Received April 19, 2018, accepted May 15, 2018, date of publication May 18, 2018, date of current version July 25, 2018. Digital Object Identifier 10.1109/ACCESS.2018.2838149
A Performance Evaluation Algorithm of Stochastic Hybrid Systems Based on Fuzzy Health Degree and Its Application to Quadrotors ZHIYAO ZHAO , XIAOYI WANG, JIPING XU, AND JIABIN YU School of Computer and Information Engineering, Beijing Technology and Business University, Beijing 100048, China
Corresponding author: Xiaoyi Wang (
[email protected]) This work was supported in part by the National Key Research and Development Program of China under Grant 2017YFC1600605, in part by the Innovation Ability Construction Fund of Science and Technology: Youth Scholar Development Plan under Grant PXM2018_014213_000033, and in part by the Support Project of High-Level Teachers in Beijing Municipal Universities in the Period of 13th Five-year Plan under Grant CIT&TCD201804014.
ABSTRACT A stochastic hybrid system (SHS) is a type of model describing real-world systems, who are both dynamic and hybrid. System performance evaluation is a key technology related to mission accomplishment and safety, where a rational selection of performance indicators is an important guarantee of truly and accurately evaluating system performance. This paper proposes a performance evaluation algorithm of SHS. First, the concept of fuzzy health degree is proposed and the SHS model studied in this paper is established. Then, the procedure of the algorithm is presented in detail, including hybrid state estimation based on a modified interacting-multiple-model algorithm, discretization of continuous variables, and quantitative calculation of fuzzy health degree. In order to validate the effectiveness of the proposed algorithm, an experiment of a quadrotor with sensor anomalies is made where the system performance is quantitatively described by using the fuzzy health degree. Comparative studies are also made and discussed. INDEX TERMS Performance evaluation, stochastic hybrid system, fuzzy health degree, quadrotor, sensor anomaly.
I. INTRODUCTION
An effective way of evaluating system performance in-situ is to rely on the Prognostics and Health Management (PHM) framework [1]. As shown in Figure 1, PHM is an approach utilizing system models and measurements to perform functions, including performance evaluation, performance prediction, intelligent decision-making, etc [2]. In the PHM framework, the plant and its PHM system itself form a closed-loop system, where the PHM system can be divided into a monitoring level, a prediction level and a management level. In all three levels and related techniques, system performance can be always characterized by some physical and mathematical quantities, namely performance indicators. Obviously, a rational selection of performance indicators is an important guarantee of truly and accurately evaluating system performance. In current research, the existing performance indicators can be mainly classified into four types [3]: process variables, data features, residuals and reliable probabilities. VOLUME 6, 2018
Process variables are intuitive parameters which are able to reflect the dynamic feature and status of a dynamical system or process. They can change rapidly when an unexpected behavior occurs. Thus, process variables are the most common performance indicators in practice. References [4] and [5] used the resistance and capacity to perform health monitoring and residual life prediction of Lithium-Ion batteries. References [6]–[8] used the terminal voltage and temperature for performance evaluation and prediction of electronic products and nuclear systems. Although process variables are capable of characterizing system performance, the variation of process variables might be not explicitly and monotonously correlated to the dynamic performance of some practical systems [9], [10]. Thus, current research uses signal processing approaches and tools to generate artificial quantities for purpose of performance quantification, including data features [11] and residuals [12]. Data features are derived values from raw sensor signals, intended to be informative, non-redundant and facilitated to
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FIGURE 1. PHM framework.
indicating the characteristics of system performance. In rotating machinery and its components, features are always extracted by data-driven approaches [13]. Details of various features and related extraction methods including timedomain methods, frequency-methods and time-frequency methods can be seen in [14]–[16]. Residuals are quantitative inconsistencies between the actual and expected behavior of a dynamical system or process [17]. Since any such an inconsistency may be caused by an anomaly, residuals are commonly used in the study of fault detection and isolation [12]. Related works are presented in [18] and [19]. Considering the tight relation between PHM and system reliability, there exists research using reliable probabilities to measure system performance. The relevant phrases include real-time reliability [20], performance reliability [21] and dynamic reliability [22]. These methods take the time-dependent behavior of the system into account by dealing with the system model, sensor data and environmental disturbance, and provide the probability of the system being in a certain status at a certain time. Another critical factor of accurately evaluating system performance is the accurate modeling of the studied system. In practice, some real-world systems are [23]: 1) dynamic, i.e., their state changes (deterministically and/or stochastically) in time; 2) hybrid, i.e., they are characterized by both discrete modes and continuous variables (e.g., components’ health conditions, like functioning, failed, and continuous physical process variables, like temperatures and pressures). In current literatures, Stochastic Hybrid System (SHS) [24], [25] is always used to describe such a hybrid dynamic process with interacting continuous dynamics and discrete dynamics. At each time, the evolution of SHS is confined in a continuous state space indexed by the discrete mode (continuous dynamics), or jumps from one discrete mode to another (discrete dynamics) and starts a new continuous evolution in another continuous state space. Studies covering performance evaluation of SHS include hybrid state estimation [26], fault 37582
detection [27] and dynamic reliability (probabilistic reachability) analysis [28], [29]. In these studies, process variables, residuals and reliable probabilities are used as performance indicators, respectively. In the above research on both general dynamical systems and SHS, system performance is viewed as a binary value {0, 1} in most situations, obtained by a comparison between the performance indicator and its failure threshold. Even if the reliable probabilities are used as the performance indicator, the failure threshold of the selected performance indicator is binary. The value ‘‘1’’ means that the system performance is normal, while the value ‘‘0’’ indicates an anomalous system performance. Actually, for complex dynamical systems such as SHS, system performance evolving from normal to failed goes through a series of degradation states. It is inappropriate and limited to characterize the system degradation with a fixed and binary failure threshold. In order to deal with this limitation, [30] and [31] applied fuzzy reliability theory to PHM, and presented an implementation process to achieve performance evaluation and prediction of general dynamical systems. Considering the effectiveness of the fuzzy reliability as the performance indicator, this paper proposes a definition of fuzzy health degree for SHS on the basis of fuzzy reliability, and presents a performance evaluation algorithm of SHS based