Discrete-time Nonlinear Control Systems Research Articles

A robust predictive actuator fault-tolerant control scheme for Takagi-Sugeno fuzzy systems

Abstract The paper deals with the problem of robust predictive fault-tolerant control for nonlinear discrete-time systems described by the Takagi-Sugeno models. The proposed approach is based on a triple stage procedure, i.e. it starts from fault estimation while the fault is compensated with a robust controller. The robust controller is designed without taking into account the input constraints related with the actuator saturation that may change due to its faulty behaviour. Thus, to check the compensation feasibility, the robust invariant set is developed, which takes into account the input constraints. If the current state does not belong to the robust invariant set, then suitable predictive control actions are performed in order to enhance the invariant set. This appealing phenomenon makes it possible to enlarge the domain of attraction, which makes the proposed approach an efficient solution for the fault-tolerant control. The final part of the paper shows an illustrative example regarding the application of the proposed approach to the twin-rotor system.

A new compensation for fuzzy static output-feedback control of nonlinear networked discrete-time systems

This paper investigates the problem of static output-feedback control of discrete-time nonlinear networked control systems with data packet dropouts on the basis of the fuzzy-model-based Takagi-Sugeno (T-S) approach. The nonlinear system is modeled via T-S fuzzy model. Both imperfect sensor-to-controller (S/C) and controller-to-actuator (C/A) links are considered via a novel data model. Some slack matrices are introduced to reduce the conservativeness. By constructing fuzzy Lyapunov function, the sufficient conditions are derived to guarantee the resulting closed-loop system to be stochastically stable as well as the predefined H∞ performance index. Finally, some simulation results are given to demonstrate the effectiveness of the proposed results.

A Unified Approach to Adaptive Neural Control for Nonlinear Discrete-Time Systems With Nonlinear Dead-Zone Input.

In this paper, an effective adaptive control approach is constructed to stabilize a class of nonlinear discrete-time systems, which contain unknown functions, unknown dead-zone input, and unknown control direction. Different from linear dead zone, the dead zone, in this paper, is a kind of nonlinear dead zone. To overcome the noncausal problem, which leads to the control scheme infeasible, the systems can be transformed into a m -step-ahead predictor. Due to nonlinear dead-zone appearance, the transformed predictor still contains the nonaffine function. In addition, it is assumed that the gain function of dead-zone input and the control direction are unknown. These conditions bring about the difficulties and the complicacy in the controller design. Thus, the implicit function theorem is applied to deal with nonaffine dead-zone appearance, the problem caused by the unknown control direction can be resolved through applying the discrete Nussbaum gain, and the neural networks are used to approximate the unknown function. Based on the Lyapunov theory, all the signals of the resulting closed-loop system are proved to be semiglobal uniformly ultimately bounded. Moreover, the tracking error is proved to be regulated to a small neighborhood around zero. The feasibility of the proposed approach is demonstrated by a simulation example.

On Feedback Passivity of Discrete-Time Nonlinear Networked Control Systems With Packet Drops

We analyze the feedback passivity of a networked control system in which the control packets may be dropped by the communication channel. Specifically, we consider a discrete-time switched nonlinear system with relative degree zero that switches between two modes. At the instants when the communication link transmits the packet successfully, the system evolves in closed-loop and the increase in storage function is bounded below the energy supplied by the control input. At the instants when a packet drop occurs, the system evolves in open loop according to the free dynamics of the closed-loop mode and the increase in storage function is not necessarily bounded by the supplied energy. The literature on passivity of switched systems seems to consider only the case when all the modes are passive, which is not the case here. We prove that if the ratio of time steps for which the system evolves in closed-loop versus in open loop is lower bounded by a critical number, the system is locally feedback passive in a suitably defined sense. This generalized definition of feedback passivity is useful since it preserves two important properties of classical passivity—that feedback passivity implies asymptotic stabilizability for zero state detectable systems and that feedback passivity is preserved in parallel and negative feedback interconnections.

