Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> It is argued that, for a cautious design, a nonlinear analysis is needed to reveal when and why linear tools fail. Emerging nonlinear tools that can be used to overcome the limitations of nonlinear designs are discussed. It is shown that some of these tools can be made adaptive and applied to nonlinear systems with unknown parameters. The emergence of robust designs for nonlinear 'interval' plants is pointed out. > <s> BIB001 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> Presents an adaptive control scheme for flexible joint robot manipulators. Asymptotic stability is insured regardless of the joint flexibility value, i.e., the results are not restricted to weak joint elasticity. Moreover, the joint flexibility is not assumed to be known. Joint position and velocity tracking errors are shown to converge to zero with all the signals in the system remaining bounded. > <s> BIB002 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> A new adaptive control scheme developed by Krstic et al. (1992) is employed to design decentralized adaptive regulators. The interconnected system to be regulated consists of N coupled subsystems having arbitrarily relative degrees. Global stability is established for the closed-loop system and perfect regulation is ensured. > <s> BIB003 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> In this paper, the integrator backstepping approach together with an adaptive law using parameter projection is employed to design robust decentralized adaptive controllers. The techniques used allow us to relax the relative degree limitations on subsystems and the structural conditions on interactions between subsystems required in some earlier results. Further, no a priori knowledge of the unmodeled dynamics is required. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. <s> BIB004 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> We propose for the first time a decentralized adaptive control scheme without requiring a priori information on subsystem high-frequency-gain signs. This is achieved by introducing Nussbaum-type functions in local adaptive controllers, and developing a new analysis approach which is quite different from the conventional one used in centralized adaptive control involving a Nussbaum-type function. With the help of integrator backstepping, the decentralized control scheme can also be applied to high-relative-degree systems. The results of this paper enlarge the class of uncertain large-scale systems for which decentralized adaptive control can be designed. <s> BIB005 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> The problem of decentralized output-feedback tracking with disturbance attenuation is addressed for a new class of large-scale and minimum-phase nonlinear systems. Common assumptions like matching and growth conditions are not required for the underlying decentralized system with a diagonal structure. An observer-based decentralized controller design is presented. The proposed decentralized output-feedback laws achieve asymptotic tracking and internal Lagrange stability when the disturbance inputs disappear, and, guarantee external stability in the presence of disturbance inputs. These external stability properties include Sontag's ISS and iISS conditions and standard L"2-gain property. <s> BIB006 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> A global decentralized robust adaptive output-feedback dynamic compensator is proposed for stabilization, tracking, and disturbance attenuation of the decentralized generalized output-feedback canonical form. This represents the largest class for which decentralized robust adaptive output-feedback tracking and disturbance attenuation results are currently available. The system is allowed to contain unknown parameters multiplying output-dependent nonlinearities, and, also, unknown nonlinearities satisfying certain bounds. Under the assumption that a constant matrix can be found for each subsystem to achieve a certain property, it is shown that reduced-order observers and backstepping controllers can be designed to achieve practical stabilization of the tracking error in each subsystem in the presence of bounded disturbance inputs. Sufficient conditions under which asymptotic tracking and stabilization can be achieved are also obtained. Signal gains from disturbance inputs to tracking errors are presented in the input-to-output-practical-stability and integral-input-to-output-practical-stability frameworks. A particular case in which the standard L"2-gain disturbance attenuation is achieved is also provided. <s> BIB007 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> A novel decentralised PID controller design procedure based on backstepping principles is presented to operate multiple-input multiple-output (MIMO) dynamic processes. First, a control Lyapunov function (CLF) and virtual control variable based on the backstepping method are derived recursively for each loop and a multivariable controller is obtained. Appropriate selection of auxiliary control variables can then eliminate coupling effects between subprocesses, and a decentralised controller can be derived. By analysing the regulation dynamics, external disturbances are rejected via appropriate tuning of the backstepping design parameters. Sufficient stability conditions are then derived using the small-gain theorem for two-input two-output and MIMO closed-loop systems, respectively. Simulation results for the control of the Shell benchmark problem and experimental results with a real-time water-level control system are provided to demonstrate the effectiveness and practicality of the proposed design procedure. <s> BIB008 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> This work is aimed at looking into the determination of optimal control strategies of large scale non-linear systems. The basic idea consists of the decomposition of the global system into interconnected subsystems. We consider non-linear systems whose subsystems are linear with respect to their state and control; in other words, non-linearities are located in the interconnection terms. Then, for each subsystem, local optimal feedback gains are easily expressed in terms of interconnection vectors. Such optimisation has been successfully achieved using a neural net yielding the optimal gain at each step time. Some numerical examples are considered in order to illustrate the proposed approach. <s> BIB009 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> Adaptive Backstepping Control Design.