Fault and Switching Instants Estimation for Switched Linear Systems

C H A P T E R

12 Fault and Switching Instants Estimation for Switched Linear Systems Khaled Laboudi, Nadhir Messai, Noureddine Manamanni, Mamadou Mboup Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France

1 INTRODUCTION In order to improve the reliability and the safety of dynamical systems, great attention has been paid in the past few decades to fault-tolerant control (FTC) techniques [1, 2]. Generally speaking, FTC techniques can be classified into two classes; passive and active FTC [3]. Passive FTC considers the faults as a special kind of uncertainty, and design controllers that are robust against a set of predefined faults. Active FTC techniques exploit the information of the fault detection and isolation (FDI) module and adapt the control law after the occurrence of a fault in order to satisfy the control objectives with minimum performance degradation. Therefore, the study of fault estimation has become a challenging research topic owing to its importance in active FTC. Switched systems are a class of hybrid systems that have a wide rang of applications, including chemical processes, traffic control, power converters, and so on [4, 5]. They consist of a finite number of subsystems and a switching signal, which specifies when and how the switching takes place among them [4, 6]. During the past two decades, a great deal of attention has been paid to the stability and stabilization of these systems [4, 7–15]. Many approaches, such the one using common Lyapunov functions [16], multiple Lyapunov functions [17], and switched Lyapunov functions [11, 18] have been developed in order to deal with both arbitrary and restricted switching. New Trends in Observer-based Control https://doi.org/10.1016/B978-0-12-817038-0.00012-3

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© 2019 Elsevier Inc. All rights reserved.

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Unlike the stability techniques, FDI and FTC techniques for switched systems received less attention, and few results are available. In this context [19], we have shown that the classical fault detection (FD) methods can be extended to switched linear systems (SLS), and that the roust fault detection filter (RFDF) can also be applied by means of switching Lyapunov functions. Moreover, an FD scheme that can achieve higher fault detectability than the standard one has been proposed in [20]. A robust hybrid observer for SLS with unknown inputs has been designed for the diagnosis tasks [21]. The proposed observer has, then, been integrated in a dedicated switched robust observer scheme, allowing the detection and the isolation of sensor faults [22]. In addition, based on common Lyapunov functions and a dwell-time, a Luenberger-like observer allowing the estimation of the state of an SLS with unknown inputs has been built in [21]. However, to the best of our knowledge, the problem of fault estimation for SLS is still an open research problem. The state estimation problem for switched systems were widely studied by many authors using different kinds of approaches, and have received considerable attention over the past two decades. The main difference is related to the knowledge of the discrete state; some approaches consider only continuous state uncertainty with known discrete states [23, 24]; whereas others assume that both the discrete state and the continuous state are unknown [25, 26]. For the last case, several approaches such as diagnosis require real-time knowledge of the operating mode. Because of the difficulty of implementing specialized sensors that indicate the operating mode, estimation techniques are therefore necessary. The aim of such methods is to detect any switching, and to accurately recognize the current operating mode. Different algebraic methods are used in [27–30] to detect all switching. In [27], an explicit algorithm has been developed and consists of computing the switching times online. The approach introduced in this chapter is dedicated to an SLS with two modes. This approach has been extended in [28, 29] for more general cases where the SLS contains several modes. An explicit algorithm based on distributions and filters has been proposed in [30] in order to compute online the switching times as follows: first, the original system is rewritten as an auxiliary impulsive model; subsequently, an annihilating algebraic manipulation is provided in order to obtain a desired differential algebraic relation in which the unknown switching times occur explicitly, and finally, by using appropriate filters, the switching times are estimated without using a derived measure. In this work, the problem of discrete state, continuous state, and fault estimation for SLS is addressed. The discrete state estimation consists of two steps. The first step is switchings detection and the second one is the reconstruction of the discrete state. The principle of switching detection consists of detecting dynamic changes of different outputs corresponding to switching times. We aim to show how the new algebraic technique for numerical differentiation [31, 32, 32a, 33] can lead to the algorithm for III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

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switching detection of SLS. After that, a generalization of the coordinate transformation, classically used for linear systems [34, 35], is introduced to decouple a subset of the SLS states from the considered faults. Then, based on the switched Lyapunov function technique, conditions to design an unknown input switched observer are obtained via a linear matrix inequality (LMI) formulation. Finally, the faults are estimated using the designed observer and a robust exact differentiator simultaneously. This work is organized as follows: the problem statement is given in Section 2. The problem of discrete state estimation is presented in Section 3. Section 4 addresses the design of the unknown input switched observer and gives the LMI for its convergence. Then, the procedure allowing the estimation of the considered faults is presented in Section 5. Finally, a numerical example illustrates the efficiency of the proposed approach in Section 6. A conclusion ends the chapter.

2 PROBLEM STATEMENT Generally, a fault occurring in an SLS can affect the continuous or discrete part. In this work we focus on SLS where faults (actuators faults) affect only the continuous part. So we consider in this document the SLS (12.1) in continuous time with actuators faults and measurement noise, defined by the following system:  x˙ (t) = Aλ x(t) + Bλ u(t) + Fλ f (t) (12.1) λ : y(t) = Cλ x(t) + η(t) where x ∈ Rn and u ∈ Rm are, respectively, the continuous state and the input, f ∈ Rl is the fault, which can be considered as an unknown input. The system’s output is defined by the vector y ∈ Rp and the vector η ∈ Rp represents measurement bounded noise. Moreover, the function λ, called a switching signal or a discrete state, is a piecewise constant function of time, which is a right-sided continuous index function. In this chapter, λ is supposed to be unknown, and takes its values from the following discrete set I = {1, . . . , N}, with N being the number of modes that compose the SLS. For all t ∈ Ic  [tc−1 , tc ), λ(t) = i (i ∈ I) where {tc }∞ c=1 is a sequence of switching times and the ith subsystem (12.2) is active.  x˙ (t) = Ai x(t) + Bi u(t) + Fi f (t) (12.2) i : y(t) = Ci x(t) + η(t) where Ai ∈ AI = {A1 , . . . , AN }, Bi ∈ BI = {B1 , . . . , BN }, Ci ∈ CI = {C1 , . . . , CN }, and Fi ∈ FI = {F1 , . . . , FN } are known matrices of appropriate size. For this system, we assume that the unknown switching sequence {tc }∞ c=1 is arbitrary and independent of the system state variables. We also suppose

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FIG. 12.1 General structure of the proposed hybrid observer.

that no zeno phenomenon occurs, meaning a finite amount of switching can occur in a finite-time interval. In the sequel of the chapter, the index time t will be omitted when there is no ambiguity. As we mentioned in the introduction, our objective is to estimate both hybrid states (discrete and continuous) and faults in an SLS with measurement noise. Fig. 12.1 shows the structure of the proposed approach. The first step consists of real-time detection of any switching occurring at times {tc }∞ c=1 by using the system’s output(s) defined by the vector y of dimension p (p = 1 for a single-output system, and p ≥ 2 for a multioutput system). For a single or multioutput switched system, switching detection from one output yi , i = 1, . . . , p is ensured by a single Estimatori , i = 1, . . . , p, whereas, for a multioutput switched system, the detection from all outputs is ensured by a set of p estimators that operate in parallel. The principle of detection of each estimator is based on an algebraic method that will be introduced in Section 3.1.1. In the case of a detection from all outputs of multioutput system, we have developed a decision block based on an algorithm that minimizes both false detections and nondetections. This algorithm analyzes, in real time, the logical output si , i = 1, . . . , p of any Estimatori , i = 1, . . . , p and then decides whether or not a switch occurred at a given time. Moreover, for a single-output system, there is only one estimator, then the decision block is not necessary. The system’s discrete state is reconstructed by using a known switching sequence, and switching

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times estimated by the algebraic approach. A hybrid observer based on partial information (available from the estimated discrete state, the output, and the command) estimates the system’s continuous state. Then the latter is used jointly with the derivative of the output system estimated by a highorder differentiator in order to estimate faults affecting the system. Let us consider the following assumptions. Assumption 1. rang(Ci ) = p. Assumption 2. rang(Fi ) = l. Assumption 3. rang(Ci Fi ) = l; (p ≥ l). T  Assumption 4. rang(Oi ) = n with Oi = Ci Ci Ai · · · Ci An−1 . i Assumptions 1–3 are necessary to decouple a subset of the state of faults system. This allows us to make a projection of the state in another base. This operation is commonly known as coordinate transformation. Assumption 4 is an observability condition for any subsystem, and it should be checked for observer synthesis.

3 DISCRETE STATE ESTIMATION As we mentioned in the previous section, the discrete state estimation is divided into two steps: (i) switching detection by using an outputs vector, and (ii) reconstruction of the system’s discrete state.

