RHONN identifier-control scheme for nonlinear discrete-time systems with unknown time-delays

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Abstract

This work presents a neural identifier-control scheme for uncertain nonlinear discrete-time systems with unknown time-delays. This scheme is based on a neural identifier to get a model of the system and a discrete-time block control technique based on sliding modes to generate the control law. The neural identifier is based on a Recurrent High Order Neural Network (RHONN) trained with an Extended Kalman Filter (EKF) based algorithm. Applicability is shown using real-time test results for linear induction motors. Also, a Lyapunov analysis is added in order to prove the semi-globally uniformly ultimately boundedness (SGUUB) of the proposed neural identifier-control scheme.

Introduction

Presence of time-delay is a frequent problem in engineering applications, time-delays make system analysis and control design more complex; also, time-delays can cause instability and poor performance in the system [1], [2], [3]. A system inherits time-delay from its components, moreover, a system internal process, actuators, and controllers can induce delays in the system [3], [4], [5]. In fact, it is well known that any transportation of materials or information has an associated time-delay. However, only when the delay is long enough to affect the system behavior then such system is treated as a time-delay system (TDS) [3], [4], [5]. It has to be noted that for some systems even small delays can destabilize them [6]. Furthermore, a TDS can be classified as deterministic or stochastic [5].

System identification is a methodology which uses the input and output data of a system to obtain a mathematical model of that system [7], [8], [9], [10]. There are several techniques to achieve system identification. Among these techniques, we find some based on neural networks, fuzzy logic, auxiliary model and hierarchical identification. Moreover, system identification techniques have been implemented as an integral part of adaptive controllers which are typically designed for systems whose dynamics vary with time [10], [11].

The following characteristics make neural networks suitable for system identification [10], [11]:

  • Neural networks are systems with mathematical models. This means that getting a close model to the system to be identified is possible by adjusting the neural network parameters through a training process.

  • There is no need to establish the structure of the model of the system to be identified.

  • Linear and nonlinear systems can be identified in real-time.

  • Online and offline training are possible, which allows online and offline identification.

Recurrent High Order Neural Networks (RHONNs) are dynamic neural networks capable of capture the dynamic response of complex nonlinear systems [11], [12] and also those with time-delays [13] due to their characteristics like flexible model that allows to incorporate a priori information about the system to be identified, approximation capabilities, robustness against noise, online and offline training and their dynamical behavior which is the result of their recurrent connections and high order connections [11], [12].

The most popular algorithms for training recurrent neural networks ones based on backpropagation [12], [14], [15]. However, even if they give great results in several occasions, they present some problems such as slow convergence, high complexity, bifurcations, instability, and limited applicability due to high computational costs [12], [14], [15] and it is not able to discover contingencies spanning long temporal intervals due to the vanishing gradient [16], [17]. On the other hand, training algorithms which are based on the Extended Kalman Filter (EKF) improve learning convergence, reduce the epoch number and the number of required neurons, and are capable of online and offline training. Moreover, EKF training methods for training feedforward and recurrent Neural Networks have proven to be reliable and practical [11], [18].

Neural block control is a methodology which has been used showing good results [11], [19], [20], [21] for controlling nonlinear systems. It uses a neural identifier of the block controllable form of a system, then, based on that model a discrete control law is designed. The designed control law combine discrete-time block-control and sliding modes technique [11].

In this subsection, firstly it is state the main differences and advantages of our neural RHONN identifier-control scheme and other schemes for time-delay systems. Several methodologies have been proposed for identification and controlling of linear and nonlinear time-delay systems, to name a few:

  • In [6], the authors present two continuous-time H control schemes are presented for power systems with multiple time delays and it includes simulation results.

  • In [22], the authors propose an optimal robust control for continuous-time systems. The paper only includes a stability analysis, it does not include simulations or experimental results.

  • In [23], the authors propose a time-delay fuzzy logic system to deal with disturbances, uncertainties, and time-delay of a continuous-time system, this method only presents simulation results, and its sliding mode control law needs the knowledge of the delay.

  • In [24], the authors present a sliding mode controller with neural identification, the identifier is based on a multilayer perceptron trained with backpropagation. Furthermore, delays are not considered and only includes simulation results.

  • In [25], the authors present a discrete-time sliding mode control based on neural networks. This method only works for SISO systems, it does not consider delays, and, it only includes simulation results.

  • In [26], the authors present a continuous-time fuzzy neural-based control for nonlinear time-varying delay systems. An exact mathematical model is not needed, and, it needs two time-average delays to simplify the controller design. This work only presents simulation results.

  • In [27], the authors present a tracking control for fuzzy delta operator systems with time-varying delays. The authors approximate the time-varying delay to get a transformation of the model, then, an H state-feedback controller is designed. Applicability is shown using simulation results.

  • In [28], the authors proposed a tracking control for nonlinear discrete-time systems. A delayed fuzzy observer is used, then, the tracking problem is converted into a stabilization one, then, the gains for the controller and the observer are obtained through a delay nonquadratic Lyapunov function. Only simulation results are presented.

