#### 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 as /, /, and / 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.

#### 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 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 norm and 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, 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 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, problem, problem, problem, problem, and 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 and problems are given. Zhang et al. in [27] also give an optimal solution to 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: Section 2 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 problem and problem are given in Section 4. The solution for problem is given in Section 5. The solutions for various 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 of by real (complex) matrices is denoted as (). For a matrix we use to denote its transpose and for its complex conjugate transpose. For a Hermitian matrix , represents the largest eigenvalue of and represents the smallest eigenvalue value of . For , denotes the largest singular value of and () denotes the smallest singular value of if (). The identity matrix is denoted as and the zero matrix is denoted as , with the subscripts dropped if they can be inferred from context.

Discrete transfer matrices and -transforms of signals are represented using bold characters and sometimes in dependence of the variable . A state-space realization of a transfer matrix is denoted as

such that We define and denote as the inverse of if is square and invertible. Now suppose is square and is nonsingular, then we have from [29]

We use to denote the set of 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. () is the set of all stable proper transfer matrices.

For we define the norm of as For we define the norm of as

Similar to the definitions of continuous system in [16, 17], we define the index of a discrete transfer matrix on the whole unit circle as

The index of over a finite frequency range is defined as In particular the index defined at is

If no superscript is added to the symbol, such as , then it represents all possible definitions. In many literatures index is also called 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. *Let and be two matrices
with appropriate dimensions, then *

The following transfer matrix factorizations will be frequently used in this paper and can be found from [29].

Lemma 2 (Left Coprime Factorization). *Let be a proper
real rational transfer matrix. A left coprime
factorization (LCF) of is a
factorization
**
where and are
left-coprime over . Let
**
be a detectable state-space realization of and let be a matrix
with appropriate dimensions such that is stable, then
a left coprime factorization of is given by
*

Lemma 3 (Spectral Factorization). *Let be a proper
real rational transfer matrix and let
**
be a detectable realization of . Suppose D has full row rank and has full row
rank for all Let be the
stabilizing solution to the following algebraic Riccati equation:
**
such that is stable and
let . Then the following spectral factorization holds
**
where and
*

#### 3. Problem Formulation

Consider a discrete time invariant system with disturbance and possible faults as: where is the state vector, is the output measurement, represents the unknown/uncertain disturbance and measurement noise, and denotes the process, sensor or actuator fault vector. and 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 and are:

(i)unknown signal with bounded energy or bounded power;(ii)white noise. Different assumptions on and 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. * is detectable.

This is a standard assumption for all fault-detection problems.

*Assumption 2. * has full row
rank.

This means that 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 has full row rank. In the case of some simplified model where does not have full row rank, we can simply add some columns to make it full row rank. For example, suppose that does not have full row rank, then let for a small . Then has full row rank.

*Assumption 3. * has full row
rank for all or, equivalently, the transfer function matrix
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 has full column
rank. We believe that this assumption is extremely restrictive. The assumption implies that measurement contains
directly the independent information on every faulty component of . In particular, this implies that 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 taking -transform of
(16) we have the system input/output equation
where , , and are , and transfer
matrices, respectively and their state-space realizations are
Since the state-space realization of , , and share the same and matrices,
applying Lemma 2 we can find an
LCF for the system (20)
where
and is a matrix
such that is stable.

It has been shown in [2] that, without loss of generality, the fault detection
filter can take the following general form:
where is the residual
vector for detection, is a free stable transfer
matrix to be designed. The filter structure is shown in Figure 1. Replacing in (23)
by the right-hand
side of (19) and (21) we have
Denote the transfer matrices from and to by and , respectively, then

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 an index is a good
measurement for a transfer function's smallest gain, is a reasonable
performance criterion for measuring fault detection sensitivity if is modeled as
unknown energy or power bounded signals. If is modeled as
unknown energy or power bounded signals, then norm is a
widely accepted worst case measure and is a good
indicator of disturbance rejection performance. On the other hand, if and/or are white
noise, the norms of and/or 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.

