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Interval observer and unknown input observer-based sensor fault estimation for high-speed railway traction motor
Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor
Journal Pre-proof
Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor Yang Tian, Ke Zhang, Bin Jiang, Xing-Gang Yan PII: DOI: Reference:
S0016-0032(19)30866-X https://doi.org/10.1016/j.jfranklin.2019.11.062 FI 4297
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
27 March 2018 2 April 2019 25 November 2019
Please cite this article as: Yang Tian, Ke Zhang, Bin Jiang, Xing-Gang Yan, Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.062
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Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor✩ Yang Tiana , Ke Zhanga,∗, Bin Jianga , Xing-Gang Yanb a College
of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 211106 P.R. China b School of Engineering and Digital Arts, University of Kent, Canterbury, Kent CT2 7NT, United Kingdom.
Abstract In this paper, fault estimation for high-speed railway traction motor with sensor fault and disturbances is investigated based on the interval observer and unknown input observer (IO-UIO). First, the proposed method, which can completely eliminate the external disturbance, is studied based on the disturbance isolation characteristic of the unknown input observer. Then, an interval observer is constructed to deal with the nonlinear part, which sandwiched the actual system between the upper and lower bounds. Moreover, the Metzler matrix is constructed using an equivalent transformation and through the unified design based on the concept of the augmented state to form a global fault augmented model. Finally, simulation results are presented to illustrate the effectiveness and advantages of the proposed IO-UIO. Keywords: Railway traction motor, Metzler matrix, Equivalent transformation, Interval observer, Unknown input observer
✩ This work was supported in part by the National Natural Science Foundation of China under Grants 61490703, 61673207, in part by Qing Lan Project of Jiangsu Province, and in part by China Scholarship Council. ∗ Corresponding author Email addresses: [email protected] (Yang Tian), [email protected] (Ke Zhang), [email protected] (Bin Jiang), [email protected] (Xing-Gang Yan)
Preprint submitted to Journal of LATEX Templates
December 11, 2019
1. Introduction The development of high-speed railways and the progress in science and technology over the past decade have resulted in the increased requirements for the safety and stability of high-speed trains. These necessities are closely related to the safety of people’s life and property. High-speed railway has developed rapidly in many countries worldwide [1]. At present, the world’s most representative high-speed railways include the German ice train, Japan Shinkansen, French TGV trains and Chinese CRH trains. Compared with air transport, the high-speed railway has several advantages, such as low cost, large operating area, environmental protection and all-weather operation. The development of high-speed railway has not only a stimulating effect on equipment technology and modern manufacturing but also presents great impacts on information communication and construction modernisation. Modern control systems have become more and more complicated. Therefore, fault detection, fault diagnosis and fault-tolerant techniques have received more attention. Fault detection is the first step of fault diagnosis, which determines the occurrence of a system fault[2–5]. The fault should be detected before it becomes serious. Fault isolation is to determine the specific location of the fault[6, 7]. Then, fault estimation is performed to identify the form and size of the fault and is an important prerequisite for fault tolerance control[8–10]. The nonlinear system has been an area of active research during the last two decades[11–13]. However, the quality of actuators and sensors in a control system has been deteriorating with time. Moreover, the system sensors are used to measure system output, and these sensors indirectly affect the performance of the control system. The fault detection and diagnosis (FDD) and fault tolerance of the actuator faults are easier to determine than sensor faults (i.e. [14–16]). Only a few studies have reported on the FDD of sensor faults. Traction motor is the core equipment of high-speed trains, and this part is closely related to the safe operation of the train. Safety of the traction motor becomes very important as the high-speed train accelerates. Therefore, investi-
2
gating the fault diagnosis of a traction motor is of great practical significance. Induction motors play an important role in the industry. A motor fault may not only cause economic loss but also casualties. Furthermore, during the past few decades, the focus of fault diagnosis research on induction motor was to explore how to efficiently monitor the induction motor to avoid motor failure. Many methods have been put forward for fault diagnosis. An adaptive observer has been proposed in [17] to detect system faults. A fast adaptive fault estimation algorithm was proposed in ([18–20]) to enhance the accuracy of fault estimation. In this process, the estimators consist of proportional and integral items. As for fault detection, to use some observers or combined observers have been discussed by researchers([21, 22]). Vector augmentation was performed using the robust observer to estimate the fault [23]. A sliding mode observer was applied to a high-speed train in [24], where linear matrix inequalities (LMIs) are used to ensure the robustness with respect to the interference. An unknown input observer (UIO) was proposed in [25], which eliminated the unknown input item. Methods for nonlinear system observers have been proposed, such as Lyapunov method, coordinate transformation method[26], extended Kalman filter, interval observer (IO) and feedback linearisation technology. Recently, the interval observer approach has been extensively investigated, and some important achievements have been obtained. In [27], a class of interval observer was successfully designed for nonlinear systems, and the state of the error system could be guaranteed to be non-negative when the condition of the Metzler matrix was satisfied. In [28] and [29], the designed method of interval observer was studied for continuous and discrete linear time-invariant systems with bounded disturbances, based on the Jordan canonical form, and the appropriate of linear time-varying transformation. In [30], the designed method of interval observer was studied based on the Sylvester equation and the linear time-invariant transformation. In [27], the system control strategy based on an interval observer was studied. In this paper, a fault estimation observer is proposed for high-speed railway traction motor with sensor fault and disturbances. The main contributions of 3
this study are as follows. Firstly, an interval observer is designed based on the unknown input observer (UIO), which eliminates the effects of the external disturbance on the fault diagnosis, and only uses partial information of the measurement output interval to ensure the real-time tracking of the original system. This process is different from the classical observer design, because the error system is asymptotically convergent to zero in the proposed design. Secondly, the observer is designed for the augmented system composed of a state vector and a fault. Then, a method for constructing the Metzler matrix for systems which do not satisfy the condition of the Metzler matrix is proposed. This technique is successfully applied to high-speed train motor system based on the observable standard form. Thirdly, the system parameters are found to enhance fault estimation performance. The remainder of this paper is organised as follows: Preliminaries and problem formulation are presented in Section 2. Section 3 contains the design of interval observer and unknown input observer (IO-UIO)-based FD for the highspeed train and the coordinate transformation. Simulation results are provided to illustrate the effectiveness of the proposed methods in Section 4. Finally, a conclusion is made in Section 5.
2. Preliminaries and Problem Formulation 2.1. Preliminaries R denotes a set of real numbers. R ≥ 0 denotes a set of non-negative real
numbers, i.e., R ≥ 0 := [0, ∞). I denotes the identity matrix in Rn×n . For
vectors xa = [xa,1 , ..., xa,n ]T ∈ Rn and xb = [xb,1 , ..., xb,n ]T ∈ Rn , the inequality
xa ≤ xb , if and only if xa,i ≤ xb,i , for all i = 1, ..., n. For matrices A = (ai,j )m×n ∈ Rm×n and B = (bi,j )m×n ∈ Rm×n , the inequality A > B(A > B) denotes that ai,j > bi,j (ai,j > bi,j ), i = 1, ..., n, j = 1, ..., m. For a square matrix Q ∈ Rn×n , let the matrix Q+ ∈ Rn×n denote Q+ = max{qi,j , 0})n,n i,j=1,1
and Q− = Q+ − Q. The superscripts + and − are defined appropriately for other cases when they appear in the following sections.
4
The following definitions are given: a matrix A ∈ Rn×n is called Hurwitz if all its eigenvalues have a negative real part, and it is called Metzler if all its off-diagonal elements are non-negative. The following lemmas will be used in the sequel. Lemma 1[31]. The eigenvalues of a given matrix A ∈ Rn×n belong to the disk region D(α, τ ) with centre α+j0 and radius τ if and only if a symmetric positive definite matrix P ∈ Rn×n exists such that the following condition holds:
−P
P(A − αIn ) −τ 2 P
∗
<0
(1)
Lemma 2[28]. Consider a vector variable x ∈ Rn satisfying x 6 x 6 x with
x, x ∈ Rn , and a constant matrix A ∈ Rm×n , the following condition holds: A+ x − A− x 6 Ax 6 A+ x − A− x
(2)
2.2. Problem Formulation Consider the ideal mathematical model of the traction motor in the d − q axis synchronous rotating reference frame [4] as follows: x˙ (t) = Ax (t) + Bu (t) + fa (x (t))
where x (t) =
h
x1
x2
(3)
y (t) = Cx (t)
x3
x4
x5
iT
=
h
iqs
ids
λqr
λdr
ωm
iT
∈ R5 ,
is the state variable, u(t) ∈ R2 is the input variable and y(t) ∈ R3 is the output variable. A is the matrix of asynchronous motor system, whilst B is the input
5
distribution matrix of the system which are as follows:
−γ
ωs A = αLm 0 0
−ωs
αβ
0
−γ
0
αβ
0
−α
−ωs
αLm
ωs
α
0
0
0
0
0 0 ,B = 0 0
1 δLs
0
0
1 δLs
0
0
0
0
0
0
,
C is the output distribution matrix, and fa (x (t)) is the nonlinear part of the system given as follows:
1
C= 0 0
0
0
1
0
0
0
0 0 , fa (x (t)) = 0 1 0 0
where δ =1− µ=
−np βx5 x4 np βx5 x3 np βx5 x4 −np βx5 x3
µ (x4 x1 − x3 x2 ) − J1 TL
,
L2m Rr Lm ,α = ,β = , Ls Lr Lr δLs Lr
L2 Rr Rs 3 Lm np ,γ = m 2 + , 2 JLr δLs Lr δLs
iqs and ids denote the stator currents in the d − q frame; λqr and λdr denote the rotor flues in the d − q frame; ωm is the mechanical angular speed; ωs is the synchronous rotating speed; Ls , Lr and Lm denote stator self inductance, rotor self inductance and state-rotor mutual inductance, respectively; TL is the load torque; np is the pole pair number; and J is the moment inertia. The CRH5 adopts three-phase squirrel-cage asynchronous motor named 6FJA3257A, the parameters of which are shown in Table 1. The actual mathematical model of the traction motor can be represented as follows: x˙ (t) = Ax (t) + Bu (t) + fa (x (t)) + Ed d (t)
y (t) = Cx (t) + Ef f (t) 6
(4)
where d(t) ∈ R1 is the disturbance vector, and f (t) ∈ R1 represents the sensor fault. Ed and Ef are the disturbance distribution matrix, and fault distribution matrix, respectively. A, B, Ef , C and Ed are constant real matrices of appropriate dimensions. In addition, matrices Ef and Ed are of full column rank given as follows:
1
h Ef = 0 , Ed = 0 0 0 0 0
− J1
iT
, d (t) = ∆TL .
The following table records the relevant parameters of the asynchronous motor: Table 1: Asynchronous motor 6FJA3257A No. 1 2 3 4 5 6 7 8 9
parameter
value
Stator resistance Rs 105.1 mΩ Stator inductance Ls 31.7 mH Rotor resistance Rr 91.9 mΩ Rotor inductance Lr 31.7 mH Mutual inductance Lm 29.9 mH The moment inertia J 15 kg · m2 The load torque TL 4500 N · m The pole pairs number np 3 The Motor rotation speed ws 2 ∗ pi ∗ 60 rad/s
Remark 1. Available results of the interval observer for the induction motor are very few. This paper mainly focuses on how to estimate the sensor fault of high-speed railway traction motor and how to improve the performance of fault estimation. A combined observer is designed for the system, where an IO is introduced to solve nonlinear part and an UIO is introduced to eliminate the effect of external disturbance on the fault estimation. When the system can not meet the IO conditions, we use coordinate transformation method to construct Metzler matrix.
