An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements

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An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurementsR Liu Xiaoyu a,b, Wei Shanbi a,b,∗, Chai Yi a,b a State

Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China b College of Automation Chongqing University, Chongqing 400044, China Received 11 March 2019; received in revised form 29 April 2019; accepted 30 September 2019 Available online xxx

Abstract This paper considers the fault estimation problem for a class of nonlinear system with quantised measurements. An iterative learning observer scheme is constructed in this paper, which combined with a logarithmic quantiser of output signals, and the number of quantisation levels of output signals are finite. Compared with the existing approaches of observer-based fault estimation, the proposed iterative learning observer scheme in this paper improve the fault estimation performance in the current iteration by considers both state error and fault estimation consequence of previous iteration. Meanwhile, the designed observer achieves stability and convergence, since Lyapunov stability theory is employed. Moreover, the extension from nominal system to system with parameter uncertainties subjecting to Bernoulli-distributed white sequences with known conditional probabilities is also addressed. Finally, an illustrative example is provided to verify the theoretical results. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

R

Fault Estimation for Nonlinear Uncertain System CTAN. Corresponding author at: State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China. E-mail addresses: [email protected] (L. Xiaoyu), [email protected] (W. Shanbi), [email protected] (C. Yi). ∗

https://doi.org/10.1016/j.jfranklin.2019.09.040 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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1. Introduction In academic and industrial area, the demand on reliability, safety and maintainability is ever-increasing, hence, the researches on fault diagnosis [1–4] and fault tolerant control [5–7] have received more attentions. While, Fault estimation [8,9] becomes the most critical factor and one of the basic researches in this field, since the precise magnitude and shape of the faults is provided by it and higher system performance can be guaranteed. Up to date, there are already considerable research results in the field of fault estimation have been reported in the literature, see [10–12] and the references therein [13–16]. While, in reality, most of the industrial system are repeated systems [17,18]. Therefore the conventional fault estimation methods mentioned above regrettably ignore the learning experience and performance from previous iteration. Fortunately with the development of information processing technology, tremendous research efforts have been devoted to design and analysis the fault estimation scheme though computer-based learning techniques, including iterative learning scheme based approaches [19,20], Fault tracking approximator (FTA) and iterative learning algorithm is utilized to obtain the estimation of the fault functions of time-delay systems [20]. In Ref. [21], for the purpose of avoid periodically occurring fault estimation in nonlinear time-varying systems, an iterative learning observers are constructed by using previous output estimation errors and inputs. The fault tracking approximator, which motivated by predictive and iterative learning control theory, uses iterative algorithms to detect and identify nonlinear system faults, even there exists model uncertainty [22]. Iterative learning scheme-based fault estimation observer is designed for multiphase batch processes with delays, disturbances, and actuator faults [23,24], and also a class of differential time-delay batch processes with actuator faults [25,26], then nonlinear systems with randomly changed trial length, period intermittent fault and time-delay [27–29]. Unfortunately, to the best of the authors knowledge, iterative learning scheme-based fault estimation problem has not been fully investigated, not to mention the case where the systems with quantised measurements. The ideal situation of system data measurement and transmission is infinite precision. While, in practical situations, the ideal situation may not be implementable, considering the presence of signal quantisation or capacity-limited feedback, such as digital computers with A/D and D/A converters, network medium with limited capacities. The presence of these factors may cause low resolution of the transmitted data and large quantisation errors. It is, therefore, the main purpose of this paper to consider the quantised measurements in fault estimation problem for a class of nonlinear systems. Compared with the existing results, the main contributions of this technical note are highlighted as follows: (1) Considering the specific quantised measurements problem of discrete-time system, a logarithmic quantiser is introduced to construct an iterative learning scheme-based fault estimator and inherits the advantages of extension to network systems with networked delay, data loss, multi-sensors, etc. (2) The proposed method satisfies stability and convergence by a lmi condition, in spite of the quantisation levels of measurements signals are finite. The rest of this paper is organized as follows: Nonlinear system with quantised measurements and the problem formulation is introduced in Section 2. In Section 3, iterative learning scheme-based fault estimation is proposed to achieve desired fault estimation results. In Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Section 4, convergence analysis is used to solve the problem. Then, iterative learning observer is realized by an optimal problem and is extended to general case with randomly occurring parameter uncertainties in Section 5. The procedures of engineering implementation are shown as example in Section 6. Simulation results is provided to illustrating the effectiveness of the proposed method in Section 7, and in Section 8 concluding remarks are given. 2. Problem statement and preliminaries Consider the following nonlinear system: x (t + 1 ) = Ax (t ) + Bu (t ) + Bgg(x (t ), t ) + B f f (t ) y (t ) = Cx (t )

