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The Flexibility of DAE Formulations

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Surveys in Differential-Algebraic Equations III

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

There has been extensive research on DAEs and their applications. One major reason given for the usefulness of DAEs is that they are the initial way that many complex systems are most naturally modeled. But there are other ways that DAE formulations are useful. This survey focuses on a number of problems where the extra flexibility of a DAE formulation permits the solution of a problem that would be hard to solve otherwise.

MSC: 34A09, 65L80, 93B07, 49J15, 34A40

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References

  1. Aplevich, J.D.: Implicit Linear Systems. Lecture Notes in Control and Information Sciences, vol. 152. Springer, Berlin (1991)

    Google Scholar 

  2. Baumann, W.T.: Feedback control of multi-input nonlinear systems by extended linearization. IEEE Trans. Autom. Control 33, 40–46 (1988)

    Article  MathSciNet  Google Scholar 

  3. Becerra, V.M.: \(\mathcal{P}\mathcal{S}\mathcal{O}\mathcal{P}\mathcal{T}\) Optimal Control Solver User Manual Release 2 build, Reading RG6 6AY, University of Reading School of Systems Engineering United Kingdom (2010)

    Google Scholar 

  4. Bellena, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2013)

    Google Scholar 

  5. Berger, T., Reis, T.: Controllability of linear differential-algebraic systems - a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations I, DAE Forum, pp. 1–62. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  6. Betts, J.T.: Parametric Tool Path Trajectory Optimization, Technical Report SSGTECH-98-006, The Boeing Company (1998)

    Google Scholar 

  7. Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd edn. SIAM, Philadelphia (2010)

    Book  MATH  Google Scholar 

  8. Betts, J.T., Biehn, N., Campbell, S.L.: Convergence of nonconvergent IRK discretizations of optimal control problems with state inequality constraints. SIAM J. Sci. Comput. 23, 1981–2007 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Betts, J.T., Campbell, S.L., Engelsone, A.: Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice. Optim. Eng. 8, 1–19 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Betts, J.T., Campbell, S.L., Thompson, K.C.: Direct transcription solution of optimal control problems with control delays. In: Numerical Analysis and Applied Mathematics ICNAAM2011. AIP Conference Proceedings, Halkidiki, vol. 138, pp. 38–41 (2011)

    Google Scholar 

  11. Betts, J.T., Campbell, S.L., Thompson, K.C.: Optimal control software for constrained nonlinear systems with delays. In: Proceedings of 2011 IEEE Multi Conference on Systems and Control, Denver, pp. 444–449 (2011)

    Google Scholar 

  12. Betts, J.T., Campbell, S.L., Thompson, K.C.: Optimal control of a delay partial differential equation. In: Biegler, L., Campbell, S., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints, pp. 213–232. SIAM, Philadelphia (2012)

    Chapter  Google Scholar 

  13. Betts, J.T., Campbell, S.L., Thompson, K.C.: Simulation and optimization in systems with delays. In: Society for Modeling and Simulation Series 2013 Proceedings, San Diego, pp. 1084–1085 (2013)

    Google Scholar 

  14. Betts, J.T., Campbell, S.L., Thompson, K.C.: Direct transcription solution of optimal control problems with differential algebraic equations with delays. In: Proceedings of 14th IASTED International Symposium on Intelligent Systems and Control (ISC 2013), Marina del Rey, pp. 166–173 (2013)

    Google Scholar 

  15. Betts, J.T., Campbell, S.L., Thompson, K.C.: Solving optimal control problems with control delays using direct transcription, preprint (2015)

    Google Scholar 

  16. Betts, J.T., Campbell, S.L., Thompson, K.C.: LobattoIIIA methods, direct transcription, and DAEs with delays. Numer. Algorithms 69, 291–300 (2015)