7
3. Main Results 3.1. IO-UIO Based Fault Estimation Design In this subsection, an observer is designed to estimate the incipient sensor faults. Interval information is applied to deal with the nonlinear part of the system. The augmented fault estimator is introduced, and the sensor fault vector is considered as the auxiliary state vector. The original system (4) can be rewritten in the following form: x(t) ˙ A 0 x(t) B E = + u(t) + d d(t) ˙ f (t) 0 0 f (t) 0 0 0 fa (x (t)) + f˙(t) + I 0 r h i x(t) y(t) = C Ef f (t)
(5)
For convenience, state the following parameters are defined: augmented vecx(t) A 0 , augmented state distribution matrix A¯ = , augtor x(t) = f (t) 0 0 B ¯= , augmented output distribution mented input distribution matrix B 0 h i fa (x (t)) , matrix C¯ = C Ef , augmented nonlinear vector f¯a (¯ x (t)) = 0 Ed ¯d = , and augmented Fault augmented disturbance distribution matrix E 0 0 . distribution matrix I¯r = Ir Then, system(5) can be rewritten as follows: ¯x(t) + Bu(t) ¯ ¯d d(t) + f¯a (¯ x ¯˙ (t) = A¯ +E x (t)) + I¯r f˙(t)
y(t) = C¯ x ¯(t)
8
(6)
where f˙(t) is the derivative of the fault f (t), and Ir is an r × r-dimensional unit matrix. ¯ C¯ = I, an equivalent augmented system state space Combining with T¯ + H description form can be obtained as follows: ¯x(t) + Bu(t) ¯ ¯d d(t) + I¯r f˙(t) + f¯a (¯ ¯ y(t) ¯ C¯ x x ¯˙ (t) = A¯ +E x (t)) + H ˙ −H ¯˙ (t) ¯x(t) + Bu(t) ¯ ¯d d(t) + I¯r f˙(t) + f¯a (¯ ¯ y(t) ¯ C( ¯ A¯ ¯x(t) + Bu(t) ¯ = A¯ +E x (t)) + H ˙ − H ¯d d(t) + I¯r f˙(t) + f¯a (¯ +E x (t))) ¯x(t) + T¯Bu(t) ¯ ¯d d(t) + T¯f¯a (¯ ¯ y(t) = T¯A¯ + T¯E ˙ x (t)) + T¯I¯r f˙(t) + H y(t) = C¯ x ¯(t) (7)
¯ and T¯ are the design parameters with appropriate dimensions. whereH
¯ C) ¯ is observable and rank(C¯ E ¯d ) = rank(E ¯d ). Assumption 1. The pair (T¯A, Lemma 3[32]. Under Assumption 1, the expression described by (8) is a UIO for system (7). For system (7), the problem of fault estimation is converted to IO-UIO design to reconstruct the fault vector. An observer is proposed to achieve fault estimation as follows: ¯ ¯ (ˆ ˆ ¯(t) + T¯Bu(t) + T¯f¯a (¯ x (t)) + T¯I¯r f˙+ (t) − L y (t) − y(t)) z¯˙ (t) = T¯A¯x ¯ x ˆ ¯(t) = z¯(t) + Hy(t) ˆ fˆ(t) = I¯rT x ¯(t) ˆ yˆ(t) = C¯ x ¯(t)
(8)
¯ C¯ = I. z¯(t) is an unknown input augmented variable, whilst x ˆ where T¯ + H ¯(t) and yˆ ¯(t) denote the augmented state vector and output vector. fˆ(t) is the sensor ¯ , T¯ and L ¯ are the gain matrices of the fault estimate value, whilst matrices H unknown input observer. Assumption 2. Given that the arbitrary x ¯− (t) 6 x ¯ (t) 6 x ¯+ (t), vector func-
9
tions f¯a− (¯ x+ (t) , x ¯− (t)) and f¯a+ (¯ x+ (t) , x ¯− (t)) exist and satisfy the following: f˙− (t) 6 f˙(t) 6 f˙+ (t) f¯a− x ¯+ (t) , x ¯− (t) 6 f¯a (¯ x (t)) 6 f¯a+ x ¯+ (t) , x ¯− (t)
where x ¯− (t) and x ¯+ (t) denote the lower and upper bounds of the augmented state vector, f˙− (t) and f˙+ (t) denote the lower and upper of the fault derivative respectively. Remark 2. There are many methods to solve nonlinear systems, such as local linearization. However, due to the particularity of high-speed railway traction motor, we can not find a suitable constant satisfying condition of Lipschitz[16]. Then an IO[28] is constructed to solve nonlinear part when Assumption 2 is satisfied. The idea of this method is to only use interval information while avoiding the limitations of other methods . ¯ exists such that the matrix (T¯A¯ − L ¯ C) ¯ is Metzler Assumption 3. Matrix L and satisfies Hurwitz. The interval observer is designed based on an unknown input observer. Under Assumption 2, the IO-UIO is described as follows: The upper bound observer is as follows: + ¯ ¯ yˆ+ (t) − y(t) ˆ ¯+ (t) + T¯Bu(t) + T¯f¯a+ (¯ x (t)) + T¯I¯r f˙+ (t) − L z¯˙ (t) = T¯A¯x ¯ + (t) x ˆ ¯+ (t) = z¯+ (t) + Hy ˆ fˆ+ (t) = I¯rT x ¯+ (t) + ˆ yˆ (t) = C¯ x ¯+ (t)
(9)
10
The lower bound observer is as follows: ¯ ¯ yˆ− (t) − y(t) ˆ z¯˙ − (t) = T¯A¯x ¯− (t) + T¯Bu(t) + T¯f¯a− (¯ x (t)) + T¯I¯r f˙− (t) − L ˆ− ¯ − (t) x ¯ (t) = z¯− (t) + Hy ˆ ¯− (t) fˆ− (t) = I¯rT x − ˆ yˆ (t) = C¯ x ¯− (t)
(10)
where z¯+ (t) and z¯− (t) represent the state of the upper and lower bounds of the ¯ C¯ = I. observers, respectively, and T¯ + H Theorem 1. Under Assumptions 2 and 3, if the initial conditions are satisfied, and the constraint x ¯− (0) 6 x ¯(0) 6 x ¯+ (0) , then the solutions of systems (7), (9) and (10) satisfy the following: ˆ− (t) 6 x ˆ x ¯ ¯(t) 6 x ¯+ (t), ∀t > 0. Proof : The augmented system error dynamic equation is considered. For the ˆ upper bound observer, let e¯+ ¯+ (t) − x ¯(t), ey + (t) = yˆ+ (t) − y(t) = C¯ e¯+ x (t) = x x (t), and ef + (t) = fˆ+ (t) − f (t).
Then, from (7) and (9), the system’s error state equation is expressed as follows: ¯ ¯ yˆ+ (t) − y(t) ˆ ˆ e¯˙ + ¯˙ + (t) − x ¯˙ (t) = T¯A¯x ¯+ (t) + T¯Bu(t) −L x (t) = x
¯ y(t) ¯x+ (t) − T¯Bu(t) ¯ ¯d d(t) + T¯f¯a+ (¯ x (t)) + H ˙ − T¯A¯ − T¯E
¯ y(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) − T¯f¯a (¯ x (t)) − H ˙ ¯e+ (t) − L ¯ yˆ+ (t) − y(t) − T¯E ¯d d(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) = T¯A¯ x + T¯ f¯a+ (¯ x (t)) − f¯a (¯ x (t)) ¯ C¯ e¯+ (t) − T¯E ¯d d(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) = T¯A¯ − L x
(11)
+ T¯f¯a+ (¯ x (t)) − T¯f¯a (¯ x (t))
When the following conditions are satisfied, the disturbance can be com-
11
pletely eliminated, ¯d = 0 T¯E ¯d and C¯ are known. Thus, the unknown matrix H ¯ can be obtained where E ¯d = 0 and T¯ + H ¯ C¯ = I. according to T¯E The simplified error dynamic system can be obtained from (11) as follows: ¯ ¯ ¯ ¯ ¯+ ¯ ¯+ x (t)) − f¯a (¯ e¯˙ + x (t)) + T¯I¯r f˙+ (t) − f˙(t) . x (t) = T A − LC e x (t) + T fa (¯ ef + (t) = I¯T e¯+ (t) r
x
(12)
Similarly, for the lower bound observer, by denoting system the augmented
ˆ ¯(t) − x ¯− (t), augmented output estimation error ey − (t) = state error e¯− x (t) = x y(t) − yˆ− (t) = C¯ e¯− x (t), and the error of lower bound of fault and true value is
ef − (t) = f (t) − fˆ− (t). Then, the system’s error state equation is expressed as follows: ¯ ¯ ¯ ¯ ¯− ¯ ¯ x (t)) − f¯a− (¯ e¯˙ − x (t)) + T¯I¯r f˙(t) − f˙− (t) . x (t) = T A − LC e x (t) + T fa (¯ ef − (t) = I¯T e¯− (t) r
x
By Assumption 2, we have
(13) e¯+ x (0)
e¯− x (0)
> 0,
> 0. From Assumption 3, the
¯− dynamics of the estimation errors are cooperative. Therefore, e¯+ x (t) > x (t) > 0, e 0 for all t > 0.
2
Remark 3. The using condition of the interval observer is that the initial condition x− (0) 6 x(0) 6 x+ (0), Assumption 2 and Assumption 3 are satisfied. As for Assumption 2, we can get the state space expression of nonlinear term:
fa (x (t)) =
−np βx5 x4 np βx5 x3 np βx5 x4 −np βx5 x3
µ (x4 x1 − x3 x2 ) − J1 TL 12
.
The upper bound x ¯+ (t) and lower bound x ¯− (t) of the state vectors under stable operation are selected. Firstly, in the simulation part, we will provide two types of faults. For constant faults, its derivative is 0, the upper and lower bounds satisfy the Assumption 2. The second one is the graded fault as an exponential function whose derivative is still an exponential function. Therefore, there exists upper and lower bounds. Secondly, for the nonlinear term of the traction motor system f¯a , we take (¯ x+ (t), x ¯− (t)), and bring them into the nonlinear term model to obtain the nonlinear upper and lower bounds. As for Assumption 3, ¯ ensures that matrix (T¯A¯ − L ¯ C) ¯ firstly, we assume that the existence of matrix L is Metzler. In the later chapter of the paper, a solution to the problem of not meeting Metzler is proposed. And in the simulation part of the paper, ¯ that the authors successfully used the LMI method to solve the parameters L ¯ L ¯ C) ¯ is located in the left half of the coordinate axis. satisfy the solution of (T¯A− Compared with the Luenberger Observer, the observer we studied can eliminate the impact of external disturbances on fault estimation when the hypothetical ¯d = 0 is satisfied. Consequently, this method effectively simplifies condition T¯E the algorithm and improves the performance of fault estimation. In dealing with the nonlinear part of the system, the method proposed in this paper involves the designing of an interval observer using part of the measurement information to replace a single point of measurement with a reasonable interval. Remark 4. Hurwitz criterion and Nygist criterion are effective tools for judging stability, however, as for the plants with structured uncertainty, Kharitonov theorem has been employed by some researchers([33–35]). In the actual system, the variation range of feature polynomial coefficients due to the perturbation of parameter values does not happen to be Super cube shape, in addition, the variation of each coefficient ai will not be independent of each other, so the application value of Haritonov theorem is limited. As a comparison, we use the positive system to construct an interval observer to diagnose the fault of the traction motor. The system can be ensured in the middle of the interval when the conditions of the Hurwitz and Metzler are satisfied and the nonlinear term satisfies the Assumption 2 by finding the upper and lower bounds of the system 13
state variables. The method of constructing interval observer is inspired by the nonlinear IO design in [27]. ¯ of the fault estimation for high-speed Theorem 2. The observer gain matrix L train can be obtained by solving the following linear matrix inequality (LMI), and the H∞ performance level γ and a circular region D(α, τ ) are provided. If a
¯ ∈ R(n+r)×(n+r) and matrix Y¯ ∈ R(n+r)×p symmetric positive definite matrix Q satisfy the following contitions:
¯ Q ¯ T¯A¯ − Y¯ C¯ − αQ ¯ −Q ¯ −τ 2 Q
∗
¯ T¯I¯r ϕ −Q
∗ ∗
I¯r
−γI
0
∗
−γI
<0
<0
(14)
(15)
¯ T¯A¯ + T¯A¯ T Q ¯ − Y¯ C¯ − Y¯ C¯ T , the eigenvalues of (T¯A¯ − L ¯ C) ¯ belong where ϕ = Q ¯=Q ¯ −1 Y¯ . to D(α, τ ), and the observer gain matrix is given by L
¯ L ¯ C) ¯ into A and P¯ into P in Lemma Proof : Constraint (14): Substituting (T¯A− 1, we can obtain (14). Constraint (15): Consider the following Lyapunov function: V (t) = e¯Tx (t)P¯ e¯x (t)
(16)
Its derivative with respect to time is ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ T Q)¯ ¯ ex + e¯T (t)Q ¯ T¯µ(t) + µT (t)T¯T Q¯ ¯ ex V˙ (t) = e¯Tx (t)(Q( x
where µ(t) = f¯a+ (¯ x (t)) − f¯a (¯ x (t)) + I¯r f˙+ (t) − f˙(t) Let us define: J1 =
Z
∞
tf
1 [ eTf (t)ef (t) − γµT (t)µ(t)]d(t) γ
14
(17)
(18)
one can obtain: J1 6 =
Z
Z
∞
tf ∞
[V˙ (t) +
1 T e (t)ef (t) − γµT (t)µ(t)]d(t) γ f
[V˙ (t) +
1 T ¯ ¯T e¯ (t)Ir Ir e¯x (t) − γµT (t)µ(t)]d(t) γ x
tf
(19)
From (17), it follows that: 1 V˙ (t) + e¯Tx (t)I¯r I¯rT e¯x (t) − γµT (t)µ(t) γ ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ T Q)¯ ¯ ex (t) + e¯Tx (t)Q ¯ T¯µ(t) = e¯Tx (t)(Q( ¯ ex + 1 e¯Tx (t)I¯r I¯rT e¯x (t) − γµT (t)µ(t) + µT (t)T¯T Q¯ γ
(20)
= ζ T (t)Ξζ(t) where
Ξ=
ζ(t) =
e¯x (t) µ(t)
,
¯ + 1 I¯r I¯rT ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ TQ Q( γ
¯ T¯ −Q
*
−γI
Through the Schur complement, Ξ < 0 becomes
¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ TQ ¯ −Q ¯ T¯ Q(
I¯r
∗
−γI
0
∗
∗
−γI
<0
(21)
¯ L. ¯ Thus, (12) and (13) are stable and which is equivalent to (15), where Y¯ = Q satisfy the H∞ performance ||ef (t)||2 < γ||µ(t)||2 if condition (15) holds.
