J.J. Benjamin Biemond

Tracking Control for Hybrid Systems With State-Triggered Jumps

This paper addresses the tracking problem in which the controller should stabilize time-varying reference trajectories of hybrid systems. Despite the fact that discrete events (or jumps) in hybrid systems can often not be controlled directly, as, e.g., is the case in impacting mechanical systems, the controller should still stabilize the desired trajectory. A major complication in the analysis of this hybrid tracking problem is that, in general, the jump times of the plant do not coincide with those of the reference trajectory. Consequently, the conventional Euclidean tracking error does not converge to zero, even if trajectories converge to the reference trajectory in between jumps, and the jump times converge to those of the reference trajectory. Hence, standard control approaches can not be applied. We propose a novel definition of the tracking error that overcomes this problem and formulate Lyapunov-based conditions for the global asymptotic stability of the hybrid reference trajectory. Using these conditions, we design hysteresis-based controllers that solve the hybrid tracking problem for two exemplary systems, including the well-known bouncing ball problem.

Tracking control for hybrid systems via embedding of known reference trajectories

We study the problem of designing controllers to track time-varying state trajectories for plants modeled as hybrid dynamical systems, which are systems with both continuous and discrete dynamics. The reference trajectories are given by functions that may exhibit jumps. The class of controllers considered are also modeled as hybrid systems. These are designed to guarantee stability of tracking and that the difference between the plants state and the reference trajectory converges to zero. Using recently developed tools for the study of asymptotic stability in hybrid systems, we recast the tracking problem as the problem of stabilizing a closed set and derive conditions for the design of tracking controllers for hybrid reference trajectories with the property that the jump times of the plant coincide with those of the given reference trajectories. The approach is illustrated in examples.

A complete bifurcation analysis of planar conewise affine systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in continuous, piecewise affine systems defined on a conic partition of the plane. To guarantee completeness of the obtained limit sets, new conditions for the existence or absence of closed orbits are combined with the study of return maps. With these results a complete bifurcation analysis of a class of planar conewise affine systems is presented.

Constructing distance functions and piecewise quadratic Lyapunov functions for stability of hybrid trajectories

Characterising the distance between hybrid trajectories is crucial for solving tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. When the Euclidean distance function is used, the socalled “peaking phenomenon” for hybrid systems arises, which forms a major obstacle as trajectories cannot be stable in the sense of Lyapunov using such a distance. Therefore, in this paper, a novel and systematic way of designing appropriate distance functions is proposed that overcomes this hurdle and enables the derivation of sufficient Lyapunov-type conditions, using minimal or maximal average dwell-time arguments, for the stability of a hybrid trajectory. A constructive design method for piecewise quadratic Lyapunov functions is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. Finally, we illustrate our results with an example.

Estimation of basins of attraction for controlled systems with input saturation and time-delays

Abstract Basins of attraction are instrumental to study the effect of input saturation in control systems, as these sets characterise the initial conditions for which the control strategy induces attraction to the desired equilibrium. In this paper, we describe these sets when the open-loop system is exponentially unstable and the system is controlled by a single actuator with both constant time-delays and saturation. Estimates of the basin of attraction are provided and the allowable time-delay in the control loop is determined with a novel piecewise quadratic Lyapunov-Krasovskii functional that exploits the piecewise affine nature of the system. As this approach leads to sufficient, but not to necessary conditions for attractivity, we present simulations of an exemplary system to show the applicability of the results.

Dynamical collapse of trajectories

Friction induces unexpected dynamical behaviour. In the paradigmatic pendulum and double-well systems with friction, modelled with differential inclusions, distinct trajectories can collapse onto a single point. Transversal homoclinic orbits display collapse and generate chaotic saddles with forward dynamics that is qualitatively different from the backward dynamics. The space of initial conditions converging to the chaotic saddle is fractal, but the set of points diverging from it is not: friction destroys the complexity of the forward dynamics by generating a unique horseshoe-like topology.

Stability Analysis of Equilibria of Linear Delay Complementarity Systems

We present stability criteria for equilibria of a class of linear complementarity systems, subjected to discrete and distributed delay. We present necessary and sufficient conditions for local exponential stability, inferred from the spectrum location of a corresponding system of delay differential algebraic equations. Subsequently, we obtain sufficient LMI-based conditions for global asymptotic stability using Lyapunov–Krasovskii functionals.

Structural stability of equilibrium sets for a class of discontinuous vector fields

A class of discontinuous vector fields is investigated, where equilibria are generically positioned in an interval in the phase space, and equilibria are not isolated points. The dynamics near such an equilibrium set is studied, and it is shown that the structural stability of trajectories near the equilibrium sets is determined by the local dynamics near the endpoints of this interval. Based on this result, sufficient conditions for structural stability of equilibrium sets in planar systems are given, and two new bifurcations are identified. The results are illustrated by application to a controlled mechanical system with dry friction.