Diagnosis of Sensor Failure Detection and ... - Semantic Scholar
I2MTC 2009 - International Instrumentation and Measurement Technology Conference Singapore, 5-7 May 2009
Diagnosis of Sensor Failure Detection and Information Rebuilding Using Polynomial Chaos Theory Huimin Li
Antonello Monti
Dept. of Electrical Engineering University of South Carolina, Columbia, SC, USA [email protected]
E.ON Energy Research Center RWTH Aachen University Aachen, Germany [email protected]
Abstract—A new strategy applying Polynomial Chaos Theory (PCT) to diagnose sensor failure is presented in this paper. Using PCT and uncertainty of parameters, the original state space model of a system is expanded to PCT domain. Variations of sensor output due to parameter uncertainty are calculated by a PCT estimator and presented as PCT output boundaries based on the probability density function (PDF) of uncertain parameters. Along with PCT boundaries, an algorithm is used to diagnose and declare failed sensors. Meanwhile, outputs of failed sensors are frozen and missing information is rebuilt with bestcase estimations from PCT estimator. Effectiveness of this method is also verified with two examples. Keywords- uncertain systems; induction motor drive; stochastic differential equations
I.
INTRODUCTION
Sensor Failure and Information Rebuilding (SFDR) is an extremely important research aspect in measurement and control. Much effort has been made in this area. In [1]-[5], Artificial Intelligence (AI) theory (such as Neural Network, one of the most popular options among AI) is applied to detect sensor failure and rebuild the information from failed sensors. Kalman filter [5], Extended Kalman Filter [6] are also favorite options in this area. CARMA model is used in [7]. On the other hand, observer-based method, for instance, Unknown Input Observer (UIO) is used in [8] and sliding mode observer is adopted in [9]-[11]. The basic ideas of all these methods are declaring sensor failure based on a pre-determined threshold and the residual between the normal data and the measurement data; rebuilding is achieved by replacing the failed data with model-based prediction. However, in many cases, indirect measurements are very sensitive to uncertainties during dynamic simulations. A proposal based on fuzzy logic is presented in [5] where the measurement uncertainty is taken into consideration in detecting sensor failure. In [12] and [13], parameter uncertainties are analyzed in simulating system dynamic trajectory. The approach proposed here is based on PCT. The nature of PCT is to approximate a random process by a complete and orthogonal polynomial basis in terms of certain random variables [14]. In the rest of the paper, the authors will show
how this approach makes it possible detecting the sensor failure by counting also for uncertainty propagations. The work is organized as following: a brief generic formulation of system model in PCT domain is introduced in section II, while the failure detection and information rebuilding algorithm are discussed in section III. Section IV shows a validation of this proposed idea through two examples and the simulation results are also portrayed and analyzed in section IV. Conclusions in section V complete the paper. II. GENERIC FORM OF PCT EXPANSION AND PCO PCT concept was first introduced by Wiener in 1938 as “Homogeneous Chaos” [15]. PCT expansion is a method that uses a polynomial based stochastic space to represent the evolution of the uncertainty propagation into the system [16]. By choosing certain polynomial base ψ, uncertain parameter X can be expanded into PCT domain as shown in equation (1). ∞
X (θ ) = ∑ a n (θ ) Ψn (ξ (θ )) n =0
Where:
θ
(1)
Random event;
a n The coefficients of the expansion; Ψn The selected polynomial basis;
ξ
Radom variables with a PDF according to the selected
Ψn . The dimension of the expansion is limited for practical purpose. The delimited dimension P is selected based on the number of independent uncertain parameters ( nv ) and the maximum order of the polynomial basis ( n p ). The infinite upper limit in equation (1) will be replaced by the selected dimension P as in equation (2):
P=
(nv + n p )! nv !n p !
−1
(2)
This work was supported by the US Office of Naval Research under the grant N00014-07-1-0603
978-1-4244-3353-7/09/$25.00 ©2009 IEEE Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
Dependency on the random variables ξ can then be eliminated by taking a Galerkin projection for the state space equation of the system. In this paper, only one uncertain parameter is considered, so equation (3) gives the method used in this paper for the projection: b
dx k = dt
∫ X (θ )Ψ dξ k
a
b
∫Ψ
2 k
failure). Last but not least, measurement uncertainty may also cause misjudging of the sensor failure, and this concerning is solved by introducing maximum tolerant band BW2. BW2 can be calculated from the nominal value of the sensor, the PCT tails and the standard of the sensor under detection. Fig. 1 illustrates the main process of diagnosing SFDR using PCT.
(3)
dξ
a
Projection method for multiple uncertain parameters along with PCT fundamental principles are analyzed in [16]. Meanwhile, detailed description of PCT application can be found in [14]. Solving state space equations of the system (after Galerkin Projection) yields to system state space model in PCT domain. The model is presented by using the uncertain variables as the state variables, and matrices (including parameter uncertainty) as the coefficients. The generic form of the system in PCT domain can be presented as in equation (4). . x& pct (t ) = A pct x pct (t ) + B pct u pct (t ) (4) y pct (t ) = C pct x(t ) + D pct u pct (t ) The system order in PCT domain is determined by a few of factors, such as the original system order, the number of the independent uncertainties considered during the expansion and the polynomial base used to perform the expansion. Quantitatively, equation (5) can be used to calculate the system order in PCT domain.
n pct =
(nv + n p )! nv !n p !
× ns
(5)
A detailed description of this expansion will be given in section IV by application examples. III.