(1)

where t ∈ [0, T] is the discrete-time index, x(t) ∈ n is the state vector, y(t) ∈ p is the output vector, u(t) ∈ m represents the input vector, f(t) ∈ q stands for the fault signal. The function g(x(t), t) ∈ r is a known nonlinear function. A ∈ n × n , B ∈ n × r , Bg ∈ n × r , Bf ∈ n × q and C ∈ p × n are all constant matrices with appropriate dimensions and n > p ≥ q. Note that f(t) ∈ q could represent various types of faults. If B = B f ,f(t) ∈ q represents actuator faults, otherwise f(t) ∈ q represents sensor faults or process faults. The system (1) is a repetitive system with a running cycle T, which has periodic characteristics, xk (t ) = x (t ), yk (t ) = y (t ), k ∈ Z+ . Each cycle is regarded as a repetitive operation cycle. f(t) ∈ q is considered as intermittent fault signal, f (t ) = f (t + T ). For system (1), the following define and assumptions are made available. Definition 1. The quantizers is modeled by qy = q(y(t ))

(2)

where q(y(t )) = [y1 (t ) · · · yn (t )]. q(·) is logarithmic quantizers. In terms of [30], q(·) satisfies ⎧ 1 1 ⎪ us < v < us ⎪ ⎨u s , 1+θ 1−θ q(v) = (3) ⎪ v=0 ⎪ ⎩0, −q(−v), v < 0 where θ =

1−ρ . 1+ρ

us is taken from the set U :  U = {±us : us = ρ s , s = 0, ±1, ±2, · · · } {0}, 0 < ρ < 1, us > 0,

(4)

Definition 2. Quantization density of quantizer q(·) is defined as ηq = l imsupε→0

#gq [ε] −l nε

(5)

where #gq [ε] stands the mount of Quantitative Level in [ε, 1ε ] Remark 1. ηq =

2 ln( ρ1

) where q(·) is Logarithmic quantizer. The smaller ρ is, the smaller

quantization density ηq is. By using the sector bound approach in [30], q(y(t)) can be written as q(y(t )) = (I + q (t ))y(t )

(6)

Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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where q (t ) = diag[1q (t ), 2q (t ), . . . , nq (t )], and |iq (t )| < θ , i ∈ 1, 2, . . . , n. If iq (t ) = 0, the system output is not affected by data quantization. Assumption 1. The pairs (A, B) and (A, C) are stabilizable and detectable, respectively. Assumption 2. The desired initial state value at each iteration is xk (0 ) = x (0 ) with the definition that k is the iteration index. Assumption 3. For the nonlinear term g(x(t), t), there exists a known positive constant parameter δ which leads to satisfy the Lipschitz conditions. For example: ||g(x1 (t ), t ) − g(x2 (t ), t )|| ≤ δ||x1 (t ) − x2 (t )||, ∀x1 (t ), x2 (t ) ∈ n

(7)

where δ is called Lipschitz constant and g(0, t ) = 0 if the set S = n is globally Lipschitz. In order to achieve the derivation of the iterative learning observer, two lemmas are introduced in this paper at first. Lemma 1. Consider that G and H are constant matrices with appropriate dimensions, there exists matrix of adequacy dimensions E(t) that satisfied the condition ET (t)E(t) ≤ I, for any positive scalar ε, the following inequality is verified [31]. GE (t )H + H T E T (t )GT ≤ ε −1 F F T + ε H T H

(8)

Lemma 2 (Schur complement theorem) [32,33]. Consider that there are two symmetric ma  trices R and Q, the inequality SQT RS > 0 is equal to Eq. (9). ⎧ ⎨R ≥ 0 Q − S R+ S T ≥ 0 ⎩ S(I − RR+ ) ≥ 0

(9)

