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This paper presents a methodology for controlling nonlinear time-varying minimum-phase underactuated systems affected by matched and unmatched perturbations. The proposed control structure consists of an integral sliding mode control coupled together with a global nonlinear

Much research in recent years has focused on the stabilization and control of mechanical systems operating under uncertain conditions such as external disturbances, uncertain parameters of the plant, and parasitic dynamics. These problems are present in real-world applications revealing, for example, instability, limit cycles, steady-state error, poor repeatability, or imprecisions.

In spite of the rich and diverse literature on the matter (see, e.g., [

Sliding modes are long recognized as a powerful control method to

In relevant works, Osuna et al. [

The aforementioned literature includes a linear

In this paper, an ISM control combined together with a nonlinear

This paper contributes to the following:

Presenting the design of a new controller by combining ISM control and a nonlinear

Developing a rigorous stability analysis, with a global solution to

Detailing a procedure to implement the Cao-Xu controller in a 1-DOF FJR manipulator

Presenting a comparative analysis of both controllers by means of numerical simulations with a trajectory tracking task

This paper is organized as follows. Section

This work aims to design a controller for nonlinear time-varying minimum-phase underactuated systems being affected by matched and unmatched perturbations. Let us denote

The block diagram of the closed-loop system with the proposed controller is depicted in Figure

Block diagram of the closed-loop system with the proposed controller, which consists of a combination of a

The next three definitions are provided in order to clarify the

The space

Let

Functions in

The steps required to design the controller

Consider a nonautonomous nonlinear system of the form

The functions

Let us assume the next structure for the controller

Let us define the sliding surface

Equation (

The product

In order to drive the trajectories of (

Once the trajectories reach the sliding surface, the dynamics of

By substituting (

Notice that, in (

Let us consider the nonautonomous nonlinear systems of the form

In the forthcoming analysis, the following assumptions are considered [

The functions

Assumption

By considering the full-information case, the static state-feedback controller

Let us consider the following hypothesis [

For some positive

Provided that Hypothesis

Consider the system (

Summarizing, the following result is obtained.

Consider system (

The conditions for the trajectories of (

In order to support the applicability and performance of the proposed controller (

The tracking control problem for this system can be established as follows.

In order to apply the proposed controller stated in Theorem

Block diagram of the methodology used for applying the proposed controller (

The first step to follow consists of designing a control law

Let us consider the following change of coordinates:

By using (

Now, let us define the reference signal

Using (

The expression (

Let us select the matrix

From (

Since the full-information case is considered, the performance output

Note that the functions

Now, let us define the smooth functions

Using (

Let us consider the system (

The performance of the proposed controller was compared with that corresponding to the nonlinear controller developed by Cao and Xu [

Let us consider the unperturbed version of the 1-DOF FJR manipulator described in (

Under the same line of reasoning presented in Section

By a proper selection of the gains

Now, we will prove that if

Now, by substituting (

A proper selection of the gains

Consider now the perturbed system (

Thus, the controller presented in [

The switching surface must also fulfill the following inequality:

This section presents the numerical simulation results obtained using the proposed controller and the Cao-Xu controller. In the following the proposed method and the Cao-Xu controller will be referred to as the iSMH controller and the Cao-Xu controller, respectively. Simulations were performed using the software MATLAB-SIMULINK®, using the solver ode45 Dormand-Prince. The parameters of the model (

The numerical simulations considered two trajectories to follow:

A sinusoidal desired link reference signal described by

A combination of sinusoids for the desired link reference signal; that is,

In order to provide a set of realistic gains for tuning the controllers, the parameters corresponding to an existing platform of Quanser [

The controller gains for both, the iSMH and Cao-Xu controllers, was tuned to obtain a similar behavior for the link and rotor position errors. These gains fulfill the constraints inherent of each controller. Furthermore, it is assumed that the magnitude of the maximum allowable voltage was 10 [volts]. In all simulations, the matched perturbation

For the first simulation, using the reference signal (

The link position error

First simulation. Link position error

In Figure

First simulation. Behavior of the sliding surfaces

For the second simulation, using the reference signal (

The link position error and joint position error for the iSMH and Cao-Xu controllers are depicted in Figure

Second simulation. Link position error

In Figure

Second simulation. Behavior of the sliding surfaces

The previous simulation results show that, although similar responses may be obtained using any of the controllers, the iSMH requires less energy to achieve the control objective, while the oscillatory behavior is significantly less than that of the Cao-Xu controller. The reason for such behavior is that the proposed controller attenuates the effect of the unmatched uncertainty

Effect of using the

The previous numerical simulations validate the proposed methodology for controlling underactuated nonlinear time-varying systems affected by matched and unmatched perturbations. Besides, it was observed that it was convenient to use the combination of the ISM control with a

It is important to remark that in practice, a saturation function can also be used instead of the signum function given in (

In this paper, a controller for time-varying minimum-phase underactuated systems affected by matched and unmatched perturbations was presented. The proposed controller uses a combination of an ISM controller with

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

This work was supported in part by CONACYT Project Cátedras 1537. C. Chavez would like to thank Programa para el Desarrollo Profesional Docente, Universidad Autónoma de Baja California, and the program of high quality scholarship for doctor studies.