Fault tolerant control based on stochastic distributions via MLP neural networks

doi:10.1016/j.neucom.2006.10.030

Neurocomputing

Volume 70, Issues 4–6, January 2007, Pages 867-874

https://doi.org/10.1016/j.neucom.2006.10.030 Get rights and content

Abstract

An optimal fault tolerant control (FTC) scheme using output probability density functions (PDFs) is studied for the general stochastic continuous time systems. Being different from the classical FTC problems, the measured information is the stochastic distribution of the system output rather than its value. The control objective is to use the output PDFs to design control schemes that can compensate the fault and attenuate the disturbance. A multi-layer perceptron (MLP) neural network is applied to approximate the output PDFs, with which nonlinear principal component analysis (NLPCA) can be used to reduce the model order. For the established continuous-time weighting system with disturbances and uncertainties which is used to link the input and the weights, an LMI-based feasible FTC method is presented to assure that the fault can be well measured and compensated, where the H_∞ performance index for the uncertain error systems is optimized.

Introduction

In some practical industrial processes (including network-based monitor processes, chemical engineering processes and paper making processes), the measured information may be the stochastic distribution of the system output rather than its deterministic value. This problem emerges along with the development of the instrumental techniques, as well as the image and data processing techniques. How to use the stochastic information for the quality control and fault detection forms a new challenge in control area. For instance, in [5], [7], [8], the quality control based on the output probability distribution functions has been studied via PI or PID control schemes. In [9], iterative learning methods have been introduced to improve the modeling processing.

On the other hand, the fault detection and diagnosis (FDD) problem using the measured statistic information has shown its special significance in complex processing and networked remote monitor systems. Apart from the data processing technique, filters have been widely used to generate residual signals to detect and estimate the fault in dynamical FDD problems [1], [3]. However, in most existing results only Gaussian variables were concerned and their mean and/or variance of estimation errors were optimized [14], [15], [16], [17], [19], while non-Gaussian variables with the asymmetric distributions exist in many cases since nonlinear mappings can lead to non-Gaussian outputs. For example, in the molecular weight distribution (MVD) problem, the shape of output probability density function (PDF) may be asymmetric and have two or three peaks [18]. For both of the above cases, new filtering and control methods are required based on the measured stochastic property of system outputs for non-Gaussian systems, which can be regarded as generalizations of the classical stochastic control and filtering theory.

One existing obstacle is to formulate the probability distributions of the stochastic output, since usually it is an infinite-dimensional problem to control (estimate) the system dynamics using PDFs, which has been known as the stochastic distribution control (estimation) problem. B-spline expansions have been used effectively to approximate the conditional output PDFs [2], [11], [12], [13], [18]. For the FDD problems, B-spline neural networks have been applied to the PDF modeling processing and a new framework has been established for the actuator faults [6], [20]. However, this procedure led to a challenge on selection of the basis functions (see also [9]). Recently, the multi-layer perception (MLP) neural network models have been applied to the shape control and fault detection problems for the output PDFs [10], [19]. However, fault detection is only the first step in reliable control procedures.

Fault tolerant control (FTC) is to use the input and the measured information to design the controllers such that the systems are able to work normally even if the fault occurs. One effective approach is to design the filter to estimate the fault and then to construct the controller to compensate or reject the estimation of the fault [16], [17]. Comparing with the fault detection and diagnosis problems, the more important problem is to use the measured output PDFs to provide fault tolerant control strategies for the general (non-Gaussian) systems.

In this paper, MLP models are applied to investigate fault tolerant control (FTC) problems. It is shown that the concerned problem can be transformed into a nonlinear FTC problem in the deterministic context. Model errors and parametric uncertainty can be merged into the disturbance input of the weighting system. It is noted that up to now, less available results have been given even for the deterministic model and new filtering algorithms are required. To improve the performance for the fault diagnosis and disturbance attenuation, the H_∞ technique will be introduced for the involved nonlinear plants in the presence of both faults and system disturbances [4]. The control objective is to use the output PDFs to design control schemes that can both compensate the fault and attenuate the disturbance. Linear matrix inequalities will be used to formulate the solutions of the considered FTC problems.

This work is organized as follows. In Section 2, the MLP neural network model for the square root of PDF and the relevant continuous-time nonlinear weighting dynamic model are given. In Section 3, the FTC filtering is designed to track the fault and compensate it. In Section 4, a simulation example is given to show the efficient of our result. The conclusion is given in Section 5.

Section snippets

MLP neural network model for the square root of PDFs

Consider a continuous-time dynamic stochastic system, where u(t)∈R^m is the input, y(t)∈[a, b] is the output of the concerned. F is supposed to be an actuator fault to be diagnosed and compensated. The output PDF is denoted by γ(z, u(t), F), which is supposed to be continuous or piecewise continuous. The following result can be seen from [10], [19] and references therein.

