Optimal control of sets of solutions to formally guarantee constraints of disturbed linear systems

Optimal control finds an optimal input trajectory which steers an initial state to a desired final state while satisfying given state and input constraints. However, most efficient approaches are restricted to a single initial state. In this paper, we present a new approach, which combines reachability analysis with optimal control. This enables us to solve the optimal control problem for a whole set of initial states by optimizing over the set of all possible solutions. At the same time, we are able to provide formal guarantees for the satisfaction of state and input constraints. Taking the effects of sets of disturbances into account ensures that the resulting controller is robust against them, which is a big advantage over many existing approaches. We show the applicability of our approach with a vehicle-platoon example.

[1]  Calin Belta,et al.  A Fully Automated Framework for Control of Linear Systems from Temporal Logic Specifications , 2008, IEEE Transactions on Automatic Control.

[2]  Ian R. Manchester,et al.  LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification , 2010, Int. J. Robotics Res..

[3]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[4]  Antoine Girard,et al.  SpaceEx: Scalable Verification of Hybrid Systems , 2011, CAV.

[5]  Matthias Althoff,et al.  Convex Interpolation Control with Formal Guarantees for Disturbed and Constrained Nonlinear Systems , 2017, HSCC.

[6]  Francesco Borrelli,et al.  Constrained Optimal Control of Linear and Hybrid Systems , 2003, IEEE Transactions on Automatic Control.

[7]  A. Girard,et al.  Efficient reachability analysis for linear systems using support functions , 2008 .

[8]  David K. Smith,et al.  Dynamic Programming and Optimal Control. Volume 1 , 1996 .

[9]  Rolf Findeisen,et al.  Parameterized Tube Model Predictive Control , 2012, IEEE Transactions on Automatic Control.

[10]  Matthias Althoff,et al.  Formal verification of maneuver automata for parameterized motion primitives , 2014, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems.

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

[12]  Rolf Findeisen,et al.  Homothetic tube model predictive control , 2012, Autom..

[13]  Ian M. Mitchell,et al.  Optimization Techniques for State-Constrained Control and Obstacle Problems , 2006 .

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

[15]  Matthias Althoff,et al.  Guaranteeing Constraints of Disturbed Nonlinear Systems Using Set-Based Optimal Control in Generator Space , 2017 .

[16]  Paulo Tabuada,et al.  Linear Time Logic Control of Discrete-Time Linear Systems , 2006, IEEE Transactions on Automatic Control.

[17]  David Q. Mayne,et al.  Robust model predictive control of constrained linear systems with bounded disturbances , 2005, Autom..

[18]  Russ Tedrake,et al.  Funnel libraries for real-time robust feedback motion planning , 2016, Int. J. Robotics Res..

[19]  Oded Maler,et al.  Recent progress in continuous and hybrid reachability analysis , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[20]  Matthias Althoff,et al.  An Introduction to CORA 2015 , 2015, ARCH@CPSWeek.

[21]  Alberto Bemporad,et al.  Predictive Control for Linear and Hybrid Systems , 2017 .

[22]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[23]  Ufuk Topcu,et al.  Reactive controllers for differentially flat systems with temporal logic constraints , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[24]  Stefan Kowalewski,et al.  Networked Cooperative Platoon of Vehicles for Testing Methods and Verification Tools , 2014, ARCH@CPSWeek.

[25]  Matthias Althoff,et al.  Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars , 2010 .

[26]  A. Agung Julius,et al.  Safety controller synthesis using human generated trajectories: Nonlinear dynamics with feedback linearization and differential flatness , 2012, 2012 American Control Conference (ACC).

[27]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[28]  Munther A. Dahleh,et al.  Maneuver-based motion planning for nonlinear systems with symmetries , 2005, IEEE Transactions on Robotics.

[29]  Bruce H. Krogh,et al.  Computational techniques for hybrid system verification , 2003, IEEE Trans. Autom. Control..

[30]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..