Differential analysis of nonlinear systems: revisiting the pendulum example F. Forni, R. Sepulchre Abstract— Differential analysis aims at inferring global prop- where k 0 is the damping coefficient and u is the torque erties of nonlinear behaviors from the local analysis of the input. The≥ specific aim of the paper is therefore to understand linearized dynamics. The paper motivates and illustrates the as much as possible of the global behavior of model (1) from use of differential analysis on the nonlinear pendulum model, its linearized dynamics ((δ#; δv) T ) an archetype example of nonlinear behavior. Special emphasis is 2 (#;v)X put on recent work by the authors in this area, which includes a δ#_ 0 1 δ# 0 differential Lyapunov framework for contraction analysis [24], = + (2) δv_ cos(#) k δv δu and the concept of differential positivity [25]. − − =:A(#;k) I. INTRODUCTION where any solution| (δ#{z( ); δv( ))}lives at each time instant The purpose of this tutorial paper is to revisit the role · · t in the tangent space T , where (#( ); v( )) is a of linearization in nonlinear systems analysis and to present (#(t);v(t)) solution to (1). X · · recent developments of this differential approach to systems The nonlinear pendulum model is an archetype example and control theory. Linearization is often considered as a of nonlinear systems analysis. As a control system, it is one synonym of local analysis, whereas nonlinear systems anal- of the simplest examples of nonlinear mechanical models ysis aims at a global understanding of the system behavior. and many of its properties extend to more complex electro- The focus of the paper is therefore on system properties that mechanical models such as models of robots, spacecrafts, allow to address non-local questions through the local-in- or electrical motors. As a dynamical system, it is one the nature analysis of a differential approach. Such properties simplest models to exhibit a rich and possibly complex global have been sporadically studied in the control community, behavior, owing to the interplay between small oscillations perhaps most importantly through the contraction property and large oscillations, two markedly distinct behaviors for advocated in the seminal paper of Lohmiller and Slotine [40], which everyone has a clear intuition developed since child- but they play at best a secondary role in the main textbooks hood. of nonlinear control. While it is not the aim of the present At the onset, it is worth observing that the pendulum tutorial to provide a comprehensive survey of the role of is a nonlinear model for two related but distinct reasons: differential analysis in systems and control (a partial account the vector field is nonlinear due to the sinusoidal nature of which can be found in Section VI of [24]; see also the of the gravity torque but also the state-space is nonlinear other paper of this tutorial session [2]), we will illustrate due to the angular nature of the pendulum position. In fact some questions that have stimulated a renewed interest for it could be argued that the nonlinearity of the space is differential analysis in the recent years. The interested reader more fundamental than the nonlinearity of the vector field is also referred to the two-part invited session of CDC2013 in that example, and this feature of the pendulum extends for a sample of recent developments in that area. arXiv:1409.4712v1 [cs.SY] 16 Sep 2014 to most nonlinear models encountered in engineering. The Owing to the tutorial nature of the paper, the discussion differential analysis, which linearizes both the space and the will be exclusively restricted to the classical (adimensional) vector field, is perhaps especially relevant for such models. nonlinear pendulum model #_ = v Σ: (#; v) := S R ; v_ = sin(#) kv + u 2 X × − − (1) F. Forni and R. Sepulchre are with the University of Cambridge, Department of Engineering, Trumpington Street, Cambridge CB2 v 1PZ, and with the Department of Electrical Engineering and ϑ Computer Science, University of Liege,` 4000 Liege,` Belgium, [email protected]|[email protected]. The research is supported by FNRS. The paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded ϑ by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its Fig. 1. The natural state-space of the pendulum (left). The measure of the authors. angle # (right). The paper is organized as follows. Section II revisits tracking, and observer design all involve the stabilization of the pendulum example from a classical nonlinear control the incremental dynamics. Only in linear system theory is the perspective, pointing to some limitations of nonlinear control error between two arbitrary solutions