*Problem 1 ( Problem). * Let an uncertain system be described by
(16)–(20) and let be a given
disturbance rejection level. Find a stable transfer matrix in (23)–(25) such that and is maximized,
that is,

*Problem 2 (Problem). *Let an uncertain system be described by
(16)–(20) and let be a given
disturbance rejection level. Find a stable transfer matrix in (23)–(25) such that and is maximized,
that is,

*Problem 3 ( Problem). *Let an uncertain system be described by
(16)–(20) and let be a given
disturbance rejection level. Find a stable transfer matrix in (23)–(25) such that and is maximized, that is,

*Remark 2. *A more conventional formulation of the above
problems is to optimize the following:
where and can be , , or . The problem that is classical
and optimal solution is available [2]. The case for and has been solved
recently in [26] for continuous-time systems. A discrete solution has
also been obtained recently in [27] for the cases of and .

Before we proceed to the solutions of the above
problems, we will first establish some preliminary results.

Lemma 4. *Suppose Assumption 3 is satisfied and let be any left
coprime factorization over . Then has no
transmission zero on the unit circle or, equivalently, for any appropriately
dimensioned matrix ,
**
has full row rank for all
.*

*Proof. *The result
follows by noting that
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 Assumptions
1–3 are satisfied and let be any left
coprime factorization over . Then there is a square transfer matrix such that and
**
In particular,
if a state-space representation of is given as in
(22), then a state space representation of is given
by
**
with
**
where is the
stabilizing solution to the Riccati equation
**
such that is stable and *

*Proof. *Since
Assumptions 1–3 are satisfied, Lemmas 3 and 4 can be applied to to get , where satisfies the
following Riccati equation

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 Assumptions 1–3 are satisfied. Let be defined as
in (32). Let
**
for and denote . Then the
fault-detection Problems 1–3
are equivalent to Problems 4–6
below, respectively.*

*Problem 4. *

*Problem 5. *

*Problem 6. *

*Proof. *We will first
show that Problems 1 and 2 are equivalent to Problems 4 and
5,
respectively.

Note that by Lemma 6 there exists such that and Therefore,
that is, We can,
therefore without loss of generality, take in the form of for some . Hence , so that is equivalent
to Moreover, , hence Problem 1 is equivalent to Problem 4 and Problem 2 is equivalent
to Problem 5.

Next we show that Problem 3 is equivalent to Problem
6. Note that in
Problem 3, we have . Hence,
such that Since and , we can let for some . Therefore, so that is equivalent
to Moreover, , hence Problem 3 is equivalent to Problem 6.

We will provide optimal solutions for each of the above problems in the following sections.

#### 4. Fault-Detection Filter Design

In this section, we give a complete solution for the fault-detection filter design problem, that is, Problem 1 or Problem 4.

Theorem 1. *Suppose Assumptions 1–3 are satisfied. Let
**
be any left coprime factorization over and let be a square
transfer matrix such that and . Then
**
and an optimal fault-detection filter for Problem
1 is
given by
**
where
*

*Proof. *Note that by
Lemma 6, we only need to solve Problem 4:
From Lemma 1 we know that for every frequency ,
so that
By letting , we have and , which means that is an optimal
solution achieving the maximum.

*Remark 3. *The optimal fault-detection filter given in
Theorem 1 does not depend on and matrices.

*Remark 4. *Note that the solution given in the above theorem
does not depend on the specific definitions of index. Hence,
the solution provided here is an optimal solution for all indices.
However, it should be pointed out that this optimal filter is not the only
optimal solution for some index
criterion. For example, let where is a low-pass
filter with a very small bandwidth so that and . Then this is also an
optimal solution for
even though this is obviously a bad fault-detection
filter because the low-pass filter 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 Assumptions 1–3 are satisfied. Let be the
stabilizing solution to the Riccati equation
**
such that is stable and
let . Define
**
Then
**
and an optimal fault-detection
filter has the following state space representation
**
where
**In other words, the optimal fault-detection
filter is the following observer:
*

*Proof. *Note that
where is a matrix
with appropriate dimensions such that is stable. Note
from Theorem 1 and Lemma 5 that
Then

Similarly, we have

*Remark 5. *Note that the optimal fault-detection filter is independent
of the choice of matrix.

*Remark 6. *It is easy to see that our optimal filter given in
Theorems 1 and
2
is also optimal for the so-called problem
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:

#### 5. Fault-Detection Filter Design

In this section, we give an optimal solution for the problem stated in Section 3 as Problem 2. Similar to the solution for problem given in Theorems 1 and 2, we have the following parallel results for the problem.

Theorem 3. *Suppose Assumptions 1–3 are satisfied. Let
**
be any left coprime factorization over and let be a square
transfer matrix such that and . Then **
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
Lemma 6,
we only need to solve Problem 5:
Note that
By letting , we have and , which means that is an optimal
solution achieving the maximum.