2
Remark 5. The algorithm which satisfies the robust H∞ performance index can effectively restrain the influence of µ on fault estimation error ef (t).
15
3.2. Transformation of Coordinates ¯ cannot be determined. However, the corresponding observer gain matrix L ¯ C) ¯ be a Metzler matrix from the nonlinear system This value can make (T¯A¯ − L of the investigated traction motor for high-speed trains. A non-singular constant transformation matrix is given to solve the problem, and a new system equation is obtained by equivalent transformation ξ¯ (t) = P x ¯ (t). ¯˙ = P AP ¯ + P Bu(t) ¯ −1 ξ(t) ¯ ¯d d(t) + P f¯a (¯ ξ(t) + PE x (t)) + P I¯r f˙(t)
¯ ¯ −1 ξ(t) y(t) = CP
(22)
An equivalent augmented system state space description form can be obtained as follows: ¯˙ = P T¯AP ¯ + P T¯Bu(t) ¯ −1 ξ(t) ¯ ¯d d(t) + P T¯f¯a (¯ ξ(t) + P T¯E x (t)) ¯ y(t) + P T¯I¯r f˙(t) + P H ˙ y(t) = CP ¯ ¯ −1 ξ(t)
(23)
¯d = 0 is considered, the simplified form can be obtained from (23) When T¯E as follows: ¯˙ = P T¯AP ¯ + P T¯Bu(t) ¯ −1 ξ(t) ¯ ¯ y(t) ξ(t) + P T¯f¯a (¯ x (t)) + P T¯I¯r f˙(t) + P H ˙
¯ ¯ −1 ξ(t) y(t) = CP
(24)
The interval observer is constructed as follows:
+ ¯˙+ (t) = P T¯AP ¯+ (t) + P T¯Bu(t) ¯ (t) − y(t)) ¯ −1 ξˆ ¯ ¯ CP ¯ −1 ξˆ ξˆ − P L( ¯ y(t) + P T¯f¯a+ (¯ x (t)) + P T¯I¯r f˙+ (t) + P H ˙ ¯+ (t) + P T¯Bu(t) ¯ ¯ y(t) ¯ + P T¯f¯a+ (¯ x (t)) + P T¯I¯r f˙+ (t) + P H ˙ + P Ly(t) = Rξˆ
− ¯˙− (t) = P T¯AP ¯− (t) + P T¯Bu(t) ¯ (t) − y(t)) ¯ −1 ξˆ ¯ ¯ C¯ P −1 ξˆ ξˆ − P L( ¯ y(t) + P T¯f¯a− (¯ x (t)) + P T¯I¯r f˙− (t) + P H ˙ ¯− (t) + P T¯Bu(t) ¯ ¯ y(t) ¯ = Rξˆ + P T¯f¯a− (¯ x (t)) + P T¯I¯r f˙− (t) + P H ˙ + P Ly(t) (25)
16
ˆ+ (t), ξ¯ ˆ− (t) denote the upper and ¯ C)P ¯ −1 and ξ¯ where R is equal to P (T¯A¯ − L lower bounds of the augmented observer vector after the transformation of co¯ ordinates. Moreover, f¯aξ (¯ x (t)) denotes the nonlinear part after the coordinates
have been transformed. The following variables are defined: ϕ¯+ ξ¯− (t) , ξ¯+ (t) = T¯f¯a+ ϕ¯− ξ¯− (t) , ξ¯+ (t) = T¯f¯a−
P¯i ξ¯− (t) − Pi ξ¯+ (t) , P¯i ξ¯+ (t) − Pi ξ¯− (t) P¯i ξ¯− (t) − Pi ξ¯+ (t) , P¯i ξ¯+ (t) − Pi ξ¯− (t)
where ξ¯+ and ξ¯− represent the upper and lower bounds of the augmented state vector after the coordinates have been transformed. Under Assumption 2, the following can be obtained: ¯ f¯aξ− ξ¯− (t) , ξ¯+ (t) = P¯ ϕ¯− ξ¯− (t) , ξ¯+ (t) − Pϕ¯+ ξ¯− (t) , ξ¯+ (t) 6 P T¯f¯a P −1 ξ¯ (t) 6 P¯ ϕ¯+ ξ¯− (t) , ξ¯+ (t) − Pϕ¯− ξ¯− (t) , ξ¯+ (t) ¯ = f¯aξ+ ξ¯− (t) , ξ¯+ (t) .
(26)
where P¯ = max {0, P }, P = P¯ − P , P¯i = max 0, P −1 and Pi = P¯i − P −1 . ¯
¯
The upper bound error e¯ξ+ (t) and lower bound error e¯ξ− (t) are defined as
follows:
+
¯ ¯ (t) − ξ¯ (t) e¯ξ+ (t) = ξˆ − ¯ ¯ (t) e¯ξ− (t) = ξ¯ (t) − ξˆ
Then, the error dynamic equation can be obtained as follows: ¯ ¯ e¯˙ ξ+ (t) = R¯ eξ+ (t) + P T¯ f¯a+ (¯ x (t)) − f¯a (¯ x (t)) + P T¯I¯r (f˙+ (t) − f˙(t))
¯ ¯ ˙ ξ− e¯ (t) = R¯ eξ− (t) + P T¯ f¯a (¯ x (t)) − f¯a− (¯ x (t)) − P T¯I¯r (f˙(t) − f˙− (t)) (27)
¯ C) ¯ and a Metzler matrix R have the Theorem 3. Let the matrix (T¯A¯ − L
¯ If two vectors e¯1 and e¯2 exist such that the pairs same eigenvalues for some L.
17
¯ C, ¯ e¯1 ) and (R, e¯2 ) are observable, then (T¯A¯ − L P = O2−1 O1 . ¯ N = P L. ¯ satisfies Sylvester equation: P T¯A¯ − RP = N C, where
O1 =
e¯1 .. . ¯ C¯ e¯1 T¯A¯ − L
n−1
, O2 =
e¯2 .. . e¯2 Rn−1
.
¯ C, ¯ e¯1 ) and (R, e¯2 ) are observable, matrices O1 Proof : Since the pairs (T¯A¯ − L and O2 are nonsingular. The following equation can be obtained through the transformation: ¯ C)O ¯ −1 = O2 RO−1 O1 (T¯A¯ − L 1 2 ¯ C)O ¯ −1 and O2 RO−1 are canonical observability forms. The where O1 (T¯A¯ − L 1 2 ¯ C) ¯ is observable. vectors e¯1 and e¯2 can be found if the pair (T¯A,
2
Remark 6. The Metzler matrices are constructed as follows: Let R be the lower triangular matrix, with the main diagonal elements as the eigenvalues of (T¯A¯ −
¯ C), ¯ and the elements below the main diagonal are positive. Moreover, the L matrix R after transformation satisfies the property of Hurwitz and is a matrix ¯− (t) 6 ξ(t) ¯ 6 Metzler at the same time. Combined with the above process, ξˆ ˆ ξˆ¯+ (t), t > 0. Then, the fault estimate is obtained by fˆ+ (t) = I¯rT x ¯+ (t), fˆ− (t) = ˆ I¯rT x ¯− (t). 3.3. Enhancement of the Accuracy of IO-UIO When the interval observer is used to estimate the fault, the response time and the accuracy of the interval, that is, the difference between the upper and lower bounds of the interval observer, are considered. This subsection focuses on finding a performance index related to the interval accuracy based on the CRH5 motor system, which makes the interval size adjustable. Since the equivalent transformation does not affect the experimental results, the system can be conveniently investigated based on the previous transforma18
tion before. The dynamic error equation can be derived from (12) and (13). x ¯(t) =
h
x1
x2
x3
x4
x5
x6
iT
=
h
iqs
ids
λqr
λdr
ωm
f
iT
Then, the following can be obtained: ¯ C)¯ ¯ ex + ψ T¯(f¯+ − f¯− ) + ψ T¯I¯r (f˙+ − f˙− ) ψ e¯˙ x = ψ(T¯A¯ − L a a where ψ =
h
0
0 0
0
0
(28)
i 1 .
Equation (28) can be simplified as follows: e˙ f = −l61 ex1 + l62 ex2 − l61 ef
(29)
where l61 represents the first column of the sixth row of the observer gain ma¯ and l62 represents the second column of the sixth row of this matrix. trix L, Moreover, the interval size of the fault estimate is related to the values of ex1 , ex2 and ef and the interval size can be regulated by l61 and l62 . Remark 7. Equation (29) is not generic and is extracted from the characteristics of CRH5. Hence, this equation will no longer be applied in changing other systems or system parameters. However, the method is general, and a performance index can be found according to the system model.
4. Simulation Results In this section, numerical and simulation results are used to verify the feasibility of the proposed method for CRH5 (6FJA3257A). First part of this section discusses the fault estimation with regional pole placement. In the second part, we changed the numerical value of the performance index. Then, we compare the influence of the performance parameters on the fault estimation precision from the residual of the upper and lower bounds of the fault estimation curve in the simulation diagram. According to the high-speed train traction motor system, the state vectors
19
under stable operation can be obtained as follows: x(t) =
h
−150.3
−222 0.3958
−6.3345 124.9053
iT
Set the upper and lower bounds of the state vectors: h iT x+ (t) = −150.0 −221 0.40 −6.32 125.5 h iT x− (t) = −150.6 −223 0.39 −6.34 124.5
it is obviously satisfying to Assumption 2.
We illustrate the simulation results with the fault of q-axis stator current sensor. The fault model is as follows: 0, f (t) = 10,
0s 6 t < 10s t > 10s
Part 1. Fault estimation with regional pole placement. We can solve (14) and (15) to obtain the observer gain matrix by choosing the region pole D(−20, 50). Remark 8. When constraints (14) and (15) using the LMI toolbox of Matlab ¯ C) ¯ are located in the left half (2014b) are satisfied, the eigenvalues of (T¯A¯ − L plane of the coordinate virtual axis and within the pole configuration circle. The LMI region is as follows: D = { z ∈ C: Re(z) < 0},
fD (z) = z + z¯ < 0.
where fD (z) is the characteristic function of D. We choose D(−20, 50) to improve the performance of the system.
20
¯ Y¯ and L ¯ can be calculated as follows: Parameters Q,
0.5649
0.0193 −0.0810 ¯ Q= −1.4510 0 −0.2635
−11.7195
28.1386 75.9143 ¯ Y = 21.5699 0 129.4668
0.0193
−0.0810
−1.4510
0
0.0855
0.2357
−0.0766
0
0.2357
0.7229
0.1294
0
−0.0766
0.1294
3.7438
0
0
0
0
2.0965
0.3055
0.8062
0.6428
0
−212.4609
0
21.6346
757.2506 0 −145.1688 0 ¯= ,L 34.7317 0 87.8535 0 0 −0.4252
−9.4913 23.3952 545.5703 0 83.6815
−0.2635
0.3055 0.8062 . 0.6428 0 3.4594
−669.2785
0
0 −0.0031 ¯ = H 0 0 0
0
0
0 0 0 −0.0031 0 0 1 0 0 0
¯d = 0, because From subsection A, matrix T¯ can be calculated through T¯E ¯d = E
h
0
0 0 0
21
−1/15 0
iT
23.6974 0 −28.8060 0 −111.9553 0 0 41.9045 −1.3582 0
¯ C¯ = I, H ¯ can be obtained when T¯ and C¯ are known. Through T¯ + H
0
then,
1
0 0.0031 T¯ = 0 0 0
0
0 0 0
0
1
0 0 0
0
0
1 0
0.0031 0 1 0
0 0
0
0 0
0 0.0031 0 0 0 0 0 1
When the numerical values of the motor in Table I are substituted into the system, the following can be obtained:
−64.5606
−376.9911 965.3409
376.9911 0.0884 A¯ = 0 0 0
−64.5606
0
0
−2.9556
0.0884
376.9911
0
0
0
0
−64.5606
376.9911 −0.1093 ¯ ¯ TA = 1.1543 0 0
0
0 0
0 0 −376.9911 0 0 . −2.9556 0 0 0 0 0 0 0 0 965.3409
−376.9911 965.3409 −64.5606
0
−1.1543
0
−0.1093
376.9911
0
0
0
0
0
0 0
0 0 −376.9911 0 0 . 0 0 0 0 0 0 0 0 0 965.3409
¯ C) ¯ is A lower triangular matrix R with the same eigenvalues as (T¯A¯ − L
¯ considered, and the matrices e¯1 , e¯2 satisfy Theorem 2. The eigenvalues of (T¯A− ¯ C) ¯ are -50.6767, -25.0914 ± 34.6127i, -36.5993 ± 27.4065i and -41.9045. L e¯1 =
h
e¯2 =
1 h
8 1
−2 1
−2
0 0 1 0
22
−1 i −1
i
R=
−50.6767 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
1.0000 + 0.0000i
−25.0914 + 34.6127i
0.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
−25.0914 − 34.6127i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
−36.5993 + 27.4065i
0.0000 + 0.0000i
0.0000 + 0.0000i
1.0000 + 0.0000i
−36.5993 + 27.4065i
0.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
−41.9045 + 0.0000i
Then, the transformation matrix can be obtained by Theorem 2,
P =
−50.6767 − 0.0000i
0.0000 − 0.0000i
0.0000 − 0.0000i
0.9999 − 0.0001i
−25.0914 + 34.6127i
0.0000 − 0.0000i
1.0000 − 0.0006i
1.0000 + 0.0001i
−25.0914 − 34.6127i
1.0000 − 0.0000i
1.0000 + 0.0000i
1.0000 − 0.0000i
0.9999 + 0.0005i
1.0001 − 0.0001i
1.0000 + 0.0000i
1.0001 − 0.0000i
1.0000 − 0.0000i
1.0000 − 0.0000i
0.0001 + 0.0001i
0.0000 − 0.0000i
0.0000 + 0.0000i
0.0001 − 0.0004i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0011 − 0.0011i
0.0000 + 0.0000i
0.0000 + 0.0001i
−36.5993 + 27.4065i
0.0000 − 0.0000i
0.0000 + 0.0000i
0.9989 + 0.0006i
−36.5993 − 27.4065i
0.0000 − 0.001i
1.0001 + 0.0001i
1.0000 − 0.0000i
−41.9045 + 0.0000i
.
