1 Introduction

In recent decades, the fault estimation of sampled-data systems by the observer design method has received more and more attention^[1-5]. However, for a well-known reason, if the sampling process is relatively slow, the stability and stabilization condition of the continuous-time observers may be lost which makes the fault estimation results unfeasible. In many practice, this unfeasibility cannot be solved. So, a new model by transforming the sampled-data systems into descriptor ones is proposed.

Descriptor systems, which can be also defined as the singular systems or differential-algebraic equation systems, are widely used to describe plenty of practical systems such as electrical networks^[6], constrained mechanical systems^[7], aircraft modeling^[8] and so on. A lot of models have been established to guarantee the feasibility of the fault estimation in descriptor systems by the observer designing^[9-11]. However, little existing work focuses on the fault estimation problem of sampled-data descriptor systems, and the observability of some descriptor systems in practical works may not be realized^[12-14]. So there is a great incentive for us to develop a novel sampled-data descriptor observer for the estimation of system faults, and furthermore to improve the detectability of the designed system.

Recently, a novel design method of observation is proposed in Ref.[15] to investigate sensor fault reconstruction and sensor compensation for discrete-time systems. Based on the results in Ref.[15], a new approach of sensor fault estimation for sampled-data systems is considered. The sampled-data systems are firstly discretized to obtain a discrete-time model. Then a fault estimation model is established by designing a discrete-time observer of the discretized descriptor system, the necessary and sufficient condition for the existence and convergence of the proposed observer is given and proved. In the paper, many results have been proposed to design observers for discrete-time systems by using the linear matrix inequality (LMI) techniques, so the observer design is formulated as an LMI feasible problem, which can be easily solved by standard convex optimization algorithms^[16].

Autonomous land vehicle control system is a class of norm bounded parameter uncertainty of generalized continuous piecewise affine systems^[17].In view of the lack of an autonomous land vehicle system in previous switching control strategies and system delays, the case of unknown external interference and noise on the adverse effects of the performance of the control system, a method with a linear matrix inequality LMI algorithm in the papercan be put forward^[18-20].Then discussion is needed for the detectability of descriptor system since a pair of parameters T and N are introduced^[21-23].

This paper is organized as follows. In Section 2, a model of the sampled-data system is proposed to be studied, and then we discretize the system into a descriptor one by an efficient way. In Section 3, sensor fault estimation observer is presented for the evaluated discrete-time linear descriptor systems which is presented and formulated as an LMI formulation. The detectability of the designed observer is discussed. In Section 4, two simulation examples are given to demonstrate the validity of our results. Finally, several conclusions are drawn in Section 5.

2 Problem Statement

Consider the following sampled-data system:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}}\left( t \right) = {\mathit{\boldsymbol{A}}_c}\mathit{\boldsymbol{x}}\left( t \right) + {\mathit{\boldsymbol{B}}_c}\mathit{\boldsymbol{u}}\left( t \right) + {\mathit{\boldsymbol{D}}_c}\mathit{\boldsymbol{d}}\left( t \right)}\\ {\mathit{\boldsymbol{y}}\left( {k{T_s}} \right) = \mathit{\boldsymbol{Cx}}\left( {k{T_s}} \right) + \mathit{\boldsymbol{Ffx}}\left( {k{T_s}} \right)} \end{array}} \right. $

(1)

where x(t)∈$\mathbb{R}^{n}$ is the state, u(t)∈$\mathbb{R}^{p}$ is the control input, y(kT_s)∈$\mathbb{R}^{m}$ is the sampled output at time instant kT_s, and f(kT_s)∈$\mathbb{R}^{m}$ denotes the sensor fault. In system(1), T_s denotes the sampling interval, A_c, B_c, D_c and F are known constant matrices of appropriate dimensions.

Additional remark: At first, the following system representation is considered:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}}\left( t \right) = {\mathit{\boldsymbol{A}}_c}\mathit{\boldsymbol{x}}\left( t \right) + {\mathit{\boldsymbol{B}}_c}\mathit{\boldsymbol{u}}\left( t \right) + {\mathit{\boldsymbol{D}}_c}\mathit{\boldsymbol{d}}\left( t \right)}\\ {\mathit{\boldsymbol{y}}\left( {k{T_s}} \right) = \mathit{\boldsymbol{Cx}}\left( {k{T_s}} \right) + \mathit{\boldsymbol{Ffx}}\left( {k{T_s}} \right)} \end{array}} \right. $

In this situation, the following descriptor system will be obtained:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{E\bar x}}(k + 1) = \mathit{\boldsymbol{\bar A\bar x}}(k) + \mathit{\boldsymbol{\bar B}}{\boldsymbol{u}}(k) + \mathit{\boldsymbol{\bar Dd}}(k)}\\ {\mathit{\boldsymbol{y}}(k) = \mathit{\boldsymbol{\bar C\bar x}}(k)} \end{array}} \right. $

where:

$ \mathit{\boldsymbol{E}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_n}}&0\\ 0&0 \end{array}} \right],\mathit{\boldsymbol{\bar A}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{A}}&0\\ 0&0 \end{array}} \right] $

$ \mathit{\boldsymbol{\bar B}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{B}}\\ 0 \end{array}} \right],\mathit{\boldsymbol{\bar D}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{D}}\\ 0 \end{array}} \right],\mathit{\boldsymbol{\bar C}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{C}}&{{\mathit{\boldsymbol{I}}_m}} \end{array}} \right] $

To obtain TE + NC = I_n+m, we must choose:

$ \mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_n}}&0\\ { - \mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{I}}_m}} \end{array}} \right],\mathit{\boldsymbol{N}} = \left[ {\begin{array}{*{20}{c}} 0\\ {{\mathit{\boldsymbol{I}}_m}} \end{array}} \right] $

Unfortunately, this may lead to the loss of observability. It can be seen that $\mathit{\boldsymbol{\bar CT\bar A}} = 0$, which means that $\left({\mathit{\boldsymbol{T\bar A}}\; \; \; \mathit{\boldsymbol{\bar C}}} \right)$ is completely not observable. As a result, the observer is stable only if ${\mathit{\boldsymbol{T\bar A}}}$ is stable.

In many practical systems, ${\mathit{\boldsymbol{T\bar A}}}$ is not stable. Therefore, we cannot use the descriptor system approach to these systems.

Denoting x(k) = x(kT_s), u(k) = u(kT_s), and y(k) = y(kT_s), the discretized representation of System(1) can be derived as:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{x}}\left( {k + 1} \right) = \mathit{\boldsymbol{Ax}}\left( k \right) + \mathit{\boldsymbol{Bu}}\left( k \right) + \mathit{\boldsymbol{Dd}}\left( k \right) + \mathit{\boldsymbol{O}}\left( {\mathit{\boldsymbol{T}}_s^2} \right)}\\ {\mathit{\boldsymbol{y}}\left( k \right) = \mathit{\boldsymbol{Cx}}\left( k \right)} \end{array}} \right. $

(2)

where:

$ \mathit{\boldsymbol{A}} = {e^{{A_c}\tau }} $

$ \mathit{\boldsymbol{B}} = \int_0^T {{e^{{A_c}\tau }}{\mathit{\boldsymbol{B}}_c}{\rm{d}}\tau } $

$ \mathit{\boldsymbol{D}} = \int_0^T {{e^{{A_c}\tau }}{\mathit{\boldsymbol{D}}_c}{\rm{d}}\tau } $

In this paper, the sampling interval T_s assumed to be sufficiently small such that the term in the order of O(T_s²) can be omitted.

However, if we consider the System (2), the pair ($\left({\mathit{\boldsymbol{T\bar A}}\; \; \; \mathit{\boldsymbol{\bar C}}} \right)$ TA, C) may be observable.

So, we have to combine the descriptor system approach to the dedicated observer scheme, as used in Professor Wu's paper (IEEE Transactions on Fuzzy Systems) 2015^[16].

The following Lemma ^[17] is needed.

