# Estimation and Fault Diagnosis of Lithium-Ion Batteries: A Fractional-Order System Approach

Authors: Shulan Kong; Mehrdad Saif; Guozeng Cui
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This study investigates estimation and fault diagnosis of fractional-order Lithium-ion battery system. Two simple and common types of observers are designed to address the design of fault diagnosis and estimation for the fractional-order systems. Fractional-order Luenberger observers are employed to generate residuals which are then used to investigate the feasibility of model based fault detection and isolation. Once a fault is detected and isolated, a fractional-order sliding mode observer is constructed to provide an estimate of the isolated fault. The paper presents some theoretical results for designing stable observers and fault estimators. In particular, the notion of stability in the sense of Mittag-Leffler is first introduced to discuss the state estimation error dynamics. Overall, the design of the Luenberger observer as well as the sliding mode observer can accomplish fault detection, fault isolation, and estimation. The effectiveness of the proposed strategy on a three-cell battery string system is demonstrated.


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## Body

## 1. Introduction

Lithium-ion (Li-ion) batteries are an integral part of electrified vehicles and have found many other areas of applications in consumer products, space, marine, and other applications. They however remain a costly and safety critical subsystem in many areas that they are used in. As a result, reliability, safety, and efficient operation of Li-ion batteries in high power applications, such as in electrified vehicles, and challenging problems such as modeling, state estimation, monitoring, diagnostics, and pyrognostics capabilities are critical areas for research [1–8].With the development of natural science and complex engineering applications, it has been found that many real-world physical systems show fractional-order dynamical behavior. And fractional calculus theories have been increasingly applied to many practical systems of a variety of scientific and engineering fields. The first application of the fractional calculus was accomplished by Abel in 1823 while investigating the solution of the famous problem of tautochrone [9]. For almost two hundred years, fractional-order models have been used to describe biology systems [10] in bioengineering, financial markets [11–13] in economics, and diffusion wave [14–17] and lead-acid battery or Li-ion battery [18–21] in physical engineering. Due to inherent properties of fractional-order calculus, fractional-order differential systems sometimes have a more appropriate mathematical description than integer-order differential systems for managing lithium-ion batteries. In integer-order models of Li-ion batteries, constant phase elements are either approximated by ideal capacitors, or a number of resistor-capacitor networks, or relaxation times [21]. The resulting models are commonly not capable of predicting battery dynamics in both the time and frequency domains over the entire operating range. They can not capture the key and accuracy battery characteristics. To address the above problems, a fractional-order model of replacing the ideal capacitor in the first-order resistor-capacitor networks model to a fractional element is explored for Li-ion cells. By using experimental data from time and frequency domains, Alavi et al. [22] found that this model can reproduce a Li-ion battery’s behavior better than its integer counterpoint, thanks to an additional degree of freedom, namely, the fractionation order. A fractional-order state space model of lithium-ion battery was proposed and the experiment results showed that it has a better fitness than the classical equivalent circuit models based on the integer differential equations in [20]. In addition, a comprehensive review of fractional-order techniques for Li-ion batteries was referred to [21]. Consequently, the fractional-order modeling methodology may not only improve prediction accuracy but also preserve some physical meanings underlying model parameters.Catastrophic failures and/or explosion of Li-ion batteries in various applications ranging from cell phones to aircrafts and automobiles are all familiar new stories of the past few years. It is therefore imperative that, at the onset of failures, such events are detected and preventive measures to be taken [23–25]. Model based fault detection and isolation strategies could prove invaluable in this direction. For such techniques to be effective, availability of accurate representative model of the system is necessary. Accordingly, significant progress towards fault diagnosis and estimation of Li-ion batteries modeled as integer-order differential systems have recently been made in [1–7, 26–28] and the references therein. In particular a synthesized design of reduced-order Luenberger observers and Learning observers was presented for the purpose of simultaneous fault isolation and estimation of a three-cell battery string in [1]. As discussed in the previous paragraph, Li-ion battery systems do exhibit the fractional phenomena. The fractional characteristic could give a better description of the mathematical model of the battery system and can potentially lead to more effective fault detection and estimation strategies. Inspired by the abovementioned works and observation, a fractional-order system approach will be considered for estimation and fault diagnosis of fractional-order Li-ion battery system instead of the integer-order system [1].Observer-based fault diagnosis of fractional-order systems remains problematic due to the lack of well-established and reliable techniques even though the design of observers for fractional-order systems has been reported [29–33]. However, little progress on fault detection and estimation for fractional-order models has been made besides [34, 35]. Reference [35] aims to provide sufficient conditions of the asymptotical convergence for the fractional-order state estimation errors and fault estimation errors via a frequency-distributed fractional integrator equivalent model [36] and employs an indirect Lyapunov method [30, 37, 38]. Recently observer-based tracking control of fractional-order systems is addressed by using Mittag-Leffler function based Lyapunov methods to prove the boundedness of state estimation error in [33]. As such, it is of practical and theoretical importance to focus studies on observer-based fault diagnosis of Li-ion batteries expressed by the fractional-order differential systems.The main goal of this paper is to propose a strategy for fault detection, isolation, and estimation via fractional-order differential models. The goal is not only in detecting, and if possible isolating and estimating the fault in Li-ion battery system, but also in theoretical development for designing stable observers and estimating faults. Firstly, sources of faults and uncertainties are separated by reorganizing the original system. Towards this, the system is divided into two subsystems. This will allow us to give detectability conditions under which a stable fractional-order observer exists for the second subsystem. A fault-detection residual is then generated by constructing a fractional-order Luenberger observer. The residual is used to characterize the feasibility of fault detection and isolation. Once a fault occurs in the system, it is possible to detect and isolate it. After that, we attempt to further explore the isolated fault. In this step, a fractional-order sliding-mode observer is designed to provide an estimate of the isolated fault for the first subsystem containing the isolated fault as well as the systems uncertainties. Mittag- Leffler stability of fractional-order estimation error system is defined and the estimation error is proved to be Mittag-Leffler stable. As a result, the synthesized design of the fractional-order Luenberger observers and the fractional-order sliding-mode observer can lead to simultaneous fault isolation and estimation.The main contribution of the paper is twofold. On one hand, two simple and common types of fractional-order observer-based strategies are derived to satisfactorily accomplish the task of fault detection, isolation, and estimation for the fractional-order Li-ion battery system inspired by the ideas proposed in [1, 39, 40]. On the other hand, Mittag-Leffler stability of fractional-order estimation error system is defined and some other theoretical results are then presented. It is believed that the obtained results can perfectly embody accuracy and practicability properties of fractional calculus describing a real process in physical systems.The rest of the paper is organized as follows. In Section2, a fractional-order system based on a Li-ion battery model is formulated to achieve fault diagnosis and estimation and to answer the listed three questions. A strategy for fault detection and isolation is presented in Section 3 and fault estimation strategy is given in Section 4. In Section 5 the proposed strategy is applied to a three-cell battery string system to show the effectiveness of the proposed approach. Finally, some conclusions are drawn in Section 6.