3.1 Switching Detection for SLS In a switched system, the switching time represents the moment when the system’s discrete state changes in value, leading to a modification of the continuous state’s dynamic, and therefore a change of the system’s output y. Nevertheless, a switching can lead to a smooth dynamic’s change for an output yi , and therefore the related Estimatori will not be able to detect that switching. Thus, for a multioutput system, the parallel combination of all operating Estimatori , i = 1, . . . , p can avoid nondetection of smooth dynamics change for one or more outputs yi . The techniques developed in this work allow us to identify the dynamics’ changes of a switching system, which correspond to the switching times. Among the existing methods for switching detection in the literature, we can cite change point detection techniques [36]. Despite their robustness in the case of noisy data, these methods are exposed to timedelay in detection. The main contribution of this research focuses on real-time detection of switching, and robustness against noise by using differential algebra, and is based on works in [32, 33] dedicated to the detection of discontinuities, called change point’s detection or abrupt change’s detection.

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In this section, we focus on switching detection by using the output vector y. First, we consider one output yi of vector y whether a singleoutput SLS (SOSLS) or multioutput SLS (MOSLS), thus we can synthesize an estimator based on an algebraic method, allowing us to obtain a linear algebraic relation between the known data and the unknown switching times. Second, the principle is applied for an MOSLS by using all outputs’ estimators, and a decision algorithm (see Section 1). 3.1.1 Switching Detection From One Output of a Switched System We consider one output yi from the system, and in order to simplify our calculations, we consider first a noiseless output, then the detection is given taking into account that noise. The main idea of the approach is to consider, at a given time τ and for an output yi , a time interval or sliding window IτT = [τ − T, τ ] (T > 0 is a predefined length of the interval considered small enough in this work), and to deduce if any switching occurred or not in that interval. Moreover, the output is restricted at any time τ over the sliding window IτT = [τ −T, τ ]. Then, to obtain a linear system depending on the researched switching times, we derive N times the output yi before applying algebraic operations based on formulas given in Appendix. We suppose first that between two successive switchings, there is a minimal dwell-time TD such that T ≤ TD , meaning that the system waits at least a duration TD between two successive switchings. Second, the first switching of the system occurs at a time t1 such that t1 > T. Thus, a restriction of the output yi in the interval IτT can be obtained as follows: yiτ (t) = H(t)yi (τ − T + t),

t ∈ [0, T]

(12.3)

where H represents the Heaviside function (see Eq. A.1). The output yi restricted into the interval IτT is used in our developments. Preliminary Developments

When a switching occurs at time tc = τ − T + τc in the interval IτT (with τc in [0, T]), the signal yiτ becomes not smooth, despite its dynamic change in the interval IτT . Thus, by using Eq. (A.2) (see Appendix), its Nth-order derivative is obtained as follows:  (1) N−1  dN−1 (yiτ (t)) dN (yiτ (t)) (k) = (t) + δ (t − τ ) (12.4) μ c N−1−k dtN dtN−1 k=0

(1) dN−1 (yiτ (t)) represents the first-order derivative of the regular part N−1 dt dN−1 (yiτ (t)) and δ (k) is the kth-order derivative of Dirac function. dtN−1

where of

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Eq. (12.4) is therefore necessary for the detection of switching occurring at τc in the interval IτT . Because the interval IτT is supposed to be small enough, it implies that the signal yiτ can be considered as a linear one in

N−1 i (1) d (yτ (t)) the interval IτT . Therefore, the term in Eq. (12.4) becomes null N−1 dt

for all N ≥ 2. Thus, the Nth-order derivative of the output yiτ becomes: dN (yiτ (t)) dt

N

=

N−1 



μN−k−1 δ (k) (t − τc )

(12.5)

k=0

In order to simplify the calculation, we transform Eq. (12.5) toward an operational area by using formulas (A.3)–(A.5) presented in Appendix. Thus, we obtain: sN yiτ (s) − with (s) =

N−1  k=0



N−1 k k=0 μN−k−1 s

sN−k−1 (yiτ )(k) (0) = (s)



(12.6)

e−τc s . This equation contains unknown

parameters (such as μN−k−1 and (yiτ )(k) (0)) that should be removed to allow estimator synthesis. For this, we derive N times Eq. (12.6) in order N−1−j (yi )(j) (0) corresponding to a (N − to suppress the expression N−1 τ j=0 s 1)th-order polynomial in s. Thus, we obtain: dN (sN yiτ (s))

=

dN ((s))

(12.7) dsN dsN The latter equality is available only for all N ≥ 2, otherwise small values of N can lead to nondetection of switching with a smooth dynamic’s change. Thus, to avoid this case, we consider from now N = 3, and we replace yiτ (s) by yiτ to simplify the expression’s notations. Eq. (12.7) becomes: d3 (s3 yiτ )

=

d3 ()

(12.8) ds ds3 We present below a useful proposition for the rest of our development. Proposition 1. For N = 3, the expression  in Eq. (12.6) satisfies the following differential equation: 3

τc3 + 3

d 2 d2  d3  τc + 3 2 τc + 3 = 0 ds ds ds

(12.9)

Proof . When N = 3, (s) = (μ2 + μ1 s + μ0 s2 )e−τc s . The derivative of  is given by: d = (μ1 + 2μ0 s)e−τc s − τc ds

(12.10)

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Then the second-order derivative of  is deduced from the last equation as follows: d2  ds

2

= 2μ0 e−τc s − (μ1 + 2μ0 s)e−τc s τc −

d τc ds

(12.11)

According to Eq. (12.10), we can write: (μ1 + 2μ0 s)e−τc s = Then, by replacing this expression in Eq. (12.11), we obtain: d2  2

= 2μ0 e−τc s − 2

d ds

d τc − τc2 ds

(12.12)

ds From Eq. (12.12), the third-order derivative of  is given by: d3  ds

3

= −2μ0 e−τc s τc − 2

d2  ds

2

τc −

+ τc .

d 2 τ ds c

(12.13) 2

2 + 2 d From Eq. (12.12), we can replace the term 2μ0 e−τc s = d  ds τc + τc ds2 in Eq. (12.13), in order to obtain Eq. (12.9). In Eq. (12.9), the switching time appears with successive derivatives of . The goal is now to replace these derivatives by successive derivatives of yiτ . We can observe that for all integers k ≥ 0, Eq. (12.8) becomes:

dk+3 (s3 yiτ )

=

dk+3 ()

(12.14) dsk+3 dsk+3 Then, the (k + 3)th-order derivative of Eq. (12.9) for all k ≥ 0 is obtained as follows: dk+3 () k+3

(τc )3 + 3

dk+4 ()

(τc )2 + 3

dk+5 ()

dsk+3

(τc )3 + 3

dk+4 (s3 yiτ ) dsk+4

(τc )2 + 3

k+5

τc = −

dk+6 ()

(12.15)

ds ds ds dsk+6 From Eqs. (12.14), (12.15), we can deduce that for all k ≥ 0: dk+3 (s3 yiτ )

k+4

dk+5 (s3 yiτ ) dsk+5

τc = −

dk+6 (s3 yiτ ) dsk+6

(12.16)

When a switching occurs at time τc in the interval IτT , the switching time τc appears explicitly in Eq. (12.16). Therefore, switching detection at time τc can be obtained from Eq. (12.16). Let us multiply Eq. (12.16) by s1υ (υ > 3) in order to obtain integrals in time domain, and let vik (s, τ ) = Thus we obtain for all k ≥ 0: vik (τc )3 + vik+1 (3(τc )2 ) + vik+2 (3τc ) = −vik+3

k+3 3 i 1 d (s yτ ) . sυ dsk+3

(12.17)

Because the previous equation is available for all k ≥ 0, it is also true for k + 1 and for k + 2. Accordingly, from the following system of three equations, we can deduce:

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⎧ i i i i 3 2 ⎪ ⎨ vk (τc ) + vk+1 (3(τc ) ) + vk+2 (3τc ) = −vk+3 i i i 3 2 vk+1 (τc ) + vk+2 (3(τc ) ) + vk+3 (3τc ) = −vik+4 ⎪ ⎩ i vk+2 (τc )3 + vik+3 (3(τc )2 ) + vik+4 (3τc ) = −vik+5

(12.18)

From Eq. (12.18), the following matrix linear system is established: ˆ τ = Qi (s, τ ) Pik (s, τ ) (12.19) k ⎤ vi vik+1 vik+2  T ⎢ ik ⎥ i with Pk (s, τ ) = ⎣vk+1 vik+2 vik+3 ⎦, Qik = − vik+3 vik+4 vik+5 , vik+2 vik+3 vik+4   ˆ c = (τˆc )3 3(τˆc )2 3τˆc T an estimation of the vector c = and  T (τc )3 3(τc )2 3τc from the output yi . Also, τˆc represents the estimation of a switching time τc in the interval IτT , from the output yi . Therefore, the value’s estimation of the switching time tc = τ − T + τc is given by ˆtc = τ − T + τˆc . Now, let us transform the system (12.19) in time domain, by expressing vik as a function of time for all k ≥ 0 and υ > 3. Thus, with Eq. (A.6) given in Appendix, vik ’s equivalent in time domain (for t ∈ [0, T]) is obtained as follows:   (−1)k+1 t d3  k+3 i υ−1 (t − ρ) (12.20) ρ yiτ (ρ)dρ vk (t, τ ) = (υ − 1)! 0 dρ 3 ⎡

where yiτ represents the restriction of the output yi in the interval IτT . Thus we obtain: ˆ τ = Qi (t, τ ) Pik (t, τ ) k

