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## Biographical review

This section is dedicated to a biographical review of regular LPV and D-LPV systems. Both cases are considered in this review, because some ideas primarily developed for regular LPV system were extended to D-LPV systems. However, to the best of the author’s knowledge, despite the existence of many works dedicated to regular LPV systems, only a few works are dedicated to D-LPV systems.

Note also that, despite in the past the gain scheduling functions were computed by considering Takagi-Sugeno fuzzy rules as defined in (Takagi and Sugeno, 1985), in the present Takagi-Sugeno (TS) and polytopic LPV systems are described by the same form. The community of people working on TS models uses the name “TS FUZZY systems”, even if with the recent modeling approaches (for example sector nonlinearity transformation), the obtained model is no longer “fuzzy” because the weighting functions are completely deterministic that corresponds to LPV or quasi-LPV systems as detailed in (Rodrigues et al., 2014). Therefore, in this review some works published as TS, which correspond to previous remark, are considered as LPV systems.

Considering the previous remarks, the biographical review is organized in two subsections. The first subsection is dedicated to systems with measurable scheduling functions and the second to systems with unmeasurable scheduling functions. This consideration is done because despite both have the same structure, the methodologies to analyze systems with unmeasurable scheduling functions are different compared to systems with measurable scheduling functions, as will be detailed in subsequent Sections. Also, each subsection is dedicated to first review works related to regular LPV systems and then those related to D-LPV systems, in order to illustrate the contrasts of both cases.

### Measurable scheduling functions case

LPV systems were introduced by (Shamma and Athans, 1988, 1990) as mathematical models to design and to guarantee a suitable closed-loop performance for a given plant in different operating conditions, such that the scheduling parameter captured the nonlinearities of the plant. The term LPV was adopted to distinguish these systems from both linear time invariant (LTI) and linear time varying (LTV) systems. The distinction with respect to LTI systems is clear, because LPV systems are non-stationary. On the other hand, LPV systems are distinguished from LTV systems in the perspective taken on both analysis and synthesis. LPV system can be seen as a family of LTV systems, where each model is parameter varying according to the scheduling functions. Therefore, properties as stability, disturbances rejection, tracking, among others, hold for a family of LTV systems, rather than a single LTV system (Shamma, 2012).

In last years, significant progress has been made for LPV systems. For example, in the presence of un-certainties or disturbances, LPV robust control techniques have proved to have better performance than robust LTI controllers (Sato, 2011). Indeed, many solutions for LTI systems given in the Linear Matrix Inequalities (LMI) framework have been extended to LPV systems, e.g. a stabilization method for an arm-driven inverted pendulum was proposed in (Kajiwara et al., 1999); the proposed LPV controller was shown to outperform classical robust control techniques H• and µ-synthesis. However, the method does not guarantee that the closed loop system exhibit the robust performance considered for the operation conditions. To handle this problem, a parametrized H• was presented in the work of (Bruzelius et al., 2002); the control showed good performance when applied to a turbo fan jet engine. Other H• controller for systems affected by time-varying parametric uncertainties can be consulted in (Scherer, 2004). In order to improve the performance of H• controllers, a switching controller designed with multiple Lya-punov functions was proposed by (Xu et al., 2011b). Similarly, in (Xu et al., 2011a) an LPV control for switched systems with slow-varying parameters was proposed by adopting the blending method proposed by (Shin et al., 2002); the blending method separates the entire parameter set into overlapped subsets and an LPV controller for the whole region is blended by regional controllers. The resulting methodology was applied to an F-16 Aircraft Model. Nevertheless, the method is applicable under the assumption that the scheduling parameters can be measured on-line, which often is difficult to satisfy in practice. To solve this problem, a robust compensator designed for stable polytopic LPV plants, which considers prior and non-real-time knowledge of the dependent parameter, was proposed by (Xie et al., 2003). A bibliographical review of identification LPV methods for real application is given in (Giarre et al., 2006).