## 2. Problem Formulation

Inspired by the idea of [20], we are ready to present a fractional-order model for a general system in [1]. Consider the following fractional-order pseudostate space description(1)Dαxt=Axt+But+Fft+Gdt,0<α<1,yt=Cxtwhere Dα denotes the Caputo fractional derivative of order α on [0,t], α is the fractional commensurate order, x∈Rn is the pseudostate vector, u∈Rm is the input vector, y∈Rp is the measurable output vector, f(t) is a bounded s-dimensional fault vector, d(t) represents bounded uncertainties/disturbances with dimension of q, and A,B,F,G, and C are n×n, n×m, n×s, n×q, and p×n constant matrices, respectively.DefineF=F1F2⋯Fs and G=G1G2⋯Gq for any i∈{1,2,…,s} and j∈{1,2,…,q}, and Fi and Gj are the i-th and j-th column of F and G. Denote also Gi′ as a matrix consisting of Fi and all columns of G, that is, Gi′=FiG1G2⋯Gq for any i∈{1,2,…,s}.For the above system, the following assumptions are made.Assumption 1.
At most a fault occurs at one time in system (1).Assumption 2.
MatrixG is of full column rank, rank(C)=p, and rank(CGi′)=rank(Gi′)=q+1 where rank(·) denotes the rank of a matrix.Assumption 3.
For eachi=1,2,…,s and for every complex number σ,(2)rankσIn-AGi′C0=n+q+1with |arg⁡(σ)|≤απ/2, In is the n×n identity matrix, and arg⁡(σ) stands for the argument of σ.Remark 4.
Assumption2 implies that Gi′ is of full column rank and q+1≤p. Then F has no zero columns.The purpose of this paper is to study the fault diagnosis problem. The fault diagnosis problem addressed here is to detect, isolate, and estimate the fault. A systematic study is performed to answer the three following questions under Assumptions1–3:(1) How can a fault be detected or what are the possible conditions for detecting a fault when it occurs?(2) How can the detected fault be isolated?(3) How can the isolated fault be dealt with or estimated?


## 3. Fault Detection and Isolation

In the section, we will reorganize system (1) and develop an approach to fault diagnosis. To present the conditions for fault detection and isolation, a particular system structure is presented, which is the structure to be built on the following results.Lemma 5.
R a n k ( C G i ′ ) = q + 1 if and only if there exist nonsingular matrices Ti and Si such that(3)Ti-1Gi′=Gi10,Si-1CTi=Ci1100Ci22,where Gi1 and Ci11 are invertible, rank(Gi1)=rank(Ci11)=q+1.Proof.
Based on Lemma 1 presented in [41], it is obtained that rank(CGi′)=rank(Gi′) if and only if there are nonsingular matrices Ti and Si such that(4)Ti-1Gi′=Gi10,Si-1CTi=Ci1100Ci22,where Gi1 and Ci11 have the same number of rows, Gi1 is of full row rank, and Ci11 is invertible.
Considering Assumption2 and(5)rankGi′=rankTi-1Gi′=rankGi1T0T=rankGi1,where T means the transpose of a matrix, we can obtain that Gi1 is of full column rank and rank(Gi1)=q+1. It is worth noting that Gi1 is also of full row rank, and then Gi1 is invertible. Because Ci11 is invertible, and Gi1 and Ci11 have the same number of rows, then rank(Ci11)=q+1.Remark 6.
Lemma5 is equivalent to the existence of state and output transformations. In order to effectively isolate every fault, the original system (1) can be reorganized using Lemma 5.


### 3.1. Bank of Reformulated Systems

This subsection deals with transformation of the system into a proper form containing two subsystems. Definef(t)=f1(t)f2(t)⋯fs(t)T, F~i is defined as the matrix after removing the column Fi from F, and f-i(t) is the matrix after deleting the fi(t) from f(t). Then, system (1) is reformulated as a bank of s systems as follows:(6)Dαxt=Axt+But+F~if-it+Fifit+Gdt,yt=Cxt,i=1,2,…,s.According to Lemma5, for i∈{1,2,…,s}, there exist nonsingular matrices Ti and Si such that state and output transformations are described as(7)x=Tiξi1ξi2,y=Siηi1ηi2and (6) can be correspondingly transformed as(8)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+F¯i1f¯it+Gi1fitdtDαξi2t=Ai21ξi1t+Ai22ξi2t+Bi2ut+F¯i2f¯itηi1t=Ci11ξi1tηi2t=Ci22ξi2t,where Gi1 and Ci11 are (q+1)×(q+1) matrices, ξi1(t) and ξi2(t) are (q+1)-dimensional and (n-q-1)-dimensional vectors, and(9)Ti-1ATi=Ai11Ai12Ai21Ai22,Ti-1B=Bi1Bi2,Ti-1F~i=F¯i1F¯i2,Ti-1Gi′=Gi10,Si-1CTi=Ci1100Ci22, where the dimensions of matrices are defined as follows: Ai11 is (q+1)×(q+1), Ai12 is (q+1)×(n-q-1), Ai21 is (n-q-1)×(q+1), Ai22 is (n-q-1)×(n-q-1), F¯i1 is (q+1)×(s-1), F¯i2 is (n-q-1)×(s-1), and Ci22 is (p-q-1)×(n-q-1).System (8) can be rewritten as two subsystems(10)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+F¯i1f¯it+Gi1fitdtηi1t=Ci11ξi1t,i=1,2,…,sand(11)Dαξi2t=Ai21ξi1t+Ai22ξi2t+Bi2ut+F¯i2f¯itηi2t=Ci22ξi2t,i=1,2,…,s,where all variables and matrices here are the same as before.Remark 7.
It is important to separate a single fault source and disturbances simultaneously by introducing state and output transformations. As well known, the key of the fault isolation is to isolate the single fault source.


