American Institute of Mathematical Sciences

Advanced
2014, 4(2): 151-179. doi: 10.3934/naco.2014.4.151

Auxiliary signal design for failure detection in differential-algebraic equations

Jason R. Scott 1, and  Stephen Campbell 1,

1. 

Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205, United States, United States

Received  January 2014 Revised  April 2014 Published  May 2014

Fault detection and identification (FDI) are important tasks in most modern industrial and mechanical systems and processes. Many of these systems are most naturally modeled by differential-algebraic equations (DAE). This paper addresses active fault detection in DAE. A technique is presented to calculate an auxiliary test signal guaranteeing detection, assuming bounded additive noise. An efficient real time detection algorithm is also provided as are example simulations. The extension to model uncertainty is discussed.
Keywords: Active fault detection, DAE., auxiliary signal design, differential-algebraic equations, failure detection.
Mathematics Subject Classification: Primary: 49N90; Secondary: 49J21, 34A0.
Citation: Jason R. Scott, Stephen Campbell. Auxiliary signal design for failure detection in differential-algebraic equations. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 151-179. doi: 10.3934/naco.2014.4.151
References:
[1]

I. Andjelkovic, K. A. Sweetingham and S. L. Campbell, Active fault detection in nonlinear systems using auxiliary signals, in American Control Conference, (2008), 2142-2147.

[2]

I. Andjelkovic and S. L. Campbell, Direct optimization determination of auxiliary test signals for linear problems with model uncertainty, in 50th IEEE CDC-ECC, (2011), 909-914.

[3]

R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.

[4]

G. Besançon, I. Rubio-Scola and D. Georges, Input selection in observer design for non-uniformly observable systems, in 9th IFAC Symposium on Nonlinear Control Systems, (2013).

[5]

K. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996.

[6]

A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere, New York, 1975.

[7]

S. L. Campbell and R. Nikoukhah, Auxiliary Signal Design for Failure Detection, Princeton University Press, Princeton, New Jersey, 2004.

[8]

S. L. Campbell, Least squares completions for nonlinear differential algebraic equations, Numerical Mathematics, 65 (1993), 77-94. doi: 10.1007/BF01385741.

[9]

D. Choe, S. L. Campbell and R. Nikoukhah, A comparison of optimal and suboptimal auxiliary signal design approaches, in IEEE Conference on Control Applications, (2005).

[10]

D. Garg, M. A. Patterson, W. W. Hager, A. V. Rao, D. A. Benson and G. T. Huntington, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843-1851. doi: 10.1016/j.automatica.2010.06.048.

[11]

D. Garg, W. W. Hager and A. V. Rao, Pseudospectral methods for solving infinite-horizon optimal control problems, Automatica, 47 (2011), 829-837. doi: 10.1016/j.automatica.2011.01.085.

[12]

D. Garg, M. A. Patterson, C. L. Darby, C. Francolin, G. T. Huntington, W. W. Hager and A. V. Rao, Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems via a radau pseudospectral method, Computational Optimization and Applications, 49 (2011), 335-358. doi: 10.1007/s10589-009-9291-0.

[13]

M. Gerdin, T. Glad and L. Ljung, Parameter estimation in linear differential-algebraic equations, in 13th IFAC Symposium on System Identification, 2003.

[14]

M. Gerdts, Parameter identification in higher DAE systems, Technical Report, Department of Mathematics, Universität Hamburg, 2005.

[15]

R. Isermann, Fault Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance, Springer, Berlin, Germany, 2006.

[16]

R. Kircheis and S. Körkel, Parameter estimation for DAE models in a multiple experiment context, 82nd Annual Meeting of the International Association of Applied Mathematics and Mechanics, 11 (2011), 715-716.

[17]

H. H. Niemann, Active fault diagnosis in closed-loop uncertain systems, in 6th IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, (2006), 587-592.

[18]

H. H. Niemann, A setup for active fault diagnosis, IEEE Transactions on Automatic Control, 51 (2006), 1572-1578. doi: 10.1109/TAC.2006.878724.

[19]

M. A. Patterson and A. V. Rao, Exploiting sparsity in direct collocation pseudospectral methods for solving continuous-time optimal control problems, Journal of Spacecraft and Rockets, 49 (2012), 364-377.

[20]

R. J. Patton, P. M. Frank and R. N. Clark, Issues of Fault Diagnosis for Dynamic Systems, Springer, Berlin, Germany, 2000.

[21]

N. K. Poulsen and H. H. Niemann, Active fault diagnosis-a stochastic approach, in 7th IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, 2009.

