBf+LDfCIDDdDf]
and L is a matrix
such that A+LC is stable.
It has been shown in [2] that, without loss of generality, the fault detection
filter can take the following general form:
(23)r=Q(My−Nuu)=Q[M−Nu][yu],
where r is the residual
vector for detection, Q∈ℛℋ∞ny×ny is a free stable transfer
matrix to be designed. The filter structure is shown in Figure 1. Replacing y in (23)
by the right-hand
side of (19) and (21) we have
(24)r=Q[NdNf][df]=QNdd+QNff.
Denote the transfer matrices from d and f to r by Grd and Grf, respectively, then
(25)Grd=QNd,Grf=QNf.
In general a good fault-detection filter must make a
tradeoff between two conflicting performance objectives: robustness to
disturbance rejection and sensitivity to faults. To achieve good robustness to
disturbance, the influence of disturbance must be minimized at the output of
the residual signals. On the other hand, the residual signal should be as
sensitive as possible to the faults. Therefore, we need to choose certain
performance criteria for measuring these two aspects so that the
fault-detection filter design has satisfactory fault detection sensitivity and guaranteed
disturbance rejection effect.
Since anℋ− index is a good
measurement for a transfer function's smallest gain, ∥Grf∥− is a reasonable
performance criterion for measuring fault detection sensitivity if f(k) is modeled as
unknown energy or power bounded signals. If d(k) is modeled as
unknown energy or power bounded signals, then ℋ∞ norm is a
widely accepted worst case measure and ∥Grd∥∞ is a good
indicator of disturbance rejection performance. On the other hand, if d(k) and/or f(k) are white
noise, the ℋ2 norms of Grd and/or Grf seem to be more
suitable criteria. See [29] for
more detailed discussions and motivations on
various performance measures.
We will now formulate several fault-detection filter
design problems.Figure 1
General fault-detection filter structure.Problem 1 (ℋ−/ℋ∞ Problem).
Let an uncertain system be described by
(16)–(20) and let γ>0 be a given
disturbance rejection level. Find a stable transfer matrix Q∈ℛℋ∞ny×ny in (23)–(25) such that ∥Grd∥∞≤γ and ∥Grf∥− is maximized,
that is,
(26)maxQ∈ℛℋ∞ny×ny{∥QNf∥−:∥QNd∥∞≤γ}.Problem 2 (ℋ2/ℋ∞Problem).
Let an uncertain system be described by
(16)–(20) and let γ>0 be a given
disturbance rejection level. Find a stable transfer matrix Q∈ℛℋ∞ny×ny in (23)–(25) such that ∥Grd∥∞≤γ and ∥Grf∥2 is maximized,
that is,
(27)maxQ∈ℛℋ∞ny×ny{∥QNf∥2:∥QNd∥∞≤γ}.Problem 3 (ℋ−/ℋ2 Problem).
Let an uncertain system be described by
(16)–(20) and let γ>0 be a given
disturbance rejection level. Find a stable transfer matrix Q∈ℛℋ∞ny×ny in (23)–(25) such that ∥Grd∥2≤γ and ∥Grf∥− is maximized, that is,
(28)maxQ∈ℛℋ∞ny×ny{∥QNf∥−:∥QNd∥2≤γ}.Remark 2.
A more conventional formulation of the above
problems is to optimize the following:(29)maxQ∈ℛℋ∞ny×ny∥QNf∥X∥QNd∥Y,
where X and Y can be 2, ∞, or −. The problem that X=Y=2 is classical
and optimal solution is available [2]. The case for X=−,∞ and Y=∞ has been solved
recently in [26] for continuous-time systems. A discrete solution has
also been obtained recently in [27] for the cases of X=∞ and Y=∞.
Before we proceed to the solutions of the above