### 3.2. Fault Detection Based on a Luenberger Observer

The observer technique is used to detect and isolate faults in this subsection. We propose to construct an observer for the second subsystem (11) for each i.Lemma 8.
Consider (11) satisfying Assumptions 2 and 3; then(12)rankσIn-q-1-Ai22Ci22=n-q-1.Proof.
By performing elementary column and row operations on matrix and using Lemma5, the following equivalence is shown:(13)σIn-AGi′C0~Gi1000Ci11000σIn-q-1-Ai2200Ci22, where “~" means equivalence. It is worth noting that(14)rankσIn-AGi′C0=rankσIn-q-1-Ai22Ci22+2q+1, and(15)rankσIn-AGi′C0=n+q+1, And then (12) is obtained.
Meanwhile, we can obtain a conclusion in Theorem9 similar to Lemma 1 presented in [42].Theorem 9.
With Assumptions2 and 3, the unobservable polynomial of the pair (Ai22,Ci22) is equal to the invariant zero polynomial of (C,A,Gi′).Following Lemmas 2 and 3 in [31], Theorem 10 can be stated.Theorem 10.
System (11) satisfying Assumptions 2 and 3 is detectable or the pair (Ai22,Ci22) is detectable.Considering system (11), ξi1 and ηi2 can be obtained from the measured output y. In fact, let(16)Si-1=Si1Si2, And then they are computed by(17)ξi1=Ci11-1Si1y,ηi2=Si2y. Moreover, based on Theorem 2 presented in [43] and Theorem 10 in this work, we are ready to design a Luenberger observer for system (11) and select Li2 such that (Ai22-Li2Ci22) is a stable matrix; that is, the spectrum of (Ai22-Li2Ci22) can be assigned anywhere in the complex of region of asymptotic stability |arg⁡(σ)|>απ/2. A Luenberger observer is designed as follows:(18)Dαξ^i2t=Ai21ξi1t+Ai22ξ^i2t+Bi2ut+Li2ηi2t-η^i2t,η^i2t=Ci22ξ^i2t,i=1,2,…,s,where  ^  indicates estimate.For eachi=1,2,…,s, the state estimation error dynamics of the second subsystem (11) can be obtained by subtracting (18) from (11)(19)Dtαei2t=Ai22-Li2Ci22ei2t+F¯i2f¯it,where ei2(t)=ξi2(t)-ξ^i2(t).We now define a fault-detection residual as(20)rit=ηi2t-η^i2t=Ci22ei2t,and then we have Theorem 11.Theorem 11.
For system (1) satisfying Assumptions 1-3 with a bank of Luenberger observers in the form of (18), we consider only one single fault scenario. If there exist i,j and i≠j such that limt→∞ri(t)=0 and limt→∞rj(t)=0, i,j∈{1,2,…,s}, then the system is healthy. If there are i,j such that limt→∞ri(t)=0 and limt→∞rj(t)≠0 for j≠i, then fi(t)≠0; that is, fi(t) is the only one fault.Proof.
Iflimt→∞ri(t)=0 and limt→∞rj(t)=0 when i≠j, it follows that f¯i(t)≡0 and f¯j(t)≡0 from (20) and the stability of (Ai22-Li2Ci22) in (19), it implies that the system is free from faults, and thus system (1) is healthy.
Iflimt→∞ri(t)=0, it holds that f¯i(t)≡0 from (20) and the stability of (Ai22-Li2Ci22) of (19). If limt→∞rj(t)≠0 when j≠i, it follows that F¯j2f¯j(t)≠0 from (19) and (20). Moreover, f¯j(t) has one and only one nonzero element, which is fi(t). Thus, fi(t)≠0 and fj(t)=0 for all j,j≠i.Remark 12.
Theorem11 implies that the residual ri(t) is not sensitive to the ith fault fi(t) but is sensitive to all other faults.


### 3.3. Fault Isolation Based on Fault-Detection Residuals

Based on the analysis presented in the above subsections, the following fault-detection and isolation algorithm was suggested.Step 1.
Design a bank of Luenberger observers of the form (18) for i=1,2,…,s.Step 2.
Compute thes fault-detection residuals ri(t),i=1,2,…,s.Step 3.
Choose a thresholdr0 (which can be chosen as small as possible theoretically because limt→∞ri(t)=0 when fi(t)≠0).Step 4.
If all residualsri(t),i=1,2,…,s are below the threshold, then, the system is healthy. If there is only one residual, say ri0(t) with i0∈{1,2,…,s}, which is below the threshold, then it is claimed that a fault has been detected and the fault is isolated as fi0(t).Remark 13.
Based on Theorem11, a fault can be detected when system (1) has only one fault by computing at most s fault-detection residuals ri(t) for the bank of Luenberger observers (18). Performing the above algorithm, the first two questions listed in Section 2 are addressed.


