Linear Fractional Transformation Model Research Articles

Linear Fractional Transformation Modeling of Multibody Dynamics Around Parameter-Dependent Equilibrium

This paper proposes a new Linear Fractional Transformation (LFT) modeling approach for uncertain Linear Parameter Varying (LPV) multibody systems with parameter-dependent equilibrium. Traditional multibody approaches, which consist in building the nonlinear model of the whole structure and linearizing it around equilibrium after a numerical trimming, do not allow to isolate parametric variations with the LFT form. Although additional techniques, such as polynomial fitting or symbolic linearization, can provide an LFT model, they may be time-consuming or miss worst-case configurations. The proposed approach relies on the trimming and linearization of the equations at the substructure level, before assembly of the multibody structure, which allows to only perform operations that preserve the LFT form throughout the linearization process. Since the physical origin of the parameters is retained, the linearized LFT-LPV model of the structure exactly covers all plants, in a single parametric model, without introducing conservatism or fitting errors. An application to the LFT-LPV modeling of a robotic arm is proposed; in its nominal configuration, the model obtained with the proposed approach matches the model provided by the software Simscape Multibody, but it is enhanced with parametric variations with the LFT form; a robust LPV synthesis is performed using Matlab robust control toolbox to illustrate the capacity of the proposed approach for control design.

Efficiency of the Bond Graph Approach for Robust Diagnosis Micro Electro Mechanic System

Improving the operational safety of systems is essentially based on fault detection and isolation algorithms, these algorithms mainly consist in deriving the different kinds of faults while minimizing false alarms, nondetections and delays in fault detection. The choice of robust diagnosis by the bond graph approach (BG) is based on the use of a robust analytical relations redundant (ARR) generation algorithm from linear fractional transformation models (LFT), where the uncertain part of the ARR is used to generate the adaptive residue thresholds. These relationships not only allow the detection and isolation of defects on the various elements of the system, but also the location since the industrial system is modeled element by element.The BG-LFT modeling and the robust diagnosis of an accelerometer using micro-electromechanical are presented in this paper. The interaction of the various phenomena is taken into account, thanks to the energy properties of the bond graph tool. The residuals and the adaptive thresholds for normal operation and in abnormal operation are determined. The results suggest that the use of the Bond Graph model for normal operation, the evolutions of the residuals converge towards zero, whereas a fault caused by an element (for example, the stiffness springs [Formula: see text] or [Formula: see text] are modeled, respectively, by capacitive elements [Formula: see text]and [Formula: see text] shows that the residues [Formula: see text], [Formula: see text] and [Formula: see text] become different from zero. These variations explain that these residues are sensitive to these elements, which are confirmed by results presented in Table 2. The analysis of the sensitivity of the residuals to parametric and structural faults is carried out to determine the detectable values of the faults and therefore to monitor the performance of the diagnosis.

A RobustH∞ Fault Tolerant Controller for Uncertain Systems Described by Linear Fractional Transformation Model

In this article, a robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fault tolerant control law is addressed for a class of the uncertain dynamical systems represented via linear fractional transformation. To this objective, a state-feedback controller law is utilized for achieving the control objective. Thus, a linear matrix inequality based performance condition would be derived to guarantee that the disturbance suppression is accomplished in the uncertain system. Hence, the gains of the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller would be suitably determined by checking the feasibility of such a linear matrix inequality problem. The proposed control technique is numerically simulated in two dynamical uncertain systems (i.e., a typical control system and a mechanical robotic arm). Considering the disturbance rejections and transient responses, the results illustrate the efficiency of the recommended robust technique compared with the existing control methods.