(12.21)

ˆ τ represents the estimation of τ for all t ∈ [0, T]. where At any instant τ , the value vik is calculated for all t ∈ [0, T]. Therefore, without loss of generality, we consider thereafter t = T, and we can deduce for all k ≥ 0 and υ > 3 the following equation:   (−1)k+1 T d3  k+3 υ−1 (T − ρ) (12.22) ρ yiτ (ρ)dρ vik (T, τ ) = (υ − 1)! 0 dρ 3 The latter equation can be written as:  T hk (ρ)yiτ (ρ)dρ vik (τ ) =

(12.23)

0

= where vi (τ ) is an abbreviation of vik (T, τ ) and hk (ρ)  k+1 3k   (−1) d k+3 υ−1   ρ ∈ [0, T] (T − ρ) d3 k+3 (T − ρ)υ−1 , (υ−1)! dρ 3 ρ . By developing dρ 3 ρ 0 otherwise

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the expression of hk (ρ) can be simplified as follows:  (−1)k+1 (h (ρ) − h2 (ρ) + h3 (ρ) − h4 (ρ)) ρ ∈ [0, T] hk (ρ) = (υ−1)! 1 0 otherwise

(12.24)

with h1 (ρ) = (k + 3)(k + 2)(k + 1)ρ k (T − ρ)υ−1 , h2 (ρ) = 3(k + 3)(k + 2) (υ − 1)ρ k+1 (T − ρ)υ−2 , h3 (ρ) = 3(k + 3)(υ − 1)(υ − 2)ρ k+2 (T − ρ)υ−3 , and h4 (ρ) = (υ − 1)(υ − 2)(υ − 3)ρ k+3 (T − ρ)υ−4 . Now let us present the proposed principle of switching detection. To simplify notations, let us replace Pik (T, τ ) and Qik (T, τ ), respectively, by Pik (τ ) and Qik (τ ). Detection Principle

The principle of switching detection is based on the mathematical results obtained in the previous section. The calculations were done for each instant τ and were based on all samples of the interval IτT . Thus, for each instant τ , we calculate all the elements of the system’s matrices Pik (τ ) and Qik (τ ) (Eq. 12.21) by applying the formula (12.23) in order to obtain vik for each instant τ . Therefore, the switching detection procedure can start only after the instant τ = T (so that τ −T ≥ 0). That is why we supposed the first switching (at time t1 ) can occur only after the instant τ = T (t1 > T). When the time interval IτT does not contain any switching, the signal yiτ stays smooth (μ0 = μ1 = μ2 = 0) and Eq. (12.4) becomes [(yiτ (t))(3) ](t). Therefore, the Pik (τ ) = 0 and Qik (τ ) = 0.

d3 (yiτ (t)) dt3

=

matrix linear system (12.21) degenerates into

Nevertheless, when a switching appears at time τc , it implies μ1 = 0, μ2 = 0, and μ0 = 0 (for a switching with rupture) or μ0 = 0 (for a switching without rupture). Moreover, we have Pik (τ ) = 0 and Qik (τ ) = 0, thus from the matrix linear system (12.21) we obtain det(Pik (τ )) = 0. Therefore, at each instant τ , the value of det(Pik (τ )) allows us to deduce if any switching occurs or not in the time interval IτT , and the switching time can be estimated. This approach is summarized in the following proposition. Proposition 2. Let us consider one output yi of vector y, and a minimal dwelltime TD . Let IτT = [τ − T, τ ] be the time interval (sliding window) whose length 0 < T < TD is considered small enough. The time interval IτT contains a switching at time tc if: Jki (τ ) > 0

(12.25)

where Jki is a function deduced from the matrix linear system (12.21), defined as follows: Jki (τ ) = | det(Pik (τ ))|

(12.26)

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In addition, tc ’s estimation (ˆtc ) is given by: ˆtc = τ − T + τˆc (12.27)   T ˆ τ = (τˆc )3 3(τˆc )2 3τˆc = (Pi (τ ))−1 Qi (τ ). ˆ τ (3, 1) and where τˆc = 13 k k Proof . When a switching is detected at τ > T in the time interval IτT from output yi , it implies det(Pik (τ )) = 0, meaning Jki (τ ) = | det(Pik (τ ))| > 0. Therefore, the time interval IτT contains a switching at time tc = τ − T + τc , where τc ∈ [0, T]. Moreover, we can deduce from Eq. (12.21) that   ˆ τ = (τˆc )3 3(τˆc )2 3τˆc T = (Pi (τ ))−1 Qi (τ ) and τˆci = 1 ˆ k 3 τ (3, 1), where τˆc k represents the estimation of τc . Accordingly, the estimated switching time from the output yi is ˆtc = τ − T + τˆc . Remark 1. The condition (12.25) is available only for a noiseless output yi . In case of a noisy signal, the function Jki (τ ) can never be null, therefore a detection threshold γJi should be considered. Thereby, for a noisy signal, the condition (12.25) from Proposition 2 should be replaced by the following one: Jki (τ ) > γJi

(12.28)

Remark 2. Theoretically, when a switching appears at time tc , it is detected in the interval IτT for all τ ∈ [tc , tc + T[. However, in practice, the detection of the first switching at time tc from the output yi is done with an unknown delay Tci . So the switching is detected as long as the time tc is included into sliding window IτT , meaning for all τ ∈ [tc + Tci , tc + T[. In addition, for each detection, the switching time estimation from the output yi is given by Eq. (12.27). In real time, the first instant τ that verifies the condition given in Eq. (12.28) is considered as a switching detected from the output yi . Thus, switching times detection can be obtained as follows:  ˆt1 = min(τ ∈ R+ | Ji (τ ) > γ i ) J k ˆtc = min(τ ∈ R+ | Ji (τ ) > γ i and τ > ˆtc−1 + T) c = 2, 3, . . . (12.29) J k where ˆt1 represents the time detection of the first switching by the Estimatori , and ˆtc−1 is the time detection of the previous switching. In the following section, we extend the detection principle for to MOSLS. 3.1.2 Switching Detection From All Outputs of a Multioutput SLS In the previous section, we presented an estimator that detects switching from one output yi for an SOSLS or MOSLS. Nevertheless, for an MOSLS, switching detection from all outputs is more precise than switching detection from only one output. Indeed, for a detection from one output, we can have a nondetection case, meaning a switching occurs at time tc while the estimator does not detect it in the time interval [tc , tc + T[. This is generally

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due to noise and/or a smooth dynamics change for an output yi . Moreover, a false detection is also possible, meaning the estimator detects a switching that does not exist. In order to reduce false and nondetection probabilities of one or more switching (for an MOSLS), switching detection is done by using all the outputs of the system. T  The system’s vector y is composed of p outputs (y = y1 · · · yi · · · yp ). Thus, we define an Estimatori for any output yi , i = 1, . . . , p, by using the same approach presented in Section 3.1.1. The detection condition given in Eq. (12.28) is used for any Estimatori , i = 1, . . . , p. A decision algorithm is developed in order to decide if a switching occurred or not, by analyzing all the estimator’s outputs. The decision’s principle is based on two criteria: (1) a logical decision by majority, and (2) it must be verified that there is no switching decided by the decision algorithm in the time interval IτT . Thanks to condition (2), when a switching occurs at time τ , the algorithm will detect it only one time in the interval [τ , τ + T[. Indeed, when an Estimatori detects a switching at time τ , the switching remains detected during the whole interval [τ , τ + T[ (Remark 2). Accordingly, it allows the algorithm to consider only the first detected switching in the interval. Moreover, contrary to the previous section, where the synthesized estimator provides direct detection of the different switching, here, each Estimatori ’s logical output si is connected to the decision algorithm. When the latter has not yet detected a switching, each output si takes the value 1 if Jki (τ ) > γJi . But once the decision algorithm detects a first switching, each output si takes the value 1 (at time τ ) only if Ji (τ ) > γ i and τ > ˆtc−1 + T, k

J

where ˆtc−1 is detection time by the decision algorithm of the previous switching. In other words, the expression of the output si is defined as follows:  si = 1 if Jki (τ ) > γJi and (Hyp is true or τ > ˆtc−1 + T) (12.30) si = 0 otherwise where Hyp represents the hypothesis: “the decision algorithm has not yet detected a switching.” To sum up, the proposed algorithm performs three tasks presented as follows: • Calculating Jki (τ ) function (given by Eq. 12.26) deduced from the system (12.21), at any instant τ and for each Estimatori , i = 1, . . . , p. • Calculating all logical outputs si of all estimators thanks to Eq. (12.30). p p • Calculating at any instant τ the expression S = i=1 si and if S ≥ 2 , then the algorithm detects that the instant τ corresponds to a switching time. The algorithm’s implementation is done on sliding windows (of length T), the principle is presented in the following section.