## 3.1. Bank of Reformulated Systems

This subsection deals with transformation of the system into a proper form containing two subsystems. Definef(t)=f1(t)f2(t)⋯fs(t)T, F~i is defined as the matrix after removing the column Fi from F, and f-i(t) is the matrix after deleting the fi(t) from f(t). Then, system (1) is reformulated as a bank of s systems as follows:(6)Dαxt=Axt+But+F~if-it+Fifit+Gdt,yt=Cxt,i=1,2,…,s.According to Lemma5, for i∈{1,2,…,s}, there exist nonsingular matrices Ti and Si such that state and output transformations are described as(7)x=Tiξi1ξi2,y=Siηi1ηi2and (6) can be correspondingly transformed as(8)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+F¯i1f¯it+Gi1fitdtDαξi2t=Ai21ξi1t+Ai22ξi2t+Bi2ut+F¯i2f¯itηi1t=Ci11ξi1tηi2t=Ci22ξi2t,where Gi1 and Ci11 are (q+1)×(q+1) matrices, ξi1(t) and ξi2(t) are (q+1)-dimensional and (n-q-1)-dimensional vectors, and(9)Ti-1ATi=Ai11Ai12Ai21Ai22,Ti-1B=Bi1Bi2,Ti-1F~i=F¯i1F¯i2,Ti-1Gi′=Gi10,Si-1CTi=Ci1100Ci22, where the dimensions of matrices are defined as follows: Ai11 is (q+1)×(q+1), Ai12 is (q+1)×(n-q-1), Ai21 is (n-q-1)×(q+1), Ai22 is (n-q-1)×(n-q-1), F¯i1 is (q+1)×(s-1), F¯i2 is (n-q-1)×(s-1), and Ci22 is (p-q-1)×(n-q-1).System (8) can be rewritten as two subsystems(10)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+F¯i1f¯it+Gi1fitdtηi1t=Ci11ξi1t,i=1,2,…,sand(11)Dαξi2t=Ai21ξi1t+Ai22ξi2t+Bi2ut+F¯i2f¯itηi2t=Ci22ξi2t,i=1,2,…,s,where all variables and matrices here are the same as before.Remark 7.
It is important to separate a single fault source and disturbances simultaneously by introducing state and output transformations. As well known, the key of the fault isolation is to isolate the single fault source.


## 3.2. Fault Detection Based on a Luenberger Observer

The observer technique is used to detect and isolate faults in this subsection. We propose to construct an observer for the second subsystem (11) for each i.Lemma 8.
Consider (11) satisfying Assumptions 2 and 3; then(12)rankσIn-q-1-Ai22Ci22=n-q-1.Proof.
By performing elementary column and row operations on matrix and using Lemma5, the following equivalence is shown:(13)σIn-AGi′C0~Gi1000Ci11000σIn-q-1-Ai2200Ci22, where “~" means equivalence. It is worth noting that(14)rankσIn-AGi′C0=rankσIn-q-1-Ai22Ci22+2q+1, and(15)rankσIn-AGi′C0=n+q+1, And then (12) is obtained.
Meanwhile, we can obtain a conclusion in Theorem9 similar to Lemma 1 presented in [42].Theorem 9.
With Assumptions2 and 3, the unobservable polynomial of the pair (Ai22,Ci22) is equal to the invariant zero polynomial of (C,A,Gi′).Following Lemmas 2 and 3 in [31], Theorem 10 can be stated.Theorem 10.
System (11) satisfying Assumptions 2 and 3 is detectable or the pair (Ai22,Ci22) is detectable.Considering system (11), ξi1 and ηi2 can be obtained from the measured output y. In fact, let(16)Si-1=Si1Si2, And then they are computed by(17)ξi1=Ci11-1Si1y,ηi2=Si2y. Moreover, based on Theorem 2 presented in [43] and Theorem 10 in this work, we are ready to design a Luenberger observer for system (11) and select Li2 such that (Ai22-Li2Ci22) is a stable matrix; that is, the spectrum of (Ai22-Li2Ci22) can be assigned anywhere in the complex of region of asymptotic stability |arg⁡(σ)|>απ/2. A Luenberger observer is designed as follows:(18)Dαξ^i2t=Ai21ξi1t+Ai22ξ^i2t+Bi2ut+Li2ηi2t-η^i2t,η^i2t=Ci22ξ^i2t,i=1,2,…,s,where  ^  indicates estimate.For eachi=1,2,…,s, the state estimation error dynamics of the second subsystem (11) can be obtained by subtracting (18) from (11)(19)Dtαei2t=Ai22-Li2Ci22ei2t+F¯i2f¯it,where ei2(t)=ξi2(t)-ξ^i2(t).We now define a fault-detection residual as(20)rit=ηi2t-η^i2t=Ci22ei2t,and then we have Theorem 11.Theorem 11.
For system (1) satisfying Assumptions 1-3 with a bank of Luenberger observers in the form of (18), we consider only one single fault scenario. If there exist i,j and i≠j such that limt→∞ri(t)=0 and limt→∞rj(t)=0, i,j∈{1,2,…,s}, then the system is healthy. If there are i,j such that limt→∞ri(t)=0 and limt→∞rj(t)≠0 for j≠i, then fi(t)≠0; that is, fi(t) is the only one fault.Proof.
Iflimt→∞ri(t)=0 and limt→∞rj(t)=0 when i≠j, it follows that f¯i(t)≡0 and f¯j(t)≡0 from (20) and the stability of (Ai22-Li2Ci22) in (19), it implies that the system is free from faults, and thus system (1) is healthy.
Iflimt→∞ri(t)=0, it holds that f¯i(t)≡0 from (20) and the stability of (Ai22-Li2Ci22) of (19). If limt→∞rj(t)≠0 when j≠i, it follows that F¯j2f¯j(t)≠0 from (19) and (20). Moreover, f¯j(t) has one and only one nonzero element, which is fi(t). Thus, fi(t)≠0 and fj(t)=0 for all j,j≠i.Remark 12.
Theorem11 implies that the residual ri(t) is not sensitive to the ith fault fi(t) but is sensitive to all other faults.


## 3.3. Fault Isolation Based on Fault-Detection Residuals

Based on the analysis presented in the above subsections, the following fault-detection and isolation algorithm was suggested.Step 1.
Design a bank of Luenberger observers of the form (18) for i=1,2,…,s.Step 2.
Compute thes fault-detection residuals ri(t),i=1,2,…,s.Step 3.
Choose a thresholdr0 (which can be chosen as small as possible theoretically because limt→∞ri(t)=0 when fi(t)≠0).Step 4.
If all residualsri(t),i=1,2,…,s are below the threshold, then, the system is healthy. If there is only one residual, say ri0(t) with i0∈{1,2,…,s}, which is below the threshold, then it is claimed that a fault has been detected and the fault is isolated as fi0(t).Remark 13.
Based on Theorem11, a fault can be detected when system (1) has only one fault by computing at most s fault-detection residuals ri(t) for the bank of Luenberger observers (18). Performing the above algorithm, the first two questions listed in Section 2 are addressed.


