An LPV Approach to Active FTC of a Two-Link Manipulator Ron J Patton, and Supat Klinkhieo Department of Engineering, University of Hull, HU6 7RX, UK (Tel: +44 (0)1482-46-5117; e-mail: r.j.patton@hull.ac.uk ) Abstract: This work is motivated by the challenge to develop an adaptive strategy for systems that are complex, have actuator faults and are difficult to control using linear methods. The novelty lies in combined use of LPV fault estimation and LPV fault compensation to meet active FTC performance requirements. The paper proposes a new design approach for systems which can be characterized via sets of LMIs and can be obtained using efficient interior-point algorithms. A polytopic LPV estimator is synthesized for generating actuator fault estimates used in an active FTC scheme to schedule the nominal system state feedback gain as a function of fault effect factors, thereby maintaining the system performance over a wide operating range within a proposed polytopic model. The method is demonstrated through a nonlinear two-link manipulator system with torque input faults at each joint. Keywords: LPV Systems, Fault Tolerant Control, Fault estimation, Active FTC, Robotic Control 1. INTRODUCTION There is a significant interest in the control of time-varying systems ([1] and [2]). LPV modelling methods have gained a great deal of interest, especially for applications related to vehicle, robust and aerospace control ([3] and [4]). The LPV approach is appealing when nonlinear plants can be modelled as time-varying systems with on-line measurable (or estimated) state-dependent parameters. Bokor and Balas (2004) ([5]) introduced the concept of the use of fault detection filters for LPV systems and many other investigators have followed different aspects of this approach ([6], [7], [8] and [9]). Recently, the idea of extending the control approach using LPV to encompass Fault-tolerant Control (FTC) schemes has been the subject of a number of studies ([10], [11], [12], [13] and [14]). Most FTC studies are based on LPV focus on active approaches and others on the passive fault tolerance. Active methods use control system adaption or reconfiguration (or both) subject to detectable faults, whilst passive FTC has no provision for actively reacting to a fault once it occurs [15]. Ganguli, Marcos and Balas (2002) ([4]) use LPV ideas for the active FTC problem based on actuator faults in aircraft. This paper proposes a new design of an active FTC and polytopic LPV estimator for systems which can be characterized via a set of LMIs and can be obtained using efficient interior-point algorithms ([17]). A polytopic LPV estimator is synthesized for providing actuator fault estimation which is used in an FTC scheme to schedule the state feedback gain. The gain is calculated using LMIs in the fault-free case in order to maintain the system performance over a wide operating range within a proposed polytopic model. The active FTC controller is a function of the fault effect factors as defined by [15] and [16] which can be derived on-line (in this case) from the residual vector of a polytopic LPV estimator mechanism. This work uses results from [17] and has mainly been motivated by: (a) The work of Weng et al (2008) ([14]) on LPV fault estimation for rate bounded time-delay systems. (b) The use of fault effect factors as described by Chen et al (1999) ([15]) and Chen and Patton (2001) ([16]). The work [14] is limited only to fault estimation and does not include the full FTC problem, whilst the work of [15] and [16] pre-dates the development of the LPV approach to control and FTC in particular. The new contribution is the combined use of fault estimation and fault compensation for FTC within an LPV framework. The proposed method is demonstrated through a nonlinear two-link manipulator system with a fault in the torque inputs at each manipulator joint. The system can be represented by a polytopic model. Section 2 overviews the LPV concept. Section 3 gives a statement of the mathematical problem to be solved. Section 4 details the polytopic LPV estimator design strategy that is to be used in the active FTC scheme. Section 5 describes the polytopic model structure of the two-link manipulator as a tutorial example, based on the LPV estimator theory. This example (with fault estimation) is used for active FTC design, via the polytopic LPV controller synthesis in Section 6. Section 7 gives concluding comments. 2. OVERVIEW OF LPV APPROACH An LPV system is a mathematical description of the linear parameter-varying nature of a nonlinear system. LPV systems have state-space matrices that are fixed with some vector of varying parameters ([1] and [3]). From a practical point of view, a nonlinear system can be reduced to an LPV representation by using the linearization along trajectories of the parameters. In other words, the idea in LPV is to obtain smooth semi–linear models that can vary or be scheduled using a parameter, for example an altitude and/or speed of an aircraft, so that the LPV model will mimic the actual nonlinear plant ([14], [18] and [19]). Instead of choosing a combination of predefined linear models, the models change parametrically. The LPV model has the structure of a time- varying linear system with the parameter-dependent matrix quadruple )] ( ), ( ), ( ), ( [ θ θ θ θ D C B A , where: nxn A ℜ ∈ ) ( θ , nxm B ℜ ∈ ) ( θ , pxn C ℜ ∈ ) ( θ and pxm D ℜ ∈ as: