The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

The stability of mechanical systems with positional non-conservative forces

A classical problem is discussed, namely, the influence of the structure of the applied forces on the stability of the equilibrium position of an autonomous mechanical system. Several propositions extending the Thomson-Tate-Chetayev theorems to systems with non-conservative positional forces are proved.     

Nonholonomic mechanical systems and stabilization of motion

Theoretical results on the stability and stabilization of the steady-state motion of nonholonomic systems are systematized. A set of theorems on stability and controllability is formulated. Numerous applications of these theoretical results are pointed out. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 117–158, 2005.     

The stability and stabilization of non-linear,non-stationary mechanical systems

Mechanical systems acted upon by extremely non-linear positional forces are considered. The decomposition method is used to determine the sufficient conditions for asymptotic stability of an equilibrium. Problems of stabilizing the equilibrium of non-linear, non-stationary systems with specified potential forces by adding forces of different structure are studied. For systems with a non-stationary, homogeneous, positive-definite potential, the possibility of stabilization by linear dissipative forces, uncharacteristic of linear systems, is established. For systems with an even number of coordinates n ≥ 4, in the presence of dissipative forces with complete dissipation, the possibility of vibrational stabilization by adding circular and gyroscopic forces with coefficients fluctuating about zero is demonstrated.     

Problems of practical stability and stabilization of motion in discrete-time systems

The paper considers some problems of practical stability of motion for systems of difference equations. Stability theorems and criteria are given for cases with various phase constraints. Stabilization of discretetime systems to practical stability levels is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 118–124, 1988.     

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.     

Matching and stabilization of discrete mechanical systems

Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Partial stability and stabilization of dynamical systems

We prove the theorem on necessary and sufficient conditions of partial instability and the theorem on partial stabilization of nonlinear dynamical systems. We obtain sufficient conditions of controllability for systems linear with respect to control. We also study the problem of control and stabilization of an angular motion of a solid body by rotors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 186–193, February, 1995.     

The Ziegler effect in a non-conservative mechanical system

The destabilization of the stable equilibrium of a non-conservative system under the action of an infinitesimal linear viscous friction force is considered. In the case of low friction, the necessary and sufficient conditions for stability of a system with several degrees of freedom and, as a consequence, the conditions for the existence of the destabilization effect (Ziegler's effect) are obtained. Criteria for the stability of the equilibrium of a system with two degrees of freedom, in which the friction forces take arbitrary values, are constructed. The results of the investigation are applied to the problem of the stability of a two-link mechanism on a plane, and the stability regions and Ziegler's areas are constructed in the parameoter space of the problem.     

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.     

Indefinite damping in mechanical systems and gyroscopic stabilization

This paper deals with gyroscopic stabilization of the unstable system Mẍ + Dẋ + Kx = 0, with positive definite mass and stiffness matrices M and K, respectively, and an indefinite damping matrix D. The main question is for which skew-symmetric matrices G the system Mẍ + (D + G)ẋ + Kx = 0 can become stable? After investigating special cases we find an appropriate solution of the Lyapunov matrix equation for the general case. Examples show the deviation of the stability limit found by the Lyapunov method from the exact value.      

The stability of the equilibrium position of scleronomous mechanical systems

The problem of the stability of the equilibrium position of a scleronomic mechanical system is considered. The comparison method enables this problem to be reduced to the problem of the stability of scalar differential equations. The stability conditions are found for certain types of scalar comparison equations (Sections 1–4), and the sufficient conditions for the stability of the equilibrium positions of various scleronomous mechanical systems are determined from these (Sections 5–9).     

Structural transformations of non-conservative systems

The method of structural mappings of gyroscopic systems [1, 2] is developed for systems involving non-conservative positional forces. This technique, considered in the aspect of the legitimate use of the precessional equations of the precessional equations of the applied theory of gyroscopes, enables the difficulties associated with the presence of non-conservative structures in the initial equations to be overcome, and in many cases enables of the Thomson—Tait—Chetayev theorems to be used directly.     

Finite-time stability and stabilization of nonlinear stochastic hybrid systems

This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which exist impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.     

On stability and stabilization of some discrete dynamical systems

The paper formulates effective and nonimprovable stability conditions for a linear difference system involving 2 integer delays. The used technique combines algorithm of the discrete D‐decomposition method with some procedures of the polynomial theory. Contrary to the related existing results, the derived conditions are fully explicit with respect to both delays, which enables their simple applicability in various scientific and engineering areas. As an illustration, we show their importance in delayed feedback controls of discrete dynamical systems, with a particular emphasis put on stabilization of unstable steady states of the discrete logistic map.