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A procedure to design an asymptotically stable second-order sliding mode observer for a class of single input single output (SISO) nonlinear systems in normal form is presented. The observer converges to the system state in spite of the existence of bounded disturbances and parameter uncertainties affecting the system dynamics. At the same time, the observer estimates the disturbances without the use of an additional filter to recover the equivalent control. The observer design is modular; each module of the observer is applied to each equation of state of the plant. Because of this, the proposed observer can be applied to a broader class of dynamic systems. The performance of the observer is illustrated in numerical and experimental form.

A state observer is a system that provides an estimate of the state variables, parameters, and disturbances of a given real system, based on measurements of the inputs and outputs of that system. From the viewpoint of dynamic systems, the observer design problem arises from the need for information to carry out control, monitoring, and modeling. In this way, it is a major problem in implementing a control system in general [

In many cases, the design of observers depends largely on the model of the plant and on the knowledge of information of the nominal values on its parameters. These observers need an accurate model of the plant; if there are external disturbances or parameter uncertainties, the performance is diminished, causing stability problems in the closed loop system. Therefore, the problem of observation of systems with unknown inputs has been one of the most important problems in control theory during the last decades.

Currently, there are different observers that exhibit some degree of robustness and can be applied to linear and nonlinear systems. Because the observers design problem is related to the controllers design problem, different robust control techniques have been used to design robust observers; see, for example, [

An observer for Lipschitz nonlinear systems, with not only unknown inputs but also measurement noise when the observer matching condition is not satisfied, is presented in [

In [

Other important proposals are [

Reference [

In this paper, an observer for SISO nonlinear systems in normal form, with parameter uncertainties and external disturbances, is presented. The observer is based on the results presented in [

The organization of this paper is as follows. Section

Consider a nonlinear SISO system described by

Consider that system (

The problem is to design an observer for system (

Consider that system (

The proposed observer has the form

As we can see, for each state of system (

The stability analysis of system (

Take the first two equations of system (

For system (

The proof of this theorem is in [

Therefore, a set of constants

Now, the dynamics of variables

Now, to analyze block

Finally, we analyze block

It is important to note that the proposed observer, unlike others such as those presented in [

This section illustrates the performance of the observer through two examples. The first is a numerical simulation where the observer is used to estimate the state vector of a scaled version of the Rössler system under chaotic behavior.

The second example is an experiment where the observer is included in a control system; the observer estimates and compensates the velocities and disturbance terms due to parametric uncertainties and unmodeled dynamics in a mechanical system; thus, the closed loop system became robust.

Consider a scaled version of the Rössler system modeled by

Using the transformation

Figure

Behavior of the plant and the observer in

Figure

Behavior of the plant and the observer in

Disturbance and identified disturbance.

In practice, each stage of the observer has an actual sliding mode instead of an ideal sliding mode, which produces high frequency components with small amplitude in the observed variables; these components produce a cumulative effect. Because of this fact, a greater content of high frequency components (see Figure

In this subsection, we use the proposed observer to estimate velocities and disturbances in a mechanical system; the estimated disturbances are used in the controller to compensate the real ones in the plant. As a result, we obtain a robust closed loop system, like those where a sliding mode controller is applied, but with a control signal free from high frequency components.

Consider the pendulum shown in Figure

The pendulum used in the experiments.

To design a controller, we define the error variables

Now, we propose the additional term

For the plant given by (

In order to observe the effect of disturbance compensation, in the first twenty seconds of the experiments, the control input does not include the disturbance identified; after this time, it is incorporated. The experimental results are shown in Figures

Measured angular position

Estimated velocity

Estimated disturbances

Control input

Reference signal

Tracking error.

Figure

Now, we analyze the closed loop performance. In Figure

Finally, Figures

This paper presents the design of a robust observer for a class of SISO nonlinear systems with full relative degree. This observer guarantees, analytically, the asymptotic convergence to the state of the plant in spite of the parametric uncertainties and bounded external disturbances. In addition, the observer identifies the disturbances in the plant without the use of an additional low-pass filter. This observer can be extended directly to other types of multiple input multiple output (MIMO) systems such as mechanical systems modeled by Lagrange equations and to those that can be brought to a normal form.

The performance of the observer has been illustrated in a particular control system but it is possible to make a generalization of this application to mechanical systems and other kinds of dynamic systems. Moreover, the example of the observer for the Rössler system opens the possibility to use the observer for the synchronization of chaotic systems.

The observer also presents certain practical problems. The use of several cascading stages can degrade the performance of the final stages due to the accumulation of numerical errors, because the variables recovered at each stage are used in the following stages. In order to minimize this problem, the observer needs to be implemented in real-time systems with small sampling times. Under these conditions, no practical problems occur even in third-order systems.

It is important to note that the experimental results are very similar to those obtained with other more complex control techniques such as

The authors declare that there is no conflict of interests regarding the publication of this paper.