Fault-Reconstruction-Based Cascaded Sliding Mode Observers for Descriptor Linear Systems

Abstract

This paper develops a cascaded sliding mode observer method to reconstruct actuator faults for a class of descriptor linear systems. Based on a new canonical form, a novel design method is presented to discuss the existence conditions of the sliding mode observer. Furthermore, the proposed method is extended to general descriptor linear systems with actuator faults. Finally, the effectiveness of the proposed technique is illustrated by a simulation example.

1. Introduction

With the development and applications of modern control techniques, the safety and reliability of control systems are becoming increasingly important. Therefore, the fault diagnosis has become one of the most important techniques to ensure the safety and reliability of control systems [1, 2]. During the last two decades, many significant results have been obtained for the analysis and observer design of fault diagnosis of the regular systems, such as unknown input observers [3, 4], eigenstructure assignment method [5], 𝐻_∞ filtering [6–9], parity space approach [10], and parameter identification approach [11].

Just like regular systems, the fault diagnosis for descriptor systems has recently attracted increasing attention due to their importance in real-world systems. In [12], a parametric approach is proposed to design unknown input observers to realize fault detection of descriptor linear multivariable systems with unknown disturbances. By directly identifying parity space, a model-free approach for fault detection is developed, which can be applied if the model of descriptor systems is unknown [13]. In [14], the factorization approach for robust residual generation is extended to descriptor systems, and then a post-filter is added to ensure the robustness of fault diagnosis. In [15], 𝐻_∞ filter is utilized for providing disturbance rejection and robustness properties of the fault detection and isolation schemes of linear time-invariant descriptor systems. In [16], several sufficient conditions of existence of unknown input observers are obtained for Takagi-Sugeno descriptor systems, which are affected by unknown inputs. Unfortunately, although these methods can successfully generate residuals, they fail to reconstruct fault signals.

Recently, fault reconstruction is a promising alternative for fault detection. Instead of generating residuals, a number of methods, such as sliding mode observers (SMOs) [17–23], descriptor observer method [24–26], and PI observer [27–29], can be used to reconstruct fault signals. The sliding mode control is employed in the situations including state estimation and fault detection, since it is insensitive to matched uncertainties, nonlinearity, or disturbances [30]. Edwards et al. [17] firstly used the concept of the equivalent output error injection signals to reconstruct faults. Tan and Edwards [19] extended this work for robust reconstruction of sensor and actuator faults by minimizing the effect of uncertainty on the reconstruction in an 𝐿₂ sense. Some well-studied works, aiming at reducing the system constraints associated with the results in [17, 19], have recently appeared in the literature [18, 20–23]. In order to relax the matching conditions, the cascaded sliding mode observer method was proposed to deal with a class of systems with relative degree higher than one [20, 21]. In [22], the auxiliary outputs are defined such that the conventional sliding mode observer in [17] can be used for systems without the observer matching condition. In order to obtain those auxiliary outputs, high-order sliding-mode observers are constructed to act as exact differentiators using a super-twisting algorithm. Inspired by Floquet et al. [22], high-gain approximate differentiators and high-order sliding-mode robust differentiators were proposed to generate auxiliary outputs for the design of sliding mode observers [18, 23].

Although there are many achievements in regular systems, few results have been reported to the descriptor case despite its importance in real-world systems. In [31, 32], the sliding mode observer method was employed to detect and isolate faults and to reconstruct the faults for descriptor systems. However, the uncertainty was not considered in these results. In [33], the sliding mode observer was proposed to minimize the effect of uncertainly on the reconstruction of faults for descriptor systems. Unfortunately, the fault detection filter based sliding mode observer has to satisfy the strict condition in [31–33], which severely limits the applicability of these approaches for a wide range of practical systems.

Motivated by the above discussion, in this paper, we develop a cascaded sliding mode observer method to reconstruct actuator faults for a class of descriptor linear systems. The main contribution of this paper can be summarized as follows: (1) we present a novel cascaded sliding mode observer method to reconstruct actuator faults for a class of descriptor linear systems; (2) in the design process, we remove this restrictive assumption and extend the cascaded sliding mode observer approach of Tan et al. [20, 21] to descriptor systems; (3) a novel cascaded sliding mode observer is designed for reconstructing actuator faults for a class of descriptor linear systems.

The paper is organized as follows. In Section 2, the problem is formulated, and appropriate coordinate transformations are introduced to exploit the system structure. In Section 3 the design algorithm of cascaded sliding mode observer for linear descriptor systems is given. In Section 4, a design method of cascaded sliding mode observer and fault reconstruction for general descriptor systems are presented. In Section 5, an example is given to support the effectiveness of the proposed approach. Finally, the conclusions are drawn.