equivalent to the error that call for a differential viewpoint. Section III revisits the between one solution and the zero equilibrium solution. pendulum example from a dynamical system perspective, Feedback linearization makes the dynamics and the in- summarizing the main geometric properties of its limit sets. cremental dynamics equivalent. But if the compensation of Section IV introduces the differential analysis, starting with nonlinear terms by feedback is not possible, regulation theory the classical role of linearization in the local analysis of becomes challenging even for the nonlinear pendulum, and hyperbolic limit sets, and gradually moving to the differential requires incremental stability properties. This is a main Lyapunov framework recently advocated by the authors [24]. motivation for contraction theory, which seeks to exploit the The section concludes on a short discussion on horizontal stability properties of the linearized dynamics, that is, the contraction, a property that purposely excludes specific di- incremental dynamics for infinitesimal differences, in order rections in the tangent space from the contraction analysis. to infer incremental stability properties. Section V illustrates on the pendulum example the novel concept of differential positivity [25], which is a projective B. Energy-based Lyapunov control form of differential contraction owing to the positivity of the linearized dynamics. We will illustrate how differential Inherited from classical methods from physics, methods positivity provides a novel tool for the differential analysis based on the conservation/dissipation of energy are central of limit cycles and, more generally, of one-dimensional in nonlinear control. The undamped pendulum preserves the attractors. sum of kinetic and potential energy 2 II. A NONLINEAR CONTROL PERSPECTIVE v E := + cos(#) (6) A. Feedback linearization and incremental dynamics 2 The pendulum is feedback linearizable: the control input during its motion, while it dissipates energy when the damp- ing is nonzero k > 0. For open systems, dissipativity theory u = sin(#) + w (3) relates the energy dissipation to an external power supply [68], [69]: the energy is an internal storage that satisfies the transforms the nonlinear pendulum model into the linear balance system #_ = v E_ u#_ (7) (4) ≤ v_ = kv + w : − meaning that its rate of growth cannot exceed the mechanical Achieving linearity by feedback has been a cornerstone of power supplied to the system. Using y := #_ to denote the nonlinear control theory and is a key property for a regulation output of the system, dissipativity with the supply uy is a theory of nonlinear systems [33]. passivity property. Passivity is closely related to Lyapunov Exploiting linearity, it is straightforward to see that solving stability. The static output feedback u = y adds damping tracking or regulation problems on (4) become trivial tasks in the system and is often sufficient to achieve− asymptotic in comparison to the fully nonlinear case. The combination stability of the minimum energy equilibrium. For open of the nonlinear cancellation in (3) with a linear stabilizing systems, the supply rate measures the effect of exogenous feedback and a feedforward injection would guarantee the signals on the internal energy of the system. asymptotic tracking of any suitable reference trajectory in Passivity based control is a building block of nonlinear (#∗( ); v∗( )) R . · · 2 !X control theory and has led to far reaching generalizations in A key difference between the nonlinear pendulum dy- the theory of port-Hamiltonian systems [65], [46], leading to incremental namics and the linear dynamics (4) is in the an interconnection theory for the energy-based stabilization property: if (#1( ); v1( )) and (#2( ); v2( )) are two solutions · · · · of electro-mechanical systems. For instance, the fundamental of (1) for two different inputs u1( ) and u2( ), the increment · · interconnection property that the feedback interconnection of (∆#( ); ∆v( )) = (#1( ) #2( ); v1( ) v2( )) satisfies · · · − · · − · passive systems provides a direct solution to the PI control of ∆_# = ∆v passive systems because a PI controller is a passive system. ∆Σ : ∆_v = (sin(#1) sin(#2)) k∆v + ∆u But a bottleneck of passivity theory is the generalization − − − (5) from stabilization to tracking control. Fundamentally, this whose right-hand side differs from the original one. Even is because the dissipativity relationship seems of no direct the definition of the angular error ∆# calls for some caution use to analyze the stability properties of the incremental because of the nonlinear nature of angular variables. dynamics. The energy