- Adaptive Backstepping Control.- Adaptive Control of Time-Varying Nonlinear Systems.- Multivariable Adaptive Control.- Decentralized Stabilization of Interconnected Systems.- Nonsmooth Nonlinearities.- Nonsmooth Nonlinearities.- Backstepping Control of Systems with Backlash Nonlinearity.- Inverse Control of Systems with Backlash Nonlinearity.- Stabilization of Interconnected Systems with Backlash Nonlinearity.- Adaptive Control of Nonlinear Systems with Dead-Zone Nonlinearity.- Adaptive Control of Systems with Input Saturation.- Control of a Hysteretic Structural System in Base Isolation Scheme.- Control of a Piezo-Positioning Mechanism with Hysteresis. <s> BIB010 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> In this paper, we propose a new totally decentralized adaptive output tracking scheme for a class of nonlinear interconnected systems by using a backstepping approach. In the design, a smooth function is introduced to compensate for the effects of interactions from other subsystems. It is shown that the designed local adaptive controllers can ensure all the signals in the closed loop system are bounded. A root mean square type of bound is obtained for the tracking error as a function of design parameters. This implies that the transient tracking error performance can be adjusted by choosing suitable design parameters. <s> BIB011 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> In this paper, a decentralized adaptive control scheme is proposed to address output tracking of a class of interconnected time-delay subsystems with the input of each loop preceded by an unknown dead-zone. Each local controller is designed using the backstepping technique and consists of a new robust control law and new updating laws. Unknown time-varying delays are compensated by using appropriate Lyapunov-Krasovskii functionals. Furthermore, by introducing a new smooth dead-zone inverse, the proposed backstepping design is able to eliminate the effects resulting from dead-zone nonlinearities in the input. It is shown that the proposed controller can guarantee not only stability, but also good transient performance. <s> BIB012 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> An adaptive backstepping neural-network control approach is extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections. The novel contribution is to remove the common assumptions on interconnections such as matching condition, bounded by upper bounding functions. Differentiation of the interconnected signals in backstepping design is avoided by replacing the interconnected signals in neural inputs with the reference signals. Furthermore, two kinds of unknown modeling errors are handled by the adaptive technique. All the closed-loop signals are guaranteed to be semiglobally uniformly ultimately bounded, and the tracking errors are proved to converge to a small residual set around the origin. The simulation results illustrate the effectiveness of the control approach proposed in this correspondence. <s> BIB013 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> So far there is still no result available for backstepping based decentralized adaptive stabilization of unknown systems with interactions directly depending on subsystem inputs, even though such interactions commonly exist in practice. In this paper, we provide a solution to this problem by considering both input and output dynamic interactions. To clearly illustrate our approaches, we will start with linear systems and then extend the results to nonlinear systems. Each local controller, designed simply based on the model of each subsystem by using the standard adaptive backstepping technique without any modification, only employs local information to generate control signals. It is shown that the designed decentralized adaptive backstepping controllers can globally stabilize the overall interconnected system asymptotically. The L"2 and L"~ norms of the system outputs are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters. <s> BIB014 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L"2 and L"~ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example. <s> BIB015 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> In this paper, an adaptive fuzzy-decentralized robust output-feedback-control approach is proposed for a class of large-scale strict-feedback nonlinear systems with the unmeasured states. The large-scale nonlinear systems in this paper are assumed to possess the unstructured uncertainties, unmodeled dynamics, and unknown high-frequency-gain sign. Fuzzy-logic systems are used to approximate the unstructured uncertainties, K-filters are designed to estimate the unmeasured states, and a dynamical signal and a special Nussbaum gain function are introduced into the control design to solve the problem of unknown high-frequency-gain sign and dominate unmodeled uncertainties, respectively. Based on the backstepping design and adaptive fuzzy-control methods, an adaptive fuzzy-decentralized robust output-feedback-control scheme is developed. It is proved that the proposed adaptive fuzzy-control approach can guarantee that all the signals in the closed-loop system are uniformly and ultimately bounded, and the tracking errors converge to a small neighborhood of the origin. The effectiveness of the proposed approach is illustrated by using simulation results. <s> BIB016 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> A novel neural network (NN)-based nonlinear decentralized adaptive controller is proposed for a class of large-scale, uncertain, interconnected nonlinear systems in strict-feedback form by using the dynamic surface control (DSC) principle, thus, the “explosion of complexity” problem which is observed in the conventional backstepping approach is relaxed in both state and output feedback control designs. The matching condition is not assumed when considering the interconnection terms. Then, NNs are utilized to approximate the uncertainties in both subsystem and interconnected terms. By using novel NN weight