Lemma 1  Given a scalar γ>0, the discrete-time system described by:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{x}}\left( {k + 1} \right) = \mathit{\boldsymbol{Ax}}\left( k \right) + \mathit{\boldsymbol{Bu}}\left( k \right)}\\ {\mathit{\boldsymbol{y}}\left( k \right) = \mathit{\boldsymbol{Cx}}\left( k \right)} \end{array}} \right. $

(3)

Which is stable and its transfer function H_yu(z) = C(zI-A)^-1B, $\left\|\boldsymbol{H}_{y u}(z)\right\|_{\infty} < \gamma$, if and only if there exists a matrix P >0 such that:

$ \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}} + {\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{C}} - \mathit{\boldsymbol{P}}}&{{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PB}}}\\ {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PA}}}&{{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}} - {\gamma ^2}\mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $

(4)

3 Sensor Fault Estimation Observer Design

In this section, a descriptor system is firstly constructed for System(6). Then, an observer is designed to simultaneously estimate the state x(k) and the sensor fault f(k).

Defining:

$ \mathit{\boldsymbol{\bar x}}\left( k \right) = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{x}}\left( k \right)}\\ {\mathit{\boldsymbol{f}}\left( k \right)} \end{array}} \right] $

(5)

Then the System (2) is rewritten as the following descriptor system:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{E\bar x}}(k + 1) = \mathit{\boldsymbol{\bar A\bar x}}(k) + \mathit{\boldsymbol{\bar B}}{\boldsymbol{u}}(k) + \mathit{\boldsymbol{\bar Dd}}(k)}\\ {\mathit{\boldsymbol{y}}(k) = \mathit{\boldsymbol{\bar C\bar x}}(k)} \end{array}} \right. $

(6)

where:

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{E}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_n}}&0\\ 0&0 \end{array}} \right],\mathit{\boldsymbol{\bar A}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{A}}&0\\ 0&0 \end{array}} \right]}\\ {\mathit{\boldsymbol{\bar B}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{B}}\\ 0 \end{array}} \right],\mathit{\boldsymbol{\bar D}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{D}}\\ 0 \end{array}} \right],\mathit{\boldsymbol{\bar C}} = \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{C}}&\mathit{\boldsymbol{F}} \end{array}} \right]} \end{array} $

(7)

For the descriptor System(6), the following observer is constructed:

$ \begin{array}{l} \mathit{\boldsymbol{\hat {\bar x}}}\left( {k + 1} \right) = \mathit{\boldsymbol{T\bar A\hat {\bar x}}}(k) + \mathit{\boldsymbol{T\bar Bu}}(k) + \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{L}}(\mathit{\boldsymbol{y}}(k) - \mathit{\boldsymbol{\bar C\hat {\bar x}}}(k)) + \mathit{\boldsymbol{Ny}}(k + 1) \end{array} $

(8)

where $\hat{\bar{\boldsymbol {x}}}(k) \in \mathbb{R}^{n+m}$ denotes the estimation of the descriptor state $\overline{\boldsymbol{x}}(k), \quad \boldsymbol{T} \in \mathbb{R}^{(n+m) \times(n+m)}$, N ∈$\mathbb{R}^{(n+m) \times m}$, and L ∈$\mathbb{R}^{(n+m) \times m}$ are matrices to be designed.

Suppose that matrices T and N are chosen such that:

$ \mathit{\boldsymbol{TE}} + \mathit{\boldsymbol{NC}} = {\mathit{\boldsymbol{I}}_{n + m}} $

(9)

where the matrix $\boldsymbol{T}\in {{\mathbb{R}}^{(n+m)\times (n+m)}}$, N ∈ $\mathbb{R}^{(n+m) \times m}$ may be singular and rank(T)=rank(N)=r≤n.Then the following theorem is presented to design observer (8).

Based on Lemma 1, the following Theorem is proposed to synthesize a robust sensor fault estimation observer.

