An Adaptive Approach to Active Fault-Tolerant Control

In this Faults or process failures may drastically change system behaviour leading to performance degradation and instability. The reliability and fault-tolerance of a control system can be achieved through the design of either an active or passive Fault Tolerant Control (FTC) scheme. This paper proposes a new approach to fault compensation for FTC using fault estimation by which the faults acting in a dynamical system are estimated and compensated within an adaptive control scheme with required stability and performance robustness. The FTC scheme has an augmented state observer (ASO) in the control system, which has an intrinsic robustness in terms of the stability and performance of the estimation error. The design concepts are illustrated using the notion that the friction forces in a mechanical system can be estimated and compensated to give good control performance and stability. The example given is that of a non-linear inverted pendulum with Stribeck friction. INTRODUCTION The reliability, robustness and fault-tolerance of the control of uncertain systems are issues that have increasing importance as modern systems grow in complexity. In keeping with these developments robust methods for detecting and isolating faults have been developed that can robustly discriminate between the effects of uncertainty and the effects of faults acting within a system or on the actuators and/or sensors. This is the subject of robust fault detection and isolation (FDI) that has been based on the use of a variety of approaches, e.g. unknown input de-coupling, H∞ [1], LPV [2], sliding mode estimation [3, 4] and non-linear geometric approaches [5]. Whilst robust FDI is concerned with the robust decision problem (detection, isolation and perhaps possible fault causes etc), the subject of FTC is concerned with the design and implementation of control schemes that are either active or passive in their method of reacting and compensating for faults [6, 7]. During recent years there has been a substantial literature on the subject of FTC as reported in the review papers [6, 7] and book [8]. FTC can be motivated by different purposes depending on the application under consideration; for example; safety in flight control, efficiency and quality improvements in industrial processes, reliable of mechatronic systems, robotics, etc. The main design challenges are: (i) the number of possible faults acting on the system and their diagnosability, (ii) the system reconfigurability in terms of available redundancy etc, and (iii) the global stability of the system [8]. A fault can make the system deviate far from its normal operating conditions and can lead to severe change in system behaviour. Even bounded faults can cause the closed-loop system to deviate rapidly from its required operation and hence the *Address correspondence to this author at the Department of Engineering, University of Hull, UK; Tel: 0044 1482 41 5117; Fax: 0044 1482 46 6664; E-mail: [email protected] fault accommodation time is a critical parameter. The requirement for rapid reaction to faults can mean that an FDI procedure, if used, may slow down the accommodation process. This paper is concerned with the active approach to FTC, following the classification of FTC systems as shown in Fig. (1), involving fault estimation, fault compensation and adaptive control. The controller is included within the structure of an ASO system in which the actuator faults are estimated via additional state components. The controller is designed using linear output feedback. However, the control system is adaptive as the on-line fault estimates are updated continuously and used to compensate the faults acting within the control channels. The compensation is achieved within the ASO estimation error system with the consequence that the control signal has a time-varying component, the adaptive part of the control. The use of on-line compensation means that the fault isolation task of FDI is not required. The ideal residual generation problem of fault detection of FDI is replaced by robust fault estimation. It is of interest here to note that, asFig. (1). Classification of FTC Systems (taken from [6]). An Adaptive Approach to Active Fault-Tolerant Control The Open Automation and Control Systems Journal, 2009, Vol. 2 55 suming a residual generator can be robust against modelling uncertainties, the residual signal or vector can be equivalent to a fault estimator under certain conditions which are described in the well known book by [9]. In this work we obviate the use of the residual generation process and turn directly to a robust fault estimation problem embedded within an adaptive control scheme. In the linear system case our control scheme makes use of observer-based output feedback. However, in the non-linear case both fault(s) and modelling uncertainties (unknown inputs) is/are estimated and compensated using an augmented state space disturbance observer structure, the ASO, with additional states corresponding to estimates of both faults and uncertainties. This paper focuses on the special case of actuator faults within an on-line fault estimation and compensation system using the ASO concept. The important principle is that the faults and modelling uncertainties both act to disturb the system dynamics. For clarity, we consider the case in which the fault compensation mechanism causes the