## 4. Fault Estimation Based on Sliding Mode Observers

In this section, fault estimation based on sliding mode observers is addressed and will address the third question posed above. It is assumed that a fault has been detected and isolated asfi0(t).


### 4.1. Design of a Sliding Mode Observer for Fault Estimation

When the faultfi0(t) occurs, limt→∞ri0(t)=0 and f¯i0(t)=0. For convenience, the isolated fault fi0(t) is still denoted as fi(t). The i0-th is denoted as i-th. In fact, the first subsystem (10) and the state estimation error dynamics (19) with fault fi0(t) can be described as follows:(21)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+Gi1fitdtηi1t=Ci11ξi1t,i=1,2,…,sand(22)Dtαei2t=Ai22-Li2Ci22ei2t.For system (21), the following equations are established:(23)rankCi11Gi1=rankGi1=q+1,(24)rankσIq+1-Ai11Gi1Ci110=q+1+rankGi1based on Lemma 5 and the equivalence of the following two matrices: (25)σIq+1-Ai11Gi1Ci110~0Gi1Ci110.The stability of the fractional-order observer for subsystem (21) can be formulated by the two conditions (23) and (24) according to [35]. Parallel to the ith Luenberger observer of the form (18), a sliding mode observer can be designed to estimate the fault for (21) as follows:(26)Dαξ^i1t=Ai11ξ^i1t+Ai12ξ^i2t+Bi1ut+Li1ηi1t-η^i1t+Gi1v^itη^i1t=Ci11ξ^i1t,i=1,2,…,s,where v^i(t) is the estimate of vi(t)=fi(t)d(t)T. It is defined as(27)v^=ρiKi1eηiKi1eηi,Ki1eηi≠0;0,Ki1eηi=0with eηi=ηi1-η^i1 and constant matrix Ki1∈R(q+1)×(q+1), ρi should be chosen to be a large enough constant, and Li1 can be selected such that the spectrum of (Ai11-Li1Ci11) is a stable matrix.Letei1(t)=ξi1(t)-ξ^i1(t) be the estimation error of state ξi1(t), then the fractional-order state estimation error can be described as follows:(28)Dαei1=Ai11-Li1Ci11ei1+Ai12ei2+Gi1v-v^,and eηi=ηi1-η^i1=Ci11ei1.


### 4.2. Stability of the Designed Sliding Mode Observer

Before embarking on the fundamental theorem, we first give some preliminary results on Caputo fractional derivatives.Definition 14 (see [9]).
The two-parameter Mittag-Leffler function is defined by(29)Eα,βz=∑k=0∞zkΓαk+β,α>0,β>0,z∈C, where Γ(·) is the Gamma function with (30)Γx=∫0∞tx-1e-tdt. When β=1, one has Eα(z)=Eα,1(z).Letei(t)=ei1(t)ei2(t)T, ei(0)=00T. According to the definition in [44], [0,0]T is the equilibrium point of the following dynamic system: (31)Dαei1t=Ai11-Li1Ci11ei1t+Ai12ei2t(32)Dtαei2t=Ai22-Li2Ci22ei2t.Then we present the definition of stability for the system formed by (28) and (22) in the sense of Mittag-Leffler similar to those in [44, 45] as follows.Definition 15.
The solutionei(t) of the system formed by (28) and (22) is said to be Mittag-Leffler stable with respect to the equilibrium point ei(0) of system (31) if (33)eit≤meit0Eα-λt-t0αb,α∈0,1, where t0=0 is the initial time, λ>0,b>0,m(0)=0,m(ei)≥0, and m(ei) is locally Lipschitz with Lipschitz constant.Lemma 16 (see [46]).
Letx(t)∈Rn be a differentiable vector; then, for any time instant,(34)12DαxTtPxt≤xTtPDαxt,∀α∈0,1,t≥0,where P∈Rn×n is a positive definite constant matrix.The following theorem is to present the stability of the designed observers in the sense of Mittag-Leffler.Theorem 17.
Consider (1) satisfying Assumptions 1–3, f(t) and d(t) are bounded, and the Luenberger observers and the sliding mode observer are designed in (18), (26), and (27). For i=1,2,…,s, if there exist matrices Li1,Li2,Ki1 and positive definite matrices P1,P2,Qi1,Qi2 such that(35)-Qi1=Ai11-Li1Ci11TP1+P1Ai11-Li1Ci11+P12(36)-Qi2=Ai22-Li2Ci22TP2+P2Ai22-Li2Ci22+Ai12TAi12(37)P1Gi1=Ci11TKi1T, then the observers given by (18), (26), and (27) can ensure that the state estimation error ei(t) is Mittag-Leffler stable.Proof.
Consider a Lyapunov function candidate as follows:(38)Vit,eit=ei1TtP1ei1t+ei2TtP2ei2t,where P1 and P2 are positive definite matrices.
Applying Caputo fractional-order operationDα to Vi with respect to t along (22) and (28), by Lemma 16, it is obtained that(39)DαVi=Dαei1TP1ei1+Dαei2TP2ei2≤2ei1TP1Dαei1+2ei2TP2Dαei1=Ai11-Li1Ci11ei1+Ai12ei2+Gi1evTP1ei1+ei1TP1Ai11-Li1Ci11ei1+Ai12ei2+Gi1ev+ei2TAi22-Li2Ci22TP2+P2Ai22-Li2Ci22ei2≤ei1TAi11-Li1Ci11TP1+P1Ai11-Li1Ci11+P12ei1+ei2TAi22-Li2Ci22TP2+P2Ai22-Li2Ci22+Ai12TAi12ei2+2ei1TPi1Gi1ev, where ev=v-v^.
IfKi1 can be chosen such that P1Gi1=Ci11TKi1T, then (40)ei1TP1Gi1v^=Ki1Ci11ei1ρi=Ki1eη1ρi,ei1TP1Gi1v=Ki1Ci11ei1Tv=Ki1eη1Tv.
Considering the following inequality:(41)ei1TPi1Gi1v≤Ki1eη1v, and applying (35) and (36), (39) can be further manipulated as follows: (42)DαVi≤ei1T-Qi1ei1+ei2T-Qi2ei2-2Ki1eη1ρi-v.
LetQi=Qi100Qi2 and P=P100P2, λmin(Qi) denoted the minimum eigenvalue of Qi, and λmin(P) and λmax(P) denoted the minimum and maximum eigenvalues of P. Then λmin(Qi)>0, λmin(P)>0, and λmax(P)>0.
Choosingρi large enough (this can be performed because v is bounded), one obtains(43)DαVi≤-λminQiei2.And the Lyapunov function candidate is satisfied by(44)λminPei2≤Vi≤λmaxPei2.
It follows from (43) and (44) that (45)DαVi≤-λminQiλmaxPVi. There exists a nonnegative function M(t) such that(46)DαVi+Mt=-λminQiλmaxPVi.
Taking Laplace transform on both sides of (46) leads to(47)sαVis-sα-1Vi0+Ms=-λminQiλmaxPVis,where Vi(s) = £{Vi(t,ei(t))} and £ is the operator for Laplace transform. Based on (47), Vi can be expressed as follows: (48)Vis=sα-1Vi0-Mssα+λminQi/λmaxP. From the existence and uniqueness theorem [9] for fractional-order differential equations and the inverse Laplace transform, the unique solution of (46) can be obtained as follows: (49)Vit=Vi0Eα-λminQiλmaxPtα-Mt∗tα-1Eα,α-λminQiλmaxPtα, where ∗ denotes the convolution operator.
Therefore,Vi(t)≤Vi(0)Eα(-λmin(Qi)/λmax(P)tα) from the facts that both tα-1 and Eα,α(-λmin(Qi)/λmax(P)tα) are nonnegative. With the aid of (44), the estimation error ei(t) satisfies the following inequality: (50)eit≤Vi0λminPEα-λminQiλmaxPtα1/2, where Vi(0)/λmin(P)≥0.
Letm=Vi(0)/λmin(P)=Vi(0,ei(0))/λmin(P)≥0, and then we have (51)eit≤mEα-λminQiλmaxPtα1/2, where m=0 holds if and only if ei(0)=0. Because V(t,ei) is locally Lipschitz with respect to ei and V(0,ei(0))=0 if and only if ei(0)=0, and it follows that m=Vi(0,ei(0))/λmin(P) is also Lipschitz with respect to ei(0) and m(0)=0. Therefore, the solution ei(t) of the system formed by (28) and (22) is Mittag-Leffler stable.Remark 18.
Theorem17 implies that the state estimation errors ei1 and ei2 are bounded when Eα(-λmin(Qi)/λmax(P)tα) is bounded. The conclusion is the same as that for integer-order systems [1].Remark 19.
The third question in Section2 has been addressed. To our knowledge the stability in the sense of Mittag- Leffler is first introduced here to study the state estimation error dynamics.