Relaxed [formula omitted] control design of discrete-time Takagi–Sugeno fuzzy systems: A multi-samples approach

This paper investigates the problem of fuzzy H∞ control design for a class of discrete-time nonlinear control systems via a multi-samples approach. A new fuzzy H∞ controller, which is parameter-dependent on not only the current-time but also the one-step-past membership functions, is developed to derive less conservative design conditions for ensuring both the asymptotic stability and the prescribed H∞ performance index of the obtained closed-loop system. Thanks to the usage of both a new fuzzy Lyapunov function and an extension of the homogenous matrix polynomial methodology, linear matrix inequality (LMI)-based H∞ control design conditions are obtained for minimizing the normal level in a systematic and relaxed way. In addition, further efforts are made by developing an efficient slack variable approach. Finally, two numerical examples are provided to illustrate the effectiveness of the results given in this paper.

Fault detection for a class of non‐linear networked control systems in the presence of Markov sensors assignment with partially known transition probabilities

In this study, the problem of fault detection for a class of discrete-time non-linear networked control systems is investigated. An event modelled as a Markov chain taking matrix values in a certain set with partially known transition probabilities is utilised to characterise the phenomenon of the sensors assignment. A full-order mode-dependent fault detection filter is constructed and the corresponding fault detection problem is converted into an H ∞ filtering problem of a Markov jump system with partially known transition probabilities. Sufficient conditions for the existence of the fault detection filter are formulated as a linear matrix inequality-based convex optimisation problem. If the convex optimisation problem has a feasible solution, the corresponding fault detection filter parameters are determined. A numerical example with four cases of transition probability matrices is presented to show the usefulness of the developed method.

Supervised Model Predictive Control for Discrete-time Nonlinear Systems with Time-varying Delay

The present paper deals with a decoupled multimodel predictive control based on multi-observer for the control of discrete-time nonlinear systems with time-varying delay. For each local model, a controller based on partial predictor/observer is synthesized. A switching algorithm is established to yield the adequate partial controller ensuring the closed-loop desired performances. Simulation results are given to illustrate the significance of the proposed decoupled multimodel predictive control strategy.

Interval type-2 fuzzy control for nonlinear discrete-time systems with time-varying delays

This paper is concerned with the problem of fault-tolerant control for discrete-time interval type-2 (IT2) fuzzy time delay system with actuator faults under imperfect premise matching. The time-varying delay and actuator fault are first taken into account for the IT2 fuzzy discrete-time systems. The novel IT2 state-feedback controller and the IT2 fuzzy model share different lower and upper membership functions. The fault-tolerant controller is designed to compensate for the effect of faults by stabilizing the closed-loop system under the actuator failures. Furthermore, the standard IT2 state-feedback controller is designed such that the closed-loop system is asymptotically stable and has an H∞ performance. The obtained conditions of the fault-tolerant controller and the standard IT2 controller can be expressed by the convex optimization problems. A numerical example is presented to show the effectiveness of the proposed results.

Event-triggered controller design of nonlinear discrete-time networked control systems in T-S fuzzy model

This article is concerned with event-triggered fuzzy control design for a class of discrete-time nonlinear networked control systems (NCSs) with time-varying communication delays. Firstly, a more general mixed event-triggering scheme (ETS) is proposed. Secondly, considering the effects of the ETS and communication delays, based on the T-S fuzzy model scheme and time delay system approach, the original nonlinear NCSs is reformulated as a new event-triggered networked T-S fuzzy systems with interval time-varying delays. Sufficient conditions for uniform ultimately bound (UUB) stability are established in terms of linear matrix inequalities (LMIs). In particular, the quantitative relation between the boundness of the stability region and the triggering parameters are studied in detail. Thirdly, a relative ETS is also provided, which can be seen as a special case of the above proposed mixed ETS. As a difference from the preceding results, sufficient conditions on the existence of desired fuzzy controller are derived to ensure the asymptotic stability of the closed-loop system with reduced communication frequency between sensors and controllers. Moreover, a co-design algorithm for simultaneously determining the gain matrices of the fuzzy controller and the triggering parameters is developed. Finally, two illustrative examples are presented to demonstrate the advantage of the proposed ETS and the effectiveness of the controller design method.

