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 & 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. Google Scholar

[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. Google Scholar

[3]

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

[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). Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[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). Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

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