Lemma 1

$\sqrt{γ (z, u (t), F)}$ can be approximated by the MLP neural network model as $\sqrt{γ (z, u, F)} = P (W (u, F)) (L (Θ (u, F)) z + b (u, F)) + d (u, F) + ω_{0},$ where $P (x) = {\begin{matrix} x, & x ⩾ 0 \end{matrix}$

Fault tolerant control via diagnostic filter

In order to estimate the fault based on the changes of output distributions, we construct the following vector as the measurement output and the residual signal $ε (t) = \int_{a}^{b} σ (z) (\sqrt{γ (z, u, F)} - \sqrt{\hat{γ} (z)}) d z,$ where $\sqrt{\hat{γ} (z)} = B_{0} V_{0} + B (z) E \hat{x} (t) + h (E \hat{x} (t)) b_{n} (z)$ and σ(z)⩾0 is a weight function to tune the difference described as (8).

To guarantee that the system stability still retains when the faults occur, apart from a stabilization law (which can be designed via the approach in [6], [10]) and a fault detection strategy

Example 1

In many practical processes such as the particle distribution control problem, the shapes of measured output PDF have 2 or 3 peaks (see [7], [18]). Suppose these output PDFs can be approximated using square root B-spline models described by (1) with n=3, z∈[0, 1.5] and $b_{i} (z) = {\begin{matrix} | \sin 2 π z |, & z \in [\begin{matrix} 0.5 (i - 1); & 0.5 i \end{matrix}] \\ 0, & others \end{matrix} (i = 1, 2, 3) .$ It is assumed the identified weighting system is formulated by (6) with the following coefficient matrices $A = [\begin{matrix} - 2.2 & 2 \\ 0 & - 2 \end{matrix}], D = [\begin{matrix} - 0.1 & 0 \\ 0 & - 0.5 \end{matrix}], H = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}] G = [\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}], E = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] .$ The bound of

Conclusions

Motivated from some industrial processes and along with the development of image processing and networked control techniques, in this paper a new FTC scheme is provided for stochastic dynamic systems, where non-Gaussian stochastic variables are concerned and the on-line measurement output is the distribution of system output rather than the output itself. By using of multi-layer perception (MLP) neural networks, the output PDFs can be formulated in terms of the weightings, and consequently the

Acknowledgement

This work is supported partially by the NSF of China (No. 60474050), the program for NCET of China, the China Post-Doctoral Foundation (20040350655) and the Jiangsu Post-Doctoral Foundation of China. The authors also would like to thank Professor H. Wang and Prof. D.S. Huang for their helpful discussions.

Yumin Zhang was born in 1971. He received the B.S. and M.S. degrees in mathematics from Shandong Normal University in 1993 and Huazhong University of Science and Technology (HUST) in 1996, respectively, and Ph.D. degree in system engineering from HUST, PR China in 2003. From 2004 to 2006, he worked as a research fellow in the Research Institute of Automation, Southeast University, P.R. China. Now he is a Professor in College of Automation Engineering, Qingdao University, P.R. China. His

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Lei Guo was born in 1966. He received the Ph.D. degree in control engineering from Southeast University (SEU), PR China, in 1997. From 1999 to 2004, he has worked at Hong Kong University, IRCCyN (France), Glasgow University, Loughborough University and UMIST, UK. Now he is a Professor in School of Instrumantal Science, Beihang University. He also holds a visiting professor position in the University of Manchester, UK and an invitation fellowship in Okayama University, Japan. His research interests include robust control, stochastic systems, fault detection, filter design, and nonlinear control with their applications.

Haisheng Yu was born in 1963. He received the B.S. and M.S. degrees in control engineering from Harbin University of Civil Engineering and Architecture in 1985 and Tsinghua University in 1988, respectively, PR China. Now he is a Professor in College of Automation Engineering, Qingdao University. His research interests include computer control and intelligent systems, applied nonlinear control, motion control systems.

Keyou Zhao was born in 1945. He had held a lecturer and an associate professor positions respectively in 1983 and 1986 respectively, at Department of Mathematics, Shandong University. Since 1987 he has been with Qingdao University, and has held a professor position from 1993 till now. His current research interests include robust and nonlinear control, constrained control, induction motor control.

¹: Present address: School of Instrumantal Science, Beihang University, Beijing 100083.

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Fault tolerant control based on stochastic distributions via MLP neural networks

Abstract

Introduction

Section snippets

MLP neural network model for the square root of PDFs

Fault tolerant control via diagnostic filter

Example 1

Conclusions

Acknowledgement

Ann. Rev. Control

J. Process Control

Automatica

Neurofuzzy Adaptive Modeling and Control

H∞ Output feedback control for delay systems with nonlinear and parametric uncertainties

IEE Proc.: Control Theory Appl.

Applying constrained nonlinear generalized PI strategy to PDF tracking control through square root B-spline models

Int. J. Control

Fault detection and diagnosis for general stochastic systems using B-spline expansions and non-linear observations

IEEE Trans. Circuits Systems—I: Regular Papers

PID controller design for output PDFs of stochastic systems using linear matrix inequalities

IEEE Trans. Systems Man Cybernet B

H_∞ Output feedback control for delay systems with nonlinear and parametric uncertainties