T-S Fuzzy Model Based Control Strategy for the Networked Suspension Control System of Maglev Train

The control problem for the networked suspension control system of maglev train with random induced time delay and packet dropouts is investigated. First, Takagi-Sugeno (T-S) fuzzy models are utilized to represent the discrete-time nonlinear networked suspension control system, and the parameters uncertainties of the nonlinear model have also been taken into account. The controllers take the form of parallel distributed compensation. Then, a sufficient condition for the stability of the networked suspension control system is derived. Based on the criteria, the state feedback fuzzy controllers are obtained, and the controller gains can be computed by using MATLAB LMI Toolbox directly. Finally, both the numerical simulations and physical experiments on the full-scale single bogie of CMS-04 maglev train have been accomplished to demonstrate the effectiveness of this proposed method.

Near optimal output feedback control of nonlinear discrete-time systems based on reinforcement neural network learning

In this paper, the output feedback based finite-horizon near optimal regulation of nonlinear affine discrete-time systems with unknown system dynamics is considered by using neural networks (NNs) to approximate Hamilton-Jacobi-Bellman (HJB) equation solution. First, a NN-based Luenberger observer is proposed to reconstruct both the system states and the control coefficient matrix. Next, reinforcement learning methodology with actor-critic structure is utilized to approximate the time-varying solution, referred to as the value function, of the HJB equation by using a NN. To properly satisfy the terminal constraint, a new error term is defined and incorporated in the NN update law so that the terminal constraint error is also minimized over time. The NN with constant weights and time-dependent activation function is employed to approximate the time-varying value function which is subsequently utilized to generate the finite-horizon near optimal control policy due to NN reconstruction errors. The proposed scheme functions in a forward-in-time manner without offline training phase. Lyapunov analysis is used to investigate the stability of the overall closed-loop system. Simulation results are given to show the effectiveness and feasibility of the proposed method.

Robust and resilient state-dependent control of discrete-time nonlinear systems with general performance criteria

A novel state-dependent control approach for discrete-time nonlinear systems with general performance criteria is presented. This controller is robust for unstructured model uncertainties, resilient against bounded feedback control gain perturbations in achieving optimality for general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state-dependent linear matrix inequality (LMI) at each time step, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control to provide a novel robust control design. The effectiveness of the proposed technique is demonstrated by simulation of the control of inverted pendulum.

Control of discrete-time nonlinear systems actuated through counter-convecting transport dynamics

We consider the stabilisation of discrete-time nonlinear systems that are actuated through a pair of transport partial difference equation (PdE) systems that convect in the opposite directions from one another. An explicit feedback law that compensates the discrete PdE dynamics is designed. Global asymptotic stability of the closed-loop system is proved with the aid of a Lyapunov function. The feedback design is illustrated through an example. The proposed design in this paper allows the delay to be arbitrarily long and time-varying. Furthermore, our predictor feedback law in discrete time is explicit as the predictor state is computed by an algebraic equation.

Adaptive Control of Nonlinear Discrete-Time Systems by Using OS-ELM Neural Networks

As a kind of novel feedforward neural network with single hidden layer, ELM (extreme learning machine) neural networks are studied for the identification and control of nonlinear dynamic systems. The property of simple structure and fast convergence of ELM can be shown clearly. In this paper, we are interested in adaptive control of nonlinear dynamic plants by using OS-ELM (online sequential extreme learning machine) neural networks. Based on data scope division, the problem that training process of ELM neural network is sensitive to the initial training data is also solved. According to the output range of the controlled plant, the data corresponding to this range will be used to initialize ELM. Furthermore, due to the drawback of conventional adaptive control, when the OS-ELM neural network is used for adaptive control of the system with jumping parameters, the topological structure of the neural network can be adjusted dynamically by using multiple model switching strategy, and an MMAC (multiple model adaptive control) will be used to improve the control performance. Simulation results are included to complement the theoretical results.