problems, we will first establish some preliminary results.Lemma 4.
Suppose Assumption3 is satisfied and let Gd=M−1Nd be any left
coprime factorization over ℛℋ∞. Then Nd has no
transmission zero on the unit circle or, equivalently, for any appropriately
dimensioned matrix L,
(30)[A+LC−ejθIBd+LDdCDd]
has full row rank for all
θ∈[0,2π].Proof.
The result
follows by noting that(31)[A+LC−ejθIBd+LDdCDd]=[IL0I][A−ejθIBdCDd]
and the fact that all coprime factors have the same
unstable transmission zeros [29].An immediate consequence of the above result is the
following spectral factorization formula.Lemma 5.
Suppose Assumptions1–3 are satisfied and let Gd=M−1Nd be any left
coprime factorization over ℛℋ∞. Then there is a square transfer matrix V∈ℛℋ∞ny×ny such that V−1∈ℛℋ∞ny×ny and
(32)VV~=NdNd~.
In particular,
if a state-space representation of Nd is given as in
(22), then a state space representation of V is given
by
(33)V=[A+LC[(A+LC)PC′+(Bd+LDd)Dd′]Rd−1/2C0Rd1/2]
with
(34)V−1=[A−(APC′+BdDd′)Rd−1C−(APC′+BdDd′)Rd−1−LRd−1/2CRd−1/2],
where P≥0 is the
stabilizing solution to the Riccati equation
(35)APA′−P−(APC′+BdDd′)(DdDd′+CPC′)−1×(DdBd′+CPA′)+BdBd′=0
such that A−(APC′+BdDd′)(DdDd′+CPC′)−1C is stable and Rd:=DdDd′+CPC′.Proof.
Since
Assumptions1–3 are satisfied, Lemmas 3 and  4 can be applied to Nd to get VV~=NdNd~, where P≥0 satisfies the
following Riccati equation
(36)ALCPALC′−P−(ALCPC′+BLDDd′)Rd−1×(DdBLD′+CPALC′)+BLDBLD′=0,ALC:=A+LC,BLD:=B+LD.
It is easy to show that the above algebraic Riccati
equation can be simplified to (35). The rest of the proof follows from some simple
algebraic manipulations.The following lemma is the key to the solutions of all
the above problems.Lemma 6.
Suppose Assumptions1–3 are satisfied. Let V,V−1∈ℛℋ∞ be defined as
in (32). Let
(37)Q=ΨV−1
for Ψ∈ℛℋ∞ and denote N˜f=V−1Nf∈ℛℋ∞. Then the
fault-detection Problems 1–3
are equivalent to Problems 4–6
below, respectively.Problem 4.
(38)maxΨ∈ℛℋ∞ny×ny{∥ΨN˜f∥−:∥Ψ∥∞≤γ}.Problem 5.
(39)maxΨ∈ℛℋ∞ny×ny{∥ΨN˜f∥2:∥Ψ∥∞≤γ}.Problem 6.
(40)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−:∥Ψ∥2≤γ}.Proof.
We will first
show that Problems1 and 2 are equivalent to Problems 4 and
5,
respectively.
Note that by Lemma6 there exists V∈ℛℋ∞ such that VV~=NdNd~ and V−1∈ℛℋ∞. Therefore,
(41)∥QNd∥∞2=supθ∈[0,2π]σ¯(Q(ejθ)Nd(ejθ)Nd~(ejθ)Q~(ejθ))=supθ∈[0,2π]σ¯(Q(ejθ)V(ejθ)V~(ejθ)Q~(ejθ))=∥QV∥∞2,
that is, ∥QNd∥∞=∥QV∥∞. We can,
therefore without loss of generality, take Q in the form of Q=ΨV−1 for some Ψ∈ℛℋ∞. Hence ∥QNd∥∞=∥QV∥∞=∥Ψ∥∞,  so that ∥QNd∥∞≤γ is equivalent
to ∥Ψ∥∞≤γ. Moreover, QNf=ΨV−1Nf=ΨN˜f, hence Problem 1 is equivalent to Problem 4 and Problem 2 is equivalent
to Problem 5.
Next we show that Problem3 is equivalent to Problem
6. Note that in
Problem 3, we have QNd∈ℛℋ2. Hence,
(42)∥QNd∥22=12π∫02πTrace{Q(ejθ)Nd(ejθ)Nd~(ejθ)Q~(ejθ)}dθ=12π∫02πTrace{Q(ejθ)V(ejθ)V~(ejθ)Q~(ejθ)}dθ=∥QV∥22
such that ∥QNd∥2=∥QV∥2. Since QV∈ℛℋ2 and V,V−1∈ℛℋ∞, we can let Q=ΨV−1 for some Ψ∈ℛℋ2. Therefore, ∥QNd∥2=∥QV∥2=∥Ψ∥2 so that ∥QNd∥2≤γ is equivalent
to ∥Ψ∥2≤γ. Moreover, QNf=ΨV−1Nf=ΨN˜f, hence Problem 3 is equivalent to Problem 6.We will provide optimal solutions for each of the
above problems in the following sections.