.
Matrix P is proved to be nonsingular, and the simulation results are displayed as follows. Figure 1 illustrates the fault estimation response with the region pole placement. The simulation results show that the fault estimate curve finally
23
tracks the actual failure, and the real fault is between the upper bound fault observation value and the lower bound observation value. Fault estimation
500
0
-500
fault lower estimation upper estimation
20
-1000
10
-1500
-10
0
-20
-2000
9
10
11
12
13
14
-2500
-3000
0
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 1: Simulation results with region pole placement.
Figure 2 shows the fault estimation response without region pole placement. The fault tracking speed in Figure 1 is faster than in Figure 2. Thus, the performance of the fault estimation can be improved after considering a circular domain constraint to the solution. Fault estimation
400
fault upper estimation lower estimation
350 300
20
Partial enlargement:
20
250
10
10 200
0 0
-10
150 -10 100
9 9
10
10
11
11
12 12
13 13
14
50 0 -50
0
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 2: Simulation results without region pole placement.
24
Part 2. The method mentioned in subsection 3.3 will be validated in this subsection. Firstly, we study the influence of the numerical value of the parameter l61 on the interval size when other parameters remain unchanged. The
0
-50 9.7
error of interval 23.7 9.8
9.9
10
10.1
10.2
10.4
15 10.5
Time (s) L61= -0.5252
50
0
-50 9.7
20 10 0 10 10.3
f(t) and estimate of f(t)
L61= -0.2252
50
9.8
9.9
10
error of interval 3.9
20 10 0 10
10.1
10.3
10.2
15 10.4
10.5
f(t) and estimate of f(t)
f(t) and estimate of f(t)
f(t) and estimate of f(t)
simulation results are as follows. The small figure inside Figure 3 is the locally L61= -0.4752
50
0 20 error of interval 9.5 -50 9.7
9.8
9.9
10
10.1
10
10.2
0 10 10.3
10.4
15 10.5
10.2
20 10 0 10 10.3 10.4
15 10.5
Time (s) L61= -0.55
50
0 error of interval 2.6 -50 9.7
Time (s)
9.8
9.9
10
10.1
Time (s)
Figure 3: Simulation results with different parameter l61 .
enlarged figure. Figure 3 shows that the interval size decreases with the decrease of the parameter l61 . Hence, the smaller the parameter l61 , the better the effect of fault estimation will be. Next, we study the influence of the numerical value of the parameter l62 on the interval size when the other parameters remain unchanged. The simulation results are as follows, in which, the small figure inside the figure 4 is error of the upper and lower bounds. Figure 4 illustrates that the interval size decreases with increase in parameter l62 . Therefore, the bigger the parameter l62 , the better the effect of fault estimation will be. Remark 9. Through (29) we can see that parameters l61 and l62 affect the fault estimation error ef . We can obtain conclusions from Figure 3 and Figure 4: the smaller the parameter l61 , the better the effect of fault estimation will be and the bigger the parameter l62 , the better the effect of fault estimation will ¯ to be. This can provide a reference in the process of solving the parameter L achieve better fault estimation.
25
20
-50 9.7
9.8
9.9
error of interval 10 14.4 0 10 10.1 10.2 10 10.3
10.4
15 10.5
Time (s) L62= -1.1582
50
0
-50 9.7
f(t) and estimate of f(t)
0
f(t) and estimate of f(t)
f(t) and estimate of f(t) f(t) and estimate of f(t)
L62= -1.6582
50
9.8
9.9
20 error of interval 10 5.3 0 10 10.1 10.2 1010.3
10.4 1510.5
L62= -1.3582
50
0
-50 9.7
9.8
9.9
20 error of interval 10 9.6 0 10 10 10.1 10.2 10.3
10.4
15 10.5
20 error of interval 10 1.1 0 10 10.1 10.2 10.3 10.4
15 10.5
Time (s) L62= -0.99
50
0
-50 9.7
Time (s)
9.8
9.9
10
Time (s)
Figure 4: Simulation results with different parameter l62 .
5. Conclusion In this paper, an IO-UIO based fault estimation scheme for high-speed railway traction motor with sensor fault and disturbances is investigated. For the nonlinear part of the system, an interval observer is introduced to solve efficiently only a part of the information in the interval. Therefore the design of unknown input observer is proposed, and this observer can completely eliminate the external disturbance. Moreover, a FD algorithm is proposed to estimate the faults accompanied by disturbance. We constructed a Metzler matrix to match the properties of the interval observers by transforming the coordinates. In addition, we found a performance index that can adjust the size of the interval observer. Finally, the effectiveness and advantages of the proposed IO-UIO method for high-speed railway traction motor is shown through simulation results.
26
References References [1] Y. J. Wang, Impediment to progress and inevitable trend of the high-speed railway development in china, (15) (2014) 256–258. [2] H. H. T. Liu, P. Shi, B. Jiang, Fault detection, diagnosis, and fault tolerant control with flight applications, Journal of the Franklin Institute 350 (9) (2013) 2371–2372. [3] M. Y. Chen, D. H. Zhou, Fault detection and isolation in a class of nonlinear systems, Acta Automatica Sinica 30 (6) (2004) 1011–1016. [4] Y. Guo, Z. Mao, B. Jiang, Observer based fault diagnosis for induction motor with sensor faults and disturbances, Chinese Automation Congress (2017) 5084–5089. [5] D. Hu, A. Sarosh, Y. F. Dong, A novel KFCM based fault diagnosis method for unknown faults in satellite reaction wheels, ISA Transactions 51 (2) (2012) 309–316. [6] W. Chen, M. Saif, A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems, International Journal of Systems Science 38 (12) (2007) 943–955. [7] Y. Zhiguo, G. Vansteenkiste, Improving the observer-based fdi design for efficient fault isolation, International Journal of Control 68 (1) (1997) 197– 218. [8] C. Deng, G. H. Yang, Adaptive fault-tolerant control for a class of nonlinear multi-agent systems with actuator faults, Journal of the Franklin Institute 354 (12) (2017) 4784–4800. [9] Q. Hu, B. Xiao, Y. Zhang, Fault-tolerant attitude control for spacecraft under loss of actuator effectiveness, Control & Decision 34 (3) (2015) 263– 268. 27
[10] C. Deng, G. H. Yang, Distributed adaptive fault-tolerant containment control for a class of multi-agent systems with non-identical matching nonlinear functions, IET Control Theory & Applications 10 (3) (2016) 273– 281. [11] S. Tong, L. Zhang, Y. Li, Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones, IEEE Transactions on Systems Man & Cybernetics Systems 46 (1) (2015) 37–47. [12] H. M. Wang, D. Ye, G. H. Yang, Actuator fault diagnosis for uncertain ts fuzzy systems with local nonlinear models, Nonlinear Dynamics 76 (4) (2014) 1977–1988. [13] P. S. Teh, H. Trinh, Design of unknown input functional observers for nonlinear systems with application to fault diagnosis, Journal of Process Control 23 (8) (2013) 1169–1184. [14] X. G. Yan, C. Edwards, Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems, International Journal of Systems Science 39 (4) (2008) 349–359. [15] T. Wang, W. Xie, Y. Zhang, Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults, ISA Transactions 51 (3) (2012) 386. [16] A. M. Pertew, H. J. Marquez, Q. Zhao, LMI-based sensor fault diagnosis for nonlinear lipschitz systems , Automatica 43 (8) (2007) 1464–1469. [17] D. Wang, K. Y. Lum, Adaptive unknown input observer approach for aircraft actuator fault detection and isolation, International Journal of Adaptive Control & Signal Processing 21 (1) (2007) 31–48. [18] B. Jiang, S. Marcel, Adaptive observer design for robust fault estimation, International Journal of Systems Science 33 (9) (2002) 767–775.
28
[19] M. Chen, P. Shi, C. C. Lim, Adaptive neural fault-tolerant control of a 3-dof model helicopter system, IEEE Transactions on Systems Man & Cybernetics: Systems 46 (2) (2016) 260–270. [20] X. G. Yan, C. Edwards, Adaptive sliding-mode-observer-based fault reconstruction for nonlinear systems with parametric uncertainties, IEEE Transactions on Industrial Electronics 55 (11) (2008) 4029–4036. [21] M. Deng, T. Maekawa, N. Bu, Actuator fault detection on a process control experimental system using combined observers, Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems & Control Engineering 225 (6) (2011) 807–813. [22] B. L. Walcott, S. H. Zak, Combined observer-controller synthesis for uncertain dynamical systems with applications, Systems Man & Cybernetics IEEE Transactions on 18 (1) (1988) 88–104. [23] K. Zhang, V. Cocquempot, B. Jiang, Adjustable parameter-based multiobjective fault estimation observer design for continuous-time/discretetime dynamic systems, International Journal of Control Automation & Systems 15 (3) (2017) 1077–1088. [24] K. Zhang, B. Jiang, X. G. Yan, Z. Mao, Sliding mode observer based incipient sensor fault detection with application to high-speed railway traction device, Isa Transactions 63 (2016) 49–59. [25] Y. M. Fu, G. R. Duan, S. M. Song, Design of unknown input observer for linear time-delay systems, International Journal of Control Automation & Systems 2 (4) (2004) 530–535. [26] D. Bestle, M. Zeitz, Canonical form observer design for non-linear timevariable systems, International Journal of Control 38 (2) (1983) 419–431. [27] D. Efimov, T. Raissi, A. Zolghadri, Control of nonlinear and Lpv systems: Interval observer-based framework, IEEE Transactions on Automatic Control 58 (3) (2013) 773–778. 29
[28] F. Mazenc, O. Bernard, Interval observers for linear time-invariant systems with disturbances , Automatica 47 (1) (2011) 140–147. [29] F. Mazenc, T. N. Dinh, S. I. Niculescu, Interval observers for discretetime systems, International Journal of Robust & Nonlinear Control 24 (17) (2015) 2867–2890. [30] D. Efimov, W. Perruquetti, T. Rassi, A. Zolghadri, Interval observers for time-varying discrete-time systems, IEEE Transactions on Automatic Control 58 (12) (2013) 3218–3224. [31] K. Zhang, B. Jiang, P. Shi, Fast fault estimation and accommodation for dynamical systems, IET Control Theory & Applications 3 (2) (2009) 189– 199. [32] J. Chen, R. J. Patton, Robust model-based fault diagnosis for dynamic systems, Springer US, 1999. [33] O. Toker, A kharitonov-like theorem for robust stability independent of delay of interval quasipolynomials, International Journal of Robust & Nonlinear Control 20 (6) (2010) 623–638. [34] P. Chen, Y. Z. Lu, Automatic design of robust optimal controller for interval plants using genetic programming and kharitonov theorem, International Journal of Computational Intelligence Systems 4 (5) (2011) 826–836. [35] H. Chapellat, S. P. Bhattacharyya, A generalization of kharitonov’s theorem; robust stability of interval plants, Automatic Control IEEE Transactions on 34 (3) (1989) 306–311.
