Skip to main content
SpringerLink
Log in

Smoothing and regularization strategies for optimization of hybrid dynamic systems

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

Mathematical programming has become a valuable tool in process engineering. However, optimization of hybrid dynamic systems with autonomous mode transitions still constitutes a major challenge for theoretical treatments and engineering application. Among the existing approaches for addressing this obstacle, reformulation strategies appear to be most promising. In this study, a modified smoothing strategy and an extended penalization approach to approximate the non-smooth dynamic optimization problem by a smooth one are presented. As a result, a local solution can be gained by a NLP solver after a discretization of the smoothed problem. This solution converges to that of the original non-smooth problem when the value of the introduced reformulation parameter goes to zero. Heuristic rules to select parameter values for both strategies are proposed based on their inherent features. Results from two case studies indicate the capability of the proposed approaches to efficiently obtain physically meaningful solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Akesson J (2010) JModelica User’s Guide. Modelon AB. http://www.jmodelica.org/page/236

  • Andersson M (1994) Object-oriented modeling and simulation of hybrid systems. PhD thesis, Dept. of Automatic Control, Lund Inst. of Technology, Sweden

  • Anitescu M, Tseng P, Wright SJ (2007) Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties. Math Program 110:337–371

    Article  MATH  MathSciNet  Google Scholar 

  • Bard JF (1988) Convex two-level optimization. Math Program 40:15–27

    Article  MATH  MathSciNet  Google Scholar 

  • Barton PI, Allgor RJ, Feehery WF, Galan S (1998) Dynamic optimization in a discontinuous world. Ind Eng Chem Res 37:966–981

    Article  Google Scholar 

  • Barton PI, Banga JR, Galan S (2000) Optimization of hybrid discrete/continuous dynamic systems. Comput Chem Eng 24:2171–2182

    Article  Google Scholar 

  • Barton PI, Lee CK (2002) Modeling, simulation, sensitivity analysis, and optimization of hybrid systems. ACM Trans Model Comput Simul 12(4):256–289

    Article  Google Scholar 

  • Barton PI, Lee CK, Yunt M (2006) Optimization of hybrid systems. Comput Chem Eng 30:1576–1589

    Article  Google Scholar 

  • Baumrucker BT, Biegler LT (2009) MPEC strategies for optimization of a class of hybrid dynamic systems. J Process Control 19:1248–1256

    Article  Google Scholar 

  • Bemporad A, Morari M (1999) Control of systems integrating logic, dynamics, and constraints. Automatica 35:402–427

    Article  MathSciNet  Google Scholar 

  • Biegler L (2010) Nonlinear programming—concepts, algorithms, and applications to chemical processes, vol 10. Mos-SIAM Series on Optimization, SIAM, Philadelphia

  • Biegler LT (2007) An overview of simultaneous strategies for dynamic optimization. Chem Eng Process 46:1043–1053

    Article  Google Scholar 

  • Cassandras CG, Pepyne DL, Wardi Y (2001) Optimal control of a class of hybrid systems. IEEE Trans Autom Control 46:398–415

    Article  MATH  MathSciNet  Google Scholar 

  • Cellier FE (1979a) Combined continuous/discrete system simulation by use of digital computers: Techniques and tools. PhD thesis, ETH Zuerich, Switzerland

  • Cellier FE (1979b) Combined continuous/discrete system simulation languages. In: Zeigler BP, Elzas MS, Klir GJ, Oeren TI (ed) Methodology in systems modelling and simulation, 201–220

  • Cellier FE, Elmqvist H, Otter M, Taylor JH (1993) Guidelines for modelling and simulation of hybrid systems. In Proceedings of 12th IFAC World Congress, 8: 391–397

  • Cervantes A, Biegler LT (1998) Large-scale DAE optimization using a simultaneous NLP formulation. AIChE J 44(5):1038–1050

    Article  Google Scholar 

  • Chen C, Mangasarian OL (1996) A class of smoothing functions for nonlinear and mixed complementarity problems. Comput Optim Appl 5:97–138

    Article  MATH  MathSciNet  Google Scholar 

  • Chen X (2000) Smoothing methods for comlementarity problems and their applications: A survey. J Oper Res Soc Jpn 43(1):32–47

    Article  MATH  Google Scholar 

  • Chen X (2012) Smoothing methods for nonsmooth, nonconvex minimization. Math Program 134(1):71–99

    Article  MATH  MathSciNet  Google Scholar 

  • Chen Y, Florian M (1995) The nonlinear bilevel programming problem: formulations, regularity and optimality conditions. Optimization 32(3):193–209

    Article  MATH  MathSciNet  Google Scholar 

  • De la Fluente RL, Flores-Tlacuahuac A (2009) Integrated design and control using a simultaneous mixed-integer dynamic optimization approach. Ind Eng Chem Res 48:1933–1943

    Article  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming. Kluwer Academie Publishers, Dordrecht

