Sliding Mode Reference Coordination of Constrained Feedback Systems

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 764348, 11 pages http://dx.doi.org/10.1155/2013/764348 Research Article Sliding Mode Reference Coordination of Constrained Feedback Systems Alejandro Vignoni,1 Fabricio Garelli,2 and Jesús Picó1 1 2 Institut d’Automàtica i Informàtica Industrial, Universitat Politècnica de València, Camı̀ de Vera s/n, 46022 València, Spain CONICET and Facultad de Ingenierı̀a, Universidad Nacional de La Plata (UNLP), Calle 48 esq. 116 s/n, 1900 La Plata, Argentina Correspondence should be addressed to Alejandro Vignoni; Received 4 September 2013; Revised 20 November 2013; Accepted 21 November 2013 Academic Editor: Rongni Yang Copyright © 2013 Alejandro Vignoni 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. This paper addresses the problem of coordinating dynamical systems with possibly different dynamics (e.g., linear and nonlinear, different orders, constraints, etc.) to achieve some desired collective behavior under the constraints and capabilities of each system. To this end, we develop a new methodology based on reference conditioning techniques using geometric set invariance and sliding mode control: the sliding mode reference coordination (SMRCoord). The main idea is to coordinate the systems references. Starting from a general framework, we propose two approaches: a local one through direct interactions between the different systems by sharing and conditioning their own references and a global centralized one, where a central node makes decisions using information coming from the systems references. In particular, in this work we focus in implementation on multivariable systems like unmanned aerial vehicles (UAVs) and robustness to external perturbations. To show the applicability of the approach, the problem of coordinating UAVs with input constraints is addressed as a particular case of multivariable reference coordination with both global and local configuration. 1. Introduction Coordination of multiagents and, in particular, formation control of multiple unmanned aerial vehicles (UAVs) is a very up-to-date topic, and it supports many practical applications, such as surveillance, weather forecasting, damage assessment, and search and rescue [1–3]. Recently, the consensus problem was addressed using algebraic graph theory and properties of the Laplacian Matrix for single integrators (see [1, 4] and references therein). This approach was extended to a chain of integrators in [5], and to nonlinear multiagent systems in [6]. Sliding mode control is an important topic in nonlinear systems. Research in nonlinear systems goes from stability analysis (e.g., with fuzzy polynomial models and sum of squares tools [7]) to estimation (e.g., with second order sliding mode observers for kinetic rates [8]) to control (e.g., with bounded L2 gain performance of Markovian jump singular time-delay systems [9]). The use of sliding mode (SM) techniques is generally used for control of swarms and multi-agent systems to achieve consensus. In those situations, a master-slave or leaderfollower configuration is implemented, and the discontinuous action is a control signal. In [10], higher-order sliding mode controllers are used in such configuration to maintain the formation shape. In [11], finite-time sliding mode estimators are used to achieve consensus in decentralized formation control with virtual leader. Also in [12], a discontinuous control input is chosen to be proportional to the gradient of a positive semidefinite disagreement function defined by the graph Laplacian matrix, leading to a sliding mode consensus algorithm. Recently in [13], consensus is achieved in connected and also in fully connected swarms of idealized and identical first-order dynamic systems enforced by sliding modes. Also in [14], finite-time consensus algorithms for a swarm of self-propelling agents based on sliding mode control and graph algebraic theories are developed. In [15], an LMI approach to multiagent systems control is performed 2 under time delay and uncertainties. Finally, in [16], exact formation control is achieved with binary information of the position of the other agents. In a recent proposal from one of the coauthors, sliding mode control has been used in a nontraditional way: the sliding mode reference conditioning (SMRC) technique [17]. This technique combines reference conditioning and sliding mode ideas and has been used in the beginning to bound cross-coupling interactions in multivariable linear systems [18, 19] and for set-point seeking in nonlinear systems with state dependent constraints [20]. In [21], an SMRC auxiliary loop has been implemented to reduce hypoglycemia in a closed-loop glucose control for DM type 1 patients. Also in [22], a geometric invariance and sliding mode ideas have been proposed for redundancy resolution in robotic systems. And in [23], an integrated solution based on the same ideas has been proposed for robotic trajectory tracking, path planning and speed autoregulation. In the previous work [24–27], the authors use sliding mode reference conditioning to enforce coordination in multi-agent systems. In [24], sliding mode reference coordination in SISO systems is imposed with a global supervisory approach and two layers of SMRC in a hierarchical structure. In [25], the coordination arises from the local interaction between the systems, and the information flows in only one level, as the swarm is assumed to have no leader. In [26], the authors make a multivariable reference coordination for ideal unconstrained systems. A preliminary unifying work is done in [27] with focus on SISO systems. The present work is a unified approach of the sliding mode reference coordination (SMRCoord) integrating both local and global approaches with focus on implementation of multivariable systems and robustness of perturbation to the coordination goals. The rest of the paper is organized as follows. Next section presents the problem of coordination in a general form. Then in Section 3, some previous results in set invariance and reference conditioning used thereafter to propose coordination strategy are described. In Section 4, a decentralized version is proposed to solve the coordination problem with only one hierarchical level and no leader. Meanwhile in Section 5, the supervised global coordination method is proposed. In Section 6, the proposed strategies are used to coordinate UAVs showing the result obtained by simulations. Finally, in Section 7, some conclusions are presented and open issues for future study are considered. 2. Problem Statement In this section, a general form of the coordination problem is presented, together with definitions and assumptions regarding constrained systems and systems coordination (...truncated)