closed-loop system to behave in an almost linear sense. However, further analysis and design shows that the approach easily handles cases of multiple faults and the co-existence of different types of faults (actuator, sensor or multiplicative faults) and unknown input signals (arising from the effect of modelling uncertainty in the estimation and control). The controller is adaptive as the fault/uncertainty estimation signal becomes a component in the state estimate feedback control, thereby cancelling bounded uncertainty effects due to either faults or unknown input signals acting on the observer state estimation error. The ASO includes a compensation gain matrix which must be designed using Lyapunov LMI-poleplacement approach, based on knowledge of the fault bounds. The paper discusses this theory via a theorem and corresponding proof. The adaptive compensation FTC concept is illustrated by considering the friction force as a special type of input or actuator fault in a mechatronic system, the inverted pendulum. It is very reasonable to consider the friction as a fault in the system as it is an unwanted effect which causes the performance of the system to change. Patton, Putra and Klinkieo [10] tackled the friction force compensation problem using a sliding mode observer together with a sliding mode controller. The friction force could alternatively be described as having an uncertain effect in the system but we prefer here to consider the whole issue of friction compensation as an FTC problem. The friction (fault) estimation and compensation is handled in this paper using the ASO and the results demonstrate excellent performance of the adaptive controller in removing the effect of the friction force to yield very precise positioning control. Extensive recent research has focused on detailed modeling of friction phenomena in order to use robust on-line friction compensation procedures [11-14]. However, the friction modeling problem remains a very difficult complex systems challenge and although complex modeling techniques are used no efficient method exists to ensure satisfactory robustness. It can easily be seen that the adaptive control approach used in this study obviates the need for explicit friction modeling. This offers significant advantages over well known model-based friction compensation methods in which detailed modeling of friction phenomena is essential and for which robustness with respect to friction characteristics is very difficult to achieve. Section II outlines the proposed approach to fault estimation and compensation using the ASO concept. The controller has two components (a) state estimate feedback together with (b) the component arising from the fault compensation. Some analysis and proof of stability for this control system structure is given, showing that the controller fault compensation gain must lie in a defined interval derived via a Lyapunov stability condition, designed via the Matlab LMI toolbox. Section III describes the results arising from the inverted pendulum friction compensation problem. Section IV provides a concluding discussion with suggestions of further research. AUGMENTED STATE OBSERVER The idea of estimation of uncertain effects in an observer-based FDI scheme was discussed extensively in [9], in which the uncertain effects (modeling errors, unknown disturbances etc) are combined into unknown input signals. These authors also discussed the related problem of sensitizing the FDI observer estimation error to specific faults and de-sensitising the error dynamics to other faults – effectively a dual of the unknown input de-coupling problem. This problem is also discussed in [15]. As discussed in Section I, a side-step from this is to consider a problem of fault or unknown input estimation, using robust estimation techniques. The unknown input estimation problem was considered in [9, 16, 17] using the ASO concept and this has motivated the current FTC study. The work by Patton and Chen did not make use of fault compensation within a control loop. Here we use the ASO concept in an FTC scheme, as outlined in Section I. ACTUATOR FAULT ESTIMATION Considering the state space representation of faulty system Cx y f F Bu Ax x a a = + + = & (1) where n x ! " is the state vector, p y ! " the output observation vector, m u ! " the input vector and B A, and C are known matrices of appropriate dimensions. a F is the fault distribution matrix for the actuator fault m a f ! " corresponding to the th i column of B (in the case of the th i actuator fault). This idea is considered even further here in the context of “an observer-based adaptive controller” of the form: 56 The Open Automation and Control Systems Journal, 2009, Vol. 2 Patton and Klinkhieo a f x f K x K u ˆ ˆ + = (2) A suitable analysis and design is required to stabilize the faulty system (1) around the origin in the presence of unwanted but bounded actuator fault signals. n m x R K ! " and m m f R K ! " are the controller and actuator fault compensation gains, respectively. The vectors x̂ and a f̂ are the state and actuator fault estimation signals, respectively obtained from the ASO with dynamics derived as follows: Substitute (2) into (1), giving a a a f x f F f BK x BK Ax x + + + = ˆ ˆ & (3) ) ˆ ( ˆ ) ( ˆ x C y L x BK A x x x ! + + = & (4) ) ˆ ( ˆ x C y L f f a ! = (5) Eqs. (4) and (5), can be re-arranged as: [ ] { y L L f x C L L BK A y L L f x C L C L BK A x C y L L f x BK A f x