Auxiliary signal design for robust active fault detection of linear discrete-time systems

doi:10.1016/j.automatica.2011.06.009

Automatica

Volume 47, Issue 9, September 2011, Pages 1887-1895

https://doi.org/10.1016/j.automatica.2011.06.009 Get rights and content

Abstract

In this paper, algorithms are proposed to design auxiliary signals for active fault detection based on a multi-model formulation of discrete-time systems. Two different scenarios are considered for this problem; the first one assumes there is no a priori information on initial conditions and no exogenous input signal, while the second allows for having a priori information and the possibility of having a known input in addition to the test signal. Approaches are proposed for solving these two types of problems which are capable of solving the problems efficiently. This is achieved by using a recursive approach based on the use of special Riccati equations. These algorithms can be used for systems of higher dimension and on longer time horizons than the existing methods.

Introduction

There exist two basic approaches for solving fault detection problems, passive approaches and active methods. Traditionally most fault detection approaches were passive in that system performance was monitored and decisions made. Recently, there has been more interest in active approaches. The term “active” here is not the same as “active fault tolerant” where the system is taking action to compensate for a fault (Mahmoud, Jiang, & Zhang, 2003). Here, active refers to acting upon the system on a periodic basis or at critical times using a test signal, called the auxiliary signal, in order to exhibit abnormal behaviors which would otherwise remain undetected during normal operation (Campbell & Nikoukhah, 2004).

It is usual in parameter identification approaches to inject a signal into the system, in order to excite the system. However, most of the fault detection methods are passive. In recent years, several active fault detection methods have been proposed (Niemann, 2006a, Niemann, 2006b, Poulsen and Niemann, 2009, Simandl and Puncochar, 2009). In Kerestecioglu (1993) and Zhang (1989) the use of auxiliary signals in the context of fault detection of stochastic systems is introduced. Another stochastic approach is proposed by Blackmore, Rajamanoharan, and Williams (2008) in a multi-model setup. That method uses a Bayesian approach and assumes a Bhattacharyya bound, an upper bound on the probability of model selection error, as the optimization cost function for two model case. For more than two models, the paper proposes a new upper bound. It does not assume a bound on uncertainties, but probability functions of noise are taken into account. Also, the model of the system can only include additive uncertainties.

A different formulation of active fault detection is the robust approach, where the uncertainties are assumed bounded and guaranteed detection is considered. The auxiliary signal design problem for robust multi-model fault detection studied in Nikoukhah, Campbell, Horton, and Delebecque (2002), and later in Campbell and Nikoukhah (2004) assumes that there are two candidate models. The objective is to find an auxiliary signal which minimizes a norm that can guarantee on-line identification of the correct model. In practical applications the Euclidean norm is often used which corresponds to the energy of the signal. The approach of Campbell and Nikoukhah (2004) has been extended to discrete-time problems (Nikoukhah, Campbell, & Delebecque, 2000), continuous-time systems with a priori information (Nikoukhah & Campbell, 2006), incipient faults (Nikoukhah & Campbell, 2008), sampled systems (Choe et al., 2009, Nikoukhah and Campbell, 2005), and nonlinear systems (Andjelkovic, Sweetingham, & Campbell, 2008).

Most previous papers have dealt with auxiliary signals for use in fault detection of continuous-time systems. However, many physical problems have a discrete-time nature and the measurements are taken at discrete instances. In this paper an algorithm is presented for computing optimal discrete-time test signals. Certain min–max optimization problems have to be solved to obtain the solution. This problem is considered in Campbell and Nikoukhah (2004) and Nikoukhah et al. (2000) and a preliminary method suggested. The problem with that method is that it needs to convert the dynamic optimization problem to a static one. If the dimensions of the system are large or the time horizon is long, the algorithm has to deal with very large matrices and the accuracy of the calculations are not reliable in some cases. This paper proposes an approach which is more efficient in these cases by solving the min–max optimization problem using Riccati equations which are capable of finding the solution recursively. This new algorithm makes it possible to carry out computations for large problems using much less memory, while the existing methods consume large amounts of memory. The use of recursions is not new, especially in robust filtering and control, but their application to this discrete-time fault detection problem is. This is the first of the two problems considered in this paper.

A priori information often exists about initial conditions and models may be subject to a known input. The second contribution of this paper is to develop a methodology for finding the optimal test signal when a priori information exists about the initial states or some known exogenous signal enters the system in addition to the auxiliary signal. This problem is solved for continuous-time systems (Nikoukhah & Campbell, 2005), but never considered in the discrete-time case. The initial condition is not known. Rather we assume that the initial condition lies in a known region. This often occurs in applications. This problem is especially important, as it allows us to consider some new types of faults (Nikoukhah & Campbell, 2005). For example, bias faults can be considered as a known constant input of the faulty model, and a jump in the state of the system can now be modeled as a non-zero initial condition of the fault model. Using this additional information, we get a smaller auxiliary signal.

