Distributed Optimal Control of Multiscale Dynamical Systems: A Tutorial

Many complex systems, ranging from renewable resources [1] to very-large-scale robotic systems (VLRS) [2], can be described as multiscale dynamical systems comprising many interactive agents. In recent years, significant progress has been made in the formation control and stability analysis of teams of agents, such as robots, or autonomous vehicles. In these systems, the mutual goals of the agents are, for example, to maintain a desired configuration, such as a triangle or a star formation, or to perform a desired behavior, such as translating as a group (schooling) or maintaining the center of mass of the group (flocking) [2]-[7]. While this literature has successfully illustrated that the behavior of large networks of interacting agents can be conveniently described and controlled by density functions, it has yet to provide an approach for optimizing the agent density functions such that their mutual goals are optimized.

[1]  Kaustubh Pathak,et al.  An integrated path-planning and control approach for nonholonomic unicycles using switched local potentials , 2005, IEEE Transactions on Robotics.

[2]  J. March.  Introduction to the Calculus of Variations , 1999 .

[3]  C. Hargraves,et al.  DIRECT TRAJECTORY OPTIMIZATION USING NONLINEAR PROGRAMMING AND COLLOCATION , 1987 .

[4]  Anil V. Rao,et al.  Optimal Reconfiguration of Spacecraft Formations Using the Gauss Pseudospectral Method , 2008 .

[5]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[6]  Vijay Kumar,et al.  Modeling and control of formations of nonholonomic mobile robots , 2001, IEEE Trans. Robotics Autom..

[7]  Tucker R. Balch,et al.  Behavior-based formation control for multirobot teams , 1998, IEEE Trans. Robotics Autom..

[8]  Hua Chen,et al.  Distributed Density Estimation Using Non-parametric Statistics , 2007, 27th International Conference on Distributed Computing Systems (ICDCS '07).

[9]  James N. Sanchirico,et al.  Optimal spatial management of renewable resources: matching policy scope to ecosystem scale , 2005 .

[10]  Tapan K. Sengupta,et al.  Galerkin finite element methods for wave problems , 2005 .

[11]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[12]  Hongyan Wang,et al.  Social potential fields: A distributed behavioral control for autonomous robots , 1995, Robotics Auton. Syst..

[13]  Peter E. Caines,et al.  The NCE (Mean Field) Principle With Locality Dependent Cost Interactions , 2010, IEEE Transactions on Automatic Control.

[14]  Tao Li,et al.  Asymptotically Optimal Decentralized Control for Large Population Stochastic Multiagent Systems , 2008, IEEE Transactions on Automatic Control.

[15]  Silvia Ferrari,et al.  Distributed optimal control for multi-agent trajectory optimization , 2014, Autom..

[16]  Ashutosh Kumar Singh,et al.  Moment closure techniques for stochastic models in population biology , 2006, 2006 American Control Conference.

[17]  Silvia Ferrari,et al.  A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks , 2015, Neurocomputing.

[18]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[19]  João Pedro Hespanha,et al.  A cooperative homicidal chauffeur game , 2007, CDC.

[20]  Márk Jelasity,et al.  Gossip-based aggregation in large dynamic networks , 2005, TOCS.

[21]  A. Astolfi.  On the stabilization of nonholonomic systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[22]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[23]  Robert F. Stengel,et al.  Optimal Control and Estimation , 1994 .

[24]  Wenjie Lu,et al.  An Information Potential Approach to Integrated Sensor Path Planning and Control , 2014, IEEE Transactions on Robotics.

[25]  A. Bloch,et al.  Control and stabilization of nonholonomic dynamic systems , 1992 .

[26]  Naomi Ehrich Leonard,et al.  Virtual leaders, artificial potentials and coordinated control of groups , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[27]  S. Peng,et al.  Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control , 1999 .

[28]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[29]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[30]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[31]  K.M. Passino,et al.  Stability analysis of social foraging swarms , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[32]  Chien Chern Cheah,et al.  Region-based shape control for a swarm of robots , 2009, Autom..

[33]  João Pedro Hespanha,et al.  A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model , 2007, Bulletin of mathematical biology.

[34]  Mario Innocenti,et al.  Autonomous formation flight , 2000 .

[35]  Wolfram Burgard,et al.  Finding and Optimizing Solvable Priority Schemes for Decoupled Path Planning Techniques for Teams of Mobile Robots , 2002, PuK.

[36]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[37]  J. Schwartz,et al.  On the Complexity of Motion Planning for Multiple Independent Objects; PSPACE- Hardness of the "Warehouseman's Problem" , 1984 .

[38]  Steven M. LaValle,et al.  Optimal motion planning for multiple robots having independent goals , 1998, IEEE Trans. Robotics Autom..

[39]  Vanessa Hertzog,et al.  Computational Fluid Mechanics And Heat Transfer , 2016 .

[40]  Zhong-Ping Jiang,et al.  A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots , 2004, IEEE Transactions on Robotics and Automation.

[41]  C. R. Ethier,et al.  An efficient characteristic Galerkin scheme for the advection equation in 3-D , 2002 .

[42]  Claude Samson,et al.  Time-varying Feedback Stabilization of Car-like Wheeled Mobile Robots , 1993, Int. J. Robotics Res..

[43]  M. Heinkenschloss,et al.  Large-Scale PDE-Constrained Optimization , 2003 .