Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems

Abstract In this study, an active control policy design is proposed for a fractional order financial system, which considers multiple conflicting objectives. An active control template is used as a nonlinear state feedback mechanism and the controller gains are selected within a multi-objective optimization (MOO) framework to satisfy the conditions of asymptotic stability, which are derived analytically. The MOO obtains a set of solutions on the Pareto optimal front for the multiple conflicting objectives that are considered. We demonstrate that there is a trade-off between the multiple design objectives where better performance for one objective can only be obtained at the cost of degrading the performance for the other objectives. The multi-objective controller design was compared using three different MOO techniques, i.e., non-dominated sorting genetic algorithm-II, epsilon variable multi-objective genetic algorithm, and multi-objective evolutionary algorithm with decomposition. The robustness of the same control policy designed with the nominal system settings was also investigated with gradual decrease in the commensurate and incommensurate fractional orders of the financial system.

[1]  Guoliang Cai,et al.  Dynamic analysis and control of a new hyperchaotic finance system , 2012 .

[2]  Guanchun Liu,et al.  Modeling and Application of a New Nonlinear Fractional Financial Model , 2013, J. Appl. Math..

[3]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[4]  Wei-Ching Chen,et al.  Dynamics and control of a financial system with time-delayed feedbacks , 2008 .

[5]  Qingfu Zhang,et al.  MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition , 2007, IEEE Transactions on Evolutionary Computation.

[6]  Igor Podlubny,et al.  Modeling of the national economies in state-space: A fractional calculus approach , 2012 .

[7]  Shantanu Das,et al.  Continuous Order Identification of PHWR Models Under Step-back for the Design of Hyper-damped Power Tracking Controller with Enhanced Reactor Safety , 2013, ArXiv.

[8]  Mikkel T. Jensen,et al.  Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms , 2003, IEEE Trans. Evol. Comput..

[9]  Xavier Blasco Ferragud,et al.  Well-Distributed Pareto Front by Using the e-MOGA Evolutionary Algorithm , 2007, IWANN.

[10]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[11]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[12]  Guoliang Cai,et al.  Modified function lag projective synchronization of a financial hyperchaotic system , 2012 .

[13]  Marco Laumanns,et al.  Combining Convergence and Diversity in Evolutionary Multiobjective Optimization , 2002, Evolutionary Computation.

[14]  S. Rostek.  Option Pricing in Fractional Brownian Markets , 2009 .

[15]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[16]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[17]  R. Vilela Mendes.  A fractional calculus interpretation of the fractional volatility model , 2008 .

[18]  Mohammad Saleh Tavazoei,et al.  Chaotic attractors in incommensurate fractional order systems , 2008 .

[19]  Wen Chen,et al.  Modeling of Macroeconomics by a Novel Discrete Nonlinear Fractional Dynamical System , 2013 .

[20]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[21]  Junhai Ma,et al.  Chaos and Hopf bifurcation of a finance system , 2009 .

[22]  Enrico Scalas,et al.  Coupled continuous time random walks in finance , 2006 .

[23]  Junwei Wang,et al.  Chaos Control of a Fractional-Order Financial System , 2010 .

[24]  Yi Chai,et al.  Control and Synchronization of Fractional-Order Financial System Based on Linear Control , 2011 .

[25]  J. A. Tenreiro Machado,et al.  Identifying economic periods and crisis with the multidimensional scaling , 2011 .

[26]  J. A. Hołyst,et al.  Observations of deterministic chaos in financial time series by recurrence plots, can one control chaotic economy? , 2001 .

[27]  D. Guégan,et al.  Chaos in economics and finance , 2009, Annu. Rev. Control..

[28]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[29]  S. Bhalekar,et al.  Synchronization of different fractional order chaotic systems using active control , 2010 .

[30]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[31]  Xavier Blasco,et al.  Well-Distributed Pareto Front by Using the \(\epsilon \hskip-0.9em \nearrow \hskip-0.4em-MOGA\) Evolutionary Algorithm , 2007 .

[32]  J. A. Tenreiro Machado,et al.  Power Law Analysis of Financial Index Dynamics , 2012 .

[33]  Alberto Herreros,et al.  Design of PID-type controllers using multiobjective genetic algorithms. , 2002, ISA transactions.

[34]  Yufeng Xu,et al.  Synchronization of variable-order fractional financial system via active control method , 2013 .

[35]  Kalyanmoy Deb,et al.  Evaluating the -Domination Based Multi-Objective Evolutionary Algorithm for a Quick Computation of Pareto-Optimal Solutions , 2005, Evolutionary Computation.

[36]  Zhen Wang,et al.  Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay , 2011, Comput. Math. Appl..

[37]  DebK.,et al.  A fast and elitist multiobjective genetic algorithm , 2002 .

[38]  W. Chang.  PID control for chaotic synchronization using particle swarm optimization , 2009 .

[39]  Saptarshi Das,et al.  Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise , 2012 .

[40]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[41]  Zhen Wang,et al.  Control of an uncertain fractional order economic system via adaptive sliding mode , 2012, Neurocomputing.

[42]  Xavier Blasco Ferragud,et al.  Controller Tuning by Means of Multi-Objective Optimization Algorithms: A Global Tuning Framework , 2013, IEEE Transactions on Control Systems Technology.

[43]  H. Hochman,et al.  Pareto Optimal Redistribution , 1969 .

[44]  W. Brock,et al.  Heterogeneous beliefs and routes to chaos in a simple asset pricing model , 1998 .

[45]  Shantanu Das,et al.  Master-slave chaos synchronization via optimal fractional order PIλDμ controller with bacterial foraging algorithm , 2012, Nonlinear Dynamics.

[46]  Saptarshi Das,et al.  Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system , 2012, ArXiv.

[47]  Tönu Puu,et al.  Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in Economics , 2000 .

[48]  S. Das.  Functional Fractional Calculus , 2011 .

[49]  Fernando B. M. Duarte,et al.  Dynamics of the Dow Jones and the NASDAQ stock indexes , 2010 .

[50]  Sara Dadras,et al.  Control of a fractional-order economical system via sliding mode , 2010 .

[51]  Ursula Faber,et al.  Controlling Chaos Suppression Synchronization And Chaotification , 2016 .

[52]  Guanrong Chen,et al.  Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching , 2013, Comput. Math. Appl..

[53]  YangQuan Chen,et al.  Fractional order control - A tutorial , 2009, 2009 American Control Conference.

[54]  Yang Tang,et al.  Multiobjective synchronization of coupled systems. , 2011, Chaos.

[55]  Xavier Blasco Ferragud,et al.  Applied Pareto multi-objective optimization by stochastic solvers , 2009, Eng. Appl. Artif. Intell..

[56]  Jesus M. Gonzalez-miranda,et al.  Synchronization And Control Of Chaos: An Introduction For Scientists And Engineers , 2004 .