Unknown Nonlinear Discrete-time Systems Research Articles

Nonlinear discrete-time controller with unknown systems identification based on fuzzy rules emulated network

This article introduces an adaptive controller for a class of unknown nonlinear discrete-time systems based on multi-input fuzzy rules emulated network (MIFREN). By the estimation of any nonlinear systems from MIFREN, this network is assigned to identify the unknown system under control. The proposed control law is introduced by the result of nonlinear system identification based on MIFREM and the defined sliding condition. Without the need of any off-line learning phase, all control parameters including the learning rate for MIFREN are selected to guarantee the bonded signals such as the model error, tuned weight vector, the tracking error and the sliding signal via the defined Lyapunov functions and proposed theorems. The performance of the proposed control algorithm is demonstrated and the main theorem is validated by computer simulation results.

Optimal control of unknown affine nonlinear discrete-time systems using offline-trained neural networks with proof of convergence

The optimal control of linear systems accompanied by quadratic cost functions can be achieved by solving the well-known Riccati equation. However, the optimal control of nonlinear discrete-time systems is a much more challenging task that often requires solving the nonlinear Hamilton-Jacobi-Bellman (HJB) equation. In the recent literature, discrete-time approximate dynamic programming (ADP) techniques have been widely used to determine the optimal or near optimal control policies for affine nonlinear discrete-time systems. However, an inherent assumption of ADP requires the value of the controlled system one step ahead and at least partial knowledge of the system dynamics to be known. In this work, the need of the partial knowledge of the nonlinear system dynamics is relaxed in the development of a novel approach to ADP using a two part process: online system identification and offline optimal control training. First, in the system identification process, a neural network (NN) is tuned online using novel tuning laws to learn the complete plant dynamics so that a local asymptotic stability of the identification error can be shown. Then, using only the learned NN system model, offline ADP is attempted resulting in a novel optimal control law. The proposed scheme does not require explicit knowledge of the system dynamics as only the learned NN model is needed. The proof of convergence is demonstrated. Simulation results verify theoretical conjecture.

Feedback-Linearization-Based Neural Adaptive Control for Unknown Nonaffine Nonlinear Discrete-Time Systems

A new feedback-linearization-based neural network (NN) adaptive control is proposed for unknown nonaffine nonlinear discrete-time systems. An equivalent model in affine-like form is first derived for the original nonaffine discrete-time systems as feedback linearization methods cannot be implemented for such systems. Then, feedback linearization adaptive control is implemented based on the affine-like equivalent model identified with neural networks. Pretraining is not required and the weights of the neural networks used in adaptive control are directly updated online based on the input-output measurement. The dead-zone technique is used to remove the requirement of persistence excitation during the adaptation. With the proposed neural network adaptive control, stability and performance of the closed-loop system are rigorously established. Illustrated examples are provided to validate the theoretical findings.

A new discrete-time sliding-mode control with time-varying gain and neural identification

In this paper, we present a new sliding mode controller for a class of unknown non-linear discrete-time systems. We make the following two modifications. (1) The neural identifier which is used to estimate the unknown non-linear system uses the projection and the dead-zone approaches to assure non-singularity in the controller and stability of identification error. (2) We propose a new sliding mode controller with time-varying gain to reduce chattering. A necessary condition is given to make the switching function decrease exponentially. We prove that the closed-loop system with the sliding mode controller and the neural identifier is stable.

Nonparametric control algorithms for nonlinear fading memory systems

We develop an algorithm to control an unknown nonlinear fading memory discrete-time system. Our approach is based on nonparametric regression techniques rather than traditional feedback control. We discuss a procedure that, given a desired periodic output and a tolerance, produces an acceptable output from any system in a wide class. The algorithm uses as data past inputs (which are selected by the algorithm) and corresponding (possibly noisy) output observations and needs no extra parametric knowledge or restrictions on the system. We also present an algorithm that produces an output that converges to the desired output when stricter conditions are imposed on the system. Our approach, in its current form, however, cannot be used to control open-loop unstable systems.

Stability of a Trajectory‐Based Control Law for an Unknown Nonlinear Non‐Minimum Phase System

ABSTRACTWe investigate the stability of an unknown nonlinear discrete‐time non‐minimum phase system under a trajectory‐based control law. The system can be regarded as a first‐order approximation to a continuous‐time system. Hence, one of the parameters in the discrete‐time system equation can be regarded as the “sampling interval”. We show that, subject to certain conditions, as long as the sampling interval is neither too short nor too long, the closed‐loop system is stable in a certain sense.

Synthesis of the sliding-mode neural network controller for unknown nonlinear discrete-time systems

This paper develops a sliding-mode neural network controller for a class of unknown nonlinear discrete-time systems using a recurrent neural network (RNN). The control scheme is based on a linearized expression of the nonlinear system using a linear neural network (LNN). The control law is proposed according to the discrete L yapunov theory. With a modified real-time recurrent learning algorithm, the RNN as an estimator is used to estimate the unknown part in the control law in on-line fashion. The stability of the control system is guaranteed owing to the on-line learning ability of the RNN algorithm. The proposed control scheme is applied to numerical problems and simulation results that it is very effective.

