Elsevier

Neurocomputing

Volume 70, Issues 4–6, January 2007, Pages 960-974
Neurocomputing

Quasi-sliding mode control strategy based on multiple-linear models

https://doi.org/10.1016/j.neucom.2006.07.011Get rights and content

Abstract

In this paper, a multiple discrete quasi-sliding mode (QSM) control scheme is proposed for a general class of nonlinear discrete time systems with unknown dynamical equations, provided that input–output data is available for system identification. The self-organizing map (SOM) is employed to divide the state space into local regions such that it associates the operating region where a local linear model is the winner with a local quasi-sliding mode controller (QSMC). Switching of the controllers is done synchronously with the active local linear model that tracks the different operating conditions. The simulation results show that the proposed controller outperforms tracking the desired trajectory in noisy environments either with a global controller or simpler controllers based on multiple models.

Introduction

The identification of unknown nonlinear dynamical systems has received considerable attention in recent years since it is an indispensable step towards controller design for nonlinear systems [27]. Specifically, the concept of multiple models with switching has been an area of interest in control theory in order to simplify both the modeling and the controller design [22], [26]. Local modeling derives a model based on neighboring samples in the operating space. If a function f to be modeled is complicated, there is no guarantee that any given global representation will approximate f equally well across all space. In this case, the dependence on representation can be reduced using local approximation where the domain of f is divided into local regions and a separate model is used for each region [8], [17], [38], [42].

In a number of local modeling applications, a self-organizing map (SOM) has been utilized to divide the operating regions into local regions [31], [32], [42]. The SOM is particularly appropriate for switching, because it converts complex, nonlinear statistical relationships of high-dimensional data into simple geometric relationships that preserve the topology in the feature space [18]. Thus the role of the SOM is to discover patterns in the high-dimensional state space and divide it into a set of regions represented by the weights of each processing element (PE). Under some mild conditions, it has been shown that multiple models can uniformly approximate any system on a closed subset of the state space provided a sufficient number of local models are given [38], [42]. Generally, control using multiple models is categorized in two approaches: global model-based control using local models and multiple model-based control with switching.

Global controller design with the aid of multiple models has been extensively reported in the literature [9], [14], [21], [34], [36]. Gain scheduling has been perhaps the most common systematic approach to control nonlinear systems in practice due to simple design and tuning [28], [33], [37], [43]. The multiple model adaptive control approach differs from gain scheduling mainly by the use of an estimator-based scheduling algorithm used to weight the local controllers. Murray-Smith and Hunt [21] utilized an extended RBF network where each local model is a linear function of the input and they reported great success for control problems. The overall controller is designed based on the local models and a validity function to guarantee smooth interpolation. Similarly, Foss et al. [9] and Gawthrop and Ronco [14] employed model predictive controllers and self-tuning predictive controllers, respectively, using multiple models. Palizban et al. [29] attempted to control nonlinear systems with the linear quadratic optimal control technique using multiple-linear models and provided the stability condition for the closed loop system. Ishigame et al. [16] proposed the sliding mode control scheme based on fuzzy modeling composing a weighted average of linear systems to stabilize an electric power system.

In contrast, Narendra et al. [26] proposed the multiple model approach in the context of adaptive control with switching where local model performance indices have been used to select the local controller. Subsequently, Narendra and Balakrishnan [23] proposed different switching and tuning schemes for adaptive control that combines fixed and adaptive models yielding a fast and accurate response. Principe et al. [32] proposed a SOM-based local linear modeling strategy and predictive multiple model switching controller to control a wind tunnel and showed improved performance with decreased control effort over both the existing controller and an expert human-in-the-loop control. Later Narendra and Xiang [25] proved that the adaptive control using multiple models is globally stable and that the tracking error converges to zero in the deterministic case. Diao and Passino [7] applied multiple model-based adaptive schemes to the fault tolerant engine control problem. A linear robust adaptive controller and multiple nonlinear neural network-based adaptive controllers were exploited by Chen and Narendra [4]. Thampi et al. [40], [41] have also shown the applicability of the multiple model approach based on the SOM for flight control.

The control of nonlinear systems considered in this paper has been an important research topic and many approaches have been proposed. While classical control techniques have produced many highly reliable and effective control systems, great attention has been devoted to the design of variable structure control systems (VSCS). Variable structure systems (VSS) are a special class of nonlinear systems characterized by a discontinuous control action, which changes structure upon reaching a set of switching hyperplanes. During the sliding mode, the VSCS has invariance properties, yielding motion that is remarkably good in rejecting certain disturbances and parameter variations [10], [20], [35], [39].

However, sliding mode control systems (SMCS) that were originally conceived for continuous-time systems may not perform well—or may even lead the system to instability—when direct digital implementation is attempted. Thus, many researchers have either addressed the limitations when direct implementation is done or have proposed designs that take the sampling process into account. Milosavljevic [20] was among the first researchers to formally state that the sampling process limits the existence of a true sliding mode. In light of this, definitions of quasi-sliding mode (QSM) have been suggested and the conditions for the existence of such modes have been investigated. Sarpturk et al. [35] specifically addressed the stability issue and gave necessary and sufficient convergence and sliding conditions. Discrete sliding mode tracking controller based on an input–output model in the presence of modeling uncertainty and disturbances have been considered earlier [3], [5], [11], [19], [30]. Furuta [11] designed a discrete VSS type self-tuning controller using an adaptive parameter estimator where the control input included a linear feedback term and a switching term with the equivalent control region. Lee and Oh [19] suggested a modified discrete VSS type self-tuning controller and improved the stability of the controller by modifying the sector with separate gains and the equivalent control algorithm. Recently, Chen et al. [5] proposed a discrete robust adaptive QSM tracking controller for the input–output system without knowing the upper and lower bounds of the unknown parameters, which overcome the unpractical assumptions of [11], [19] since the bounds of the unknown parameters can hardly ever be known in practice. On the other hand, Gao et al. [12] presented an algorithm that drives the system state to the vicinity of a switching hyperplane in the state space, rather than to a sector of a different shape [11]. They specified desired properties of the controlled systems and proposed a reaching law-based approach for designing the discrete-time sliding mode control law. Later, modified quasi-sliding mode control (QSMC) strategy with a reaching law approach was proposed by Bartoszewicz [1] to guarantee better robustness and improved performance.

Most of the VSCS proposed in the literature have been developed mainly based on the state-space model with the assumption that all state variables are measurable or on the input–output model for a linear system. But in some control problems, we are allowed to access only the input and the output of the nonlinear plant. In this case, an observer could be used to estimate the unmeasurable state variables if the state equations are known. Otherwise, this is not possible. This is where the multiple model-based control framework can be very attractive for nonlinear control problems since it is capable of not only utilizing this robust control technique for nonlinear systems but also applying it for unknown systems. Thus, it is the purpose of this work to provide a new technique to design a sliding mode control law for unknown discrete-time nonlinear systems so that the amount of guesswork1 is reduced, while attainable performance is increased. In this way, one of the difficulties in designing a SMC (that requires the complete knowledge of the plant to be controlled) can be removed as well as the problems that arise due to the uncertainties of the plant model and measurement noise can be alleviated by incorporating the robustness provided by the sliding mode technique into the multiple modeling approach. In addition, we examined the effect by the modeling error due to the quantization of state space as well as by measurement noise to the proposed multiple model-based sliding mode control performance. It is shown that the switching scheme does not create an issue to be considered in order to guarantee BIBO stability of the overall system.

Simulation results using the proposed strategy for identification and control of nonlinear systems are presented to demonstrate the versatility of the algorithm. Results show that the switching linear models are a promising alternative for system identification when compared with a single global model. The overall system with the controller tracks the desired trajectory very well. Additionally, it offers excellent robustness under noise condition during the control process when compared with a global nonlinear controller and other multiple controller approaches.

Section snippets

Multiple model-based system identification

The idea of multiple modeling is to approximate a nonlinear system with a set of relatively simple local models valid in certain operating regions, such that the dynamic space is decomposed in the appropriate switching among very simple linear models. Multiple models are very appealing for modeling complex nonlinear systems due to the intrinsic simplicity, since we often cannot derive appropriate models from first principles, and are not capable of deriving accurate and complete equations for

Discrete-time QSMC for local linear models

Once we identify the plant using multiple models, it is necessary to associate these models with corresponding controllers. In doing so, controllers can be designed a priori corresponding to each of the local models. In addition, if the nonlinear system can be adequately described by a linear model within a sufficiently small neighborhood of an operating point, the corresponding controller is easily designed through the linearized plant [22], [26].