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Algorithm 1 A L G O R I T H M P R E S E N TAT I O N First step: Initialization Set for each output yi a detection threshold γJi and choose observation windows T = MTs , with Ts the sampling step and M + 1 the number of samples in the windows T; Initialize the parameter δ such as δ > T; Define the length tf of the sampled output y: tf = length(y); Second step: switching detection for j = M + 2 : tf if δ > T Step 1: y(jTs ) reading; for i = 1 : p si resetting (si = 0); Calculating Jki (τ ) function (given by Eq. 12.26) deduced from the system (12.21); if the condition (12.28) is verified then the Estimatori detects a switching at the instant τ = jTs , thus si = 1; end if end for p Calculating sum S = i=1 si ; p if S ≥ 2 , meaning the majority of estimators has detected a switching then the instant τ = jTs corresponds to a time of switching, and the parameter δ is resettled (δ = 0); else the instant τ = jTs does not correspond to a time of switching; end if else Step 2: The instant τ = jTs does not correspond to a time of switching, and si = 0. Thus the parameter δ increases by one sampling step Ts : δ = δ + Ts ; end if end for

switching detection times by the decision algorithm is obtained as follows: ⎧  ⎨ˆt1 = min τ ∈ R+ | S = p si ≥ p 2 i=1  ⎩ ˆtc = min τ ∈ R+ | S = p si ≥ p and τ > ˆtc−1 + T c = 2, 3, . . . 2 i=1 (12.31) where ˆt1 represents the time of the first switching detected by this algorithm, and ˆtc−1 is a time of the previous switching detected by this algorithm. III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

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Discussion

The principle of the decision algorithm consists of deciding if a switching has occurred or not, by analyzing the output si , i = 1, . . . , p of any estimator. It is composed of two execution phases: initialization and detection. First Step: Initialization In this step, the parameters T and γJi are defined, and δ is initialized. The parameter T corresponds to the length of the time interval (or windows) IτT . Thus, there is, at most, one switching during the time interval of length T. For each i = 1, 2, 3, the parameter γJi corresponds to the detection threshold, which depends on noise and parameter T. We have mentioned in Section 3.1.1 that when an estimator detects a switching at time τ , it is detected throughout the time interval [τ , τ + T[. Thus, knowing that the parameter δ is initialized such as δ > T, the algorithm takes into account only the first detection of that switching by the Estimatori . Second Step: Detection Assuming that no switching has occurred in the time interval [0, T], the algorithm starts the detection procedure at time τ = T + Ts (j = M + 2). While the condition δ > T is verified, the algorithm starts the initialization step and waits for switching occurrence. After y(jTs ) reading, the algorithm calculates for each Estimatori the function Jki (jTs ) deduced from the system (12.21). When the condition (12.28) is verified for an Estimatori , its output si takes the value 1 (otherwise it remains to 0). p Therefore, if the sum of all outputs si is greater or equal to 2 (meaning the majority of estimators has detected a switching), the algorithm concludes that a switching occurred at time τ = jTs and then resets the parameter δ to 0. Subsequently, the M + 1 following samples are ignored by the algorithm because that switching remains detected throughout the time interval [τ , τ + T[. For this, the algorithm increments the parameter δ by one sampling step (at each iteration) and verifies if δ > T. In the M + 1 samples following that detection, δ < T and therefore the algorithm attacks the second phase. p At time τ = jTs during step 1 (δ > T), when the outputs’ sum S = i=1 si p is less than 2 (meaning a minority of Estimatori has detected a switching), the algorithm considers that the instant τ = jTs does not correspond to a switching time. Thus at next iteration, the algorithm keeps the same value for δ (meaning δ > T) and consequently stays in step 1 in order to verify if a switching occurred or not.

3.2 Reconstruction of Discrete State The approach of discrete state estimation consists of two steps: • Switching times’ estimation by using the method presented in Section 3.1. III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

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421

• Reconstruction of the discrete state based on known switching sequences and initial modes. Based on system (12.29) for a single-output system or system (12.31) for a multioutput system, an estimation of the discrete state of the system (12.1) is given by:  ˆ = 0) = λ(t = 0); 0 ≤ t < ˆt1 λˆ = λ(t (12.32) λˆ = λ(tˆc−1 ) = λ(tc−1 ); tˆc−1 ≤ t < tˆc where ˆtc represents the instant of switching detection and ˆtc−1 represents the instant of previous switching detection. Then, the estimated discrete state is injected in a state observer (see Fig. 12.1) in order to estimate the system’s continuous state. The state observer synthesis is developed in the following.

4 CONTINUOUS STATE ESTIMATION 4.1 Coordinate Transformation In order to estimate both of the states and the faults, a coordinate transformation is introduced in order to decouple a sub of the states from the faults. We consider first a noiseless output, then the estimation is given taking into account that noise. Thus, under Assumption 2, one can find for each i ∈ I an arbitrary matrix Ni ∈ Rn×(n−l) such that Ti = Ni Fi ∈ Rn×n is a nonsingular matrix. Thus we can transform the continuous state x in another base and deduce a new state x¯ defined by: T  (12.33) x¯ = Tλ−1 x = x¯ 1 x¯ 2 where x¯ 1 ∈ n−l and x¯ 2 ∈ Rl are, respectively, the substates, which are (respectively, are not) affected by the considered faults. The result of this coordinate transformation allows us to obtain the equivalent of the system (12.1) given in a new base in the following form: ⎧ ¯ λ x¯ (t) + B¯ λ u(t) + F¯ λ f (t) ⎪ x˙¯ (t) = A ⎪ ⎨ ¯ λ x¯ (t) y(t) = C (12.34) ⎪ ⎪ + ⎩x¯ (t ) = T−1+ T − x¯ (t− ) c c λ(t ) λ(tc ) c

x¯ (t− c )

x¯ (t+ c )

and are, respectively, state vectors of the transformed where system before and after the switching instant tc . On the other hand, the matrices Tλ(t−c ) and Tλ(t+c ) are, respectively, the transformation matrices of the active mode of the system before and after the switching time (i.e., if

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FIG. 12.2 Jump of the state of the transformed system at a switching instant.

i and j are, respectively, the active mode before and after the switching instant tc , then Tλ(t−c ) = Ti and Tλ(t+c ) = Tj ). The other matrices of Eq. (12.34) are given by:      ¯ 12 ¯ 11 A A B¯ 1λ λ λ −1 −1 ¯ λ = T ATλ = ¯λ = T B = ¯λ = T−1 Fλ = 0 , and A , B , F λ λ λ ¯ 21 A ¯ 22 Il B¯ 2λ A λ λ   C¯ λ = Cλ Tλ = Cλ Nλ Cλ Fλ . The F¯ λ matrix corresponds to the fault matrix in the new base. From the structure of this matrix, we remark that a part of the transformed state is decoupled from the fault. Note that the last equation of Eq. (12.34) is introduced in order to update the state, taking into account the jumps generated by the coordinate transformation at each switching instants (Fig. 12.2). Indeed, by using −1 x(t+ Eq. (12.33), we obtain x¯ (t+ c ) = Tλ(t+ c ) where tc represents switching c ) time. Moreover, the state x from initial system (12.1) is continuous what− x¯ (t− ever the switching time, therefore x(t+ c ) = x(tc ) = Tλ(t− c ). Thus we c ) obtain the third part of the system (12.34). The first equation from Eq. (12.34) allows us to deduce the dynamics of the two substates x¯ 1 and x¯ 2 in the following form:  ¯ 11 x¯ 1 (t) + A ¯ 12 x¯ 2 (t) + B¯ 1 u(t) x˙¯ 1 (t) = A λ λ λ (12.35) ¯ 21 x¯ 1 (t) + A ¯ 22 x¯ 2 (t) + B¯ 2 u(t) + f (t) x˙¯ 2 (t) = A λ

λ

λ

Unlike the dynamic of x¯ 2 , the dynamic of x¯ 1 is independent of faults. Therefore, in order to synthesize the hybrid observer, allowing us to estimate in the first time x¯ 1 and to deduce, in a second time, the state x of the initial system (12.1), one exploits the dynamic of x¯ 1 given by: ⎧ ¯ 11 x¯ 1 (t) + A ¯ 12 x¯ 2 (t) + B¯ 1 u(t) ⎪ ⎨ x˙¯ 1 (t) = A λ λ λ (12.36) y(t) = Cλ Nλ Cλ Fλ x¯ (t) ⎪ ⎩x¯ (t+ ) = S − x¯ (t− ) 1 c λ ,λ c

  −1  where Sλ− ,λ = In−l 0(n−l)×l Tλ(t+ ) Tλ(t−c ) = S1λ− ,λ S2λ− ,λ is a matrix of c

order (n − l) × n, S1λ− ,λ is of order (n − l) × (n − l), and S2λ− ,λ is a matrix of