The traditional problem and the more general problem of having a priori information on initial conditions and known inputs to the system, look very similar and we express them both in one problem formulation. The first problem may appear to be a special case of the second one. However, we will show that one cannot just use the more general case algorithm to solve the other. Two routines are proposed to solve the two problems. In giving a solution to these problems, we extend some mathematical results which consider a general discrete-time dynamic optimization problem. In addition to the finite time horizon problem, a solution is given to the infinite time horizon case, which is helpful in approximating the solution of the problem on long time intervals. In some applications, an approximation of the solution is enough for a long time interval and we do not really need the exact solution. Such an approximate solution reduces the computational burdens compared to the main method which tries to solve the problem using Riccati equations. Note that a long interval for theoretical and computational purposes may be a short interval for practical purposes.

Section snippets

Mathematical background

In order to solve the active fault detection problems in this paper, we need the solution of the following dynamic optimization problem.

Consider the cost functional $C = {(x (0) - {\hat{x}}_{0})}^{T} P_{0}^{- 1} (x (0) - {\hat{x}}_{0}) + \sum_{t = 0}^{j - 1} ω^{T} (t) Γ ω (t),$ and the optimization problem $J (j, a, b) = min_{x (0), ω} C$ subject to $x (t + 1) = A x (t) + M ω (t) + a (t),$ $b (t) = C x (t) + N ω (t),$ for integer values of $t \in [0, τ - 1]$ . Matrices $A$ , $C$ , $M$ , $N$ may depend on $t$ . We suppose that $N$ has full row rank, $P_{0} > 0$ and $Γ$ is symmetric and invertible but not necessarily sign definite.

Problem formulation

We now describe the fault detection problem. Consider the linear discrete-time system with additive noise and model uncertainties, for $i = 0, 1$ and $t \in [0, τ]$ , $x_{i} (t + 1) = A_{i} x_{i} (t) + B_{i} v (t) + M_{i, x} ν_{i} (t) + {\bar{M}}_{i} u_{i} (t),$ $y (t) = C_{i, y} x_{i} (t) + D_{i, y} v (t) + N_{i, y} ν_{i} (t) + {\bar{N}}_{i, y} u_{i} (t),$ $z_{i} (t) = C_{i, z} x_{i} (t) + D_{i, z} v (t) + N_{i, z} ν_{i} (t) + {\bar{N}}_{i, z} u_{i} (t) .$ Here $x_{i} (t) \in ℜ^{n_{x}}$ is the state vector, $v (t) \in ℜ^{n_{v}}$ is the auxiliary input vector to be designed, $y (t) \in ℜ^{n_{y}}$ is the output of the system, $ν_{i} (t) \in ℜ^{n_{ν_{i}}}$ represents the additive uncertainty, and $u_{i} (t) \in ℜ^{n_{u_{i}}}$ is the known input,

A solution for the active fault detection problem

We now present a solution for the active fault detection problem described in Section 3.

Numerical examples and discussion

In this section, we give two examples. The first example is without any known input other than the test signal $v$ . This example will show the computational advantage of the new approach. In Example 5.2 we inject a known input into the system in addition to the test signal $v$ . This example will illustrate the difference an additional known input can make.

Conclusions

A complete solution to the discrete-time robust multiple-model active fault detection problem is given based on a completely recursive approach. The new method is capable of dealing with two different scenarios. One scenario is when there is neither a priori information on initial conditions nor an exogenous input signal in addition to the auxiliary test signal. The second scenario is when either or both are present. The first scenario has already been considered in Nikoukhah et al. (2000).

Alireza Esna Ashari received his B.S. and M.S. in Control Engineering from Electrical Engineering Faculty of University of Tehran, Tehran, Iran. He received his Ph.D. from University of Paris-Est, working at INRIA (the French national institute for research in computer science and control), Paris-Rocquencourt center, France.

