Abstract
Hybrid systems, which exhibit both discrete state and continuous state behavior, have become the modeling framework of choice for a wide variety of applications that require detailed dynamic models with embedded discontinuities. Economic, environmental, safety and quality considerations in these applications, such as the automated design of safe operating procedures and the formal verification of embedded systems, strongly motivate the development of algorithms and tools for the global optimization of hybrid systems. For safety critical tasks such as formal verification, it is crucial that the global solution be found within epsilon optimality.
Recently, a deterministic algorithm has been proposed for the global optimization of linear time varying hybrid systems in the continuous time domain, provided the transition times and the sequence of modes are fixed. This work is extended by presenting a method for determining the optimal mode sequence when the transition times are fixed. A reformulation of the problem by introducing binary decision variables while retaining the linearity of the underlying dynamic system is proposed. This allows recently developed convexity theory for linear time varying continuous systems to be employed to construct a convex relaxation of the resulting mixed-integer dynamic optimization problem, which enables the global solution to be found in a finite number of iterations using nonconvex outer approximation.
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References
Adjiman C.S., Androulakis I.P. and Floudas C.A. (2000), “Global optimization of mixed-integer nonlinear problems,” AIChE J. Vol. 46, 1769–1797.
Adjiman C.S., Dallwig S., Floudas C.A. and Neumaier A. (1998), “A global optimization method, aBB, for general twice-differentiable constrained NLPs - I. Theoretical advances,” Comput. Chem. Eng. Vol. 22, 1137–1158.
Alur R., Courcoubetis C., Halbwachs N., Henzinger T.A., Ho P.H., Nicollin X., Olivero A., Sifakis J. and Yovine S. (1995), “The algorithmic analysis of hybrid systems,” Theor. Comput. Sci. Vol. 138, 3–34.
Avraam M.P., Shah N. and Pantelides C.C. (1998), “Modelling and optimisation of general hybrid systems in the continuous time domain,” Comput. Chem. Eng. Vol. 22, S221–S228.
Bagajewicz M.J. and Manousiouthakis V. (1991), “On the generalized Benders decomposition,” Comput. Chem. Eng. Vol. 15, 691–700.
Barton P.I., Allgor R.J., Feehery W.F. and Galan S. (1998), “Dynamic optimization in a discontinuous world,” Ind. Eng. Chem. Res. Vol. 37, 966–981.
Barton P.I., Banga J.R. and Galan S. (2000), “Optimization of hybrid discrete/continuous dynamic systems,” Comput. Chem. Eng. Vol. 24, 2171–2182.
Barton P.I. and Lee C.K. (2002), “Modeling, simulation, sensitivity analysis and optimization of hybrid systems,” ACM Trans. Model. Comput. Simul. Vol. 12, 1–34.
Barton P.I. and Lee C.K. (2003), “Design of process operations using hybrid dynamic optimization,” in Grossmann I.E. and McDonald C.M., Editors, “Proceedings of the 4th International Conference on Foundations of Computer-Aided Process Operations,” 89–102.
Barton P.I. and Pantelides C.C. (1994), “Modeling of combined discrete/continuous processes,” AIChE J. Vol. 40, 966–979.
Bemporad A., Borrelli F. and Morari M. (2000), “Optimal controllers for hybrid systems: Stability and piecewise linear explicit form,” in “Proc. 39th IEEE Conf. Decis. Control,” Vol. 2, 1810–1815.
Branicky M.S., Borkar V.S. and Mitter S.K. (1998), “A unified framework for hybrid control: Model and optimal control theory,” IEEE Transactions on Automatic Control Vol. 43, 31–45.
Clabaugh J.A., Tolsma J.E. and Barton P.I. (1999), “ABACUSS II: Advanced modeling environment and embedded simulator,” http://yoric.mit.edu/abacuss2/abacuss2.html.
Coddington E.A. and Levinson N. (1955), Theory of Ordinary Differential Equations, McGraw Hill.
Cuzzola F.A. and Morari M. (2001), “A generalized approach for analysis and control of discrete-time piecewise affine and hybrid systems,” in Benedetto M.D.D. and Sangiovanni-Vincentelli A., Editors, “Hybrid Systems: Computation and Control,” Vol. 2034 of Lect. Notes in Comput. Sci., 189–203, Springer-Verlag, Berlin.