Linear Fractional Transformation co-modeling of high-order aeroelastic systems for robust flutter analysis

This article presents a new paradigm for robust flutter modeling and analysis of high-order uncertain and linear aeroelastic systems. The fundamental idea is to couple the state-of-art in robust worst-case analysis (Linear Fractional Transformation modeling and µ analysis) with the state-of-practice in aeroelasticity (fluid-structure-interaction solvers). The issue with the latter is that, although capable of providing different levels of fidelity, they are less efficient in coping with the analysis of systems subject to uncertainties. In fact, while they have the advantage of capturing directly the physical uncertainty, the analyses can only be applied to a defined parameter combination, and due to their computational cost, it is usually only possible to consider a limited set of cases. To tackle this lack of robustness, in recent works the application of analytic worst-case methods has been proposed, but the intimately related problem of constructing accurate uncertain models has not been fully addressed. In this article, a co-modeling framework is presented that leverages the main features of both fluid-structure interaction solvers and robust control-based methods. The key idea is to combine these two typically distinct steps in a single one, enabling in this way to obtain an uncertainty description which is flexible and reconciles the physical sources of uncertainty with the uncertain parameters used in the LFT model. An exemplification of the developed framework on an unconventional aircraft configuration is provided. Results show its potential to provide valuable physical insights into the problem when analyzing complex systems.

Parametric and linear parameter varying modeling and optimization of uncertain crane systems

This research investigates uncertain crane motor system unstructured, structured, and linear parameter varying uncertainty modeling, control, and optimization frameworks. The proposed approaches effectively address the uncertain or time-varying plant components that typically exhibit significant variations under normal physical operational conditions with external influence, implying a complex stabilization and performance synthesis problem with a need for sophisticated quantitative frameworks, suitable for real-time implementations, as compared to traditional proportional-integral-derivative controller implementations. The crane system motor inductor component uncertainty is modeled analytically for the proposed three frameworks by using the uncertain state-space approach and the corresponding Multi-Input Multi-Output Linear Fractional Transformation modeling is used to formulate robust optimization problems for superior tracking performances under operational disturbances. The uncertain crane controller synthesis numerical results clearly indicate the effectiveness of the proposed modeling and optimization frameworks on desired tower crane stability and performance levels.

Study of Flexible Aircraft Body Freedom Flutter with Robustness Tools

Body freedom flutter is a dynamic instability featuring strong coupling between rigid-body and elastic modes of the aircraft. Flexible configurations with adverse structural and geometric properties have been found susceptible to this phenomenon. Features that complicate its study are the presence of multiple modal instabilities and the different influence that system parameters have on each of them. The robust analysis framework based on linear fractional transformation modeling and structured singular value analysis is used in this work to study the body freedom flutter problem in a systematic way. The analyses performed showcase the potential of these methods, not only in supplying a characterization of the system in terms of its robustness but also in gaining further understanding of the body freedom flutter problem and reconciling the results with physical features. It is also shown that the robust modeling analysis framework complements the conventional, state-of-practice methods while allowing the study of highly coupled systems (of which the flexible aircraft is an example) to be addressed in an incremental and methodological manner. For this study, a simplified wing model is augmented including the short-period approximation aircraft model and the rigid–elastic coupling terms. The proposed model captures properties and trends of both restrained wing flutter and body freedom flutter instabilities.

Aeroelastic modeling and stability analysis: A robust approach to the flutter problem

SummaryIn this paper, a general approach to address modeling of aeroelastic systems, with the final goal to apply μ analysis, is discussed. The chosen test bed is the typical section with unsteady aerodynamic loads, which enables basic modeling features to be captured and so extend the gained knowledge to practical problems treated with modern techniques. The aerodynamic operator has a nonrational dependence on the Laplace variable s, and hence, 2 formulations for the problem are available: frequency domain or state‐space (adopting rational approximations). The study attempts to draw a parallel between the 2 consequent linear fractional transformation modeling processes, emphasizing critical differences and their effect on the predictions obtained with μ analysis. A peculiarity of this twofold formulation is that aerodynamic uncertainties are inherently treated differently and therefore the families of plants originated by the possible linear fractional transformation definitions are investigated. One of the main results of the paper is to propose a unified framework to address the robust modeling task, which enables the advantages of both the approaches to be retained. On the analysis side, the application of μ analysis to the different models is shown, emphasizing its capability to gain insight into the problem.