Theorem 1  Given a scalar γ>0, the dynamic System (8) is a robust sensor fault estimation observer with$\left\|\boldsymbol{H}_{e d}(z)\right\|_{\infty} < \gamma$. if and only if there exist matrices: $\boldsymbol{P} \in \mathbb{R}^{(n+m) \times(n+m)}$ and $\boldsymbol{W}\in {{\mathbb{R}}^{(n+m)\times m}}$ such that:

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{P}} > 0\\ \left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{P}}}&{{{\left( {\mathit{\boldsymbol{T\bar A}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} - {{\mathit{\boldsymbol{\bar C}}}^{\rm{T}}}{\mathit{\boldsymbol{W}}^{\rm{T}}}}\\ {\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{T\bar A}}} \right) - \mathit{\boldsymbol{W\bar C}}}&{ - \mathit{\boldsymbol{P}}} \end{array}} \right] < 0 \end{array} \right. $

(10)

and the gain matrix L in observer (8) is determined by L = P^-1W.

Proof   Using Eq. (9), the descriptor System (6) is written as:

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\bar x}}\left( {k + 1} \right) = \mathit{\boldsymbol{T\bar A\bar x}}\left( k \right) + \mathit{\boldsymbol{T\bar Bu}}\left( k \right) + }\\ {\mathit{\boldsymbol{\bar Dd}}(k) + \mathit{\boldsymbol{Ny}}(k + 1)} \end{array} $

(11)

The augmented state estimation error is defined as:

$ \mathit{\boldsymbol{e}}(k) = \mathit{\boldsymbol{x}}(k) - \mathit{\boldsymbol{\hat x}}(k) $

(12)

Subtracting Eq. (8) from Eq.(11), the error dynamic equation is obtained as:

$ \mathit{\boldsymbol{e}}\left( {k + 1} \right) = \left( {\mathit{\boldsymbol{T\bar A}} - \mathit{\boldsymbol{L\bar C}}} \right)\mathit{\boldsymbol{e}}\left( k \right) + \mathit{\boldsymbol{\bar Dd}}\left( k \right) $

(13)

Applying Lemma 1 to Eq.(13), we obtain that $\left\|\boldsymbol{H}_{e d}(z)\right\|_{\infty} < \gamma$ if and only if:

$ \left[ {\begin{array}{*{20}{c}} {\Delta + {\mathit{\boldsymbol{I}}_n} - \mathit{\boldsymbol{P}}}&{{{(\mathit{\boldsymbol{T\bar A}} - \mathit{\boldsymbol{L\bar C}})}^{\rm{T}}}\mathit{\boldsymbol{PT\bar D}}}\\ *&{{{(\mathit{\boldsymbol{T\bar D}})}^{\rm{T}}}\mathit{\boldsymbol{PT\bar D}} - {\gamma ^2}{\mathit{\boldsymbol{I}}_q}} \end{array}} \right] < 0 $

(14)

where $\Delta ={{(\boldsymbol{T\bar{A}}-\boldsymbol{L\bar{C}})}^{\bf{T}}}\boldsymbol{P}(\boldsymbol{T\bar{A}}-\boldsymbol{L\bar{C}})$.

By the well-known Schur complement Lemma in Ref. [22], Inequality (14) is equivalent to:

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{P}} > 0\\ \left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{P}} + {\mathit{\boldsymbol{I}}_n}}&0&{{{\left( {\mathit{\boldsymbol{T\bar A}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} - {{\mathit{\boldsymbol{\bar C}}}^{\rm{T}}}{\mathit{\boldsymbol{L}}^{\rm{T}}}\mathit{\boldsymbol{P}}}\\ * &{ - {\gamma ^2}{\mathit{\boldsymbol{I}}_q}}&{{{\left( {\mathit{\boldsymbol{T\bar D}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}}}\\ * & * &{ - \mathit{\boldsymbol{P}}} \end{array}} \right] < 0 \end{array} \right. $

(15)

Letting W=PL, Eq.(15) comes to Eq.(10). Then the gain matrix L is determined by L=P^-1W.