Simulation results

To examine the effectiveness of the proposed controller design methodology, discrete-time systems have been considered assuming the following: the only state available for measurements is yk=xk(1) and the nonlinear function f is completely unknown. By assuming that the function f is unknown, we confront a worst case (least prior knowledge) control design. Our objective is to design multiple sliding mode controller for unknown nonlinear plants that guarantees global stability and forces the

Conclusion

In this paper, the MQSMC strategy has been proposed for a general class of nonlinear unknown discrete-time systems via SOM, which divides the state space into a set of operating regions. Contrary to what is assumed in the field of sliding mode controller design, the plant dynamics under control are assumed to be unknown. This is a challenge in the conventional design framework with the ambiguities introduced by the noise on the measured quantities. Thus, we have taken the concept of

Acknowledgment

This work was partially supported by NASA grant NAG-1-02068.

Jeongho Cho received the B.S. degree in control and instrumentation engineering from the Soonchunhyang University, Korea, in 1995 and the M.S. degree in electrical engineering from Dongguk University, Korea, in 1997 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 2001 and 2004, respectively. From 2005 to 2006, he worked as a postdoctoral research associate in the department of biomedical engineering at the University of

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  • Cited by (0)

    Jeongho Cho received the B.S. degree in control and instrumentation engineering from the Soonchunhyang University, Korea, in 1995 and the M.S. degree in electrical engineering from Dongguk University, Korea, in 1997 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 2001 and 2004, respectively. From 2005 to 2006, he worked as a postdoctoral research associate in the department of biomedical engineering at the University of Florida, concentrating on biomedical signal modeling and on EEG signal analysis for the control of epileptic seizure. Since 2006, he has been a senior engineer in Samsung Electronics where he is working on developing multi-functional printers. His research interests include time-series prediction and nonlinear system identification, with applications to navigation and control.

    Jose C. Principe is currently a Distinguished Professor of electrical and biomedical engineering at the University of Florida, Gainesville, since 2002. He joined the University of Florida in 1987, after an eight-year appointment as Professor at the University of Aveiro, in Portugal. He holds degrees in electrical engineering from the University of Porto (B.S.), Portugal, University of Florida (M.S. and Ph.D.), USA and a Laurea Honoris Causa degree from the Universita Mediterranea in Reggio Calabria, Italy. His interests lie in nonlinear non-gaussian optimal signal processing and modeling in biomedical engineering. He created the Computational NeuroEngineering Laboratory in 1991 to synergistically focus the research in biological information processing models. Dr. Principe is a Fellow of the IEEE, past President of the International Neural Network Society, and Editor in Chief of the Transactions of Biomedical Engineering since 2001, as well as a former member of the Advisory Science Board of the FDA. He holds five patents and has submitted seven more. Dr. Principe was supervisory committee chair of 47 Ph.D. and 61 Master students, and he is author of more than 400 refereed publications (3 books, 4 edited books, 14 book chapters, 116 journal papers and 276 conference proceedings).

    Deniz Erdogmus received the B.S. in Electrical & Electronics Engineering (EEE), and the B.S. in Mathematics both in 1997, and the M.S. in EEE in 1999 from the Middle East Technical University, Turkey. He received his Ph.D. in Electrical & Computer Engineering from the University of Florida, Gainesville in 2002. He worked as a research engineer at TUBITAK-SAGE, Turkey from 1997 to 1999, focusing on the design of navigation, guidance, and flight control systems. He was also a research assistant and a postdoctoral research associate at UF from 1999 to 2004, concentrating on signal processing, adaptive systems, machine learning, and information theory, specifically with applications in biomedical engineering. Currently, he is holding an Assistant Professor position jointly at the Computer Science and Electrical Engineering Department and the Biomedical Engineering Department of the Oregon Health and Science University. Dr. Erdogmus has over 35 articles in international scientific journals and numerous conference papers and book chapters. He has also served as associate editor and guest editor for various journals. He is a member of Tau Beta Pi, Eta Kappa Nu, and IEEE.

    Mark A. Motter was born in Columbia, Pennsylvania. He served in the United States Navy from 1973 until 1979, honorably discharged at the rank of Electronics Technician First Class. He then began his formal engineering education at Old Dominion University in Norfolk, Virginia, receiving his BSEE, magna cum laude, and MSEE, in 1983 and 1985, respectively. He received Ph.D. in Electrical and Computer Engineering from the University of Florida in 1998. Since 1985 Dr. Motter has been employed at NASA Langley Research Center. Currently, he is a controls research engineer in the Electronics Systems Branch. His current research project is investigating the implementation of self-organizing and other biologically inspired flight control approaches, using fully autonomous unmanned aerial vehicles. He is a senior member of the IEEE, a registered Professional Engineer, and a member of the Academy of Model Aeronautics.

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