2. Problem Statement and System Analysis

Consider a descriptor linear system described by 𝐸𝑥̇=𝐴𝑥+𝐵𝑢+𝐷𝑓𝑦=𝐶𝑥, (2.1) where 𝑥∈𝑅^𝑛 is the state variable, 𝑢∈𝑅^𝑘 is the input vector, 𝑦∈𝑅^𝑝 is the output variable, and 𝑓∈𝑅^𝑞 is unknown but bounded so that 𝑓≤𝛽, (2.2) where the positive scalar 𝛽 is known. The signal 𝑓 models the actuator fault within the system. 𝐴∈𝑅^𝑚×𝑛, 𝐵∈𝑅^𝑚×𝑘, 𝐶∈𝑅^𝑝×𝑛, and 𝐷∈𝑅^𝑚×𝑞 are known constant real matrices. Without loss of generality, it is assumed that rank(𝐷)=𝑞, rank(𝐶)=𝑝, and 𝐸 is full row rank.

In [32], a sliding mode observer is given in the following form: 𝑧̇=𝐹𝑧+𝑇₁𝐵𝑢+𝐾₁𝑦+𝐾₂𝑦+𝐺_𝑛𝜐𝑥=𝑧+𝑇₂𝑦𝑦=𝐶𝑥, (2.3) where 𝑧∈𝑅𝑛 is the state vector of the SMO, 𝑥 is the estimation of the state vector 𝑥, and 𝜐 is the discontinuous output error injection vector defined by 𝑃𝜐=−𝜂⁰𝑒_𝑦𝑃₀𝑒_𝑦𝑒_𝑦≠00other, (2.4) where 𝑒_𝑦=𝑦−𝑦, 𝜂>0, 𝐹, 𝑇₁, 𝑇₂, 𝐾₁, 𝐾₂, 𝐺_𝑛, and 𝑃₀ are parameters to be designed.

For the descriptor system (2.1), the sufficient conditions for the existence of the sliding mode observer (2.3) are as follows: (𝑠)rank𝐸𝐷𝐶0=𝑛+𝑞rank𝑠𝐸−𝐴𝐷𝐶0=𝑛+𝑞,Re≥0. (2.5)(2.6)

It is well known that condition (2.5) is quite restrictive and may not apply to a wide range of systems. In the following, we give two more relaxed conditions: 𝐸𝐶rank=𝑛,rank𝐸𝐷𝐶0=𝑛+𝑙, (2.7)(2.8) where 𝑙≤𝑞.

Before presenting the main results, some lemmas are given as follows.

Lemma 2.1. If the conditions (2.7) and (2.8) hold, there exists a nonsingular matrix 𝑈 such that rank𝐸𝐷₁𝐶0=𝑛+𝑙,rank𝐸𝐷₂𝐶0=𝑛, (2.9)(2.10) where [𝐷₁𝐷₂]=𝐷𝑈, and 𝐷₁∈𝑅^𝑚×𝑙, 𝐷₂∈𝑅^{𝑚×(𝑞−𝑙)}.

Proof. if 𝑙 is equal to 𝑞, the conclusion is obviously true. So the following is to prove the case that 𝑙 is less than 𝑞.
Obviously, there exists a nonsingular matrix 𝑈₁ so that 𝐷𝑈₁=[𝐷₁𝐷₂] and (2.9) hold.
Then, rank𝐸𝐷𝐶0=rank𝐸𝐷₁𝐷₂𝐶00=𝑛+𝑙. (2.11) So there exists a matrix 𝑌=^𝑌1_𝑌2 so that 𝐷₂0=𝐸𝐷₁𝑌𝐶0𝑌¹₂. (2.12) Thus, we have 𝐷₂=𝐸𝑌₁+𝐷₁𝑌₂ and 𝐶𝑌₁=0.
Setting 𝑈₂=𝐼−𝑌₂0𝐼 (2.13) and 𝑈=𝑈₁𝑈₂, we have rank𝐸𝐷₂𝐶0=rank𝐸𝐸𝑌₁𝐶𝐶𝑌₁𝐸𝐶=rank=𝑛. (2.14)

Lemma 2.2. If the following conditions rank𝐸𝐷₁𝐶0=𝑛+𝑙,rank𝑠𝐸−𝐴𝐷₁(𝑠)𝐶0=𝑛+𝑙,Re≥0 (2.15) hold, there exist two nonsingular matrices 𝑃 and 𝑄 such that 𝑃𝐸𝑄=0𝐸₁₂𝐼_𝑛−𝑝𝐸₂₂𝐴,𝑃𝐴𝑄=¹¹𝐴₁₂𝐴₂₁𝐴₂₂𝑃𝐷₁=𝐷_𝐼0,𝐶𝑄=0𝐼_𝑝, (2.16) where 𝐸₁₂∈𝑅^{(𝑚−𝑛+𝑝)×𝑝}, 𝐸₂₂∈𝑅^{(𝑛−𝑝)×𝑝}, 𝐴₁₁∈𝑅^{(𝑚−𝑛+𝑝)×(𝑛−𝑝)}, 𝐴₁₂∈𝑅^{(𝑚−𝑛+𝑝)×𝑝}, 𝐴₂₁∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}, 𝐴₂₂∈𝑅^{(𝑛−𝑝)×𝑝}, 𝐷_𝐼=[0𝐼_𝑙]^𝑇∈𝑅^{(𝑚−𝑛+𝑝)×𝑙}, and the subblock 𝐴₁₁ has the structure 𝐴₁₁=𝐴₁₁₁𝐴₁₁₂, (2.17) in which 𝐴₁₁₁∈𝑅^{(𝑚−𝑛+𝑝−𝑙)×(𝑛−𝑝)}, 𝐴₁₁₂∈𝑅^{𝑙×(𝑛−𝑝)}, and the pair (𝐴₂₁,𝐴₁₁₁) is detectable.
It can be established easily by Lemma 2 in [33], and hence the proof is omitted.