Recently, several Model Predictive Control (MPC) schemes have been proposed, e.g. (Lee et al., 2011) proposed an MPC method to stabilize LPV systems with delayed state; as a result, sufficient conditions which guarantee the asymptotic stability were obtained. A Quasi-min-max algorithm was developed in (Park et al., 2011); first, an off-line method for designing a robust state observer was obtained by considering LMI techniques then, an on-line optimization algorithm was applied to implement a robust stable controller with input constraints. Another difficulty in dealing with LPV plants is that the one-step ahead state prediction set is non-convex and depends quadratically on the scheduling vector. The problem has been solved by using the polytopic framework, which exploits the symmetrical property existing between any pairs of elements of the quadratic form. An improvement of these method can be consulted in (Casavola et al., 2012), where an ellipsoidal MPC strategy for discrete-time polytopic LPV systems subject to bounded disturbances, input and state constraints, was presented.

Indeed, LPV systems have proved to handle successfully real application problems, some of them are mentioned as follows: an semi-active suspension controller was proposed in (Poussot-Vassal et al., 2008); the authors developed a LPV method in order to guarantee internal stability and some performance criteria for the semi-active suspension. (Abbas and Herbert, 2011) proposed a method for frequency-weighted discrete-time LPV model reduction with guaranteed stability of the reduced-order model, when both input and output weighting filters are used; the method was applied to a beam-head assembly of an industry-grade prototype gantry robot. A gain scheduling controller for the electronic throttle body in ride-by-wire racing motorcycles was proposed by (Corno et al., 2011), which considered model-based gain-scheduled position control system for throttle position tracking. An LPV system identification algorithm to model a leakage detection in high pressure natural gas transportation networks can be consulted in (Lopes et al., 2011). Many other successfully applications have been proposed, e.g. (Pfifer and Hecker, 2011) proposed an algorithm for modeling an industrial, highly nonlinear missile model, which allowed to prove the robust stability for a large region of the flight envelope. (Diaz-Salas et al., 2011) proposed a Magneto-Rheological (MR) LPV model that provided a control-oriented dynamical description suitable to design control strategies and to enhance comfort, steering, and road grid. Other MR model used to attenuate the vibration of a two-story model structure can be found in (Shirazi et al., 2011), by considering an output feedback controller for MR damper. In (Rotondo et al., 2013), a method to design a quasi-linear parameter varying (quasi-LPV) modeling, identification and control of a Twin Rotor MIMO by considering a state feedback gain-scheduling controller was proposed. A survey about LPV methods for vehicle dynamic control can be consulted in (Sename et al., 2013b). (Bolea et al., 2014) proposed a method to design LPV controllers applied to real-time control of open-flow irrigation canals. Military application to a modern air defense missile model can be consulted in (Tekin and Pfifer, 2013).

#### Unmeasurable gain scheduling function case

The idea of considering unmeasurable scheduling functions was studied first by (Zak, 1999) to stabilize a multiple model system by considering linear controllers. Following the same idea, a robust controller was studied in (Shi and Nguang, 2003). (Bergsten and Palm, 2000; Bergsten et al., 2002) proposed a method to analyze and design sliding mode observers, which proved to deal effectively with model/plan mis-matches. The system was transformed such that the mismatches represented the difference between the unmeasurable and the estimate scheduling function. Indeed, the uncertain system transformation proved to deal successfully with the unmeasurable scheduling problem, e.g. the observer design applied to state estimation was treated in (Yoneyama, 2009). The design of robust observers for fault diagnosis purposes was studied in (Ichalal et al., 2008, 2010), (Theilliol and Aberkane, 2011). Nevertheless, only few con-tributions have been proposed for D-LPV systems, although , we can cite (Nagy-Kiss et al., 2011), where the authors proposed a state observer by transforming the D-LPV system with unmeasurable scheduling functions into an equivalent uncertain system. In (Nagy Kiss et al., 2011), an unknown input observer was developed by considering the original system as a perturbed system, where the perturbation vector represents a bounded uncertainty given by the measurable and the unmeasurable scheduling functions. In both previous works, the observers were successfully evaluated by using a nonlinear model of a waste-water treatment plant. In the same context, based on the perturbed system technique, a fault detection scheme was proposed in (Hamdi et al., 2012a) with application to an electrical system.