From Eq. (13), it is known that there exists a stable sensor fault estimation observer if the pair $\left(\boldsymbol{T\bar{A}}\ \ \ \boldsymbol{\bar{C}} \right)$ is detectable. Therefore, the detectability of $\left(\boldsymbol{T\bar{A}}\ \ \ \boldsymbol{\bar{C}} \right)$ is studied in the following.

For the convenience of discussion, the definition of detectability for the descriptor systems is introduced in Ref.[1].

Definition 1   The descriptor System (6) is called R-detectable, if the following condition holds:

$ {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {z\mathit{\boldsymbol{E}} - \mathit{\boldsymbol{\bar A}}} \\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = n + m,\forall z \in \mathbb{C},\left| z \right| \geqslant 1 $

(16)

To discuss the detectability of the (TA  C), the following Theorem is presented.

Theorem 2  The pair $ \left(\boldsymbol{T\bar{A}}\ \ \ \boldsymbol{\bar{C}} \right)$ is detectable, if System(2) is R-detectable and the matrix T satisfying Eq.(9) is of full rank. Furthermore, the detectability of $\left(\boldsymbol{T\bar{A}}\ \ \ \boldsymbol{\bar{C}} \right)$ is independent of the choice of matrix T.

Proof   It is obvious that:

$ {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {z{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{z}}{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}}\\ {z\mathit{\boldsymbol{\bar C}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] $

(17)

Using Eq. (9), we obtain:

$ \begin{array}{l} {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{z}}{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}}\\ {\mathit{\boldsymbol{z\bar C}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = \\ \;\;\;\;\;\;\;{\rm{rank}}\left( {\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{T}}&\mathit{\boldsymbol{N}}&0\\ 0&{{\mathit{\boldsymbol{I}}_m}}&0\\ 0&0&{{\mathit{\boldsymbol{I}}_m}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{zE}} - \mathit{\boldsymbol{\bar A}}}\\ {z\mathit{\boldsymbol{\bar C}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right]} \right) \end{array} $

(18)

Since T is of full rank, we have:

$ {\rm{rank}}\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{T}}&\mathit{\boldsymbol{N}}&0\\ 0&{{\mathit{\boldsymbol{I}}_m}}&0\\ 0&0&{{\mathit{\boldsymbol{I}}_m}} \end{array}} \right] = n + 3m $

(19)

Applying Sylvester's equality to Eq. (18) gives:

$ \begin{array}{l} {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{z}}{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}}\\ {\mathit{\boldsymbol{z\bar C}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = \\ \;\;{\rm{ }}\;\;\;\;\;{\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{zE}} - \mathit{\boldsymbol{\bar A}}}\\ {\mathit{\boldsymbol{z\bar C}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{zE}} - \mathit{\boldsymbol{\bar A}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] \end{array} $

(20)

Considering Eq. (17) and Eq.(20), it becomes:

$ {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {z{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{zE}} - \mathit{\boldsymbol{\bar A}}}\\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] $

(21)

If the descriptor system (6) is R-detectable, we have:

$ \begin{array}{*{20}{c}} {{\rm{rank}}\left[ {\begin{array}{*{20}{c}} {z{\mathit{\boldsymbol{I}}_{n + m}} - \mathit{\boldsymbol{T\bar A}}} \\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = {\rm{rank}}\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{zE}} - \mathit{\boldsymbol{\bar A}}} \\ {\mathit{\boldsymbol{\bar C}}} \end{array}} \right] = } \\ {n + m,\forall \mathit{\boldsymbol{z}} \in \mathbb{C},\left| \mathit{\boldsymbol{z}} \right| \geqslant 1} \end{array} $

(22)

which means that the pair $\left(\boldsymbol{T\bar{A}}\ \ \ \boldsymbol{\bar{C}} \right)$ is detectable.