    MATH  Google Scholar 

  • Dempe S, Dutta J (2012) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math Program 131(1–2):37–48

    Article  MATH  MathSciNet  Google Scholar 

  • Diehl M, Bock HG, Schloeder JP, Findeisen R, Nagy Z, Allgoewer F (2002) Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J Process Control 12(4):577–585

    Article  Google Scholar 

  • Ferris MC, Pang JS (1997) Engineering and economic applications of complementarity problems. SIAM Rev 39:669–713

    Article  MATH  MathSciNet  Google Scholar 

  • Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer Academic Publisher, Dordrecht

    Book  Google Scholar 

  • Fischer A (1992) A special Newton-type optimization method. Optimization 24:269–284

    Article  MATH  MathSciNet  Google Scholar 

  • Flegel M (2005) Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints. PhD thesis, Julius-Maximilians-Universität Würzburg, Germany

  • Fletcher R, Leyffer S (2004) Solving mathematical programs with complementarity constraints as nonlinear programs. Optim Method Softw 19:15–40

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher R, Leyffer S, Ralph D, Scholtes S (2006) Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J Optim 17:259–286

    Article  MATH  MathSciNet  Google Scholar 

  • Flores-Tlacuahuac A, Biegler LT (2007) Simultaneous mixed-integer dynamic optimization for integrated design and control. Comput Chem Eng 31:588–600

    Article  Google Scholar 

  • Goebel R, Sanfelice RG, Teel AR (2009) Hybrid dynamical systems. IEEE Control Syst Mag 29:28–93

    Article  MathSciNet  Google Scholar 

  • Grossmann C, Kleinmichel H (1979) Verfahren der nichtlinearen Optimierung. Teubner, Leipzig

  • Grossmann IE (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3:227

    Article  MATH  MathSciNet  Google Scholar 

  • Herskovits J, Filho MT, Leontiev A (2013) An interior point technique for solving bilevel programming problems. Optim Eng 14:381–394

    Article  MATH  MathSciNet  Google Scholar 

  • Hoheisel T, Kanzow C, Schwartz A (2013) Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math Program 137(1–2):257–288

    Article  MATH  MathSciNet  Google Scholar 

  • Hu XM, Ralph D (2004) Convergence of a penalty method for mathematical programming with complementarity constraints. J Optim Theory Appl 123:365–390

    Article  MathSciNet  Google Scholar 

  • Jiang H, Ralph D (1999) Smooth SQP methods for mathematical programs with nonlinear complementarity constraints. SIAM J Optim 10:779–808

    Article  MathSciNet  Google Scholar 

  • Kardani A, Dussault JP, Benchakroun A (2009) A new regularization scheme for mathematical programs with complementarity constraints. J Phys Chem A 20:78–103

    Google Scholar 

  • Kennedy J, Eberhart R (2002) Particle swarm optimization. In Neural Networks, 1995. Proceedings. IEEE 4:1942–1948

    Google Scholar 

  • Kravanja S, Sorsak A, Kravanja Z (2003) Efficient multilevel MINLP strategies for solving large combinatorial problems in engineering. Optim Eng 4:97–151

    Article  MATH  MathSciNet  Google Scholar 

  • Leyffer S (2003) Mathematical programs with complementarity constraints. SIAG/OPT Views-and-News 14:15–18

    Google Scholar 

  • Leyffer S, Lopez-Calva G, Nocedal J (2006) Interior methods for mathematical programs with complementarity constraints. SIAM J Optim 17:52–77

    Article  MATH  MathSciNet  Google Scholar 

  • Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Mehrmann V, Wunderlich L (2009) Hybrid systems of differential-algebraic equations—analysis and numerical solution. J Process Control 19(8):1218–1228

    Article  Google Scholar 

  • Michalik C, Hannemann R, Marquardt W (2009) Incremental single shooting: a robust method for the estimation of parameters in dynamical systems. Comput Chem Eng 33:1298–1305

    Article  Google Scholar 

  • Morari M, Baric M (2006) Recent developments in the control of constrained hybrid systems. Comput Chem Eng 30:1619–1631

    Article  Google Scholar 

  • Mynttinen I, Li P (2011) A reformulation scheme for parameter estimation of hybrid systems. In: 21st European symposium on computer-aided process engineering, 778–782

  • Nakamura K, Fusaoka A (2005) On transfinite hybrid automata. In: Proceeding: 8th International Workshop, HSCC 2005, 495–510

  • Nocedal J, Wright SJ (1999) Numerical optimization. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Pappala VS, Erlich I (2008) A new approach for solving the unit commitment problem by adaptive particle swarm optimization. In: Power and energy society general meeting, 2008 IEEE, 1–6

  • Park T, Barton PI (1996) State event location in differential-algebraic models. ACM Trans Model Comput Simul 6:137–165

    Article  MATH  Google Scholar 

  • Pfeiffer A, Arnold M (2010) Sensitivity analysis of discontinuous multidisciplinary models: Two examples. Non-smooth problems in vehicle systems dynamics. Springer, Berlin, pp 239–251