SENSOR FAILURE DETECTION AND INFORMATION REBUILDING ALGORITHM In this section, an algorithm based on PCT and PDF of the uncertain parameters is designed to diagnose the system when performing SFDR. Boundaries in PCT domain, produced by PDF tails, are actually the plant’s output due to the parameter uncertainty. The idea is to exploit PCT to make the sensor diagnostic less sensitive to parameter uncertainty. It should be underlined, anyway, that the proposed algorithm could still fail if the system changes so dramatically that the model becomes unreliable. The work here focuses on examining the ability of PCT in SFDR, not considering the possibility of system failure and then assuming that the model is reliable. To apply this approach, the distributions of uncertain variables are assumed to be known. Then the normal operation range (BW1) for each sensor can be calculated from PCT. This range is a safe operation bands for a sensor. However, the following two circumstances may cause the real output exceeding this range: one is the sensor fails, the other is the model is not reliable anymore (mostly caused by system
Figure 1. Diagnosis Algorism for SFDR
The main problem then is to identify the variable boundaries starting from the PCT expansion. This issue can be solved by properly choosing a proper output matrix in PCT domain. Description of this process is illustrated in the next section with the help of examples. IV.
APPLICATION OF THE APPROACH TO SFDR FOR INDUCTION MACHINE
In this section, two examples are introduced. The first example refers to an induction machine. The purpose is to test the proposed method in diagnosing a stator phase current sensor failure and rebuilding the missing data when there are sensor failures. The main uncertainty is assumed to be the rotor resistance. Then, in the second example, PCT based SFDR method is applied to a failure-tolerant field-oriented control of an induction motor (FOCIM). A. Example 1 Modeling of Induction Machine in PCT domain is a key to conduct SFDR with the proposed approach. The model of induction machine in PCT domain is established based on the uncertain rotor resistance. As mentioned in [16], the distribution of rotor resistance is assumed to be known a priori. For this specific example, a uniform distribution is adopted but different choices could be done without affecting the value of the algorithm. Legendre polynomial basis is selected to expand variables into PCT domain [17]. TABLE I gives the first three bases for Legendre. TABLE I.
LEGENDRE BASIS FOR ONE-DIMENSIONAL BASIS
First term
Second term
Third term
Ψ0 =1
Ψ1 =ξ
Ψ2 =(3ξ2-1)/2
Parameters needed to perform PCT expansion of an induction machine are summarized in TABLE II:
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
TABLE II. PCT Expansion Parameter Calculation Description
Symbol and Value Symbol
Value
Number of the uncertain variable
nv
1
Maximum Order of Polynomial Basis
np
1
Order of the original system
ns
2
Number of terms for each variable after expanding
P
2
n pct
8
OB pct
4
Order of the system in PCT Domain Order of the Flux Observer in PCT domain
n
C ipct
1 2 3 4 5 6 7 8
(6)
The second step is to perform Galerkin Projection of the original system to remove the dependency on the random variable as in (7). 1
i&sdk =
P
∫ (∑ i
−1 n =0
sdn
Φ n )Φ k dξ
(7)
1
RULES OF SFDR
BW1
BW2
Sensor Status
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
NF NA NF F NA NA NF NF
BW1 is the range delimited by PCT boundaries. BW2 is the maximum range for the non-failure operation of a sensor. “e” is the difference of the rotor mechanical speed between two time steps. “NF” means no failure; “NA” means circumstance is not reasonable in practice or case is not under consideration in this paper; “F” means sensor fails. For “BW1” and “BW2”, “0” means within the range and “1” means out of the range; For “e”, “0” means very low (not more than 2%) and induction machine runs in steady state and “1” means very large (larger than 2%) and occurs during machine startup and shutdown or speed sensor fails.
Simulation results of this example are given in the following along with analysis of various cases. Firstly, PCT diagnosis boundaries (BW1) and sensor fault tolerance band (BW2) are given in Fig. 2 and Fig. 3 respectively. PCT Boundaries (BW1)
2 ∫ Φ k dξ
3
−1
Upper1 Mean Lower1
2
(8)
1
Current (A)
Then, the complete induction machine model in PCT domain can be calculated as in equation (8).
λ&r = Arpct λrpct + Brpct ispct
1
(9)
e
1
i&s = Aipct ispct + Bi1 pct λrpct + Bi 2 pctVspct
0 1
Conditions
Case
P
All the variables that depend on rotor resistance should also be expanded [14].
0
TABLE III.
The first step to perform PCT expansion of a system is to expand the uncertain variables (rotor resistance) to PCT domain, as in equation (6).
n =0
0⎤ 0 ⎥⎥ 0⎥ ⎥ 1⎥ 0⎥ ⎥ − 1⎦⎥
0
TABLE III illustrates the rules for the failure detection of current sensors in this application.
The original model adopted for the induction machine is a standard two-axis representation [18]. We assumed that the main source of uncertainty is given by the variation of the rotor resistance.
Rr = ∑ Rrn Φ n
⎡1 1 ⎢0 1 ⎢ ⎢1 − 1 =⎢ ⎢0 0 ⎢0 0 ⎢ ⎣⎢0 0
0
-1
-2
Values for all matrices in equation (8) are given in Appendix 1 and Appendix 2. As mentioned in section III, PCT boundaries can be calculated by properly choosing the output matrix in PCT domain. In this application, PDF of uncertain parameter is uniform with Legendre polynomial basis selected, so output matrix can be chosen as shown in equation (9) to find stator current boundaries in d − q domain. Three phase boundaries can then be solved with axes transformation theory.