Since rank $\left[\begin{array}{ll}{\boldsymbol{E}} & {\boldsymbol{C}}\end{array}\right]^{\boldsymbol{T}}=n$, there exists a full rank matrix [T  N] satisfying Eq. (9). The general solution for [T  N] is given by:

$ \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{T}}&\mathit{\boldsymbol{N}} \end{array}} \right] = {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{E}}\\ \mathit{\boldsymbol{C}} \end{array}} \right]^{\rm{T}}} + \mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{I}}_{n + m}} - \left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{E}}\\ \mathit{\boldsymbol{C}} \end{array}} \right]{{\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{E}}\\ \mathit{\boldsymbol{C}} \end{array}} \right]}^{\rm{T}}}} \right) $

(23)

where $\boldsymbol{S} \in \mathbb{R}^{n \times(n+m)}$ is an arbitrary matrix, which provides us a degree of freedom to determine a nonsingular matrix T.

4 Simulations

In this section, a simulation example is given to show the effectiveness of the proposed method.

Firstly, a fault estimation observer for Sensor 1 is designed.

Case 1   Consider the following sampled-data system in the form of System(1) with the following parameters:

$ \mathit{\boldsymbol{C}} = \left[ {\begin{array}{*{20}{l}} 0&1&0&0\\ 0&0&0&1 \end{array}} \right] $

$ \mathit{\boldsymbol{F}} = \left[ {\begin{array}{*{20}{l}} 1\\ 0 \end{array}} \right] $

$ \mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} {0.1}\\ 1\\ 0\\ {0.1} \end{array}} \right] $

$ \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - 0.277}&0&{ - 32.9}&{9.81}\\ { - 0.1033}&{ - 8.525}&{3.75}&0\\ {0.3649}&0&{ - 0.639}&0\\ 0&1&0&0 \end{array}} \right] $

with the sampling interval T_s=0.1 s, Choosing:

$ \mathit{\boldsymbol{S}} = \left[ {\begin{array}{*{20}{c}} { - 1}&0&0&0&0&{ - 1}&0\\ 1&{ - 1}&0&1&0&0&{ - 1}\\ 0&{ - 1}&0&1&0&0&{ - 1}\\ 0&0&{ - 1}&{ - 1}&{ - 1}&{ - 1}&{ - 1}\\ 0&{ - 1}&0&0&{ - 1}&{ - 1}&{ - 1} \end{array}} \right] $

We obtain:

$ \mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0\\ 0&1&0&{ - 0.5}&1\\ 0&0&1&1&0\\ 0&0&0&{0.5}&{ - 1}\\ 0&{ - 1}&0&{0.5}&{ - 1} \end{array}} \right] $

$ \mathit{\boldsymbol{N}} = \left[ {\begin{array}{*{20}{r}} 0&0\\ 0&{0.5}\\ 0&{ - 1}\\ 0&{0.5}\\ 1&{ - 0.5} \end{array}} \right] $

It is obvious that T is of full rank. Then, by letting γ=0.4 and applying Theorem 1, it comes:

$ \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {0.0002}&{1.2671}\\ {0.4567}&{9.2066}\\ {0.0048}&{4.3691}\\ {0.0024}&{1.2483}\\ { - 0.4567}&{ - 9.2066} \end{array}} \right] $

The disturbance d(t) is chosen as random noise with d(t)~N(0, 1). The following sensor fault (of Sensor 1) is considered:

$ \mathit{\boldsymbol{f}}\left( k \right) = \left\{ {\begin{array}{*{20}{l}} {0,}&{k < 100}\\ {1\sin \left( {0.1k} \right),}&{k \ge 100} \end{array}} \right. $

The fault estimation result is shown in Fig. 1.

Then, a fault estimation observer for Sensor 2 is designed.

Case 2  Consider the following sampled-data system in the form of System (1) with the following parameters.