on fuzzy health degree. In this paper, the concept of fuzzy health degree is first proposed. Then, given the SHS model of a studied real-world system, an implementation process of performance evaluation algorithm is provided, including hybrid state estimation based on a modified Interacting-Multiple-Model (IMM) algorithm, discretization of continuous variables and quantitative calculation of fuzzy health degree. In order to validate the effectiveness of the proposed algorithm, an experiment of a quadrotor with sensor anomalies is made, where the system performance is quantitatively calculated by using the fuzzy health degree, and comparative studies are also made and discussed. The major contribution of this paper lies in three aspects. First, the concept of fuzzy health degree is proposed for dynamical system on the basis of the fuzzy performance reliability, and applied to SHS as a new performance indicator. From the perspective of performance indicators, the fuzzy health degree is a performance indicator defined in fuzzy-state space, which is a complement of the current health degree defined in the binary-state space. Second, a performance evaluation algorithm for SHS model is comprehensively proposed, including hybrid state estimation, discretization of continuous variables and quantitative calculation of fuzzy health degree. The proposed performance evaluation algorithm is a complement of current in-situ performance evaluation studied in the binary-state space. Third, the proposed method is applied to the performance evaluation of quadrotors, and an experiment is performed to validate the effectiveness and availability of the proposed method in real practical scenario rather than correctness verification based on simulations. VOLUME 6, 2018
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The remainder of this paper is organized as follows. Section II presents the concept of fuzzy health degree and an establishment of the SHS model studied in this paper. Section III lists the procedure of the proposed performance evaluation algorithm. Section IV uses a case study of a quadrotor with sensor anomalies to validate the effectiveness and availability of the proposed method in real practical scenario. Section V gives a conclusion, and indicates future development of the proposed method. II. FUZZY HEALTH DEGREE AND SHS MODELING A. FUZZY HEALTH DEGREE
In order to make the paper self-contained, a brief introduction of fuzzy reliability is presented in this part [30]– [32]. On this basis, the concept of fuzzy health degree is introduced. Fuzzy reliability theory extends the traditional binary state space {0, 1} into a fuzzy state space [0, 1], and models fuzzy state transitions for a component or system representing various degrees of success and failure. For a discrete domain U = {S1 , S2 , · · · SN }. In the domain U, fuzzy success states are defined as S = {Si , µS (Si ) ; Si ∈ U}, and fuzzy failure states are defined as F = {Si , µF (Si ) ; Si ∈ U}. The physical meaning of fuzzy success/failure states is a division of system performance, representing various degrees of success and failure (degradation levels). In the fuzzy state space, the system operational states can be described with the fuzzy membership values, rather than just successful and failed states through single threshold segmentation. Without loss of generality, it is usually considered that µS (Si ) = 1 − µF (Si ), Si ∈ U. Let UT = mij , i, j = 1, · · · , N , where mij represents the transition from state Si to state Sj . In the domain UT , a transition from a fuzzy success state to a fuzzy failure state is defined as [32] TSF = mij , µTSF mij ; Si , Sj ∈ U . Here, TSF is viewed as a fuzzyevent, and its corresponding membership function µTSF mij is determined as [32] ( βF|S Sj −βF|S (Si ) if βF|S Sj > βF|S (Si ) µTSF mij = 0 otherwise, where
Definition 1 [32]: For Si ,Sj ∈U, the fuzzy interval reliability over a time interval [t0 , t] is R (t0 , t) = 1 −
i=1 j=1
where πij (t0 , t) is the transition probability from state Si to state Sj over the time interval [t0 , t]. The fuzzy reliability is at time t is R (t) = R (0, t). As indicated in [30], Definition 1 is inappropriate to be used in real-time performance evaluation. Thus, [30] presented an algorithm of fuzzy performance reliability for the purpose of PHM, and [31] further provided a modified algorithm of fuzzy performance reliability for general dynamical systems. In this paper, the concept of fuzzy health degree for performance evaluation study can be defined by the definition of fuzzy performance reliability. The details are addressed in the following theorems. Theorem 1 [30]: For Si ,Sj ∈U and a time interval [t0 , t], the fuzzy health degree at time t is N N X X µTSF mij · πij (t0 , t) R (t) = 1 − i=1 j=1
" N # N X X · µS (Si ) · pSi (t0 ) − µF (Si ) · pSi (t0 ). i=1
µF (Si ) , µS (Si ) + µF (Si )
Si ∈ U.
Here, βF|S (Si ) is interpreted as the weight that state Si is attached to the fuzzy failure state compared to the fuzzy success state. In this case, only if condition βF|S Sj > βF|S (Si ) is true, a transition from state to state promotes the transition from fuzzy success to fuzzy failure. VOLUME 6, 2018
i=1
where pSi (t0 ) is the state probability of state Si at time t0 . Here, the time interval [t0 , t] is a sliding calculation interval during the real-time performance evaluation. Theorem 2 [31]: For Si ,Sj ∈U, suppose µF (SN ) 6 µF (SN −1 ) 6 · · · 6 µF (S2 ) 6 µF (S1 ). Further, suppose that the system performance is in state Si at time t0 with the probability pSi (t0 ), where Si ∈ U pSi (t0 ) > 0 N X pSi (t0 ) = 1. i=1
Then, the fuzzy health degree at time t is N i−1 X X R (t) = 1 − pSi (t0 ) · µTSF mij · pSj (t) i=2 j=1 " N # N X X · µS (Si ) · pSi (t0 ) − µF (Si ) · pSi (t0 ). i=1
βF|S (Si ) =
N X N X µTSF mij · πij (t0 , t).
i=1
B. SHS MODELING
The SHS model studied in this paper is a special class of SHS, where the continuous dynamics is described by a set of stochastic differential equations, and the discrete dynamics is modeled by a first-order Markov chain. Define an SHS with a tuple H = (Q, n, F, u, 0 w , Q, 0 v , R, 5, p, Init), which is detailed as follows. 37583
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1) HYBRID STATE SPACE (Q AND n)
Q := {q1 , q2 , · · · , qm } is a finite set of discrete modes for m ∈ N. n : Q −→ N assigns each discrete mode q ∈ Q a continuous state space Rn . Let x denote the continuous variable vector. Then, the hybrid state s = (q, x) is defined on the hybrid state space S = Q × Rn . 2) CONTINUOUS DYNAMICS (F, u, 0 w , Q, 0 v AND R)
For a discrete mode qj ∈ Q, the continuous dynamics is described by a stochastic differential equation: ( x (k) = Fj (x (k − 1) , u (k − 1)) + 0 w,j wj (k − 1) (1) y (k) = Cj x (k) + 0 v,j vj (k) where x ∈ Rn×1 is the continuous variable vector, u ∈ Rnu ×1 is the system input; F : Rn×1 × Rnu ×1 → Rn×1 is a nonlinear process equation; y ∈ Rny ×1 is the system measurement vector; Cqj ∈ Rny ×n is the measurement matrix. The item w ∼ N (0, Q) is the process noise, and 0 w is the driven matrix, satisfying 0 w w ∈ Rn×1 ; the item v ∼ N (0, R) is the measurement noise, and 0 v is the driven matrix, satisfying 0 v v ∈ Rny ×1 . 3) DISCRETE DYNAMICS (5 AND p)
FIGURE 2. Structure of SHS.
III. PROCEDURES OF PERFORMANCE EVALUATION ALGORITHM
The procedure of the performance evaluation algorithm is shown in Figure 3. First, given an SHS model H = (Q, n, F, u, 0 w , Q, 0 v , R, 5, p, Init) as shown in (1)-(4), the hybrid state is estimated in real-time by a modified IMM algorithm. Meanwhile, each continuous variable is discretized into several discrete fuzzy states. Then, in order to quantitatively evaluate system performance, the fuzzy health degree of each continuous variable is calculated, and the fuzzy health degree of the SHS is further obtained.