#### 6. Fault-Detection Filter Design: Case 1

From Lemma 6 we know that the problem is equivalent to Problem 6, that is, Unlike the problem studied in Section 4, we have different solutions for the problem if different 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 Assumptions 1–3 are satisfied. Then
**
Furthermore, for any given , let and
**
Then
**
is satisfied for a sufficiently small .*

*Proof. *Again note that
the equivalent Problem 6 in this case is
Take such that . Then and Let , then , so that

*Remark 7. *We should point out that an optimal filter
designed using Theorem 4
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. Fault-Detection Filter Design: Case 2

In this section, we will consider another special case where the index is defined for all frequencies but with 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. *Suppose has full column
rank. Then an optimal solution to Problem 6**
has the form and
**
where is a positive
scalar and is an all-pass stable
transfer matrix.*

*Proof. *We will first
show
where C is a nonnegative scalar.

Suppose there exists a such that does not hold.
Let denote the set
of all values such
that
is achieved.
Let such that
Then there exists a weighting function such that and
Therefore, and is not an
optimal solution. Hence, it must be true that for every

Next we show that

Suppose there exists a such that for some , that is,
Then a can be selected
such that
Since for every Let , then and
Therefore,
is not optimal
and by contradiction the assumption is false. So holds for every

Since for every , and that has full column
rank implies , has the form
where is an all-pass
stable transfer matrix and is a positive
scalar. Let , then

Lemma 8. *Suppose has full column
rank. Then Problem 6*
is equivalent to the following problem.
*Problem 7. *

*Proof. *From Lemma 7
we know that the optimal solution to Problem 6
has
the form and
Let , where is and is Then so and . Since Problem 6
needs to maximize with the
constraint , it is equivalent to find a with the smallest norm such that Denote 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
**
is strictly
minimum phase and has full column
rank. Let is chosen such
that and , then the optimal solution to problem
**
is given by
**
where is the solution
to the algebraic Ricatti equation
*

*Proof. *The equation is equivalent
to ,
so Problem 7 is equivalent to finding an with the smallest norm such that Hence the
conclusion in [32] can be applied to to get the
optimal . is then
obtained by taking transpose of

Theorem 5. *Suppose Assumptions 1–3 are satisfied. Let has all zeros
inside the unit circle and has full column
rank. Let
**
be any left coprime factorization over and let be a square
transfer matrix such that and . Let be the optimal
solution to Problem 7. Then
**
and an optimal fault detection filter is given by
**
where
*

*Proof. *Note that by
Lemma 6, we only need to solve
Problem 6
Since has all zeros
inside the unit circle and , is strictly
minimum phase. From Lemmas 7–9 we know that the optimal solutionto Problem 6 is
given by
where is the optimal
solution to Problem 7 and is a unitary
matrix. Take , then an optimal
solution is given by

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 Assumptions 1–3 are satisfied. Let has all zeros
inside the unit circle and has full column
rank. Let be the
stabilizing solution to the Riccati equation
**
such that is stable. Let and define
**
Let is chosen such
that and . Let is the solution
to the algebraic Ricatti equation
**
and define
**
Then
**
where
**
and an optimal fault-detection
filter has the following state-space representation:
**
where
**
where , and *

*Proof. *Note that
where is a matrix
with appropriate dimensions such that is stable. From
Theorem 1
From Theorem 2
From Lemma 9
Therefore,
where and

*Remark 8. *Note that the optimal fault-detection filter
is independent
of the choice of
matrix.

*Remark 9. *Note that the strictly minimum phase assumption
for is not needed.
In general, if does not have
any zeros on the unit circle, one can always factorize
so that is strictly
minimum phase and is a stable
all-pass matrix. Then the solution can be computed by using
in place of . In the case when
has zeros on
the unit circle, approximation factorization can also be carried out to obtain
an approximation solution.

#### 8. Fault-Detection Filter Design: Case 3

When Problem 3 is considered with the index defined over a finite frequency range , the solution becomes much more complicated. We will now state this as a separate problem as below.

*Problem 8 (Interval Problem). *Let an
uncertain system be described by (16)–(20) and let be a given
disturbance rejection level. Find a stable transfer matrix
in (23)–(25) such that
and is maximized, that is,
or, equivalently, let and solve

*Remark 10. *It is not hard to see that there is no rational
function solution to the above problem. This is because an optimal must satisfy almost every
where for any . Hence, an analytic optimal solution seems to be
impossible. Nevertheless, it is intuitively feasible to find some rational
approximations so that a rational has the form of
a bandpass filter with the pass-band close to and .