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order (n − l) × l. On the other hand x¯ 1 (t+ c ) is the state vector of the reduced transformed system (12.36) after the switching instant tc . Note that the last equation of Eq. (12.36) is introduced in order to update the substate x¯ 1 , taking into account the jumps generated by the coordinate transformation at each switching instants. It is deduced from the third part of the system (12.34). To eliminate the substate x¯ 2 , and synthesize the observer for the esti1 and 3, mation of x¯ 1 , one can find for each i ∈ I, under Assumptions   an arbitrary matrix Qi ∈ Rp×(p−l) such that Ui = Ci F Qi ∈ Rp×p is a T  nonsingular matrix. Thus, by considering that (Uλ )−1 = Uλ1 Uλ2 , where Uλ1 ∈ Rl×p and Uλ2 ∈ R(p−l)×p , it yields:    1C F 1Q U U 0 I λ λ λ λ λ (Uλ )−1 Uλ = (12.37) = l 2 2 0 I p−l Uλ Cλ Fλ Uλ Qλ Then, using Eq. (12.37) and pre- and postmultiplying the output y given in Eq. (12.36) by (Uλ )−1 , leads to:  Uλ1 y(t) = Uλ1 Cλ Nλ x¯ 1 (t) + x¯ 2 (t) (12.38) Uλ2 y(t) = Uλ2 Cλ Nλ x¯ 1 (t) From this system, we can establish the following equation: x¯ 2 (t) = Uλ1 y(t) − Uλ1 Cλ Nx¯ 1 (t) Finally, using Eqs. (12.36), (12.39), one obtains: ⎧ ˜ ⎪ ⎨ x˙¯ 1 (t) = Aλ x¯ 1 (t) + B˜ λ u(t) + E˜ λ y(t) y¯ (t) = C˜ λ x¯ 1 (t) ⎪ ⎩ ¯ (t− x¯ 1 (t+ c ) = Sλ− ,λ x c )

(12.39)

(12.40)

¯ 11 − A ¯ 12 U1 Cλ Nλ , B˜ λ = B¯ 1 , C ˜ λ = U2 Cλ Nλ , E˜ λ = A ¯ 12 U1 , and ˜λ = A where A λ λ λ λ λ λ λ 2 y¯ = Uλ y.

4.2 Observer Design We consider the observer structure associated with the reduced transformed system (12.40), defined as follows: ⎧ ˙ ⎪ ⎪ u(t) + Cobs y(t) xˆ¯ 1 (t) + Bobs xˆ¯ 1 (t) = Aobs ⎪ λˆ λˆ λˆ ⎪ ⎪ ⎨xˆ¯ (t+ ) = S − ˆ ¯ (tc ) 1 c λˆ −,λˆ x (12.41)  ⎪ xˆ¯ 1 (t) ⎪ ⎪ ⎪ ⎪ ⎩ xˆ (t) = Tλˆ U1 y(t) − U1 C N x¯ˆ 1 (t) ˆ ˆ λˆ

λˆ λ

λ

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where xˆ¯ 1 , xˆ , and λˆ correspond to the estimation of x¯ 1 , x, and λ, respectively. The estimate λˆ of the discrete state is given by Eq. (12.32) and take its ˆ = i value in the discrete set I = {1, . . . , N}. For all t ∈ Iˆc  [ˆtc−1 , ˆtc ), λ(t) ∞ ˆ (i ∈ I) where {tc }c=1 is a sequence of switching times estimation given by Eq. (12.31) for an MOSLS or by Eq. (12.29) for an SOSLS and the observer evolves in the ith mode defined as follows: ⎧ ˙ obs obs ˆ¯ ⎪ xˆ¯ 1 (t) = Aobs ⎪ i x1 (t) + Bi u(t) + Ci y(t) ⎪ ⎪ ⎨xˆ¯ (t+ ) = S xˆ¯ (t− ) 1 c j,i (12.42)   c ⎪ ˆ¯ 1 (t) ⎪ x ⎪ ⎪ ⎩ xˆ (t) = Ti Ui1 y(t) − Ui1 Ci Ni xˆ¯ 1 (t) where j is the previous mode of i, xˆ¯ (t− c ) is the state vector estimation of the transformed system (12.34) before the switching time estimation ˆtc ; on the other hand, xˆ¯ 1 (t+ c ) is the state vector estimation of the reduced transformed system (12.36) after the switching time estimation ˆtc . obs obs 2 ˜ ˜ ˜ ˜ For each i ∈ I, Aobs i = Ai − Li Ci , Bi = Bi , Ci = (Li Ui + Ei ), and Li are observer gains that are chosen such that the estimation error between the estimated substate xˆ¯ 1 and the real substate x¯ 1 converges asymptotically to zero. This asymptotic convergence allows also an asymptotic convergence of the estimated state xˆ to the real state x of the SLS (12.1). The online reconstruction of the discrete state λ is made with a certain delay Tc occurring at any switching time tc . Consequently, this will induce a detection delay of the active mode after any discrete transition. Thus, between two switching times tc and tc+1 we have two phases (see Fig. 12.3). The first one represents the phase between the switching occurrence at time tc and its time detection ˆtc = tc + Tc . During this phase, the system and the observer do not evolve necessarily in the same mode. Therefore, the observer cannot follow the trajectory of the state x¯ 1 of the transformed reduced system (12.40) and also of the state x of system (12.1). When the switching is detected at time ˆtc = tc + Tc , the system and the observer start evolving at that time in the same mode (second phase).

FIG. 12.3 Evolution of the observer and the system between two switching instants.

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Thus, the estimated state xˆ¯ 1 (respectively, xˆ ) starts to converge toward x¯ 1 (respectively, x). Several simulation results have shown that the switching detection is done in a short time, thus, the following hypothesis can be considered as: Assumption 5. Time-delay in switching detection is negligible, and both the system and the observer evolve in the same mode (λˆ = λ). The estimation error between the estimated substate xˆ¯ 1 and the real substate x¯ 1 is given by: ε1 (t) = xˆ¯ 1 (t) − x¯ 1 (t)

(12.43)

In the following, a proposition is introduced to give the explicit expression of the dynamics of the estimation error. Proposition 3. We consider the SLS (12.1), the reduced transformed system (12.36) and the observer (12.41). The dynamics of the estimation error (12.43) is given by:  ε˙ 1 (t) = Aobs λ ε1 (t) (12.44) ˆ− ) = G ε1 (ˆt+ λ− ,λ ε1 (tc ) c ˆ+ where Gλ− ,λ = S1λ− ,λ − S2λ− ,λ Uλ1 Cλ Nλ and ε1 (ˆt− c ) and ε1 (tc ) are, respectively, the estimation error before and after the switching time estimation ˆtc . Proof . The first equation of Eq. (12.44) represents the evolution of the estimation error dynamics in each mode and its proof is trivial. The second part is necessary for the update of estimation error at any switching time. Let tc a switching time. The estimation error after a switching time tc can be defined as follows: ˆ¯ 1 (t+ ¯ 1 (t+ ε1 (t+ c )=x c )−x c ) By using Eq. (12.40), we obtain: ε1 (t+ c )

= Sλ− ,λ





¯ (t− xˆ¯ (t− c )−x c )

=



S1λ− ,λ

S2λ− ,λ



  ! " x¯ 1 (t− xˆ¯ 1 (t− c ) c ) − x¯ 2 (t− x¯ˆ 2 (t− c ) c )

So we can rewrite the expression as follows:  1 2 ˆ¯ 2 (t− ˆ¯ 1 (t− ¯ 1 (t− ¯ 2 (t− ε1 (t+ c ) = Sλ− ,λ x c )−x c ) + Sλ− ,λ (x c )−x c ))

(12.45)

Then we obtain by using Eq. (12.39):  1 1 ¯ 1 (t− x¯ 2 (t− c ) = Uλ y(t) − Uλ Cλ Nλ x c ) 1 1 − x¯ˆ 2 (tc ) = U y(t) − U Cλ Nλ xˆ¯ 1 (t− c ) λ

λ

Therefore, by introducing these two expressions in Eq. (12.45), we obtain the second part of the system (12.44).