## 4.ℋ−/ℋ∞ Fault-Detection Filter Design

In this section,
we give a complete solution for theℋ−/ℋ∞ fault-detection
filter design problem, that is, Problem 1 or
Problem 4.Theorem 1.
Suppose Assumptions1–3 are satisfied. Let
(43)[GuGdGf]=M−1[NuNdNf]
be any left coprime factorization over ℛℋ∞ and let V∈ℛℋ∞ be a square
transfer matrix such that V−1∈ℛℋ∞ and VV~=NdNd~. Then
(44)maxQ∈ℛℋ∞ny×ny{∥QNf∥−:∥QNd∥∞≤γ}=γ∥V−1Nf∥−
and an optimal fault-detection filter for Problem
1 is
given by
(45)r=Qopt[M−Nu][yu],
where
(46)Qopt=γV−1.Proof.
Note that by
Lemma6, we only need to solve Problem 4:
(47)maxΨ∈ℛℋ∞ny×ny{∥ΨN˜f∥−:∥Ψ∥∞≤γ}.
From Lemma 1 we know that for every frequency θ∈[0,2π],
(48)σ̲(Ψ(ejθ)N˜f(ejθ))≤σ¯(Ψ(ejθ))σ̲(N˜f(ejθ))
so that
(49)∥ΨN˜f∥−≤∥Ψ∥∞∥N˜f∥−≤γ∥N˜f∥−.
By letting Ψ=γI, we have ∥Ψ∥∞=γ and ∥ΨN˜f∥−=γ∥N˜f∥−, which means that Ψ=γI is an optimal
solution achieving the maximum.Remark 3.
The optimal fault-detection filter given in
Theorem1 does not depend on Bf and Df matrices.Remark 4.
Note that the solution given in the above theorem
does not depend on the specific definitions ofℋ− index. Hence,
the solution provided here is an optimal solution for all ℋ− indices.
However, it should be pointed out that this optimal filter is not the only
optimal solution for some ℋ− index
criterion. For example, let Q=γL(z)V−1 where L∈ℛℋ∞ is a low-pass
filter with a very small bandwidth so that ∥L(z)∥∞=1 and L(1)=I. Then this Q is also an
optimal solution for
(50)maxQ∈ℛℋ∞ny×ny{∥QNf∥−[0]:∥QNd∥∞≤γ},
even though this is obviously a bad fault-detection
filter because the low-pass filter L(z) would make the
filter much less sensitive to faults.
Note also that the solution given in the above theorem
is completely general and it does not depend on specific state space
representation of those coprime factorization and spectral factorization, which
may be necessary in some fault tolerant control applications [5, 31]. On the other hand, if those specific state-space
coprime and spectral factorizations in the previous sections are used, the
optimal filter can be written in a very simple form.Theorem 2.
Suppose Assumptions1–3 are satisfied. Let P≥0 be the
stabilizing solution to the Riccati equation
(51)APA′−P−(APC′+BdDd′)(DdDd′+CPC′)−1×(DdBd′+CPA′)+BdBd′=0
such that A−(APC′+BdDd′)(DdDd′+CPC′)−1C is stable and
let Rd=DdDd′+CPC′. Define
(52)L0=−(APC′+BdDd′)Rd−1.
Then
(53)maxQ∈ℛℋ∞ny×ny{∥QNf∥−:∥QNd∥∞≤γ}=γ∥V−1Nf∥−
and an optimal ℋ−/ℋ∞ fault-detection
filter has the following state space representation
(54)r=Qopt[M−Nu][yu],
where
(55)Qopt[M−Nu]=γ[A+L0C−L0B+L0D−Rd−1/2CtRd−1/2−Rd−1/2D],V−1Nf=[A+L0CBf+L0DfRd−1/2CRd−1/2Df].
In other words, the optimalℋ−/ℋ∞ fault-detection
filter is the following observer:
(56)x^(k+1)=(A+L0C)x^(k)−L0y(t)+(B+L0D)u(k)r(k)=γRd−1/2(y(k)−Cx^(k)−Du(k)).Proof.
Note that(57)[MNu]=[A+LCLB+LDCID],
where L is a matrix
with appropriate dimensions such that A+LC is stable. Note
from Theorem 1 and Lemma 5 that
(58)Qopt=γV−1=γ[A+L0CL0−LRd−1/2CRd−1/2].
Then