5 := πij m×m ∈ Rm×m is the transition probability matrix of a first-order Markov chain, with transition probabilities as P qj (k) |qi (k − 1) = πij , ∀qi , qj ∈ Q (2) and m X
πij = 1,
i = 1, 2, · · · , m.
j=1
The meaning of (2) is the probability of being in mode qj at time k, given that the system was in mode qi at time k − 1. Following the Markov property, the system mode at time k only depends on the system mode at time k − 1, without considering former time instances. Let p = [p1 , p2 , · · · , pm ] ∈ R1×m be the vector of discrete mode probabilities. Then p (k) = p (k − 1) · 5.
(3)
4) INITIAL CONDITION (Init )
Init is the initial condition of the SHS. Let the symbol f (·) be the probability density function (pdf). The initial distribution of the hybrid state s (0) = (q (0) , x (0)) can be described as ( f x (0) qj (0) = N µj (0) , 6 j (0) j = 1, 2, · · · , m, P qj (0) = pj (0), (4) P where pj (0) > 0 for ∀qj ∈ Q and m j=1 pj (0) = 1. For clarification, taking m = 3 as an example, the structure of the proposed SHS model is depicted in Figure 2. 37584
FIGURE 3. Procedure of the proposed performance evaluation algorithm.
A. HYBRID STATE ESTIMATION
In this part, the hybrid state estimation of the SHS is performed by a modified IMM algorithm. First, the classic IMM algorithm is presented. Then, two modifications are made for the purpose of accurate hybrid state estimation and convenient calculation of fuzzy health degree. VOLUME 6, 2018
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Compute the measurement residual at time k: i h rj , y (k) − E y (k) |qj (k) , Yk−1
1) CLASSIC IMM ALGORITHM
The classic IMM algorithm is a recursive estimator [27]. In each recursive cycle, it consists of four steps: 1) modelconditional reinitialization (interacting or mixing of the estimates), where the input to the filter of each mode is obtained by mixing the estimates of all filters at the previous time under the assumption that the system is in this particular mode at the present time; 2) model-conditional filtering, performed in parallel for each mode; 3) mode probability update, based on the model-conditional likelihood functions; 4) estimate fusion, which yields the overall state estimate as the weighted sum of the updated state estimates of all filters. The detailed procedure of the classic IMM algorithm is provided as follows. • Step 1. Interaction (for j = 1, 2, · · · , m) Compute the predicted mode probability from k − 1 to k: n o X pj (k|k − 1) , P qj (k) |Yk−1 = πij pi (k − 1). i
= y (k) − Cj (k) xˆ j (k|k − 1) . Compute the residual covariance at time k: i h Sj , cov rj qj (k) , Yk−1 = Cj (k) Pj (k|k − 1) CTj (k) + 0 v,j Rj 0 Tv,j .
Kj = Pj (k|k − 1) CTj (k) S−1 j .
= xˆ j (k|k − 1) + Kj rj .
= Pj (k|k − 1) − Kj STj Kj . •
(6)
P0j (k − 1|k − 1) i h , cov xˆ 0j (k − 1|k − 1) qj (k) , Yk−1 X = {Pi (k − 1|k − 1) + [ˆx0j (k − 1|k − 1)
Step 3. Mode probability update (for j = 1, 2, · · · , m) Compute the likelihood function at time k: Lj (k) = N rj ; 0, Sj 1 T −1 1 exp − r S r (16) = q j . 2 j j (2π) Sj
Recognize the current mode at time k: pj (k)
i
•
(15)
Update the mode probability at time k: n o pj (k|k − 1) Lj (k) pj (k) , P qj (k) |Yk = P . i pi (k|k − 1) Li (k) (17)
Compute the mixing covariance at time k − 1 :
− xˆ i (k − 1|k − xˆ i (k − 1|k
(14)
Update the covariance matrix at time k: h i Pj (k|k) , cov xˆ j (k|k) qj (k) , Yk
(7)
− 1)][ˆx0j (k − 1|k − 1) − 1)]T }pi|j (k − 1) .
(13)
Update the estimated state at time k: h i xˆ j (k|k) , E x (k) |qj (k) , Yk
Compute the mixing estimate at time k − 1 : h i xˆ 0j (k − 1|k − 1) , E x (k − 1) |qj (k) , Yk−1 X = xˆ i (k − 1|k − 1) pi|j (k − 1). i
(12)
Compute the filter gain at time k:
(5)
Compute the mixing probability at time k − 1 : n o pi|j (k − 1) , P qi (k − 1) |qj (k) , Yk−1 = πij pi (k − 1) pj (k|k − 1) .
(11)
( > pT H⇒ The system is in mode qj = max pi (k) i < pT H⇒ No mode is recognized. (18)
(8)
Step 2. Model-conditional filtering (for j = 1, 2, · · · , m) Compute the predicted state from k − 1 to k: h i xˆ j (k|k − 1) , E x (k) |qj (k) , Yk−1 = Fj xˆ 0j (k − 1|k − 1) , u (k − 1) . (9)
•
Step 4. Estimate fusion (for output purpose) Compute the overall estimate at time k: h i X xˆ (k|k) , E x (k) |Yk = xˆ j (k|k) pj (k) . (19) j
Compute the predicted covariance from k − 1 to k:
Compute the overall covariance matrix at time k: h i T P (k|k) , E x (k) − xˆ (k|k) x (k) − xˆ (k|k) |Yk X = {Pj (k|k) + xˆ (k|k) − xˆ j (k|k)
Pj (k|k − 1) i h , cov xˆ j (k|k − 1) qj (k) , Yk−1 = Aj (k − 1) P0j (k − 1|k − 1) ATj (k − 1) + 0 w,j Qj 0 Tw,j . where Aj (k − 1) = VOLUME 6, 2018
j
(10)
∂Fj ∂x xˆ 0j (k−1|k−1),u(k−1)
.
T · xˆ (k|k) − xˆ j (k|k) }pj (k) .
•
(20)
Step 5. k → k + 1 37585
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2) MODIFIED IMM ALGORITHM
In the classic IMM algorithm, the transition probability is assumed to be constant over the studied time interval. However, as indicated in [33] and [34], the unchanged transition probability can mislead the mode recognition to intermittently declare a false recognition result, especially when the continuous behavior under different modes is similar. In addition, according to the classic IMM algorithm, the real-time distribution of f x (k) qj (k) , Yk cannot be approximated as a Gaussian distribution, since the mixing covariance in Step 1 will change the covariance of x under mode qj in a non-Gaussian way. This will lead to difficulty in calculating the fuzzy health degree due to a non-Gaussian distribution of f x (k) qj (k) , Yk . Considering the above deficiencies, two modifications are made to the classic IMM algorithm [35].
proposed, and the procedure of the algorithm is presented in Table 1. TABLE 1. Procedure of the modified IMM algorithm.
a: MODIFICATION 1
An update of transition probability is added to the classic IMM algorithm. Suppose the system is in mode qi at time k − 1, and in mode qj at time k, qi 6 = qj . Let 5 (k) = πij (k) m×m be the transition probability matrix. Further define an elementary matrix as 1 ··· 0 1 ← i th row . .. . 2= 1 0 ← j th row ··· 1 m×m (21) Then, the transition probability matrix 5 (k) is updated as 5 (k) = 2 · 5 (k − 1) · 2, After the update, mode qj at time k becomes currently dominant instead of mode qi , and the restriction that real discrete dynamics should be precisely known as a priori is relaxed.
