Nonlinear Systems

The theory of (controlled) invariant (co-)distributions is reviewed, emphasizing the theory of liftings of vector fields, one-forms and (co-)distributions to the tangent and cotangent bundle. In particular, it is shown how invariant distributions can be equivalently described as invariant submanifolds of the tangent and cotangent bundle. This naturally leads to the notion of an invariant Lagrangian subbundle of the Whitney sum of tangent and cotangent bundle, which amounts to a special case of the central equation of contraction analysis. The interconnection of the prolongation of a nonlinear control system (living on the tangent bundle of the state space manifold) with its Hamiltonian extension (defined on the cotangent bundle) is shown to result in a differential Hamiltonian system. The invariant submanifolds of this differential Hamiltonian system corresponding to Lagrangian subbundles are seen to result in general differential Riccati and differential Lyapunov equations. The established framework thus yields a geometric underpinning of recent advances in contraction analysis and convergent dynamics.

The spatially distributed structure of complex systems motivates the idea of distributed control. In a distributed control system, the subsystems are controlled by local controllers through information exchange with neighboring agents for coordination purposes. One of the major difficulties of distributed control is due to the complex characteristics such as nonlinearity, dimensionality, uncertainty, and information constraints. This chapter introduces small-gain methods for distributed control of nonlinear systems. In particular, a cyclic-small-gain result in digraphs is presented as an extension of the standard nonlinear small-gain theorem. It is shown that the new result is extremely useful for distributed control of nonlinear systems. Specifically, this chapter first gives a cyclic-small-gain design for distributed output-feedback control of nonlinear systems. Then, an application to formation control problem of nonholonomic mobile robots with a fixed information exchange topology is presented.

Convergent systems are systems that have a uniquely defined globally asymptotically stable steady-state solution. Asymptotically stable linear systems excited by a bounded time varying signal are convergent. Together with the superposition principle, the convergence property forms a foundation for a large number of analysis and (control) design tools for linear systems. Nonlinear convergent systems are in many ways similar to linear systems and are, therefore, in a certain sense simple, although the superposition principle does not hold. This simplicity allows one to solve a number of analysis and design problems for nonlinear systems and makes the convergence property highly instrumental for practical applications. In this chapter, we review the notion of convergent systems and its applications to various analyses and design problems within the field of systems and control.

Generally speaking, for a network of interconnected systems, synchronisation consists in the mutual coordination of the systems’ motions to reach a common behaviour. For homogeneous systems that have identical dynamics this typically consists in asymptotically stabilising a common equilibrium set. In the case of heterogeneous networks, in which systems may have different parameters and even different dynamics, there may exist no common equilibrium but an emergent behaviour arises. Inherent to the network, this is determined by the connection graph but it is independent of the interconnection strength. Thus, the dynamic behaviour of the networked systems is fully characterised in terms of two properties whose study may be recast in the domain of stability theory through the analysis of two interconnected dynamical systems evolving in orthogonal spaces: the emergent dynamics and the synchronisation errors relative to the common behaviour. Based on this premise, we present some results on robust stability by which one may assess the conditions for practical asymptotic synchronisation of networked systems. As an illustration, we broach a brief case-study on mutual synchronisation of heterogeneous chaotic oscillators.

This chapter discusses a synchronization problem, anticipating synchronization, and its application in control design for nonlinear systems with delays. The anticipating synchronization phenomena was initially reported by Voss (2000), and Oguchi and Nijmeijer (2005) then generalized it from the framework of control theory. This chapter revisits the anticipating synchronization problem and introduces a state predictor based on synchronization. Furthermore, we discuss recent progress on predictor design for nonlinear systems with delays.

Time-delays are important components of many systems from engineering, economics and the life sciences, due to the fact that the transfer of material, energy and information is mostly not instantaneous. They appear for instance as computation and communication lags, they model transport phenomena and hereditary effects and they arise as feedback delays in control loops. The aim of the chapter is to present a guided tour on stand-alone and interconnected systems with delays, thereby explaining some important qualitative properties. The focus rather lies on the main ideas as technical details are avoided. Different mechanisms with which delays can interact with the system are outlined, with the emphasis on the effects of delays on stability. It is clarified how these mechanisms affect control design problems. Not only limitations induced by delays in control loops are discussed, but also opportunities to use delays in the construction of controllers. Finally, extensions of these results toward networks of interconnected dynamical systems are discussed, with the focus on relative stability problems, in particular the synchronization problem.

We discuss the emergence of oscillations in networks of single-input–single-output systems that interact via linear time-delay coupling functions. Although the systems itself are inert, that is, their solutions converge to a globally stable equilibrium, in the presence of coupling, the network of systems exhibits ongoing oscillatory activity. We address the problem of emergence of oscillations by deriving conditions for; 1. solutions of the time-delay coupled systems to be bounded, 2. the network equilibrium to be unique, and 3. the network equilibrium to be unstable. If these conditions are all satisfied, the time-delay coupled inert systems have a nontrivial oscillatory solution. In addition, we show that a necessary condition for the emergence of oscillations in such networks is that the considered systems are at least of second order.

In this work, leader–follower synchronisation is considered for underactuated followers in an inhomogeneous multi-agent system. The goal is to synchronise the motion of a leader and an underactuated follower. Measurements of the leader’s position, velocity, acceleration and jerk are available, while the dynamics of the leader is unknown. The leader velocities are used as input for a constant bearing guidance algorithm to assure that the follower synchronises its motion to the leader. It is also shown that the proposed leader–follower scheme can be applied to multi-agent systems that are subjected to unknown environmental disturbances. Furthermore, the trajectory of the leader does not need to be known. The closed-loop dynamics are analysed and it is shown that under certain conditions all solutions remain bounded and the synchronisation error kinematics are shown to be integral input-to-state stable with respect to changes in the unactuated sway velocity. For straight-line motions, i.e. where the desired yaw rate and sway velocity go to zero, synchronisation is achieved. Simulation results are presented to validate the proposed control strategy.

In this chapter, position control strategies via force feedback are presented for standard mechanical systems in the port-Hamiltonian framework. The presented control strategies require a set of coordinate transformations, since force feedback in the port-Hamiltonian framework is not straightforward. With the coordinate transformations force feedback can be realized while preserving the port-Hamiltonian structure. The port-Hamiltonian formalism offers a modeling framework with a clear physical structure and other properties that can often be exploited for control design purposes, which is why we believe it is important to preserve the structure. The proposed control strategies offer an alternative solution to position control with more tuning freedom and exploit knowledge of the system dynamics.

The endogenous configuration space approach is a control theory-oriented methodology of robotics research, dedicated to mobile manipulators. A cornerstone of the approach is a parameterized control system with output whose input–output map constitutes the mobile manipulator’s kinematics. An endogenous configuration consists of the control function and of the vector of output function parameters representing the joint positions of the on-board manipulator. The mobile manipulator’s Jacobian is defined as the input–output map of the linear approximation to the control system. Regular and singular endogenous configurations are introduced. The regular endogenous configuration corresponds to the local output controllability of the control system, while the singular configuration coincides with a singular optimal control-parameter pair of the control system. The inverse kinematics problem is formulated as a control problem in a driftless control system. A collection of Jacobian kinematics inverses is presented, leading to Jacobian motion planning algorithms. Performance measures of the mobile manipulator are introduced.