Optimal and reinforced robustness designs of fuzzy variable structure tracking control for a piezoelectric actuator system

In this paper, a piezoelectric actuator (PEA) system is approximated by N subsystems, which are described by pulse transfer functions. The approximation error between the PEA system and the fuzzy linear pulse transfer function system is represented by additive nonlinear time-varying uncertainties in every subsystem. First, a dead-beat to the switching surface for every ideal subsystem is designed. It is called the "variable structure tracking control". The output disturbance of the ith subsystem is caused by the approximation error of fuzzy-model and the interaction dynamics resulting from other subsystems. In general, it is not small. Then, the H/sup /spl infin//-norm of the sensitivity function between the switching surface and the output disturbance is minimized. It is the "optimal robustness". Although the effect of the output disturbance is attenuated, a better performance can be reinforced by a switching control which is based on the Lyapunov redesign. This is the final step for the robustness design of control, which is "reinforced robustness". The stability of the overall system is verified by Lyapunov stability theory. Experimental work of a PEA system was carried out to confirm the validity of the proposed control.     

Robust discrete variable structure control with finite-time approach to switching surface

By using a dead-beat control technique of discrete-time systems, a robust discrete variable structure control (DVSC) is developed for the linear discrete-time systems subject to input disturbance, measurement noise and uncertainty. The proposed control includes two parts: equivalent control and switching control. Based on the internal model principle, the input disturbance and the measurement noise modeled as pulse transfer functions, are rejected by the equivalent control. The unmatched uncertainty caused by the time-invariant parameter variations is also tackled by the equivalent control. If the inverse of stable characteristic polynomial of the real closed-loop system is a finite-degree polynomial, the trajectory reaches the switching surface in a finite-time step. Due to the subjection of input disturbance or measurement noise or uncertainty, a poor response occurs. Under these circumstances, a switching control based on Lyapunov redesign is employed to improve the system performance. The stability of the closed-loop system is then verified by Lyapunov stability theory. Simulations are also given to confirm the usefulness of the proposed controller.     

Fuzzy neural-based control for nonlinear time-varying delay systems.

In this paper, a partially known nonlinear dynamic system with time-varying delays of the input and state is approximated by N fuzzy-based linear subsystems described by a state-space model with average delay. To shape the response of the closed-loop system, a set of fuzzy reference models is established. Similarly, the same fuzzy sets of the system rule are employed to design a fuzzy neural-based control. The proposed control contains a radial-basis function neural network to learn the uncertainties caused by the approximation error of the fuzzy model (e.g., time-varying delays and parameter variations) and the interactions resulting from the other subsystems. As the norm of the switching surface is inside of a defined set, the learning law starts; in this situation, the proposed method is an adaptive control possessing an extra compensation of uncertainties. As it is outside of the other set, which is smaller than the aforementioned set, the learning law stops; under this circumstance, the proposed method becomes a robust control without the compensation of uncertainties. A transition between robust control and adaptive control is also assigned to smooth the possible discontinuity of the control input. No assumption about the upper bound of the time-varying delays for the state and the input is required. However, two time-average delays are needed to simplify the controller design: 1) the stabilized conditions for every transformed delay-free subsystem must be satisfied; and 2) the learning uncertainties must be relatively bounded. The stability of the overall system is verified by Lyapunov stability theory. Simulations as compared with a linear transformed state feedback with integration control are also arranged to consolidate the usefulness of the proposed control.     

A fuzzy decentralized variable structure tracking control with optimal and improved robustness designs: theory and applications

To begin with, each subsystem of a nonlinear interconnected system was approximated by a weighted combination of L linear pulse transfer function systems (LPTFSs). For every nominal LPTFS of the mth subsystem, a dead-beat to its switching surface was first designed. The output disturbance of the mth LPTFS included the interconnections coming from the other subsystems, the approximation error of the mth subsystem, and the interactions resulting from the other LPTFSs. In general, this output disturbance was not small and contains various frequencies. Under this circumstances, the H/sup /spl infin//-norm of the weighted sensitivity function between the mth switching surface and its corresponding output disturbance was minimized. In addition, an appropriate selection of the weighted function for the sensitivity could reject the corresponding mode of the output disturbance. Although the effect of the output disturbance is attenuated and partially rejected, a better performance could be enhanced by a switching control based on the Lyapunov redesign. In addition, the stability of the overall system was verified by Lyapunov stability theory. The simulations for the LPTFSs with different delays or nonminimum phases or unstable features were arranged to evaluate the effectiveness of the proposed control. Finally, the application to the trajectory tracking of the robot arm including the fuzzy modeling was carried out to confirm the practicality of the proposed control.     