B. DISCRETIZATION OF CONTINUOUS VARIABLES
Suppose a continuous variable xl ∈ x. Given an interval xl,lower , xl,upper , the continuous variable xl can be discretized into discrete states as Si = xl ∈ xl,lower , xl,upper | δi 6 xl < δi−1 ; i = 2, · · · , N , (23) where δi = xl,upper − (i − 1) · σ, xl,upper − xl,lower , σ = N −1
i = 1, 2, · · · , N , (24)
and S1 = xl | xl ∈ / xl,lower , xl,upper .
(25)
In (8), omit the step of mixing covariance, and just let P0j (k − 1|k − 1) = Pj (k − 1|k − 1). After this modification, the distribution of continuous variables under a specific mode at time k, written as f x (k) qj (k) , Yk , can be approximated as [36] f x (k) qj (k) , Yk = N xˆ j (k|k) , Pj (k|k) . (22)
After discretization, xl is classified into a set of discrete states U = {S1 , S2 , · · · SN }. The principle of continuous variable discretization is to satisfy µF (SN ) 6 µF (SN −1 ) 6 · · · 6 µF (S2 ) 6 µF (S1 ) in Theorem 2. Thus, equations (23)-(25) are only an alternative discretization method by multi-threshold segmentation. Actually, for ‘‘monotonically non-increasing’’ membership functions as shown in Figure 4, a discretization by (23 )-(25) can guarantee µF (SN ) 6 µF (SN −1 ) 6 · · · 6 µF (S2 ) 6 µF (S1 ). For other types of membership functions, the discretization method should be adjusted according to the detailed membership function shape of continuous variables.
Without the mixing covariance step, it will not influence the performance of the IMM algorithm in our research, which will be shown in Section IV. Meanwhile, the distribution of mode probability is obtained by (17). Thus, the distribution of the hybrid state s = (q, x) of the SHS, written as f qj (k) , x (k) Yk , can be viewed as a Gaussian Mixture Model (GMM) [37]. By applying Modifications 1&2 to the classic IMM algorithm, the modified IMM algorithm is
FIGURE 4. ‘‘Monotonically non-increasing’’ membership functions.
b: MODIFICATION 2
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C. FUZZY HEALTH DEGREE CALCULATION
According to expected mission requirements of the studied read-world system and practical engineering requirements, different fuzzy membership functions µS (·) and µF (·) can be allocated to each continuous variable xl ∈ x, l = 1, 2, · · · , n [30]. Meanwhile, according to (22), for any continuous variable xl ∈ x under mode qj , we have f xl (k) qj (k) , Yk = N xˆl,j (k|k) , P(l,l),j (k|k) , where xˆl,j is the lth component of xˆ j under mode qj ; P(l,l),j is the component of Pj at the lth row and lth column. Then, the state probability of Si ∈U under mode qj at time k is obtained as Z δi−1 i 6= 1 f xl (k) qj (k) , Yk dxl δi Z pSi ,j (k) = xl,upper f xl (k) qj (k) , Yk dxl i = 1. 1 − xl,lower
According to Theorem 2, the fuzzy health degree of all continuous variable xl under mode qj at time k can be calculated as Rx1 ,j (k) , · · · , Rxl ,j (k) , · · · , Rxn ,j (k) . After this, the fuzzy health degree of the SHS under mode qj at time k is Rj (k) = 9 Rx1 ,j (k) , · · · , Rxl ,j (k) , · · · , Rxn ,j (k) , (26) where 9 (·) is a function that reflects the performance relation between the SHS and the continuous variables. Generally, the function 9 (·) could have two types: Rj (k) =
n Y
Rxl ,j (k)
(27)
wl · Rxl ,j (k)
(28)
l=1
and Rj (k) =
n X l=1
where wl is the weight coefficient of xl in the SHS, satisfying P wl = 1. Equation (27) considers that all continuous variables influence the performance of the SHS in a serial manner, and any continuous variable is able to totally influence the system performance; equation (28) considers that each continuous variable influences the performance of the SHS in a weighted manner, and the system performance is a weighted sum of performance of all continuous variables. From (26), for ∀qj ∈ Q, we have {R1 (k) , · · · , Rj (k) , · · · , Rm (k)}. Then, the fuzzy health degree of the SHS at time k is R (k) =
m X
pj (k) · Rj (k).
j=1
where pj (k) is the mode probability of mode qj obtained by (17). VOLUME 6, 2018
IV. A CASE STUDY: QUADROTOR WITH SENSOR ANOMALIES
In this section, an experiment of quadrotor with sensor anomalies is presented. Quadrotor is a multi-rotor helicopter that is lifted and propelled by four rotors [38]. It is classified as rotorcrafts, as opposed to fixed-wing aircrafts, because the lift is generated by a set of rotors (vertically oriented propellers). By changing the speed of each rotor, it is possible to specifically generate a desired total thrust; to locate for the center of thrust both laterally and longitudinally; and to create a desired total torque, or turning force. Recently, quadrotors are adopted in both military and civil applications such as search and rescue [39], package delivery [40], border patrol [41], military surveillance [42] and agricultural production [43]. A fault or failure in any part of the quadrotor may lead to catastrophic disasters. Therefore, in order to ensure safety, it is necessary for a quadrotor to have a PHM module so that it can automatically change the control strategy and mission planning after detecting an anomaly. In this section, an SHS-based modeling of quadrotors is first established. Then, following the procedure of the proposed algorithm, the performance of the studied quadrotor is evaluated and discussed. A. SHS-BASED MODELING OF QUADROTORS 1) QUADROTOR DYNAMICS
Equation (29) presents a general dynamic model of quadrotors [44]: p˙ x = vx p˙ y = vy p˙ z = vz v˙ x = −uz (cos φ sin θ cos ψ + sin φ sin ψ) /m v˙ y = −uz (cos φ sin θ sin ψ − sin φ cos ψ) /m v˙ z = −uz cos φ cos θ/m + g φ˙ = vφ + tan θ vψ cos φ + vθ sin φ θ˙ = vθ cos φ − vψ sin φ ψ˙ = sec θ vψcos φ + vθ sin φ v˙ φ = Jy − Jz vψ vθ /Jx + uφ /Jx v˙ θ = (Jz − Jx )vφ vψ /Jy + uθ /Jy v˙ ψ = Jx − Jy vφ vθ /Jz + uψ /Jz {z } | x˙ = G (x, u),
(29)
where x = (px , py , pz , vx , vy , vz , φ, θ, ψ, vφ , vθ , vψ )T ∈ R12×1 contains continuous (process) variables of the quadrotor. The components px , py , pz represent the quadrotor’s position in the earth-fixed frame; the components vx , vy , vz represent the quadrotor’s velocity in the earthfixed frame; the components φ, θ, ψ represent the angles of roll, pitch and yaw, respectively; the components vφ , vθ , vψ represent the angular velocity of φ, θ, ψ, respectively; The parameters Jx , Jy , Jz are the moments of inertia along x, y, z directions, respectively; m is the mass of the quadrotor; g is the acceleration of gravity. The positive direction of z-axis of the earth-fixed frame points to the ground. The control input 37587