    Article  MathSciNet  Google Scholar 

  17. Biehn, N., Campbell, S.L., Nikoukhah, R., Delebecque, F.: Numerically constructible observers for linear time-varying descriptor systems. Automatica 37, 445–452 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Birk, J., Zeitz, M.: Extended Luenberger observer for non-linear multivariable systems. Int. J. Control 47, 1823–1836 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Blajer, W.: Index of differential-algebraic equations governing the dynamics of constrained mechanical systems. Appl. Math. Model. 16, 70–77 (1992)

    Article  MATH  Google Scholar 

  20. Bobinyec, K., Campbell, S.L., Kunkel, P.: Maximally reduced observers for linear time varying DAEs. In: Proceedings of IEEE Multiconference on Systems and Control, Denver, pp. 1373–1378 (2011)

    Google Scholar 

  21. Bodestedt, M., Tischendorf, C.: PDAE models of integrated circuits and perturbation analysis. Math. Comput. Model. Dyn. Syst. 13, 1–17 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  23. Campbell, S.L.: Least squares completions for nonlinear differential algebraic equations. Numer. Math. 65, 77–94 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Campbell, S.L., Bobinyec, K.: Surveys in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum, Springer, 1–67 (2014)

    Google Scholar 

  25. Campbell, S.L., Holte, L.E.: Eigenvalue placement in completions of DAEs. Electron. J. Linear Algebra 26, 520–534 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Campbell, S.L., Kunkel, P.: Completions of nonlinear DAE flows based on index reduction techniques and their stabilization. J. Comput. Appl. Math. 233, 1021–1034 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Campbell, S.L., März, R.: Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations. J. Comput. Appl. Math. 202, 186–202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  29. Campbell, S.L., Nikoukhah, R.: Auxiliary Signal Design for Failure Detection. Princeton University Press, Princeton (2004)

    MATH  Google Scholar 

  30. Campbell, S.L., Delebecque, F., Nikoukhah, R.: Observer design for linear time varying descriptor systems. In: Proceedings of Control Industrial Systems (CIS97), Belfort, pp. 507–512 (1997)

    Google Scholar 

  31. Campbell, S.L., Biehn, N., Jay, L., Westbrook, T.: Some comments on DAE theory for IRK methods and trajectory optimization. J. Comput. Appl. Math. 120, 109–131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Cloosterman, M., van de Wouw, N., Heemels, W., Nijmeijer, H.: Stability of networked control systems with uncertain time-varying delays. IEEE Trans. Autom. Control 54, 1575–1580 (2009)

    Article  Google Scholar 

  33. Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Science, vol. 118. Springer, Berlin (1989)

    Google Scholar 

  34. Darby, C.L., Hager, W.W., Rao, A.V.: An hp-adaptive pseudospectral method for solving optimal control problems. Optim. Control Appl. Methods 32, 476–502 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Darouach, M.: Functional observers for linear descriptor systems. In: Proceedings of 17th Mediterranean Conference on Control & Automation, Thessaloniki, pp. 1535–1539 (2009)

    Google Scholar 

  36. Darouach, M., Benzaouia, A.: Constrained observer based control for linear singular systems. In: Proceedings of 18th Mediterranean Conference on Control & Automation, Marrakech, pp. 29–33 (2010)

    Google Scholar 

  37. Darouach, M., Boutat-Baddas, L.: Observers for a class of nonlinear singular systems. IEEE Trans. Autom. Control 53, 2627–2633 (2008)

    Article  MathSciNet  Google Scholar 

  38. Darouach, M., Boutayeb, M.: Design of observers for descriptor systems. IEEE Trans. Autom. Control 40, 1323–1327 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  39. Darouach, M., Zasadzinski, M., Hayar, M.: Reduced-order observer design for descriptor systems with unknown inputs. IEEE Trans. Autom. Control 41, 1068–1072 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  40. El-Tohami, M., Lovass-Nagy, V., Mukundan, R.: On the design of observers for generalized state space systems using singular value decomposition. Int. J. Control 38, 673–683 (1983)