Lemma 2.3. If the conditions (2.6), (2.7), and (2.8) hold, there exist nonsingular matrices 𝑃, 𝑄, and 𝑈 such that 𝑃𝐸𝑄=0𝐸₁₂𝐼_𝑛−𝑝𝐸₂₂𝐴,𝑃𝐴𝑄=¹¹𝐴₁₂𝐴₂₁𝐴₂₂𝐵𝑃𝐵=¹𝐵₂,𝐶𝑄=0𝐼_𝑝𝐷𝑃𝐷𝑈=¹¹00𝐷₂₂, (2.18)(2.19)(2.20) where 𝐸₁₂∈𝑅^{(𝑚−𝑛+𝑝)×𝑝}, 𝐸₂₂∈𝑅^{(𝑛−𝑝)×𝑝}, 𝐴₁₁∈𝑅^{(𝑚−𝑛+𝑝)×(𝑛−𝑝)}, 𝐴₁₂∈𝑅^{(𝑚−𝑛+𝑝)×𝑝}, 𝐴₂₁∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}, 𝐴₂₂∈𝑅^{(𝑛−𝑝)×𝑝}, 𝐵₁∈𝑅^{(𝑚−𝑛+𝑝)×𝑘}, 𝐵₂∈𝑅^{(𝑛−𝑝)×𝑘}, 𝐷₁₁=[0𝐼_𝑙]^𝑇∈𝑅^{(𝑚−𝑛+𝑝)×𝑙}, 𝐷₂₂∈𝑅^{(𝑛−𝑝)×(𝑞−𝑙)}, and the subblock 𝐴₁₁ has the structure 𝐴₁₁=𝐴₁₁₁𝐴₁₁₂, (2.21) where 𝐴₁₁₁∈𝑅^{(𝑚−𝑛+𝑝−𝑙)×(𝑛−𝑝)}, 𝐴₁₁₂∈𝑅^{𝑙×(𝑛−𝑝)}, and (𝐴₂₁,𝐴₁₁₁) is detectable.

Proof . By Lemma 2.1, there exists a nonsingular matrix 𝑈 such that (2.9) and (2.10) hold, where 𝐷𝑈=[𝐷₁𝐷₂].
Obviously, rank𝑠𝐸−𝐴𝐷₁(𝑠)𝐶0=𝑛+𝑙,Re≥0. (2.22)
By Lemma 2.2, there exist two nonsingular matrices 𝑃 and 𝑄 such that (2.18) and (2.19) hold and 𝑃𝐷₁=𝐷₁₁0. (2.23)
Setting 𝑃𝐷₂=^𝐷21_𝐷22, we have rank𝐸𝐷₂𝐶0=rank𝑃00𝐼𝐸𝐷₂𝐶0𝑄00𝐼=rank0𝐸₁₂𝐷₂₁𝐼_𝑛−𝑝𝐸₂₂𝐷₂₂0𝐼_𝑝0=rank𝐷₂₁+𝑛. (2.24)
Combining (2.10) and (2.24), we have rank(𝐷₂₁)=0. Obviously, 𝐷₂₁=0.

By Lemma 2.3, it can be assumed without loss of generality that system (2.1) has the following form: 0𝐸₁₂𝐼_𝑛−𝑝𝐸₂₂𝑥̇₁𝑥̇₂=𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂𝑥₁𝑥₂+𝐵₁𝐵₂𝑢+𝐷₁₁0𝑓₁+0𝐷₂₂𝑓₂𝑦=𝑥₂, (2.25) where 𝑥=[𝑥^𝑇₁𝑥^𝑇₂]^𝑇,𝑥₁∈𝑅^𝑛−𝑝, 𝑥₂∈𝑅^𝑝 and 𝑓→𝑈𝑓=𝑓^𝑇₁𝑓^𝑇₂^𝑇. (2.26)

The descriptor system (2.25) may be considered as the system with the fault 𝑓₁ and the disturbance 𝑓₂. Using the fault reconstruction method in [33], the fault 𝑓₁ can be reconstructed and the 𝐿₂ gain from the 𝑓₂ to reconstruction error of fault 𝑓₁ can be minimized. But the fault 𝑓₂ and the state 𝑥₁ cannot be estimated. Inspired by Tan et al. [20, 21], the cascaded sliding mode observer is applied to estimate both the state 𝑥 and fault 𝑓 in the following.