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T u = uz , uφ , uθ , uψ includes a total lift and moments of angles φ, θ, ψ, respectively. With the estimated xˆ by a designed observer T and an expected control target xd = px,d , py,d , pz,d , ψd , the con T trol input u = uz , uφ , uθ , uψ can be calculated by a PD controller as uz = −kP,z pz,d − pˆ z + kD,z vˆ z + mg uφ = kP,φ φd − φˆ − kD,φ vˆ φ uθ = kP,θ θd − θˆ − kD,θ vˆ θ uψ = kP,ψ ψd − ψˆ − kD,ψ vˆ ψ , (30) where −1 ˆ θd sin ψˆ −1 cos ψ =g φd sin ψˆ − cos ψˆ pˆ x − px,d vˆ · KPa + KDa x . pˆ y − py,d vˆ y
(31)
Equations (30) and (31) describe a position controller of the quadrotor. To obtain the discrete-time dynamic model of the quadrotor, equation (29) is discretized through the Euler method [45] as x (k) = x (k − 1) + T G (x (k − 1) , u (k − 1)),
For ∀qj ∈ Q, the continuous dynamics has a form of (33). Combining (1) and (33), for each state qj , j = 1, 2, 3, we have F1 (·) = F2 (·) = F3 (·) = F 0 w,1 = 0 w,2 = 0 w,3 = 0 w Q1 = Q2 = Q3 = Q. As to the observation equation, for state q1 representing the fully healthy status, we have c1 c2 C = C = 1 .. . c 12 0 v,1 = 0 v , R1 = R, where ci is the ith row vector of C1 . For state q2 representing the GPS anomalous status, the components px , py of x may be incorrectly measured, even the GPS measurements are completely lost. Then, we have c C2 = C1 \ 1 , c2
(32)
where T is the discretization time. Combining (32) with an observation equation and related noise items, we have general dynamic model of quadrotors: ( x (k) = F (x (k − 1) , u (k − 1)) + 0 w w (k − 1) (33) y (k) = Cx (k) + 0 v v (k), where the function F (x (k − 1) , u (k − 1)) = x (k − 1) + T G (x (k − 1) , u (k − 1)) ; the vector y contains system measurements, and C is the corresponding parameter matrix. Without loss of generality, let C = I12 be an identity matrix, which means all variables in x are directly measured. The items w and v are the process noise and measurement noise, satisfying that ( w (·) ∼ N (0, Q) , v (·) ∼ N (0, R) , ∀k T cov [w (k) , v (j)] = E w (k) v (j) = 0, ∀k, j,
which is interpreted that C2 is the rest part of subtracting the rows c1 and c2 from C1 . The meaning of C2 is that when a GPS anomaly occurs, the continuous dynamics under state q2 does not consider GPS measurements, despite whether the GPS can generate measurements or not. According to this principle, for states q3 , we have C3 = C1 \e3 , because the height pz is the 3rd component of x. The observation noise v and the corresponding noise driven matrix 0 v under states q2 and q3 can be also obtained following the similar variation of the matrix Cj . Besides the continuous dynamics, the discrete dynamics is a first-order Markov chain with transition probabilities as P qj (k + 1) |qi (k) = πij , ∀qi , qj ∈ Q,
where Q and R are the covariance matrices. The matrices 0 w and 0 v are the corresponding noise driven matrices. Note that (33) is a discrete-time stochastic continuous (variable) system.
and
2) SHS MODELING
The failure rate or anomaly rate of onboard components embedded on quadrotors and the related reliability test data are helpful for determining the value of the transition probability πij . Here, note that the SHS-based model can be extended for considering more kinds of unhealthy situations. For example, other sensors such as compass and gyroscope anomaly can be also modeled as new discrete modes following the principles above.
In this experiment, we considers sensor anomalies, including the GPS and barometer. Thus, in the SHS-based quadrotor model, three discrete modes should be included. According to the definition of the SHS model, we have Q = {q1 , q2 , q3 }, where q1 is a fully healthy status, q2 is a GPS anomalous status and q3 is a barometer anomalous status, respectively. 37588
3 X
πij = 1,
i = 1, 2, 3.
j=1
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TABLE 2. Quadrotor parameters.
FIGURE 5. Quadrotor with a Pixhawk autopilot.
3) MODEL CONFIGURATION
The parameters in the quadrotor dynamics are shown in Table 2. For the configuration of the modified IMM algorithm, the process noise w and the noise driven matrix 0 w are set as: w (·) ∼ N (0, Q) 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, Q = diag 0.001, 0.001, 0.001, 0.001, 0.001, 0.001 0 w = diag {0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1}. For measurement noise v and noise driven matrix 0 v , set v (·) ∼ N (0, R) 0.2, 0.2, 0.2, 0.5, 0.5, 0.5, 0.001, R = diag 0.001, 0.001, 0.001, 0.001, 0.001 0 v = diag {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}. The transition probability matrix in the discrete dynamics is set as [27]: 0.98 0.01 0.01 0.9 0 . 5 = 0.1 0.1 0 0.9
2) HYBRID STATE ESTIMATION
By the proposed modified IMM algorithm, the hybrid state of the SHS-based model is estimated. Figure 6 depicts the measured and true values of px , py , pz . It can be seen that when t ∈ [43.7, 62.0], the GPS measurement is influenced by large disturbance. After t = 62.0s, the GPS measurement is recovered and the barometer measurement is suffered from Figure 7 presents the estimated value large disturbance. of px , py , pz obtained by the modified IMM algorithm. Figure 8 presents the probability distribution of discrete modes in the SHS-based model, and Figure 9 presents the mode identification result. From Figures 6-9, we can conclude that: i) given the system measurements, the true values of the quadrotor’s continuous variables x can be estimated by the modified IMM algorithm; ii) when the GPS and barometer measurement by large disturbance, are influenced the estimated values pˆ x , pˆ y , pˆ z will be deviated from the true values px , py , pz ; iii) besides the estimated continuous variables x, the discrete mode of the quadrotor can be also identified by the modified IMM algorithm.
It can be seen that state q1 is dominant in the configuration of 5. However, the transition probability matrix will be updated in the modified IMM algorithm. Thus, the restriction that true discrete dynamics should be precisely known as a priori is relaxed. B. EXPERIMENT PROCESS AND RESULT ANALYSIS 1) EXPERIMENT SCENARIO
In our experiment, a quadrotor with a Pixhawk autopilot is used to perform a hover mission as shown in Figure 5. After the power is turned on, the pilot unlocks the quadrotor at t = 29.1s; the quadrotor enters into a hover status at a fixed position in space at t = 38.5s; a large noise is injected to the GPS measurement at t = 43.7s; the large noise is removed from the GPS measurement, and a large noise is injected to the barometer measurement at t = 62.0s. After the quadrotor enters into a hover status, the pilot no longer manipulates the sticks on the remote controller, and the quadrotor is automatically controlled by the autopilot to keep it staying at the fixed position in space. VOLUME 6, 2018
FIGURE 6. Measured and true values of px , py , pz . 37589
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FIGURE 9. Mode identification result.