-3 0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
time (s)
Figure 2. PCT boundaries (BW1) for Sensor of Phase A
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
SFDR for Hard Failure
Sensor Fault Tolerance Band (BW2) Upper2 data2 Lower2
3
6
2
4
Current (A)
1
Current (A)
Original Output Rebuilded Data
8
0 -1
2 0 -2 -4
-2
-6
-3
-8
0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
0.44
time (s)
0.48
0.5
0.52
0.54
0.56
time (s)
Figure 5. SFDR for Hard Failure of Current Sensor for Phase A
Figure 3. Maximum Tolerance Boundaries (BW2) for Sensor of Phase A
In TABLE III, there are eight cases. Case 8 indicates that the induction machine is in startup or shutdown period (e is 1). In this period, currents change dramatically and SFDR module is not executed. Cases 2 and 6 are not reasonable (Exceeding BW2 means also exceeding BW1). Moreover, cases 5 and 7 are excluded from consideration in this paper since they indicate speed sensor failure (speed sensor is assumed to be working correctly). Case 1 is the normal performance of the system. It indicates that the induction machine is in steady state (e is zero), and no current fail is detected (status of BW1 and BW2 are all zero). In this case, the SFDR module will operate but the output of the real sensors will be used. Case 3 and 4 are the most important. The former shows that the machine operates in steady state (e is zero), the output of the phase A sensor is out of BW1 while stays in BW2. In this situation, SFDR will continuously monitor the sensor output other than declaring a failure immediately till the original measurement is out of BW2. Case 4 is a little bit different: the output of the sensor exceeds both BW1 and BW2. SFDR module declares a failure and restores the failed measurement simultaneously. Simulation results of case 3 and 4 are given in Fig. 4 and Fig. 5 respectively. Diagnosis of SFDR 4
Current (A)
0.46
In Fig. 4, operation from 0.45 to 0.475 second is an example of case 3. Beginning from 0.573 second, we enter in case 4: SFDR declares a failure, freezes the failed sensor and rebuilds sensor output as portrayed with the dashed black line. Basically, it simulates a soft failure [3]. In Fig. 5, the sensor is hard failed at 0.5 second. The original data (red line) is accommodated by the PCT best-case estimation (dashed black line). Example 2: Fault-Tolerant FOC Control of Induction Machine As portrayed in the following FOCIM schematic (Fig. 6), three phase stator currents are measured by current sensors. Failed sensors may lead to an invalid control signal to the three-phase inverter. A regulator is typically used between measurements and PWM generator [19]. B.
Here, SFDR module is responsible for the detection and rebuilding of the failed sensors, which guarantees the validity of the measurements. The validated currents are used by the Hysteresis Comparator in PWM generator module (Fig. 6). Conditions for PCT expansion are the same as in the previous example. In this example, the control of the induction machine is compared between using the measured current signals and the PCT estimated signals. Simulated results for rotor speed are reported for both cases in Fig. 7.
Upper2 Upper1 Original Output Rebuilded Data Lower1 Lower2
2 0 -2 -4 0.45
0.455
0.46
0.465
0.47
0.475
time (s) SFDR for Soft Failure
Current (A)
4 2 0 -2 -4
0.66
0.665
0.67
0.675
0.68
0.685
0.69
0.695
0.7
time (s)
Figure 6. FOCIM with PCT SFDR Figure 4. Diagnosis of SFDR of Current Sensor for Phase A
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
Rotor Speed 1200
[9] 1000
800
Speed (RPM)
[10] 600
[11]
400
200
[12]
0
Control With Measured Data Control With PCT Estimated Data 0.5
1
1.5
2
2.5
3
Time (s)
[13]
Figure 7. Comparison of Rotor Speed: measured data VS rebuilt data
As illustrated in Fig. 7, PCT estimator of the induction machine can rebuild the data even if all of the currents for the stator phases of induction machine fail, keeping the motor under control. Simulation result indicates though that this will lead to a significant delay in the motor start-up time. Compensation for this drawback is currently under investigation. V.
CONCLUSION AND FUTURE WORK
The work presented here introduces a new approach to diagnosis of sensor failure and information rebuilding. Numerical results of the two examples demonstrate a good performance of this approach and also call for further research in applying this approach to more complex systems in real-time hardware in loop application. REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Narayanan S., Vian J. L., Choi J.J., Marks R.J. II, EI-Sharkawi M.A., and Thompson B.B., “Missing sensor data restoration for vibration sensors on a jet aircraft engine,” Proceedings of the International Joint Conference on Neural Networks July 20-24, 2003 . NapoLitano M.R., Silvestri G., Windon D.A. II, Casanova J. L., Innocenti M., “Sensor validation using hardware-based on-line learning neural networks,” IEEE transactions on aerospace and electronic systems, vol. 34, NO. 2, pp.456-468, April 1998. Neppach, Charles Dominic, “Application of neural networks in sensor failure detection, identification, and accommodation in a system without sensor redundancy, ” West Virginia University, Master Thesis, 1994. Zidani, F., Diallo, D., Benbouzid, M.E.H., and Berthelot, E., “Diagnosis of Speed Sensor Failure in Induction Motor Drive,” IEEE International Electric Machines & Drives Conference, Volume 2, pp.1680 - 1684 , May, 2007. Napolitano M.R., Windon D. A. II, Casanova J. L., and Innocenti M., “Kalman Filters and Neural-Networks Schemes for Sensor Validation in Flight Contrl Systems,” IEEE transactions on control systems technology, vol. 6, NO. 5, pp.596-611, Sep. 1998. Del Gobbo D., Napolitano M. R., Famouri P., and Innocenti M.,“Experimental Application of Extended Kalman Filtering for Sensor Validation,” IEEE transactions on control systems technology, vol. 9, NO. 2, pp.376-380, March, 2001. Yan D., Zhang H., “New Method of Sensor Failure Detection and Correction in Flight Control Systems,” Proceedings of the IEEE International Conference on Industrial Technology , pp. 444 – 448, 5-9 Dec. 1994. Kai Rothenhagen, Sonke Thomsen, Friedrich W. Fuchs, “Voltage Sensor Fault Detection and Reconfiguration for a Doubly Fed Induction
[14]
[15] [16]
[17]
[18] [19]
[20]
Generator,” IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, pp. 377 – 382, 2007. Chee Pin Tan; Edwards, C., “A robust sensor fault reconstruction scheme using sliding mode observers applied to a nonlinear aero,” ACC : proceedings of the 2002 American Control Conference, pp. vol. 6, 5086 – 5091, May 8-10, 2002. Chee PinTan, Christopher Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, p.p. 1815 - 1821, 2002. Chee PinTan, Christopher Edwards, “Sliding mode observers for reconstruction of simutanenous actuator and sensor faults,” Proceedings of 42th IEEE Conference on Decision and Control , pp. 1456 – 1460, Dec. 2003. Hiskens, I.A., Pai, M.A., Nguyen, T.B, “Bounding uncertainty in power system dynamic simulations,” IEEE PES Winter Meeting, vol. 2, pp. 1533 - 1537, 23-27 Jan. 2000. Hiskens, I.A., Pai, M.A.,”Trajectory sensitivity analysis of hybrid systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, Issue 2, pp. 204-220, Feb. 2000. Monti, A., Ponci, F., Lovett, T., “A polynomial chaos theory approach to the control design of a power converter,” Power Electronics Specialists Conference, 2004, PESC 04. 2004 IEEE 35th Annual, Vol.6, pp. 4809 – 4813, 20-25 June 2004. Wiener, “The Homogeneous Chaos,” Amer.J.Math., vol 60, pp.897936,1938. Smith A. H. C., “Robust and Optimal Control Using Polynomial Chaos Theory,” PHD dissertation, Dept. of Electrical Engineering, University of South Carolina, 2007. Lovett, T.E., Ponci F., Monti A., “A Polynomial Chaos Approach to Measurement Uncertainty,” IEEE transactions on instrumentation and measurement, vol.55, NO.3, pp. 729-736, June 2006. Leonhard W., Control of Electrical Drives, Springer-Verlag, 2001, Berlin, 3rd edition. Universal Bridge, Field-Oriented Control Induction Motor Drive, http://www.mathworks.com/access/helpdesk/help/toolbox/physmod/pow ersys. Ned Mohan, “Advanced Electric Drives Analysis, Control and Modeling using Simulink”.2001.
