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This paper concerns active vibration damping of a frictionless physical inverted pendulum with a radially moving mass. The motion of the inverted pendulum is restricted to an admissible set. The proposed Proportional Derivative linear controller damps the inverted pendulum (which is anchored by a torsion spring to keep it in a stable upright position), exerting a force on the radially moving mass. The controller design procedure, which follows a traditional Lyapunov-based approach, tailors the energy behavior of the system described in Euler-Lagrange terms.

Vibrating mechanical systems constitute an important class of dynamical systems. In fact buildings, bridges, car suspensions, pacemakers, wind generators, and hi-fi speakers (or even the mammalian middle ear) are common examples of this type of systems. In physical terms all vibrating systems consist of an interplay between an energy-storing component and an energy-carrying component. Thus, the behavior of the system can be described in terms of energy changes, that is, the motion of the system results from energy conversions. For theoretical and technological reasons, the control of vibrating mechanical systems is an important domain of research, which has provided technological solutions to several problems concerning oscillatory behaviors of some important classes of dynamical systems, for example, active control of vibrations is applied to attenuate undesired oscillations in buildings affected by external forces such as strong winds and earthquakes (see, e.g., [

As far as mathematical tools are concerned, the control of vibrations has mainly been tackled via frequency-domain techniques, which are essentially restricted to linear systems (see, e.g., [

This paper focuses on active control of underdamped lumped nonlinear underactuated vibrating mechanical systems following an Energy-based approach, that is, the control of the vibratory behavior is tackled via the shaping of the energy flow which characterizes the system in dynamical terms. The control of vibrations is then considered in terms of the solution of a particular asymptotic stabilizing feedback control problem, around a selected equilibrium point. A stabilizing controller is then obtained following an Energy-based Lyapunov approach, which exploits the physical properties of the involved mechanical system. In this way conservative control strategies based either on high gains or on cancelled nonlinear terms are avoided. It must be pointed out that standard nonlinear strategies such as sliding modes control and feedback linearization are frequently characterized by conservativeness (see, e.g., [

The main objective of this paper is to propose an asymptotic stabilizing controller, for the active vibration damping in a nonlinear underactuated and frictionless mechanical system, only

The rest of the paper is organized as follows. Section

The dynamic system consists of a physical pendulum of mass

Inverted pendulum with a radially moving mass. The pendulum is anchored to the pivot point using a restoring torsion spring, which maintains the system in a stable upright position (but not asymptotically stable).

In order to describe the pendulum motion, the origin of the inertial frame is chosen at point

Therefore, the corresponding Euler-Lagrange equations, that is,

It is quite obvious that system (

matrix

The operator

Note that, if

In what follows we use the symbols

The above system is an underactuated and poorly damped mechanical system, since it has two degrees of freedom and it does not have one dissipative force in the nonactuated coordinate

Before establishing the control objective we introduce a necessary assumption:

the structural parameters of the original system satisfy the following relation:

The inequality (

The control objective is then posed as follows.

Find a smooth feedback

We must emphasize that the physical restrictions included in the formulation of the control problem are necessary to guarantee that the inverted pendulum can only moves inside a fraction of the upper half plane while the mass

In what follows we tackle the solution of the stabilizing feedback control problem.

Consider the following candidate Lyapunov function:

Under assumption (A1), the modified potential energy

Taking into account the passivity properties of system (

As

Before applying the well-known LaSalle invariance theorem (see, e.g., [

Consider the closed-loop system (see (

Now we are ready to apply the LaSalle invariance theorem. Let us define a compact set

The set

Let

This section concludes with the following proposition.

Under the assumptions of Lemma

It is easy to check that if assumption (A1) is relaxed, we can assure asymptotic stability of the closed-loop system. However, we cannot assure that

In order to illustrate the proposed energy-based feedback (PD linear) control law, we perform in the following section some computer-based simulations.

In order to carry out the simulation of the closed-loop system, we set the system parameters to be

The parameter value of

Note that the admissible set is given by

We simulate the closed-loop behavior of the nonlinear mechanical system using the Matlab + Simulink

We choose as the initial conditions vector

Simulation of the closed-loop system starting from

When choosing

Closed-loop response of the nonlinear system, when the initial condition is stated as

In this work, we presented a Lyapunov-based approach for the asymptotic stabilization of a frictionless inverted physical pendulum (which is maintained in the stable upright position, in the Lyapunov sense, via the inclusion of a torsion spring which anchors the pendulum to the pivot) with a radially moving mass. The motion of the pendulum is restricted to be in an admissible set

The stability analysis of the closed-loop system has been carried out by using the well-known LaSalle invariance theorem. It is worth mentioning that if a damping force is considered in the nonactuated coordinate then asymptotically stability of this device is reinforced.

Concerning the applicability of the proposed control law, the nonlinear mechanical system chosen here models, in a simplified way, the dynamics of rigid buildings restricted to oscillate in the plane (when affected by external excitations). We are interested in the attenuation of the effects of unknown disturbances (seismic forces) on the behavior of civil structures via smooth active control. In the considered model the radially moving mass is proposed as an active control element; the potentiality of such an actuator must be clarified via the evaluation of the energy consumption characteristics of the control law.

In the fist part we estimate the bound

First of all we define the set

From (A1) we conclude that the single solution of (

Now, since

From the above inequality we have that

This research was supported by the following mexican institutions: the Centro de Investigación en Computación of the Instituto Politécnico Nacional (CIC-IPN); the Coordinación de Posgrado e Investigación of the Instituto Politécnico Nacional, under Research Grant 20020247; the Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav-IPN); CONACYT-México, under Research Grant 32681-A.