Sliding mode dynamics in continuous feedback control for distributed discrete-event scheduling

A continuous feedback control approach for real-time scheduling of discrete events is presented in this paper motivated by the need for control theoretic techniques to analyze and design such systems in distributed manufacturing applications. These continuous feedback control systems exhibit highly nonlinear and discontinuous dynamics. Specifically, when the production demand in the manufacturing system exceeds the available resource capacity then the control system “chatters” and exhibits sliding modes. This sliding mode behavior is advantageously used in the scheduling application by allowing the system to visit different schedules within an infinitesimal region near the sliding surface. In this paper, an analytical model is developed to characterize the sliding mode dynamics. This model is then used to design controllers in the sliding mode domain to improve the effectiveness of the control system to “search” for schedules with good performance. Computational results indicate that the continuous feedback control approach can provide near-optimal schedules and that it is computationally efficient compared to existing scheduling techniques.     

Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems

Two robust adaptive control schemes for single-input single-output (SISO) strict feedback nonlinear systems possessing unknown nonlinearities, capable of guaranteeing prescribed performance bounds are presented in this paper. The first assumes knowledge of only the signs of the virtual control coefficients, while in the second we relax this assumption by incorporating Nussbaum-type gains, decoupled backstepping and non-integral-type Lyapunov functions. By prescribed performance bounds we mean that the tracking error should converge to an arbitrarily predefined small residual set, with convergence rate no less than a prespecified value, exhibiting a maximum overshoot less than a sufficiently small prespecified constant. A novel output error transformation is introduced to transform the original “constrained” (in the sense of the output error restrictions) system into an equivalent “unconstrained”one. It is proven that the stabilization of the “unconstrained” system is sufficient to solve the problem. Both controllers are smooth and successfully overcome the loss of controllability issue. The fact that we are only concerned with the stabilization of the “unconstrained” system, severely reduces the complexity of selecting both the control parameters and the regressors in the neural approximators. Simulation studies clarify and verify the approach.     

Adaptive boundary control for unstable parabolic PDEs—Part II: Estimation-based designs

The certainty equivalence approach to adaptive control is commonly used with two types of identifiers: passivity-based identifiers and swapping identifiers. The “passive” (also known as “observer-based”) approach is the prevalent identification technique in existing results on adaptive control for PDEs but has so far not been used in boundary control problems. The swapping approach, prevalent in finite-dimensional adaptive control is employed here for the first time in adaptive control of PDEs. For a class of unstable parabolic PDEs we prove a separation principle result for both the passive and swapping identifiers combined with the backstepping boundary controllers. The result is applicable in any dimension. For physical reasons we restrict our attention to dimensions no higher than three. The results of the paper are illustrated by simulations.     

Adaptive neural dynamic surface control for servo systems with unknown dead-zone

An adaptive neural controller is proposed for nonlinear systems with a nonlinear dead-zone and multiple time-delays. The often used inverse model compensation approach is avoided by representing the dead-zone as a time-varying system. The “explosion of complexity” in the backstepping synthesis is eliminated in terms of the dynamic surface control (DSC) technique. A novel high-order neural network (HONN) with only a scalar weight parameter is developed to account for unknown nonlinearities. The control singularity and some restrictive requirements on the system are circumvented. Simulations and experiments for a turntable servo system with permanent-magnet synchronous motor (PMSM) are provided to verify the reliability and effectiveness.     

Adaptive robust stabilisation of uncertain nonlinear dynamical systems: an improved backstepping approach

The main purpose of this paper is to propose a design approach by which some simple adaptive robust controllers can be synthesised for a class of uncertain nonlinear dynamical systems which can be transformed into uncertain strict-feedback nonlinear systems. In this paper, an improved backstepping design approach is presented to synthesising a class of continuous adaptive robust state-feedback controllers with a rather simple structure. The improved backstepping design approach can avoid the repeated differentiation problem which appears in using the conventional backstepping algorithm. In particular, it is not required to know the nonlinear upper bound functions of uncertainties. In the light of the presented approach, the state-feedback controllers can be constructed to be linear in the state, with the time-varying control gains which can be self-tuned by the adaptive laws. Similar to the conventional backstepping algorithm, the improved backstepping approach can be extended to a rather large class of uncertain nonlinear systems, and by combining the improved backstepping approach with other control methods, it may be expected to obtain a number of interesting results.     