In reality, noise, time delay, model uncertainties, unknown input and sensor faults may come into the system inadvertently due to the complex environment and cumbersome process. They will affect the operation performance in different ways. For example, measurable precision of sensor drops greatly when there exist noise. Model uncertainties will influence the control precision and tracking accuracy. However, they rarely appear together. Otherwise, it makes the system breakdown and even disaster. As a result, for expression to be concise, this paper is addressed to analyze the impact of randomly occurring parameter uncertainties. 3. Iterative learning observer design In this section, the sate observer is designed to estimate the system states and outputs, and iterative learning law is designed for fault estimation. Based on the system (1), the observer-based fault estimator considered in this paper is proposed as:

 xˆk (t + 1 ) = Axˆk (t ) + Bu (t ) + Bgg xˆk (t ), t + B f fˆk (t ) + L q(y (t ) ) − yˆk (t ) yˆk (t ) = C xˆk (t ) (10) The order of the observer equals the number of states. In Eq. (10), xˆk (t ) and yˆk (t ) are the state estimate and output estimate of state vector x(t) and output vector y(t) at k iterations, Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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respectively. The parameter matrix L represents the observer gain. fˆk (t ) denotes the estimate of fault signal f(t) at k iterations. On the other exwide, defining that gk (t ) = g(xk (t ), t ) − g(xˆk (t ), t ), ek (t ) = xk (t ) − xˆk (t ), rk (t ) = fk (t ) − fˆk (t ). The dynamic error system is shown as the following form. ek (t ) = ek (t + 1 ) − ek (t ) = (A − I − LC)ek + B f rk (t ) + Bggk (t ) − Lq (t )Cxk (t ) yk (t ) = yk (t ) − yˆk (t )

(11)

Then, the iterative learning scheme based fault estimating law is proposed as: fˆk+1 (t ) = fˆk (t ) + K1 ek (t ) + K2 ek (t )

(12)

In which, K1 and K2 stand for iterative learning gain matrices. In order to simplify the following derivation, one can give a definition of the iterative learning error of fault estimation. rk+1 (t ) = f (t ) − fˆk+1 (t ) = rk (t ) − K1 ek (t ) − K2 ek (t )

(13)

= M1 ek (t ) + M2 rk (t ) + M3 gk (t ) + M4 x(t ) In Eq. (13), the matrices are defined as M1 = −[K1 + K2 (A − I − LC)], M2 = (I − K2 B f ), M3 = −K2 Bg and M4 = K2 LCq (t ). 4. Convergence analysis The following theorem gives the convergence of the proposed iterative learning based observer for the case that the initial state is accurately reset. Lyapunov function candidate is constructed to guarantee the stability of the error system (11) and a novel optimal function is proposed to ensure the perfect fault tracking trajectory. Theorem 1. Consider nonlinear system with randomly occurring uncertainties (1) and iterative learning fault estimation law (9) is applied as well as Assumptions 1–3 hold. According to Lemmas 1 and 2, the error dynamic system (11) is asymptotically stable while satisfying the fault estimating error convergence, if there exists positive-definite matrices P = PT , Q = QT , scalar ϕ ∈ [0, 1], λ > 0, ς 1 > 0, ς 2 > 0, εi > 0, i = 1, 2, and the symmetric definite matrix  satisfies ⎡ ⎤ −I 0 13 14 −K2 Bg 0 0 0 K2 0 ⎢∗ −P 23 PB f P Bg 0 0 −L¯ 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ 33 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ −ϕ 0 0 0 0 0 C T L¯ T ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ ∗ −λ 0 0 0 0 0 ⎥ ⎢ ⎥<0 √ =⎢ ∗ ∗ ∗ ∗ −ε1 3δBgT Q 0 0 0 ⎥ ⎢∗ ⎥ ⎢∗ ∗ ∗ ∗ ∗ ∗ 77 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I 0 0 ⎥ ⎢ ⎥ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 99 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ς2 I 13 = −K1 − K2 (A − I ) Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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14 = I − K2 B f 23 = PA − L¯ C 33 = −P + C T C + λδ 77 = AT (Q + 4ε1 )A − Q + δ 2 BgT QBg + ε2 θ 2C T C 99 = −(2ς1 − ς2 )I

(14)

In which L¯ = PL. Then the observer gain matrix can be obtained as L = P−1 L¯ . Theorem 1 presents the sufficient condition for the existence and design of iterative learning fault estimator for system (1). Proof. The first objective of this technical note is to achieve the stability and convergence of the state observer, such that to realize the desired state estimating results. Consider the Lyapunov function as V (t ) = eTk (t )Pek (t )+x T (t )Qx (t ) > 0

(15)

From Eqs. (11) and (15), one can calculating the derivative of V(t) with respect to time as Vk (t ) = eTk (t )[(A − LC)T P (A − LC) − P]ek (t ) + rkT (t )BTf PB f rk (t ) + gTk (t )BgT PBggk (t ) + x T (t )[(LC q (k))T PLC q (t ) + AT QA − Q]x(t ) + 2rkT (t )BTf P (A − LC)ek (t ) + 2gTk (t )BgT P (A − LC)ek (t ) + 2x T (t )(LC q (t ))T P (A − L1C)ek (t ) + 2gTk (t )BgT P1 B f rk (t ) + 2x T (t )(LC q (t ))T PB f rk (t ) + 2x T (t )(LC q (t ))T PBggk (t ) + 2ukT (t )BT QAx (t ) + 2 ftT (t )BTf QAx (t ) + 2gT (x (t ), t )BgT QAx (t )          1

+ 2f 

T

2

 4



+ u (t )B QBu (t ) + f     T

3

(t )BTf QBu (t ) + 2gT (x (t ), t )BgT QBu (t ) + 2gT (x (t ), t )BgT QB f T

7

 T





5

(t )BTf QB f  8

f (t ) + g  

T



 6

f (t ) 

(x (t ), t )BgT QBgg(x (t ), t )  9



(16)

For positive scalar ε1 , the following inequalities are established as: 1 = 2u (t )T BT QAx (t ) ≤

1 2 u¯ λ1max (BT QQB) + ε1 [Ax (t )]T [Ax (t )] ε1

(17)

2 = 2 f T (t )BTf QAx (t ) ≤

1 ¯2 f λ2max (BTf QQB f ) + ε1 [Ax (t )]T [Ax (t )] ε1

(18)

3 = 2gT (x (t ), t )BgT QAx (t ) 1 ≤ gT (x (t ), t )BgT QQBgg(x (t ), t ) + ε1 [Ax (t )]T [Ax (t )] ε1 1 ≤ δ 2 [x (t )]T BgT QQBgx (t ) + ε1 [Ax (t )]T [Ax (t )] ε1

(19)

Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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4 = 2 f T (t )BTf QBu (t ) 1 ¯2 ≤ f λ3max (BTf QQB f ) + ε1 u¯2 λ4max (BT B) ε1

7

(20)

5 = 2gT (x (t ), t )BgT QBu (t ) 1 ≤ gT (x (t ), t )BgT QQBgg(x (t ), t ) + ε1 [Bu (t )]T [Bu (t )] ε1 1 ≤ δ 2 [x (t )]T BgT QQBgx (t ) + ε1 u¯2 λ5max (BT B) ε1 6 = 2gT (x (t ), t )BgT QB f f (t ) 1 ≤ gT (x (t ), t )BgT QQBgg(x (t ), t ) + ε1 [B f f (t )]T [B f f (t )] ε1 1 ≤ δ 2 [x (t )]T BgT QQBgx (t ) + ε1 ε1 f¯2 λ6max (BTf B f ) ε1

(21)

(22)

7 = u (t )T BT QBu (t ) ≤ u¯2 λ7max (BT QB)

(23)

8 = f T (t )BTf QB f f (t ) ≤ f¯2 λ8max (BTf QB f )

(24)

9 = gT (x (t ), t )BgT QBgg(x (t ), t ) ≤ δ 2 x T (t )BgT QBgx (t )

(25)

One can further obtain that V (t ) ≤ ξkT (t )1 ξ (t ) + δ1 + δ2 + δ3 + δ4 + δ5 + δ6 + δ7 + δ8 ⎡

1_11 ⎢ ∗ 1 = ⎢ ⎣ ∗ ∗

1_12 BTf PB f ∗ ∗

1_13 BTf PBg Bg T P Bg ∗



(26)

⎤

1_14 −BTf PL q (k)C ⎥ ⎥ −BgT PL q (k)C ⎦



1_44

1_11 = (A − LC) P (A − LC) − P 1_12 = (A − LC)T PB f T

1_13 = (A − LC)T PBg 1_14 = −BTf PL q (k)C 1_44 = (L q (k)C)T PL q (k)C + AT (Q + 3ε1 )A + 3

1 2 T δ Bg QQBg + δ 2 BgT QBg − Q ε1

(27)

 T where ξk (t ) = ek (t ) rk (t ) g(x (t ), t ) x (t ) , δ1 = u¯2 λ1max (BT QQB), δ2 = ¯f 2 λ2max (BT QQB f ), δ3 = f¯2 λ3max (BT QQB f ), δ4 = u¯2 λ4max (BT B), δ5 = u¯2 λ5max (BT B), f f δ6 = f¯2 λ6max (BTf B f ), δ7 = u¯2 λ7max (BT QB), δ8 = f¯2 λ2_ max (BTf QB f ), u¯ = |u(t )|∞ , ¯f = | f (t )|∞ . Based on Lyapunov stability theory, the error dynamic system is stable Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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and the designed observer is converged if the inequalities V(t) > 0 and V(t) < 0 hold. Based on Lemma 2, it is obvious that the inequality V(t) < 0 holds only if the equation δ5 +δ6 +δ7 +δ8 ||ξkT (t )||2 > δ1 +δ2 +δ3 +δ4 + is true and 1 < 0. Notice that Lyapunov function is φ constructed to only ensure the stability of the output when the updating law be applied in system (1). The second objective in this paper is to obtain appropriate learning gain matrixes such that the tracking error converges to zero for all within the whole time interval t as [0, T]. To attain H∞ robustness performance and convergence of proposed method, the following performance index is introduced for the prescribed scalar γ ∈ [0, 1] at any iteration k ∈ Z+ . J1 =