Duran M.A. and Grossmann I.E. (1986), “An outer-approximation algorithm for a class of mixed-integer nonlinear programs,” Math. Programming Vol. 36, 307–339.
Falk J.E. and Soland R.M. (1969), “An algorithm for separable nonconvex programming problems,” Manage. Sci. Vol. 15, 550–569.
Feehery W.F., Tolsma J.E. and Barton P.I. (1997), “Efficient sensitivity analysis of large-scale differential-algebraic systems,” Appl. Numer. Math. Vol. 25, 41–54.
Fletcher R. and Leyffer S. (1994), “Solving mixed integer nonlinear programs by outer approximation,” Math. Programming Vol. 66, 327–349.
Floudas C.A. (1995), Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications, Oxford University Press, Oxford.
Galan S. and Barton P.I. (1998), “Dynamic optimization of hybrid systems,” Comput. Chem. Eng. Vol. 22, S183–S190.
Galan S., Feehery W.F. and Barton P.I. (1999), “Parametric sensitivity functions for hybrid discrete/continuous systems,” Appl. Numer. Math. Vol. 31, 17–48.
Horst R. and Tuy H. (1993), Global Optimization, Springer-Verlag, Berlin.
Kesavan P., Allgor R.J., Gatzke E.P. and Barton P.I. (2001), “Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs,” Submitted to: Math. Programming, available from http://yoric.mit.edu/publications.html.
Kesavan P. and Barton P.I. (2000), “Decomposition algorithms for nonconvex mixed-integer nonlinear programs,” AIChE Symp. Ser. Vol. 96, 458–461.
Kocis G.R. and Grossmann I.E. (1989), “A modelling and decomposition strategy for the MINLP optimization of process flowsheets,” Comprit. Chem. Eng. Vol. 13, 797–819.
Lee C.K., Singer A.B. and Barton P.I. (2002), “Global optimization of linear hybrid systems with explicit transitions,” Submitted to: Syst. Control Lett., available from http://yoric.mit.edu/publications.html.
McCormick G.P. (1976), “Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems,” Math. Programming Vol. 10, 147–175.
Ryoo H.S. and Sahinidis N.V. (1996), “A branch-and-reduce approach to global optimization,” J. Global Optim. Vol. 8, 107–139.
Sahinidis N.V. and Grossmann I.E. (1991), “Convergence properties of generalized Benders decomposition,” Comput. Chem. Eng. Vol. 15,481–491.
Shakernia O., Pappas G.J. and Sastry S. (2000), “Semidecidable controller synthesis for classes of linear hybrid systems,” in “Proc. 39th IEEE Conf. Decis. Control,” Vol. 2, 1834–1839.
Singer A.B. and Barton P.I. (2001), “Global solution of linear dynamic embedded optimization problems - Part I: Theory,” Submitted to: J. Optim. Theor. Appl., available from http://yoric.mit.edu/publications.html.
Singer A.B. and Barton P.I. (2003), “Global optimization with nonlinear ordinary differential equations - Part I: Theory,” Submitted to: J. Optim. Theor. Appl.
Teo K., Goh G. and Wong K. (1991), A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, Wiley, New York.
Tolsma J.E. and Barton P.I. (2000), “DAEPACK: An open modeling environment for legacy models,” Ind. Eng. Chem. Res. Vol. 39, 1826–1839.
Tolsma J.E. and Barton P.I. (2002), “Hidden discontinuities and parametric sensitivity calculations,” SIAM J. Sci. Comput. Vol. 23, 1862–1875.
Trontis A. and Spathopoulos M.P. (2001), “Target control for hybrid systems with linear continuous dynamics,” in “Proc. 40th IEEE Conf. Decis. Control,” Vol. 2, 1229–1234.
Xu X. and Antsaklis P.J. (2001), “An approach for solving general switched linear quadratic optimal control problems,” in “Proceedings of the 40th IEEE Conference on Decision and Control,” 2478–2483.
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Lee, C.K., Barton, P.I. (2004). Global Dynamic Optimization of Linear Hybrid Systems. In: Floudas, C.A., Pardalos, P. (eds) Frontiers in Global Optimization. Nonconvex Optimization and Its Applications, vol 74. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0251-3_16
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DOI: https://doi.org/10.1007/978-1-4613-0251-3_16
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