3. Design of Cascaded Sliding Mode Observer

The primary sliding mode observer for system (2.25) is 𝑧̇=𝑇₁𝐴−𝐾₁𝐶𝑧+𝑇₁𝐵𝑢+𝐾₁𝑦+𝐹𝑇₂𝑦+𝐺_𝑛𝜐𝑥=𝑧+𝑇₂𝑦𝑦=𝑥₂, (3.1) where 𝑧∈𝑅^𝑛 is the state vector of the SMO, 𝑥=[𝑥^𝑇₁𝑥^𝑇₂]^𝑇 with 𝑥₁∈𝑅^𝑛−𝑝 and 𝑥₂∈𝑅^𝑝 is the estimation of the state vector 𝑥, 𝐺_𝑛=[0𝐼]^𝑇,𝑇₁ and 𝑇₂ are defined by 𝑇₁=𝑍₁𝐼_𝑛−𝑝𝑍₂0𝑇₂=0𝐼_𝑝−𝑇₁𝐸₁₂𝐸₂₂𝑍₁=𝑍₁₁0, (3.2)(3.3)(3.4)𝑍₁∈𝑅^{(𝑛−𝑝)×(𝑚−𝑛+𝑝)}, 𝑍₁₁∈𝑅^{(𝑛−𝑝)×(𝑚−𝑛+𝑝−𝑞)}, 𝑍₂∈𝑅^{𝑝×(𝑚−𝑛+𝑝)} is full rank, 𝜐 is the discontinuous output error injection vector defined by: 𝑃𝜐=−𝜂²𝑒₂𝑃₂𝑒₂𝑒₂≠00other, (3.5)𝑒₂=𝑥₂−𝑥₂, 𝜂>0, 𝑍₁₁, 𝑍₂, 𝐾₁, 𝐹, and 𝑃₂ are parameters to be designed.

In [33], it is shown that for an appropriate choice of observer parameters an ideal sliding motion takes place on 𝑆={(𝑒₁,𝑒₂)∣𝑒₂=0} in finite time.

Define 𝑒=𝑥−𝑥 as the state estimation error, the following estimation error dynamic is obtained: 𝑒₁̇=𝐴₂₁+𝑍₁𝐴₁₁𝑒₁+𝑍₁𝐴₁₂+𝐴₂₂−𝐾₁₁𝑒₂+𝐷₂₂𝑓₂𝑒₂̇=𝑍₂𝐴₁₁𝑒₁+𝑍₂𝐴₁₂−𝐾₁₂𝑒₂−𝑍₂𝐷₁₁𝑓₁+𝜐, (3.6) where 𝑒=[𝑒^𝑇₁𝑒^𝑇₂]^𝑇,𝑒₁∈𝑅^𝑛−𝑝, 𝐾₁=[𝐾^𝑇₁₁𝐾^𝑇₁₂]^𝑇, 𝐾₁₁∈𝑅^{(𝑛−𝑝)×𝑝}, and 𝐾₁₂∈𝑅^𝑝×𝑝.

Assuming the primary sliding mode observer has been designed, and that a sliding motion has been achieved, then 𝑒₂=𝑒₂̇=0, and the error equation becomes 𝑒₁̇=𝐴₂₁+𝑍₁𝐴₁₁𝑒₁+𝐷₂₂𝑓₂𝑍⁺₂𝜐_eq=−𝐴₁₁𝑒₁+𝐷₁₁𝑓₁, (3.7) where 𝑍⁺₂ is the generalized inverse matrix of 𝑍₂, 𝜐_eq is the equivalent output error injection term that can be approximated to any degree of accuracy by replacing (3.8) with 𝜐_eq𝑃=−𝜂²𝑒₂𝑃₂𝑒₂,+𝛿 (3.8) where 𝛿 is a small positive constant.

The remaining system freedom can be used to estimate the state 𝑥₁ and reconstruct the fault 𝑓₂. Equation (3.7) can be rewritten as 𝑒₁̇=𝐴₂₁+𝑍₁𝐴₁₁𝑒₁+𝐷₂₂𝑓₂𝐷^⟂₁₁𝑍⁺₂𝜐_eq=−𝐴₁₁₁𝑒₁, (3.9)(3.10) where 𝐷^⟂₁₁=[^{𝐼𝑚+𝑝−𝑛−𝑙0}].

For any 𝐴₁₁₁, there exists a nonsingular matrix 𝑊 so that 𝑊𝐴₁₁₁=𝐴^𝑇₁₁₁₀^𝑇, where 𝐴₁₁₁∈𝑅𝑝_{×(𝑛−𝑝)} is full row rank. We have 𝐼𝑝0𝑊𝐷^⟂₁₁𝑍⁺₂𝜐_eq=−𝐴₁₁₁𝑒₁. (3.11)

The system (3.9) and (3.11) may be considered as the linear system with the 𝑞−𝑙 faults, the 𝑛−𝑝 states 𝑓₁ and the 𝑝 outputs. Using the sliding mode observer design method for the linear system in [17], we can design a secondary sliding mode observer to estimate 𝑒₁ and 𝑓₂ if the following conditions hold: rank𝐴₁₁₁𝐷₂₂=rank𝐷₂₂𝐴rank𝑠𝐼−₂₁+𝑍₁𝐴₁₁𝐷₂₂𝐴₁₁₁0=𝑛−𝑝+𝑞−𝑙,Re(𝑠)≥0. (3.12) Obviously, (3.12) are equivalent to rank𝐴₁₁₁𝐷₂₂=rank𝐷₂₂𝐴rank𝑠𝐼−₂₁+𝑍₁𝐴₁₁𝐷₂₂𝐴₁₁₁0=𝑛−𝑝+𝑞−𝑙,Re(𝑠)≥0. (3.13)(3.14) Combined with (3.4), (3.14) is equivalent to rank𝑠𝐼−𝐴₂₁𝐷₂₂𝐴₁₁₁0=𝑛−𝑝+𝑞−𝑙,Re(𝑠)≥0. (3.15)

From the above analysis, if the conditions (3.13) and (3.15) satisfy, there exists a cascaded sliding mode observer for the descriptor system (2.1).