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Finally, in order to study the convergence of the estimation error, let us introduce a switched Lyapunov function defined as: V(ε1 (t)) = ε1T (t)Pλˆ (t) ε1 (t)

(12.46)

The function Pλˆ (t) takes its value in the discrete set {P1 , . . . , PN } where P1 , . . . , PN are positive definite matrices. For all t ∈ Iˆc  [ˆtc−1 , ˆtc ), Pλˆ (t) (i ∈ I) and V(ε1 (t)) = ε1T (t)Pi ε1 (t). In the following, the main results of the work are delivered. The first one provides the LMI formulation, allowing the design of the observer (12.41), and the second one concerns the estimation of the considered faults. Theorem 1. We consider the SLS (12.1), the reduced transformed system (12.36) and the observer (12.41). If there exists for each i ∈ I matrices Hi ∈ (n−l)×(p−l) , symmetric positive definite matrices Pi ∈ (n−l)×(n−l) such that: ˜ T Pi + Pi A ˜ i − HT C ˜i − C ˜ T Hi ≤ 0 A i i  i T Pi Gi,j Pj ≥0 * Pj

(12.47) (12.48)

where j is the successor mode of i. T Then the switched observer (12.41) with the gain matrices Li = P−1 i Hi , ensures the asymptotic convergence of the estimation error (12.43), as well as that of the estimated state xˆ to the real state x. Proof . To prove the convergence of the estimation error (12.43), let us at first consider a switched Lyapunov candidate function V(ε1 (t)) of the form (12.46). Thus, for λ(t+ ) = λ(t) = i, the time derivative of the Lyapunov function leads to: ˙ 1 (t)) = ε˙ T (t)Pi ε1 (t) + εT (t)Pi ε˙ 1 (t) V(ε 1 1 Then, using Eqs. (12.44), (12.49) becomes:

 ˙ i (ε1 ) = εT (Aobs )T Pi + Pi Aobs ε1 V 1 i i

(12.49)

(12.50)

˜ ˜ Now, because Aobs i = Ai − Li Ci , it can be shown that the estimation error convergence is guaranteed if the following matrix inequalities are verified for each mode i ∈ I ˜ i − Pi Li C ˜ i − C˜ T LT Pi ˜ T Pi + Pi A A i i i

(12.51)

Then, after introducing a bijective change of variables Hi = (Pi Li )T , the inequality (12.47) is obtained. On the other hand, in order to guarantee the asymptotic stability of the estimation error, the following condition should be ensured at each switching instant [14]: V(ε1 (t+ )) ≤ V(ε1 (t)) if λ(t+ ) = λ(t).

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Thus, by considering that the mode j is the successor of the mode i we get: ε1T (t)(GTi,j Pj Gi,j − Pi )ε1 (t) ≤ 0

(12.52)

This implies that GTi,j Pj Gi,j − Pi ≤ 0 is equivalent, by the Schur complement (see Appendix), to Eq. (12.48).

5 FAULT ESTIMATION Until now, we have discussed the estimation of the state, but we have not yet dealt with the problem of the fault estimation. In fact, according to the development of Section 4.1, it is clear that the estimation of the fault is directly linked to the estimation of the substate xˆ 1 . Thus, on the basis of the previous results, the following proposition can be formulated. Proposition 4. By considering SLS (12.1), the reduced transformed system (12.40) and the gain matrices Li , i ∈ I, issued from Theorem 1, then the following estimator ensures the asymptotic reconstruction of the faults. fˆ (t) = Uλ1ˆ y˙ + θλˆ1 xˆ¯ 1 + θλˆ2 y + θ ˆ3 u λ

(12.53)

where λˆ is the discrete state estimation given by Eq. (12.32) and takes its values from the following discrete set I = {1, . . . , N}. For each i ∈ I, θi1 = 1 2 ¯ 22 1 ¯ 21 2 ¯ 22 1 ˜ −Ui1 Ci Ni Aobs i + Ai Ui Ci Ni − Ai , θi = −Ui Ci Ni (Li Ui + Ei ) − Ai Ui , and 3 1 1 2 θi = −Ui Ci Ni B¯ i − B¯ i . Proof . Using Eqs. (12.35), (12.39), one can find the following equations ¯ 21 x1 (t) + A ¯ 22 x2 (t) − B¯ 2 u(t) + f (t) and x2 (t) = U1 y(t) − U1 Cλ x2 (t) = A λ λ λ λ λ Nλ x1 (t). Thus, under the conditions (12.47), (12.48) of Theorem 1, one can rewrite ˙ ¯ 21 xˆ¯ 1 (t) − the two last expressions in the following form: fˆ (t) = xˆ¯ 2 (t) − A λ ¯ 22 xˆ¯ 2 (t) − B¯ 2 u(t) and xˆ¯ 2 (t) = U1 y(t) − U1 Cλ Nλ xˆ¯ 1 (t). Finally, the estimator A λ λ λ λ (12.53) is obtained after an algebraic manipulation. At this stage, the proposed estimator of the fault (12.53) requires the calculation of the derivative of the SLS outputs. This problem is handled, using a robust exact differentiator based on sliding modes techniques [37]. Moreover, because it has been shown that the accuracy of the derivatives increases when the order of the used differentiator gets higher [38, 39], the derivatives of the SLS output signals are herein recovered using a high-order differentiator instead of a first-order one (i.e., the accuracy is getting better for a differentiator of order r than the one given by an order r − 1). Thus, a differentiator of order r can be used to calculate the r first derivatives of the output y, including its first-order derivative that we need for the faults vector estimation. Therefore, assuming that the output and

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noise are bounded, the algorithm (with r outputs) is given by the following system: ⎧ z˙ 0 = υ0 ⎪ ⎪ r ⎪ ⎪ ⎪ υ0 = −α0 |z0 − y| (r+1) sign(z0 − y) + z1 ⎪ ⎪ ⎪ ⎪ z˙ 1 = υ1 ⎪ ⎪ ⎪ ⎨ υ = −α |z − υ | (r−1) 1 1 1 0 r sign(z1 − υ0 ) + z2 (12.54) . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ z˙ r−1 = υr−1 ⎪ ⎪ ⎪ ⎪ υ = −αr−1 |zr−1 − υr−2 |1/2 sign(zr−1 − υr−2 ) + zr ⎪ ⎪ ⎩ r−1 z˙ r = −αr sign(zr − υr−1 ) where, for all j = 0, . . . , r, αj represent positives gains. These gains are chosen in a suitable way in order to ensure finite-time convergence of the algorithm and to minimize chattering phenomenon on the estimated derivatives [39]. In [37], Levant has expressed these coefficients according to the Lipschitz constant L of the function as follows: 1

αj = α0j L r−j+1 ,

j = 0, . . . , r

(12.55)

Initialization of the algorithm (12.54) can be done as follows: z0 (0) = y(0) and for all j = 1, . . . , r, zj (0) = υj−1 (0) = 0. The vector z =  T z0 z1 · · · zr represents the output of the algorithm. The second term z1 of the vector z represents an estimation of the first-order derivative of y that we need to estimate the faults vector f . The chattering phenomenon that appears in the estimated successive derivatives can be damped by choosing low values for αj (j = 0, . . . , r). However, these values should be high enough to ensure finite-time convergence. Therefore, the values for gains αj (j = 0, . . . , r) must be chosen to minimize the chattering phenomenon and to ensure finite-time convergence of the algorithm.

6 SIMULATION RESULTS Let us consider an SLS, with three modes and the following matrices satisfying Assumptions 1–5. ⎡ ⎡ ⎤ ⎤ −1 3 −2 −3 −2 −1 −2 2 1 0 ⎢−2 0 ⎢ 0 −2 2 1 1 −2⎥ 1 −1⎥ ⎢ ⎢ ⎥ ⎥ ⎥ , A2 = ⎢ 1 −1 0 −2 −1 0 −3 1 1 −2⎥ A1 = ⎢ ⎢ ⎢ ⎥ ⎥, ⎣2 ⎣ 0 −1 −1 −2 2 ⎦ 0 1 0 −3⎦ −2 1 0 2 −1 0 1 1 1 −2

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FIG. 12.4 Switching signal λ.

⎡

⎡ ⎤ ⎤ 1 3 1 ⎢0⎥ −1 2 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 2 −3⎥ ⎥ , B1 = B2 = B3 = ⎢1⎥ , ⎣0⎦ ⎦ −1 3 0 −1 0 ⎡ ⎤T ⎤ 0 0 1 0 ⎢0 1⎥ 0 0⎥ ⎢ ⎥ ⎥ ⎥ ⎥ 1 0⎥ , and F1 = F2 = F3 = ⎢ ⎢1 1⎥ ⎣0 0⎦ 0 1⎦ 0 0 0 1   In this example, x ∈ R5 is the state, y = y1 y2 y3 ∈ R3 is the output, m = 1 is the number of known inputs, and l = 2 is the number of faults. For the simulation task, let us consider that the system switches between the three modes according to the sequence shown in Fig. 12.4. In order to show the efficiency of the proposed fault estimation technique, faults will appear at the instant t = 3.5 s and have the following formulations f1 (t) = 7 sin(t) and f2 (t) = 6 cos(t). On the other hand, the control is given by u = 1. The switching times are given as follows: t1 = 1 s, t2 = 4 s, t3 = 7.4 s, t4 = 12.5 s, t5 = 16.5 s, t6 = 20.2 s, t7 = 25 s, t8 = 29.5 s, and t9 = 34 s  T and the system’s initial state is defined by x(0) = 4 5 −3 4 2 . The software Matlab/Simulink is used for simulation, with a sampling step 1 . The measurement noise η is a white Gaussian one, generated in Ts = 3000 Matlab/Simulink with the block “band-limited white noise,” with a power given by “noise power: [0.000001].” The variation of the noise power is given by the parameter “seed” of this block, by [23341]. The signal-to-noise ratio (SNR) is 57 dB for y1 , 56 dB for the second output y2 , and 64 dB for y3 . −1 −1 ⎢1 1 ⎢ A3 = ⎢ ⎢−2 0 ⎣0 3 −1 1