## 4.1. Design of a Sliding Mode Observer for Fault Estimation

When the faultfi0(t) occurs, limt→∞ri0(t)=0 and f¯i0(t)=0. For convenience, the isolated fault fi0(t) is still denoted as fi(t). The i0-th is denoted as i-th. In fact, the first subsystem (10) and the state estimation error dynamics (19) with fault fi0(t) can be described as follows:(21)Dαξi1t=Ai11ξi1t+Ai12ξi2t+Bi1ut+Gi1fitdtηi1t=Ci11ξi1t,i=1,2,…,sand(22)Dtαei2t=Ai22-Li2Ci22ei2t.For system (21), the following equations are established:(23)rankCi11Gi1=rankGi1=q+1,(24)rankσIq+1-Ai11Gi1Ci110=q+1+rankGi1based on Lemma 5 and the equivalence of the following two matrices: (25)σIq+1-Ai11Gi1Ci110~0Gi1Ci110.The stability of the fractional-order observer for subsystem (21) can be formulated by the two conditions (23) and (24) according to [35]. Parallel to the ith Luenberger observer of the form (18), a sliding mode observer can be designed to estimate the fault for (21) as follows:(26)Dαξ^i1t=Ai11ξ^i1t+Ai12ξ^i2t+Bi1ut+Li1ηi1t-η^i1t+Gi1v^itη^i1t=Ci11ξ^i1t,i=1,2,…,s,where v^i(t) is the estimate of vi(t)=fi(t)d(t)T. It is defined as(27)v^=ρiKi1eηiKi1eηi,Ki1eηi≠0;0,Ki1eηi=0with eηi=ηi1-η^i1 and constant matrix Ki1∈R(q+1)×(q+1), ρi should be chosen to be a large enough constant, and Li1 can be selected such that the spectrum of (Ai11-Li1Ci11) is a stable matrix.Letei1(t)=ξi1(t)-ξ^i1(t) be the estimation error of state ξi1(t), then the fractional-order state estimation error can be described as follows:(28)Dαei1=Ai11-Li1Ci11ei1+Ai12ei2+Gi1v-v^,and eηi=ηi1-η^i1=Ci11ei1.