Next, the fault reconstruction method based cascaded sliding mode observer is given.

Assuming that the secondary sliding mode observer has been designed and the 𝑒₁̂ and 𝑓₂ are the estimations of 𝑒₁ and 𝑓₂, respectively. Then, the reconstruction signal of the fault 𝑓₁ is described by 𝑓₁=𝐴₁₁₂𝑒₁̂+0𝐼_𝑙𝑍⁺₂𝜐_eq, (3.16) and the estimation of the state 𝑥₁ is described by 𝑥₁−𝑒₁̂→𝑥₁. (3.17)

The reconstruction of fault is described by 𝑓=𝑈⁻¹𝑓^𝑇₁𝑓^𝑇₂^𝑇 (3.18)

Equations (3.13) and (3.15) are the sufficient conditions for the existence of the cascaded sliding mode observer, but these cannot be checked using the parameters of the original system (2.1). Now, for system (2.1), sufficient conditions for the existence of the cascaded sliding mode observer can be given by Theorem 3.1.

Theorem 3.1. There exists a cascaded sliding mode observer for system (2.1) if the following conditions hold: 𝐸𝐶(𝑠)rank=𝑛rank𝐸𝐴𝐷00𝐸0𝐷𝐶0000𝐶00=𝑛+𝑞+rank𝐸𝐷𝐶0,rank𝑠𝐸−𝐴𝐷𝐶0=𝑛+𝑞,Re≥0. (3.19)(3.20)(3.21)

Proof. If 𝑙 is equal to 𝑞, the conclusion is obviously true. So, the following is to prove the case that 𝑙 is less than 𝑞.
Substituting (2.8) and (2.25) into (3.20), we have rank𝐸𝐴𝐷00𝐸0𝐷𝐶0000𝐶00=rank0𝐸₁₂𝐴₁₁𝐴₁₂𝐷₁₁𝐼000_𝑛−𝑝𝐸₂₂𝐴₂₁𝐴₂₂0𝐷₂₂00000𝐸₁₂00𝐷₁₁000𝐼_𝑛−𝑝𝐸₁₂000𝐷₂₂0𝐼_𝑝000000000𝐼_𝑝𝐴0000=𝑛+𝑝+𝑙+rank¹¹𝐷₁₁0𝐼_𝑛−𝑝0𝐷₂₂𝐴=𝑛+𝑝+𝑙+rank𝐴¹¹¹00₁₁₂𝐼_𝑙0𝐼_𝑛−𝑝0𝐷₂₂𝐼=𝑛+𝑝+2𝑙+rank^𝑛−𝑝𝐷₂₂𝐴₁₁₁0=2𝑛+𝑞+𝑙. (3.22)
So we have 𝐼rank^𝑛−𝑝𝐷₂₂𝐴₁₁₁0=𝑛−𝑝+𝑞−𝑙. (3.23)
By Lemma 1 in [32], (3.13) holds.
Substituting (2.18), (2.19), and (2.20) into (3.21), we can obtain (3.15).

4. Cascaded Sliding Mode Observer Design and Fault Reconstruction for General Descriptor Systems

In Section 2, it is assumed that 𝐸 is full row rank. In the following, it is discussed that 𝐸 is rank deficient. Let 𝑟∶=rank(𝐸)≤min{𝑚,𝑛}.

Now, since rank(𝐸)=𝑟, there exists a regular matrix 𝑃^∗ such that (2.1) is restricted system equivalent to 𝐸^∗𝑥̇=𝐴^∗𝑥+𝐵^∗𝑢+𝐷^∗𝑓𝑦₁=−𝐵₁𝑢=𝐴₁𝑥+𝐷₁𝑓𝑦=𝐶𝑥, (4.1) where 𝑃^∗𝐸=^𝐸∗₀,𝑃^∗𝐴=^𝐴∗_𝐴1, 𝑃^∗𝐵=^𝐵∗_𝐵1,𝑃^∗𝐷=^𝐷∗_𝐷1,𝐸^∗∈𝑅^𝑟×𝑛, 𝐴^∗∈𝑅^𝑟×𝑛, 𝐵^∗∈𝑅^𝑟×𝑘, 𝐷^∗∈𝑅^𝑟×𝑝, 𝐴₁∈𝑅^{(𝑚−𝑟)×𝑛},𝐵₁∈𝑅^{(𝑚−𝑟)×𝑘}, and 𝐷₁∈𝑅^{(𝑚−𝑟)×𝑞}.