(59)Qopt[M−Nu]=γ[A+L0CL0−LRd−1/2CRd−1/2][A+LCL−(B+LD)CI−D]=γ[A+L0CL0C−LCL0−L−(L0−L)D0A+LCL−(B+LD)Rd−1/2CRd−1/2CRd−1/2−Rd−1/2D]=γ[A+L0C0L0−(B+L0D)0A+LCL−(B+LD)R−1/2C0Rd−1/2−Rd−1/2D]=γ[A+L0CL0−(B+L0D)Rd−1/2CRd−1/2−Rd−1/2D]=γ[A+L0C−L0B+L0D−Rd−1/2CRd−1/2−Rd−1/2D].
Similarly, we have(60)V−1Nf=[A+L0CL0−LRd−1/2CRd−1/2][A+LCBf+LDfCDf]=[A+L0C(L0−L)C(L0−L)Df0A+LCBf+LDfRd−1/2CRd−1/2CRd−1/2Df]=[A+L0C0Bf+L0Df0A+LCBf+LDfRd−1/2C0Rd−1/2Df]=[A+L0CBf+L0DfRd−1/2CRd−1/2Df].Remark 5.
Note that the optimal fault-detection filterQopt[M−Nu] is independent
of the choice of L matrix.Remark 6.
It is easy to see that our optimal filter given in
Theorems1 and
2
is also optimal for the so-called ℋ∞/ℋ∞ problem
(61)maxQ∈ℛℋ∞ny×ny{∥QNf∥∞:∥QNd∥∞≤γ}
and it turns out this filter is the same as the one
given by Zhang et al. in [27]
under the following equivalent optimization
criterion:
(62)maxQ∥QNf∥∞∥QNd∥∞.


## 5.ℋ2/ℋ∞ Fault-Detection Filter Design

In this section,
we give an optimal solution for theℋ2/ℋ∞ problem stated
in Section 3 as
Problem 2.
Similar to the solution for ℋ−/ℋ∞ problem given
in Theorems 1 and
2,
we have the following parallel results for the ℋ2/ℋ∞ problem.Theorem 3.
Suppose Assumptions1–3 are satisfied. Let
(63)[GuGdGf]=M−1[NuNdNf]
be any left coprime factorization over ℛℋ∞ and let V∈ℛℋ∞ be a square
transfer matrix such that V−1∈ℛℋ∞ and VV~=NdNd~. Then (64)maxQ∈ℛℋ∞ny×ny{∥QNf∥2:∥QNd∥∞≤γ}=γ∥V−1Nf∥2
and the optimal fault-detection filter for Problem 1
given in Theorems 1 and 2 is also the optimal filter for this
problem.Proof.
Note that by
Lemma6,
we only need to solve Problem 5:
(65)maxΨ∈ℛℋ∞ny×ny{∥ΨN˜f∥2:∥Ψ∥∞≤γ}.
Note that
(66)∥ΨN˜f∥2≤∥Ψ∥∞∥N˜f∥2≤γ∥N˜f∥2.
By letting Ψ=γI, we have ∥Ψ∥∞=γ and ∥ΨN˜f∥2=γ∥N˜f∥2, which means that Ψ=γI is an optimal
solution achieving the maximum.


## 6.ℋ−/ℋ2 Fault-Detection Filter Design: Case 1

From Lemma6
we know that the ℋ−/ℋ2 problem is
equivalent to Problem 6, that is,
(67)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−:∥Ψ∥2≤γ}.
Unlike the ℋ−/ℋ∞ problem studied
in Section 4,
we have different solutions for the ℋ−/ℋ2 problem if
different ℋ− definitions are
considered. In this section and the next two sections we will illustrate this
point and give solutions for all cases.Theorem 4.
Suppose Assumptions1–3 are satisfied. Then
(68)supQ∈ℛℋ∞ny×ny{∥QNf∥−[0]:∥QNd∥2≤γ}=∞.
Furthermore, for any given α>0, let 2>ϵ>0 and
(69)Qsub=γϵ(2−ϵ)z−1+ϵV−1.
Then
(70){∥QsubNf∥−[0]>α,∥QsubNd∥2≤γ}
is satisfied for a sufficiently small ϵ>0.Proof.
Again note that
the equivalent Problem6 in this case is
(71)supΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−[0]:∥Ψ∥2≤γ}.
Take Ψ(z)=(γϵ(2−ϵ))/(z−1+ϵ)I such that 2>ϵ>0. Then Ψ(1)=γ2−ϵ/ϵI and ∥Ψ∥2=γ. Let ϵ→0, then Ψ(1)→∞, so that
(72)∥ΨN˜f∥−[0]=σ̲(Ψ(1)N˜f(1))=γ2−ϵϵσ̲(N˜f(1))→∞.Remark 7.
We should point out that an optimal filter
designed using Theorem4
is not necessarily good for fault detection since
this optimal filter can be extremely narrowbanded near
0 frequency so
that any higher frequency component of fault may not be detected.