    Article  MATH  Google Scholar 

  41. Engelsone, A., Campbell, S.L., Betts, J.T.: Direct transcription solution of higher-index optimal control problems and the virtual index. Appl. Numer. Math. 57, 281–296 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Farrell, K., Grove, E.A., Ladas, G.: Neutral delay differential equations with positive and negative coefficients. Appl. Anal. 27, 182–197 (1988)

    Article  MathSciNet  Google Scholar 

  43. Ford, N.J., Lumb, P.M.: Mixed-type functional differential equations: a numerical approach. J. Comput. Appl. Math. 229, 471–479 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  44. Gao, Z., Wang, H.: Descriptor observer approaches for multivariable systems with measurement noises and application in fault detection and diagnosis. Syst. Control Lett. 55, 304–313 (2006)

    Article  MATH  Google Scholar 

  45. Gill, P.E., Murray, W., Saunders, M.A.: User’s Guide for SNOPT 7: A Fortran Package for Large-Scale Nonlinear Programming. Systems Optimization Laboratory, Stanford University, CA, 9, pp. 4305–4023 (2007)

    Google Scholar 

  46. Hadeler, K.P.: Neutral delay equations from and for population dynamics. Electron. J. Qual. Theory Differ. Equ. 11, 1–18 (2008)

    Google Scholar 

  47. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Heidelberg (1996)

    Book  MATH  Google Scholar 

  48. He, Y., Liu, G., Rees, D., Wu, M.: Stability analysis for neural networks with time-varying interval delay. IEEE Trans. Neural Netw. 18, 1850–1854 (2007)

    Article  Google Scholar 

  49. Hou, M., Müller, P.C.: Observer design for descriptor systems. IEEE Trans. Autom. Control 44, 164–168 (1999)

    Article  MATH  Google Scholar 

  50. Ilchmann, A., Mehrmann, V.: A behavioral approach to time-varying linear systems. Part 2: descriptor systems. SIAM J. Control Optim. 44, 1748–1765 (2005)

    Google Scholar 

  51. Ionescu, C., Hodrea, R., De Keyser, R.: Variable time-delay estimation for anesthesia control during intensive care. IEEE Trans. Biomed. Eng. 58, 363–369 (2011)

    Article  Google Scholar 

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

    Book  Google Scholar 

  53. Ismail, G.A.F.: Modified technique for solving advance-delay differential system. Math. Comput. Model. 41, 287–289 (2005)

    Article  MATH  Google Scholar 

  54. Jacobsen, D.H., Lele, M.M., Speyer, J.L.: New necessary conditions of optimality for control problems with state variable inequality constraints. J. Math. Anal. Appl. 35, 255–284 (1971)

    Article  MathSciNet  Google Scholar 

  55. Jansen, L., Tischendorf, C.: A unified (P)DAE modeling approach for flow networks. In: DAE-Forum Progress in Differential-Algebraic Equations - Deskriptor 2013. Springer, Heidelberg (2014)

    Google Scholar 

  56. Kidane, N., Yamashita, Y., Nishitani, H.: Observer based I/O-linearizing control of high index DAE systems. In: Proceedings of American Control Conference, Denver, CO, pp. 3537–3542 (2003)

    Google Scholar 

  57. Koenig, D., Mammar, S.: Design of proportional-integral observer for unknown input descriptor systems. IEEE Trans. Autom. Control 47, 2057–2062 (2002)

    Article  MathSciNet  Google Scholar 

  58. Krener, A.J., Isidori, A.: Linearization by output injection and nonlinear observers. Syst. Control Lett. 3, 47–52 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  59. Krener, A.J., Respondek, W.: Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23, 197–216 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  60. Kuang, J., Tian, H., Shan, K.: Asymptotic stability of neutral differential systems with many delays. Appl. Math. Comput. 217(24), 10087–10094 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  61. Kumar, A., Daoutidis, P.: Control of Nonlinear Differential Algebraic Equation Systems. Chapman and Hall/CRC, New York (1999)

    MATH  Google Scholar 

  62. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zürich (2006)