First passing the output 𝑦₁ through a nonsingular matrix 𝑃_∗ so that 𝑃_∗𝑦₁=𝑦₁₁(𝑡)=𝐴₁₁𝑥𝑦₁₂(𝑡)=𝐴₁₂𝑥+𝐷₁𝑓, (4.2) where 𝐷₁∈𝑅^𝑝×𝑞 is row full rank.

Consider a new state 𝑥_𝑓∈𝑅^𝑝 which is a filtered version of 𝑦₁₂ satisfying 𝑥̇_𝑓=−𝐴_𝑓𝑥_𝑓+𝐴_𝑓𝑦₁₂, (4.3) where −𝐴_𝑓∈𝑅^𝑝×𝑝 is a stable (filter) matrix.

Equations (4.1), (4.2), and (4.3) can be combined to form an augmented state-space system with order 𝑛 as follows: 𝐸_𝑎𝑥̇=𝐴_𝑎𝑥+𝐵_𝑎𝑢+𝐷_𝑎𝑓𝑦_𝑎=𝐶_𝑎𝑥, (4.4) where 𝐸_𝑎=^𝐸∗0_0𝐼, 𝐴_𝑎=^𝐴∗0_{𝐴𝑓𝐴12−𝐴𝑓}, 𝐵_𝑎=^𝐵∗₀, 𝐷_𝑎=^𝐷∗_{𝐴𝑓𝐷1}, 𝐶_𝑎=^𝐴110_𝐶00𝐼, 𝑥=[𝑥^𝑇𝑥^𝑇_𝑓]^𝑇, 𝑦_𝑎=[𝑦^𝑇₁₁𝑦^𝑇𝑥^𝑇_𝑓]^𝑇, 𝑛=𝑛+𝑝, and 𝑝=𝑝+𝑚−𝑟.

Obviously, the matrix 𝐸_𝑎 is full row rank so that the cascaded sliding mode observer can be designed using the method in Section 3.

In the following, the existence conditions of the cascaded sliding mode observer for general descriptor systems are given by Theorem 4.1.

Theorem 4.1. There exists a cascaded sliding mode observer for system (2.1) with rank-deficient 𝐸 if the following conditions hold: (𝑠)rank𝐸𝐴𝐷0𝐶0=𝑚+𝑝rank𝐸𝐴𝐷0𝐸00𝐶0=𝑛+rank𝐸𝐷rank𝐸𝐴0𝐷000𝐸𝐴0𝐷000𝐸00𝐷0𝐶000000𝐶000=𝑛+𝑞+rank𝐸𝐴𝐷00𝐸0𝐷0𝐶00,rank𝑠𝐸−𝐴𝐷𝐶0=𝑛+𝑞,Re≥0. (4.5)(4.6)(4.7)(4.8)