## 7.ℋ−/ℋ2 Fault-Detection Filter Design: Case 2

In this section, we will consider another special
case where theℋ− index is
defined for all frequencies but with Df full column
rank. As we have mentioned before, this is a very restrictive case. We are
interested in this case because an analytic solution is possible.Lemma 7.
SupposeDf has full column
rank. Then an optimal solution Ψopt to Problem 6(73)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−[0,2π]:∥Ψ∥2≤γ}
has the form Ψopt=UΨo and
(74)ΨoN˜f=[αInf0(ny−nf)×nf],
where α is a positive
scalar and U is an ny×ny all-pass stable
transfer matrix.Proof.
We will first
show(75)σ̲(Ψopt(ejθ)N˜f(ejθ))=Cfor everyθ∈[0,2π],
where C is a nonnegative scalar.
Suppose there exists aΨopt such that σ̲(Ψopt(ejθ)N˜f(ejθ))=C does not hold.
Let Θ1 denote the set
of all θ values such
that
σ̲(Ψopt(ejθ)N˜f(ejθ))=∥ΨoptN˜f∥−[0,2π] is achieved.
Let Θ2:=[0,2π]−Θ1 such that
(76)σ̲(Ψopt(ejθ)N˜f(ejθ))θ∈Θ1<σ̲(Ψopt(ejθ)N˜f(ejθ))θ∈Θ2.
Then there exists a weighting function W∈ℛℋ∞ny×ny such that ∥WΨopt∥2=∥Ψopt∥2 and
(77)σ̲(Ψopt(ejθ)Nf(ejθ))θ∈Θ1<σ̲(W(ejθ)Ψopt(ejθ)N˜f(ejθ))θ∈Θ1≤σ̲(W(ejθ)Ψopt(ejθ)Nf(ejθ))θ∈Θ2.
Therefore, ∥WΨoptN˜f∥−[0,2π]>∥ΨoptN˜f∥−[0,2π] and Ψopt is not an
optimal solution. Hence, it must be true that σ̲(Ψopt(ejθ)N˜f(ejθ))=C for every θ∈[0,2π].
Next we show that(78)σ¯(Ψopt(ejθ)N˜f(ejθ))=σ̲(Ψopt(ejθ)N˜f(ejθ))for everyθ∈[0,2π].
Suppose there exists aΨopt such that σ¯(Ψopt(ejθ1)N˜f(ejθ1))≠σ̲(Ψopt(ejθ1)N˜f(ejθ1)) for some θ1 , that is,
(79)σ¯(Ψopt(ejθ1)N˜f(ejθ1))>σ̲(Ψopt(ejθ1)N˜f(ejθ1))=∥ΨoptN˜f∥−[0,2π].
Then a Ψ1 can be selected
such that
(80)σ̲(Ψ1(ejθ)N˜f(ejθ))=σ̲(Ψopt(ejθ)N˜f(ejθ)),∀θ∈[0,2π],σ¯(Ψ1(ejθ1)N˜f(ejθ1))<σ¯(Ψopt(ejθ1)N˜f(ejθ1)),∥Ψ1∥2<∥Ψopt∥2.
Since σ̲(Ψopt(ejθ)N˜f(ejθ))=C for every θ∈[0,2π],∥Ψ1N˜f∥−[0,2π]=∥ΨoptN˜f∥−[0,2π]. Let Ψ2=∥Ψopt∥2/∥Ψ1∥2Ψ1, then ∥Ψ2∥2=∥Ψopt∥2 and
(81)∥Ψ2N˜f∥−[0,2π]=∥Ψopt∥2∥Ψ1∥2∥Ψ1N˜f∥−[0,2π]=∥Ψopt∥2∥Ψ1∥2∥ΨoptN˜f∥−[0,2π]>∥ΨoptN˜f∥−[0,2π]. Therefore,
Ψopt is not optimal
and by contradiction the assumption is false. So σ¯(Ψopt(ejθ)N˜f(ejθ))=σ̲(Ψopt(ejθ)N˜f(ejθ))   holds for every θ∈[0,2π].
Sinceσ¯(Ψopt(ejθ)N˜f(ejθ))=σ̲(Ψopt(ejθ)N˜f(ejθ))=C for every θ∈[0,2π], and that Df has full column
rank implies ny≥nf, ΨoptN˜f has the form
(82)ΨoptN˜f=U[αInf0(ny−nf)×nf],
where U is an all-pass
stable transfer matrix and α is a positive
scalar. Let Ψopt=UΨo, then
(83)ΨoN˜f=[αInf0(ny−nf)×nf].Lemma 8.
SupposeDf has full column
rank. Then Problem 6(84)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−[0,2π]:∥Ψ∥2≤γ}
is equivalent to the following problem.
Problem 7.
(85)minN˜f+∈ℛℋ2nf×ny{∥N˜f+∥2:N˜f+N˜f=I}.Proof.
From Lemma7
we know that the optimal solution to Problem 6
has
the form Ψopt=UΨo and
(86)ΨoN˜f=[αInf0(ny−nf)×nf].
Let Ψo=[Ψ1Ψ2], where Ψ1 is nf×ny and Ψ2 is (ny−nf)×ny. Then [Ψ1N˜fΨ2N˜f]=[αI0] so Ψ2=0 and ∥Ψo∥2=∥Ψ1∥2. Since Problem 6
needs to maximize ∥ΨN˜f∥− with the
constraint ∥Ψ∥2≤γ, it is equivalent to find a Ψ1 with the smallest ℋ2 norm such that Ψ1N˜f=I. Denote Ψ1=N˜f+, then Problem 6
is equivalent to Problem 7.In [32]
the solution to a dual problem of Problem 7 is
given. Similarly, we have the solution to
Problem 7 given by the following
lemma.Lemma 9.
Assume(87)H(z)=[ABCD]
is strictly
minimum phase and D has full column
rank. Let D+=D(D′D)−1,D⊥ is chosen such
that D⊥′D⊥=I−D(D′D)−1D′ and A0=A−B(D+)′C, then the optimal solution to problem
(88)minH+(z)∈ℛℋ2nf×ny{∥H+(z)∥2:H+(z)H(z)=I}
is given by
(89)H+(z)opt=[A+KCKRCR],
where Q≥0 is the solution
to the algebraic Ricatti equation
(90)Q=A0′QA0−A0′QC′D⊥′(I+D⊥CQC′D⊥′)−1D⊥CQA0+B(D′D)−1B′,K=−B(D+)′−A0QC′D⊥′(I+D⊥CQC′D⊥′)−1D⊥,R=(D+)′(I+CQC′D⊥′D⊥)−1.Proof.
The equationH+(z)H(z)=I is equivalent
to HT(z)(H+(z))T=I,
so Problem 7 is equivalent to finding an H(rinv)(z) with the smallest ℋ2 norm such that HT(z)H(rinv)(z)=I. Hence the
conclusion in [32] can be applied to HT(z) to get the