Td 

T [rk+1 (t )rk+1 (t ) − γ 2 rkT (t )rk (t )] ≤ 0

(28)

t=0

Using Assumption 3, there exists a positive scalar λ ∈ [0, 1] that satisfies J2 =

Td  

λδeTk (t )ek (t ) − λgTk (t )gk (t ) ≥ 0

(29)

t=0

Then the derivative of V(t) and the inequality (29) are taken into Eq. (28), the optimal function J1 is rewritten as follows. J1 < J1 +J2    = J1 + J2 + V (t ) − [V (Td ) − V (0)] T  d  T = ξk (t )2 ξk (t )dt − [V (τ ) − V (0)] ≤ 0

(30)

t=0

Based on Lemma 1 and Eq. (27), one can obtain that ⎡

1_11 + M1T M1 ⎢ ∗ 2 = ⎢ ⎣ ∗ ∗

1_12 + M1T M2 BTf PB f + M2T M2 ∗ ∗

1_13 + M1T M3 BTf PBg + M2T M3 BgT PBg + M3T M3 ∗

⎤ 1_14 + M1T M4 −BTf PL q (k)C + M2T M4⎥ ⎥ −BgT PL q (k)C + M3T M4⎦ 1_44 + M4T M4 (31)

By denoting L¯ =PL, then one can get that L =P−1 L¯ . According to Lemma 2, it is easy to see that Eq. (31) holds if the following inequalities holds ⎡

−I ⎢∗ ⎢ ⎢∗ ⎢ 3 = ⎢ ⎢∗ ⎢∗ ⎢ ⎣∗ ∗ 

0 −P ∗ ∗ ∗ ∗ ∗

−K1 − K2 (A − I ) PA − LC −P + C T C + λ ∗ ∗ ∗ ∗

I − K2 B f PB f 0 −γ 2 I ∗ ∗ ∗  4

−K2 Bg 0 0 0 −λ ∗ ∗

0 0 0 0 0 −ε1 ∗

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ √ 3δBgT Q⎦ 4_77 

Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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⎡

0 ⎢∗ ⎢ ⎢∗ ⎢ +⎢ ⎢∗ ⎢∗ ⎢ ⎣∗ ∗ 

0 0 ∗ ∗ ∗ ∗ ∗

K2 P−1 L¯ C 0 0 ∗ ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 ∗ ∗  5

0 0 0 0 0 0 ∗

⎤ K2 P−1 L¯ q (k)C −L¯ q (k)C ⎥ ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎥ ⎥ 0 ⎥ ⎦ 0 0 

4_77 = A (Q + 3ε1 )A + δ Bg QBg − Q + ε2 θ 2C T C T

2

9

T

(32)

Then the matrix 3 is well extracted into the summation of two components. One is the constant term and another is the uncertain term with nonlinear term. The uncertain term q (k) = I ∗ Qq ∗ δq , |q (k)| ≤ θ , QqT Qq ≤ 1. The nonlinear term −γ 2 I is linearized as −ϕ. Moreover, using Lemma 1 the inequality (32) holds if and only if there exists ε2 > 0 and ε3 > 0 such that ⎡ ⎤ −I 0 6_13 6_14 −K2 Bg 0 0 0 K2 0 ⎢∗ −P 6_23 PB f P Bg 0 0 −L¯ 0 0 ⎥ ⎢ ⎥ T ¯T ⎥ ⎢∗ ∗ 6_33 0 0 0 0 0 0 C L ⎥ ⎢ ⎢∗ ∗ ∗ −γ 2 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ ∗ −λ 0 0 0 0 0 ⎥ ⎥ 6 = ⎢ ⎢∗ ∗ ∗ ∗ ∗ −ε1 6_67 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ ∗ ∗ ∗ 6_77 0 0 0 ⎥ ⎢ ⎥ ⎢∗ L¯ T ⎥ ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I 0 ⎢ ⎥ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε3−1 P 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε3 P 6_13 = −K1 − K2 (A − I ) 6_14 = I − K2 B f 6_23 = PA − LC 6_33 = −P + C T C + λδ √ 6_67 = 3δBgT Q 6_77 = AT (Q + 4ε1 )A − Q + δ 2 BgT QBg + ε2 θ 2C T C