−1 −2 0 −3 1 ⎡ 0 ⎢1 ⎢ C1 = C2 = C3 = ⎢ ⎢0 ⎣0 0

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12. FAULT AND SWITCHING INSTANTS ESTIMATION FOR SLS

6.1 Discrete State Estimation The discrete state estimation is performed on two phases. The first consists of estimating switchings by the method presented in Section 3.1. The second phase consists of the reconstruction of the discrete state from the system (12.32). First, we start by showing the simulation results of the first phase. As we mentioned in Section 3.1.1, switching detection starts by calculating Jki function given by Eq. (12.26). For this, it is necessary to calculate the functions vik of matrices Pik and Qik given by Eq. (12.23). In continuous time and at a given instant τ , we have vik (τ ) = #T i 0 hk (ρ)yτ (ρ)dρ for all k ≥ 0 and υ > 3, where hk (ρ) is defined by Eq. (12.24). During the method’s implementation, we have considered the samples of the output vector y with a sampling step Ts . The time interval [0, T] is replaced by the set of samples 0, . . . , M (T = MTs ). Thus, each instant τ is equivalent to a sample nTs , and time interval IτT is replaced by a set of samples [(n − M)Ts , . . . , nTs ]. For ρ ∈ [0, T], the output  T yτ (ρ) = y1τ (ρ) y2τ (ρ) y3τ (ρ) is therefore equivalent to yn−M+m =  1 T yn−M+m y2n−M+m y3n−M+m for all m = 0, . . . , M. By applying trapezoidal methods on vik expression for all k ≥ 0 and υ > 3, we obtain the new expression: vik (τ ) = vik,n = 

M 

1 Wm hk,m yin−M+m ,

i = 1, 2, 3

(12.56)

m=0

0.5 pour m = 0 and m = M , and hk,m a numerical filter 1 pour m = 2, . . . , M − 1 given by {hk,m = hk (mTs )}M m=0 . All terms hk,m , m = 0, . . . , M are calculated m and T by 1 in Eq. (12.24). by replacing ρ by M In this example, we have considered M = 500. The system has three outputs yi , thus each Estimatori , i = 1, 2, 3 is dedicated to one output yi . We have also chosen υ = 5, k = 1, and detection thresholds γJ1 = γJ2 = γJ3 = 1 = where Wm

710−6 . At each instant τ = nTs , n > M, we have J1i (nTs ) = | det(Pi1 (nTs ))| ⎛ i ⎞ v1 vi2 vi3 ⎜ ⎟ for all i = 1, 2, 3, with Pi1 = ⎝vi2 vi3 vi4 ⎠. vi3 vi4 vi5 By applying the decision algorithm presented in Section 1, detection times are presented in Table 12.1 where tc represents the time of real switching, and ˆtic , i = 1, 2, 3 corresponds to detection time by an Estimatori . The instant ˆtc represents the detection time by the decision algorithm. We can observe that the majority of switchings are detected by all estimators. The switching that occurred at t1 = 1 s is detected by Estimator3 at III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

431

SIMULATION RESULTS

TABLE 12.1 Switching Detection FD

tc

1

ˆt1c

1.018

ˆt2c

1.099

ˆt3c ˆtc

4

7.4

12.5

16.5

20.2

25

29.5

34

4.0507

7.436

12.522

16.544

20.2257

25.0273

29.5847

34.0217

4.0433

7.432

12.525

ND

ND

25.029

ND

34.0233

1.0157

4.0487

ND

12.5237

16.539

20.2277

25.0537

29.557

ND

1.018

4.0487

7.436

12.5237

16.544

20.2277

25.029

29.5847

34.0233

3.596

FD , fault detection; ND , not detected.

ˆt3 = 1.0157 s, whereas the Estimator1 and Estimator2 detect it, respectively, 1 at ˆt11 = 1.018 s and ˆt21 = 1.099 s. Therefore, the decision algorithm considers ˆtc = ˆt1 = 1.018 s as the switching time because, at this moment, the majority 1 of estimators have detected the switching. On the other hand, the switchings occurring at t3 = 7.4 s, t5 = 16.5 s, t6 = 20.2 s, t8 = 29.5 s, and t9 = 34 s are detected by only two estimators (meaning the majority of estimators), therefore the decision algorithm concludes that these instants correspond to switching times. For example, the switching at t5 = 16.5 s is only detected by Estimator1 (at ˆt15 = 16.544 s) and Estimator3 (at ˆt35 = 16.539 s). Thus, the decision algorithm considers the instant ˆtc = ˆt15 = 16.544 s as the switching time, because at this moment, the majority of estimators have detected the switching. On the other hand, at τ = 3.596 s, there is a false detection by Estimator2 . This false detection is therefore ignored by the decision algorithm because the majority of estimators did not detect it. Once all switchings are detected and based on a known switching sequence, the estimated switching signal λˆ is deduced by estimator (12.32). Thus, the switching signal λ and its estimation λˆ are given in Fig. 12.5. In the following, we proceed to estimate the system continuous state.

ˆ FIG. 12.5 Switching signal λ and its estimation λ. III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

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12. FAULT AND SWITCHING INSTANTS ESTIMATION FOR SLS

6.2 Continuous State Estimation Driven by the development given in Section 4.1, one introduces the nonsingular transformation matrices that allow decoupling the faults from a subset of the transformed state. Thus, the following matrices Ni and Qi (i = 1, 2, 3) are selected: ⎡ ⎡ ⎤T ⎤T −5 −1 2 −5 −1 2 −5 3 0 −2 4 ⎦ , N2 = ⎣−2 2 1 −4 −3⎦ , N1 = ⎣−4 −2 −2 0 −1 1 4 −2 0 0 3 3 −3 5 ⎡ ⎤T 1 −4 −3 2 −3  T N3 = ⎣1 2 −4 4 −4⎦ , Q1 = 5 5 2 , 4 −2 −5 −3 5  T  T Q2 = −3 0 3 , and Q3 = −2 −1 5 Finally, by solving the inequalities (12.47), (12.48) of Theorem 1, the following observer gain matrices are obtained: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ −5.4897 1.0212 23.7127 L1 = ⎣−0.0425⎦ , L2 = ⎣−2.3719⎦ , and L3 = ⎣−4.6914⎦ 6.7638 −1.1462 −1.8971  T The observer’s initial state is defined by xˆ¯ 1 (0) = 20 20 −20 . From Figs. 12.6–12.8, we present the evolution of the state x and its xˆ . These figures represent simulation results based on matrix gains obtained after the LMI resolve in Theorem 1, and they show the effectiveness of the proposed observer (12.41). In fact, we notice that even with noise presence, the estimated state xˆ converges toward the real state x. However, after any switching, the observer diverges in a finite-time interval because of timedelay detection Tc at time tc (c = 1, 2, . . .). Once a switching is detected at time τ = ˆtc , the estimated discrete state λˆ converges toward λ (λˆ λ) and the state observer starts converging.

6.3 Fault Estimation In this paragraph, we present simulation results for faults estimation, based on the estimation of output vector y’s derivative. The latter has been estimated by the differentiator equation (12.54) based on sliding modes techniques. We choose r = 5, L = 10, and α0i = 5 * (r − i + 1), i = 0, . . . , 5. Thus, the different gains αi i = 0, . . . , 5 are calculated by using Eq. (12.55). Fig. 12.9 presents the evolution of faults and their estimations. These simulation results show that when the estimated state xˆ converges toward the real state x of the SLC (12.1), the estimated faults vector fˆ converges toward the real faults vector f . Thus, chattering phenomenons

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

(B)

FIG. 12.6 (A) State x1 and its estimation xˆ 1 . (B) State x2 and its estimation xˆ 2 .

(A)

SIMULATION RESULTS

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

(A)

(B)

FIG. 12.7 (A) State x3 and its estimation xˆ 3 . (B) State x4 and its estimation xˆ 4 .

433

434

12. FAULT AND SWITCHING INSTANTS ESTIMATION FOR SLS

FIG. 12.8 State x5 and its estimation xˆ 5 .

(A)

(B) FIG. 12.9 (A) Fault f1 and its estimation fˆ1 . (B) Fault f2 and its estimation fˆ2 .

appear at any switching time, because of time-delay detection (Tc ) of switching and dynamic changes at times tc . On the other hand, we notice the appearance of noise in the estimated faults, because of the amplification of noise during output y’s derivation by

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

435

APPENDIX

* the differentiator. In fact, the SNR defined by SNRdb = 10 * log10

f

fˆ −f

+

and calculated after faults apparition at time τ = 3.5 s equals SNR = −0.3644 dB for fault f1 and SNR = 0.8014 dB for fault f2 .

7 CONCLUSION This work considers the problem of switching detection and the problem of fault estimation for a class of SLS. An estimator based on an algebraic approach has been proposed in order to detect switching times from one output of an SLS with a model not necessarily known, and by using only the noisy outputs. The detection’s principle consists in detecting dynamic changes of the output, corresponding to switching times. Then, the method has been extended for a multioutput SLS in which an estimator was affected to each output. A decision algorithm has been designed to analyze all estimators’ outputs, and to decide if any switching occurred or not by using the principle of majority vote. After that, a switched unknown input observer has been proposed to estimate the state and reconstruct the fault. Through a coordinate transformation, and some nonrestrictive classical hypotheses, the states are decoupled from the faults. Then, based on switched Lyapunov functions, LMI conditions were delivered in order to obtain the observer gain matrices, ensuring the convergence of the estimation error. The faults are, then, estimated using the designed observer and a robust exact differentiator simultaneously. Finally, an illustrative example has shown the performance and the limitation of this approach.