Proof. Define a nonsingular matrix as follows: 𝑃₁=diag𝐼00𝑃_∗𝑃^∗,𝐼00𝑃_∗𝑃^∗,𝐼. (4.9)
We have rank𝑃^∗𝐸𝐸𝐷=rank^∗𝐷^∗0𝐷₁=𝑟+𝑝rank𝐸𝐴𝐷0𝐸00𝐶0=rank𝑃₁𝐴𝐸𝐴𝐷0𝐸00𝐶0=rank¹¹00𝐼_𝑝𝐸^∗0𝐸𝐶0+𝑟,rank^𝑎𝐶_𝑎𝐸=rank^∗00𝐼_𝑝𝐴₁₁0𝐶00𝐼_𝑝𝐴=rank¹¹00𝐼_𝑝𝐸^∗0𝐶0. (4.10)(4.11)(4.12)
Combining (4.6), (4.10), (4.11), and (4.12), we have 𝐸rank^𝑎𝐶_𝑎=𝑛. (4.13)
Define a nonsingular matrix as follows: 𝑃₂=diag𝐼00𝑃_∗𝑃^∗,𝐼00𝑃_∗𝑃^∗,𝐼00𝑃_∗𝑃^∗,𝐼,𝐼. (4.14) We have rank𝑃₂𝐸𝐴0𝐷000𝐸𝐴0𝐷000𝐸00𝐷0𝐶000000𝐶000=𝑟+𝐸𝑝+rank^∗𝐴^∗𝐷^∗00𝐴₁₂𝐷₁00𝐸^∗0𝐷^∗000𝐷₁𝐴𝐶000₁₁0000𝐶000𝐴₁₁𝐸00rank^𝑎𝐴_𝑎𝐷_𝑎00𝐸_𝑎0𝐷_𝑎𝐶_𝑎0000𝐶_𝑎00=2𝐸𝑝+rank^∗𝐴^∗𝐷^∗00𝐴₁₂𝐷₁00𝐸^∗0𝐷^∗000𝐷₁𝐴₁₁000𝐶0000𝐴₁₁000𝐶00,rank𝑃₁𝐸𝐴𝐷00𝐸0𝐷0𝐶00=rank𝐸𝐴𝐷00𝐴₁₁000𝐴₁₂𝐷₁00𝐸0𝐷000𝐷₁0𝐶^∗𝐴00=rank¹²𝐷₁0𝐸0𝐷00𝐷₁𝐴₁₁𝐶00_∗𝐸00+𝑟=rank^𝑎𝐷_𝑎𝐶_𝑎0+𝑟. (4.15)(4.16)(4.17) Combining (4.7), (4.15), (4.16), and (4.17), we have 𝐸rank^𝑎𝐴_𝑎𝐷_𝑎00𝐸_𝑎0𝐷_𝑎𝐶_𝑎0000𝐶_𝑎=𝐸00𝑛+𝑞+rank^𝑎𝐷_𝑎𝐶_𝑎0. (4.18)
Define a nonsingular matrix as follows: 𝑃₃=diag𝐼00𝑃_∗𝑃^∗,𝐼. (4.19)
We have rank𝑠𝐸^∗−𝐴^∗𝐷^∗𝐶^∗0=rank𝑃₃𝑠𝐸^∗−𝐴^∗𝐷^∗𝐶0=rank𝑠𝐸−𝐴𝐷−𝐴₁₁0−𝐴₁₂𝐷₁𝐶^∗0=𝑛+𝑞; (4.20) thus, rank𝑠𝐸^∗−𝐴^∗0𝐷^∗−𝐴₁₁00−𝐴₁₂0𝐷₁𝐶^∗000𝐼_𝑝0=𝑛+𝑞+𝑝. (4.21)
Define 𝑃₄=𝐼000000𝐴_𝑓0𝑠+𝐴_𝑓0𝐼000000𝐼00000𝐼, (4.22)
Then, rank𝑠𝐸^∗−𝐴^∗0𝐷^∗−𝐴₁₁00−𝐴₁₂0𝐷₁𝐶^∗000𝐼_𝑝0=rank𝑃₄𝑠𝐸^∗−𝐴^∗0𝐷^∗−𝐴₁₁00−𝐴_𝑓𝐴₁₂0𝐴_𝑓𝐷₁𝐶^∗000𝐼_𝑝0=rank𝑠𝐸^∗−𝐴^∗0𝐷^∗−𝐴_𝑓𝐴₁₂𝑠+𝐴_𝑓𝐴_𝑓𝐷₁−𝐴₁₁𝐶00_∗000𝐼_𝑝0=rank𝑠𝐸_𝑎−𝐴_𝑎𝐷_𝑎𝐶_𝑎0. (4.23)
Hence, rank𝑠𝐸_𝑎−𝐴_𝑎𝐷_𝑎𝐶_𝑎0=𝑛+𝑞,Re(𝑠)≥0. (4.24)
Combining (4.12), (4.16), and (4.24), we get the conclusion by Theorem 3.1.

Corollary 4.2. There exists a cascaded sliding mode observer for linear system if the following conditions hold: ()(𝐷)(𝑠)rank𝐶𝐴𝐷𝐶𝐷𝐶𝐷0=rank𝐶𝐷+rank,rank𝑠𝐼−𝐴𝐷𝐶0=𝑛+𝑞,Re≥0. (4.25)(4.26)

Proof. Obviously, since 𝐸=𝐼 for linear systems, (4.5), (4.6), and (4.8) hold. Hence, the following is to prove that (4.7) holds.
In [21], the canonical form of the linear system is given as follows: 𝐴𝐴=¹𝐴₂𝐴₃𝐴₄𝐷,𝐶=0𝑇,𝐷=¹00𝐷₂,𝐴₃=𝐴_3𝑎𝐴_3𝑏𝐴_3𝑐𝐴_3𝑑𝐷₁=𝐷₁₁0,𝐷₂=0𝐷₂₂, (4.27) where 𝐴₁∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}, 𝐴_3𝑎∈𝑅^{(𝑝−𝑙)×(𝑞−𝑙)}, 𝑇∈𝑅^𝑝×𝑝 is orthogonal, 𝐷₁∈𝑅^{(𝑛−𝑝)×𝑙}, 𝐷₁₁∈𝑅^{(𝑞−𝑙)×(𝑞−𝑙)}, and 𝐷₂₂∈𝑅^𝑙×𝑙 are invertible.
Substituting (4.27) into (4.25), we obtain rank𝐴𝑇00𝑇𝐶𝐴𝐷𝐶𝐷𝐶𝐷0=rank_3𝑎𝐷₁₁+2𝑙. (4.28)
Combining (4.25) and (4.28), we obtain rank𝐴_3𝑎=𝑞−𝑙. (4.29)
Substituting (4.27) into (4.7), we obtain 𝐼rank𝐸𝐴0𝐷000𝐸𝐴0𝐷000𝐸00𝐷0𝐶000000𝐶000=𝑛+rank^𝑛−𝑝0𝐴₁𝐴₂𝐷₁0000𝐼_𝑝𝐴₃𝐴₄0𝐷₂0000𝐼_𝑛−𝑝000𝐷₁0000𝐼_𝑝000𝐷₂0𝐼_𝑝000000000𝐼_𝑝𝐴0000=2𝑛+𝑝+rank³𝐷₂𝐼00_𝑛−𝑝0𝐷₁0000𝐷₂=3𝑛−𝑞+𝑙+2rank𝐷₂₂+rank𝐴_3𝑎+rank𝐷₁₁𝐼=3𝑛+𝑙+𝑞,rank𝐸𝐴𝐷00𝐸0𝐷0𝐶00=rank^𝑛−𝑝0𝐷₁00𝐼_𝑝0𝐷₂0𝐼_𝑝00+𝑛=2𝑛+𝑙. (4.30) Obviously, (4.7) holds.