Fuzzy Mixed H2/H∞ Optimization-Based Decentralized Model Reference Control and Application to Piezo-Driven XY Table Systems

In this paper, a decentralized model reference control via fuzzy mixed H²/H^infin optimization design was developed. Each subsystem contained L linear pulse transfer function systems (LPTFSs). The reference model for every LPTFS was first designed to shape the response of the ith closed-loop subsystem. Then the H² -norm of the output error (i.e., the difference between the output of the reference model and the system) and weighted control input of the jth LPTFS was minimized to obtain a control such that smaller energy consumption with bounded tracking error of the jth LPTFS was achieved. However, an output disturbance of the jth LPTFS caused by the interactions among the LPTFSs, the interconnections among the subsystems, modeling errors, and external loads deteriorated system performance or even resulted in instability. In this situation, the H ^infin-norm of weighted sensitivity between output disturbance and output error of the jth LPTFS was minimized to attenuate its effect. A nonlinear control based on output error for every LPTFS was also established to improve robust performance. The stability of the overall system was then verified by Lyapunov stability criterion. The application to piezo-driven XY table system (PD-XY-TS) was carried out to confirm the usefulness of the proposed control     

Robust discrete-time sliding mode control for tracking and model following of uncertain nonlinear systems

This paper proposes a discrete-time controller for robust tracking and model following of a class of nonlinear, multi-input multi-output, systems. For this purpose, a discrete-time sliding mode controller (DTSMC) is used to ensure the stability, robustness and an output tracking against the modelling uncertainties, even at relatively large sampling periods. In this way, Takagi–Sugeno (T–S) fuzzy modelling is used to decompose the nonlinear system to a set fuzzy-blended locally linearised subsystems. Implementation of the second Lyapunov theory for mismatched uncertain nonlinear T–S fuzzy models results in a set of linear matrix inequalities, which is used to design the sliding surface. A new method is then proposed to reach the quasi-sliding mode and stay thereafter. Simulation studies show that the proposed method guarantees the stability of closed-loop system and achieves small tracking error in the presence of parametric uncertainties at large sampling periods.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

A new robust observer-based adaptive type-2 fuzzy control for a class of nonlinear systems

In this paper, a novel robust observer-based adaptive controller is presented using a proposed simplified type-2 fuzzy neural network (ST2FNN) and a new three dimensional type-2 membership function is presented. Proposed controller can be applied to the control of high-order nonlinear systems and adaptation of the consequent parameters and stability analysis are carried out using Lyapunov theorem. Moreover, a new adaptive compensator is presented to eliminate the effect of the external disturbance, unknown nonlinear functions approximation errors and sate estimation errors. In the proposed scheme, using the Lyapunov and Barbalat's theorem it is shown that the system is stable and the tracking error of the system converges to zero asymptotically. The proposed method is simulated on a flexible joint robot, two-link robot manipulator and inverted double pendulums system. Simulation results confirm that in contrast to other robust techniques, our proposed method is simple, give better performance in the presence of noise, external disturbance and uncertainties, and has less computational cost.     

Truncated adaptation design for decentralised neural dynamic surface control of interconnected nonlinear systems under input saturation

This paper studies decentralised neural adaptive control of a class of interconnected nonlinear systems, each subsystem is in the presence of input saturation and external disturbance and has independent system order. Using a novel truncated adaptation design, dynamic surface control technique and minimal-learning-parameters algorithm, the proposed method circumvents the problems of ‘explosion of complexity’ and ‘dimension curse’ that exist in the traditional backstepping design. Comparing to the methodology that neural weights are online updated in the controllers, only one scalar needs to be updated in the controllers of each subsystem when dealing with unknown systematic dynamics. Radial basis function neural networks (NNs) are used in the online approximation of unknown systematic dynamics. It is proved using Lyapunov stability theory that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. The tracking errors of each subsystems, the amplitude of NN approximation residuals and external disturbances can be attenuated to arbitrarily small by tuning proper design parameters. Simulation results are given to demonstrate the effectiveness of the proposed method.     

Stability and asynchronous stabilization for a class of discrete-time switched nonlinear systems with stable and unstable subsystems

The stability analysis and asynchronous stabilization problems for a class of discrete-time switched nonlinear systems with stable and unstable subsystems are investigated in this paper. The Takagi-Sugeno (T-S) fuzzy model is used to represent each nonlinear subsystem. Through using the T-S fuzzy model, the studied systems are modeled into the switched T-S fuzzy systems. By using the switching fuzzy-basis-dependent Lyapunov functions (FLFs) approach and mode-dependent average dwell time (MDADT) technique, the stability conditions for the open-loop switched T-S fuzzy systems with unstable subsystems and asynchronous stabilization conditions for the closed-loop switched T-S fuzzy systems with unstable subsystems are obtained. Both the stability results and asynchronous stabilization results are derived in terms of linear matrix inequalities (LMIs). Finally two numerical examples are provided to illustrate the effectiveness of the results obtained.     