(33)

In Eq. (8), there exists nonlinear terms ε3−1 P and ε3 P, we use the following constraint and approximation P > ς1 I , ε3 +

1 ≥2 ε3

(34)

then a new LMI is constructed of α and β = αε3 . From Eq. (14), we have ε13 P ≥ 2P − ε3 P = (2ς1 − ς2 )I and ε3 P ≥ ε3 ς1 I = ς2 I . Hence, the terms (2ς1 − ς2 )I and ς 2 I are used to replace the blocks ε13 P and ε3 P in Eq. (8), respectively. Hence, Theorem 1 is obtained by employing Lemma 2. This completes the proof.  5. Iterative learning observer realization and extension Theorem 1 can be rewritten as following corollary. Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Corollary 1. If there exists symmetric positive definite matrixes P = PT > 0, Q = QT > 0, scalars ς 1 > 0, ς 2 > 0, ϕ ∈ [0, 1], εi > 0, i = 1, 2, Theorem 1 can be satisfied as following optimal functions subject to the LMI in Eq. (14). Min{ϕ }, s.t. < 0

(35)

Then fault estimation algorithm (13) using iterative learning scheme can realize ek (t) and rk (t) uniformly bounded. Namely, the monotonic convergence of tracking error (15) and the H∞ performance of error argued system (28) are achieved. In reality, parameter uncertainties usually enter systems in an unknown way and such variations are unknown but with known bounds due to simplified modeling, ever-changing environments and accidentally operation. Hence, consider the following nonlinear uncertain system:: x (t + 1 ) = (A + α(t )A )x (t ) + Bu (t ) + B f f (t ) + Bgg(x (t ), t ) y (t ) = (C + β(t )C)x (t )

(36)

α(t)A(t) and β(t)C(t) denote the uncertain state parameter matrix and output parameter matrix respectively. In this technical note, the random variables α(t) and β(t) are defined to describe the parameter variations of a random nature. For system (36), the following define and assumptions are made available. The corresponding observer-based fault estimator is proposed as:

xˆ(t + 1 ) = Axˆ(t ) + Bu (t ) + B f fˆ(t ) + Bgg xˆ(t ), t + L(q(y (t ) ) − yˆ(t ) ) (37) yˆ(t ) = C xˆ(t ) Assumption 4. The matrix A(t) and C(t) represent the norm bounded parameter uncertainties of the following structure. A(t ) = G1 F1 (t )N1 , C (t ) = G2 F2 (t )N2

(38)

where Gi and Ni known matrices with adequate dimensions, the unknown matrices Fi (t) satisfy the conditions Fi (t )FiT (t ) ≤ I , i = 1, 2. The stochastic variables α(t) and β(t) are Bernoulli distributed white sequences taking on values of either zero or one with Prob{α(t ) = 1} = υ1 , Prob{α(t ) = 0} = 1 − υ1 Prob{β(t ) = 1} = υ2 , Prob{β(t ) = 0} = 1 − υ2

(39)

In which, ν 1 ∈ [0, 1] and ν 2 ∈ [0, 1] are known constants. It is assumed that α(t) and β(t) are independent of each other. The dynamic error system will be obtained as ek (t ) = ek (t + 1 ) − ek (t ) = (A − I − LC)ek (t ) + B f rk (t ) + Bgg(x (t ), t ) + L[α(t )A − q (t )C − β(t )C − q (t )C]xk (k)yk (t ) = yk (t ) − yˆk (t )

(40)

Corollary 2. If there exists symmetric positive definite matrixes P = P > 0, Q = Q > 0, scalars ς 1 > 0, ς 2 > 0, ϕ ∈ [0, 1], εi > 0, i = 1, 2, Theorem 1 can be satisfied as following optimal functions subject to the LMI. T

Min{ϕ }, s.t. < 0

T

(41)

Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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⎡ −I 0 13 14 15 0 0 0 0 0 0 K2 ⎢ ∗ −P 23 PB f ¯ ¯ G2 L L 0 0 0   0 28 210 ⎢ ⎢ ∗ ∗ 33 0 0 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ −γ 2 I 0 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ ∗ −λ 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ ∗ ∗ −ε  0 0 0 0 0 1 67 ⎢ ∗ ∗ ∗ ∗ 77 0 0 0 0 0 =⎢ ⎢∗ ∗ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ I 0 0 0 0 1 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ I 0 0 0 2 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ I 0 0 3 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ I 0 4 ⎢ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1212 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ¯ ¯ 13 = −K1 − K2 (A − I ), 23 = PA − LC, 28 = L υ1 G1 , 210 = L υ2 G2 , √ 33 = P + C T C + λ, 15 = −K2 Bg, 14 = I − K2 B f , 67 = 3δBgT Q

