Backstepping controller synthesis and characterizations of incremental stability

Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in either parametric-strict-feedback or strict-feedback form, and they require these control systems to be smooth. In this paper, we propose a design technique that is applicable to larger classes of control systems, including a class of non-smooth control systems. Moreover, we propose a recursive way of constructing contraction metrics (for smooth control systems) and incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems, the approximation of stochastic hybrid systems, source-code model checking for nonlinear dynamical systems and so on. The effectiveness of the proposed results in this paper is illustrated by synthesizing controllers rendering two non-smooth control systems incrementally stable. The first example aims to show how to recursively construct the incremental Lyapunov functions. The second example aims to show the key role of the computed incremental Lyapunov function in constructing a finite abstraction that is equivalent to the system under study.

[1]  Rodolphe Sepulchre,et al.  Analysis of Interconnected Oscillators by Dissipativity Theory , 2007, IEEE Transactions on Automatic Control.

[2]  Paulo Tabuada,et al.  Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations , 2007, SIAM J. Control. Optim..

[3]  Rodolphe Sepulchre,et al.  Global State Synchronization in Networks of Cyclic Feedback Systems , 2012, IEEE Transactions on Automatic Control.

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

[5]  Paulo Tabuada,et al.  Approximately bisimilar symbolic models for nonlinear control systems , 2007, Autom..

[6]  Munther A. Dahleh,et al.  A Framework for Robust Stability of Systems Over Finite Alphabets , 2008, IEEE Transactions on Automatic Control.

[7]  Rupak Majumdar,et al.  A Lyapunov approach in incremental stability , 2011, IEEE Conference on Decision and Control and European Control Conference.

[8]  Yan Li,et al.  Compact Modeling of Nonlinear Analog Circuits Using System Identification via Semidefinite Programming and Incremental Stability Certification , 2009, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[9]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[10]  Paulo Tabuada,et al.  Verification and Control of Hybrid Systems , 2009 .

[11]  I. N. Kar,et al.  Contraction based adaptive control of a class of nonlinear systems , 2009, 2009 American Control Conference.

[12]  Antoine Girard,et al.  Controller synthesis for safety and reachability via approximate bisimulation , 2010, Autom..

[13]  Nathan van de Wouw,et al.  Global nonlinear output regulation: Convergence-based controller design , 2007, Autom..

[14]  Paulo Tabuada,et al.  PESSOA : A tool for embedded control software synthesis , 2009 .

[15]  Nathan van de Wouw,et al.  From convergent dynamics to incremental stability , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[16]  Nathan van de Wouw,et al.  Tracking and synchronisation for a class of PWA systems , 2008, Autom..

[17]  Indra Narayan Kar,et al.  Design of Asymptotically Convergent Frequency Estimator Using Contraction Theory , 2008, IEEE Transactions on Automatic Control.

[18]  P. Olver Nonlinear Systems , 2013 .

[19]  N. Wouw,et al.  Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach , 2005 .

[20]  Mario di Bernardo,et al.  Contraction Theory and Master Stability Function: Linking Two Approaches to Study Synchronization of Complex Networks , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[21]  A. Teel,et al.  A Smooth Lyapunov Function from a Class-kl Estimate Involving Two Positive Semideenite Functions , 1999 .

[22]  Paulo Tabuada,et al.  Backstepping Design for Incremental Stability , 2010, IEEE Transactions on Automatic Control.

[23]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[24]  John M. Lee Introduction to Smooth Manifolds , 2002 .

[25]  Pierre Rouchon,et al.  An intrinsic observer for a class of Lagrangian systems , 2003, IEEE Trans. Autom. Control..

[26]  Silke Wagner,et al.  Control software model checking using bisimulation functions for nonlinear systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[27]  Nathan van de Wouw,et al.  Convergent systems vs. incremental stability , 2013, Syst. Control. Lett..

[28]  Antoine Girard,et al.  Synthesis of switching controllers using approximately bisimilar multiscale abstractions , 2011, HSCC '11.

[29]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

[30]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[31]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..

[32]  Alessio Franci,et al.  Phase-locking between Kuramoto oscillators: Robustness to time-varying natural frequencies , 2010, 49th IEEE Conference on Decision and Control (CDC).

[33]  Insup Lee,et al.  Robust Test Generation and Coverage for Hybrid Systems , 2007, HSCC.

[34]  Jérôme Jouffryo,et al.  INTEGRATOR BACKSTEPPING USING CONTRACTION THEORY: A BRIEF METHODOLOGICAL NOTE. , 2002 .

[35]  I. Holopainen Riemannian Geometry , 1927, Nature.

[36]  A. Teel,et al.  A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions , 2000 .

[37]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[38]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[39]  Nathan van de Wouw,et al.  Reconfigurable control of piecewise affine systems with actuator and sensor faults: Stability and tracking , 2011, Autom..

[40]  George J. Pappas,et al.  Approximations of Stochastic Hybrid Systems , 2009, IEEE Transactions on Automatic Control.

[41]  Titu Andreescu,et al.  Problems in Real Analysis: Advanced Calculus on the Real Axis , 2009 .

[42]  Paulo Tabuada,et al.  Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems , 2008, IEEE Transactions on Automatic Control.

[43]  Vicentiu D. Radulescu,et al.  Problems in Real Analysis , 2009 .