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 the equilibrium positions of non-linear non-autonomous mechanical systems

The problem of the stability of the equilibrium positions for a certain class of non-linear mechanical systems under the action of time-dependent quasipotential and dissipative-accelerating forces is considered. A method is proposed for constructing Lyapunov functions for these systems. Sufficient conditions for the stability of an equilibrium position both with respect to all of the variables as well as with respect to some of the variables are determined using the direct Lyapunov method.     

The brachistochrone motion of a mechanical system with non-holonomic, non-linear and non-stationary constraints

Further to previous studies /1, 2/ of the brachistochrone motion of non-holonomic mechanical systems with linear homogeneous constraints, consideration is given here to non-holonomic, non-linear and non-stationary mechanical systems. The problem is to formulate the differential equations of the brachistochrone motion of non-holonomic, non-linear and non-stationary mechanical systems and to determine the additional forces which must be introduced in order to implement motion of this type.     

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.     

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)     

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.     

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 stability of the parametric oscillations of second-order non-linear systems

The parametric oscillations of strongly non-linear systems with one degree of freedom are considered using a more general definition of these oscillations than the generally accepted definition. Stability criteria, that are verifiable using the signs of the derivatives of the amplitude-frequency characteristics, are found for the two families of periodic solutions corresponding to the fundamental parametric resonance. A condition is indicated under which the latter are monotonic and, as a result, one of the families is stable and the other is unstable. It is shown that, in a system with a concave non-monotonic elastic characteristic, the stable family loses stability for fairly large amplitudes and this effect is not revealed by the well-known analytical methods of non-linear mechanics.     

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.      

On the eventual stability of perturbed non-linear systems

Summary In this paper, general types of necessary conditions are obtained for the set x=0 of a system of differential equation to be eventually uniformly stable. These results are then employed to study the effect of certain general types of perturbation on the set x=0. Entrata in Redazione il 6 febbraio 1976.     

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.     

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.     

The stability of a specific class of mechanical systems

A special class of mechanical systems is considered, the linearized equations of which either belong to the class of time-varying systems, reducible to stationary systems using constructive Lyapunov transformations or to systems close to these. A method of decomposing of the matrices of a system, which differs from the traditional method, is proposed for investigating of the stability of motion. It is shown that the conclusions concerning the stability are more complete in the case of this decomposition of the system matrix. A number of problems on the stability of motion of various mechanical systems is considered as examples.     

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.     

The global asymptotic stability and stabilization in nonlinear cascade systems with delay

We study certain sufficient conditions for the local and global uniform asymptotic stability, as well as the stabilizability of the equilibrium in cascade systems of delay differential equations. As distinct from the known results, the assertions presented in this paper are also valid for the cases, when the right-hand sides of equations are nonlinear and depend on time or arbitrarily depend on the historical data of the system. We prove that the use of auxiliary constant-sign functionals and functions with constant-sign derivatives essentially simplifies the statement of sufficient conditions for the asymptotic stability of a cascade. We adduce an example which illustrates the use of the obtained results. It demonstrates that the proposed procedure makes the study of the asymptotic stability and the construction of a stabilizing control easier in comparison with the traditional methods.     

Input-to-state stability and stabilization for switched nonlinear positive systems

Input-to-state stability (ISS) analysis and stabilization are concerned in this paper for switched nonlinear positive systems (SNPS), where the deterministic and random switching are both included. For general SNPS, switched affine nonlinear positive systems (SANPS) and switched linear positive systems (SLPS) with deterministic and some kinds of random ”slow” switching, some criterions on ISS are provided. From the criterions for SANPS and SLPS, the ISS properties can be judged just by the differential, algebraic and switching characteristics of the systems. Further, based on the criterions for SANPS and SLPS, some state feedback controllers are designed such that the closed-loop systems be positive, ISS or ISS in some stochastic senses. Four simulation examples verify the validity of our results.