With uncertainty by choice of zero dynamics

We propose an approach to output stabilization of multiply connected control systems with uncertainty based on structural decomposition of the original system and asymptotic invariance methods. The proposed approach solves the stabilization problem for minimum-phase systems. Bounds are obtained on the rate of convergence of the stabilization algorithms. Conditions are derived expanding the class of vector controlled systems with uncertainty that are stabilizable by asymptotic invariance methods.     

On stabilization of an elastic system by fast-oscillating time-variant feedback

We study a class of elastic systems described by a (hyperbolic) second-order partial differential equation. Our working example is the equation of a vibrating string subject to a destabilizing linear disturbance. Our main goal is to establish conditions for stabilization and asymptotic stabilization of the equilibrium configuration of the string by applying to it fast oscillating controlled force. In the first situation studied we assume that the string is subject to damping; after that we consider the same system without damping. We extend the tools of high-order averaging and of chronological calculus for studying the stability of this distributed parameter system.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complete damping of a linear autonomous differential-difference system by a controller of the same type

For a spectrally controllable linear autonomous system of retarded type with commensurable delays, we construct a dynamic state feedback ensuring a complete damping of the original system. The closed-loop system is of retarded type as well. The results are illustrated by an example.     

Attitude stabilization of a rigid body in conditions of decreasing dissipation

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.     

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

Finite spectrum assignment and complete damping of a differential system of the neutral type by a single controller

For a linear autonomous system of the neutral type with commensurable delays, we construct a dynamic feedback that ensures a complete damping of the original system and an asymptotically stable finite spectrum of the closed-loop system. The results are illustrated by examples.     

Mathematical modelling and stabilization of large flexible spacecraft subject to orbital perturbation

In this paper the problem of modelling of large flexible spacecraft and their stabilization under the influence of orbital (radial) perturbation is considered. A complete dynamics of the spacecraft consisting of a rigid bus and a flexible beam is derived using Hamilton's principle. The equations of motion consist of a coupled system of partial differential equations governing the vibration of the flexible beam and ordinary differential equations describing the translational and rotational motions of the rigid bus. The asymptotic stability of the system is proved using Lyapunov's approach. Simple feedback controls are suggested for the stabilization of the system. For illustration, numerical simulations are carried out, giving interesting results.     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

Stability and Stabilization Criteria for a Class of Uncertain Neutral Systems with Time-Varying Delays

In this paper, the robust asymptotic stability and stabilization for a class of uncertain neutral system with time-varying delays are considered. Based on the Lyapunov-Krasovskii functional theory, some stability and stabilization criteria are derived. Delay-dependent and delay-independent criteria are proposed for the stability and stabilization of the considered systems. State and output feedbacks are considered to stabilize the uncertain neutral system. A linear matrix inequality approach and a genetic algorithm are used to solve the stability and stabilization problems. Finally, some numerical examples are shown to illustrate the use of the obtained results.The research reported here was supported by the National Science Council of Taiwan under Grant NSC 91-2213-E-214-016.Communicated by C. T. Leondes     

Spectral reduction, complete damping, and stabilization of a delay system by a single controller

For a spectrally controllable linear autonomous differential-difference system of retarded type, we construct a dynamic state feedback that provides the complete damping of the original system and a finite spectrum of the closed-loop system. The closed-loop system can be made asymptotically stable by an appropriate choice of the spectrum. The results are illustrated by examples.     

多时滞线性切换系统的稳定性及其反馈镇定

研究了具有多时滞线性切换系统的稳定性及其反馈镇定问题,利用完备性条件、矩阵分解与二次Lyapunov泛函,给出了多时滞切换系统渐近稳定的充分条件和切换律设计方法.在此基础上,研究了这类系统的镇定控制问题,设计了保证系统时滞独立渐近镇定的控制器.     

Dimension Reduction methods for Infinite Dimensional Systems

From experiments and also from computer simulation of dynamical systems it is well known that for many dynamical phenomena in physics or engineering, which are modelled by infinite dimensional dynamical systems, the asymptotic behavior can be accurately described by replacing the original infinite dimensional system by a low dimensional system represented by so‐called essential variables. Such a dimension reduction of a dynamical system turns out to be central, both for a qualitative and quantitative understanding of its behaviour. In [1] Approximate Inertial Manifolds are presented, which perform extremely well for nonlinear evolution equations, but don't work as expected for the dynamics of a fluid conveying tube. By comparing the results for different internal damping values it can be seen that the larger gaps and the location of the cluster point in the spectrum for the weaker damping improve the approximation quality considerably.     

Boundary stabilization of non-classical micro-scale beams

In this paper, the problem of boundary stabilization of a vibrating non-classical micro-scale Euler–Bernoulli beam is considered. In non-classical micro-beams, the governing Partial Differential Equation (PDE) of motion is obtained based on the non-classical continuum mechanics which introduces material length scale parameters. In this research, linear boundary control laws are constructed to stabilize the free vibration of non-classical micro-beams which its governing PDE is derived based on the modified strain gradient theory as one of the most inclusive non-classical continuum theories. Well-posedness and asymptotic stabilization of the closed loop system are investigated for both cases of complete and incomplete boundary control inputs. To illustrate the performance of the designed controllers, the closed loop PDE model of the system is simulated via Finite Element Method (FEM). To this end, new strain gradient beam element stiffness and mass matrices are derived in this work.     

一类含有小迟滞的奇摄动方程组的渐近解

本文研究了一类含有小迟滞的奇摄动方程组的渐近解.利用原问题的退化形式和伸长变量,依据边界层特有的性质,获得了边界层的渐近解.推广了奇摄动方程组初值问题和小迟滞问题的研究结果.     

Approach to the stabilization of unstable periodic solutions of autonomous systems of partial differential equations

We suggest an approach to the stabilization of unstable periodic solutions of autonomous systems of partial differential equations based on the introduction of a derivative system in which each periodic solution of the original system is stationary. By using the introduction of an additional space into the derivative system, we suggest to stabilize its stationary solution corresponding to a periodic solution of the original system. This approach permits effectively obtaining a complete ordered set of functions corresponding to an unstable cycle of the original system. We consider an example of stabilization of an unstable cycle in the Kuramoto-Tsuzuki system.     

Passivity-based control and stability analysis for hydro-turbine governing systems

Low-frequency oscillations can occur in hydropower systems under in the new context of power system and the classical controller for hydro-turbine governing systems need to be enhanced with the purpose of improving its stability. We propose a controller based on passivity theory with the aim of damping oscillations in a power system. Passivity-based control arises as a natural choice for hydro-turbine governing system since its open-loop dynamic has a port-Hamiltonian structure, which allows designing a controller that preserves the passive structure in closed-loop via interconnection and damping reassignment. The proposed controller considers the complete non-linear model of the system and guarantees global asymptotic stability in the sense of Lyapunov. Time-domain simulations demonstrate the robustness and proper performance of the proposed methodology under different operative conditions when is compared with the classical controllers.     

A variational approach to a shape design problem for the wave equation

The problem of determining the optimal damping set for the stabilization of the wave equation may be not well-posed. By means of a vector variational reformulation and use of gradient Young measures, we present a general methodology to relax this kind of problems. From the optimal Young measure associated with the relaxed problem, we obtain information concerning minimizing sequences for the original problem as well as continuity properties of the relaxed cost function. To cite this article: A. Münch et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.