Adaptive Fuzzy Control for Nonlinear Time‐Delay Systems with Dynamical Uncertainties

In this paper, an adaptive fuzzy backstepping robust control approach is proposed for a class of SISO nonlinear strict‐feedback systems. The nonlinear systems addressed in this paper are assumed to possess three uncertainties: (i) the unstructured uncertainties; (ii) the time delays; and (iii) the dynamics uncertainties. In adaptive backstepping recursive design, fuzzy logic systems are used to approximate the unstructured uncertainties. A nonlinear damping technique and Lyapunov–Krasovskii functions are introduced to cancel the effects of the dynamics uncertainties and deal with the time delays, respectively. Combining the backstepping technique and a small gain approach, a stable adaptive fuzzy robust control approach is developed. It is proved that all the signals of the closed‐loop system are semi‐golablly uniformaly ultimately bounded (SUUB). The effectiveness of the proposed approach is illustrated by a simulation example.     

Stabilization of max-plus-linear systems using model predictive control: The unconstrained case

Max-plus-linear (MPL) systems are a class of event-driven nonlinear dynamic systems that can be described by models that are “linear” in the max-plus algebra. In this paper we derive a solution to a finite-horizon model predictive control (MPC) problem for MPL systems where the cost is designed to provide a trade-off between minimizing the due date error and a just-in-time production. In general, MPC can deal with complex input and states constraints. However, in this paper we assume that these are not present and it is only required that the input should be a nondecreasing sequence, i.e. we consider the “unconstrained” case. Despite the fact that the controlled system is nonlinear, by employing recent results in max-plus theory we are able to provide sufficient conditions such that the MPC controller is determined analytically and moreover the stability in terms of Lyapunov and in terms of boundedness of the closed-loop system is guaranteed a priori.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

Adaptive backstepping control for a class of nonlinear systems using neural network approximations

In this paper, an adaptive neural network (NN) backstepping technique is developed for tracking control of a class of nonlinear systems. NNs are used to compensate for the unknown nonlinear functions in the system. A systematic backstepping approach is established to synthesize the adaptive NN control scheme that ensures the boundedness of all the signals in the closed‐loop system, and yields a small tracking error. The issue of transient performance is also addressed under an analytical framework. The effectiveness of the proposed scheme is demonstrated by computer simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

一类纯反馈力学系统的自适应模糊动态面控制

针对一类纯反馈形式的不稳定力学系统,提出自适应模糊动态面控制方法。在一般动态面控制的设计框架下,引入模糊系统逼近模型的未知函数,设计自适应律在线估计模糊系统权参数和模型未知参数,通过Lyapunov方法证明得出闭环系统半全局稳定。该策略避免了传统反演设计存在的“微分爆炸”现象,并且解决了纯反馈系统控制设计中通常存在的循环设计问题。仿真结果表明,控制系统能够克服不确定性,且能够简单有效地实现跟踪控制。     

Adaptive backstepping controller design for nonlinear uncertain systems using fuzzy neural systems

In this article, we propose an adaptive backstepping control scheme using fuzzy neural networks (FNNs), ABC_FNN, for a class of nonlinear non-affine systems in non-triangular form. The nonlinear non-affine system contains the uncertainty, external disturbance or parameters variations. Two kinds of FNN systems are used to estimate the unknown system functions. According to the FNN estimations, the adaptive backstepping control (ABC_FNN) signal can be generated by backstepping design procedure such that the system output follows the desired trajectory. To ensure robustness and performance, a proportional-integral-surface function and robust controller are designed to improve the control performance. Based on the Lyapunov stability theory, the stability of a closed-loop system is guaranteed and the adaptive laws of the FNN parameters are obtained. This approach is also valid for nonlinear affine system with uncertainty or disturbance. The uncertainty and disturbance terms are estimated by FNNs and treated by the ABC_FNN scheme. Finally, the effectiveness of the proposed ABC_FNN is demonstrated through the simulation of controlling a nonlinear non-affine system and the continuously stirred tank reactor plant to demonstrate the performances of our approach.     

Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large‐scale systems using DSC technique

In this paper, an adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large‐scale stochastic nonlinear systems without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. Using the designed fuzzy state observer, and by combining the adaptive backstepping technique with dynamic surface control technique, an adaptive fuzzy decentralized output feedback control approach is developed. It is shown that the proposed control approach can guarantee that all the signals of the resulting closed‐loop system are semi‐globally uniformly ultimately bounded in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. A simulation example is provided to show the effectiveness of the proposed approaches. Copyright © 2011 John Wiley & Sons, Ltd.     

-bounded robust control of nonlinear cascade systems

In this paper, we consider the L^∞-bounded robust control problem for a class of nonlinear cascade systems with disturbances. Sufficient conditions are provided under which a hard bound is imposed on the system performance measure. The backstepping approach is used for controller design. A practical example is provided to illustrate the method.     