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⎤ 0 0 ⎥ ⎥ T ¯T ⎥ C L ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 813 ⎥ ⎥ L¯ T ⎥ ⎥ 1013 ⎥ ⎥ GT2 L¯ ⎥ ⎥ 0 ⎦ −ζ2 I

77 = AT (Q + 3ε1 )A + δ 2 BgT QBg − Q + σ1 N1T N1 + σ2 θ 2C T C + σ3 N2T N2 + σ4 θ 2 N2T N2 813 = υ1 GT1 L¯ , 1013 = υ2 GT2 L¯ T , 1212 = −(2ζ1 − ζ2 )I (42) Then fault estimation algorithm (13) using iterative learning scheme can realize ek (t) and rk (t) uniformly bounded. Namely, the monotonic convergence of tracking error (15) and the H∞ performance of error argued system (28) are achieved. As the discussion above, Corollary 2 can be employed for fault estimator design of nonlinear uncertain system (36) directly. 6. Engineering implementation The procedures of engineering implementation are divided into two parts. Offline Calculation: Step 1: Determine F1 (t), M1 , N1 , F2 (t), M2 , N2 and the distribution of α(t), β(t) according to system uncertainties; Step 2: Determine the boundary parameter θ of q(t) according to sampling uncertainties; Step 3: Determine optimization solutions of P, Q, L, K1 , K2 by using Eqs. (41) and (42); Online estimation: Step 1: Set Initial State x0 , fˆ0 (t ) = 0, t ∈ [0, Td ]; Step 2: k > 0, make iterative estimation fˆk (t ) by using Eqs. (10)–(12), t ∈ [0, Td ], and store in memory; Step 3: if k + 1 < kd ,then back to step 2, stop iteration while k > kd 7. Illustrative example In this section, a numerical example has been performed to demonstrate the validity and effectiveness of the proposed approach. Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Consider the following nonlinear systems (36) with randomly occurring parameter uncertainties according to Eq. (38), where the state variables at k iteration is denoted by xk (t ) = [x1,k (t ), x2,k (t ), x3,k (t ), x4,k (t )]T . The sampling period is T = 0.1. ⎡ ⎤ ⎡ ⎤ 1 0.1 0 0 0 ⎢ 0 ⎢ 0.0247 ⎥ 1 0.1 0⎥ ⎥ ⎢ ⎥ A=⎢ ⎣−2.16 −1.36 0.58 1.6⎦ B = ⎣ ⎦ 0 −0.1 0 0 1 −0.0476 ⎡ ⎤ ⎡ ⎤ 0 0.1 ⎢ 0.05 ⎥ ⎢0.1⎥

⎥ ⎢ ⎥ Bf = ⎢ ⎣ 0 ⎦ Bg = ⎣0.1⎦, g[xk (t ), t ] = sin x1,k (t ) . −0.05 0.1 ⎧ 0.5 sin (2πt ), t ∈ [0, 1s ) ⎪ ⎪ ⎪ ⎪ ⎨sin (2πt ), t ∈ [1s, 2s ) f (t ) = 1.5 sin (2πt ), t ∈ [2s, 3s ) ⎪ ⎪ 2 sin (2πt ), t ∈ [3s, 4s ) ⎪ ⎪ ⎩ 2.5 sin (2πt ), t ∈ [4s, 5s ) The constant matrices of uncertainties for A(t) and C(t) are given as    T 1 0 , M1 = 0 0.1 0 0.1 , M2 = 0 1    0.2 0.1 0.1 0.2 . N1 = 0.02 0.01 0.01 0.01 , N2 = 0.1 0.2 0.2 0.1 And the probability distribution is described as Prob{α(t ) = 1} = 0.6, Prob{α(t ) = 0} = 0.4, t ∈ [0, Td ] and Prob{β(t ) = 1} = 0.9, Prob{β(t ) = 0} = 0.1, t ∈ [0, Td ].In which, •(t ) = 1 denotes the fault occurring and •(t ) = 0 represents there is no fault inversely (• is α or β.). For definiteness and without loss of generality, considering that F1 (t ) = 0.5 sin (2πt ) and F2 (t ) = cos (πt ).  T The initial desired value of state variables is set to be xd (0) = 0 0 0 0 and the  T controller that is employed as constant uk (t ) = 1 1 1 1 . Solving the optimization problem Corollary 2, results of the observer and fault estimator gain matrices are shown as follows. ⎡ ⎤ 36.7062 −9.6213 6.0758 −29.9866 ⎢ −9.6213 7.4207 −1.0072 6.6564 ⎥ ⎥, P=⎢ ⎣ 6.0758 −1.0072 2.0966 −4.9638 ⎦ −29.9866 6.6564 −4.9638 26.6870 ⎡ ⎤ 6.5915 2.8728 0.7777 −10.2895 ⎢ 2.8728 1.2442 0.3570 −4.5039 ⎥ ⎥, Q=⎢ ⎣ 0.7777 0.3570 0.1436 −1.2467 ⎦ −10.2895 −4.5039 −1.2467 16.1969 ⎡ ⎤ −0.2208 0.4916 ⎢ 0.0263 0. 5884 ⎥ ⎥, L=⎢ ⎣ 1.0709 −1.2681⎦ −0.0170 0.2801 Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Fig. 1. The tracking trajectory of abrupt fault for nonlinear system with constant parameter uncertainties.