He is currently a postdoctoral fellow at INRIA, Rennes center. His current research interests include fault detection and isolation, optimization, optimal control and robust

References (19)

R. Nikoukhah et al.
Auxiliary signal design for active failure detection in uncertain linear systems with a priori information
Automatica
(2006)
R. Nikoukhah et al.
On the detection of small parameter variations in linear uncertain systems
European Journal of Control
(2008)
R. Nikoukhah et al.
A multi-model approach to failure detection in uncertain sampled-data systems
European Journal of Control
(2005)
M. Simandl et al.
Active fault detection and control: unified formulation and optimal design
Automatica
(2009)
Andjelkovic, I., Sweetingham, K., & Campbell, S.L. (2008). Active fault detection in nonlinear systems using auxiliary...
L. Blackmore et al.
Active estimation for jump markov linear systems
IEEE Transactions on Automatic Control
(2008)
S.L. Campbell et al.
D. Choe et al.
Optimal piecewise-constant signal design for active fault detection
International Journal of Control
(2009)
Esna Ashari, A., Nikoukhah, R., & Campbell, S.L. (2009). Asymptotic behavior and solution approximation of active...

There are more references available in the full text version of this article.

Cited by (35)

Input design for active fault diagnosis
2019, Annual Reviews in Control
Citation Excerpt :
The more general case of simultaneous incipient faults is studied by Fair and Campbell (2009b) for two faults and by Fair and Campbell (2009a) for more than two faults; see also Nikoukhah, Campbell, and Drake (2010). Nikoukhah and Campbell (2006a) and Ashari, Nikoukhah, and Campbell (2011) consider the problem of incorporating information about the initial system state through bounding the total uncertainty from the initial state and the disturbances and noise. This approach enables accounting for known input signals to the system (for example control signals) and handling a broader class of faults, including those modeled by a bias.
Reliably diagnosing faults and malfunctions has become increasingly challenging in modern technical systems because of their growing complexity as well as increasingly stringent requirements on safety, availability, and high-performance operation. Traditional methods for fault detection and diagnosis rely on nominal input–output data, which can contain insufficient information to support reliable conclusions. Recent years have witnessed a growing interest in active fault diagnosis, which addresses this issue by injecting input signals specifically designed to reveal the fault status of the system. This paper provides an overview of state-of-the-art methods for input design for active fault diagnosis and discusses the primary considerations in the formulation and solution of the input-design problem. We also discuss the primary challenges and suggest avenues for future research in this rapidly evolving field.
Event-based H<inf>∞</inf> fault estimation for networked time-varying systems with randomly occurring nonlinearities and (x, v)-dependent noises
2018, Neurocomputing
Citation Excerpt :
Over the past few years, the fault diagnosis and fault-tolerant control problems have attracted much attention from a variety of communities owing to the increasing requirements for safety and reliability in engineering practices, such as chemical, aerospace and automotive industries [1–6].
In this paper, the problem of finite-horizon H_∞ fault estimation is investigated for a class of networked time-varying stochastic systems with randomly occurring nonlinearities and state- and disturbance-dependent noises (also called (x, v)-dependent noises). An event-triggered scheme is proposed to reduce data transmission burden where the current measurement is transmitted only when the certain condition is satisfied. The aim of the addressed problem is to design a fault estimator, in the presence of randomly occurring nonlinearities and (x, v)-dependent noises, such that faults can be estimated through measurement outputs. By employing the stochastic analysis method, the sufficient conditions are derived to guarantee that the error dynamics of estimations satisfies a prescribed H_∞ performance constraint. Moreover, the parameters of fault estimator can be calculated via the recursive linear matrix inequality (RLMI) approach. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.
A Survey of Active Fault Diagnosis Methods
2018, undefined
During the last three decades, there has been a growing interest in active fault diagnosis (AFD) and several important results have been reported in the literature. This survey aims to provide a compact up-to-date overview of main research directions and to propose a basic classification of AFD methods. The contributions are presented almost in the chronological order as they appeared in the literature and they are divided into two groups that differ in the way the uncertainties are modeled. The survey is concluded with the description of main trends, some application results, and future challenges in the area of AFD.
Stochastic Nonlinear Model Predictive Control with Active Model Discrimination: A Closed-Loop Fault Diagnosis Application
2017, undefined
This paper addresses the problem of model predictive control with multiple models for nonlinear systems subject to stochastic disturbances. The multiple models can represent various operating conditions such as system faults or failures, or arise from model structure uncertainty. The paper presents a stochastic nonlinear model predictive control (SNMPC) approach with endogenous learning for active discrimination between the competing models based on closed-loop system observations. The system learning is endogenized through explicit inclusion of a model discrimination measure into the stochastic optimal control problem, which facilitates probabilistic discrimination between the predictions of multiple models. The control approach uses a Bayesian estimation algorithm for recursive estimation of the probabilities that represent the degree to which each model predicts the online system observations. The performance of the proposed SNMPC approach with active model discrimination is demonstrated for closed-loop fault diagnosis.
Robust Fault Diagnosis by Optimal Input Design for Self-sensing Systems
2017, IFAC-PapersOnLine
This paper presents a methodology for model based robust fault diagnosis and a methodology for input design to obtain optimal diagnosis of faults. The proposed algorithm is suitable for real time implementation. Issues of robustness are addressed for the input design and fault diagnosis methodologies. The proposed technique allows robust fault diagnosis under suitable conditions on the system uncertainty. The designed input and fault diagnosis techniques are illustrated by numerical simulation.
Active fault diagnosis: A multi-parametric approach
2017, Automatica
Citation Excerpt :
A significant reduction in the time required for diagnosis may be obtained by suitably modifying the inputs of the process. This type of approaches, named “active FDI”, has received increasing attention in recent years (Ashari, Nikoukhah, & Campbell, 2012a, 2011; Cheong & Manchester, 2015; Nikoukhah, 1998; Poulsen & Niemann, 2008; Punčochář & Šimandl, 2014; Punčochář, Širokỳ, & Šimandl, 2015; Scott, Findeisen, Braatz, & Raimondo, 2014; Streif, Petzke, Mesbah, Findeisen, & Braatz, 2014; Tabatabaeipour, 2015; Xu, Olaru, Puig, Ocampo-Martínez, & Niculescu, 2014; Yang, Hamelin, & Sauter, 2014). The focus of this paper is on deterministic active FDI.
This paper considers the design of an input signal for minimizing the time and energy required to detect and isolate faults in the outputs of a system. Faults are represented by discrete switches between affine models with bounded disturbances and bounded measurement errors. Within this framework, previous work has demonstrated that a minimally harmful input guaranteeing fault diagnosis can be obtained by solving a Mixed Integer Quadratic Program (MIQP). A closed-loop approach allows to reduce the length and/or norm of this input by solving an MIQP at each time instant with the newly available measurements. However, solving such programs online can be computationally demanding. In this paper, we employ multi-parametric (mp) programming to move most of the computation offline, thus allowing the application of the closed-loop approach to fast processes. Still, the mp-MIQP complexity becomes quickly prohibitive as the number of faulty models increases. In order to overcome this problem, we propose a strategy based on mp-optimization and graph theory that takes into account only two models at a time. While this approach is suboptimal compared to the case in which all the models are considered simultaneously, simulations show that, in practice, the performance is comparable.

