A Lyapunov approach in incremental stability Majid Zamani and Rupak Majumdar Abstract— The notion of incremental stability was proposed by several researchers as a strong property of dynamical and control systems. Incremental stability describes the convergence of trajectories with respect to themselves, rather than with respect to an equilibrium point or a particular trajectory. Similarly to stability, Lyapunov functions play an important role in the study of incremental stability. In this paper, we propose new notions of incremental Lyapunov functions which are coordinate independent and provide the description of incremental stability in terms of the proposed Lyapunov func- tions. Moreover, we develop a backstepping design approach providing a recursive way of constructing controllers, enforcing incremental stability, as well as incremental Lyapunov func- tions. The effectiveness of the proposed method is illustrated by synthesizing a controller rendering a single-machine infinite- bus electrical power system incrementally stable. I. I NTRODUCTION Incremental stability is a the requirement that all trajec- tories of a dynamical system converge to each other, rather than to an equilibrium point or a particular trajectory. While it is well-known that for linear systems such a property is equivalent to stability, it can be a much stronger property for nonlinear systems. The study of incremental stability goes back to the work of Zames in the 60’s [1]; see [2] for a historical discussion and a broad list of applications of incremental stability. Similarly to stability, Lyapunov functions play an im- portant role in the study of incremental stability. Angeli [3] proposed the notions of incremental Lyapunov function and incremental input-to-state Lyapunov function, and used these notions to prove charactrizations of incremental global asymptotic stability (δ-GAS) and incremental input-to-state stability (δ-ISS). Both proposed notions of Lyapunov func- tions in [3] are not coordinate independent, in general. In this paper, we propose new notions of incremental Lyapunov functions and incremental input-to-state Lyapunov functions that are coordinate invariant. Moreover, we use these new notions of Lyapunov functions to describe notions of incre- mental stability, proposed in [2]. Since the proposed notions of Lyapunov functions in this paper are coordinate invariant, they potentiate the develop- ment of synthesis tools for incremental stability. As an exam- This research was sponsored in part by the grant NSF-CNS-0953994 and a contract from Toyota Corp. M. Zamani is with the Department of Electri- cal Engineering, University of California, Los Ange- les, CA 90095, USA. Email: zamani@ee.ucla.edu, URL: http://www.ee.ucla.edu/∼zamani. R. Majumdar is with Max Planck Institute for Software Systems, Kaiser- slautern, Germany and the Department of Computer Science, University of California, Los Angeles, CA 90095. Email: rupak@mpi-sws.org, URL: http://www.cs.ucla.edu/ rupak. ple, we develop a backstepping design method for incremen- tal stability for strict-feedback 1 form systems. The proposed approach was inspired by the incremental backstepping ap- proach provided in [2]. While the approach in [2], provides a recursive way of constructing contraction metrics, the proposed approach in this paper provide a recursive way of constructing incremental Lyapunov functions, identified as a key property for the construction of finite abstractions in [5], [6], [7]. See [8], for a broad list of applications of incremental Lyapunov functions. Like the original backstepping method, the proposed approach in this paper provides a recursive way of constructing controllers as well as incremental Lyapunov functions. Our design approach is illustrated by designing a controller rendering a single-machine infinite-bus electrical power system incrementally stable. II. CONTROL SYSTEMS AND STABILITY NOTIONS A. Notation The symbols R, R + and R + 0 denote the set of real, positive, and nonnegative real numbers, respectively. The symbol I n denotes the identity matrix in R n×n . Given a vector x ∈ R n , we denote by x i the i–th element of x, and by ‖x‖ the Euclidean norm of x; we recall that ‖x‖ =  x 2 1 + x 2 2 + ... + x 2 n . Given a measurable function f : R + 0 → R n , the (essential) supremum of f is denoted by ‖f ‖ ∞ ; we recall that ‖f ‖ ∞ := (ess)sup{‖f (t)‖,t ≥ 0}. f is essentially bounded if ‖f ‖ ∞ < ∞. For a given time τ ∈ R + , define f τ so that f τ (t)= f (t), for any t ∈ [0,τ ), and f (t)=0 elsewhere; f is said to be locally essentially bounded if for any τ ∈ R + , f τ is essentially bounded. A function f : R n → R is called radially un- bounded if f (x) →∞ as ‖x‖→∞. A continuous function γ : R + 0 → R + 0 , is said to belong to class K if it is strictly increasing and γ (0) = 0; function γ is said to belong to class K ∞ if γ ∈K and γ (r) →∞ as r →∞. A continuous function β : R + 0 × R + 0 → R + 0 is said to belong to class KL if, for each fixed s, the map β(r,s) belongs to class K ∞ with respect to r and, for each fixed nonzero r, the map β(r,s) is decreasing with respect to s and β(r,s) → 0 as s →∞. If φ : R n → R n is a global diffeomorphism, and if X : R n → R n is a continuous map, we denote by φ ∗ X the map defined by (φ ∗ X)(y)= ∂φ ∂x   x=φ −1 (y) X ◦ φ −1 (y). A function d : R n × R n → R + 0 is a metric on R n if for any x,y,z ∈ R n , the following three conditions are satisfied: i) d(x,y)=0 if and only if x = y; ii) d(x,y)= d(y,x); and iii) d(x,z) ≤ d(x,y)+ d(y,z). 1 See equation (III.8) or [4] for a definition. 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 302