APPENDIX 1: THIS PAPER
PARAMETERS
OF INDUCTION MACHINE USED IN
Nominal Voltage: Vn = 460V; Number of pole pairs P=4 (2 pairs); Rated Power Pn =2.4KW; Rated Frequency f=60Hz; Stator Resistance Rs =1.77 ohm; Nominal Rotor Resistance Rr0=1.34 ohm; Uncertainty on Rotor Resistance Rr1=75%Rr0; Stator Inductance Ls=0.3826 H; Rotor Inductance Lr =0.3808 H; Mutual Inductance Lm=0.3687 H; Rotor Inertia J=0.025 Kg.m^2, R Rotor Drag coefficient H=1e-4N.m.s (Test induction motor in [20]) Poles of Flux Observer: -100+700i, -100-700i, -900+100i, 900-100i. Poles of Controller: p1=-700; p2=-1000; p3= -10+50i; p4=-1050i.
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
APPENDIX 2: MATRCES PCT DOMAIN λ rpct
Arpct
Aipct
Brpct
OF INDUCTION MACHINE MODEL IN
⎡V sd 0 ⎤ ⎡i sd 0 ⎤ ⎡λ rd 0 ⎤ ⎢V ⎥ ⎢i ⎥ ⎢λ ⎥ rd 1 sd 1 = ⎢λ ⎥, i spct = ⎢i ⎥ , V spct = ⎢Vsd 1 ⎥ ⎢ sq 0 ⎥ ⎢ sq 0 ⎥ ⎢ rq 0 ⎥ ⎢V ⎥ ⎢i ⎥ ⎢λ ⎥ rq 1 sq 1 ⎣ sq1 ⎦ ⎣ ⎦ ⎦ ⎣
⎡ Rr 0 ⎢ − L r ⎢ ⎢ − Rr1 ⎢ Lr =⎢ ⎢ω r − ω e ⎢ ⎢ ⎢ 0 ⎣
R r1 3 Lr R − r0 Lr
−
0
ω r − ωe
ωe − ωr 0 R − r0 Lr Rr1 − Lr
Bi1 pct
⎤ ⎥ ⎥ ωe − ω r ⎥ ⎥ Rr1 ⎥ ⎥ − 3 Lr ⎥ Rr 0 ⎥ − ⎥ Lr ⎦ 0
Bi 2 pct
⎡ Lm R r 0 ⎢− σL L s r ⎢ ⎢ − Lm R r 1 ⎢ σLs Lr =⎢ ω L ⎢− e m ⎢ σLs Lr ⎢ ⎢ 0 ⎣
⎡ 1 ⎢σL ⎢ s ⎢ 0 ⎢ =⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣
Lm R r 1 3σLs Lr L R − m r0 σLs Lr
−
−
0 1
σLs 0 0
ω e Lm σLs Lr 0
0
−
ω e Lm σLs Lr
−
0 0 1
σLs 0
Lm Rr 0
σLs Lr
Lm Rr1
σLs Lr
⎤ ⎥ ⎥ ω e Lm ⎥ σLs Lr ⎥ L R ⎥ − m r1 ⎥ 3σLs Lr ⎥ L R ⎥ − m r0 ⎥ σLs Lr ⎦ 0
⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 1 ⎥ σLs ⎥⎦
(1 − σ ) Rr1 ⎡ Rs 1 − σ ⎤ − 0 ωe ⎢− ( σL + σL Rr 0 ) ⎥ 3σLr s r ⎢ ⎥ Rs 1 − σ ⎢ − (1 − σ ) Rr1 ⎥ −( + 0 ωe Rr 0 ) ⎢ ⎥ σLr σLs σLr =⎢ ⎥ R (1 − σ ) Rr1 1−σ ⎢ ⎥ − ωe −( s + − 0 Rr 0 ) 3σLr σLs σLr ⎢ ⎥ ⎢ ⎥ R (1 − σ ) Rr1 1−σ − ωe − −( s + 0 Rr 0 ) ⎥ ⎢ σ σ σ L L L r s r ⎣ ⎦
⎡ Lm R r 0 ⎢ L ⎢ r ⎢ Lm Rr1 ⎢ L =⎢ r ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣
Lm R r 1 3 Lr Lm Rr 0 Lr 0
ω r − ωe
0 0 Lm R r 0 Lr Lm R r 1 Lr
⎤ ⎥ ⎥ 0 ⎥ ⎥ Lm Rr 1 ⎥ ⎥ 3 Lr ⎥ Lm R r 0 ⎥ ⎥ Lr ⎦ 0
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
Diagnosis of Sensor Failure Detection and Information Rebuilding Using Polynomial Chaos Theory Huimin Li
Antonello Monti
Dept. of Electrical Engineering University of South Carolina, Columbia, SC, USA [email protected]
E.ON Energy Research Center RWTH Aachen University Aachen, Germany [email protected]
Abstract—A new strategy applying Polynomial Chaos Theory (PCT) to diagnose sensor failure is presented in this paper. Using PCT and uncertainty of parameters, the original state space model of a system is expanded to PCT domain. Variations of sensor output due to parameter uncertainty are calculated by a PCT estimator and presented as PCT output boundaries based on the probability density function (PDF) of uncertain parameters. Along with PCT boundaries, an algorithm is used to diagnose and declare failed sensors. Meanwhile, outputs of failed sensors are frozen and missing information is rebuilt with bestcase estimations from PCT estimator. Effectiveness of this method is also verified with two examples. Keywords- uncertain systems; induction motor drive; stochastic differential equations
I.