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Journal Pre-proof
Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor Yang Tian, Ke Zhang, Bin Jiang, Xing-Gang Yan PII: DOI: Reference:
S0016-0032(19)30866-X https://doi.org/10.1016/j.jfranklin.2019.11.062 FI 4297
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
27 March 2018 2 April 2019 25 November 2019
Please cite this article as: Yang Tian, Ke Zhang, Bin Jiang, Xing-Gang Yan, Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.062
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Interval Observer and Unknown Input Observer-based Sensor Fault Estimation for High-speed Railway Traction Motor✩ Yang Tiana , Ke Zhanga,∗, Bin Jianga , Xing-Gang Yanb a College
of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 211106 P.R. China b School of Engineering and Digital Arts, University of Kent, Canterbury, Kent CT2 7NT, United Kingdom.
Abstract In this paper, fault estimation for high-speed railway traction motor with sensor fault and disturbances is investigated based on the interval observer and unknown input observer (IO-UIO). First, the proposed method, which can completely eliminate the external disturbance, is studied based on the disturbance isolation characteristic of the unknown input observer. Then, an interval observer is constructed to deal with the nonlinear part, which sandwiched the actual system between the upper and lower bounds. Moreover, the Metzler matrix is constructed using an equivalent transformation and through the unified design based on the concept of the augmented state to form a global fault augmented model. Finally, simulation results are presented to illustrate the effectiveness and advantages of the proposed IO-UIO. Keywords: Railway traction motor, Metzler matrix, Equivalent transformation, Interval observer, Unknown input observer
✩ This work was supported in part by the National Natural Science Foundation of China under Grants 61490703, 61673207, in part by Qing Lan Project of Jiangsu Province, and in part by China Scholarship Council. ∗ Corresponding author Email addresses: [email protected] (Yang Tian), [email protected] (Ke Zhang), [email protected] (Bin Jiang), [email protected] (Xing-Gang Yan)
Preprint submitted to Journal of LATEX Templates
December 11, 2019
1. Introduction The development of high-speed railways and the progress in science and technology over the past decade have resulted in the increased requirements for the safety and stability of high-speed trains. These necessities are closely related to the safety of people’s life and property. High-speed railway has developed rapidly in many countries worldwide [1]. At present, the world’s most representative high-speed railways include the German ice train, Japan Shinkansen, French TGV trains and Chinese CRH trains. Compared with air transport, the high-speed railway has several advantages, such as low cost, large operating area, environmental protection and all-weather operation. The development of high-speed railway has not only a stimulating effect on equipment technology and modern manufacturing but also presents great impacts on information communication and construction modernisation. Modern control systems have become more and more complicated. Therefore, fault detection, fault diagnosis and fault-tolerant techniques have received more attention. Fault detection is the first step of fault diagnosis, which determines the occurrence of a system fault[2–5]. The fault should be detected before it becomes serious. Fault isolation is to determine the specific location of the fault[6, 7]. Then, fault estimation is performed to identify the form and size of the fault and is an important prerequisite for fault tolerance control[8–10]. The nonlinear system has been an area of active research during the last two decades[11–13]. However, the quality of actuators and sensors in a control system has been deteriorating with time. Moreover, the system sensors are used to measure system output, and these sensors indirectly affect the performance of the control system. The fault detection and diagnosis (FDD) and fault tolerance of the actuator faults are easier to determine than sensor faults (i.e. [14–16]). Only a few studies have reported on the FDD of sensor faults. Traction motor is the core equipment of high-speed trains, and this part is closely related to the safe operation of the train. Safety of the traction motor becomes very important as the high-speed train accelerates. Therefore, investi-
2
gating the fault diagnosis of a traction motor is of great practical significance. Induction motors play an important role in the industry. A motor fault may not only cause economic loss but also casualties. Furthermore, during the past few decades, the focus of fault diagnosis research on induction motor was to explore how to efficiently monitor the induction motor to avoid motor failure. Many methods have been put forward for fault diagnosis. An adaptive observer has been proposed in [17] to detect system faults. A fast adaptive fault estimation algorithm was proposed in ([18–20]) to enhance the accuracy of fault estimation. In this process, the estimators consist of proportional and integral items. As for fault detection, to use some observers or combined observers have been discussed by researchers([21, 22]). Vector augmentation was performed using the robust observer to estimate the fault [23]. A sliding mode observer was applied to a high-speed train in [24], where linear matrix inequalities (LMIs) are used to ensure the robustness with respect to the interference. An unknown input observer (UIO) was proposed in [25], which eliminated the unknown input item. Methods for nonlinear system observers have been proposed, such as Lyapunov method, coordinate transformation method[26], extended Kalman filter, interval observer (IO) and feedback linearisation technology. Recently, the interval observer approach has been extensively investigated, and some important achievements have been obtained. In [27], a class of interval observer was successfully designed for nonlinear systems, and the state of the error system could be guaranteed to be non-negative when the condition of the Metzler matrix was satisfied. In [28] and [29], the designed method of interval observer was studied for continuous and discrete linear time-invariant systems with bounded disturbances, based on the Jordan canonical form, and the appropriate of linear time-varying transformation. In [30], the designed method of interval observer was studied based on the Sylvester equation and the linear time-invariant transformation. In [27], the system control strategy based on an interval observer was studied. In this paper, a fault estimation observer is proposed for high-speed railway traction motor with sensor fault and disturbances. The main contributions of 3
this study are as follows. Firstly, an interval observer is designed based on the unknown input observer (UIO), which eliminates the effects of the external disturbance on the fault diagnosis, and only uses partial information of the measurement output interval to ensure the real-time tracking of the original system. This process is different from the classical observer design, because the error system is asymptotically convergent to zero in the proposed design. Secondly, the observer is designed for the augmented system composed of a state vector and a fault. Then, a method for constructing the Metzler matrix for systems which do not satisfy the condition of the Metzler matrix is proposed. This technique is successfully applied to high-speed train motor system based on the observable standard form. Thirdly, the system parameters are found to enhance fault estimation performance. The remainder of this paper is organised as follows: Preliminaries and problem formulation are presented in Section 2. Section 3 contains the design of interval observer and unknown input observer (IO-UIO)-based FD for the highspeed train and the coordinate transformation. Simulation results are provided to illustrate the effectiveness of the proposed methods in Section 4. Finally, a conclusion is made in Section 5.
2. Preliminaries and Problem Formulation 2.1. Preliminaries R denotes a set of real numbers. R ≥ 0 denotes a set of non-negative real
numbers, i.e., R ≥ 0 := [0, ∞). I denotes the identity matrix in Rn×n . For
vectors xa = [xa,1 , ..., xa,n ]T ∈ Rn and xb = [xb,1 , ..., xb,n ]T ∈ Rn , the inequality
xa ≤ xb , if and only if xa,i ≤ xb,i , for all i = 1, ..., n. For matrices A = (ai,j )m×n ∈ Rm×n and B = (bi,j )m×n ∈ Rm×n , the inequality A > B(A > B) denotes that ai,j > bi,j (ai,j > bi,j ), i = 1, ..., n, j = 1, ..., m. For a square matrix Q ∈ Rn×n , let the matrix Q+ ∈ Rn×n denote Q+ = max{qi,j , 0})n,n i,j=1,1
and Q− = Q+ − Q. The superscripts + and − are defined appropriately for other cases when they appear in the following sections.
4
The following definitions are given: a matrix A ∈ Rn×n is called Hurwitz if all its eigenvalues have a negative real part, and it is called Metzler if all its off-diagonal elements are non-negative. The following lemmas will be used in the sequel. Lemma 1[31]. The eigenvalues of a given matrix A ∈ Rn×n belong to the disk region D(α, τ ) with centre α+j0 and radius τ if and only if a symmetric positive definite matrix P ∈ Rn×n exists such that the following condition holds:
−P
P(A − αIn ) −τ 2 P
∗
<0
(1)
Lemma 2[28]. Consider a vector variable x ∈ Rn satisfying x 6 x 6 x with
x, x ∈ Rn , and a constant matrix A ∈ Rm×n , the following condition holds: A+ x − A− x 6 Ax 6 A+ x − A− x
(2)
2.2. Problem Formulation Consider the ideal mathematical model of the traction motor in the d − q axis synchronous rotating reference frame [4] as follows: x˙ (t) = Ax (t) + Bu (t) + fa (x (t))
where x (t) =
h
x1
x2
(3)
y (t) = Cx (t)
x3
x4
x5
iT
=
h
iqs
ids
λqr
λdr
ωm
iT
∈ R5 ,
is the state variable, u(t) ∈ R2 is the input variable and y(t) ∈ R3 is the output variable. A is the matrix of asynchronous motor system, whilst B is the input
5
distribution matrix of the system which are as follows:
−γ
ωs A = αLm 0 0
−ωs
αβ
0
−γ
0
αβ
0
−α
−ωs
αLm
ωs
α
0
0
0
0
0 0 ,B = 0 0
1 δLs
0
0
1 δLs
0
0
0
0
0
0
,
C is the output distribution matrix, and fa (x (t)) is the nonlinear part of the system given as follows:
1
C= 0 0
0
0
1
0
0
0
0 0 , fa (x (t)) = 0 1 0 0
where δ =1− µ=
−np βx5 x4 np βx5 x3 np βx5 x4 −np βx5 x3
µ (x4 x1 − x3 x2 ) − J1 TL
,
L2m Rr Lm ,α = ,β = , Ls Lr Lr δLs Lr
L2 Rr Rs 3 Lm np ,γ = m 2 + , 2 JLr δLs Lr δLs
iqs and ids denote the stator currents in the d − q frame; λqr and λdr denote the rotor flues in the d − q frame; ωm is the mechanical angular speed; ωs is the synchronous rotating speed; Ls , Lr and Lm denote stator self inductance, rotor self inductance and state-rotor mutual inductance, respectively; TL is the load torque; np is the pole pair number; and J is the moment inertia. The CRH5 adopts three-phase squirrel-cage asynchronous motor named 6FJA3257A, the parameters of which are shown in Table 1. The actual mathematical model of the traction motor can be represented as follows: x˙ (t) = Ax (t) + Bu (t) + fa (x (t)) + Ed d (t)
y (t) = Cx (t) + Ef f (t) 6
(4)
where d(t) ∈ R1 is the disturbance vector, and f (t) ∈ R1 represents the sensor fault. Ed and Ef are the disturbance distribution matrix, and fault distribution matrix, respectively. A, B, Ef , C and Ed are constant real matrices of appropriate dimensions. In addition, matrices Ef and Ed are of full column rank given as follows:
1
h Ef = 0 , Ed = 0 0 0 0 0
− J1
iT
, d (t) = ∆TL .
The following table records the relevant parameters of the asynchronous motor: Table 1: Asynchronous motor 6FJA3257A No. 1 2 3 4 5 6 7 8 9
parameter
value
Stator resistance Rs 105.1 mΩ Stator inductance Ls 31.7 mH Rotor resistance Rr 91.9 mΩ Rotor inductance Lr 31.7 mH Mutual inductance Lm 29.9 mH The moment inertia J 15 kg · m2 The load torque TL 4500 N · m The pole pairs number np 3 The Motor rotation speed ws 2 ∗ pi ∗ 60 rad/s
Remark 1. Available results of the interval observer for the induction motor are very few. This paper mainly focuses on how to estimate the sensor fault of high-speed railway traction motor and how to improve the performance of fault estimation. A combined observer is designed for the system, where an IO is introduced to solve nonlinear part and an UIO is introduced to eliminate the effect of external disturbance on the fault estimation. When the system can not meet the IO conditions, we use coordinate transformation method to construct Metzler matrix.
