Automatic Control

International Review of

Automatic Control (IREACO) Theory and Applications

Contents Analytic Prognostic Model for a Dynamic System by Abdo Abou Jaoude, Khaled El-Tawil, Seifedine Kadry, Hassan Noura, Mustapha Ouladsine

568

Modeling Issues and Structural Properties of a Class of Nonlinear Bioprocesses by Monica Roman, Dan Selişteanu

578

A Finite Time Convergent Chattering Free Second Order Sliding Mode Observer for Non Stationary Systems by Mohamed Mihoub, Ahmed S. Nouri, Ridha Ben Abdennour

588

Robust Fault Detection in Uncertain Hybrid Systems, a PI Observer Approach by Ezzeddine Khadri, Moncef Tagina

594

Identification of Hybrid Switching Systems with Unknown Number of Models and Unknown Orders by K. Halbaoui, D. Boukhetala, F. Boudjema

604

A New Neural Adaptive Control Based on Neural Emulation of Complex Square MIMO Systems by A. Atig, F. Druaux, D. Lefebvre, K. Abderrahim, R. Ben Abdennour

612

Induction Motor Robust Control: a Quantitative Feedback Theory Approach by Faraz Dara, Asghar Akbari Foroud

624

Photovoltaic Array, Fuel Cell and Electrolyzer Connection to Grid by Direct Non-Linear Controlled H-Bridge Multilevel Inverter by M. Nazari, M. Abedi, G. B. Gharehpetian, H. Toodeji

633

Decreasing Ferroresonance Oscillation in Potential Transformers Including Nonlinear Core Losses by Connecting Metal Oxide Surge Arrester in Parallel to the Transformer by Hamid Radmanesh, Mehrdad Rostami

641

ANN’s Sensorless Induction Motor Fuzzy Logic Speed Control by Ben Hamed Mouna, Sbita Lassaâd

651

Implementation of Five Level Inverter Based UPFC System Using PIC Microcontroller by S. Muthukrishnan, A. Nirmalkumar

658

(continued)

Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved

Design of a Robust Multivariable Controller for an AC-VSC- HVDC System by R. K. Mallick, P. K. Dash

663

Self-Tuning Position Tracking Control of an Electro-Hydraulic Servo System in the Presence of Internal Leakage and Friction by Mohd F. Rahmat, Zulfatman, Abdul R. Husain, Kashif Ishaque, Mukhtar Irhouma

673

Nonlinear Damping Control Schemes Using the Unified Power Flow Controller and Fuzzy Theory by Tsao-Tsung Ma

684

Design and Simulation of Nonlinear Power Controllers for Variable-Speed Wind Energy Conversion Systems by M. Rajabzadeh, H. Khomami, M. Alizadeh Bidgoli

692

Parameter Identification and Synchronization of a Rotational Mechanical System with a Centrifugal Governor by E. G. Razmjou, A. Ranjbar, M. Hosseini

702

An Enabling Vision-Based Approach for Non-Calibrated, Robot-Positioning Task by Luis A. Raygoza-Pérez, Emilio J. González-Galván, Ambrocio Loredo-Flores, Jorge J. Pastor, Eric T. Baumgartner

710

An Intelligent Speed Controller for Space Vector Modulated Induction Motor Drive by R. Arulmozhiyal, K. Baskaran

723


International Review of Automatic Control (I.RE.A.CO.), Vol. 3, N. 6 November 2010

Analytic Prognostic Model for a Dynamic System Abdo Abou Jaoude, Khaled El-Tawil, Seifedine Kadry, Hassan Noura, Mustapha Ouladsine Abstract – To ensure high availability of industrial systems, recent developments in system design technology like in aerospace, defense, petro-chemistry and automobiles, are represented earlier in literature by simulated models during the conception step. These developments have facilitated the integration of diagnostic-prognostic models in these industrial systems. The monitoring of degradation indicators is used indirectly in failure prognostic. It is just a measurment of an unwanted situation. Hence, the diagnostic is not only a failure detection procedure but it also indicates the actual state and the historic of the system. The subsequent prognostic model leads to a predictive maintenance. The Remaining Useful Life is estimated from a predefined threshold of degradation. We will present here a procedure to create a failure prognostic model based on a physical dynamic system. But instead of degradation abaci largely used in prognostic studies, we will adopt here analytic laws of degradation such as Paris’ law for fatigue degradation and Miner’s law for cumulative damage. Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Analytic Laws, Degradation, Diagnostic, Fatigue, Miner’s Law, Paris’ Law, Prognostic

I.

