Robust decentralized direct adaptive output feedback fuzzy control for a class of large-sale nonaffine nonlinear systems

Abstract

In the previous work of Huang et al., a decentralized direct adaptive fuzzy H_∞ tracking controller of large-scale nonaffine nonlinear systems is obtained predicated upon the assumption that the mismatching error dynamics stay squared integrable. In this note, we focus in the absence of the conservative assumption upon developing a robust decentralized direct adaptive output feedback fuzzy controller. By combination of a state observer, a fuzzy inference system and robust control technique, the previous controller design is modified and no a priori knowledge of bounds on lumped uncertainties is required. All the signals of the closed-loop large-scale system are proved to be uniformly ultimately bounded. The effectiveness of the developed scheme is demonstrated through the simulation results of interconnected inverted pendulums.

Introduction

In recent years, intelligent control system design, along with classical controller syntheses [5], [34], [35], [39], has attracted much considerable attention in the control community because heuristic knowledge or linguistic information can be applied to nonlinear control problems. Conceptually, there are two distinct configurations that have been formulated in the construction of a fuzzy adaptive control system: indirect and direct schemes, which can systematically incorporate plant knowledge and control knowledge, respectively [29]. Many results on indirect adaptive fuzzy or neural control have been obtained for single-input single-output (SISO) nonlinear systems in the past over a decade [2], [18], [19], [20], [22], [24], [27], [36], [37], [38]. In the meanwhile, direct adaptive intelligent control algorithms were propounded for nonlinear systems [3], [4], [29], [30], [8]. As shown in the above literature, more restrictions are often required for the stability and robustness of direct adaptive control systems. In [30], direct adaptive fuzzy control (DAFC) design was firstly presented for nonlinear systems whose control gains, however, are assumed positive constants rather than a nonlinear functions. To relax the constant assumption on control gains, many scholars developed various schemes for uncertain nonlinear systems [2], [7], [8], [15], [21], [23], [28], [30], [31]. The authors in the paper [2] proposed a resolution with the help of two additional assumptions in which control gains can be decomposed into two parts: known and unknown dynamics. However, the dynamics known are still difficult to determine in practice because many physical systems are of complexity and uncertainty. The algorithm proposed in [7] presupposes that the first derivative of the input gain is available beforehand, whereas the upper and lower bounds of the ideal control signal in [23] need to be accessible in order to implement its update law. It should be pointed out that the vast majority of these researches are conducted for nonlinear systems in affine form. In [15], [21], a few direct adaptive fuzzy or neural control approaches were investigated for nonaffine nonlinear systems, which relaxes stringent assumptions on plants, to a great degree. Furthermore, some scholars extended the DAFC design of SISO nonlinear systems to that of multiple-input multiple-output (MIMO) ones [18], [19], [20]. However, all the aforementioned adaptive fuzzy or neural control techniques are focused only upon affine or nonaffine nonlinear systems whose state variables are available for measurement. If system states are unmeasurable, output feedback or observer-based control schemes should be utilized in reality. Several adaptive fuzzy or neural output feedback control algorithms were presented for nonlinear systems using direct [1], [32], [33] or indirect [17] methodologies. Nevertheless, the foregoing direct approaches have the following limitations. Firstly, none of them is applicable to large-scale nonlinear systems, and still less to nonaffine ones. Secondly, the scheme proposed in [32] is questionable not only because the supervisory control u_s is always missing in the observation error dynamics but also because the given control v is not robust in the presence of the designed filter. Thirdly, both the observation error and the control gain are required to be online measurable in order to implement the control and adaptation laws in [33]. However, the filtering of the observation error dynamics was not accommodated at all and the famous strictly positive real (SPR) condition cannot be utilized. As a result, the proposed algorithm is deep down dubious. Therefore, how to design observer-based adaptive intelligent controllers in essence remains open to question for nonaffine systems.

With the physical inability for subsystem information exchange, the lack of computing capacities and the difficulty in measurement within a large-scale system, the above centralized controller design will technically become impracticable for interconnected nonlinear systems [25]. The research in decentralized control systems has been motivated by emerging applications of novel actuation devices to active control of industrial automation, cooperating robotic systems, power systems and aerospace processes. The last thirty years or so have witnessed a large quantity of results concerned. In the bibliographies [6], [16], decentralized control algorithms were proposed for several classes of large-scale linear systems. By using classical adaptive control technique [25] or neural network theory [10], [14], some researchers investigated decentralized adaptive control methodologies of uncertain large-scale nonlinear systems. However, these control designs take full state feedback such that they cannot be applied to interconnected nonlinear systems in which all state variables are unmeasurable. Recently, observer-based indirect [11], [26], direct [9], [12], [26] and hybrid [13] adaptive intelligent control algorithms have been developed for large-scale nonlinear systems, but many major problems are in great need of further solution to date. Firstly, the DAFC designs in [9], [26] needs to presuppose that the control input gains are positive constants rather than nonlinear functions. Secondly, the H_∞ tracking-based control methodologies propounded in [11], [12], [26] are subject to this precondition that the mismatching error stays squared integrable, but this is exceptionally difficult to verify in practice. Thirdly, the control schemes proposed in [13], [26] are not numbered among strict or real decentralized controller design, because communication or coordination is required between the individual subsystems. In [40], an output regulation problem for a class of large-scale nonlinear systems with neutrally stable uncertain ecosystem was tackled via a sort of decentralized error output feedback adaptive control strategy. In contradistinction to the current vast amount of work on adaptive fuzzy or neural control for conventional nonlinear systems, only very few results are accessible in terms of decentralized output feedback for large-scale nonlinear systems. Hence, great importance should be attached to the research into intelligent controller synthesis of uncertain correlating nonaffine systems.

In this paper, to at least partially resolve the before-mentioned problems, we develop a novel observer-based robust decentralized DAFC algorithm of large-scale uncertain nonlinear systems by using a fuzzy system, implicit function theorem and the SPR condition. The main contributions of the work are characterized by the following three novel features: (1) the proposed DAFC design via output feedback is applicable to large-scale nonlinear systems whose state variables are unmeasurable; (2) the controlled plants are assumed to be nonaffine such that typical restrictions on them can be removed; (3) no a priori knowledge of upper bounds on lumped uncertainties is required to implement the presented DAFC architecture. Furthermore, each subsystem is able to adaptively compensate for interconnections and disturbances with unknown bounds, and the derived control and update laws guarantee not only that all the signals of the resulting closed-loop large-scale system keep uniformly ultimately bounded (UUB) but also that the tracking errors converge to tunable neighborhoods of the origin. The validity of the suggested approach is supported by simulation results of interactive inverted pendulums.

The rest of this paper is organized as follows. In Section 2, the problem under investigation is stated. Section 3 introduces an observer-based decentralized DAFC scheme. The overall design methodology is proposed and the stability of the closed-loop large-scale system is analyzed using Lyapunov’s approach in Section 4. The designed DAFC algorithm is applied to controlling double inverted pendulums in Section 5. Finally, in Section 6 we conclude this paper.