FIGURE 7. Estimated and true values of px , py , pz .
where px,d , py,d , pz,d is the expected hover position in space. For 1x ∈ [0, +∞), its fuzzy success membership function is defined as: 1 1x ∈ [0, 2] 1 − 4 x µS (1x ) = 1x ∈ (2, 4] −2 0 1x ∈ (4, +∞) and its fuzzy failure membership function is µF (1x ) = 1 − µS (1x ). Further, discrete fuzzy states of 1x can be obtained by discretizing 1x as following Si = {1x ∈ [0, 4] | δi 6 1x < δi−1 ; i = 2, · · · , N }, where σ =
4 , N −1
N = 100, δi = 4 − (i − 1) · σ, i = 1, 2, · · · , N ,
and S1 = {1x | 1x ∈ / [0, 4]}.
FIGURE 8. Distribution of mode probability.
3) FUZZY HEALTH DEGREE CALCULATION
In order to calculate the fuzzy health degree, related membership functions and discretization method of continuous variables should be defined. In our experiment, the mission of the quadrotor is a persistent surveillance at a fixed position. Thus, the performance evaluation of the quadrotor in our experiment should concentrate on a consistency between the expected hover position and the true quadrotor position. Define a real-time three-axis distance 1x (k) = px (k) − px,d 1y (k) = py (k) − py,d 1z (k) = pz (k) − pz,d , 37590
After discretization, 1x is classified into a set of discrete states U= {S1 , S2 , · · · SN }, and it satisfies that µF (SN ) 6 µF (SN −1 ) 6 · · · 6 µF (S2 ) 6 µF (S1 ). Similarly, the fuzzy membership functions of 1y , 1z and the discretization methods can be defined. According to the above configuration and the result of real-time continuous variable estimation under mode qj , the fuzzy health degrees of {1x , 1y , 1z } can be calculated by Theorem 2, written as {R1x ,j (k) , R1y ,j (k) , R1z ,j (k) |k = 0, 1, · · · }. Then, by (27) and (28), we have Rsys1,j (k) = R1x ,j (k) · R1y ,j (k) · R1z ,j (k) and Rsys2,j (k) = w1x R1x ,j (k) + w1y R1y ,j (k) + w1z R1z ,j (k), VOLUME 6, 2018
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where w1x = w1y = w1z = 1 3 . For the performance of the SHS, we have Rsys1 (k) =
3 X
pj (k) · Rsys1,j (k)
j=1
and Rsys2 (k) =
3 X
pj (k) · Rsys2,j (k).
j=1
Figure 10 depicts the values of Rsys1 and Rsys2 . It can be seen that when sensor anomaly occurs, the values of Rsys1 and Rsys2 will decrease to some extent. The difference lies in how we view the relation between system performance and continuous variable performance. For the fuzzy health degree Rsys1 , a deviation from the expected hover position in any direction can be totally reflected in the variation of Rsys1 . In other words, if any 1x , 1y or 1z is greater than 4, Rsys1 will decrease to 0, because all continuous variables influence the performance of the SHS in a serial manner. However, for the fuzzy health degree Rsys2 , a deviation from the expected hover position in a specific direction can be only reflected in the variation of Rsys2 with a specific weight. For example, if 1x is greater than 4 and 1y , 1z ∈ [0, 2], Rsys2 will only decrease to 2 3 , because all continuous variables influence the performance of the SHS in a weighted manner.
quadrotor model. Then, Extended Kalman Filter (EKF) is used to estimate the continuous variables x of the quadrotor instead of the modified IMM algorithm. The estimation result is shown in Figure 11. Comparing Figure 11 with Figure 7, it indicates that the values of px , py , pz estimated by EKF fluctuate heavily, while the estimates of px , py , pz obtained by the modified IMM algorithm vary steadily. The reason is that when a specific system measurement, for example the GPS measurement, is polluted by large noise, a discrete mode switch will occur in the SHS-based model. In such a case, the estimate of {px , py } will be calculated by an integral of {vx , vy } rather than filtered from the noisy measurement. This processing manner can ensure the estimation accuracy in a short-time horizon. A further quantitative comparison result is present inTable 3, where the maximum and mean absolute errors of px , py , pz are calculated. It can be seen that the modified IMM algorithm can result in a more satisfactory estimation. In addition, based on the estimates of px , py , pz obtained by EKF, the fuzzy health degree can be also calculated and depicted in Figure 12. Comparing Figure 12 with Figure 10, it can be seen that the values of Rsys1 and Rsys2 in Figure 12 are very different from the values in Figure 10. Furthermore, the values of Rsys1 and Rsys2 in Figure 12 fluctuate more heavily than those in Figure 10 due to the fluctuant estimates of px , py , pz by EKF. This comparative study demonstrates that the SHS-based model is more advantageous in performance evaluation than the general dynamic model, and the fuzzy health degree calculated by the estimate from the SHS-based model is more suitable as a performance indicator.
FIGURE 10. Fuzzy health degree of the SHS.
4) COMPARATIVE STUDIES
In order to validate the advantage of the proposed performance evaluation algorithm, two comparative studies are presented here. In the first comparative study, we use a general dynamic model (shown in (33)) to replace the SHS-based VOLUME 6, 2018
FIGURE 11. Estimated values of px , py , pz by EKF.
The second comparative study is to demonstrate the advantage of the fuzzy health degree over the non-fuzzy health degree as a performance indicator. Reference [28] defined 37591
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TABLE 3. Accuracy comparison of estimates.
FIGURE 13. Non-fuzzy health degree based on the estimates px , py , pz obtained by the modified IMM algorithm.
that the fuzzy health degree is more advantageous than the non-fuzzy health degree for evaluating system performance. V. CONCLUSION
FIGURE 12. Fuzzy health degree based on the estimates px , py , pz by EKF.