Thus, we need to generate this physical model of the dynamic system [5].

Introduction

Each system or component of a system passes by three periods during its functioning life. The last period wich is a clear degradation period shows the deterioration of the system and that its lifetime becomes shorter. It is important for industrialists to predict the remaining lifetime in order to prevent costly sudden failure. The strategy of preventive and corrective maintenance to increase the system availability is expensive due to the frequent replacement of expensive accessories during the remaining useful lifetime (RUL) [1]. Moreover, this strategy is not efficient because most of equipments failures are not related only to the number of hours of functioning. The prognostic [2] is the capacity to predict the RUL of a component or of a system in service. RUL can be expressed in hours of functionning, in Kilometers run or in utilisation numbers. An earlier prognostic work [3] on vehicle suspension system used a non-analytical model and was based on abaci of degradation under a class of increasing functions. This methodology constructed a degradation trajectories of a mission (to be accomplished with success) with a use profile, under an environmental context and resources conditions defects. A procedure is presented in this paper to create a failure prognostic model based on a physical dynamic system. But instead of using degradation abaci [4] we will introduce here a degradation indicator D deduced from analytic laws of degradation like Paris’ law for fatigue degradation and Miner’s law for cumulative fatigue damage.

II.

Prognostic: State of the Art

The prognostic is a quite new area of interest. Reference [6] defines prognostics as the ability to “predict and prevent” possible fault or system degradation before failures occur. If we can effectively predict the condition of machines and systems, maintenance actions can be taken ahead of time. As a result, minimum downtime can be achieved. Prognosis has been defined by [7] as “prediction of when a failure may occur” i.e. a means to calculate remaining useful life of an asset. In order to make a good and reliable prognosis it must have good and reliable diagnosis. Various approaches to prognostics have been developed that range in fidelity from simple historical failure rate models to high-fidelity physics-based models [8]. The required information (depending on the type of prognostics approach) include: engineering model and data, failure history, past operating conditions, current conditions, identified fault patterns, transitional failure trajectories, maintenance history, system degradation and failure modes. A number of different methods have been applied to study prognosis of degraded components. In general, prognostics approaches can be classified into three primary categories: (1) model driven, (2) data driven and (3) probability based prognostic techniques. Fig. 1 [6] summarizes the range of prognosis approaches applied to

Manuscript received and revised October 2010, accepted November 2010

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Abdo Abou Jaoude, Khaled El-Tawil, Seifedine Kadry, Hassan Noura, Mustapha Ouladsine

different systems and their implementation and operation.

relative

cost

explicit relationship between the condition variables and the lifetimes (current lifetime and failure lifetime) via failure mechanism modeling. Two examples of research along this line are [19] for machines considered as energy processors subject to vibration monitoring and [20] for bearings with vibration monitoring. In [21] and [22] the problem of forecasting machine failure is illustrated for a high power fan bearing and a railroad diesel engine. Engel et al. [23] discussed some practical issues regarding accuracy, precision and confidence of the RUL estimates. Lesieutre et al. [24] developed a hierarchical modeling approach for system simulation to assess the RUL.

for

II.2.

The main advantage of model based approaches is their ability to incorporate physical understanding of the monitored system. In addition, in many situations, the changes in feature vector are closely related to model parameters and a functional mapping between the drifting parameters and the selected prognostic features can be established [25]. Moreover, if the understanding of the system degradation improves, the model can be adapted to increase its accuracy and to address subtle performance problems. Consequently, they can significantly outperform data-driven approaches (next section). But, this closed relation with a mathematical model may also be a strong weakness: it can be difficult, even impossible to catch the system's behavior. Further, some authors think that the monitoring and prognostic tools must evolve as the system doews.

Fig. 1. Prognostic approaches

II.1.