[22]

I. Okay, S. L. Campbell and P. Kunkel, Completions of implicitly defined time varying vector fields, Linear Algebra and its Applications, 431 (2009), 1422-1438. doi: 10.1016/j.laa.2009.05.006.

[23]

I. Rubio-Scola, G. Besançon and D. Georges, Online observability optimization for state affine systems with output injection and observer design, in 21st IEEE Mediterranean Conference on Control and Automation, 2013.

[24]

I. Rubio-Scola, G. Besançon and D. Georges, Input optimization for observability of state affine systems, in 5th IFAC Symposium on System Structure and Control, 2013.

show all references

References:
[1]

I. Andjelkovic, K. A. Sweetingham and S. L. Campbell, Active fault detection in nonlinear systems using auxiliary signals, in American Control Conference, (2008), 2142-2147.

[2]

I. Andjelkovic and S. L. Campbell, Direct optimization determination of auxiliary test signals for linear problems with model uncertainty, in 50th IEEE CDC-ECC, (2011), 909-914.

[3]

R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.

[4]

G. Besançon, I. Rubio-Scola and D. Georges, Input selection in observer design for non-uniformly observable systems, in 9th IFAC Symposium on Nonlinear Control Systems, (2013).

[5]

K. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996.

[6]

A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere, New York, 1975.

[7]

S. L. Campbell and R. Nikoukhah, Auxiliary Signal Design for Failure Detection, Princeton University Press, Princeton, New Jersey, 2004.

[8]

S. L. Campbell, Least squares completions for nonlinear differential algebraic equations, Numerical Mathematics, 65 (1993), 77-94. doi: 10.1007/BF01385741.

[9]

D. Choe, S. L. Campbell and R. Nikoukhah, A comparison of optimal and suboptimal auxiliary signal design approaches, in IEEE Conference on Control Applications, (2005).

[10]

D. Garg, M. A. Patterson, W. W. Hager, A. V. Rao, D. A. Benson and G. T. Huntington, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843-1851. doi: 10.1016/j.automatica.2010.06.048.

[11]

D. Garg, W. W. Hager and A. V. Rao, Pseudospectral methods for solving infinite-horizon optimal control problems, Automatica, 47 (2011), 829-837. doi: 10.1016/j.automatica.2011.01.085.

[12]

D. Garg, M. A. Patterson, C. L. Darby, C. Francolin, G. T. Huntington, W. W. Hager and A. V. Rao, Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems via a radau pseudospectral method, Computational Optimization and Applications, 49 (2011), 335-358. doi: 10.1007/s10589-009-9291-0.

[13]

M. Gerdin, T. Glad and L. Ljung, Parameter estimation in linear differential-algebraic equations, in 13th IFAC Symposium on System Identification, 2003.

[14]

M. Gerdts, Parameter identification in higher DAE systems, Technical Report, Department of Mathematics, Universität Hamburg, 2005.

[15]

R. Isermann, Fault Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance, Springer, Berlin, Germany, 2006.

[16]

R. Kircheis and S. Körkel, Parameter estimation for DAE models in a multiple experiment context, 82nd Annual Meeting of the International Association of Applied Mathematics and Mechanics, 11 (2011), 715-716.

[17]

H. H. Niemann, Active fault diagnosis in closed-loop uncertain systems, in 6th IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, (2006), 587-592.

[18]

H. H. Niemann, A setup for active fault diagnosis, IEEE Transactions on Automatic Control, 51 (2006), 1572-1578. doi: 10.1109/TAC.2006.878724.

[19]

M. A. Patterson and A. V. Rao, Exploiting sparsity in direct collocation pseudospectral methods for solving continuous-time optimal control problems, Journal of Spacecraft and Rockets, 49 (2012), 364-377.

[20]

R. J. Patton, P. M. Frank and R. N. Clark, Issues of Fault Diagnosis for Dynamic Systems, Springer, Berlin, Germany, 2000.

[21]

N. K. Poulsen and H. H. Niemann, Active fault diagnosis-a stochastic approach, in 7th IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, 2009.

[22]

I. Okay, S. L. Campbell and P. Kunkel, Completions of implicitly defined time varying vector fields, Linear Algebra and its Applications, 431 (2009), 1422-1438. doi: 10.1016/j.laa.2009.05.006.

[23]

I. Rubio-Scola, G. Besançon and D. Georges, Online observability optimization for state affine systems with output injection and observer design, in 21st IEEE Mediterranean Conference on Control and Automation, 2013.

[24]

I. Rubio-Scola, G. Besançon and D. Georges, Input optimization for observability of state affine systems, in 5th IFAC Symposium on System Structure and Control, 2013.

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