K1 = [−1.0055 0.012 −1.0012 0.0051] K2 = [0.0019 10.004 0 −10.004] In the simulation, the randomly occurring uncertainties (38) with probability distribution (39) are addressed for demonstrating the effectiveness of the iterative learning fault estimator. To further illustrate the effectiveness of the proposed fault estimation approach in a class of nonlinear uncertain systems, the maximum value of absolute error Ek is introduced to evaluate the effectiveness of fault estimating performance in different iterations. The definition of Ek is shown as follows.     (43) Ek = sup  f (t ) − fˆk (t ) t∈[0,Td ]

We do simulation with ρ = 0.7. The fault estimation results in the nonlinear system with constant randomly occurring parameter uncertainties are shown in Figs. 1 and 2. Fig. 1 shows the fault estimating results and actual fault signals with time-varying fault,meanwhile,also shows the comparison with a existing technique of He [34] and a standard approach of Feng [35]. The fault estimating results are more close to actual fault with iterations increase. It can be concluded that the proposed fault estimation observer and algorithm have an excellent performance to estimate the actual fault and better than both existing technique and standard approach. The variation trend of maximum absolute error and comparison with He and Feng is exhibited in Fig. 2. It can be seen that decreases with iterations increase and converges to zero. One can conclude that satisfactory estimation performance has been achieved and better than other two approaches. It should be pointed that the state estimating error and fault estimating results in previous iterations are utilized in current iteration to improve the estimation performance. Table 1 shows the performance of the maximal tracking error with different quantization density of quantizer is presented in Table 1. It shows that the maximal tracking error decreases with quantization density from 0.046 to 0.009. It is obvious that tracking error is determined by quantisation level. It should be noted that the tracking residual error originates from Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Fig. 2. The tracking trajectory of intermittent fault for nonlinear system with time-varying parameter uncertainties. Table 1 The tracking trajectory of abrupt fault for nonlinear system with different quantization density, ρ θ Ek

0.7 0.1765 0.046

0.8 0.1111 0.039

0.85 0.0811 0.036

0.9 0.0526 0.026

0.95 0.0256 0.023

0.99 0.005 0.009

measuring link and cannot be eliminated by the iterative learning scheme with the integral calculus, completely. 8. Conclusion In this paper, we have investigated the observer-based fault estimation method using iterative leaning scheme for nonlinear systems where the data quantization are coexisting in measurement. For analysis of iterative leaning scheme, considering the effects of data quantization is more practical. Furthermore, a new scheme has been proposed to describe the randomly occurring parameter uncertainties and data quantization in a unified framework. The problem of observer-based fault estimation using iterative leaning scheme has been transformed into an optimal problem. Hence, the proposed scheme achieves a high generality and it could be integrated into both constant uncertain case and time-varying uncertain case. Acknowledgment This work was funded by the National Natural Science Foundation of China (61374135, U1637107). All data generated or analysed during this study are included in this published article. References [1] D. Wu, Y. Li, Fault diagnosis of variable pitch for wind turbines based on the multi-innovation forgetting gradient identification algorithm, Nonlinear Dyn. 79 (3) (2015) 2069–2077. Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040

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Please cite this article as: L. Xiaoyu, W. Shanbi and C. Yi, An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements, Journal of the Franklin Institute, https:// doi.org/ 10. 1016/j.jfranklin.2019.09.040