^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.

The ball and the beam system (

The

Due to its importance several works related to the control of the

In this paper we propose a novel inverse Lyapunov-based procedure in combination with the energy shaping method to stabilize the

The rest of this paper is organized as follows. In Section

Consider the ^{1}

The ball and beam system.

The control objective is to bring all the states of system (

A brief description of the inverse Lyapunov method, inspired in the previous work of Ortega and García-Canseco [

Let us propose a candidate Lyapunov function for the closed-loop system energy function, of the form

In fact

Fixing the following auxiliary variable as ^{2}

Physically, we are choosing a convenient dissipation function,

On the other hand,

If we are able to shape the candidate Lyapunov function (

In this section we explain how to take the original expression of

Defining, ^{3}

From the above we have that

This implies that

Note that this equation has two unknown parameters, given by

Notice that

In this section we obtain the unknown control variables

Now, to obtain ^{4}

Notice that relation (

Hence, the needed controller, defined by (^{5}

We end this section introducing the following important remark.

Notice that we can always compute

To illustrate the geometrical estimation of the bound “

Level curves of the function

Since the obtained

The rest of the stability proof is based on LaSalle’s invariant theorem [

We finish this section by presenting the main proposition of this paper.

Consider system (

To show the effectiveness of the proposed nonlinear control strategy we have carried out some numerical simulations by means of the Matlab program. The original system p,arameters, with their respective physical restrictions, were set as
^{6}

Figure

Closed-loop response of the

In order to provide an intuitive idea of how good our nonlinear control strategy (

Closed-loop response of the

A comparative study between our control strategy and other control strategies presented in the literature for solving the stabilization of the

In this work we proposed a novel procedure to stabilize the

This research was supported by the Centro de Investigación en Computación of the Instituto Politecnico Nacional (CIC-IPN) and by the Secretaría de Investigación y Posgrado of the Instituto Politecnico Nacional (SIP-IPN), under Research Grants 20113116 and 20113280. A first version of this work was presented in the AMCA 2011 Conference.

Evidently,

The set

For simplicity, we use

Symbol

After some simple algebraic manipulations it is easy to show that

For this particular case, the condition of Remark