a concept of (non-fuzzy) health degree in the binary-state space. Here, we calculate the defined non-fuzzy health degree Hsys based on the estimates of px , py , pz obtained by the modified IMM algorithm. The result is shown in Figure 13. Comparing Figure 13 with Figure 10, it demonstrates that when a sensor anomaly occurs, the value of Hsys rapidly decreases to 0 in a short-time horizon [43s, 53s]. The deficiency is that Hsys cannot comprehensively indicate the variation of system performance, especially when the system works in a degraded condition. Since the fuzzy health degree uses fuzzy state transitions representing various degrees of success and failure, it can not only indicate the performance degradation of the quadrotor, but also describe the detailed variation of the degradation. Even though Hsys and Rsys1 decrease to 0 at about t = 55s, Rsys1 can also reflect the slight performance variation before this time index. In addition, after t = 55s, Hsys has reached 0, while Rsys2 can also vary between the value range [0.3, 0.6], reflecting an accurate degradation variation. This comparative study demonstrates 37592
This paper proposes a performance evaluation algorithm of SHS based on the fuzzy health degree. The concept of fuzzy health degree is proposed, and the procedure of the performance evaluation algorithm is presented in detail, including hybrid state estimation based on a modified IMM algorithm, discretization of continuous variables and quantitative calculation of fuzzy health degree. A case study of quadrotor with sensor anomalies is presented to validate the effectiveness of the proposed method. The advantages of the proposed performance evaluation method and the required SHS-based modeling include: 1) the SHS-based modeling concerns the safety issues of a real-world system. The discrete and continuous dynamics in SHS can model different health statuses and corresponding dynamic behavior. The simulation results show that the performance of the SHS-based model behaves better than the general dynamic model. 2) The fuzzy health degree introduced in this paper gives a quantitative indicator of system performance, which is beneficial for understanding the working condition of real-world systems. Furthermore, it is more suitable to evaluate system performance compared with the non-fuzzy health degree. In future research, the proposed method can be extended in three aspects: 1) other anomalies such as propulsion system anomaly and communication breakdown can be added into the SHS-based quadrotor model to extend the applicability of the proposed method. 2) Since the health degree is calculated on the system process variables, the health evaluation result is sensitive to external disturbances, which bring fluctuations to process variables. In order to solve this problem, the wind model should be added in the SHS-based quadrotor model, or a performance evaluation algorithm of a homogeneous quadrotor team should be established. The two manners can both VOLUME 6, 2018
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effectively evaluate the amplitude and form of the external disturbance, and eliminate its influence on fuzzy health degree calculation. 3) After evaluating the performance of the quadrotor or detecting the anomaly occurrence, a failsafe mechanism is required to provide a suitable failsafe measure to guarantee the safety of the quadrotors. Thus, scientific methods should be studied to design a logically correct failsafe mechanism in future research. REFERENCES [1] P. W. Kalgren, C. S. Byington, and M. J. Roemer, ‘‘Defining PHM, a lexical evolution of maintenance and logistics,’’ in Proc. IEEE Autotestcon Conf., Sep. 2006, pp. 353–358. [2] J. W. Sheppard, T. J. Wilmer, and M. A. Kaufman, ‘‘IEEE standards for prognostics and health management,’’ IEEE Aerosp. Electron. Syst. Mag., vol. 24, no. 9, pp. 34–41, Sep. 2009. [3] L. Liao, ‘‘Discovering prognostic features using genetic programming in remaining useful life prediction,’’ IEEE Trans. Ind. Electron., vol. 61, no. 5, pp. 2464–2472, May 2014. [4] G. Bai, P. Wang, and C. Hu, ‘‘A self-cognizant dynamic system approach for prognostics and health management,’’ J. Power Sources, vol. 278, pp. 163–174, Mar. 2015. [5] J. Remmlinger, M. Buchholz, T. Soczka-Guth, and K. Dietmayer, ‘‘On-board state-of-health monitoring of lithium-ion batteries using linear parameter-varying models,’’ J. Power Sources, vol. 239, no. 10, pp. 689–695, 2013. [6] B. Ji, V. Pickert, W. Cao, and B. Zahawi, ‘‘In situ diagnostics and prognostics of wire bonding faults in IGBT modules for electric vehicle drives,’’ IEEE Trans. Power Electron., vol. 28, no. 12, pp. 5568–5577, Dec. 2013. [7] E. Zio and F. Di Maio, ‘‘A data-driven fuzzy approach for predicting the remaining useful life in dynamic failure scenarios of a nuclear system,’’ Rel. Eng. Syst. Safety, vol. 95, no. 1, pp. 49–57, Jan. 2010. [8] J. Dai, D. Das, and M. Pecht, ‘‘Prognostics-based risk mitigation for telecom equipment under free air cooling conditions,’’ Appl. Energy, vol. 99, no. 2, pp. 423–429, 2012. [9] K. Javed, R. Gouriveau, N. Zerhouni, and P. Nectoux, ‘‘Enabling health monitoring approach based on vibration data for accurate prognostics,’’ IEEE Trans. Ind. Electron., vol. 62, no. 1, pp. 647–656, Jan. 2015. [10] D. Liu, J. Zhou, H. Liao, Y. Peng, and X. Peng, ‘‘A health indicator extraction and optimization framework for lithium-ion battery degradation modeling and prognostics,’’ IEEE Trans. Syst., Man, Cybern. Syst., vol. 45, no. 6, pp. 915–928, Jun. 2015. [11] W. Yan, H. Qiu, and N. Iyer, ‘‘Feature extraction for bearing prognostics and health management (PHM)—A survey,’’ in Proc. 62nd Meeting Soc. Mach., Virginia Beach, VA, USA, 2008, pp. 1–11. [12] V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S. N. Kavuri, ‘‘A review of process fault detection and diagnosis: Part I: Quantitative model-based methods,’’ Comput. Chem. Eng., vol. 27, no. 3, pp. 293–311, 2003. [13] A. Soualhi, H. Razik, G. Clerc, and D. D. Doan, ‘‘Prognosis of bearing failures using hidden Markov models and the adaptive neuro-fuzzy inference system,’’ IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2864–2874, Jun. 2014. [14] J. Lee, F. J. Wu, W. Y. Zhao, M. Ghaffari, L. X. Liao, and D. Siegel, ‘‘Prognostics and health management design for rotary machinery systemsreviews, methodology and applications,’’ Mech. Syst. Signal Process., vol. 42, nos. 1–2, pp. 314–334, Jan. 2014. [15] H. Li, D. Pan, and C. L. P. Chen, ‘‘Intelligent prognostics for battery health monitoring using the mean entropy and relevance vector machine,’’ IEEE Trans. Syst., Man, Cybern. Syst., vol. 44, no. 7, pp. 851–862, Jul. 2014. [16] X.-J. Li and G.-H. Yang, ‘‘Fault detection in finite frequency domain for Takagi–Sugeno fuzzy systems with sensor faults,’’ IEEE Trans. Cybern., vol. 44, no. 8, pp. 1446–1458, Aug. 2014. [17] P. M. Frank and X. Ding, ‘‘Survey of robust residual generation and evaluation methods in observer-based fault detection systems,’’ J. Process Control, vol. 7, no. 6, pp. 403–424, 1997. [18] Z. Gao, C. Cecati, and S. X. Ding, ‘‘A survey of fault diagnosis and fault-tolerant techniques—Part I: Fault diagnosis with model-based and signal-based approaches,’’ IEEE Trans. Ind. Electron., vol. 62, no. 6, pp. 3757–3767, Jun. 2015. VOLUME 6, 2018