optimal H(rinv)(z)opt. H+(z)opt is then
obtained by taking transpose of H(rinv)(z)opt.Theorem 5.
Suppose Assumptions1–3 are satisfied. Let Gf has all zeros
inside the unit circle and Df has full column
rank. Let
(91)[GuGdGf]=M−1[NuNdNf]
be any left coprime factorization over ℛℋ∞ and let V∈ℛℋ∞ be a square
transfer matrix such that V−1∈ℛℋ∞ and VV~=NdNd~. Let (N˜f+)opt=(V−1Nf)opt+ be the optimal
solution to Problem 7. Then
(92)maxQ∈ℛℋ2ny×ny{∥QNf∥−[0,2π]:∥QNd∥∞≤γ}=γ∥(V−1Nf)opt+∥2
and an optimal fault detection filter is given by
(93)r=Qopt[M−Nu][yu],
where
(94)Qopt=γ∥(V−1Nf)opt+∥2[(V−1Nf)opt+V−10].Proof.
Note that by
Lemma6, we only need to solve
Problem 6(95)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−[0,2π]:∥Ψ∥2≤γ}.
Since Gf has all zeros
inside the unit circle and V−1∈ℛℋ∞, N˜f is strictly
minimum phase. From Lemmas 7–9 we know that the optimal solutionto Problem 6 is
given by
(96)Ψopt=Uγ∥(N˜f+)opt∥2[(N˜f+)opt0],
where (N˜f+)opt=(V−1Nf)opt+ is the optimal
solution to Problem 7 and U is a unitary
matrix. Take U=I, then an optimal
solution is given by
(97)Qopt=γ∥(N˜f+)opt∥2[(N˜f+)optV−10]=γ∥(V−1Nf)opt+∥2[(V−1Nf)opt+V−10].Again the solution given in the above theorem is
general and it does not depend on specific state-space representation of
those coprime factorization and spectral factorization. If specific state-space
coprime and spectral factorization in the previous section are used, the
optimal filter can be written in an explicit form.Theorem 6.
Suppose Assumptions1–3 are satisfied. Let Gf has all zeros
inside the unit circle and Df has full column
rank. Let P≥0 be the
stabilizing solution to the Riccati equation
(98)APA′−P−(APC′+BdDd′)(DdDd′+CPC′)−1×(DdBd′+CPA′)+BdBd′=0
such that A−(APC′+BdDd′)(DdDd′+CPC′)−1C is stable. Let Rd=DdDd′+CPC′ and define
(99)L0=−(APC′+BdDd′)Rd−1.
Let D+=Rd−1/2Df(Df′Rd−1Df)−1,D⊥ is chosen such
that D⊥′D⊥=I−Rd−1/2Df(Df′Rd−1Df)−1Df′Rd−1/2 and A0=A+L0C−(Bf+L0Df)(D+)′Rd−1/2C. Let Q≥0 is the solution
to the algebraic Ricatti equation
(100)Q=A0′QA0′−A0′QC′D⊥′(I+D⊥CQC′D⊥′)−1D⊥CQA0+(Bf+L0Df)(Df′Rd−1Df)−1(Bf+L0Df)′
and define
(101)K0=−(Bf+L0Df)(D+)′−A0QC′Rd−1/2D⊥′(I+D⊥Rd−1/2CQC′Rd−1/2D⊥′)−1D⊥,R0=(D+)′(I+Rd−1/2CQC′Rd−1/2D⊥′D⊥)−1.
Then
(102)maxQ∈ℛℋ2ny×ny{∥QNf∥−[0,2π]:∥QNd∥∞≤γ}=γ∥(V−1Nf)opt+∥2,
where
(103)(V−1Nf)opt+=[A+K0CK0−L0R0CR0]
and an optimal ℋ−/ℋ2 fault-detection
filter has the following state-space representation:
(104)r=Qopt[M−Nu][yu],
where
(105)Qopt=γ∥(V−1Nf)opt+∥2[(V−1Nf)opt+V−10],Qopt[M−Nu]=γ∥(V−1Nf)opt+∥2×[[A+L0C0−L0B+L0DKL5A+K0CKL7KL8−R0Rd−1/2CR0CR0Rd−1/2−R0Rd−1/2D]0],
where KL5=−(K0−L0)Rd−1/2C,KL7=(K0−L0)Rd−1/2, and KL8=−(K0−L0)Rd−1/2D.Proof.
Note that(106)[MNu]=[A+LCLB+LDCID],
where L is a matrix
with appropriate dimensions such that A+LC is stable. From
Theorem 1(107)V−1=[A+L0CL0−LRd−1/2CRd−1/2].
From Theorem 2(108)V−1Nf=[A+L0CBf+L0DfRd−1/2CRd−1/2],V−1[M−Nu]=[A+L0C−L0B+L0D−Rd−1/2CRd−1/2−Rd−1/2D].
From Lemma 9(109)(V−1Nf)opt+=[A+K0CK0−L0R0CR0].
Therefore,
(110)Qopt[M−Nu]=γ∥(V−1Nf)opt+∥2[(V−1Nf)opt+0]V−1[M−Nu]=γ∥(V−1Nf)opt+∥2[[A+K0CK0−L0R0CR0]0]×[A+L0C−L0B+L0D−Rd−1/2CRd−1/2−Rd−1/2D]=γ∥(V−1Nf)opt+∥2×[[A+L0C0−L0B+L0DMN5A+K0CMN7MN8−R0Rd−1/2CR0CR0Rd−1/2−R0Rd−1/2D]0].
where MN5=−(K0−L0)Rd−1/2C,MN7=(K0−L0)Rd−1/2, and MN8=−(K0−L0)Rd−1/2D.Remark 8.
Note that the optimal fault-detection filterQopt[M−Nu]
is independent
of the choice of
Lmatrix.Remark 9.
Note that the strictly minimum phase assumption
forN˜f is not needed.
In general, if N˜f does not have
any zeros on the unit circle, one can always factorize N˜f=N˜fminN˜fa
so that N˜fmin is strictly
minimum phase and N˜fa is a stable
all-pass matrix. Then the solution can be computed by using	N˜fmin
in place of	N˜f. In the case when
N˜f has zeros on
the unit circle, approximation factorization can also be carried out to obtain
an approximation solution.