Remark 4.3. For the linear system, the rank conditions (4.25) and (4.26) are identical to the ones in [21], it is obvious that the conclusion of the paper is more general compared with [21].

5. Simulation

A machine infinite bus system linear model is described as follows [29]: 𝑥̇₁=𝑥₄𝑥̇₂=𝑥₅𝑥̇₃=𝑥₆𝑥̇₄=1𝑀₁𝑢₁−𝑌₁₂𝑉₁𝑉₂𝑥₁−𝑥₂−𝑌₁₅𝑉₁𝑉₅𝑥₁−𝑥₇−𝐷₂𝑥₄𝑥̇₅=1𝑀₂𝑢₂−𝑌₂₁𝑉₁𝑉₂𝑥₂−𝑥₁−𝑌₂₅𝑉₂𝑉₅𝑥₂−𝑥₇−𝐷₂𝑥₅𝑥̇₆=1𝑀₃𝑢₃−𝑌₃₄𝑉₃𝑉₄𝑥₃−𝑌₃₅𝑉₃𝑉₅𝑥₃−𝑥₇−𝐷₃𝑥₆0=𝑃_𝑐ℎ−𝑌₅₁𝑉₅𝑉₁𝑥₇−𝑥₁−𝑌₅₂𝑉₅𝑉₂𝑥₇−𝑥₂−𝑌₅₃𝑉₅𝑉₃𝑥₇−𝑥₃−𝑌₅₄𝑉₅𝑉₄𝑥₇, (5.1) where 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₇ are the generator angles, 𝑥₄, 𝑥₅, and 𝑥₆ are the generator speeds. 𝑢₁, 𝑢₂, and 𝑢₃ are the mechanical power, 𝑃_𝑐ℎ is unknown load, the nominal values of inertia 𝑀₁, 𝑀₂ and 𝑀₃, of damping 𝐷₁, 𝐷₂, and 𝐷₃, of admittance 𝑌₁₅, 𝑌₂₅, 𝑌₃₅, 𝑌₅₁, 𝑌₅₂, 𝑌₅₃, and 𝑌₅₄ and of potential 𝑉₁, 𝑉₂, 𝑉₃, 𝑉₄, and 𝑉₅ are shown in 𝑀₁=0.014𝑀₂=0.026𝑀₃=0.02𝐷₁=0.057𝐷₂𝐷=0.15₃=0.11𝑌₁₅=0.5𝑌₂₅=1.2𝑌₃₄=0.7𝑌₄₅𝑌=1₃₄=0.7𝑌₁₂=1𝑉_𝑖=1(𝑖=1-5). (5.2)

It is assumed that the available measurements are the generator angles 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₇. In order to illustrate the effectiveness of the design method, it is assumed that there exist faults on the actuator  𝑢₁–𝑢₃. It is easy to verify that the existence conditions of sliding mode observer in [32] do not hold, but the existence conditions of cascaded sliding mode observer hold.

In the following simulation, the cascaded sliding mode observer in Section 3 is designed to reconstruct the actuator faults.

Considering system (5.1) affected by the inputs 𝑢₁=1, 𝑢₂=1, and 𝑢₃=2+sin(5𝑡), the unknown load 𝑃_𝑐ℎ=sin(𝑡) and an uncertain admittance 𝑌_𝑖𝑗=𝑌_𝑖𝑗+Δ𝑌_𝑖𝑗, (5.3) where Δ𝑌_𝑖𝑗=𝛿_𝑖𝑗sin(𝜔_𝑖𝑗𝑡), |𝛿_𝑖𝑗|<0.1, |𝜔_𝑖𝑗|<1𝑟𝑑/𝑠,𝑖=1,…,5,𝑗=1,…,5.

Figures 1, 2, and 3 show faults and reconstruction signals. Although there exists unknown input and parameter uncertainty in the system, the cascaded sliding mode observer faithfully reconstructs the faults.

Figure 1 

Fault signal 𝑓1 and its reconstruction signal 𝑓1.

Figure 2 

Fault signal 𝑓2 and its reconstruction signal 𝑓2.

Figure 3 

Fault signal 𝑓3 and its reconstruction signal 𝑓3.

6. Conclusions and Future Works

This paper proposes a fault reconstruction method for a class of descriptor systems using cascaded sliding mode observer. The method can effectively relax the restrictions on the existence of a sliding mode observer, which allows the applicability of our proposed method to a wider range of systems. In our future work, the proposed actuator fault reconstruction schemes can be extended to some sensor fault reconstruction problems by using a suitable output filtering technique. Another interesting future research topic is to extend the current results to fault estimation of nonlinear systems based on T-S fuzzy models [34–36].

Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities under Grant no. HIT.NSRIF.2012031.