update laws with quadratic error terms as well as proposed control inputs, it is demonstrated using Lyapunov stability that the system states errors converge to zero asymptotically with both state and output feedback controllers, even in the presence of NN approximation errors in contrast with the uniform ultimate boundedness result, which is common in the literature with NN-based DSC and backstepping schemes. Simulation results show the effectiveness of the approach. <s> BIB017 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> In this paper, two adaptive neural network (NN) decentralized output feedback control approaches are proposed for a class of uncertain nonlinear large-scale systems with immeasurable states and unknown time delays. Using NNs to approximate the unknown nonlinear functions, an NN state observer is designed to estimate the immeasurable states. By combining the adaptive backstepping technique with decentralized control design principle, an adaptive NN decentralized output feedback control approach is developed. In order to overcome the problem of “explosion of complexity” inherent in the proposed control approach, the dynamic surface control (DSC) technique is introduced into the first adaptive NN decentralized control scheme, and a simplified adaptive NN decentralized output feedback DSC approach is developed. It is proved that the two proposed control approaches can guarantee that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded, and the observer errors and the tracking errors converge to a small neighborhood of the origin. Simulation results are provided to show the effectiveness of the proposed approaches. <s> BIB018 </s> Decentralised control for complex systems - an invited survey <s> Backstepping approach-based decentralised control <s> This paper focuses on the problem of neural-network-based decentralized adaptive output-feedback control for a class of nonlinear strict-feedback large-scale stochastic systems. The dynamic surface control technique is used to avoid the explosion of computational complexity in the backstepping design process. A novel direct adaptive neural network approximation method is proposed to approximate the unknown and desired control input signals instead of the unknown nonlinear functions. It is shown that the designed controller can guarantee all the signals in the closed-loop system to be semiglobally uniformly ultimately bounded in a mean square. Simulation results are provided to demonstrate the effectiveness of the developed control design approach. <s> BIB019
Backstepping is a technique developed by Petar V. Kokotovic and others at the beginning of 1990s BIB001 BIB002 . Compared with variable structure techniques and adaptive control, the history of the backstepping approach is relatively short. This approach requires that the considered system has a special structure called strict-feedback form described as followṡ . . . where x ∈ R n is vector and z i are scalars for i = 1, 2, . . . , k, and the variables x and z i with i = 1, 2, . . . , k form the system state and u ∈ R is system input. The functions f i (·) vanish at the origin and g i (·) are non-zero over the domain of interest for i = 0, 1, 2, . . . , k. It should be noted that a backstepping controller can not be explicitly expressed in terms of equations because the controller is developed step by step using recursive design. This may limit its application. However, backstepping design methods can provide a systematic procedure to design stabilising controllers, following a step by step recursive algorithm, whereby the feedback control laws and Lyapunov functions can be constructed systematically. Moreover, this approach can be employed to relax the matching condition when compared with the traditional Lyapunov design approach. A major advantage of backstepping is that it has the flexibility to avoid cancellation of useful nonlinearities and achieve desirable regulation and tracking properties BIB010 . Backstepping techniques require the considered system to have a special structure and thus most work considers parametric uncertainty instead of unstructured uncertainty. It is well known that for uncertain/unknown parameters, adaptive methods are much powerful as pointed out in the last section. Therefore large scale interconnected systems considered in this area usually have two characteristics: 1 all the isolated nominal subsystems have or can be transformed to the strict feedback form 2 the considered systems involve parametric uncertainty. These two features require adaptive updated laws to estimate the uncertain parameters and thus the approaches usually involve both adaptive and backstepping techniques BIB015 BIB014 BIB011 BIB004 BIB012 BIB005 . A decentralised PID controller is proposed using backstepping techniques in BIB008 where the considered system model is expressed in the frequency domain. Recently, some authors have combined backstepping techniques with intelligent control such as fuzzy control and neural network to deal with interconnected systems BIB017 BIB018 BIB019 BIB016 . As one of the advantages of backstepping approach is that it has the flexibility to achieve regulation and tracking properties, the relevant decentralised strategies have also received much attention. The first decentralised control based on backstepping without any requirement on subsystem relative degree was reported in BIB003 where perfect regulation is obtained. Global decentralised output regulation is considered in BIB005 where the reference input and disturbances are the sum of finitely sinusoidal functions with unknown amplitudes and frequencies. Observer-based decentralised disturbance attenuating output-feedback tracking controllers are designed in BIB006 ; BIB007 . By combining a neural network approach BIB009 , decentralised backstepping controllers are proposed in BIB018 and BIB013 where the tracking error only converges to a bounded region of the origin. The result developed in BIB011 expressed the tracking error as a function of design parameters which makes it possible to adjust the tracking transient performance. Compared with decentralised variable structure control and decentralised adaptive control, the domain of backstepping-based decentralised control is more immature and the obtained results are fewer.