Advantage and Drawback of this Approach

Model Based Approaches

The model-based methods assume that an accurate mathematical model can be constructed from first principles. This approach to prognostic requires specific failure mechanism knowledge and theory relevant to the monitored machine. The existing papers propose different model based solution for the industrial problems. Bartelmus and Zimroz [9] estimated through a demodulation process, the vibration signal for a planetary gearbox in good and bad conditions. Kacprzynski et al. [10] proposed fusing the physics of failure modeling with relevant diagnostic information for helicopter gear prognostic. Chelidze and Cusumano [11] proposed a general method for tracking the evolution of a hidden damage process given a situation where a slowly evolving damage process is related to a fast, directly observable dynamic system. Luo et al. [12] introduced an integrated prognostic process based on data from modelbased simulations under nominal and degraded conditions. Oppenheimer and Loparo [13] applied a physical model for predicting the machine condition in combination with a fault strengths-to-life model, based on a crack growth law, to estimate the RUL. Adams [14] proposed to model damage accumulation in a structural dynamic system as first/second order nonlinear differential equations. Chelidze [15] modeled degradation as a "slow-time" process, which is coupled with a "fast-lime", observable subsystem. The model was used to track battery degradation (voltage) of a vibrating beam system. Li et al. [16] and [17] introduced two defect propagation models via failure mechanism modeling for RUL estimation of bearings. Ray and Tangirala [18] used a nonlinear stochastic model of fatigue crack dynamics for real-time computation of the time-dependent damage rate and accumulation in mechanical structures. A different way of applying model-based approaches to prognostic is to derive the

II.3.

Data Driven Approaches

Data-driven approaches use real data (like on-line gathered with sensors or operator measures) to approximate and track features revealing the degradation of components and to forecast the global behavior of a system. Indeed, in many applications, measured input/output data is the major source for a deeper understanding of the system degradation. Data-driven approaches can be divided into two categories: artificial intelligence (AI) techniques (neural networks, fuzzy systems, decision trees, etc.), and statistical techniques (multivariate statistical methods, linear and quadratic discriminators, partial least squares, etc.). II.4.

Artificial Intelligence Techniques

Within the field of maintenance problems, Artificial Neural Networks (ANNs) and neuro-fuzzy systems (NFs) have successfully been used to support the detection, diagnostic and prediction processes, and research works emphasize on the interest of using it [26], [27], [28], [29], [30], [49]: ANNs and NFs are a general and flexible modeling tool, especially for prediction problems. Let's point out the principle arguments of this assumption (non exhaustive list). International Review of Automatic Control, Vol. 3, N. 6


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II.5.

estimation approach for a nonlinear model with temperature measurements of gas turbines [43]. The online detection procedure presented can track small variations in parameters for early warning.

Statistical Techniques

Statistical techniques require, due to the algorithms involved, quantitative data measurements. This information related to the sources is combined and the result is a stochastic estimation of the future state. Following paragraphs give a non exhaustive list of these techniques. Yan et al. employed the logistic regression model to calculate the probability of failure for given condition variables [31]. A predetermined level of failure probability was used in combination with an ARMA (autoregressive moving average) time series model to trend the condition and to estimate the RUL. Lin and Makis [32] introduced a partially observable continuousdiscrete stochastic process model to describe the hidden evolution process of the machine state associated with the observation process and to estimate the RUL. HHM (Hidden Markov Model) and PIM (Proportional Intensity Model) are two statistical models in survival analysis that enable having trending models for the fault propagation process to estimate the future states. HMM describes the failure mechanism which depends both on age and condition variables. In [33] and [34], HMM and PIM are considered as powerful tools for RUL estimation. Vlok et al. [46] applied PIM with covariate extrapolation to estimate bearing residual life. Wang [35] used the residual delay time concept and stochastic filtering theory to derive the residual life distribution and used an AR process to model a vibration signal for prognostic [36]. Phelps et al. [37] proposed to track sensor-level test failure probability vectors instead of the physical system or sensor parameters for prognostics. A Kalman filter with an associated IMM (interacting multiple model) was used to perform the tracking. In [38], a prognostic process for transmission gears is proposed by modeling the vibration signal as a Gaussian mixture. By adaptively identifying and tracking the changes in the parameters of Gaussian mixture, it is possible to predict gear faults. Swanson [39] proposed to use a Kalman filter to track the dynamics of the mode frequency of vibration signals in tensioned steel band (with seeded crack growth). Wang et al. [40] proposed a stochastic process, called a "gamma process", with hazard rate as the residual life prediction criterion. The condition information considered was expert judgment based on vibration analysis. Goode et al. [41] used the statistical process control (SPC) to separate the whole machine life into two intervals, the I-P (Installation-Potential failure) interval in which the machine is running correctly and the P-F (Potential failure-Functional failure) in which the machine is running with a problem. Based on two Weibull distributions assumed for the I-P and P-F time intervals respectively, failure prediction was derived in the two intervals and the RUL was estimated. In [42], Garga et al. proposed a signal analysis approach for prognostics of an industrial gearbox. The main features used included the root mean square (RMS) value, Kurtosis and Wavelet magnitude of vibration data. Zhang and Ganesan proposed a parameter

II.6.

