Jan Maximilian Montenbruck

Practical synchronization with diffusive couplings

We investigate the problem of synchronizing nonidentical or perturbed nonlinear systems. In the considered setup, the systems are incapable to synchronize under diffusive couplings. Instead, assuming the quad property for each system, we derive conditions under which the synchronization error can be kept arbitrarily small by a proper choice of the interconnection structure. This motivates the definition of practical synchronization as an alternative synchronization notion for nonidentical or perturbed dynamical systems. The presented results are intimately related to synchronization of passive systems, but it is shown that the stronger quad assumption is essential in our framework. The proposed concept of practical synchronization translates directly into a notion of robust synchronization. Beyond that, the results open the way for an investigation of synchronization phenomena on unbalanced graphs, leading to the concept of cluster synchronization.

Compensating Drift Vector Fields With Gradient Vector Fields for Asymptotic Submanifold Stabilization

We derive sufficient conditions on a drift vector field to let an asymptotically stable invariant submanifold of an input-affine system without drift remain asymptotically stable for the system with drift. In doing so, we use the same feedback laws modulo control gain tuning, such that no new feedback laws need to be designed for the system with drift. Our main assumption is that the vector field of the input-affine system without drift assumes the form of a gradient vector field for given feedback laws. We show how one can assess the performance of the system with drift only via knowledge about the system without drift. Finally, we find that our results are relevant in synchronization problems and backstepping controllers for mechanical systems.

A robust nonlinear controller for nontrivial quadrotor maneuvers: Approach and verification

This paper presents a nonlinear control approach for quadrotor Micro Aerial Vehicles (MAVs), which combines a backstepping-like regulator based on the solution of a certain class of global output regulation problems for the rigid body equations on SO(3), a robust controller for the system with bounded disturbances, as well as a trajectory generator using a model predictive control method. The proposed algorithm is endowed with strong convergence properties so that it allows the quadrotor MAVs to reach almost all the desired attitudes. The control approach is implemented on a high-payload-capable quadcopter with unstructured dynamics and unknown disturbances. The performance of our algorithm is demonstrated through a series of experimental evaluations and comparisons with another control method on normal and aggressive trajectory tracking tasks.

On the Necessity of Diffusive Couplings in Linear Synchronization Problems With Quadratic Cost

We show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals. This holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left half-plane, whilst for eigenvalues in the open right half-plane, we present a counterexample in which the strong solution to the associated algebraic Riccati equation is not diffusive. For nonidentical systems satisfying the internal model principle for synchronization, we show that a certain part of the coupling must be diffusive. For equally chosen weights in the cost functional, we show that the dimension of the associated algebraic Riccati equation can be reduced significantly.

Practical and Robust Synchronization of Systems with Additive Linear Uncertainties

Abstract We investigate the synchronization of systems with additive uncertainties. In doing so, we establish a setup of diffusively coupled nonlinear systems that are perturbed by unknown linear functions, each. By assuming bounded solutions of the nominal uncoupled systems, we derive sufficient conditions for boundedness of the solutions of the coupled systems with uncertainties. Next, using the QUAD condition, we derive conditions for the synchronization error to remain bounded. Subsequently, we investigate the impact of the coupling strength on this bound and find that the bound can be made arbitrarily small for sufficiently large gains, thus establishing criteria for practical synchronization. Finally, we consider classes of uncertainties which consist of matrices whose maximal singular value is smaller than a specific value and show practical synchronization for all uncertainties belonging to that class. Therefore, we establish conditions for robust synchronization with respect to such a class. Our theoretical results are validated with a numerical example composed of perturbed Van der Pol oscillators.

Synchronization of diffusively coupled systems on compact Riemannian manifolds in the presence of drift

Abstract Recently, it has been shown that the synchronization manifold is an asymptotically stable invariant set of diffusively coupled systems on Riemannian manifolds. We regionally investigate the stability properties of the synchronization manifold when the systems are subject to drift. When the drift vector field is quad (i.e. satisfies a certain quadratic inequality) and the underlying Riemannian manifold is compact, we prove that a sufficiently large algebraic connectivity of the underlying graph is sufficient for the synchronization manifold to remain asymptotically stable. For drift vector fields which are quad or contracting, we explicitly characterize the rate at which the solution converges to the synchronization manifold. Our main result is that the synchronization manifold is asymptotically stable even for drift vector fields which are only locally Lipschitz continuous, as long as the algebraic connectivity of the underlying graph is sufficiently large.

Practical cluster synchronization of heterogeneous sytems on graphs with acyclic topology

We study the problem of practical cluster synchronization in networks of heterogeneous dynamical systems. The considered framework involves groups of identical dynamical systems, interacting with each other through linear couplings. The control objective studied in this paper is to achieve synchronization up to a possibly small error of all identical systems. Based on the two assumptions that all systems satisfy the QUAD condition and that the global coupling structure is acyclic, a constructive procedure for a coupling design ensuring practical cluster synchronization is proposed. For establishing the desired result, first, the synchronization of identical systems under external disturbances is studied. The main contribution is the extension of this result to the complete heterogeneous network. The theoretical results are illustrated and tested numerically on an exemplary network composed of several Van der Pol oscillators and Chuas circuits.

Linear systems with quadratic outputs

We study linear systems with quadratic outputs. Our goal is to analyze and control the evolution of such quadratic outputs under a given linear control system. In this spirit, we first attempt to find a differential equation governing the evolution of the outputs in order to consequently be able to control this evolution. We derive tangible algebraic conditions for being able to find such a differential equation and explicitly determine controls which bilinearize that differential equation.

Robust nonlinear control approach to nontrivial maneuvers and obstacle avoidance for quadrotor UAV under disturbances

Abstract In this paper, we present an onboard robust nonlinear control approach for quadrotor Unmanned Aerial Vehicles (UAVs) in the environments with disturbances and obstacles. The complete framework consists of an attitude controller based on the solution of global output regulation problems for SO( 3 ), a backstepping-like position controller, a 6 -dimensional wrench observer to estimate the unknown force and torque disturbances, and an online trajectory planner based on a model predictive control method with obstacle avoiding constraints. We prove the strong convergence properties of the proposed method both in theory and via real-robot experiments. The control approach is onboard implemented on a quadrotor UAV, and has been validated through intensive experiments and compared with other nonlinear control methods for waypoint navigation and large-tilted path following tasks in the presence of external disturbances, e.g. wind gusts. The presented approach has also been evaluated in the scenarios with randomly located obstacles.

Asymptotic stabilization of submanifolds embedded in Riemannian manifolds

We study control problems in which systems whose states spaces are Riemannian manifolds must be steered towards embedded submanifolds of their state spaces in a stable fashion, i.e., feedback laws rendering the given submanifold asymptotically stable are sought. Bringing the closed loop to the form of a gradient system with drift, we find that the gradient part can be scaled in a fashion so as to guarantee asymptotic stability of the desired submanifold. Under this circumstance, we show that solutions of the system can be brought to an arbitrarily small neighborhood of the submanifold in arbitrary time. Thereafter, we derive an algebraic condition under which the control vector fields can be brought to the desired form. Lastly, we recast our algebraic condition in terms of vertical and horizontal spaces for the case that the submanifold is an equivalence class.