7
3. Main Results 3.1. IO-UIO Based Fault Estimation Design In this subsection, an observer is designed to estimate the incipient sensor faults. Interval information is applied to deal with the nonlinear part of the system. The augmented fault estimator is introduced, and the sensor fault vector is considered as the auxiliary state vector. The original system (4) can be rewritten in the following form: x(t) ˙ A 0 x(t) B E = + u(t) + d d(t) ˙ f (t) 0 0 f (t) 0 0 0 fa (x (t)) + f˙(t) + I 0 r h i x(t) y(t) = C Ef f (t)
(5)
For convenience, state the following parameters are defined: augmented vecx(t) A 0 , augmented state distribution matrix A¯ = , augtor x(t) = f (t) 0 0 B ¯= , augmented output distribution mented input distribution matrix B 0 h i fa (x (t)) , matrix C¯ = C Ef , augmented nonlinear vector f¯a (¯ x (t)) = 0 Ed ¯d = , and augmented Fault augmented disturbance distribution matrix E 0 0 . distribution matrix I¯r = Ir Then, system(5) can be rewritten as follows: ¯x(t) + Bu(t) ¯ ¯d d(t) + f¯a (¯ x ¯˙ (t) = A¯ +E x (t)) + I¯r f˙(t)
y(t) = C¯ x ¯(t)
8
(6)
where f˙(t) is the derivative of the fault f (t), and Ir is an r × r-dimensional unit matrix. ¯ C¯ = I, an equivalent augmented system state space Combining with T¯ + H description form can be obtained as follows: ¯x(t) + Bu(t) ¯ ¯d d(t) + I¯r f˙(t) + f¯a (¯ ¯ y(t) ¯ C¯ x x ¯˙ (t) = A¯ +E x (t)) + H ˙ −H ¯˙ (t) ¯x(t) + Bu(t) ¯ ¯d d(t) + I¯r f˙(t) + f¯a (¯ ¯ y(t) ¯ C( ¯ A¯ ¯x(t) + Bu(t) ¯ = A¯ +E x (t)) + H ˙ − H ¯d d(t) + I¯r f˙(t) + f¯a (¯ +E x (t))) ¯x(t) + T¯Bu(t) ¯ ¯d d(t) + T¯f¯a (¯ ¯ y(t) = T¯A¯ + T¯E ˙ x (t)) + T¯I¯r f˙(t) + H y(t) = C¯ x ¯(t) (7)
¯ and T¯ are the design parameters with appropriate dimensions. whereH
¯ C) ¯ is observable and rank(C¯ E ¯d ) = rank(E ¯d ). Assumption 1. The pair (T¯A, Lemma 3[32]. Under Assumption 1, the expression described by (8) is a UIO for system (7). For system (7), the problem of fault estimation is converted to IO-UIO design to reconstruct the fault vector. An observer is proposed to achieve fault estimation as follows: ¯ ¯ (ˆ ˆ ¯(t) + T¯Bu(t) + T¯f¯a (¯ x (t)) + T¯I¯r f˙+ (t) − L y (t) − y(t)) z¯˙ (t) = T¯A¯x ¯ x ˆ ¯(t) = z¯(t) + Hy(t) ˆ fˆ(t) = I¯rT x ¯(t) ˆ yˆ(t) = C¯ x ¯(t)
(8)
¯ C¯ = I. z¯(t) is an unknown input augmented variable, whilst x ˆ where T¯ + H ¯(t) and yˆ ¯(t) denote the augmented state vector and output vector. fˆ(t) is the sensor ¯ , T¯ and L ¯ are the gain matrices of the fault estimate value, whilst matrices H unknown input observer. Assumption 2. Given that the arbitrary x ¯− (t) 6 x ¯ (t) 6 x ¯+ (t), vector func-
9
tions f¯a− (¯ x+ (t) , x ¯− (t)) and f¯a+ (¯ x+ (t) , x ¯− (t)) exist and satisfy the following: f˙− (t) 6 f˙(t) 6 f˙+ (t) f¯a− x ¯+ (t) , x ¯− (t) 6 f¯a (¯ x (t)) 6 f¯a+ x ¯+ (t) , x ¯− (t)
where x ¯− (t) and x ¯+ (t) denote the lower and upper bounds of the augmented state vector, f˙− (t) and f˙+ (t) denote the lower and upper of the fault derivative respectively. Remark 2. There are many methods to solve nonlinear systems, such as local linearization. However, due to the particularity of high-speed railway traction motor, we can not find a suitable constant satisfying condition of Lipschitz[16]. Then an IO[28] is constructed to solve nonlinear part when Assumption 2 is satisfied. The idea of this method is to only use interval information while avoiding the limitations of other methods . ¯ exists such that the matrix (T¯A¯ − L ¯ C) ¯ is Metzler Assumption 3. Matrix L and satisfies Hurwitz. The interval observer is designed based on an unknown input observer. Under Assumption 2, the IO-UIO is described as follows: The upper bound observer is as follows: + ¯ ¯ yˆ+ (t) − y(t) ˆ ¯+ (t) + T¯Bu(t) + T¯f¯a+ (¯ x (t)) + T¯I¯r f˙+ (t) − L z¯˙ (t) = T¯A¯x ¯ + (t) x ˆ ¯+ (t) = z¯+ (t) + Hy ˆ fˆ+ (t) = I¯rT x ¯+ (t) + ˆ yˆ (t) = C¯ x ¯+ (t)
(9)
10
The lower bound observer is as follows: ¯ ¯ yˆ− (t) − y(t) ˆ z¯˙ − (t) = T¯A¯x ¯− (t) + T¯Bu(t) + T¯f¯a− (¯ x (t)) + T¯I¯r f˙− (t) − L ˆ− ¯ − (t) x ¯ (t) = z¯− (t) + Hy ˆ ¯− (t) fˆ− (t) = I¯rT x − ˆ yˆ (t) = C¯ x ¯− (t)
(10)
where z¯+ (t) and z¯− (t) represent the state of the upper and lower bounds of the ¯ C¯ = I. observers, respectively, and T¯ + H Theorem 1. Under Assumptions 2 and 3, if the initial conditions are satisfied, and the constraint x ¯− (0) 6 x ¯(0) 6 x ¯+ (0) , then the solutions of systems (7), (9) and (10) satisfy the following: ˆ− (t) 6 x ˆ x ¯ ¯(t) 6 x ¯+ (t), ∀t > 0. Proof : The augmented system error dynamic equation is considered. For the ˆ upper bound observer, let e¯+ ¯+ (t) − x ¯(t), ey + (t) = yˆ+ (t) − y(t) = C¯ e¯+ x (t) = x x (t), and ef + (t) = fˆ+ (t) − f (t).
Then, from (7) and (9), the system’s error state equation is expressed as follows: ¯ ¯ yˆ+ (t) − y(t) ˆ ˆ e¯˙ + ¯˙ + (t) − x ¯˙ (t) = T¯A¯x ¯+ (t) + T¯Bu(t) −L x (t) = x
¯ y(t) ¯x+ (t) − T¯Bu(t) ¯ ¯d d(t) + T¯f¯a+ (¯ x (t)) + H ˙ − T¯A¯ − T¯E
¯ y(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) − T¯f¯a (¯ x (t)) − H ˙ ¯e+ (t) − L ¯ yˆ+ (t) − y(t) − T¯E ¯d d(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) = T¯A¯ x + T¯ f¯a+ (¯ x (t)) − f¯a (¯ x (t)) ¯ C¯ e¯+ (t) − T¯E ¯d d(t) + T¯I¯r f˙+ (t) − T¯I¯r f˙(t) = T¯A¯ − L x
(11)
+ T¯f¯a+ (¯ x (t)) − T¯f¯a (¯ x (t))
When the following conditions are satisfied, the disturbance can be com-
11
pletely eliminated, ¯d = 0 T¯E ¯d and C¯ are known. Thus, the unknown matrix H ¯ can be obtained where E ¯d = 0 and T¯ + H ¯ C¯ = I. according to T¯E The simplified error dynamic system can be obtained from (11) as follows: ¯ ¯ ¯ ¯ ¯+ ¯ ¯+ x (t)) − f¯a (¯ e¯˙ + x (t)) + T¯I¯r f˙+ (t) − f˙(t) . x (t) = T A − LC e x (t) + T fa (¯ ef + (t) = I¯T e¯+ (t) r
x
(12)
Similarly, for the lower bound observer, by denoting system the augmented
ˆ ¯(t) − x ¯− (t), augmented output estimation error ey − (t) = state error e¯− x (t) = x y(t) − yˆ− (t) = C¯ e¯− x (t), and the error of lower bound of fault and true value is
ef − (t) = f (t) − fˆ− (t). Then, the system’s error state equation is expressed as follows: ¯ ¯ ¯ ¯ ¯− ¯ ¯ x (t)) − f¯a− (¯ e¯˙ − x (t)) + T¯I¯r f˙(t) − f˙− (t) . x (t) = T A − LC e x (t) + T fa (¯ ef − (t) = I¯T e¯− (t) r
x
By Assumption 2, we have
(13) e¯+ x (0)
e¯− x (0)
> 0,
> 0. From Assumption 3, the
¯− dynamics of the estimation errors are cooperative. Therefore, e¯+ x (t) > x (t) > 0, e 0 for all t > 0.
2
Remark 3. The using condition of the interval observer is that the initial condition x− (0) 6 x(0) 6 x+ (0), Assumption 2 and Assumption 3 are satisfied. As for Assumption 2, we can get the state space expression of nonlinear term:
fa (x (t)) =
−np βx5 x4 np βx5 x3 np βx5 x4 −np βx5 x3
µ (x4 x1 − x3 x2 ) − J1 TL 12
.
The upper bound x ¯+ (t) and lower bound x ¯− (t) of the state vectors under stable operation are selected. Firstly, in the simulation part, we will provide two types of faults. For constant faults, its derivative is 0, the upper and lower bounds satisfy the Assumption 2. The second one is the graded fault as an exponential function whose derivative is still an exponential function. Therefore, there exists upper and lower bounds. Secondly, for the nonlinear term of the traction motor system f¯a , we take (¯ x+ (t), x ¯− (t)), and bring them into the nonlinear term model to obtain the nonlinear upper and lower bounds. As for Assumption 3, ¯ ensures that matrix (T¯A¯ − L ¯ C) ¯ firstly, we assume that the existence of matrix L is Metzler. In the later chapter of the paper, a solution to the problem of not meeting Metzler is proposed. And in the simulation part of the paper, ¯ that the authors successfully used the LMI method to solve the parameters L ¯ L ¯ C) ¯ is located in the left half of the coordinate axis. satisfy the solution of (T¯A− Compared with the Luenberger Observer, the observer we studied can eliminate the impact of external disturbances on fault estimation when the hypothetical ¯d = 0 is satisfied. Consequently, this method effectively simplifies condition T¯E the algorithm and improves the performance of fault estimation. In dealing with the nonlinear part of the system, the method proposed in this paper involves the designing of an interval observer using part of the measurement information to replace a single point of measurement with a reasonable interval. Remark 4. Hurwitz criterion and Nygist criterion are effective tools for judging stability, however, as for the plants with structured uncertainty, Kharitonov theorem has been employed by some researchers([33–35]). In the actual system, the variation range of feature polynomial coefficients due to the perturbation of parameter values does not happen to be Super cube shape, in addition, the variation of each coefficient ai will not be independent of each other, so the application value of Haritonov theorem is limited. As a comparison, we use the positive system to construct an interval observer to diagnose the fault of the traction motor. The system can be ensured in the middle of the interval when the conditions of the Hurwitz and Metzler are satisfied and the nonlinear term satisfies the Assumption 2 by finding the upper and lower bounds of the system 13
state variables. The method of constructing interval observer is inspired by the nonlinear IO design in [27]. ¯ of the fault estimation for high-speed Theorem 2. The observer gain matrix L train can be obtained by solving the following linear matrix inequality (LMI), and the H∞ performance level γ and a circular region D(α, τ ) are provided. If a
¯ ∈ R(n+r)×(n+r) and matrix Y¯ ∈ R(n+r)×p symmetric positive definite matrix Q satisfy the following contitions:
¯ Q ¯ T¯A¯ − Y¯ C¯ − αQ ¯ −Q ¯ −τ 2 Q
∗
¯ T¯I¯r ϕ −Q
∗ ∗
I¯r
−γI
0
∗
−γI
<0
<0
(14)
(15)
¯ T¯A¯ + T¯A¯ T Q ¯ − Y¯ C¯ − Y¯ C¯ T , the eigenvalues of (T¯A¯ − L ¯ C) ¯ belong where ϕ = Q ¯=Q ¯ −1 Y¯ . to D(α, τ ), and the observer gain matrix is given by L
¯ L ¯ C) ¯ into A and P¯ into P in Lemma Proof : Constraint (14): Substituting (T¯A− 1, we can obtain (14). Constraint (15): Consider the following Lyapunov function: V (t) = e¯Tx (t)P¯ e¯x (t)
(16)
Its derivative with respect to time is ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ T Q)¯ ¯ ex + e¯T (t)Q ¯ T¯µ(t) + µT (t)T¯T Q¯ ¯ ex V˙ (t) = e¯Tx (t)(Q( x
where µ(t) = f¯a+ (¯ x (t)) − f¯a (¯ x (t)) + I¯r f˙+ (t) − f˙(t) Let us define: J1 =
Z
∞
tf
1 [ eTf (t)ef (t) − γµT (t)µ(t)]d(t) γ
14
(17)
(18)
one can obtain: J1 6 =
Z
Z
∞
tf ∞
[V˙ (t) +
1 T e (t)ef (t) − γµT (t)µ(t)]d(t) γ f
[V˙ (t) +
1 T ¯ ¯T e¯ (t)Ir Ir e¯x (t) − γµT (t)µ(t)]d(t) γ x
tf
(19)
From (17), it follows that: 1 V˙ (t) + e¯Tx (t)I¯r I¯rT e¯x (t) − γµT (t)µ(t) γ ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ T Q)¯ ¯ ex (t) + e¯Tx (t)Q ¯ T¯µ(t) = e¯Tx (t)(Q( ¯ ex + 1 e¯Tx (t)I¯r I¯rT e¯x (t) − γµT (t)µ(t) + µT (t)T¯T Q¯ γ
(20)
= ζ T (t)Ξζ(t) where
Ξ=
ζ(t) =
e¯x (t) µ(t)
,
¯ + 1 I¯r I¯rT ¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ TQ Q( γ
¯ T¯ −Q
*
−γI
Through the Schur complement, Ξ < 0 becomes
¯ T¯A¯ − L ¯ C) ¯ + (T¯A¯ − L ¯ C) ¯ TQ ¯ −Q ¯ T¯ Q(
I¯r
∗
−γI
0
∗
∗
−γI
<0
(21)
¯ L. ¯ Thus, (12) and (13) are stable and which is equivalent to (15), where Y¯ = Q satisfy the H∞ performance ||ef (t)||2 < γ||µ(t)||2 if condition (15) holds.