Advantage and Drawback of this Approach

The strength of data-driven techniques is their ability to transform high-dimensional noisy data into lower dimensional information for diagnostic/prognostic decisions. AI techniques have been increasingly applied to machine prognostic and have shown improved performances over conventional approaches. In practice however, it isn't easy to apply AI techniques due to the lack of efficient procedures to obtain training data and specific knowledge. So far, most of the applications in the literature just use experimental data for model training. Thus, data-driven approaches are highly-dependent on the quantity and quality of system operational data. II.7.

Experience-Based Prognostic [44]

It is necessary where we cannot use the two previous approaches. It is based on a reliability function or on a Bayesian process where the parameters are taken from feedback experience or expert opinion. Its disadvantages are the incapacity to treat complex systems of many components and its exclusive binary principle (success/failure) rather than continuous states of degradation. Our proposed procedure belongs to the first prognostic approach and is based on a physical model leading to a degradation indicator. It is focused on developing and implementing effective diagnostic and prognostic technologies with the ability to detect faults in the early stages of degradation. Early detection and analysis may lead to better prediction and end of life estimates by tracking and modeling the degradation process. The idea is to use these estimates to make accurate and precise prediction of the time to failure of components. Early detection also helps in avoiding catastrophic failures. The case of fatigue degradation was chosen, and it can be mathematically formulated by analytic laws such as Paris and Miner laws.

III. Damage Evolution Law The fatigue of materials under cyclic loading and the micro-cracks (starting from a0 ) that become detectable and unstable (Fig. 2), will make the macro-cracks grow to a critical length (a = ℓ*) creating thus fractures and hence leading to failure (Fig. 3) The law of damage growth [9] is given by ParisErdogan [48] as follows: m da = C ⎡⎣ ∆K ( a ) ⎤⎦ dN

(1)

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Fig. 2. Pre-Crack fatigue damage

Fig. 3. Crack length evolution

The initial damage is: a ( 0 ) = a0 .

where: da = the increase of the crack per cycle dN ∆K ( a ) = Y ( a ) ∆σ π a = the stress intensity factor

From the general formulation, we denote: and g ( x,a ) = φ1 ( a ) φ2 ( h ( x ) ) .

Y ( a ) = the function of the component’s crack geometry

IV.

∆σ = the range of the applied stress in a cycle (σmax -

σmin )

Linear Cumulative Damage Modeling

To study the prognosis of degraded component, our idea is to predict and estimate the end of life of the component by tracking and modeling the degradation function. To facilitate the analysis, it is convenient to adopt a damage measurement D ∈ [ 0 ,1] by using the cumulative damage law of Miner (Fig. 4). This law [10] is used to estimate the lifetime of components subject to load cycles. It considers that the damage fraction d i at

C and m = constants of materials; C is a small number ( 0 < C 0, solutions of the following convex optimization problem: min τ under LMI constraints:

(24)

The condition which guaranteed the estimation error exponential convergence while satisfying the dynamic performances (24) is:

⎡ ΦTq( t ) + Φ q( t ) + I + 2α P PEq ( t ) − LP N ⎤ ⎢ ⎥ < 0, ⎢ E T P − ( L N )T ⎥ − τ I P q( t ) ⎣ ⎦

V ( t ) + 2αV ( t ) + X T ( t ) X ( t ) − τ 2 e T ( t ) e ( t ) < 0 (25)

with Φ q ( t ) = PAq0( t ) − LP C 0 C1T − LI C2T ,

The writing of this equation according to X ( t ) and e ( t ) gives:

and

X T ( t ) ( AT P + PA ) X ( t ) + +λ X T ( t ) γ 2 X ( t ) + λ −1 X T ( t ) PPX ( t ) + −e T ( t ) N T K PT PX ( t ) − X T ( t ) PK P Ne ( t ) + +2α X T ( t ) PX ( t ) +

τ =τ2

In this case, the use of the L2 approach makes it possible to synthesize the observer gains so as to attenuate the influence of this disturbance as well as the measurement noise. PI observer (9) convergence conditions are given in the following theorem:

But by taking account of relieving (12), that is to say: 2

LI = PK I ,

III.3. Case 2: the Bound on d ( t ) is Not Known

The exponential convergence of the estimation error is normally guaranteed if:

2

(28)

One, then obtains the Linear Matrix Inequalities presented in theorem 1.

