Adjoint sensitivity analysis for nonsmooth differential-algebraic equation systems.

*(English)*Zbl 1338.34034The authors present a sensitivity analysis for explicit nonsmooth differential algebraic equations (NDAEs) of index 1. The proposed method combines techniques known for smooth DAEs and nonsmooth ordinary differential equations. The sensitivity analysis is derived for generalized Mayer type functionals which include the typical least-squares parameter fitting problem. In contrast to classical Mayer type functionals these functionals depend on information at interior time points. Finally, two interesting case studies are presented.

Reviewer: Johannes Schropp (Konstanz)

##### MSC:

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

49Q12 | Sensitivity analysis for optimization problems on manifolds |

65L80 | Numerical methods for differential-algebraic equations |

34A36 | Discontinuous ordinary differential equations |

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\textit{R. Hannemann-Tamás} et al., SIAM J. Sci. Comput. 37, No. 5, A2380--A2402 (2015; Zbl 1338.34034)

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[1] | R. Baker and C. L. E. Swartz (2004), Rigorous handling of input saturation in the design of dynamically operable plants, Ind. Eng. Chem. Res. 43, pp. 5880–5887. |

[2] | I. Bauer, H. G. Bock, S. Körkel, and J. P. Schlöder (2000), Numerical methods for optimum experimental design in DAE systems, J. Comput. Appl. Math. 120, pp. 1–25. · Zbl 0998.65083 |

[3] | G. A. Bliss (1918), The problem of Mayer with variable end points, Trans. Amer. Math. Soc. 19, pp. 305–314. · JFM 46.0758.04 |

[4] | Y. Cao, S. Li, L. Petzold, and R. Serban (2003), Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution, SIAM J. Sci. Comput. 24, pp. 1076–1089. · Zbl 1034.65066 |

[5] | F. E. Cellier and E. Kofman (2006), Continuous System Simulation, Springer, New York. · Zbl 1112.93004 |

[6] | S. Engell, S. Kowalewski, C. Schulz, and O. Stursberg (2000), Continuous-discrete interactions in chemical processing plants, Proceedings of the IEEE 88, pp. 1050–1068. |

[7] | R. Fletcher and S. Leyffer (2002), Nonlinear programming without a penalty function, Math. Program. 91, pp. 239–269. · Zbl 1049.90088 |

[8] | P. Fritzson (2003), Principles of Object-Oriented Modeling and Simulation with Modelica 2.1, Wiley-IEEE Press, Piscataway, NJ. |

[9] | S. Gal\a’an, W. E. Feehery, and P. I. Barton (1999), Parametric sensitivity functions for hybrid discrete/continuous systems, Appl. Numer. Math. 31, pp. 17–47. · Zbl 0937.65137 |

[10] | J. Gerhard, W. Marquardt, and M. Mönnigmann (2008), Normal vectors on critical manifolds for robust design of transient processes in the presence of fast disturbances, SIAM J. Appl. Dyn. Syst. 7, pp. 461–490. · Zbl 1159.37429 |

[11] | A. Griewank and A. Walther (2008), Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia. · Zbl 1159.65026 |

[12] | E. Hairer, S. P. Nørsett, and G. Wanner (1993), Solving Ordinary Differential Equations I - Nonstiff Problems, Springer-Verlag, Berlin. · Zbl 0789.65048 |

[13] | R. Hannemann and W. Marquardt (2007), Fast computation of the Hessian of the Lagrangian in shooting algorithms for dynamic optimization, in Proceedings of the 8th International Symposium on Dynamics and Control of Process Systems (DYCOPS), B. Foss and J. Alvarez, eds., Cancun, Mexico, pp. 105–110. |

[14] | R. Hannemann and W. Marquardt (2010), Continuous and discrete composite adjoints for the Hessian of the Lagrangian in shooting algorithms for dynamic optimization, SIAM J. Sci. Comput. 31, pp. 4675–4695. · Zbl 1203.49050 |

[15] | R. Hannemann-Tamás (2013). Adjoint Sensitivity Analysis for Optimal Control of Non-Smooth Differential-Algebraic Equations. Ph.D. thesis, RWTH Aachen University, Aachen, Germany. |

[16] | L. Hascoët and V. Pascual (2013), The Tapenade automatic differentiation tool: Principles, model, and specification, ACM Trans. 3 Math. Software, 39, article 20. · Zbl 1295.65026 |

[17] | P. W. Hemker (1972), Numerical methods for differential equations in system simulation and in parameter estimation, in Analysis and Simulation of Biochemical Systems, H. C. Hemker and B. Hess, eds., North–Holland, Amsterdam, pp. 59–80. |

[18] | A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward (2005), SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Software 31, pp. 363–396. · Zbl 1136.65329 |

[19] | J. B. Jørgensen (2007), Adjoint sensitivity results for predictive control, state- and parameter-estimation with nonlinear models, in Proceedings of the European Control Conference 2007, pp. 3649–3656. |

[20] | R. I. Leine and H. Nijmeijer (2004), Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer-Verlag, Berlin. · Zbl 1068.70003 |

[21] | N. Mihajlovic, N. van de Wouw, M. P. M. Hendriks, and H. Nijmeijer (2006), Friction-induced limit cycling in flexible rotor systems: An experimental drill-string set-up, Nonlinear Dynam. 46, pp. 273–291. · Zbl 1170.70303 |

[22] | Modelica Association (2010). Modelica - A Unified Object-Oriented Language for Physical Systems Modeling. Language Specification. Version Modelica Association. Linköping, Sweden, 3.2. |

[23] | A. Namjoshi, A. Kienle, and D. Ramkrishna (2003), Steady-state multiplicity in bioreactors: Bifurcation analysis of cybernetic models, Chem. Eng. Sci. 58, pp. 793–800. |

[24] | U. Naumann (2012), The Art of Differentiating Computer Programs - An Introduction to Algorithmic Differentiation, Software Environ. Tools, SIAM, Philadelphia. · Zbl 1275.65015 |

[25] | D. B. Özyurt and P. I. Barton (2005), Cheap second order directional derivatives of stiff ODE embedded functionals, SIAM J. Sci. Comput. 26, pp. 1725–1743. · Zbl 1076.65067 |

[26] | T. Park and P. I. Barton (1996), State event location in differential-algebraic models, ACM Trans. Model. Comput. Simul. 6, pp. 137–165. · Zbl 0887.65075 |

[27] | J. B. Rawlings and D. Q. Mayne (2009), Model Predictive Control: Theory and Design, Nob Hill Publishing, LLC, Madison, WI. |

[28] | A. I. Ruban (1997), Sensitivity coefficients for discontinuous dynamic systems, J. Comput. Syst. Sci. Int. 36, pp. 536–542. · Zbl 0912.93022 |

[29] | O. Stursberg and T. H. Tran (2006), Algorithmic and abstraction-based design of discrete controllers for hybrid automata, at-Automatisierungstechnik 54, pp. 450–458. |

[30] | A. Wächter and L. T. Biegler (2006), On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106, pp. 25–57. · Zbl 1134.90542 |

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