The stabilization of program motions of controlled nonlinear mechanical systems

We consider a controlled nonlinear mechanical system described by the Lagrange equations. We determine the control forcesQ ₁ and the restrictions for the perturbationsQ ₂ acting on the mechanical system which allow to guarantee the asymptotic stability of the program motion of the system. We solve the problem of stabilization by the direct Lyapunov's method and the method of limiting functions and systems. In this case we can use the Lyapunov's functions having nonpositive derivatives. The following examples are considered: stabilization of program motions of mathematical pendulum with moving point of suspension and stabilization of program motions of rigid body with fixed point.     

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

On the stability of the steady-state motions of systems with quasicyclic coordinates

The stability of the steady-state motions of a system with quasicyclic coordinates under the action of potential and dissipative forces and also forces which depend on the quasicyclic velocities is investigated. The results are applied to the problem of the stability of the steadystate plane-parallel motions of a rotor on a shaft which is set up in elasticated bearings with a non-linear reaction /1/.

The stability of the stationary motions and relative equilibria of systems with a single cyclic (quasicyclic) coordinate has previously been investigated /2/ from a common point of view. The question of the stability of the stationary motions of systems with quasicyclic coordinates under the action of constant and dissipative forces has been considered in /3/. The results obtained in /2/ have been generalized /4/ to systems with several cyclic (quasicyclic) coordinates and, additionally, a third regime of uniform motions, which includes the regime considered in /3/, has also been investigated.     

Methods of stabilizing the motions of reversible dynamic systems using non-linear canonical transformations

Methods of synthesizing stabilizing and robust control laws for non-linear reversible systems which ensure asymptotic stability of programmed motions, specified figures of merit and decomposition of transients are considered. Non-linear canonical transformations of state space and the controls are obtained which simplify the synthesis and analysis of the laws of the stabilization of reversible dynamic systems.     

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

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.     

The stability of non-conservative systems and an estimate of the domain of attraction

The stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces act, is investigated. The condition for asymptotic stability is obtained using the Lyapunov function and an estimate of the domain of attraction is also found in terms of the system being considered. A precessional system is also examined. It is shown that the condition for the asymptotic stability of a system is the condition of acceptability in the sense of the stability of a precessional system. The results obtained are applied to the problem of the stabilization, using external moments, of the steady motion of a balanced gyroscope in gimbals.     

Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach

In this paper, the asymptotic stability problem for a class of neutral systems with discrete and distributed delays is considered. Based on linear matrix inequality, a delay-dependent criterion is proposed to guarantee asymptotic stability for such systems. Some numerical examples are given to illustrate our main result.     

Stability criteria for uncertain neutral systems with interval time-varying delays

This paper investigates asymptotic stability problem for neutral system with interval time-varying delays and two classes of uncertainties. Delay-dependent and delay-independent criteria are proposed to guarantee the asymptotic stability for our considered systems. Lyapunov–Krasovskii functional and Leibniz–Newton formula are applied to find the delay-dependent stability results. Linear matrix inequality (LMI) approach is used to solve the proposed conditions. Finally, some numerical examples are illustrated to show the improvement of this paper.     

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.     

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

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 influence of dissipative and constant forces on the form and stability of steady motions of mechanical systems with cyclic coordinates

Mechanical systems with cyclic coordinates subject to dissipative forces with complete dissipation and constant forces applied only to the cyclic variables are considered. Problems of the existence of steady motions in such systems and the conditions for their stability are discussed. It is shown, in particular, that if the Rayleigh function is proportional to the kinetic energy, the stability conditions for the steady motions of the system are the same as or (under certain assumptions) similar to such conditions for steady motions of a corresponding conservative system. The example of a physical pendulum is used to show that such conclusions are generally false: dissipative and constant forces may cause destabilization of stable motions of the system.     

基于LMI的时滞关联模糊大系统的分散镇定

讨论了具有时滞关联的连续时间模糊大系统的分散镇定问题 ,基于李亚普诺夫稳定性理论及大系统分散控制理论 ,给出了保证该时滞模糊大系统闭环渐近稳定的 LMI形式的充分条件 .满足这些充分条件的分散化并行分布补偿 ( DPDC)控制器参数可通过 MATLAB的 LMI工具箱有效的对其进行求取 .     

Stability analysis of a class of nonlinear positive switched systems with delays

This paper addresses the stability problem of delayed nonlinear positive switched systems whose subsystems are all positive. Both discrete-time systems and continuous-time systems are studied. In our analysis, the delays in systems can be unbounded. Two conditions are established to test the local asymptotic stability of the considered systems. The method to compute the domains of attraction is also proposed provided that the system is locally asymptotically stable. When reduced to general nonlinear positive systems, that is, the considered switched system consists of only one mode, an interesting conclusion follows that the proposed nonlinear positive system is locally asymptotically stable for all admissible delays and nonnegative nonlinearities which satisfy an extra condition at the origin, if and only if the system represented by the linear part is asymptotically stable for all admissible delays. Finally, a numerical example is presented to illustrate the obtained results.     

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.     

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.     

Delay-Dependent Criterion for Guaranteed Cost Control of Neutral Delay Systems

In this article, the guaranteed cost control problem for a class of neutral delay systems is investigated. A linear--quadratic cost function is considered as a performance measure for the closed-loop system. Based on the Lyapunov method, delay-dependent criteria, which are expressed in terms of matrix inequalities, are proposed to guarantee the asymptotic stability of the system. The matrix inequalities can be solved easily by various efficient optimization algorithms.