    Book  Google Scholar 

  63. Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential Algebraic Equations Forum. Springer, Heidelberg (2012)

    Google Scholar 

  64. Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S.: Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol. 160. Birkhauser, Basel (2012)

    Google Scholar 

  65. Lewis, F.L., Syrmos, V.: Optimal Control, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  66. Lucero, J.C.: Advanced-delay differential equation for aeroelastic oscillations in physiology. Biophys. Rev. Lett. 3, 125–133 (2008)

    Article  Google Scholar 

  67. Lucht, W., Strehmel, K.: Discretization based indices for semilinear partial differential algebraic equations. Appl. Numer. Math. 28, 371–386 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  68. Lucht, W., Strehmel, K., Eichler-liebenow, C.: Linear Partial Differential Algebraic Equations - Part I: Indexes, Consistent Boundary/Initial Conditions, Report 17. Martin-Luther-Universitat Halle, Fachbereich Mathematik und Informatik (1997)

    Google Scholar 

  69. Lucht, W., Strehmel, K., Eichler-Liebenow, C.: Indexes and special discretization methods for linear partial differential algebraic equations. BIT Numer. Math. 39, 484–512 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  70. Markovsky, I., Willems, J.C., Van Huffel, S., De Moor, B.L.M.: Exact and Approximate Modeling of Linear Systems: A Behavioral Approach. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  71. Marszalek, W., Campbell, S.L.: DAEs arising from traveling wave solutions of PDEs. J. Comput. Appl. Math. 82, 41–58 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  72. Marszalek, W., Campbell, S.L.: DAEs arising from traveling wave solutions of PDEs II. Comput. Math. Appl. 37, 15–34 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  73. Martinson, W.S., Barton, P.I.: Index and characteristic analysis of linear PDAE systems. SIAM J. Sci. Comput. 24, 905–923 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  74. Matthes, M., Tischendorf, C.: Convergence analysis of a partial differential algebraic system from coupling a semiconductor model to a circuit model. Appl. Numer. Math. 61, 382–394 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  75. Michalska, H., Mayne, D.Q.: Moving horizon observers and observer-based control. IEEE Trans. Autom. Control 40, 995–1006 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  76. Moraal, P.E., Grizzle, J.W.: Observer design for nonlinear systems with discrete-time measurements. IEEE Trans. Autom. Control 40, 395–404 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  77. Niemann, H.H.: A setup for active fault diagnosis. IEEE Trans. Autom. Control 51, 1572–1578 (2006)

    Article  MathSciNet  Google Scholar 

  78. Nikoukhah, R.: A new methodology for observer design and implementation. IEEE Trans. Autom. Control 43, 229–234 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  79. Okay, I., Campbell, S.L., Kunkel, P.: Completions of implicitly defined vector fields and their applications. In: Proceedings of 18th International Symposium on Mathematical Theory of Networks and Systems, MTNS 08, Blacksburg, VA (2008)

    Google Scholar 

  80. Okay, I., Campbell, S.L., Kunkel, P.: Completions of implicitly defined linear time varying vector fields. Linear Algebra Appl. 431, 1422–1438 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  81. Patton, R.J., Frank, P.M., Clark, R.N.: Issues of Fault Diagnosis for Dynamic Systems. Springer, Berlin (2006)

    Google Scholar 

  82. Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. In: Stepleman, R.S., et al. (eds.) Scientific Computing. North-Holland, Amsterdam (1983)

    Google Scholar 

  83. Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory: A Behavioral Approach. Springer, New York (1998)

    Book  Google Scholar 

  84. Rabier, P.J., Rheinboldt, W.C.: Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  85. Rao, A.V., Benson, D.A., Darby, C., Patterson, M.A., Francolin, C., Sanders, I., Huntington, G.T.: Algorithm 902: Gpops, a matlab software for solving multiple-phase optimal control problems using the Gauss pseudospectral method. ACM Trans. Math. Softw. 37, 22:1–22:39 (2010)

