Mathematical Problems in Engineering

Volume 2017, Article ID 4876019, 13 pages

https://doi.org/10.1155/2017/4876019

## Integral Sliding Modes with Nonlinear -Control for Time-Varying Minimum-Phase Underactuated Systems with Unmatched Disturbances

^{1}CONACYT-Instituto Politécnico Nacional-CITEDI, Av. Instituto Politécnico Nacional 1310, Nueva Tijuana, 22435 Tijuana, BC, Mexico^{2}Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional 1310, Nueva Tijuana, 22435 Tijuana, BC, Mexico

Correspondence should be addressed to Luis T. Aguilar; xm.npi@braliugal

Received 11 July 2016; Revised 19 October 2016; Accepted 21 November 2016; Published 17 January 2017

Academic Editor: Kalyana C. Veluvolu

Copyright © 2017 Roger Miranda-Colorado et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 -control for rejecting vanishing and nonvanishing matched perturbations and for attenuating the unmatched ones, respectively. It is theoretically proven that, using the proposed controller, the origin of the free-disturbance nonlinear system is asymptotically stabilized, while the matched disturbances are rejected whereas the -gain of the corresponding nonlinear system with unmatched perturbation is less than a given disturbance attenuation level with respect to a given performance output. The capability of the designed controller is verified through a flexible joint robot manipulator typically affected by both classes of external perturbations. In order to assess the performance of the proposed controller, an existing sliding modes controller based on a nonlinear integral-type sliding surface is also implemented. Both controllers are then compared for trajectory tracking tasks. Numerical simulations show that the proposed approach exhibits better performance.

#### 1. Introduction

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., [1–3]), the unmatched disturbances are still a challenging problem, faced by control engineers, that adversely affect the performance of any system to be controlled. This kind of disturbances cannot be trivially neglected since they can be aroused by unavoidable noise in the measurements or perturbing the output as well. Moreover, disturbances acting in the nonactuated part of an underactuated mechanical system (e.g., pendulums, car-like robots, biped robots, and unmanned aerial vehicles) are a typical example where the unmatched disturbances must be counteracted. Indeed, the problem becomes more complicated for the motion control of this kind of systems since unmodeled dynamics can emerge (see, e.g., [4]).

Sliding modes are long recognized as a powerful control method to* reject* vanishing and nonvanishing uniformly bounded matched disturbances and plant uncertainties. However, unmatched disturbances are not counteracted. On the other hand, nonlinear control has the capability of* attenuating* both matched and unmatched disturbances [5, 6]. There have been many results dealing with unmatched disturbances; however, integral sliding modes have begun to receive a growing interest. For example, Kumar et al. [7] solve the regulation problem for a Stewart robot using a smooth integral sliding mode (ISM) controller, which drives the position error to the origin in finite time, while the closed-loop system is demonstrated to be robust against matched disturbances only. Mahieddine et al. [8] propose a sliding mode controller that allows attenuating matched and unmatched uncertainties in nonlinear systems and also reducing the chattering in the control signal. Han et al. [9] developed output feedback-based sliding mode control schemes for linear time-delayed systems considering both matched and unmatched uncertainties. A detailed revision of ISM addressing the unmatched disturbances is presented in [10].

In relevant works, Osuna et al. [11] make a -gain analysis for hybrid mechanical systems operating under unilateral constraints and admitting sliding modes and collision phenomena. Rubagotti et al. [12] prove that the definition of a suitable sliding manifold and the generation of sliding modes upon it guarantee that matched disturbances are completely rejected while the unmatched ones are not amplified. Besides, a linear control with nonlinear compensation is proposed against matched disturbances. Cao and Xu [13] present a nonlinear ISM controller where the unmatched disturbances are not amplified, but a linear controller is also used. Castaños and Fridman [14] show the robustness properties of an integral sliding mode controller ensuring rejection of the matched disturbances, while unmatched perturbations are not amplified using also a linear control. Galvan-Guerra and Fridman [15] proposed an output ISM for linear time-variant systems, where the main goal is to eliminate the matched perturbation. In [16], an adaptive ISM control for a class of nonlinear uncertain and invariant systems is proposed to eliminate the quantization sensitivity parameters and matched perturbations, which is accomplished by using an integral sliding function from the local dynamics of the plant. Chen et al. [17] propose a nonlinear ISM fault tolerant control where an optimal control is used against matched disturbances.

The aforementioned literature includes a linear control to attenuate unmatched disturbances to easily find a suboptimal solution through an algebraic Riccati equation (ARE). On the other hand, a physical phenomenon is better described by its nonlinear dynamic equations. However, the local solution of these dynamic equations is required for the ARE to be solved. Besides, hard computational work is also entailed for verifying the Hamilton-Jacobi-Isaacs inequality and obtaining a global solution.

In this paper, an ISM control combined together with a nonlinear control is presented. The proposed controller allows rejecting matched bounded disturbances and attenuating the effect of the unmatched ones. The synthesized controller, that admits a time-varying input matrix, was applied for solving the tracking control problem for 1 degree of freedom (DOF) flexible joint robot (FJR) manipulator, which consists of a single link interconnected by an elastic revolute joint. The formulation of the nonlinear -control problem is confined to nonautonomous affine systems, and it requires a controller design that guarantees both the internal asymptotic stability of the closed-loop system and its dissipativity with respect to admissible external disturbances. In contrast to previous works, a strict Lyapunov function was proposed in this paper to ensure a global solution of the control problem, by means of the verification of the Hamilton-Jacobi-Isaacs inequality, thus avoiding a hard numerical computation of a partial differential equation or a solution of the corresponding differential Riccati equation [18, 19] where the linearization around the equilibrium point of the plant is required. In order to assess the performance of the proposed controller, we implemented a controller, introduced by Cao and Xu in [13], based on a nonlinear integral-type sliding surface. The results of both, the proposed controller and the Cao-Xu controller, were compared in a trajectory tracking task.

This paper contributes to the following:(i)Presenting the design of a new controller by combining ISM control and a nonlinear control for time-varying minimum-phase underactuated systems affected by both matched and unmatched disturbances(ii)Developing a rigorous stability analysis, with a global solution to control and verifying the Hamilton-Jacobi-Isaacs inequality(iii)Detailing a procedure to implement the Cao-Xu controller in a 1-DOF FJR manipulator(iv)Presenting a comparative analysis of both controllers by means of numerical simulations with a trajectory tracking task

This paper is organized as follows. Section 2 presents the problem statement and synthesis of ISM control and control for a class of time-varying systems. In Section 3, the combined ISM and nonlinear tracking control is developed for a single pendulum with elastic joint affected by matched and unmatched perturbation. Here, the details regarding the implementation of the Cao-Xu controller are also presented. In Section 4, the performance of both controllers is evaluated in a simulation study. Finally, conclusions are provided in Section 5.

#### 2. Controller Design

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 as the desired reference signal and as the output of the nonlinear system to be controlled. The control problem can now be defined as follows.

*Control Problem*. Given a smooth reference signal () , find a control law such that holds for the free-disturbance case (i.e., ), and the -gain of the perturbed nonlinear system is less than a positive constant level with respect to a given performance output to be controlled.

The block diagram of the closed-loop system with the proposed controller is depicted in Figure 1. The performance output is nothing other than a vector of variables, including the nonmeasurable ones, where the disturbances must be attenuated. The effect of the matched perturbation is canceled out by means of an ISM controller . However, the resulting closed-loop system is still affected by the unmatched perturbation . Then, using an -control, the -gain of the closed-loop system is made less than a given attenuation level .