## 8.ℋ−/ℋ2 Fault-Detection Filter Design: Case 3

When Problem3 is considered with the
ℋ− index defined
over a finite frequency range [θ1,θ2], the solution becomes much more complicated. We will
now state this as a separate problem as below.Problem 8 (Intervalℋ−/ℋ2 Problem).
Let an
uncertain system be described by (16)–(20) and let γ>0 be a given
disturbance rejection level. Find a stable transfer matrix Q∈ℛℋ∞ny×ny
in (23)–(25) such that	∥Grd∥2≤γ
and ∥Grf∥−[θ1,θ2] is maximized, that is,
(111)maxQ∈ℛℋ∞ny×ny{∥QNf∥−[θ1,θ2]:∥QNd∥2≤γ}
or, equivalently, let Q=ΨV−1 and solve
(112)maxΨ∈ℛℋ2ny×ny{∥ΨN˜f∥−[θ1,θ2]:∥Ψ∥2≤γ}.Remark 10.
It is not hard to see that there is no rational
function solution to the above problem. This is because an optimalΨ must satisfy Ψ(ejθ)=0 almost every
where for any θ∉[θ1,θ2]. Hence, an analytic optimal solution seems to be
impossible. Nevertheless, it is intuitively feasible to find some rational
approximations so that a rational Ψ has the form of
a bandpass filter with the pass-band close to [θ1,θ2] and ∥Ψ∥2=γ.Remark 11.
When the
condition thatDf has full column
rank is not satisfied, the rational optimal solution to the problem
(113)maxQ∈ℛℋ∞ny×ny{∥QNf∥−[0,2π]:∥QNd∥2≤γ}
may not exist. In this case, we also need to find some
rational approximate solutions. Moreover, this problem is a special case of
Problem 8 by letting θ1=0 and θ2=2π, we will only consider the solution to
Problem 8.In the following, we will describe an optimization
approach to find a good rational approximation for the two
cases above. To do
that, we will need a state-space parametrization
of a stable rational function
with a givenℋ2 norm  [33].Lemma 10.
Let(114)Ψ=[AψBψCψDψ]∈ℛℋ2ny×ny
be an
nψth order
proper stable transfer matrix. Then the state space parameters of Ψ can be
expressed as Aψ=(I+Aψk)(I−Aψk)−1(I−Cψ′Cψ)1/2 for some Aψk=−Aψk′ and some Cψ satisfies ∥Cψ∥≤1. Furthermore, ∥Ψ∥22=Trace(Dψ′Dψ+Bψ′Bψ).Proof.
Assume that(115)Ψ=[A^ψB^ψC^ψD^ψ]∈ℛℋ2
is an nψth order
observable realization, then the Observability Gramian Lo satisfies
(116)A^ψ′LoA^ψ−Lo+C^ψ′C^ψ=0.
Since Lo>0, there exists a Cholesky factorization of Lo=T′T where T is invertible.
Perform a similarity transformation on Ψ such that
(117)Ψ=[TA^ψT−1TB^ψC^ψT−1D^ψ]=[AψBψCψDψ]. Thus, Aψ′Aψ−I+Cψ′Cψ=0, so that Aψ=O(I−Cψ′Cψ)1/2 where O is an
orthogonal matrix and I−Cψ′Cψ is a nonnegative
definite. Since an orthogonal matrix O with no
eigenvalue equals −1 can be
represented as A=(I+Aψk)(I−Aψk)−1, where Aψk=−Aψk′ is a
skew-symmetric matrix, we have
(118)Aψ=(I+Aψk)(I−Aψk)−1(I−Cψ′Cψ)1/2
and ∥Cψ∥≤1. Consequently, ∥Ψ∥22=Trace(Dψ′Dψ+Bψ′Bψ).If we use directly the elements ofAψ, Bψ, Cψ, and Dψ as optimization
variables the total number of variables is nψ2+2nynψ+ny2. However, from
Lemma 10Aψ can be computed
from Cψ and Aψk so the elements Aψk, Bψ, Cψ, and Dψ are all (necessary) optimization variables. Using this technique, the total number of optimization
variables is nψ(nψ−1)/2+2nynψ+ny2 and the
reduction is nψ(nψ+1)/2.In order to carry out the subsequent optimization
effectively, we need an effective method of computingℋ− index fast and
exactly. Enlightened by the bisection method of computing ℋ∞ norm of a
transfer matrix [34],
we now present a bisection algorithm to compute the ℋ− index defined
over [θ1,θ2] .The following result shows the main idea used in our
algorithm.Lemma 11.
Suppose(119)G(z)=[ABCD]∈ℛℋ∞
and θ∈[θ1,θ2], then
(120)minθσ̲[G(ejθ)]>β
if and only if σ̲[D+C(I−A)−1]>β, and
(121)S=[HK1HK2−(A′+C′DR−1B′)−1C′β2R−1C(A′+C′DR−1B′)−1],
where HK1=A+BR−1D′C−BR−1B′(A′+C′DR−1B′)−1C′β2R−1C and HK2=BR−1B′(A′+C′DR−1B′)−1 has no eigenvalues on the segment of unit circle
between θ=θ1 and θ=θ2, where R=β2I−D′D.The detailed procedure of our algorithm for computingℋ− index is
summarized below.(1)
Give an initial
guess on lower bound and upper bound such that(122)0≤β1≤minθ∈[θ1,θ2]σ̲(G(ejθ))≤β2
and give a tolerance ϵ>0.(2)
Letβ=(1/2)(β1+β2). Compute the eigenvalues of
(123)S=[MP1MP2−(A′+C′DR−1B′)−1C′β2R−1C(A′+C′DR−1B′)−1]
where
MP1=A+BR−1D′CBR−1B′(A′+C′DR−1B′)−1C′β2R−1C and MP2=R−1B′(A′+C′DR−1B′)−1.(3)
IfS has no
eigenvalue on the segment of unit circle between θ=θ1 and θ=θ2, which means
that(124)minθ∈[θ1,θ2]σ̲(G(ejθ))>β
is true, then let β1=β; else let β2=β.(4)
Repeat steps (2)
and (3) untilβ2−β1<ϵ is satisfied.
And the approximate value of
(125)minθ∈[θ1,θ2]σ̲(G(ejθ))
is given by (1/2)(β1+β2) with tolerance ϵ.With the state-space parametrization ofΨ on ℛℋ2 space and our
bisection algorithm for computing ℋ− index, the
optimization process for solving Problem 8,
(126)max∥Ψ∥2≤γ∥ΨN˜f∥−[θ1,θ2],
can be performed as
(127)maxAψk,Bψ,Cψ,Dψ,∥Cψ∥≤1,Trace(Dψ′Dψ+Bψ′Bψ)≤γ2∥[(I+Aψk)(I−Aψk)−1(I−Cψ′Cψ)1/2BψCψDψ]N˜f∥−[θ1,θ2].Furthermore, we introduce a penalty functionΘ(Bψ,Cψ,Dψ) to ensure the
conditions Trace(Dψ′Dψ+Bψ′Bψ)≤γ2 and ∥Cψ∥≤1.Θ is defined as
(128)Θ(Bψ,Cψ,Dψ)={C,ifTrace(Bψ′Bψ+Dψ′Dψ)>γ2or∥Cψ∥>1;0else,
where C is a
large positive number. Therefore, the new optimization scheme is
(129)maxAψk,Bψ,Cψ,Dψ∥[(I+Aψk)(I−Aψk)−1(I−Cψ′Cψ)1/2BψCψDψ]N˜f∥−[θ1,θ2]−Θ(Bψ,Cψ,Dψ).For this optimization scheme we have developed a
two-stage optimization algorithm which is a combination of genetic
algorithm [35, 36] and
Nelder-Mead simplex method [10, 26]. Genetic algorithm is good at searching for the right
direction for global optimum but has slow convergence, while Nelder-Mead
simplex method is good at searching for small neighborhood. So the result
obtained by genetic algorithm is used as the starting point for the second-step
optimization by Nelder-Mead simplex method, the latter gives the final results
of the optimization process.Theoretically,Ψ can be a
transfer matrix of any order. However, in practice we try to find a Ψ with low
degree. Thus, we run the optimization process as follows: first set Ψ with a given
starting order, searching for the optimal value; then increase the order of Ψ, run the searching algorithm again and compare the
results with the former one; if higher degree Ψ gives a better
performance and the Ψ's degree
does not exceed the predefined limit, then keep increasing the degree of Ψ and redo the
searching process; else the optimization process ends.
Example 4 will
demonstrate the effectiveness of this optimization method.