$ \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - 0.277}&0&{ - 32.9}&{9.81}\\ { - 0.1033}&{ - 8.525}&{3.75}&0\\ {0.3649}&0&{ - 0.639}&0\\ 0&1&0&0 \end{array}} \right] $

$ \mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} {0.1}\\ 1\\ 0\\ {0.1} \end{array}} \right] $

$ \mathit{\boldsymbol{C}} = \left[ {\begin{array}{*{20}{l}} 0&1&0&0\\ 0&0&0&1 \end{array}} \right] $

$ \mathit{\boldsymbol{F}} = \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right] $

Choosing:

$ \mathit{\boldsymbol{S}} = \left[ {\begin{array}{*{20}{r}} { - 1}&0&0&0&0&{ - 1}&0\\ 1&{ - 1}&0&1&0&0&{ - 1}\\ 0&{ - 1}&0&1&0&0&{ - 1}\\ 0&0&{ - 1}&{ - 1}&{ - 1}&{ - 1}&{ - 1}\\ 0&{ - 1}&0&0&{ - 1}&{ - 1}&{ - 1} \end{array}} \right] $

We obtain:

$ \mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{r}} 1&{0.5}&0&0&0\\ 0&{0.5}&0&0&1\\ 0&{ - 0.5}&1&0&0\\ 0&{0.5}&0&1&{ - 1}\\ 0&0&0&{ - 1}&{ - 1} \end{array}} \right] $

$ \mathit{\boldsymbol{N}} = \left[ {\begin{array}{*{20}{r}} { - 0.5}&0\\ {0.5}&0\\ {0.5}&0\\ { - 0.5}&0\\ 0&1 \end{array}} \right] $

Let γ=0.4 and apply Theorem 1, it comes:

$ \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{r}} {1.5732}&{0.6053}\\ {0.4878}&{0.0143}\\ { - 1.1456}&{ - 0.3431}\\ {0.9857}&{0.7585}\\ { - 0.4979}&{ - 0.7441} \end{array}} \right] $

The disturbance d(t) is chosen as random noise with d(t)~N(0, 1). The following sensor fault (of Sensor 2) is considered.

$ \mathit{\boldsymbol{f}}(k) = \left\{ {\begin{array}{*{20}{c}} {0,}&{k < 100}\\ {0.5,}&{k \ge 100} \end{array}} \right. $

The fault estimation result is shown in Fig. 2.

Case 3  Finally, we consider a practical example of the tunnel diode circuit, in order to confirm the analysis and synthesis measures established in the priorsections^[23]. In the tunnel diode circuit system, x₁ (k), x₂(k), x₃(k) can be described by the following Table 1.

The practical example can be abstracted as a system which is given by the prior dynamics system(1). y(k)= C_ix(k), i=1, 2.The data we used is given in Ref.[23]. Now, we also need some more data:

$ {\mathit{\boldsymbol{D}}_{11}} = \left[ {\begin{array}{*{20}{l}} {0.3}\\ {0.5}\\ {0.6} \end{array}} \right] $

$ {\mathit{\boldsymbol{D}}_{21}} = \left[ {\begin{array}{*{20}{c}} {1.1}\\ {2.1}\\ {1.7} \end{array}} \right] $

$ {\mathit{\boldsymbol{D}}_{22}} = 1.5,{\mathit{\boldsymbol{D}}_{12}} = 0.6 $

The other parameters are obtained by Case 2. Let γ=0.9 and apply Theorem 1, it comes:

$ \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{r}} {5.5342}&{0.5435}\\ {0.9878}&{0.9843}\\ { - 0.1876}&{ - 12.2151}\\ {0.3257}&{0.6544}\\ { - 5.4349}&{ - 6.7331} \end{array}} \right] $

The disturbance d(t) is chosen as random noise with d(t)~N(0, 1). The following sensor fault (of Sensor 2) is considered.

$ \mathit{\boldsymbol{f}}\left( k \right) = \left\{ {\begin{array}{*{20}{l}} {0.7,}&{k < 100}\\ {1\cos (0.1k),}&{k \ge 100} \end{array}} \right. $

5 Conclusions

The fault estimation method for sampled-data systems with sensor faults is addressed in this paper, based on designing observers for discrete-time descriptor systems, and a novel approach of sensor fault estimation for sampled-data systems is presented. The paper has established the necessary and sufficient conditions for the convergence of the proposed observer for the discretized discrete-time descriptor systems, and expressed it in a linear matrix inequalities formulation [LMI], and the detectability of the designed system is proved as well. Then two simulation examples are used to demonstrate the effectiveness of the design procedure which can be implemented.