## 4.2. Stability of the Designed Sliding Mode Observer

Before embarking on the fundamental theorem, we first give some preliminary results on Caputo fractional derivatives.Definition 14 (see [9]).
The two-parameter Mittag-Leffler function is defined by(29)Eα,βz=∑k=0∞zkΓαk+β,α>0,β>0,z∈C, where Γ(·) is the Gamma function with (30)Γx=∫0∞tx-1e-tdt. When β=1, one has Eα(z)=Eα,1(z).Letei(t)=ei1(t)ei2(t)T, ei(0)=00T. According to the definition in [44], [0,0]T is the equilibrium point of the following dynamic system: (31)Dαei1t=Ai11-Li1Ci11ei1t+Ai12ei2t(32)Dtαei2t=Ai22-Li2Ci22ei2t.Then we present the definition of stability for the system formed by (28) and (22) in the sense of Mittag-Leffler similar to those in [44, 45] as follows.Definition 15.
The solutionei(t) of the system formed by (28) and (22) is said to be Mittag-Leffler stable with respect to the equilibrium point ei(0) of system (31) if (33)eit≤meit0Eα-λt-t0αb,α∈0,1, where t0=0 is the initial time, λ>0,b>0,m(0)=0,m(ei)≥0, and m(ei) is locally Lipschitz with Lipschitz constant.Lemma 16 (see [46]).
Letx(t)∈Rn be a differentiable vector; then, for any time instant,(34)12DαxTtPxt≤xTtPDαxt,∀α∈0,1,t≥0,where P∈Rn×n is a positive definite constant matrix.The following theorem is to present the stability of the designed observers in the sense of Mittag-Leffler.Theorem 17.
Consider (1) satisfying Assumptions 1–3, f(t) and d(t) are bounded, and the Luenberger observers and the sliding mode observer are designed in (18), (26), and (27). For i=1,2,…,s, if there exist matrices Li1,Li2,Ki1 and positive definite matrices P1,P2,Qi1,Qi2 such that(35)-Qi1=Ai11-Li1Ci11TP1+P1Ai11-Li1Ci11+P12(36)-Qi2=Ai22-Li2Ci22TP2+P2Ai22-Li2Ci22+Ai12TAi12(37)P1Gi1=Ci11TKi1T, then the observers given by (18), (26), and (27) can ensure that the state estimation error ei(t) is Mittag-Leffler stable.Proof.
Consider a Lyapunov function candidate as follows:(38)Vit,eit=ei1TtP1ei1t+ei2TtP2ei2t,where P1 and P2 are positive definite matrices.
Applying Caputo fractional-order operationDα to Vi with respect to t along (22) and (28), by Lemma 16, it is obtained that(39)DαVi=Dαei1TP1ei1+Dαei2TP2ei2≤2ei1TP1Dαei1+2ei2TP2Dαei1=Ai11-Li1Ci11ei1+Ai12ei2+Gi1evTP1ei1+ei1TP1Ai11-Li1Ci11ei1+Ai12ei2+Gi1ev+ei2TAi22-Li2Ci22TP2+P2Ai22-Li2Ci22ei2≤ei1TAi11-Li1Ci11TP1+P1Ai11-Li1Ci11+P12ei1+ei2TAi22-Li2Ci22TP2+P2Ai22-Li2Ci22+Ai12TAi12ei2+2ei1TPi1Gi1ev, where ev=v-v^.
IfKi1 can be chosen such that P1Gi1=Ci11TKi1T, then (40)ei1TP1Gi1v^=Ki1Ci11ei1ρi=Ki1eη1ρi,ei1TP1Gi1v=Ki1Ci11ei1Tv=Ki1eη1Tv.
Considering the following inequality:(41)ei1TPi1Gi1v≤Ki1eη1v, and applying (35) and (36), (39) can be further manipulated as follows: (42)DαVi≤ei1T-Qi1ei1+ei2T-Qi2ei2-2Ki1eη1ρi-v.
LetQi=Qi100Qi2 and P=P100P2, λmin(Qi) denoted the minimum eigenvalue of Qi, and λmin(P) and λmax(P) denoted the minimum and maximum eigenvalues of P. Then λmin(Qi)>0, λmin(P)>0, and λmax(P)>0.
Choosingρi large enough (this can be performed because v is bounded), one obtains(43)DαVi≤-λminQiei2.And the Lyapunov function candidate is satisfied by(44)λminPei2≤Vi≤λmaxPei2.
It follows from (43) and (44) that (45)DαVi≤-λminQiλmaxPVi. There exists a nonnegative function M(t) such that(46)DαVi+Mt=-λminQiλmaxPVi.
Taking Laplace transform on both sides of (46) leads to(47)sαVis-sα-1Vi0+Ms=-λminQiλmaxPVis,where Vi(s) = £{Vi(t,ei(t))} and £ is the operator for Laplace transform. Based on (47), Vi can be expressed as follows: (48)Vis=sα-1Vi0-Mssα+λminQi/λmaxP. From the existence and uniqueness theorem [9] for fractional-order differential equations and the inverse Laplace transform, the unique solution of (46) can be obtained as follows: (49)Vit=Vi0Eα-λminQiλmaxPtα-Mt∗tα-1Eα,α-λminQiλmaxPtα, where ∗ denotes the convolution operator.
Therefore,Vi(t)≤Vi(0)Eα(-λmin(Qi)/λmax(P)tα) from the facts that both tα-1 and Eα,α(-λmin(Qi)/λmax(P)tα) are nonnegative. With the aid of (44), the estimation error ei(t) satisfies the following inequality: (50)eit≤Vi0λminPEα-λminQiλmaxPtα1/2, where Vi(0)/λmin(P)≥0.
Letm=Vi(0)/λmin(P)=Vi(0,ei(0))/λmin(P)≥0, and then we have (51)eit≤mEα-λminQiλmaxPtα1/2, where m=0 holds if and only if ei(0)=0. Because V(t,ei) is locally Lipschitz with respect to ei and V(0,ei(0))=0 if and only if ei(0)=0, and it follows that m=Vi(0,ei(0))/λmin(P) is also Lipschitz with respect to ei(0) and m(0)=0. Therefore, the solution ei(t) of the system formed by (28) and (22) is Mittag-Leffler stable.Remark 18.
Theorem17 implies that the state estimation errors ei1 and ei2 are bounded when Eα(-λmin(Qi)/λmax(P)tα) is bounded. The conclusion is the same as that for integer-order systems [1].Remark 19.
The third question in Section2 has been addressed. To our knowledge the stability in the sense of Mittag- Leffler is first introduced here to study the state estimation error dynamics.