INTRODUCTION
Sensor Failure and Information Rebuilding (SFDR) is an extremely important research aspect in measurement and control. Much effort has been made in this area. In [1]-[5], Artificial Intelligence (AI) theory (such as Neural Network, one of the most popular options among AI) is applied to detect sensor failure and rebuild the information from failed sensors. Kalman filter [5], Extended Kalman Filter [6] are also favorite options in this area. CARMA model is used in [7]. On the other hand, observer-based method, for instance, Unknown Input Observer (UIO) is used in [8] and sliding mode observer is adopted in [9]-[11]. The basic ideas of all these methods are declaring sensor failure based on a pre-determined threshold and the residual between the normal data and the measurement data; rebuilding is achieved by replacing the failed data with model-based prediction. However, in many cases, indirect measurements are very sensitive to uncertainties during dynamic simulations. A proposal based on fuzzy logic is presented in [5] where the measurement uncertainty is taken into consideration in detecting sensor failure. In [12] and [13], parameter uncertainties are analyzed in simulating system dynamic trajectory. The approach proposed here is based on PCT. The nature of PCT is to approximate a random process by a complete and orthogonal polynomial basis in terms of certain random variables [14]. In the rest of the paper, the authors will show
how this approach makes it possible detecting the sensor failure by counting also for uncertainty propagations. The work is organized as following: a brief generic formulation of system model in PCT domain is introduced in section II, while the failure detection and information rebuilding algorithm are discussed in section III. Section IV shows a validation of this proposed idea through two examples and the simulation results are also portrayed and analyzed in section IV. Conclusions in section V complete the paper. II. GENERIC FORM OF PCT EXPANSION AND PCO PCT concept was first introduced by Wiener in 1938 as “Homogeneous Chaos” [15]. PCT expansion is a method that uses a polynomial based stochastic space to represent the evolution of the uncertainty propagation into the system [16]. By choosing certain polynomial base ψ, uncertain parameter X can be expanded into PCT domain as shown in equation (1). ∞
X (θ ) = ∑ a n (θ ) Ψn (ξ (θ )) n =0
Where:
θ
(1)
Random event;
a n The coefficients of the expansion; Ψn The selected polynomial basis;
ξ
Radom variables with a PDF according to the selected
Ψn . The dimension of the expansion is limited for practical purpose. The delimited dimension P is selected based on the number of independent uncertain parameters ( nv ) and the maximum order of the polynomial basis ( n p ). The infinite upper limit in equation (1) will be replaced by the selected dimension P as in equation (2):
P=
(nv + n p )! nv !n p !
−1
(2)
This work was supported by the US Office of Naval Research under the grant N00014-07-1-0603
978-1-4244-3353-7/09/$25.00 ©2009 IEEE Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
Dependency on the random variables ξ can then be eliminated by taking a Galerkin projection for the state space equation of the system. In this paper, only one uncertain parameter is considered, so equation (3) gives the method used in this paper for the projection: b
dx k = dt
∫ X (θ )Ψ dξ k
a
b
∫Ψ
2 k
failure). Last but not least, measurement uncertainty may also cause misjudging of the sensor failure, and this concerning is solved by introducing maximum tolerant band BW2. BW2 can be calculated from the nominal value of the sensor, the PCT tails and the standard of the sensor under detection. Fig. 1 illustrates the main process of diagnosing SFDR using PCT.
(3)
dξ
a
Projection method for multiple uncertain parameters along with PCT fundamental principles are analyzed in [16]. Meanwhile, detailed description of PCT application can be found in [14]. Solving state space equations of the system (after Galerkin Projection) yields to system state space model in PCT domain. The model is presented by using the uncertain variables as the state variables, and matrices (including parameter uncertainty) as the coefficients. The generic form of the system in PCT domain can be presented as in equation (4). . x& pct (t ) = A pct x pct (t ) + B pct u pct (t ) (4) y pct (t ) = C pct x(t ) + D pct u pct (t ) The system order in PCT domain is determined by a few of factors, such as the original system order, the number of the independent uncertainties considered during the expansion and the polynomial base used to perform the expansion. Quantitatively, equation (5) can be used to calculate the system order in PCT domain.
n pct =
(nv + n p )! nv !n p !
× ns
(5)
A detailed description of this expansion will be given in section IV by application examples. III.
SENSOR FAILURE DETECTION AND INFORMATION REBUILDING ALGORITHM In this section, an algorithm based on PCT and PDF of the uncertain parameters is designed to diagnose the system when performing SFDR. Boundaries in PCT domain, produced by PDF tails, are actually the plant’s output due to the parameter uncertainty. The idea is to exploit PCT to make the sensor diagnostic less sensitive to parameter uncertainty. It should be underlined, anyway, that the proposed algorithm could still fail if the system changes so dramatically that the model becomes unreliable. The work here focuses on examining the ability of PCT in SFDR, not considering the possibility of system failure and then assuming that the model is reliable. To apply this approach, the distributions of uncertain variables are assumed to be known. Then the normal operation range (BW1) for each sensor can be calculated from PCT. This range is a safe operation bands for a sensor. However, the following two circumstances may cause the real output exceeding this range: one is the sensor fails, the other is the model is not reliable anymore (mostly caused by system
Figure 1. Diagnosis Algorism for SFDR
The main problem then is to identify the variable boundaries starting from the PCT expansion. This issue can be solved by properly choosing a proper output matrix in PCT domain. Description of this process is illustrated in the next section with the help of examples. IV.