NN-adaptive output feedback tracking control for a class of discrete-time non-affine systems with a dynamic compensator

The problem of tracking control for a class of uncertain non-affine discrete-time nonlinear systems with internal dynamics is addressed. The fixed point theorem is first employed to ensure the control problem in question is solvable and well-defined. Based on it, an adaptive output feedback control scheme based on neural network (NN) is presented. The proposed control algorithm consists of two parts: a dynamic compensator is introduced to stabilise the linear portion of the tracking error system; a single-hidden-layer neural network (SHL NN) approximation mechanism is introduced to cancel the uncertainties resulting from the non-affine function, where the recursive weight update rules of NN estimation are derived from the discrete-time version of Lyapunov control theory. Ultimate boundedness of the error signals is shown through Lyapunov’s direct method and the discrete-time version of input-to-state stability (ISS) theory. Finally, a model of automatical underwater vehicle (AUV) is considered to show the effectiveness of the proposed control scheme.     

不确定非线性切换系统的鲁棒H∞控制

讨论了一类不确定非线性切换系统的鲁棒H∞控制问题．首先,基于多Lyapunov函数方法,设计状态反馈控制器以及切换律,使得对于所有允许的不确定性．相应的闭环系统渐近稳定又具有指定的L2-增益．该问题可解的充分条件以一组含有纯量函数的偏微分不等式形式给出,此偏微分不等式较一般Hamilton-Jacobi不等式更具可解性．所提出的方法不要求任何一个子系统渐近稳定．接着作为应用,借助混杂状态反馈策略讨论了非切换不确定非线性系统的鲁棒H∞控制问题．最后通过一个简单例子说明了控制设计方法的可行性．     

Adaptive fuzzy sliding mode algorithm-based decentralised control for a permanent magnet spherical actuator

The dynamic model of multi-degree-of-freedom permanent magnet (PM) spherical actuators is multivariate and nonlinear due to strong inter-axis couplings, which affects the trajectory tracking performance of the system. In this paper, a decentralised control strategy based on adaptive fuzzy sliding mode (AFSM) algorithm is developed for a PM spherical actuator to enhance its trajectory tracking performance. In this algorithm, the coupling terms are separated as subsystems from the entire system. The AFSM algorithm is applied in such a way that the fuzzy logic systems are used to approximate the subsystem with uncertainties. A sliding mode term is introduced to compensate for the effect of coupling terms and fuzzy approximation error. The stability of the proposed method is guaranteed by choosing the appropriate Lyapunov function. Both simulation and experimental results show that the proposed control algorithm can effectively handle various uncertainties and inter-axis couplings, and improve the trajectory tracking precision of the spherical actuator.     

Extended state observer for MIMO nonlinear systems with stochastic uncertainties

ABSTRACT

In this paper, both linear extended state observer (ESO) and nonlinear ESO with homogeneous weighted functions are proposed for a class of multi-input multi-output (MIMO) nonlinear systems composed of coupled subsystems with large stochastic uncertainties. The stochastic uncertainties in each subsystem including internal coupled unmodelled dynamics and external stochastic disturbance without known statistical characteristics are lumped together as the stochastic total disturbance (extended state) of each subsystem. The linear ESO and nonlinear ESO are designed separately for real-time estimation of not only the unmeasured state but also the stochastic total disturbance of each subsystem. The practical mean square convergence of these two classes of ESOs are developed. Some numerical simulations are presented to demonstrate the effectiveness of the ESOs with the advantages of smaller peaking values and more accurate estimation by the nonlinear ESO.     

Decentralised H ∞ filtering of interconnected discrete-time fuzzy systems with time delays

This article considers the decentralised H _∞ filtering of interconnected discrete-time fuzzy systems with time delays based on piecewise Lyapunov–Krasovskii functionals. The fuzzy system consists of J interconnected time-delay discrete-time Takagi–Sugeno fuzzy subsystems and the decentralised H _∞ filter is designed for each subsystem. It is shown that the stability with H _∞ performance of overall filtering error system can be established if a piecewise Lyapunov–Kroasovskii functional can be constructed, and moreover, the functional can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A simulation example is given to show the effectiveness of the presented approach.     

