IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 11, NOVEMBER 2009 2611 Output Integral Sliding Mode for Min-Max Optimization of Multi-Plant Linear Uncertain Systems Francisco Javier Bejarano, Leonid M. Fridman, Member, IEEE, and Alexander S. Poznyak, Member, IEEE Abstract—We consider the application of a min-max optimal control based on the LQ-index for a set of systems where only the output information is available. Here every system is affected by matched uncertainties, and we propose to use an output integral sliding mode to compensate the matched uncertainties right after the beginning of the process. For the case when the extended system is free of invariant zeros, a hierarchical sliding mode ob- server is applied. The error of realization of the proposed control algorithm is estimated in terms of the sampling step and actuator time constant. An example illustrates the suggested method of design. Index Terms—Hierarchical sliding mode observer (HSMO), output integral sliding mode (OISM). I. INTRODUCTION A. Antecedents and Motivation F OR the case of multi-plant there are two main approaches to control such systems. One is to decentralize the controls of each plant ([1], [2]). The other method is to design the same optimal control law for all the plants and make this control ro- bust with respect to perturbations. The robustification of the optimal control is one of the main problems in the modern control theory (see, e.g., [3]–[12] and references therein). In [5] and [6], a robust optimal control based in a min-max LQ-index for a multi-model system was proposed. Basically, it was considered a set of possible models for the same plant, each model is characterized by a LQ-index and the objective of the robust optimal control is to minimize the worst of the LQ-indexes. However, the exact solution of this optimal control problem requires of two basic assumptions: • the system is free from any uncertainty; • the state vector is completely available. Manuscript received February 21, 2008; revised August 04, 2008 and November 19, 2008. First published October 13, 2009; current version pub- lished November 04, 2009. This paper was presented in part at the American Control Conference, New York, NY, 2007. This work was supported by CONACyT under Project 56819, and by the Postdoctoral Grant CVU 103957. Recommended by Associate Editor A. Ferrara. F. J. Bejarano and L. M. Fridman are with the Department of Control, Di- vision of Electrical Engineering, National Autonomous University of Mexico (UNAM), Mexico City C.P. 04510, México, D.F. (e-mail: javbejarano@yahoo. com.mx; lfridman@servidor.unam.mx). A. S. Poznyak is with the Departamento de Control Automático, CIN- VESTAV-IPN,A.P. 14-740, Mexico City C.P. 07000, México D.F. (e-mail: apoznyak@ctrl.cinvestav.mx). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2009.2031718 Thus, for the case when we have output information only, we should ensure the compensation of the matched uncertainties. Furthermore, we need to reconstruct the original states to take advantage of the state feedback robust optimal control. The integral sliding mode (ISM) was proposed in [13], it is usually used to compensate the matched uncertainties from the beginning of the process. The optimal control problem in the presence of matched uncertainty was considered in [7], [9]. They proposed the use of the integral sliding mode control al- lowing to ensure the robustness of the solution from the ini- tial time moment. Some applications of ISM can be found in [9]–[11], [14]–[20]. The order of the motion equation in ISM is equal to the order of the original system. As a result, robustness of the trajectory for a system driven by a smooth control law can be guaranteed throughout an entire response of the system starting from the initial time instance. However, again, the main problem related to the implementation of this ISM concept con- sists in the requirement of the knowledge of the state vector, including the initial conditions. Thus, the ISM turns out to be useless when being applied directly and only output informa- tion is available. B. Methodology To realize the robust optimal output control for the multi-plant case three approaches must be modified and synthesized: • the min-max optimal LQ control; • the integral sliding-mode control; • the hierarchical sliding mode observation. In [21] and [22] were proposed two different forms for re- solving the problem of matched uncertainty compensation for the case of a control based on the min-max LQ-index in the context of a multi-model system. The difference between the multi-plant and multi-model systems is the following: in the multi-model case it is considered that for the same plant dif- ferent models could be realized. However, in the multi-plant systems we are considering a set of plants working simulta- neously and the min-max optimal control law is applied to all plants simultaneously. On the other hand, the robust optimal control based on a min-max LQ-index requires the knowledge of a weighting vector that minimizes a functional. This vector can be found graphically for the cases of two and three plants, evidently for the other cases, a numerical method need to be used. Hence, for finding the weighting vector we will make use of the algorithm proposed in [23]. Recently it was designed an output integral sliding mode al- lowing to robustify the optimal LQ-control for the case when 0018-9286/$26.00 © 2009 IEEE Authorized licensed use limited to: Universidad Nacional Autonoma de Mexico. 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