APPLICATION OF THE APPROACH TO SFDR FOR INDUCTION MACHINE
In this section, two examples are introduced. The first example refers to an induction machine. The purpose is to test the proposed method in diagnosing a stator phase current sensor failure and rebuilding the missing data when there are sensor failures. The main uncertainty is assumed to be the rotor resistance. Then, in the second example, PCT based SFDR method is applied to a failure-tolerant field-oriented control of an induction motor (FOCIM). A. Example 1 Modeling of Induction Machine in PCT domain is a key to conduct SFDR with the proposed approach. The model of induction machine in PCT domain is established based on the uncertain rotor resistance. As mentioned in [16], the distribution of rotor resistance is assumed to be known a priori. For this specific example, a uniform distribution is adopted but different choices could be done without affecting the value of the algorithm. Legendre polynomial basis is selected to expand variables into PCT domain [17]. TABLE I gives the first three bases for Legendre. TABLE I.
LEGENDRE BASIS FOR ONE-DIMENSIONAL BASIS
First term
Second term
Third term
Ψ0 =1
Ψ1 =ξ
Ψ2 =(3ξ2-1)/2
Parameters needed to perform PCT expansion of an induction machine are summarized in TABLE II:
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
TABLE II. PCT Expansion Parameter Calculation Description
Symbol and Value Symbol
Value
Number of the uncertain variable
nv
1
Maximum Order of Polynomial Basis
np
1
Order of the original system
ns
2
Number of terms for each variable after expanding
P
2
n pct
8
OB pct
4
Order of the system in PCT Domain Order of the Flux Observer in PCT domain
n
C ipct
1 2 3 4 5 6 7 8
(6)
The second step is to perform Galerkin Projection of the original system to remove the dependency on the random variable as in (7). 1
i&sdk =
P
∫ (∑ i
−1 n =0
sdn
Φ n )Φ k dξ
(7)
1
RULES OF SFDR
BW1
BW2
Sensor Status
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
NF NA NF F NA NA NF NF
BW1 is the range delimited by PCT boundaries. BW2 is the maximum range for the non-failure operation of a sensor. “e” is the difference of the rotor mechanical speed between two time steps. “NF” means no failure; “NA” means circumstance is not reasonable in practice or case is not under consideration in this paper; “F” means sensor fails. For “BW1” and “BW2”, “0” means within the range and “1” means out of the range; For “e”, “0” means very low (not more than 2%) and induction machine runs in steady state and “1” means very large (larger than 2%) and occurs during machine startup and shutdown or speed sensor fails.
Simulation results of this example are given in the following along with analysis of various cases. Firstly, PCT diagnosis boundaries (BW1) and sensor fault tolerance band (BW2) are given in Fig. 2 and Fig. 3 respectively. PCT Boundaries (BW1)
2 ∫ Φ k dξ
3
−1
Upper1 Mean Lower1
2
(8)
1
Current (A)
Then, the complete induction machine model in PCT domain can be calculated as in equation (8).
λ&r = Arpct λrpct + Brpct ispct
1
(9)
e
1
i&s = Aipct ispct + Bi1 pct λrpct + Bi 2 pctVspct
0 1
Conditions
Case
P
All the variables that depend on rotor resistance should also be expanded [14].
0
TABLE III.
The first step to perform PCT expansion of a system is to expand the uncertain variables (rotor resistance) to PCT domain, as in equation (6).
n =0
0⎤ 0 ⎥⎥ 0⎥ ⎥ 1⎥ 0⎥ ⎥ − 1⎦⎥
0
TABLE III illustrates the rules for the failure detection of current sensors in this application.
The original model adopted for the induction machine is a standard two-axis representation [18]. We assumed that the main source of uncertainty is given by the variation of the rotor resistance.
Rr = ∑ Rrn Φ n
⎡1 1 ⎢0 1 ⎢ ⎢1 − 1 =⎢ ⎢0 0 ⎢0 0 ⎢ ⎣⎢0 0
0
-1
-2
Values for all matrices in equation (8) are given in Appendix 1 and Appendix 2. As mentioned in section III, PCT boundaries can be calculated by properly choosing the output matrix in PCT domain. In this application, PDF of uncertain parameter is uniform with Legendre polynomial basis selected, so output matrix can be chosen as shown in equation (9) to find stator current boundaries in d − q domain. Three phase boundaries can then be solved with axes transformation theory.