## 9. Numerical Examples

In this section,
we give some numerical examples to show the
effectiveness of our approaches for
solving the fault-detection problems.Example 1.
We consider Problem1
for a third-order system:
(130)A=[−0.1964−0.3962−0.5884123−0.5428−1.0879−1.6291],Bd=[0.01000.0100],Bf=[1−3−0.510.50],C=[−0.1964−0.3962−0.5884−0.3650−2.1320−3.0951],Dd=[0.02000.02],Df=[00.701]. Let the pair (γ,β) represents the
performance of an ℋ−/ℋ∞ fault-detection
filter such that ∥Grd∥∞≤γ and ∥Grf∥−[0,2π]≥β. Using our approach an optimal fault-detection filter
has the form in Theorem 2 with
(131)L0=[−0.0500−0.0500]. Let γ=1, we have the
optimal β=γ∥N˜f∥−[0,2π]=0.7632. The singular value plots of Grd and Grf are shown in
Figures 2 and 3, respectively.Figure 2
The singular value plot ofGrd, ∥Grd∥∞=1, for Example 1.Figure 3
The singular value plot ofGrf, ∥Grf∥−[0,2π]=0.75, for Example 1.Example 2.
We consider Problem2
for the same system in Example 1.
Let the pair (γ,β) represents the
performance of an ℋ2/ℋ∞ fault-detection
filter such that ∥Grd∥∞≤γ and ∥Grf∥2[0,2π]≥β. From Theorem 3
the optimal fault-detection filter in Example 1 is
also optimal for this example. Let γ=1, the optimal β=γ∥N˜f∥2=9.7591.
Note that if the so-calledℋ∞/ℋ∞ problem is
considered for this system, the above fault-detection filter is also the
optimal ℋ∞/ℋ∞ filter. Let ∥Grd∥∞≤1, then the optimal ∥Grf∥∞ is 11.4598.Example 3.
We consider Problem3
for the system
(132)A=[−0.1964−0.3962−0.5884123−0.5428−1.0879−1.6291],Bd=[0.01000.0100],Bf=[10110.50],C=[−0.1964−0.3962−0.5884−0.3650−2.1320−3.0951],Dd=[0.02000.02],Df=[10.701].
We let the pair (γ,β) represents the
performance of an ℋ−/ℋ2 fault-detection
filter such that ∥Grd∥2≤γ and ∥Grf∥−[0,2π]≥β. Since this Gf has all zeros
inside the unit circle and Df has full column
rank, we get from Theorem 6(133)(V−1Nf)opt+=[−0.2555−1.4924−2.16660.1900−0.14001.30593.03584.51690.20000.0500−0.5723−1.6360−2.41820.1000−0.0700−0.0591−1.0962−1.57820.2000−0.14000.36502.13203.095100.2000]
and the optimal filter
(134)Qopt=γ∥(V−1Nf)opt+∥2(V−1Nf)opt+V−1=[−0.07400.3126−0.8467−1.20030.3892−2.87611.3720−5.92375.7439−0.2946−0.41100.2304−0.93590.8321−0.03520.7188−0.75092.51220.7430−0.5201−1.73401.3456−4.868200.7430].
Let γ=1 the optimal
(135)β=γ∥(V−1Nf)opt+∥2=0.7430.
The singular value plots of Grd and Grf are shown in
Figures 4 and 5, respectively.Figure 4
Singular value plot ofGrd, ∥Grd∥2=1, for Example 3.Figure 5
Singular value plot ofGrf, ∥Grf∥−=0.7430, for Example 3.Example 4.
We consider Problem8 for a system
(136)A=[−0.1964−0.3962−0.58841.00002.00003.0000−0.5428−1.0879−1.6291],Bd=[0.010.010],Bf=[1.0−0.50.5]C=[−0.1964−0.3962−0.5884],Dd=0.02,Df=0,
where θ1=0 and θ2=π/2.
As discussed in Section8
we use optimization method
to search for a good solution. Let us denote the maximum of ∥Grf∥−[θ1,θ2] as β when ∥Grd∥2≤1. In Table 1 the results obtained using our
optimization algorithm with different predefined Ψ orders are
given. It is clear that the results improve with the increasing order of Ψ. In particular, a third-order Ψ design
achieving β=22.8182 is given by
(137)Ψ=[0.1187−0.00190.4270−0.1424−0.84450.42200.32700.2363−0.3536−0.90120.24080.88650.38440.09880.80790.3714].
The singular value plots ofGrd and Grf are shown in
Figures 6 and 7 for this third-order Ψ. Figure 8
demonstrates how the smallest singular
value of Grf changes in the
frequency range of [0,π/2] with the order
of Ψ. It is seen that the improvement on the performance
with any Ψ of higher order
than 3 is insignificant.
It is interesting to note that theΨ is trying to
invert N˜f in the
frequency interval [0,π/2].Table 1
Results for
differentΨ's order.
Ψ's orderFirstSecondThirdβ14.266122.534522.8182Figure 6
Singular value plot ofGrd with a third
order Ψ, ∥Grd∥2=1, for Example 4.Figure 7
Singular value plot ofGrf with a third
order Ψ, ∥Grf∥−[0,π/2]=22.8182, for Example 4.Figure 8
Singular value plot ofGrf for different
order of Ψ: first order (solid line), second order (dotted
line), and third order (dashed line), for Example 4.


## 10. Conclusion

In this paper,
we have presented optimal solutions to various robust fault-detection problems
for linear discrete time systems in parallel with our continuous time results
in [28]. We have shown that an
optimal filter for both ℋ−/ℋ∞ and ℋ2/ℋ∞ can be obtained
by solving one Riccati equation. It is also interesting to note that we are able to give
analytic solution to an ℋ−/ℋ2 problem defined
on the entire frequency range [0,2π] when Df has full column
rank. In contrast, the corresponding continuous time problem does not make any
sense [28]. The critical reason for this difference is because
the entire frequency range in discrete time is finite (≤2π) while the entire frequency range in continuous time
is infinite. We have also shown that many design criteria considered in the
literature do not give desirable fault-detection designs.



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*Source: 829459-2007-12-26.xml*