On the stability of equilibria of nonholonomic systems with nonlinear constraints

This paper proposes a method for the stability analysis of deterministic switched systems. Two motivational examples are introduced (nonholonomic system and constrained pendulum). The finite collection of models consists of nonlinear models, and a switching sequence is arbitrary. It is supposed that there is no jump in the state at switching instants, and there is no Zeno behavior, i.e., there is a finite number of switches on every bounded interval. For the analysis of deterministic switched systems, the multiple Lyapunov functions are used, and the global exponential stability is proved. The exponentially stable equilibrium of systems is relevant for practice because such systems are robust to perturbations.

“…The stability of equilibrium of nonholonomic systems with nonlinear constraints was considered in Ref. [17]. In Ref.…”

Section: Two Examplesmentioning

confidence: 99%

Global exponential stability of switched systems

Filipović

2011

“…In this way, the algebraic criteria occurring in [14,16,[20][21][22][23] are substituted here by the criterion without the variable κ 1 . In this way, the existence of the vector e satisfying A 1 = λe, λ > 0 is proved.…”

Section: Theorem On Instability -Case Amentioning

confidence: 99%

Lyapunov-Kozlov method for singular cases

Čović

Djuric

Vesković

et al. 2011

Self Cite

Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.

“…This result is extended in [15] to a nonholonomic system with linear homogeneous constraints, in [16] to a nonholonomic system with linear nonhomogeneous constraints, and in [17] to a nonholonomic system with nonlinear constraints.…”

Section: Introductionmentioning

confidence: 99%

Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces

Vesković

Čović

Obradović

2011

Self Cite

The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative, conservative, and circulatory forces. The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity. It is assumed that the kinetic energy, the Rayleigh dissipation function, and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions. The results obtained here are partially generalized the results obtained by Kozlov et al. (Kozlov, V. V. The asymptotic motions of systems with dissipation. ). The results are illustrated by an example.