2
Remark 5. The algorithm which satisfies the robust H∞ performance index can effectively restrain the influence of µ on fault estimation error ef (t).
15
3.2. Transformation of Coordinates ¯ cannot be determined. However, the corresponding observer gain matrix L ¯ C) ¯ be a Metzler matrix from the nonlinear system This value can make (T¯A¯ − L of the investigated traction motor for high-speed trains. A non-singular constant transformation matrix is given to solve the problem, and a new system equation is obtained by equivalent transformation ξ¯ (t) = P x ¯ (t). ¯˙ = P AP ¯ + P Bu(t) ¯ −1 ξ(t) ¯ ¯d d(t) + P f¯a (¯ ξ(t) + PE x (t)) + P I¯r f˙(t)
¯ ¯ −1 ξ(t) y(t) = CP
(22)
An equivalent augmented system state space description form can be obtained as follows: ¯˙ = P T¯AP ¯ + P T¯Bu(t) ¯ −1 ξ(t) ¯ ¯d d(t) + P T¯f¯a (¯ ξ(t) + P T¯E x (t)) ¯ y(t) + P T¯I¯r f˙(t) + P H ˙ y(t) = CP ¯ ¯ −1 ξ(t)
(23)
¯d = 0 is considered, the simplified form can be obtained from (23) When T¯E as follows: ¯˙ = P T¯AP ¯ + P T¯Bu(t) ¯ −1 ξ(t) ¯ ¯ y(t) ξ(t) + P T¯f¯a (¯ x (t)) + P T¯I¯r f˙(t) + P H ˙
¯ ¯ −1 ξ(t) y(t) = CP
(24)
The interval observer is constructed as follows:
+ ¯˙+ (t) = P T¯AP ¯+ (t) + P T¯Bu(t) ¯ (t) − y(t)) ¯ −1 ξˆ ¯ ¯ CP ¯ −1 ξˆ ξˆ − P L( ¯ y(t) + P T¯f¯a+ (¯ x (t)) + P T¯I¯r f˙+ (t) + P H ˙ ¯+ (t) + P T¯Bu(t) ¯ ¯ y(t) ¯ + P T¯f¯a+ (¯ x (t)) + P T¯I¯r f˙+ (t) + P H ˙ + P Ly(t) = Rξˆ
− ¯˙− (t) = P T¯AP ¯− (t) + P T¯Bu(t) ¯ (t) − y(t)) ¯ −1 ξˆ ¯ ¯ C¯ P −1 ξˆ ξˆ − P L( ¯ y(t) + P T¯f¯a− (¯ x (t)) + P T¯I¯r f˙− (t) + P H ˙ ¯− (t) + P T¯Bu(t) ¯ ¯ y(t) ¯ = Rξˆ + P T¯f¯a− (¯ x (t)) + P T¯I¯r f˙− (t) + P H ˙ + P Ly(t) (25)
16
ˆ+ (t), ξ¯ ˆ− (t) denote the upper and ¯ C)P ¯ −1 and ξ¯ where R is equal to P (T¯A¯ − L lower bounds of the augmented observer vector after the transformation of co¯ ordinates. Moreover, f¯aξ (¯ x (t)) denotes the nonlinear part after the coordinates
have been transformed. The following variables are defined: ϕ¯+ ξ¯− (t) , ξ¯+ (t) = T¯f¯a+ ϕ¯− ξ¯− (t) , ξ¯+ (t) = T¯f¯a−
P¯i ξ¯− (t) − Pi ξ¯+ (t) , P¯i ξ¯+ (t) − Pi ξ¯− (t) P¯i ξ¯− (t) − Pi ξ¯+ (t) , P¯i ξ¯+ (t) − Pi ξ¯− (t)
where ξ¯+ and ξ¯− represent the upper and lower bounds of the augmented state vector after the coordinates have been transformed. Under Assumption 2, the following can be obtained: ¯ f¯aξ− ξ¯− (t) , ξ¯+ (t) = P¯ ϕ¯− ξ¯− (t) , ξ¯+ (t) − Pϕ¯+ ξ¯− (t) , ξ¯+ (t) 6 P T¯f¯a P −1 ξ¯ (t) 6 P¯ ϕ¯+ ξ¯− (t) , ξ¯+ (t) − Pϕ¯− ξ¯− (t) , ξ¯+ (t) ¯ = f¯aξ+ ξ¯− (t) , ξ¯+ (t) .
(26)
where P¯ = max {0, P }, P = P¯ − P , P¯i = max 0, P −1 and Pi = P¯i − P −1 . ¯
¯
The upper bound error e¯ξ+ (t) and lower bound error e¯ξ− (t) are defined as
follows:
+
¯ ¯ (t) − ξ¯ (t) e¯ξ+ (t) = ξˆ − ¯ ¯ (t) e¯ξ− (t) = ξ¯ (t) − ξˆ
Then, the error dynamic equation can be obtained as follows: ¯ ¯ e¯˙ ξ+ (t) = R¯ eξ+ (t) + P T¯ f¯a+ (¯ x (t)) − f¯a (¯ x (t)) + P T¯I¯r (f˙+ (t) − f˙(t))
¯ ¯ ˙ ξ− e¯ (t) = R¯ eξ− (t) + P T¯ f¯a (¯ x (t)) − f¯a− (¯ x (t)) − P T¯I¯r (f˙(t) − f˙− (t)) (27)
¯ C) ¯ and a Metzler matrix R have the Theorem 3. Let the matrix (T¯A¯ − L
¯ If two vectors e¯1 and e¯2 exist such that the pairs same eigenvalues for some L.
17
¯ C, ¯ e¯1 ) and (R, e¯2 ) are observable, then (T¯A¯ − L P = O2−1 O1 . ¯ N = P L. ¯ satisfies Sylvester equation: P T¯A¯ − RP = N C, where
O1 =
e¯1 .. . ¯ C¯ e¯1 T¯A¯ − L
n−1
, O2 =
e¯2 .. . e¯2 Rn−1
.
¯ C, ¯ e¯1 ) and (R, e¯2 ) are observable, matrices O1 Proof : Since the pairs (T¯A¯ − L and O2 are nonsingular. The following equation can be obtained through the transformation: ¯ C)O ¯ −1 = O2 RO−1 O1 (T¯A¯ − L 1 2 ¯ C)O ¯ −1 and O2 RO−1 are canonical observability forms. The where O1 (T¯A¯ − L 1 2 ¯ C) ¯ is observable. vectors e¯1 and e¯2 can be found if the pair (T¯A,
2
Remark 6. The Metzler matrices are constructed as follows: Let R be the lower triangular matrix, with the main diagonal elements as the eigenvalues of (T¯A¯ −
¯ C), ¯ and the elements below the main diagonal are positive. Moreover, the L matrix R after transformation satisfies the property of Hurwitz and is a matrix ¯− (t) 6 ξ(t) ¯ 6 Metzler at the same time. Combined with the above process, ξˆ ˆ ξˆ¯+ (t), t > 0. Then, the fault estimate is obtained by fˆ+ (t) = I¯rT x ¯+ (t), fˆ− (t) = ˆ I¯rT x ¯− (t). 3.3. Enhancement of the Accuracy of IO-UIO When the interval observer is used to estimate the fault, the response time and the accuracy of the interval, that is, the difference between the upper and lower bounds of the interval observer, are considered. This subsection focuses on finding a performance index related to the interval accuracy based on the CRH5 motor system, which makes the interval size adjustable. Since the equivalent transformation does not affect the experimental results, the system can be conveniently investigated based on the previous transforma18
tion before. The dynamic error equation can be derived from (12) and (13). x ¯(t) =
h
x1
x2
x3
x4
x5
x6
iT
=
h
iqs
ids
λqr
λdr
ωm
f
iT
Then, the following can be obtained: ¯ C)¯ ¯ ex + ψ T¯(f¯+ − f¯− ) + ψ T¯I¯r (f˙+ − f˙− ) ψ e¯˙ x = ψ(T¯A¯ − L a a where ψ =
h
0
0 0
0
0
(28)
i 1 .
Equation (28) can be simplified as follows: e˙ f = −l61 ex1 + l62 ex2 − l61 ef
(29)
where l61 represents the first column of the sixth row of the observer gain ma¯ and l62 represents the second column of the sixth row of this matrix. trix L, Moreover, the interval size of the fault estimate is related to the values of ex1 , ex2 and ef and the interval size can be regulated by l61 and l62 . Remark 7. Equation (29) is not generic and is extracted from the characteristics of CRH5. Hence, this equation will no longer be applied in changing other systems or system parameters. However, the method is general, and a performance index can be found according to the system model.
4. Simulation Results In this section, numerical and simulation results are used to verify the feasibility of the proposed method for CRH5 (6FJA3257A). First part of this section discusses the fault estimation with regional pole placement. In the second part, we changed the numerical value of the performance index. Then, we compare the influence of the performance parameters on the fault estimation precision from the residual of the upper and lower bounds of the fault estimation curve in the simulation diagram. According to the high-speed train traction motor system, the state vectors
19
under stable operation can be obtained as follows: x(t) =
h
−150.3
−222 0.3958
−6.3345 124.9053
iT
Set the upper and lower bounds of the state vectors: h iT x+ (t) = −150.0 −221 0.40 −6.32 125.5 h iT x− (t) = −150.6 −223 0.39 −6.34 124.5
it is obviously satisfying to Assumption 2.
We illustrate the simulation results with the fault of q-axis stator current sensor. The fault model is as follows: 0, f (t) = 10,
0s 6 t < 10s t > 10s
Part 1. Fault estimation with regional pole placement. We can solve (14) and (15) to obtain the observer gain matrix by choosing the region pole D(−20, 50). Remark 8. When constraints (14) and (15) using the LMI toolbox of Matlab ¯ C) ¯ are located in the left half (2014b) are satisfied, the eigenvalues of (T¯A¯ − L plane of the coordinate virtual axis and within the pole configuration circle. The LMI region is as follows: D = { z ∈ C: Re(z) < 0},
fD (z) = z + z¯ < 0.
where fD (z) is the characteristic function of D. We choose D(−20, 50) to improve the performance of the system.
20
¯ Y¯ and L ¯ can be calculated as follows: Parameters Q,
0.5649
0.0193 −0.0810 ¯ Q= −1.4510 0 −0.2635
−11.7195
28.1386 75.9143 ¯ Y = 21.5699 0 129.4668
0.0193
−0.0810
−1.4510
0
0.0855
0.2357
−0.0766
0
0.2357
0.7229
0.1294
0
−0.0766
0.1294
3.7438
0
0
0
0
2.0965
0.3055
0.8062
0.6428
0
−212.4609
0
21.6346
757.2506 0 −145.1688 0 ¯= ,L 34.7317 0 87.8535 0 0 −0.4252
−9.4913 23.3952 545.5703 0 83.6815
−0.2635
0.3055 0.8062 . 0.6428 0 3.4594
−669.2785
0
0 −0.0031 ¯ = H 0 0 0
0
0
0 0 0 −0.0031 0 0 1 0 0 0
¯d = 0, because From subsection A, matrix T¯ can be calculated through T¯E ¯d = E
h
0
0 0 0
21
−1/15 0
iT
23.6974 0 −28.8060 0 −111.9553 0 0 41.9045 −1.3582 0
¯ C¯ = I, H ¯ can be obtained when T¯ and C¯ are known. Through T¯ + H
0
then,
1
0 0.0031 T¯ = 0 0 0
0
0 0 0
0
1
0 0 0
0
0
1 0
0.0031 0 1 0
0 0
0
0 0
0 0.0031 0 0 0 0 0 1
When the numerical values of the motor in Table I are substituted into the system, the following can be obtained:
−64.5606
−376.9911 965.3409
376.9911 0.0884 A¯ = 0 0 0
−64.5606
0
0
−2.9556
0.0884
376.9911
0
0
0
0
−64.5606
376.9911 −0.1093 ¯ ¯ TA = 1.1543 0 0
0
0 0
0 0 −376.9911 0 0 . −2.9556 0 0 0 0 0 0 0 0 965.3409
−376.9911 965.3409 −64.5606
0
−1.1543
0
−0.1093
376.9911
0
0
0
0
0
0 0
0 0 −376.9911 0 0 . 0 0 0 0 0 0 0 0 0 965.3409
¯ C) ¯ is A lower triangular matrix R with the same eigenvalues as (T¯A¯ − L
¯ considered, and the matrices e¯1 , e¯2 satisfy Theorem 2. The eigenvalues of (T¯A− ¯ C) ¯ are -50.6767, -25.0914 ± 34.6127i, -36.5993 ± 27.4065i and -41.9045. L e¯1 =
h
e¯2 =
1 h
8 1
−2 1
−2
0 0 1 0
22
−1 i −1
i
R=
−50.6767 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
1.0000 + 0.0000i
−25.0914 + 34.6127i
0.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
−25.0914 − 34.6127i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
−36.5993 + 27.4065i
0.0000 + 0.0000i
0.0000 + 0.0000i
1.0000 + 0.0000i
−36.5993 + 27.4065i
0.0000 + 0.0000i
1.0000 + 0.0000i
1.0000 + 0.0000i
−41.9045 + 0.0000i
Then, the transformation matrix can be obtained by Theorem 2,
P =
−50.6767 − 0.0000i
0.0000 − 0.0000i
0.0000 − 0.0000i
0.9999 − 0.0001i
−25.0914 + 34.6127i
0.0000 − 0.0000i
1.0000 − 0.0006i
1.0000 + 0.0001i
−25.0914 − 34.6127i
1.0000 − 0.0000i
1.0000 + 0.0000i
1.0000 − 0.0000i
0.9999 + 0.0005i
1.0001 − 0.0001i
1.0000 + 0.0000i
1.0001 − 0.0000i
1.0000 − 0.0000i
1.0000 − 0.0000i
0.0001 + 0.0001i
0.0000 − 0.0000i
0.0000 + 0.0000i
0.0001 − 0.0004i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0011 − 0.0011i
0.0000 + 0.0000i
0.0000 + 0.0001i
−36.5993 + 27.4065i
0.0000 − 0.0000i
0.0000 + 0.0000i
0.9989 + 0.0006i
−36.5993 − 27.4065i
0.0000 − 0.001i
1.0001 + 0.0001i
1.0000 − 0.0000i
−41.9045 + 0.0000i
.