    Google Scholar 

  86. Riaza, R.: Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. Word Scientific, Hackensack, NJ (2008)

    Book  MATH  Google Scholar 

  87. Scott, J.R.: Fault detection in differential algebraic equations. Ph.D. thesis, North Carolina State University (2015)

    Google Scholar 

  88. Scott, J.R., Campbell, S.L.: Auxiliary signal design for failure detection in differential-algebraic equations. Numer. Algebra Control Optim. 4, 151–179 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  89. Scott, J.R., Campbell, S.L.: Auxiliary signal design for failure detection in high index differential-algebraic equations. In: Proceedings of IEEE Conference on Decision and Control, Los Angeles (2014)

    Book  Google Scholar 

  90. Scott, J.R., Campbell, S.L.: Asynchronous auxiliary signal design for failure detection. In: IEEE International Conference on Systems, Man, and Cybernetics (SMC2014), San Diego, CA, pp. 2785–2790 (2014)

    Google Scholar 

  91. Stengel, R.F., Ghigliazza, R., Kulkarni, N., Laplace, O.: Optimal control of innate immune response. Optim. Control Appl. Methods 23, 91–104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  92. Thompson, K.C.: Solving Nonlinear Constrained Optimization Time Delay Systems with a Direct Transcription Approach. Ph.D. thesis, North Carolina State University (2014)

    Google Scholar 

  93. Tian, H., Kuang, J., Qiu, L.: The stability of linear multistep methods for linear systems of neutral differential equations. J. Comput. Math. 19, 125–130 (2001)

    MATH  MathSciNet  Google Scholar 

  94. Tian, H., Yu, Q., Kuang, J.: Asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. SIAM J. Numer. Anal. 49, 608–618 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  95. Tian, H., Yu, Q., Kuang, J.: Asymptotic stability of linear neutral delay differential-algebraic equations and Runge-Kutta methods. SIAM J. Numer. Anal. 52, 68–82 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  96. Von Wissel, D.: DAE Control of Dynamical Systems: Example of a Riderless Bicycle. Ph.D. thesis, INRIA (1996)

    Google Scholar 

  97. von Wissel, D., Nikoukhah, R., Campbell, S.L., Delebecque, F.: Nonlinear observer design using implicit system descriptions. In: Proceedings of Computational Engineering in Systems Applications, Lille, pp. 404–409 (1996)

    Google Scholar 

  98. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  99. Wagner, Y.: A further index concept for linear PDAEs of hyperbolic type. Math. Comput. Simul. 53, 287–291 (2000)

    Article  Google Scholar 

  100. Walcott, B.L., Corless, M.J., Zak, S.H.: Comparative study of non-linear state-observation techniques. Int. J. Control 45, 2109–2132 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  101. Wang, L., Wang, Z., Zou, X.: Periodic solutions of neutral functional differential equations. J. Lond. Math. Soc. 65, 439–452 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  102. Wu, A.G., Duan, G.R.: Design of generalized PI observers for descriptor linear systems. IEEE Trans. Circuits Syst. I: Regul. Pap. 53, 2828–2837 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The writing of this paper was supported in part by NSF Grants DMS-0907832 and DMS-1209251. Some of the results discussed were supported by earlier grants from ONR, AFOSR, ARO, and NSF. The number of people, both colleagues and students who have had an impact on the author’s understanding of DAEs and their applications over the years, is too numerous to mention. This is especially true of the author’s introduction to DAEs and their applications. But in terms of the specific topics discussed in this survey, the author would like to especially thank his colleagues John Betts, Ramine Nikoukhah, Peter Kunkel, Volker Mehrmann, Roswitha Maerz and PhD students Neil Biehn, Jason Scott, Karmethia Thompson, Angela Engelsone, and Dirk von Wissel.

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  1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Stephen L. Campbell

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    Achim Ilchmann

  2. Universität Hamburg, Fachbereich Mathematik, Hamburg, Germany

    Timo Reis

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Campbell, S.L. (2015). The Flexibility of DAE Formulations. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations III. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-22428-2_1

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