Robust stabilization of the angular velocity of a rigid body

The problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed. It is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances, whereas in the case of non-matched disturbances no such a feedback law generically exists. Hence, a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense. Explicit expressions of the control laws solving the proposed robust stabilization problems are given.     

Numerical simulation of pyroclastic density currents using locally refined Cartesian grids

Pyroclastic density currents are ground hugging, hot, gas–particle flows representing the most hazardous events of explosive volcanism. Their impact on structures is a function of dynamic pressure, which expresses the lateral load that such currents exert over buildings. Several critical issues arise in the numerical simulation of such flows, which involve a rheologically complex fluid that evolves over a wide range of turbulence scales, and moves over a complex topography. In this paper we consider a numerical technique that aims to cope with the difficulties encountered in the domain discretization when an adequate resolution in the regions of interest is required. Without resorting to time-consuming body-fitted grid generation approaches, we use Cartesian grids locally refined near the ground surface and the volcanic vent in order to reconstruct the steep velocity and particle concentration gradients. The grid generation process is carried out by an efficient and automatic tool, regardless of the geometric complexity. We show how analog experiments can be matched with numerical simulations for capturing the essential physics of the multiphase flow, obtaining calculated values of dynamic pressure in reasonable agreement with the experimental measurements. These outcomes encourage future application of the method for the assessment of the impact of pyroclastic density currents at the natural scale.     

Linear Problems of Optimal Control of Multiple-Valued Trajectories

This paper deals with rapidity and rendezvous problems and with problems with multi-valued and vector quality criteria for objects whose behavior is described by linear differential inclusions that contain controls.     

Periodic motion planning for virtually constrained Euler–Lagrange systems

The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler–Lagrange systems subject to (n-1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler–Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart–pendulum system and how then to make them orbitally exponentially stable following the machinery developed in [A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164–1176]. The extension here also considers time-varying virtual constraints.     

Coordination of multi-agent Euler–Lagrange systems via energy-shaping: Networking improves robustness

In this paper, the robust coordination of multi-agent systems via energy-shaping is studied. The agents are nonidentical, Euler–Lagrange systems with uncertain parameters which are regulated (with and without exchange of information between the agents) by the classical energy-based controller where the potential energy function is shaped such that, if the parameters are known, all agents converge globally to the same desired constant equilibrium. Under parameter uncertainty, the globally asymptotically stable (GAS) equilibrium point is shifted away from its desired value and this paper shows that adding information exchange between the agents to the decentralized control policy improves the steady-state performance. More precisely, it proves that if the undirected communication graph is connected, the equilibrium of the networked controller is always closer (in a suitable metric) to the desired one than that of the decentralized controller. The result holds for all interconnection gains if the potential energy functions are quadratic, else, it is true for sufficiently large gains. An additional advantage of networking is that the asymptotic stabilization objective can be achieved by using lower gains into the loop. Some experimental results (using two nonlinear manipulators) given support to the main results of the paper.     

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula cannot be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula; however, we show in this article that this is the wrong choice and it may lead to divergence if time-dependent methods are used to march the solution to steady state. We develop, in this paper, the correct quadrature formula for these problems. This formula takes into account the degree of the polynomials involved. We show that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.     

Flocking of networked uncertain Euler–Lagrange systems on directed graphs

This paper investigates the flocking problem of networked nonlinear Euler–Lagrange systems with parametric uncertainties, and they are assumed to interact on directed graphs with a directed spanning tree. We propose an adaptive controller to achieve the flocking objective, and the resultant closed-loop networked system bears the cascade structure. Using a new similarity decomposition approach, a critical-characteristic-root based approach, and the input–output stability analysis, we demonstrate the convergence of the position/velocity synchronization errors among the uncertain nonlinear Euler–Lagrange agents. We also show that the velocities of the Euler–Lagrange systems converge to the weighted average velocity value. Simulation results are provided to demonstrate the performance of the proposed controller.     

On Continuous Dependence of Solutions of Impulsive Differential Inclusions and Impulse Control Problems

This article deals with problems of continuous dependence of solutions of impulsive differential inclusions on their initial conditions and right sides and of the use of these inclusions in control problems arising in systems with impulses.