Ellipsoidal Estimates of Reachable Sets for Control Systems with Nonlinear Terms

Abstract The problem of estimating reachable sets of nonlinear dynamical control systems with quadratic nonlinearity and with uncertainty in initial states is studied. We assume that the uncertainty is of a set-membership kind when we know only the bounding set for unknown items and any additional statistical information on their behavior is not available. We present here approaches that allow finding ellipsoidal estimates of reachable sets which use the special structure of nonlinearity of studied control system. The algorithms of constructing such ellipsoidal set-valued estimates and numerical simulation results are given.

[1]  Stanislav Mazurenko,et al.  A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion , 2012 .

[2]  Tatiana F. Filippova,et al.  SET-VALUED DYNAMICS IN PROBLEMS OF MATHEMATICAL THEORY OF CONTROL PROCESSES AND STATE ESTIMATION , 2012 .

[3]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[4]  Richard D. Braatz,et al.  Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties , 2014 .

[5]  A. Kurzhanski.  Hamiltonian techniques for the problem of set‐membership state estimation , 2011 .

[6]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[7]  R. Bellman.  Introduction To Matrix Analysis Second Edition , 1997 .

[8]  A. Kurzhanski,et al.  On the Theory of Trajectory Tubes — A Mathematical Formalism for Uncertain Dynamics, Viability and Control , 1993 .

[9]  Russian Federation,et al.  Algorithms of Estimating Reachable Sets of Nonlinear Control Systems with Uncertainty , 2015 .

[10]  Eric Walter,et al.  Ellipsoidal parameter or state estimation under model uncertainty , 2004, Autom..

[11]  V. Sinyakov.  Method for computing exterior and interior approximations to the reachability sets of bilinear differential systems , 2015 .

[12]  Vsevolod M. Kuntsevich,et al.  Ellipsoidal and Interval Estimation of State Vectors for Families of Linear and Nonlinear Discrete-Time Dynamic Systems1 , 2015 .

[13]  Narcisa C. Apreutesei,et al.  An optimal control problem for a prey-predator system with a general functional response , 2009, Appl. Math. Lett..

[14]  T. F. Filippova.  Estimates of reachable sets of control systems with nonlinearity and parametric perturbations , 2016 .

[15]  T. F. Filippova,et al.  Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system , 2010 .

[16]  Antonio Vicino,et al.  Estimation theory for nonlinear models and set membership uncertainty , 1991, Autom..