Alexander V. Roup

Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincaré maps

The verge and foliot escapement mechanism of a mechanical clock is a classical example of a feedback regulator. In this paper we analyse the dynamics of this mechanism to understand its operation from a feedback perspective. Using impulsive differential equations and Poincaré maps to model the dynamics of this closed-loop system, we determine conditions under which the system possesses a limit cycle, and we analyse the period and amplitude of the oscillations in terms of the inertias of the colliding masses and their coefficient of restitution.

Adaptive stabilization of a class of nonlinear systems with nonparametric uncertainty

We consider adaptive stabilization for a class of nonlinear second-order systems. Interpreting the system states as position and velocity, the system is assumed to have unknown, nonparametric position-dependent damping and stiffness coefficients. Lyapunov methods are used to prove global convergence of the adaptive controller. Furthermore, the controller is shown to be able to reject constant disturbances and to asymptotically track constant commands. For illustration, the controller is used to stabilize the van der Pol limit cycle, the Duffing oscillator with multiple equilibria, and several other examples.

Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincare maps

The verge and foliot escapement mechanism of a mechanical clock is a classical example of a feedback regulator. In this paper we analyze the dynamics of this mechanism to understand its operation from a feedback perspective. Using impulsive differential equations and Poincare maps to model the dynamics of this closed-loop system, we determine conditions under which the system possesses a limit cycle, and we analyze the period and amplitude of the oscillations in terms of the inertias of the colliding masses and their coefficient of restitution.

On the dynamics of the escapement mechanism of a mechanical clock

The escapement mechanism of a mechanical clock is a classical example of a feedback regulator. We analyze the dynamics of this mechanism to understand its operation from a feedback perspective. Using impulsive differential equations to model the dynamics of this closed-loop system, we show that the system possesses a periodic orbit, and we analyze the period and amplitude of this orbit in terms of the inertias of the colliding masses and their coefficient of restitution.

Adaptive stabilization and disturbance rejection for first-order systems

We use a discontinuous adaptive control algorithm to achieve simultaneous stabilization and disturbance rejection for a first-order linear system. Lyapunov methods are used to prove stability of the closed-loop system and convergence to zero of the measurement. Closed-loop performance is compared to standard high-gain methods.

Stabilization of a class of nonlinear systems using direct adaptive control

We consider adaptive stabilization for a class of nonlinear second-order systems. Interpreting the system states as position and velocity, the system is assumed to have unknown, position-dependent damping and stiffness. Lyapunov methods are used to prove global convergence of the adaptive controller. For illustration, the controller is used to stabilize the Van der Pol limit cycle as well as the multiple equilibria of the Duffing oscillator.

Adaptive Stabilization of Linear Second-Order Systems with Bounded Time-Varying Coefficients

We consider adaptive stabilization for a class of linear time-varying second-order systems. Interpreting the system states as position and velocity, the system is assumed to have unknown, non-paranetric, bounded time-varying damping and stiffness coefficients. The coefficient bounds need not be known to implement the adaptive controller. Lyapunov methods are used to prove global convergence of the system states. For illustration, the controller is used to stabilize several example systems.

Adaptive stabilization of second-order non-minimum phase systems

We consider the adaptive output convergence for a class of unstable non-minimum-phase second-order relative-degree-one linear systems with unknown constant coefficients. The coefficient uncertainty bounds are required to be known and to satisfy an inequality constraint. We illustrate the use of the controller with several example systems.

Adaptive stabilization and disturbance rejection for siso minimum-phase relative-degree-one systems

We use a discontinuous control algorithm to adap tively stabilize SISO minimum-phase relative-degreeone linear systems in the presence of external disturbances. Lyapunov methods are used to prove stability of the closed-loop system and convergence to zero of the measurement. By proving Lyapunov stability, these results complement existing convergence results that utilize adaptive algorithms with discontinuous feedback.