-3 0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
time (s)
Figure 2. PCT boundaries (BW1) for Sensor of Phase A
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
SFDR for Hard Failure
Sensor Fault Tolerance Band (BW2) Upper2 data2 Lower2
3
6
2
4
Current (A)
1
Current (A)
Original Output Rebuilded Data
8
0 -1
2 0 -2 -4
-2
-6
-3
-8
0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
0.44
time (s)
0.48
0.5
0.52
0.54
0.56
time (s)
Figure 5. SFDR for Hard Failure of Current Sensor for Phase A
Figure 3. Maximum Tolerance Boundaries (BW2) for Sensor of Phase A
In TABLE III, there are eight cases. Case 8 indicates that the induction machine is in startup or shutdown period (e is 1). In this period, currents change dramatically and SFDR module is not executed. Cases 2 and 6 are not reasonable (Exceeding BW2 means also exceeding BW1). Moreover, cases 5 and 7 are excluded from consideration in this paper since they indicate speed sensor failure (speed sensor is assumed to be working correctly). Case 1 is the normal performance of the system. It indicates that the induction machine is in steady state (e is zero), and no current fail is detected (status of BW1 and BW2 are all zero). In this case, the SFDR module will operate but the output of the real sensors will be used. Case 3 and 4 are the most important. The former shows that the machine operates in steady state (e is zero), the output of the phase A sensor is out of BW1 while stays in BW2. In this situation, SFDR will continuously monitor the sensor output other than declaring a failure immediately till the original measurement is out of BW2. Case 4 is a little bit different: the output of the sensor exceeds both BW1 and BW2. SFDR module declares a failure and restores the failed measurement simultaneously. Simulation results of case 3 and 4 are given in Fig. 4 and Fig. 5 respectively. Diagnosis of SFDR 4
Current (A)
0.46
In Fig. 4, operation from 0.45 to 0.475 second is an example of case 3. Beginning from 0.573 second, we enter in case 4: SFDR declares a failure, freezes the failed sensor and rebuilds sensor output as portrayed with the dashed black line. Basically, it simulates a soft failure [3]. In Fig. 5, the sensor is hard failed at 0.5 second. The original data (red line) is accommodated by the PCT best-case estimation (dashed black line). Example 2: Fault-Tolerant FOC Control of Induction Machine As portrayed in the following FOCIM schematic (Fig. 6), three phase stator currents are measured by current sensors. Failed sensors may lead to an invalid control signal to the three-phase inverter. A regulator is typically used between measurements and PWM generator [19]. B.
Here, SFDR module is responsible for the detection and rebuilding of the failed sensors, which guarantees the validity of the measurements. The validated currents are used by the Hysteresis Comparator in PWM generator module (Fig. 6). Conditions for PCT expansion are the same as in the previous example. In this example, the control of the induction machine is compared between using the measured current signals and the PCT estimated signals. Simulated results for rotor speed are reported for both cases in Fig. 7.
Upper2 Upper1 Original Output Rebuilded Data Lower1 Lower2
2 0 -2 -4 0.45
0.455
0.46
0.465
0.47
0.475
time (s) SFDR for Soft Failure
Current (A)
4 2 0 -2 -4
0.66
0.665
0.67
0.675
0.68
0.685
0.69
0.695
0.7
time (s)
Figure 6. FOCIM with PCT SFDR Figure 4. Diagnosis of SFDR of Current Sensor for Phase A
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
Rotor Speed 1200
[9] 1000
800
Speed (RPM)
[10] 600
[11]
400
200
[12]
0
Control With Measured Data Control With PCT Estimated Data 0.5
1
1.5
2
2.5
3
Time (s)
[13]
Figure 7. Comparison of Rotor Speed: measured data VS rebuilt data
As illustrated in Fig. 7, PCT estimator of the induction machine can rebuild the data even if all of the currents for the stator phases of induction machine fail, keeping the motor under control. Simulation result indicates though that this will lead to a significant delay in the motor start-up time. Compensation for this drawback is currently under investigation. V.
CONCLUSION AND FUTURE WORK
The work presented here introduces a new approach to diagnosis of sensor failure and information rebuilding. Numerical results of the two examples demonstrate a good performance of this approach and also call for further research in applying this approach to more complex systems in real-time hardware in loop application. REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Narayanan S., Vian J. L., Choi J.J., Marks R.J. II, EI-Sharkawi M.A., and Thompson B.B., “Missing sensor data restoration for vibration sensors on a jet aircraft engine,” Proceedings of the International Joint Conference on Neural Networks July 20-24, 2003 . NapoLitano M.R., Silvestri G., Windon D.A. II, Casanova J. L., Innocenti M., “Sensor validation using hardware-based on-line learning neural networks,” IEEE transactions on aerospace and electronic systems, vol. 34, NO. 2, pp.456-468, April 1998. Neppach, Charles Dominic, “Application of neural networks in sensor failure detection, identification, and accommodation in a system without sensor redundancy, ” West Virginia University, Master Thesis, 1994. Zidani, F., Diallo, D., Benbouzid, M.E.H., and Berthelot, E., “Diagnosis of Speed Sensor Failure in Induction Motor Drive,” IEEE International Electric Machines & Drives Conference, Volume 2, pp.1680 - 1684 , May, 2007. Napolitano M.R., Windon D. A. II, Casanova J. L., and Innocenti M., “Kalman Filters and Neural-Networks Schemes for Sensor Validation in Flight Contrl Systems,” IEEE transactions on control systems technology, vol. 6, NO. 5, pp.596-611, Sep. 1998. Del Gobbo D., Napolitano M. R., Famouri P., and Innocenti M.,“Experimental Application of Extended Kalman Filtering for Sensor Validation,” IEEE transactions on control systems technology, vol. 9, NO. 2, pp.376-380, March, 2001. Yan D., Zhang H., “New Method of Sensor Failure Detection and Correction in Flight Control Systems,” Proceedings of the IEEE International Conference on Industrial Technology , pp. 444 – 448, 5-9 Dec. 1994. Kai Rothenhagen, Sonke Thomsen, Friedrich W. Fuchs, “Voltage Sensor Fault Detection and Reconfiguration for a Doubly Fed Induction
[14]
[15] [16]
[17]
[18] [19]
[20]
Generator,” IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, pp. 377 – 382, 2007. Chee Pin Tan; Edwards, C., “A robust sensor fault reconstruction scheme using sliding mode observers applied to a nonlinear aero,” ACC : proceedings of the 2002 American Control Conference, pp. vol. 6, 5086 – 5091, May 8-10, 2002. Chee PinTan, Christopher Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, p.p. 1815 - 1821, 2002. Chee PinTan, Christopher Edwards, “Sliding mode observers for reconstruction of simutanenous actuator and sensor faults,” Proceedings of 42th IEEE Conference on Decision and Control , pp. 1456 – 1460, Dec. 2003. Hiskens, I.A., Pai, M.A., Nguyen, T.B, “Bounding uncertainty in power system dynamic simulations,” IEEE PES Winter Meeting, vol. 2, pp. 1533 - 1537, 23-27 Jan. 2000. Hiskens, I.A., Pai, M.A.,”Trajectory sensitivity analysis of hybrid systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, Issue 2, pp. 204-220, Feb. 2000. Monti, A., Ponci, F., Lovett, T., “A polynomial chaos theory approach to the control design of a power converter,” Power Electronics Specialists Conference, 2004, PESC 04. 2004 IEEE 35th Annual, Vol.6, pp. 4809 – 4813, 20-25 June 2004. Wiener, “The Homogeneous Chaos,” Amer.J.Math., vol 60, pp.897936,1938. Smith A. H. C., “Robust and Optimal Control Using Polynomial Chaos Theory,” PHD dissertation, Dept. of Electrical Engineering, University of South Carolina, 2007. Lovett, T.E., Ponci F., Monti A., “A Polynomial Chaos Approach to Measurement Uncertainty,” IEEE transactions on instrumentation and measurement, vol.55, NO.3, pp. 729-736, June 2006. Leonhard W., Control of Electrical Drives, Springer-Verlag, 2001, Berlin, 3rd edition. Universal Bridge, Field-Oriented Control Induction Motor Drive, http://www.mathworks.com/access/helpdesk/help/toolbox/physmod/pow ersys. Ned Mohan, “Advanced Electric Drives Analysis, Control and Modeling using Simulink”.2001.