    Google Scholar 

  • Prata DM, Schwaab M, Lima EL, Pinto JC (2009) Nonlinear dynamic data reconciliation and parameter estimation through particle swarm optimization: application for an industrial polypropylene reactor. Chem Eng Sci 64:3953–3967

    Article  Google Scholar 

  • Raghunathan A, Biegler LT (2005) An interior point method for mathematical programs with complementarity constraints. SIAM J Optim 15:720–750

    Article  MATH  MathSciNet  Google Scholar 

  • Raghunathan AU, Prez-Correa JR, Agosin E, Biegler LT (2006) Parameter estimation in metabolic flux balance models for batch fermentation—formulation & solution using differential variational inequalities (DVIs). Ann Oper 148:251–270

    Article  MATH  Google Scholar 

  • Ralph D, Wright SJ (2004) Some properties of regularization and penalization schemes for MPECs. Optim Method Softw 19:527–556

    Article  MATH  MathSciNet  Google Scholar 

  • Russell RD, Christiansen J (1978) Adaptive mesh selection strategies for solving boundary value problems. SIAM J Numer Anal 15:59–80

    Article  MATH  MathSciNet  Google Scholar 

  • Sager S (2009) Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J Process Control 19:1238–1247

    Article  Google Scholar 

  • Sarabia D, Capraro F, Larsen LFS, De Prada C (2009) Hybrid NMPC of supermarket display cases. Control Eng Pract 17:428–441

    Article  Google Scholar 

  • Scheel H, Scholtes S (2000) Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math Oper Res 25:1–22

    Article  MATH  MathSciNet  Google Scholar 

  • Schittkowski K (2002) Numerical data fitting in dynamical systems. Kluwer Academic Press, Dordrecht

    Book  MATH  Google Scholar 

  • Scholtes S (2001) Convergence properties of regularization schemes for mathematical programs with complementarity constraints. SIAM J Optim 11(4):918–936

    Article  MATH  MathSciNet  Google Scholar 

  • Scholtes S, Stöhr M (2001) How stringent is the linear independence assumption for mathematical programs with complementarity constraints? Math Oper Res 26(4):851–863

    Article  MATH  MathSciNet  Google Scholar 

  • Sonntag C, Stursberg O, Engell S (2006) Dynamic optimization of an industrial evaporator using graph search with embedded nonlinear programming. In: 2nd IFAC conference on analysis and design of hybrid systems, 211–216

  • Stein O, Oldenburg J, Marquardt W (2004) Continuous reformulations of discrete-continuous optimization problems. Comput Chem Eng 28:1951–1966

    Article  Google Scholar 

  • Stewart DE, Anitescu M (2010) Optimal control of systems with discontinuous differential equations. Numer Math 114:653–695

    Article  MATH  MathSciNet  Google Scholar 

  • Stortelder WJH (1998) Parameter estimation in nonlinear dynamic systems, vol 124. CWI Tracts, Amsterdam

    Google Scholar 

  • Tamimi J, Li P (2010) A combined approach to nonlinear model predictive control of fast systems. J Process Control 20(9):1092–1102

    Article  Google Scholar 

  • Till J, Engell S, Panek S, Stursberg O (2004) Applied hybrid system optimization: an empirical investigation of complexity. Control Eng Pract 12:1291–1303

    Article  Google Scholar 

  • Tiller M (2001) Introduction to physical modeling with Modelica. Springer Verlag GmbH, Berlin

    Book  Google Scholar 

  • Trenn S (2009) Distributional differential algebraic equations. PhD thesis, Technische Universität Ilmenau, Germany

  • Vassiliadis VS, Sargent RWH, Pantelides CC (1994) Solution of a class of multistage optimization problems. 2. Problems with path constraints. Ind Eng Chem Res 33:2123–2133

    Article  Google Scholar 

  • Waechter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57

    Article  MATH  MathSciNet  Google Scholar 

  • Zavala VM, Laird CD, Biegler LT (2008) Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem Eng Sci 63:4834–4845

    Article  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for several important clarifications, which considerably improved the readability of the present paper.

Author information

Authors and Affiliations

  1. Institute of Automation and Systems Engineering, Technische Universität Ilmenau, Helmholtzplatz 5, 98693, Ilmenau, Germany

    I. Mynttinen & P. Li

  2. Institute of Mathematics, Technische Universität Ilmenau, Weimarer Strasse 25, 98693, Ilmenau, Germany

    A. Hoffmann

  3. Institute of Physics, Technische Universität Ilmenau, Weimarer Strasse 25, 98693, Ilmenau, Germany

    E. Runge

Authors
  1. I. Mynttinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Runge
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mynttinen, I., Hoffmann, A., Runge, E. et al. Smoothing and regularization strategies for optimization of hybrid dynamic systems. Optim Eng 16, 541–569 (2015). https://doi.org/10.1007/s11081-014-9261-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-014-9261-y

Keywords

Search

Navigation