Controller Synthesis for Nonlinear Systems with Reachability Specifications Using Monotonicity

In this paper, we consider a control synthesis problem for a continuous-time nonlinear system. The problem under consideration consists in driving the state of the system to some target interval at a given time instant. We propose a solution based on candidate under-approximations of the backward reachable sets using multi-dimensional intervals. We show that a suitable controller can be designed by enforcing a monotonicity property of the closed-loop system on these intervals. For this purpose, we utilize the monotonicity conditions for nonlinear systems with inputs in the infinitesimal form. From these differential inequalities on the control strategy, we design some particular controllers which are time-varying, linear with respect to the state. The approach is illustrated by two examples.

[1]  Angelika Bayer,et al.  Ellipsoidal Calculus For Estimation And Control , 2016 .

[2]  Calin Belta,et al.  Formal Methods for Discrete-Time Dynamical Systems , 2017 .

[3]  P. Caines,et al.  Hierarchical hybrid control systems: a lattice theoretic formulation , 1998, IEEE Trans. Autom. Control..

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

[5]  Antoine Girard,et al.  Synthesis for Constrained Nonlinear Systems Using Hybridization and Robust Controllers on Simplices , 2012, IEEE Transactions on Automatic Control.

[6]  Calin Belta,et al.  Language-Guided Controller Synthesis for Linear Systems , 2014, IEEE Transactions on Automatic Control.

[7]  Calin Belta,et al.  Controlling a Class of Nonlinear Systems on Rectangles , 2006, IEEE Transactions on Automatic Control.

[8]  Rafael Wisniewski,et al.  Control to facet for polynomial systems , 2014, HSCC.

[9]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[10]  Antoine Girard,et al.  Contract Based Design of Symbolic Controllers for Interconnected Multiperiodic Sampled-Data Systems , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[11]  Antoine Girard,et al.  Robust controlled invariance for monotone systems: Application to ventilation regulation in buildings , 2016, Autom..

[12]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[13]  Paulo Tabuada,et al.  Verification and Control of Hybrid Systems - A Symbolic Approach , 2009 .

[14]  Paul I. Barton,et al.  Bounds on the reachable sets of nonlinear control systems , 2013, Autom..

[15]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

[16]  Antoine Girard,et al.  Control of polynomial dynamical systems on rectangles , 2013, 2013 European Control Conference (ECC).

[17]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[18]  Y. Candau,et al.  Computing reachable sets for uncertain nonlinear monotone systems , 2010 .

[19]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[20]  Elena K. Kostousova.  On Control Synthesis for Uncertain Differential Systems Using a Polyhedral Technique , 2013, LSSC.