APPENDIX 1: THIS PAPER
PARAMETERS
OF INDUCTION MACHINE USED IN
Nominal Voltage: Vn = 460V; Number of pole pairs P=4 (2 pairs); Rated Power Pn =2.4KW; Rated Frequency f=60Hz; Stator Resistance Rs =1.77 ohm; Nominal Rotor Resistance Rr0=1.34 ohm; Uncertainty on Rotor Resistance Rr1=75%Rr0; Stator Inductance Ls=0.3826 H; Rotor Inductance Lr =0.3808 H; Mutual Inductance Lm=0.3687 H; Rotor Inertia J=0.025 Kg.m^2, R Rotor Drag coefficient H=1e-4N.m.s (Test induction motor in [20]) Poles of Flux Observer: -100+700i, -100-700i, -900+100i, 900-100i. Poles of Controller: p1=-700; p2=-1000; p3= -10+50i; p4=-1050i.
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
APPENDIX 2: MATRCES PCT DOMAIN λ rpct
Arpct
Aipct
Brpct
OF INDUCTION MACHINE MODEL IN
⎡V sd 0 ⎤ ⎡i sd 0 ⎤ ⎡λ rd 0 ⎤ ⎢V ⎥ ⎢i ⎥ ⎢λ ⎥ rd 1 sd 1 = ⎢λ ⎥, i spct = ⎢i ⎥ , V spct = ⎢Vsd 1 ⎥ ⎢ sq 0 ⎥ ⎢ sq 0 ⎥ ⎢ rq 0 ⎥ ⎢V ⎥ ⎢i ⎥ ⎢λ ⎥ rq 1 sq 1 ⎣ sq1 ⎦ ⎣ ⎦ ⎦ ⎣
⎡ Rr 0 ⎢ − L r ⎢ ⎢ − Rr1 ⎢ Lr =⎢ ⎢ω r − ω e ⎢ ⎢ ⎢ 0 ⎣
R r1 3 Lr R − r0 Lr
−
0
ω r − ωe
ωe − ωr 0 R − r0 Lr Rr1 − Lr
Bi1 pct
⎤ ⎥ ⎥ ωe − ω r ⎥ ⎥ Rr1 ⎥ ⎥ − 3 Lr ⎥ Rr 0 ⎥ − ⎥ Lr ⎦ 0
Bi 2 pct
⎡ Lm R r 0 ⎢− σL L s r ⎢ ⎢ − Lm R r 1 ⎢ σLs Lr =⎢ ω L ⎢− e m ⎢ σLs Lr ⎢ ⎢ 0 ⎣
⎡ 1 ⎢σL ⎢ s ⎢ 0 ⎢ =⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣
Lm R r 1 3σLs Lr L R − m r0 σLs Lr
−
−
0 1
σLs 0 0
ω e Lm σLs Lr 0
0
−
ω e Lm σLs Lr
−
0 0 1
σLs 0
Lm Rr 0
σLs Lr
Lm Rr1
σLs Lr
⎤ ⎥ ⎥ ω e Lm ⎥ σLs Lr ⎥ L R ⎥ − m r1 ⎥ 3σLs Lr ⎥ L R ⎥ − m r0 ⎥ σLs Lr ⎦ 0
⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 1 ⎥ σLs ⎥⎦
(1 − σ ) Rr1 ⎡ Rs 1 − σ ⎤ − 0 ωe ⎢− ( σL + σL Rr 0 ) ⎥ 3σLr s r ⎢ ⎥ Rs 1 − σ ⎢ − (1 − σ ) Rr1 ⎥ −( + 0 ωe Rr 0 ) ⎢ ⎥ σLr σLs σLr =⎢ ⎥ R (1 − σ ) Rr1 1−σ ⎢ ⎥ − ωe −( s + − 0 Rr 0 ) 3σLr σLs σLr ⎢ ⎥ ⎢ ⎥ R (1 − σ ) Rr1 1−σ − ωe − −( s + 0 Rr 0 ) ⎥ ⎢ σ σ σ L L L r s r ⎣ ⎦
⎡ Lm R r 0 ⎢ L ⎢ r ⎢ Lm Rr1 ⎢ L =⎢ r ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣
Lm R r 1 3 Lr Lm Rr 0 Lr 0
ω r − ωe
0 0 Lm R r 0 Lr Lm R r 1 Lr
⎤ ⎥ ⎥ 0 ⎥ ⎥ Lm Rr 1 ⎥ ⎥ 3 Lr ⎥ Lm R r 0 ⎥ ⎥ Lr ⎦ 0
Authorized licensed use limited to: North Carolina State University. Downloaded on July 15,2010 at 17:46:58 UTC from IEEE Xplore. Restrictions apply.