As it can be concluded from the previous review, the problem of control techniques, specially observer design and its application to fault diagnosis, for D-LPV systems with unmeasurable scheduling functions has not been fully investigated so far. Moreover, all these works exemplify the relevance of the techniques for D-LPV systems with unmeasurable scheduling functions and their application to fault detection and isolation. Furthermore, the results proposed in this thesis address these topics by considering four axes: first, the observer design is addressed by considering the H• performance in order to attenuate distur-bances, noise and minimize the error generated by the unmeasurable scheduling functions. Second, fault detection and isolation is addressed by means of residual evaluation. Third, fault diagnosis is performed by extending one of the proposed H• methods. Finally, by combining H• and H approaches, a fault detection observer, sensitive to faults and insensitive to disturbances, is proposed.

The following Section presents some methods to model descriptor nonlinear systems as D-LPV systems, to illustrate how this kind of systems, with scheduling functions depending on the states, are obtained, for example considering the nonlinear sector approach.

**Modeling of D-LPV systems**

There are different methods to derive a D-LPV system from a nonlinear system, such as linearization (Johansen et al., 2000; Marcos and Balas, 2004), sector nonlinearity approach (Ohtake et al., 2003), state-space transformation (Shamma and Cloutier, 1993), function substitution (Wey, 1997; Pfifer, 2012) and identification (Tóth, 2010). Detailed information about derivation of LPV systems can be found in (Marcos, 2001). Nevertheless, the nonlinear sector approach and the linearization are usually considered, such methods are detailed below.

**Table of contents :**

**1 Introduction **

1.1 Context of the Thesis

1.2 Introduction

1.3 Main contributions

**2 Overview of D-LPV systems **

2.1 Definition of D-LPV systems

2.2 Biographical review

2.2.1 Measurable scheduling functions case

2.2.2 Unmeasurable gain scheduling function case

2.3 Modeling of D-LPV systems

2.3.1 Linearization

2.3.2 The sector nonlinearity approach

2.4 Properties of D-LPV systems

2.4.1 Stability and admissibility of D-LTI systems

2.4.2 Stability of D-LPV systems

2.4.3 Observability

2.5 State observers for D-LPV systems with unmeasurable gain scheduling

2.5.1 Definitions

2.5.2 Observer design

2.6 Descriptor System Package

2.7 Conclusions

**3 Robust H• state observer design **

3.1 Preliminary definitions

3.2 Approach 1: Descriptor observer approach

3.3 Approach 2: The uncertain system approach

3.4 Approach 3: Observer design by considering the uncertain error system approach

3.4.1 Observer design without disturbances

3.4.2 Observer design with disturbances

3.5 Comparison between the three approaches

3.6 Application to robust fault detection and fault isolation

3.6.1 Sensor fault detection and isolation

3.6.2 Comparison between the three approaches under faults

3.7 Robust fault estimation based on H• observer

3.8 Conclusions

**4 Fault detection observer design based on the H−/H• performance **

4.1 Introduction

4.2 Problem statement

4.3 Observer gain synthesis

4.3.1 Fault sensitivity condition

4.3.2 Robustness condition

4.3.3 Mixed H−/H• observer design

4.4 Illustrative Example

4.5 Conclusions

**5 Conclusions and perspectives **

**Appendix A Descriptor systems **

**Appendix B Linear Matrix Inequalities **

**Appendix C Technical results in tracking controllers and discrete-time systems **

**References **