*Remark 11. *When the
condition that has full column
rank is not satisfied, the rational optimal solution to the problem

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 and , 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 given norm [33].

Lemma 10. *Let
**
be an
th order
proper stable transfer matrix. Then the state space parameters of can be
expressed as for some and some satisfies Furthermore, *

*Proof. *Assume that
is an th order
observable realization, then the Observability Gramian satisfies
Since , there exists a Cholesky factorization of where is invertible.
Perform a similarity transformation on such that
Thus, , so that where is an
orthogonal matrix and is a nonnegative
definite. Since an orthogonal matrix with no
eigenvalue equals can be
represented as , where is a
skew-symmetric matrix, we have
and Consequently,

If we use directly the elements of , , , and as optimization variables the total number of variables is However, from Lemma 10 can be computed from and so the elements , , , and are all (necessary) optimization variables. Using this technique, the total number of optimization variables is and the reduction is

In order to carry out the subsequent optimization effectively, we need an effective method of computing index fast and exactly. Enlightened by the bisection method of computing norm of a transfer matrix [34], we now present a bisection algorithm to compute the index defined over .

The following result shows the main idea used in our algorithm.

Lemma 11. *Suppose
**
and , then
**
if and only if , and
**
where and has no eigenvalues on the segment of unit circle
between and , where .*

The detailed procedure of our algorithm for computing index is summarized below.

(1)Give an initial guess on lower bound and upper bound such that and give a tolerance .(2)Let . Compute the eigenvalues of where and (3)If has no eigenvalue on the segment of unit circle between and , which means that is true, then let ; else let .(4)Repeat steps (2) and (3) until is satisfied. And the approximate value of is given by with tolerance .

With the state-space parametrization of on space and our bisection algorithm for computing index, the optimization process for solving Problem 8, can be performed as

Furthermore, we introduce a penalty function to ensure the conditions and is defined as where is a large positive number. Therefore, the new optimization scheme is

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, can be a transfer matrix of any order. However, in practice we try to find a with low degree. Thus, we run the optimization process as follows: first set with a given starting order, searching for the optimal value; then increase the order of , run the searching algorithm again and compare the results with the former one; if higher degree gives a better performance and the 's degree does not exceed the predefined limit, then keep increasing the degree of 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 Problem 1
for a third-order system:
Let the pair represents the
performance of an fault-detection
filter such that and . Using our approach an optimal fault-detection filter
has the form in Theorem 2 with
Let , we have the
optimal . The singular value plots of and are shown in
Figures 2 and 3, respectively.

*Example 2. *We consider Problem 2
for the same system in Example 1.
Let the pair represents the
performance of an fault-detection
filter such that and . From Theorem 3
the optimal fault-detection filter in Example 1 is
also optimal for this example. Let , the optimal .

Note that if the so-called problem is
considered for this system, the above fault-detection filter is also the
optimal filter. Let , then the optimal is .*Example 3. *We consider Problem 3
for the system
We let the pair represents the
performance of an fault-detection
filter such that and . Since this has all zeros
inside the unit circle and has full column
rank, we get from Theorem 6
and the optimal filter
Let the optimal
The singular value plots of and are shown in
Figures 4 and 5, respectively.

*Example 4. *
We consider Problem 8 for a system
where and .

As discussed in Section 8
we use optimization method
to search for a good solution. Let us denote the maximum of as when . In Table 1 the results obtained using our
optimization algorithm with different predefined orders are
given. It is clear that the results improve with the increasing order of . In particular, a third-order design
achieving is given by

The singular value plots of and are shown in
Figures 6 and 7 for this third-order . Figure 8
demonstrates how the smallest singular
value of changes in the
frequency range of with the order
of . It is seen that the improvement on the performance
with any of higher order
than 3 is insignificant.

It is interesting to note that the is trying to
invert in the
frequency interval .

#### 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 and can be obtained by solving one Riccati equation. It is also interesting to note that we are able to give analytic solution to an problem defined on the entire frequency range when 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 () 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.

#### Acknowledgments

This work was supported in part by grants from NASA (NCC5-573), LEQSF (NASA/LEQSF(2001-04)-01), and the NNSFC Young Investigator Award for Overseas Collaborative Research (60328304).