APPENDIX Heaviside function is given by: H(t) =



0 si t < 0 1 si t ≥ 0

(A.1)

The derivative of order N of a function h that changes dynamics at a time τc satisfies the following relation:  (1) N−1  dN h dN−1 h (k) = + δ (t − τ ) (A.2) μ c N−k−1 dtN dtN−1 k=0

N−1 (1) N−1 where d N−1h is the derivative of the regular part of d N−1h . dt dt The transformed of the derivative of order n of a function ϕ in operational area is given by [40]: + * n n−1  d ϕ(t) n Φ(s) − sn−k−1 ϕ (k) (0) (A.3) = s TL dtn k=0

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12. FAULT AND SWITCHING INSTANTS ESTIMATION FOR SLS

The transformed operational area of a delayed function is given by: TL(ϕ(t − τr )) = e−sτr Φ(s)

(A.4)

The transformed operational area of a Dirac function is given by: TL(δ(t)) = 1

(A.5)

Let Φ the transformed in operational area of a function ϕ, we so have: * +  t r d (−1)κ 1 dκ (sr Φ(s)) −1 TL [(t − ρ)υ−1 ρ κ ]ϕ(ρ)dρ (A.6) = sυ dsκ (υ − 1)! 0 dtr

References [1] D. Efimov, A. Zolghadri, T. Raïssi, Actuator fault detection and compensation under feedback control, Automatica 47 (8) (2011) 1699–1705. [2] J.H. Richter, W.P.M.H. Heemels, N. van de Wouw, J. Lunze, Reconfigurable control of piecewise affine systems with actuator and sensor faults: stability and tracking, Automatica 47 (4) (2011) 678–691. [3] H. Noura, D. Sauter, F. Hamelin, D. Theilliol, Fault tolerant control in dynamic systems: application to a winding machine, Control Syst. Mag. 20 (1) (2000) 33–49. [4] D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA, 2003. [5] A.V.D. Schaft, H. Schumacher, An Introduction to Hybrid Dynamical Systems, vol. 251, Springer-Verlag, London, 2000, 174 pp. [6] J.P. Hespanha, Stabilization through hybrid control, Control Syst. Robot. Autom. 15 (2004) 11. [7] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (4) (1998) 475–482. [8] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Decision and Control, 1999, Proc. 38th IEEE Conference, IEEE, vol. 3, 1999, pp. 2655–2660, https://doi.org/10.1109/CDC.1999.831330. [9] R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proc. IEEE 88 (7) (2000) 1069–1082. [10] S.H. Lee, T.H. Kim, J.T. Lim, A new stability analysis of switched systems, Automatica 36 (6) (2000) 917–922. [11] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach, IEEE Trans. Autom. Control 47 (11) (2002) 1883–1887. [12] P.J. Antsaklis, X.D. Koutsoukos, Hybrid dynamical systems: review and recent progress, in: T. Samad, G. Balas (Eds.), Software-Enabled Control: Information Technology for Dynamical Systems, John Wiley & Sons, Inc., Hoboken, NJ, 2003, https://doi.org/ 10.1002/047172288X. [13] Z. Sun, Switched Linear Systems: Control and Design, Springer-Verlag, London, 2006, 329 pp. [14] G. Xie, L. Wang, Stability and stabilization of switched impulsive systems, in: American Control Conference, vol. 4, IEEE, Minneapolis, MN, 2006, pp. 4406–4411. [15] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control 54 (2) (2009) 308–322. [16] D. Liberzon, R. Tempo, Common Lyapunov functions and gradient algorithms, IEEE Trans. Autom. Control 49 (6) (2004) 990–994.

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

REFERENCES

437

[17] J. Lu, L.J. Brown, A multiple Lyapunov functions approach for stability of switched systems, in: American Control Conference (ACC), 2010, pp. 3253–3256. [18] P.J. Antsaklis, Persistent disturbance attenuation properties for networked control systems, in: 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), vol. 1, 2004, pp. 953–958. [19] A. Abdo, W. Damlakhi, J. Saijai, S.X. Ding, Design of robust fault detection filter for hybrid switched systems, in: Conference on Control and Fault-Tolerant Systems, IEEE, Nice, France, 2010, pp. 161–166. [20] A. Abdo, S.X. Ding, J. Saijai, W. Damlakhi, Fault detection for switched systems based on a deterministic method, in: Conference on Decision and Control, IEEE, 2012, pp. 568–573. [21] D.E.C. Belkhiat, N. Messai, N. Manamanni, A fault detection & isolation scheme for discrete time switched linear systems, 2011 9th IEEE International Conference on Control and Automation (ICCA), 2011, pp. 596–601. [22] D.E.C. Belkhiat, N. Manamanni, N. Messai, M. Djemai, Fault detection & isolation for a class of hybrid systems: a dedicated switched robust observer scheme, in: Mediterranean Conference on Control & Automation, IEEE, 2012, pp. 984–989. [23] A. Alessandri, P. Coletta, Design of observers for switched discrete time linear systems, in: Proceedings of the American Control Conference, vol. 4, IEEE, Denver, CO, 2003, pp. 2785–2790. [24] F.J. Bejarano, A. Pisano, E. Usai, Finite-time converging jump observer for switched linear systems with unknown inputs, Nonlinear Anal. Hybrid Syst. 5 (2) (2011) 174–188. [25] F.J. Bejarano, L. Fridman, Exact observer for switching linear systems via super twisting algorithm, IFAC Conf. Anal. Des. Hybrid Syst. 42 (17) (2009) 370–375. [26] F. Arichi, M. Djemai, B. Cherki, Active mode detection for a class of hybrid dynamic systems, in: International Conference on Systems, Signal Processing and Electronics Engineering, Dubai, 2012, pp. 210–215. [27] Y. Tian, T. Floquet, L. Belkoura, W. Perruquetti, Switching time estimation for linear switched systems: an algebraic approach, in: Conference on Decision and Control, IEEE, Shanghai, China, 2009, pp. 3909–3913. [28] Y. Tian, T. Floquet, L. Belkoura, W. Perruquetti, Algebraic switching time identification for a class of linear hybrid systems, Nonlinear Anal. Hybrid Syst. 5 (2011) 233–241. [29] Y. Tian, T. Floquet, L. Belkoura, W. Perruquetti, Algebraic switching time identification for a class of linear hybrid systems, IFAC Conf. Anal. Des. Hybrid Syst. 5 (2) (2009) 233–241. [30] D. Mincarelli, T. Floquet, L. Belkoura, Active mode and switching time estimation for switched linear systems, in: Conference on Decision and Control and European Control Conference, IEEE, Orlando, FL, 2011, pp. 1854–1859. [31] M. Mboup, C. Join, M. Fliess, Numerical differentiation with annihilators in noisy environment, Numer. Algorithms 50 (4) (2009) 439–467. [32] M. Mboup, Parameter estimation via differential algebra and operational calculus. Research Report, 2007, pp. 40. Hal-Inria-00138294. https://hal.inria.fr/inria-00138294/ document. [32a] M. Fliess, H. Sira–Ramírez, An algebraic framework for linear identification, ESAIM Contr. Optim. Calc. Var. 9 (2003) 151–168. [33] M. Mboup, Neural spike detection and localisation via volterra filtering, in: IEEE International Workshop on Machine Learning for Signal Processing, 2012, p. 4. [34] M. Hou, P.C. Muller, Design of observers for linear systems with unknown inputs, IEEE Trans. Autom. Control 37 (6) (1992) 871–875. [35] T.G. Park, K.S. Lee, Process fault isolation for linear systems with unknown inputs, IEE Proc. Control Theory Appl. 151 (6) (2004) 720–726. [36] K.M. Pekpe, G. Mourot, K. Gasso, J. Ragot, Identification of switching systems using change detection technique in the subspace framework, in: Conference on Decision and Control, vol. 4, IEEE, Nassau, Bahamas, 2004, pp. 3720–3725.

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL

438

12. FAULT AND SWITCHING INSTANTS ESTIMATION FOR SLS

[37] A. Levant, Higher-order sliding modes, differentiation and output feedback control, Int. J. Control 76 (9/10) (2003) 924–941. [38] A. Levant, Sliding order and sliding accuracy in sliding mode control, Int. J. Control 58 (6) (1993) 1247–1263. [39] A. Levant, Robust exact differentiation via sliding mode technique, Automatica 34 (3) (1998) 379–384. [40] J.L. Schiff, The Laplace Transform: Theory and Applications, vol. 85, Springer-Verlag, New York, 2001, 231 pp.

III. OBSERVER-BASED FAULT DETECTION AND TOLERANT CONTROL