## 5. Fault Detection and Estimation of the Three-Cell Battery String

In this section, we apply the proposed method for fault detection and estimation of a three-cell battery string model reported in [1] and demonstrate its effectiveness in application. Then system (1) is now parameterized as follows: (52)A=-1.6026000-1.6026000-1.6026,B=0.17955.24840.02250.17955.24840.02250.17955.24840.0225,C=100010001,G=0.110.05,F=300030003. A disturbance is selected as d(t)=0.0151sin⁡(5t). It is assumed that f1=f3=0 for all the time. When t∈[0,T), f2=0 and when t≥T, f2=0.2; that is, fault f2 occurs at time T. Furthermore, a threshold is set as r0=0.015 and take T=25s where s stands for “second.”Because there are three fault sources, system (1) has three formulations with each one corresponding to a particular fault. According to Lemma 5, nonsingular transform matrices are selected as (53)T1=10001000.051,T2=1000100.501,T3=0.101100010 and Si=Ti,i=1,2,3.Obviously, for eachi-th system, the second subsystem does not include i-th fault source fi and their observers are designed as (54)Dαξ12t=-1.6026ξ12t+0.1705Z+4.9860+0.0214I-0.15f2+3f3,η12t=ξ12t;Dαξ^12t=-1.6026ξ^12t+0.1705Z+4.9860+0.0214I+0.6ηi2t-η^i2t,η^12t=ξ^12t,Dαξ22t=-1.6026ξ22t+0.0897Z+2.6242+0.0113I-1.5f1+3f3,η22t=ξ22t;Dαξ^22t=-1.6026ξ^22t+0.0897Z+2.6242+0.0113I+0.3η22t-η^22t,η^22t=ξ^22t,Dαξ32t=-1.6026ξ32t+0.1616Z+4.7236+0.0203I+3f1-0.3f2,η32t=ξ32t;Dαξ^32t=-1.6026ξ^32t+0.1616Z+4.7236+0.0203I+0.6η32t-η^32t,η^32t=ξ^32t, where L12=L32=0.6,L22=0.3.Then, thei-th fault-detection residual is designated as (55)rit=ηi2t-η^i2t=ei2t,i=1,2,3.In what follows, we only show the first subsystem and its observer fori=2. Let (56)ξ21t=ξ211tξ221t,η21t=η211tη221t, and then the subsystem and its sliding mode observer are as follows: (57)Dαξ211t=-1.6026ξ211t+0.1795Z+5.2484+0.0225I+0.1dt,Dαξ221t=-1.6026ξ221t+0.1795Z+5.2484+0.0225I+3f2+dt,η211t=ξ211t,(58)η221t=ξ221t, and (59)Dαξ^211t=-1.6026ξ^211t+0.1795Z+5.2484+0.0225I+0.1d^t+3.6η211t-η^211t,Dαξ^221t=-1.6026ξ^221t+0.1795Z+5.2484+0.0225I+3f^2+d^t+3.5η221t-η^221t,η^211t=ξ^211t,η^221t=ξ^221t, where (60)f^2=ρh1tht,ht≠0;0,ht=0,d^=ρh2tht,ht≠0;0,ht=0 and (61)h1t=9ξ211-ξ^211+4ξ221-ξ^221,h2t=4ξ211-ξ^211+0.3+43ξ221-ξ^221,ht=h1t2+h2t2. To illustrate the performance of the above strategy, the Adams-Bashforth-Moulton predictor-corrector method is used to calculate the fractional system, and the detailed instructions of this algorithm are available in [47, 48] based on [49]. Next, the simulations are performed while parameter ρ and the commensurate order α are changed.Faultf2(t) is caused by a change of internal resistance of the battery and occurs at time instant 25 seconds. Figures 1 and 2 show the evolution of fault-detection residuals ri(t),i=1,2,3, and the chosen threshold r0 where the horizontal bold lines are thresholds with ρ=0.25,α=0.8, and ρ=100,α=0.4. It can be seen indeed in the figures that f2(t) is detected at 25s because r1 and r3 go beyond the threshold r0 after 25 seconds, and r2 is well below the threshold r0. Moreover the evolution of fault-detection residuals r1, r2, and r3 is the same when parameters ρ and α are changed.Figure 1
Fault detection and isolation whenρ=0.25,α=0.8.Figure 2
Fault detection and isolation whenρ=100,α=0.4.Figures3 and 4 show the evolution of the system states (terminal voltage) and their estimations with the fault estimate that is the second subsystem (i=2) with ρ=0.25,α=0.8 and ρ=100,α=0.4. Although parameters ρ and α are changed, it is noted that each state estimation error is Mittag-Leffler stable because the estimation curve is varying along its own state curve in a small bounded interval and the evolution of the curves has not been affected by the changing parameters. At time instant 25 seconds, ξ221(t) and its estimation start to identify the isolated fault accurately in Figures 3 and 4.Figure 3
System state and estimate with the fault estimate whenρ=0.25,α=0.8.Figure 4
System state and estimate with the fault estimate whenρ=100,α=0.4.To illustrate the advantages of our proposed method, the simulations are performed on the integer-order modeling of the three-cell battery string [1]. Figure 5 shows the evolution of state and its estimate with the fault estimate system by applying the integer-order modeling in [1]. Compared with our proposed method, it can be found that the state and its estimation with the fault estimate start to identify the isolated fault at time instant 25 seconds. But at instant 50 seconds, dynamic curve of estimation with the fault estimate jumps again. The estimation value deviates from state value at the instant too. But the evolution of curves in Figures 3 and 4 is changed smoothly except at time instant 25 seconds when fault f2 occurs. It is shown that our proposed method of fractional-order modeling has even better results than that of integer-order modeling.Figure 5
System state and estimate with the fault estimate of the integer-order modeling.


## 6. Conclusions

Fault diagnosis and estimation problem have been studied for a fractional-order system with applications to Lithium-ion battery cell. Sources of fault and system uncertainties are first separated into two subsystems through a transformation on the model of the system. A fractional-order Luenberger observer is designed to generate a fault-detection residual and faults are then easily detected and isolated. Secondly, a fractional-order sliding mode observer is constructed to provide an estimate of the isolated fault; hence, simultaneous fault detection, isolation, and estimation is accomplished. The effectiveness of the proposed strategy is demonstrated on a three-cell battery string. The proposed strategy is completely built on a new framework. Properties of fractional-order systems and Mittag-Leffler stability are used to describe the estimation error of observers by the Lyapunov directed method. The stability in the sense of Mittag-Leffler embodies properties of fractional calculus.



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*Source: 8705363-2018-10-24.xml*