[19] G. Wang and Z. Huang, ‘‘Data-driven fault-tolerant control design for wind turbines with robust residual generator,’’ IET Control Theory Appl., vol. 9, no. 7, pp. 1173–1179, 2015. [20] Z. Xu, Y. Ji, and D. Zhou, ‘‘A new real-time reliability prediction method for dynamic systems based on on-line fault prediction,’’ IEEE Trans. Rel., vol. 58, no. 3, pp. 523–538, Sep. 2009. [21] S. Lu, H. Lu, and W. J. Kolarik, ‘‘Multivariate performance reliability prediction in real-time,’’ Rel. Eng. Syst. Safety, vol. 72, no. 1, pp. 39–45, 2001. [22] F. Di Maio, A. Rai, and E. Zio, ‘‘A dynamic probabilistic safety margin characterization approach in support of integrated deterministic and probabilistic safety analysis,’’ Rel. Eng. Syst. Safety, vol. 145, pp. 9–18, Jan. 2016. [23] P. Turati, N. Pedroni, and E. Zio, ‘‘Advanced RESTART method for the estimation of the probability of failure of highly reliable hybrid dynamic systems,’’ Rel. Eng. Syst. Safety, vol. 154, pp. 117–126, Oct. 2016. [24] X. D. Koutsoukos and D. Riley, ‘‘Computational methods for verification of stochastic hybrid systems,’’ IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 38, no. 2, pp. 385–396, Mar. 2008. [25] A. Abate, M. Prandini, J. Lygeros, and S. Sastry, ‘‘Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems,’’ Automatica, vol. 44, no. 11, pp. 2724–2734, Nov. 2008. [26] W. Liu and I. Hwang, ‘‘On hybrid state estimation for stochastic hybrid systems,’’ IEEE Trans. Autom. Control, vol. 59, no. 10, pp. 2615–2628, Oct. 2014. [27] Y. Zhang and X. Rong Li, ‘‘Detection and diagnosis of sensor and actuator failures using IMM estimator,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 4, pp. 1293–1313, Oct. 1998. [28] Z. Zhao, Q. Quan, and K.-Y. Cai, ‘‘A health performance prediction method of large-scale stochastic linear hybrid systems with small failure probability,’’ Rel. Eng. Syst. Safety, vol. 165, pp. 74–88, Sep. 2017. [29] J. I. Aizpurua, V. M. Catterson, I. F. Abdulhadi, and M. S. Garcia, ‘‘A model-based hybrid approach for circuit breaker prognostics encompassing dynamic reliability and uncertainty,’’ IEEE Trans. Syst., Man, Cybern. Syst., to be published, doi: 10.1109/TSMC.2017.2685346. [30] Z. Zhao, Q. Quan, and K.-Y. Cai, ‘‘A profust reliability based approach to prognostics and health management,’’ IEEE Trans. Rel., vol. 63, no. 1, pp. 26–41, Mar. 2014. [31] Z. Zhao, Q. Quan, and K.-Y. Cai, ‘‘A modified profust-performancereliability algorithm and its application to dynamic systems,’’ J. Intell. Fuzzy Syst., vol. 32, no. 1, pp. 643–660, 2017. [32] K.-Y. Cai, Introduction to Fuzzy Reliability. Norwell, MA, USA: Kluwer, 1996. [33] S. Zhao, B. Huang, and F. Liu, ‘‘Fault detection and diagnosis of multiplemodel systems with mismodeled transition probabilities,’’ IEEE Trans. Ind. Electron., vol. 62, no. 8, pp. 5063–5071, Aug. 2015. [34] Y. Zhou, D. Wang, H. Huang, J. Li, and L. Yi, ‘‘Fuzzy logic based interactive multiple model fault diagnosis for PEM fuel cell systems,’’ in Discrete Time Systems. Rijeka, Croatia: InTech, 2011, pp. 425–446. [35] Z. Zhao, Q. Quan, and K.-Y. Cai, ‘‘A health evaluation method of multicopters modeled by Stochastic Hybrid System,’’ Aerosp. Sci. Technol., vol. 68, pp. 149–162, Sep. 2017. [36] H. A. P. Blom and Y. Bar-Shalom, ‘‘The interacting multiple model algorithm for systems with Markovian switching coefficients,’’ IEEE Trans. Autom. Control, vol. TAC-33, no. 8, pp. 780–783, Aug. 1988. [37] D. Mukherjee, Q. M. J. Wu, and T. M. Nguyen, ‘‘Gaussian mixture model with advanced distance measure based on support weights and histogram of gradients for background suppression,’’ IEEE Trans. Ind. Informat., vol. 10, no. 2, pp. 1086–1096, May 2014. [38] H. Liu, X. Wang, and Y. Zhong, ‘‘Quaternion-based robust attitude control for uncertain robotic quadrotors,’’ IEEE Trans. Ind. Informat., vol. 11, no. 2, pp. 406–415, Apr. 2015. [39] T. Tomic et al., ‘‘Toward a fully autonomous UAV: Research platform for indoor and outdoor urban search and rescue,’’ IEEE Robot. Autom. Mag., vol. 19, no. 3, pp. 46–56, Sep. 2012. [40] A.-A. Agha-Mohammadi, N. K. Ure, J. Vian, and J. P. How, ‘‘Health aware stochastic planning for persistent package delivery missions using quadrotors,’’ in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), Chicago, IL, USA, Sep. 2014, pp. 3389–3396. [41] A. R. Girard, A. S. Howell, and J. K. Hedrick, ‘‘Border patrol and surveillance missions using multiple unmanned air vehicles,’’ in Proc. 43rd IEEE Conf. Decis. Control (CDC), Nassau, Bahamas, Dec. 2004, pp. 620–625. 37593
Z. Zhao et al.: Performance Evaluation Algorithm of SHSs
[42] B. Bethke, J. How, and J. Vian, ‘‘Multi-UAV persistent surveillance with communication constraints and health mangement,’’ in Proc. AIAA Guid., Navigat., Control Conf. (GNCC), Chicago, IL, USA, 2009, pp. 1–23. [43] Y. Huang, W. C. Hoffman, B. K. Fritz, S. J. Thomson, and Y. Lan, ‘‘Development of a low-volume sprayer for an unmanned helicopter,’’ J. Agricult. Sci., vol. 7, no. 1, pp. 148–153, 2015. [44] Q. Quan, Introduction to Multicopter Design and Control. Singapore: Springer, 2017. [45] M. Braun and M. Golubitsky, Differential Equations and Their Applications. New York, NY, USA: Springer, 1983, pp. 343–350.
JIPING XU received the B.S. and M.S. degrees in automation from Beijing Technology and Business University, Beijing, China, in 2002 and 2005, respectively, and the Ph.D. degree in control theory and control engineering from the School of Automation, Beijing Institute of Technology, Beijing, in 2010. He has been an Associate Professor with Beijing Technology and Business University since 2010. His current research interests include water environment evaluation and prediction, and big data analysis.
ZHIYAO ZHAO received the B.S. degree in automation from Beijing Technology and Business University, Beijing, China, in 2011, and the Ph.D. degree in guidance, navigation, and control from the School of Automation Science and Electrical Engineering, Beihang University, Beijing, in 2017. He has been a Lecturer with Beijing Technology and Business University since 2017. His current research interests include water environment evaluation and prediction, system health management, and stochastic hybrid systems. XIAOYI WANG received the B.S. degree in automation from the Department of Automation, Shenyang College of Technology, Shenyang, China, in 2000, the M.S. degree in optics from Shanxi University, Taiyuan, China, in 2003, and the Ph.D. degree in control theory and control engineering from the School of Automation, Beijing Institute of Technology, Beijing, China, in 2006. He has been a Professor with Beijing Technology and Business University since 2013. His current research interests include water environment modeling, optimization and decision making, and optimal control.
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JIABIN YU received the B.S. degree from Beijing Technology and Business University, Beijing, China, in 2007, the M.S. degree in automation from the Beijing Institute of Technology in 2009, and the Ph.D. degree in control theory and control engineering from the Institute of Automation, Chinese Academy of Sciences, in 2012. He has been an Associate Professor with Beijing Technology and Business University since 2017. His current research interests include water environment evaluation and prediction, motor control, and complex system design.
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