State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities.

This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data.

[1]  Sinan Kilicaslan,et al.  Tracking control of elastic joint parallel robots via state-dependent Riccati equation , 2015 .

[2]  Stephen P. Banks,et al.  Nonlinear optimal tracking control with application to super-tankers for autopilot design , 2004, Autom..

[3]  F. Ornelas‐Tellez,et al.  Optimal Tracking for State‐Dependent Coefficient Factorized Nonlinear Systems , 2014 .

[4]  Moharam Habibnejad Korayem,et al.  Nonlinear optimal control via finite time horizon state-dependent Riccati equation , 2014, 2014 Second RSI/ISM International Conference on Robotics and Mechatronics (ICRoM).

[5]  Wei-Min Lu,et al.  Nonlinear optimal control: alternatives to Hamilton-Jacobi equation , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[6]  B. Baigzadeh,et al.  Simulation of the Performance of Novel Nonlinear Optimal Control Technique on Two Cart-inverted Pendulum System , 2010 .

[7]  G. Oriolo,et al.  Robotics: Modelling, Planning and Control , 2008 .

[8]  Edgar N. Sánchez,et al.  Optimal tracking for a class of nonlinear systems based on the state-dependent Riccati equation , 2013, 2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE).

[9]  S. R. Nekoo,et al.  Finite-time state-dependent Riccati equation for time-varying nonaffine systems: rigid and flexible joint manipulator control. , 2015, ISA transactions.

[10]  B K Sarkar,et al.  GA-optimized feedforward-PID tracking control for a rugged electrohydraulic system design. , 2013, ISA transactions.

[11]  Chenguang Yang,et al.  Adaptive fuzzy-based motion generation and control of mobile under-actuated manipulators , 2014, Eng. Appl. Artif. Intell..

[12]  F. Ornelas‐Tellez,et al.  Optimal tracking control for an AC-DC power converter , 2013, 2013 IEEE International Autumn Meeting on Power Electronics and Computing (ROPEC).

[13]  Ozan Tekinalp,et al.  Development of a State Dependent Ricatti Equation Based Tracking Flight Controller for an Unmanned Aircraft , 2013 .

[14]  Moharam Habibnejad Korayem,et al.  ANALYSIS OF MANIPULATORS USING SDRE: A CLOSED LOOP NONLINEAR OPTIMAL CONTROL APPROACH , 2010 .

[15]  Harvey Thomas Banks,et al.  A state‐dependent Riccati equation‐based estimator approach for HIV feedback control , 2006 .

[16]  Weijie Sun,et al.  Adaptive PID formation control of nonholonomic robots without leader's velocity information. , 2014, ISA transactions.

[17]  Javier Moreno-Valenzuela,et al.  Operational space trajectory tracking control of robot manipulators endowed with a primary controller of synthetic joint velocity. , 2011, ISA transactions.

[18]  Rafael Castro-Linares,et al.  Trajectory tracking for non-holonomic cars: A linear approach to controlled leader-follower formation , 2010, 49th IEEE Conference on Decision and Control (CDC).

[19]  A. Heydari,et al.  Closed‐form solution to finite‐horizon suboptimal control of nonlinear systems , 2015 .

[20]  S. Rafee Nekoo,et al.  Nonlinear closed loop optimal control: a modified state-dependent Riccati equation. , 2013 .

[21]  J. Cloutier,et al.  Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method , 1998 .

[22]  Ming-Kun Chang,et al.  T–S fuzzy model-based tracking control of a one-dimensional manipulator actuated by pneumatic artificial muscles , 2011 .

[23]  M. Spong,et al.  Robot Modeling and Control , 2005 .

[24]  Hamid Khaloozadeh,et al.  An iterative procedure for optimal nonlinear tracking problem , 2002, 7th International Conference on Control, Automation, Robotics and Vision, 2002. ICARCV 2002..

[25]  G. Cook,et al.  Suboptimal control for the nonlinear quadratic regulator problem , 1975, Autom..

[26]  Dennis S. Bernstein,et al.  A numerical comparison of frozen-time and forward-propagating Riccati equations for stabilization of periodically time-varying systems , 2014, 2014 American Control Conference.

[27]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[28]  D. Subbaram Naidu,et al.  Nonlinear optimal tracking with incomplete state information using finite-horizon State Dependent Riccati Equation (SDRE) , 2014, 2014 American Control Conference.

[29]  Edgar N. Sánchez,et al.  PSO Optimal Tracking Control for State-Dependent Coefficient Nonlinear Systems , 2013, WCSC.

[30]  Liang Xu,et al.  Adaptive backstepping trajectory tracking control of robot manipulator , 2012, J. Frankl. Inst..

[31]  Robert J. Schilling,et al.  Fundamentals of robotics - analysis and control , 1990 .

[32]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[33]  Tayfun Çimen,et al.  State-Dependent Riccati Equation (SDRE) Control: A Survey , 2008 .

[34]  Sung Woo Kim,et al.  Hardware-In-the-Loop Simulations of spacecraft attitude synchronization using the State-Dependent Riccati Equation technique , 2013 .

[35]  Behdad Geranmehr,et al.  The nonlinear suboptimal diving control of an autonomous underwater vehicle , 2014, 2014 Second RSI/ISM International Conference on Robotics and Mechatronics (ICRoM).

[36]  Ilya V. Kolmanovsky,et al.  Predictive energy management of a power-split hybrid electric vehicle , 2009, 2009 American Control Conference.

[37]  Behdad Geranmehr,et al.  Nonlinear suboptimal control of fully coupled non-affine six-DOF autonomous underwater vehicle using the state-dependent Riccati equation , 2015 .

[38]  John J. Craig,et al.  Introduction to Robotics Mechanics and Control , 1986 .

[39]  S. C. Beeler,et al.  State Estimation and Tracking Control of Nonlinear Dynamical Systems , 2003 .

[40]  David E. Cox,et al.  State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities , 2004 .

[41]  Zoran Gajic,et al.  Solving the Matrix Differential Riccati Equation: A Lyapunov Equation Approach , 2010, IEEE Transactions on Automatic Control.

[42]  Tayfun Çimen,et al.  APPROXIMATE NONLINEAR OPTIMAL SDRE TRACKING CONTROL , 2007 .

[43]  T. C. Kuo,et al.  Robust output tracking control for nonlinear time-varying robotic manipulators , 2005 .