Modelling flexible robot dynamics using discrete-time dynamic recurrent neural networks

The identification of a general class of multi-input multi-output (MIMO) discrete-time nonlinear systems expressed in state space form is studied in this paper using dynamic recurrent neural networks (DRNNs). A discrete-time DRNN, which is represented by a set of parameterized nonlinear difference equations and has a universal approximation capability, is proposed for modelling unknown discrete-time nonlinear systems. A dynamic back propagation learning algorithm which carries out the modelling task using the input-output data is discussed. The proposed scheme is used for modelling the nonlinear dynamics of a single-link robot with a flexible joint.

Linearization and state estimation of unknown discrete-time nonlinear dynamic systems using recurrent neurofuzzy networks

Model-based methods for the state estimation and control of linear systems have been well developed and widely applied. In practice, the underlying systems are often unknown and nonlinear. Therefore, data based model identification and associated linearization techniques are very important. Local linearization and feedback linearization have drawn considerable attention in recent years. In this paper, linearization techniques using neural networks are reviewed, together with theoretical difficulties associated with the application of feedback linearization. A recurrent neurofuzzy network with an analysis of variance (ANOVA) decomposition structure and its learning algorithm are proposed for linearizing unknown discrete-time nonlinear dynamic systems. It can be viewed as a method for approximate feedback linearization, as such it enlarges the class of nonlinear systems that can be feedback linearized using neural networks. Applications of this new method to state estimation are investigated with realistic simulation examples, which shows that the new method has useful practical properties such as model parametric parsimony and learning convergence, and is effective in dealing with complex unknown nonlinear systems.

Nonparametric output prediction for nonlinear fading memory systems

The authors construct a class of elementary nonparametric output predictors of an unknown discrete-time nonlinear fading memory system. Their algorithms predict asymptotically well for every bounded input sequence, every disturbance sequence in certain classes, and every linear or nonlinear system that is continuous and asymptotically time-invariant, causal, and with fading memory. The predictor is based on k/sub n/-nearest neighbor estimators from nonparametric statistics. It uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system, the bounds on the input, or the properties of noise. Under additional smoothness conditions the authors provide rates of convergence for the time-average errors of their scheme. Finally, they apply their results to the special case of stable linear time-invariant (LTI) systems.

Output tracking of a class of unknown nonlinear discrete-time systems using neural networks

In this paper, an adaptive controller based on neural networks is derived for controlling a class of unknown nonlinear discrete-time systems. A two-layered neural network is used to characterize the input-output behavior of the unknown systems. The Widrow-Hoff delta rule is the learning algorithm used to minimize the error signal between the actual response and that of the neural networks. The control signal is generated on-line using another two-layered neural network, so that the plant results in zero asymptotic tracking errors with respect to a desired reference signal. It is proved that the control objective is achieved by the closed-loop system and that the system remains closed-loop stability. The effectiveness of the proposed control scheme is also demonstrated by a simulation example.

Output tracking of a class of unknown nonlinear discrete-time systems using neural networks

Output tracking of a class of unknown nonlinear discrete-time systems using neural networks

Dynamic Recurrent Neural Networks for Control of Unknown Nonlinear Systems

A scheme of dynamic recurrent neural networks (DRNNs) is discussed in this paper, which provides the potential for the learning and control of a general class of unknown discrete-time nonlinear systems which are treated as “black boxes” with multi-inputs and multi-outputs (MIMO). A model of the DRNNs is described by a set of nonlinear difference equations, and a suitable analysis for the input-output dynamics of the model is performed to obtain the inverse dynamics. The ability of a DRNN structure to model arbitrary dynamic nonlinear systems is incorporated to approximate the unknown nonlinear input-output relationship using a dynamic back propagation (DBP) learning algorithm. An equivalent control concept is introduced to develop a model based learning control architecture with simultaneous on-line identification and control for unknown nonlinear plants. The potentials of the proposed methods are demonstrated by simulation results.

Model Matching Control of Unknown Nonlinear Systems Using Recurrent Neural Networks

A scheme of multilayered recurrent neural networks (MRNNs) is discussed in this paper, which provides the potential for the learning and control of a general class of unknown discrete-time nonlinear systems which are treated as "black boxes" with multi-inputs and multi-outputs (MIMO). A model of the MRNN is described by a set of nonlinear difference equations, and a suitable analysis for the input-output dynamics of the model is performed to obtain the inverse dynamics. The ability of a MRNN structure to model arbitrary dynamic nonlinear systems is incorporated to approximate the unknown nonlinear input-output relationship using a dynamic back propagation (DBP) learning algorithm. An equivalent control concept is introduced to develop a model based learning control architecture with simultaneous on-line identification and control for the unknown nonlinear plant. The potentials of the proposed methods are demonstrated by simulation results.