Robust adaptive fuzzy backstepping output feedback tracking control for nonlinear system with dynamic uncertainties

In this paper, an adaptive fuzzy output feedback control approach based on backstepping design is proposed for a class of SISO strict feedback nonlinear systems with unmeasured states, nonlinear uncertainties, unmodeled dynamics, and dynamical disturbances. Fuzzy logic systems are employed to approximate the nonlinear uncertainties, and an adaptive fuzzy state observer is designed for the states estimation. By combining backstepping technique with the fuzzy adaptive control approach, a stable adaptive fuzzy...     

Adaptive neural control of nonlinear MIMO systems with unknown time delays

In this paper, a novel adaptive NN control scheme is proposed for a class of uncertain multi-input and multi-output (MIMO) nonlinear time-delay systems. RBF NNs are used to tackle unknown nonlinear functions, then the adaptive NN tracking controller is constructed by combining Lyapunov-Krasovskii functionals and the dynamic surface control (DSC) technique along with the minimal-learning-parameters (MLP) algorithm. The proposed controller guarantees uniform ultimate boundedness (UUB) of all the signals in the closed-loop system, while the tracking error converges to a small neighborhood of the origin. An advantage of the proposed control scheme lies in that the number of adaptive parameters for each subsystem is reduced to one, triple problems of “explosion of complexity”, “curse of dimension” and “controller singularity” are solved, respectively. Finally, a numerical simulation is presented to demonstrate the effectiveness and performance of the proposed scheme.     

Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties

We pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters. Extremum seeking controllers are developed to drive system states to the desired set-points that extremize the value of an objective function. The proposed adaptive extremum seeking controller is “inverse optimal” in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action. Simulation studies are provided to verify the effectiveness of the proposed approach.     

Output recurrent wavelet neural network-based adaptive backstepping controller for a class of MIMO nonlinear non-affine uncertain systems

In this paper, an adaptive backstepping control problem is proposed for a class of multiple-input-multiple-output nonlinear non-affine uncertain systems. An output recurrent wavelet neural network (ORWNN) is used to approximate the unknown nonlinear functions to develop the proposed adaptive backstepping controller. The proposed ORWNN combines the advantages of wavelet-based neural network, fuzzy neural network, and output feedback layer to achieve higher approximation accuracy and faster convergence. According to the estimation of ORWNN, the control scheme is designed by backstepping approach such that the system outputs follow the desired trajectories. Based on the Lyapunov approach, our approach guarantees that the system outputs converge to a small neighborhood of the references signals, that is, all signals of the closed-loop system are semi-globally uniformly ultimately bounded. Finally, simulation results including double pendulums system and two inverted pendulums on carts system are shown to demonstrate the performance and effectiveness of our approach.     

Robust backstepping control of nonlinear systems using neuralnetworks

A controller is proposed for the robust backstepping control of a class of general nonlinear systems using neural networks (NNs). A tuning scheme is proposed which can guarantee the boundedness of tracking error and weight updates. Compared with adaptive backstepping control schemes, we do not require the unknown parameters to be linear parametrizable. No regression matrices are needed, so no preliminary dynamical analysis is needed. One salient feature of our NN approach is that there is no need for the off-line learning phase. Three nonlinear systems, including a one-link robot, an induction motor, and a rigid-link flexible-joint robot, were used to demonstrate the effectiveness of the proposed scheme     

Approximation-based adaptive fuzzy control for a class of non-strict-feedback stochastic nonlinear systems

This paper considers the problem of adaptive fuzzy control of a class of single-input/single-output （SISO） nonlinear stochastic systems in non-strict-feedback form. Fuzzy logic systems are used to approximate the uncertain nonlinearities and backstepping technique is utilized to construct an adaptive fuzzy controller. The proposed controller guarantees that all the signals in the resulting closed-loop system are bounded in probability. The main contribution of this note lies in providing a control strategy for a class of nonlinear systems in non- strict-feedback form. Simulation result is used to test the effectiveness of the suggested approach.     

Adaptive neural output feedback tracking control for a class of nonlinear systems

In this paper, an adaptive neural output feedback control scheme based on backstepping technique and dynamic surface control (DSC) approach is developed to solve the tracking control problem for a class of nonlinear systems with unmeasurable states. Firstly, a nonlinear state observer is designed to estimate the unmeasurable states. Secondly, in the controller design process, radial basis function neural networks (RBFNNs) are utilised to approximate the unknown nonlinear functions, and then a novel adaptive neural output feedback tracking control scheme is developed via backstepping technique and DSC approach. It is shown that the proposed controller ensures that all signals of the closed-loop system remain bounded and the tracking error converges to a small neighbourhood around the origin. Finally, two numerical examples and one realistic example are given to illustrate the effectiveness of the proposed design approach.