View all citing articles on Scopus

He is currently a postdoctoral fellow at INRIA, Rennes center. His current research interests include fault detection and isolation, optimization, optimal control and robust control.

Ramine Nikoukhah received his Ph.D. in 1988, from the Massachusetts Institute of Technology, in Electrical Engineering and Computer Science. He is currently Software Development Director at Altair Development France on leave from his position of Directeur de Recherche at INRIA, Paris-Rocquencourt center.

His current professional interests are in the areas of modeling and simulation of hybrid dynamical systems. He has also worked in the fields of systems, control and signal processing (in particular failure detection). He has published over 100 journal and conference papers, and co-authored 4 books.

As a member of the original team that developed the scientific software package Scilab, he has contributed to a number of its toolboxes. He is in particular the creator of the modeling and simulation environment Scicos.

Stephen L. Campbell received his B.A. in mathematics from Dartmouth College, Hanover, N.H., in 1967, and his M.S. and Ph.D. degrees in Mathematics from Northwestern University, Evanston, Illinois, in 1969 and 1972. He is now Distinguished Professor of Mathematics at North Carolina State University, Raleigh, NC. His current research concerns the numerical solution and analytic behavior of implicit systems of differential and difference equations, such as those arising in electrical circuits and constrained mechanical systems, and their application to control, systems theory, and computer simulation. He is a member of the Center for Research in Scientific Computing and the Operations Research Program at NCSU.

S. Campbell is a Fellow of the IEEE, a SIAM Fellow, and a member of the editorial board for the SIAM book series Advances in Design and Control. He has served as a member of the IEEE Conference Editorial Board and the SIAM J. Sci. Computing editorial board. He is the co-author of “Numerical Solution of Initial Value Problems in Differential Algebraic Equations” and seven other books on implicit dynamical systems, software for scientific problem solving, and applications to control.

^☆: This work was supported in part by NSF Grant ECS-0620986 and DMS-0907832. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Michele Basseville under the direction of Editor Torsten Söderström.

View full text

Auxiliary signal design for robust active fault detection of linear discrete-time systems☆

Abstract

Introduction

Section snippets

Mathematical background

Problem formulation

A solution for the active fault detection problem

Numerical examples and discussion

Conclusions

Automatica

European Journal of Control

European Journal of Control

Automatica

Active estimation for jump markov linear systems

IEEE Transactions on Automatic Control

Optimal piecewise-constant signal design for active fault detection

International Journal of Control