Predictive and robust fault-tolerant control for Takagi-Sugeno systems

The paper deals with the problem of robust predictive fault-tolerant control for nonlinear discrete-time systems described by the Takagi-Sugeno models. The proposed approach consists of three steps, i.e. it starts from fault estimation, the fault is compensated with a robust controller, and finally, when the compensation is not successful then a suitable predictive action is performed. This appealing phenomenon makes it possible to enlarge the domain of attraction, which makes the proposed approach an efficient solution for the fault-tolerant control. The final part of the paper shows an illustrative example regarding the application of the proposed approach to the two-tank system.

H∞ fault detection for a class of T-S fuzzy model-based nonlinear networked control systems

In this paper, a robust H ∞ fault detection filter is designed for a class of discrete-time nonlinear networked control systems via T-S fuzzy model with multiple bounded state delay and random packet dropout induced by the limited bandwidth of communication networks. Our aim in this paper is to analyze and design a robust H ∞ full-order fault detection filter, such that the filtering error dynamics is asymptotically mean-square stable with a prescribed H ∞ performance level. Sufficient conditions for the existence of the desired filter are presented. Finally, an example is given to illustrate the effectiveness and applicability of the proposed new design method.

SOS Based Robust <inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$ </tex-math></inline-formula> Fuzzy Dynamic Output Feedback Control of Nonlinear Networked Control Systems

In this paper, a methodology for designing a fuzzy dynamic output feedback controller for discrete-time nonlinear networked control systems is presented where the nonlinear plant is modelled by a Takagi-Sugeno fuzzy model and the network-induced delays by a finite state Markov process. The transition probability matrix for the Markov process is allowed to be partially known, providing a more practical consideration of the real world. Furthermore, the fuzzy controller's membership functions and premise variables are not assumed to be the same as the plant's membership functions and premise variables, that is, the proposed approach can handle the case, when the premise of the plant are not measurable or delayed. The membership functions of the plant and the controller are approximated as polynomial functions, then incorporated into the controller design. Sufficient conditions for the existence of the controller are derived in terms of sum of square inequalities, which are then solved by YALMIP. Finally, a numerical example is used to demonstrate the validity of the proposed methodology.

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.

Measurement feedback disturbance decoupling in discrete-time nonlinear systems

The paper studies the disturbance decoupling problem by the dynamic measurement feedback for discrete-time nonlinear control systems. To address the problem the algebraic approach, called the algebra of functions, is applied, which allows the system description also depend on non-differentiable functions. A necessary and sufficient condition is given in terms of controlled and (h,f)-invariant functions. Also, algorithms are derived, which find invariant functions and the required feedback. The algorithms are implemented in Mathematica software which is made available over the internet.

Stationary policies for lower bounds on the minimum average cost of discrete-time nonlinear control systems

SUMMARY The paper deals with the control problem of discrete-time nonlinear systems. The main contribution of this note is to present conditions that assure the existence of stationary policies that generate lower bounds for the minimal long-run average cost. These lower bounds coincide with the optimal solution when a mild convergence assumption holds. To illustrate the results, the paper presents an application for the simultaneous state-feedback control problem, and the derived strategy is used to design a real-time simultaneous control for two direct current motor devices. The dynamics of these two devices are written in terms of a nonlinear algebraic matrix recurrence, which in turn represents a particular case for our general nonlinear approach. The optimal gain for the corresponding simultaneous state-feedback problem is obtained, and such a gain was implemented in a laboratory testbed to control simultaneously the two direct current motors. Copyright © 2013 John Wiley & Sons, Ltd.