.
Matrix P is proved to be nonsingular, and the simulation results are displayed as follows. Figure 1 illustrates the fault estimation response with the region pole placement. The simulation results show that the fault estimate curve finally
23
tracks the actual failure, and the real fault is between the upper bound fault observation value and the lower bound observation value. Fault estimation
500
0
-500
fault lower estimation upper estimation
20
-1000
10
-1500
-10
0
-20
-2000
9
10
11
12
13
14
-2500
-3000
0
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 1: Simulation results with region pole placement.
Figure 2 shows the fault estimation response without region pole placement. The fault tracking speed in Figure 1 is faster than in Figure 2. Thus, the performance of the fault estimation can be improved after considering a circular domain constraint to the solution. Fault estimation
400
fault upper estimation lower estimation
350 300
20
Partial enlargement:
20
250
10
10 200
0 0
-10
150 -10 100
9 9
10
10
11
11
12 12
13 13
14
50 0 -50
0
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 2: Simulation results without region pole placement.
24
Part 2. The method mentioned in subsection 3.3 will be validated in this subsection. Firstly, we study the influence of the numerical value of the parameter l61 on the interval size when other parameters remain unchanged. The
0
-50 9.7
error of interval 23.7 9.8
9.9
10
10.1
10.2
10.4
15 10.5
Time (s) L61= -0.5252
50
0
-50 9.7
20 10 0 10 10.3
f(t) and estimate of f(t)
L61= -0.2252
50
9.8
9.9
10
error of interval 3.9
20 10 0 10
10.1
10.3
10.2
15 10.4
10.5
f(t) and estimate of f(t)
f(t) and estimate of f(t)
f(t) and estimate of f(t)
simulation results are as follows. The small figure inside Figure 3 is the locally L61= -0.4752
50
0 20 error of interval 9.5 -50 9.7
9.8
9.9
10
10.1
10
10.2
0 10 10.3
10.4
15 10.5
10.2
20 10 0 10 10.3 10.4
15 10.5
Time (s) L61= -0.55
50
0 error of interval 2.6 -50 9.7
Time (s)
9.8
9.9
10
10.1
Time (s)
Figure 3: Simulation results with different parameter l61 .
enlarged figure. Figure 3 shows that the interval size decreases with the decrease of the parameter l61 . Hence, the smaller the parameter l61 , the better the effect of fault estimation will be. Next, we study the influence of the numerical value of the parameter l62 on the interval size when the other parameters remain unchanged. The simulation results are as follows, in which, the small figure inside the figure 4 is error of the upper and lower bounds. Figure 4 illustrates that the interval size decreases with increase in parameter l62 . Therefore, the bigger the parameter l62 , the better the effect of fault estimation will be. Remark 9. Through (29) we can see that parameters l61 and l62 affect the fault estimation error ef . We can obtain conclusions from Figure 3 and Figure 4: the smaller the parameter l61 , the better the effect of fault estimation will be and the bigger the parameter l62 , the better the effect of fault estimation will ¯ to be. This can provide a reference in the process of solving the parameter L achieve better fault estimation.
25
20
-50 9.7
9.8
9.9
error of interval 10 14.4 0 10 10.1 10.2 10 10.3
10.4
15 10.5
Time (s) L62= -1.1582
50
0
-50 9.7
f(t) and estimate of f(t)
0
f(t) and estimate of f(t)
f(t) and estimate of f(t) f(t) and estimate of f(t)
L62= -1.6582
50
9.8
9.9
20 error of interval 10 5.3 0 10 10.1 10.2 1010.3
10.4 1510.5
L62= -1.3582
50
0
-50 9.7
9.8
9.9
20 error of interval 10 9.6 0 10 10 10.1 10.2 10.3
10.4
15 10.5
20 error of interval 10 1.1 0 10 10.1 10.2 10.3 10.4
15 10.5
Time (s) L62= -0.99
50
0
-50 9.7
Time (s)
9.8
9.9
10
Time (s)
Figure 4: Simulation results with different parameter l62 .
5. Conclusion In this paper, an IO-UIO based fault estimation scheme for high-speed railway traction motor with sensor fault and disturbances is investigated. For the nonlinear part of the system, an interval observer is introduced to solve efficiently only a part of the information in the interval. Therefore the design of unknown input observer is proposed, and this observer can completely eliminate the external disturbance. Moreover, a FD algorithm is proposed to estimate the faults accompanied by disturbance. We constructed a Metzler matrix to match the properties of the interval observers by transforming the coordinates. In addition, we found a performance index that can adjust the size of the interval observer. Finally, the effectiveness and advantages of the proposed IO-UIO method for high-speed railway traction motor is shown through simulation results.
26
References References [1] Y. J. Wang, Impediment to progress and inevitable trend of the high-speed railway development in china, (15) (2014) 256–258. [2] H. H. T. Liu, P. Shi, B. Jiang, Fault detection, diagnosis, and fault tolerant control with flight applications, Journal of the Franklin Institute 350 (9) (2013) 2371–2372. [3] M. Y. Chen, D. H. Zhou, Fault detection and isolation in a class of nonlinear systems, Acta Automatica Sinica 30 (6) (2004) 1011–1016. [4] Y. Guo, Z. Mao, B. Jiang, Observer based fault diagnosis for induction motor with sensor faults and disturbances, Chinese Automation Congress (2017) 5084–5089. [5] D. Hu, A. Sarosh, Y. F. Dong, A novel KFCM based fault diagnosis method for unknown faults in satellite reaction wheels, ISA Transactions 51 (2) (2012) 309–316. [6] W. Chen, M. Saif, A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems, International Journal of Systems Science 38 (12) (2007) 943–955. [7] Y. Zhiguo, G. Vansteenkiste, Improving the observer-based fdi design for efficient fault isolation, International Journal of Control 68 (1) (1997) 197– 218. [8] C. Deng, G. H. Yang, Adaptive fault-tolerant control for a class of nonlinear multi-agent systems with actuator faults, Journal of the Franklin Institute 354 (12) (2017) 4784–4800. [9] Q. Hu, B. Xiao, Y. Zhang, Fault-tolerant attitude control for spacecraft under loss of actuator effectiveness, Control & Decision 34 (3) (2015) 263– 268. 27
[10] C. Deng, G. H. Yang, Distributed adaptive fault-tolerant containment control for a class of multi-agent systems with non-identical matching nonlinear functions, IET Control Theory & Applications 10 (3) (2016) 273– 281. [11] S. Tong, L. Zhang, Y. Li, Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones, IEEE Transactions on Systems Man & Cybernetics Systems 46 (1) (2015) 37–47. [12] H. M. Wang, D. Ye, G. H. Yang, Actuator fault diagnosis for uncertain ts fuzzy systems with local nonlinear models, Nonlinear Dynamics 76 (4) (2014) 1977–1988. [13] P. S. Teh, H. Trinh, Design of unknown input functional observers for nonlinear systems with application to fault diagnosis, Journal of Process Control 23 (8) (2013) 1169–1184. [14] X. G. Yan, C. Edwards, Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems, International Journal of Systems Science 39 (4) (2008) 349–359. [15] T. Wang, W. Xie, Y. Zhang, Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults, ISA Transactions 51 (3) (2012) 386. [16] A. M. Pertew, H. J. Marquez, Q. Zhao, LMI-based sensor fault diagnosis for nonlinear lipschitz systems , Automatica 43 (8) (2007) 1464–1469. [17] D. Wang, K. Y. Lum, Adaptive unknown input observer approach for aircraft actuator fault detection and isolation, International Journal of Adaptive Control & Signal Processing 21 (1) (2007) 31–48. [18] B. Jiang, S. Marcel, Adaptive observer design for robust fault estimation, International Journal of Systems Science 33 (9) (2002) 767–775.
28
[19] M. Chen, P. Shi, C. C. Lim, Adaptive neural fault-tolerant control of a 3-dof model helicopter system, IEEE Transactions on Systems Man & Cybernetics: Systems 46 (2) (2016) 260–270. [20] X. G. Yan, C. Edwards, Adaptive sliding-mode-observer-based fault reconstruction for nonlinear systems with parametric uncertainties, IEEE Transactions on Industrial Electronics 55 (11) (2008) 4029–4036. [21] M. Deng, T. Maekawa, N. Bu, Actuator fault detection on a process control experimental system using combined observers, Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems & Control Engineering 225 (6) (2011) 807–813. [22] B. L. Walcott, S. H. Zak, Combined observer-controller synthesis for uncertain dynamical systems with applications, Systems Man & Cybernetics IEEE Transactions on 18 (1) (1988) 88–104. [23] K. Zhang, V. Cocquempot, B. Jiang, Adjustable parameter-based multiobjective fault estimation observer design for continuous-time/discretetime dynamic systems, International Journal of Control Automation & Systems 15 (3) (2017) 1077–1088. [24] K. Zhang, B. Jiang, X. G. Yan, Z. Mao, Sliding mode observer based incipient sensor fault detection with application to high-speed railway traction device, Isa Transactions 63 (2016) 49–59. [25] Y. M. Fu, G. R. Duan, S. M. Song, Design of unknown input observer for linear time-delay systems, International Journal of Control Automation & Systems 2 (4) (2004) 530–535. [26] D. Bestle, M. Zeitz, Canonical form observer design for non-linear timevariable systems, International Journal of Control 38 (2) (1983) 419–431. [27] D. Efimov, T. Raissi, A. Zolghadri, Control of nonlinear and Lpv systems: Interval observer-based framework, IEEE Transactions on Automatic Control 58 (3) (2013) 773–778. 29
[28] F. Mazenc, O. Bernard, Interval observers for linear time-invariant systems with disturbances , Automatica 47 (1) (2011) 140–147. [29] F. Mazenc, T. N. Dinh, S. I. Niculescu, Interval observers for discretetime systems, International Journal of Robust & Nonlinear Control 24 (17) (2015) 2867–2890. [30] D. Efimov, W. Perruquetti, T. Rassi, A. Zolghadri, Interval observers for time-varying discrete-time systems, IEEE Transactions on Automatic Control 58 (12) (2013) 3218–3224. [31] K. Zhang, B. Jiang, P. Shi, Fast fault estimation and accommodation for dynamical systems, IET Control Theory & Applications 3 (2) (2009) 189– 199. [32] J. Chen, R. J. Patton, Robust model-based fault diagnosis for dynamic systems, Springer US, 1999. [33] O. Toker, A kharitonov-like theorem for robust stability independent of delay of interval quasipolynomials, International Journal of Robust & Nonlinear Control 20 (6) (2010) 623–638. [34] P. Chen, Y. Z. Lu, Automatic design of robust optimal controller for interval plants using genetic programming and kharitonov theorem, International Journal of Computational Intelligence Systems 4 (5) (2011) 826–836. [35] H. Chapellat, S. P. Bhattacharyya, A generalization of kharitonov’s theorem; robust stability of interval plants, Automatic Control IEEE Transactions on 34 (3) (1989) 306–311.
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