Mixed H/sub 2//H/sub /spl infin// design for a decentralized discrete variable structure control with application to mobile robots

In this paper, a decentralized discrete variable structure control via mixed H/sub 2//H/sub /spl infin// design was developed. In the beginning, the H/sub 2/-norm of output error and weighted control input was minimized to obtain a control such that smaller energy consumption with bounded tracking error was assured. In addition, a suitable selection of this weighted function (connected with frequency) could reduce the effect of disturbance on the control input. However, an output disturbance caused by the interactions among subsystems, modeling error, and external load deteriorated system performance or even brought about instability. In this situation, the H/sub /spl infin//-norm of weighted sensitivity between output disturbance and output error was minimized to attenuate the effect of output disturbance. Moreover, an appropriate selection of this weighted function (related to frequency) could reject the corresponding output disturbance. No solution of Diophantine equation was required; the computational advantage was especially dominated for low-order system. For further improving system performance, a switching control for every subsystem was designed. The proposed control (mixed H/sub 2//H/sub /spl infin// DDVSC) was a three-step design method. The stability of the overall system was verified by Lyapunov stability criterion. The simulations and experiments of mobile robot were carried out to evaluate the usefulness of the proposed method.     

Robust stability of a class of hybrid nonlinear systems

In this paper, we analyze the discrete behavior to identify all kinds of cycles of hybrid nonlinear systems and then study the continuous behavior along each kind of cycle. Based on these analysis, we construct some continuous functions to bound Lyapunov functions along all subsystems and identify a subsequence of time points where the Lyapunov functions are non-increasing. We use these results to derive some new sufficient conditions for the robust stability of a class of hybrid nonlinear systems with polytopic uncertainties. These conditions do not require the Lyapunov functions to be non-increasing along each subsystem nor the whole sequence of the switching. Furthermore, they do not require the knowledge of continuous trajectory     

An integro-differential formula on the Wiener kernels and its application to sandwich system identification

It is well known that a nonlinear system with a white Gaussian noise input can be characterized in terms of kernels using the celebrated Wiener theory. In a practical use of the method, however, one may encounter difficulty in obtaining higher order kernels except for the first few because of, for instance, the excessive computational requirement. In this paper, we give an integro-differential formula on the kernels and as its application, an algorithm to identify a cascade system of a linear, a memoryless nonlinear, and linear subsystems, which we call a sandwich system as a whole. According to the formula, kernels up to the second order for different power levels of the input noise are required to identify the subsystems. Impulse response functions of the two linear subsystems can be obtained under appropriate normalization conditions, while the nonlinear subsystem is estimated in the form of a truncated Hermite polynomial expansion. As illustrated examples, two such systems are identified using the algorithm.     

超空泡航行体的自适应模糊滑模控制研究

针对超空泡航行体动态中的非线性项和未建模动态,以及由空泡形状改变引起的扰动问题,提出了一种基于自适应模糊滑模的纵平面运动控制器设计方法。该方法利用自适应模糊系统逼近超空泡航行体模型中的非线性不确定项;利用滑模控制对干扰的鲁棒性,克服逼近误差和干扰;最后用Lyapunov定理证明了闭环系统的稳定性。仿真结果表明,设计的控制器可有效克服滑行力的计算误差以及水动力参数的不确定性。     

Robust fault diagnosis scheme in a class of nonlinear system based on UIO and fuzzy residual

In this study, a novel robust fault diagnosis scheme is developed for a class of nonlinear systems when both fault and disturbance are considered. The proposed scheme includes both component and sensor fault with nonlinear system that transferred to nonlinear Takagi-Sugeno (T-S) model. It considers a larger category of nonlinear system when fuzzification is used for only nonlinear distribution matrices. In fact the proposed method covers nonlinear systems could not transform to linear T-S model. This paper studies the problem of robust fault diagnosis based on two fuzzy nonlinear observers, the first one is a fuzzy nonlinear unknown input observer (FNUIO) and the other is a fuzzy nonlinear Luenberger observer (FNLO). This approach decouples the faulty subsystem from the rest of the system through a series of transformations. Then, the objective is to design FNUIO to guarantee the asymptotic stability of the error dynamic using the Lyapunov method; meanwhile, FNLO is designed for faulty subsystem to generate fuzzy residual signal based on a quadratic Lyapunov function and some matrices inequality convexification techniques. FNUIO affects only the fault free subsystem and completely removes any unknown inputs such as disturbances when residual signal is generated by FNLO is affected by component or sensor fault. This novelty and using nonlinear system in T-S model make the proposed method extremely effective from last decade literature. Sufficient conditions are established in order to guarantee the convergence of the state estimation error. Thus, a residual generator is determined on the basis of LMI conditions such that the estimation error is completely sensitive to fault vector and insensitive to the unknown inputs. Finally, an numerical example is given to show the highly effectiveness of the proposed fault diagnosis scheme.