  • In [29], the authors propose a robust controller for trajectory tracking of linear systems with time-delays and disturbances. In order to obtain their controller, the system with delays is transformed into one without delays and an observer is used to compensate the disturbances. Simulation results are presented.

  • In [30], the authors study the delay-dependent H filtering problem for a class of discrete-time Markovian jump linear systems with time-varying delay and incomplete transition descriptions. Applicability is presented using simulation examples.

  • In [31], the authors study the problem of H filtering for a class of two-dimensional Markovian jump linear systems described by the FornasiniMarchesini local state-space model with state-delay and deficient mode information in the Markov Chain. Applicability is presented using simulation examples.

  • In [32], the authors study the problem of delay-dependent H dynamic output feedback control for a class of discrete-time Markovian jump linear systems with defective mode information. Simulation examples are presented to show applicability.

  • In [33], the authors study the problem of quantized H filtering for a class of continuous-time Markovian jump linear systems with deficient mode information. Simulation examples are included.

  • In [34], the authors study the problem of H model approximation for a class of two-dimensional (2-D) discrete-time Markovian jump linear systems with state-delays. Applicability of this work is presented using simulation examples.

Our methodology offers some interesting advantages compared with the above-mentioned works:

  • It works for discrete-time systems. The current trend towards digital rather than analog control of dynamic systems is mainly due to advantages offered by digital signals [35], [36], [37], such as:

    • Digital systems can tolerate considerable variation in signals values.

    • Digital implementation permits the use of a wide variety of hardware options, including computers, microprocessors, digital signal processors, and field programmable gate arrays.

    • Lower computational cost.

  • It does not require previous knowledge of the system model. The mathematical model of the system is considered unknown.

  • The knowledge about the delay, its estimation or its bounds is not necessary.

  • Measurements, estimations or bounds of disturbances are not necessary either.

  • Once the training process begins, the mathematical structure of the recurrent high order identifier gives a mathematical model that is close to the actual system model. This identifier model can be used to design a control law.

  • It is presented the stability analysis of our proposal, in terms of Lyapunov methodology.

  • There are presented experimental results from real-time applications.

In order to clarify the differences of this approach with respect to our previous works about RHONN identifiers, next we are going to compare it with those works:

  • In [13], it is presented the RHONN identifier and its stability analysis, also, real-time results are shown using a linear induction motor and a tracked robot. This work does not include a control scheme.

  • In [38], time-delays are not considered, only simulation results are included and there is not stability analysis.

  • In [39], it is presented only the RHONN identifier for TDS. It shows simulation and real-time results, but, the stability analysis is not included.

In this work, we extend those previous works by adding a control methodology and presenting the stability analysis of the RHONN identifier – control scheme for TDS and real-time results.

The following points present the main contribution of this work:

  • An RHONN-identifier control scheme for TDS which allows the presence of disturbances, noise, and delays and that does not require the knowledge of the plant to be controlled.

  • A real-time implementation of the proposed scheme, presenting, different cases with time-delays.

  • A Lyapunov analysis of the RHONN-identifier control scheme for TDS to prove the semi-globally uniformly boundedness.

The organization of this paper is as follows: Section 2 is dedicated to the RHONN and Kalman filter training and, Section 3, in Section 3.1 the kind of TDS that we work with is introduced, then, in Section 3.2 the neural identification process using RHONN is described and in Section 3.3 the neural block controller design is explained. Results are explained in Section 4, conclusions are included in Section 5 and the Lyapunov analysis to prove the semi-globally uniformly ultimately boundedness (SGUUB) of the proposed neural identifier-control scheme is presented in the Appendix A.

Section snippets

Recurrent high order neural network

A neural network is an information processing system made up interconnecting simple processing units called neurons, this system imitates the capacity of the human brain to solve complex problems. In this systems, knowledge is acquired through a learning process and synaptic weights (inter-neuron connection strength) store the knowledge [15], [40]. Neural networks can be implemented using electronic components or in software [15].

The ability of neural networks to approximate function make them

Neural block control

This section introduces the elements of neural block control scheme shown in Fig. 1.

Results

The following results have been obtained using the presented RHONN identifier – control scheme. The RHONN identifier (12) is trained online using the EKF algorithm (6). The control law was generated using the block control technique using the RHONN identifier model.

To prove the performance of the proposed RHONN identifier-control scheme, the linear induction motor (LIM) prototype shown in Fig. 2 is used. This prototype is based on a LIM LabVolt®1

Conclusions

The proposed neural identifier-control scheme shows a good performance in the real-time tests, with a small sample time equal to 0.0003s and as it can be seen in the graph the tracking of the references are achieved. Due to the characteristics of the prototype, most of the signals are noisy, even so, the tracking objectives are both within a bounded error.

The work presents three cases with different references and different time-delay scenarios, and it can be seen that the identification errors

Acknowledgments

The authors thank the support of CONACYT Mexico, through Projects CB256769 and CB258068 (“Project supported by Fondo Sectorial de Investigación para la Educación).

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