P

V ( t ) + 2αV ( t ) < 0

q ( t ) ∈ {1,...,h}

The equation (28) is not a LMI in the variables P, KP, KI, λ andτ. This problem can be solved by using the Schur complement as well as the following variables changes:

P

2 X ( t ) − τ 2 e ( t )

Ψ q( t ) = AT P + PA + λγ 2 I + λ −1 PP + 2α P + I

By using the activation functions property (2), the inequality (27) is respected if:

(20)

The lemma1 application makes it possible to bound the Lyapunov derivative function in the form: V ( t ) < X T ( t ) ( AT P + PA ) X ( t ) + +λ d T ( t ) d ( t ) + λ −1 X T ( t ) PPX ( t ) +

T

⎡ X ( t ) ⎤ ⎡ X ( t ) ⎤ ∑ q(t )=1 p ( q ( t ) ) ⎢ e ( t ) ⎥ Λ q(t ) ⎢ e ( t ) ⎥ < 0 ⎣ ⎦ ⎣ ⎦ − PK P N ⎤ ⎡ Ψ q(t ) where Λ q ( t ) = ⎢ ⎥ −τ 2 I ⎥⎦ ⎢⎣ − N T K PT P

(19)

(30)

q ( t ) = 1,...,h

For a given scalar α >0, the observer gains are given ) KP = P−1LP and KI = P−1LI. by Demonstration 2: the theorem demonstration is carried out by considering a step identical to that of theorem 1, that is to say:

(26)

+ X T ( t ) X ( t ) − τ 2 e T ( t )e ( t ) < 0


International Review of Automatic Control, Vol. 3, N. 6

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E. Khadri, M. Tagina

V ( t ) = X T ( t ) ( AT P + PA ) X ( t ) + + e T ( t ) E T PX ( t ) + X T ( t ) PEe ( t )

estimate and the residual signal, where the latter one is used by the DO. A mode transition is detected when the residual of CO surpasses its predetermined threshold at some time. Once the new mode is identified correctly by the DO, the CO is switched to the new one. The DO is designed to estimate the mode of the hybrid system and provide the mode information to the CO. The DO consists of a bank of parallel mode isolators (MIs), each of which is a RO and has a similar structure but different parameters. The output of the DO is the mode estimation. When a mode transition is detected, the DO is enabled to identify the new mode.

(31)

By observing the condition (23), guaranteeing the estimation error exponential convergence while satisfying the dynamic performances (24), one can write the following equation: X T ( t ) ( AT P + PA ) X ( t ) + e T ( t ) ( E T P ) X ( t ) + + X T ( t ) ( PE ) e ( t ) + X T ( t )( 2α P ) X ( t ) + (32) + X T ( t ) X ( t ) + e T ( t ) ( −τ 2 ) e ( t ) < 0

IV.2. Robust Fault Detection Strategy

That is to say still (33): T

In this section, a robust fault detection scheme for the uncertain hybrid systems under consideration is proposed to incorporate the RHO proposed in Section 4 into the well known unknown input observer scheme [22]. A robust state observer (RSO) is a RHO designed for the hybrid system with unknown uncertainty and faults, which is utilized to estimate the continuous state ˆx ( t ) and discrete mode qˆ ( t ) of the hybrid system under consideration. The RSO consists of a CO and a DO, where the CO is a switching system which consists of h modes, and the DO consists of a bank of parallel MIs. The residual rs ( t ) = y ( t ) − Cx ( t ) is robust to all unknown uncertainties and faults. The fault detector is a CO designed for the fault-free hybrid system with unknown uncertainties, which is utilized to detect if a fault has occurred. The mode of fault detector depends on the mode estimation of the RSO and the detection of the fault is made from the residual analysis. The residual r f ( t ) = y ( t ) − Cx ( t ) is

⎡ X ( t ) ⎤ ⎡ A P + PA + I + 2α P PE ⎤ ⎡ X ( t ) ⎤ ⎢ T ⎥⎢ ⎥⎢ ⎥= hv

Fig. 3. Hybrid automat of the system

T

T

e1

e2

h1