LFT Modeling and μ Analysis of the Aircraft Landing Benchmark

The landing of a civil transport aircraft is one of the most challenging phases. The reason mainly lies in the need to satisfy different concurrent requirements in the face of a wide range of system variations and environmental perturbations. Automatic control laws have been developed in the past to address this complex scenario, using an involved iterative process of design, tuning and validation. Due to the uncertainties and perturbations present in this phase, robust synthesis techniques (e.g standard and structured H∞ design) provide an effective framework to accomplish these tasks. This compels on the other hand to rely on procedures to generate the proper mathematical description of the plant and the availability of analysis tools to obtain time efficient robust predictions and insight into the system being analyzed. This work presents the application of Linear Fractional Transformation modeling and μ analysis to the Aircraft Landing Benchmark proposed by ONERA and Airbus.

Dynamic IQC-Based Control of Uncertain LFT Systems With Time-Varying State Delay.

This paper presents a new exact-memory delay control scheme for a class of uncertain systems with time-varying state delay under the integral quadratic constraint (IQC) framework. The uncertain system is described as a linear fractional transformation model including a state-delayed linear time-invariant (LTI) system and time-varying structured uncertainties. The proposed exact-memory delay controller consists of a linear state-feedback control law and an additional term that captures the delay behavior of the plant. We first explore the delay stability and the L2 -gain performance using dynamic IQCs incorporated with quadratic Lyapunov functions. Then, the design of exact-memory controllers that guarantee desired L2 -gain performance is examined. The resulting delay control synthesis conditions are formulated in terms of linear matrix inequalities, which are convex on all design variables including the scaling matrices associated with the IQC multipliers. The IQC-based exact-memory control scheme provides a novel approach for delay control designs via convex optimization, and advances existing control methods in two important ways: 1) better controlled performance and 2) simplified design procedure with less computational cost. The effectiveness and advantages of the proposed approach have been demonstrated through numerical studies.

An overview of the RFCS project V&amp;V framework: optimization-based and linear tools for worst-case search

This article presents the application of nonlinear (simulation-based) and linear (structured singular value) worst-case tools to the VEGA launcher Verification and Validation process, during atmospheric ascent. The simulation-based worst-case evaluation is performed by minimizing a set of cost functions that capture the launcher’s performance objectives, using the Worst-Case Analysis Optimization Tool and a high-fidelity nonlinear simulator of VEGA. The linear worst-case search uses the structured singular value ( $$\mu $$ ) and a linear fractional transformation model representing the yaw rigid motion of the VEGA launcher but numerically evaluated using time simulation data from the VEGA simulator. To facilitate the analysis of the worst-case results as well as the comparison between the two analysis tools, a selection of the most critical uncertainties is performed using sensitivity analysis based on selected nonlinear simulator time responses. It is highlighted that the presented analysis tools are complementary to traditional Monte Carlo approaches in that they strive to identify worst-case uncertainty combinations as opposed to providing probabilistic guarantees on performance metric satisfaction. In addition, as it will be shown, these approaches require only a fraction of the time required to perform a Monte Carlo campaign.

LPV modeling and LFT Uncertainty Identification for Robust Analysis: application to the VEGA Launcher during Atmospheric Phase

In this article, the development of a proper Linear Fractional Transformation model for the robust analysis of the atmospheric controllers of the European VEGA launcher is presented. By proper it is meant a model that: (i) captures the key launcher’s behavior during the ascent atmospheric phase, (ii) has a complexity in terms of number of parameters and repetitions that does not limit the applicability of robust analyses, (iii) can be easily updated with very different trajectory profiles, and (iv) arises from a methodology that is well connected with industrial practice and can be understood and used relatively easy. This last requirement is fundamental to transfer the LFT methodology into the VEGA program, which is critical if robust analysis is to be used within their verification and validation process. The potential of the proposed methodology and the validity of the developed LFT models for robust analysis have been successfully verified using simulation data from three of the four VEGA qualification flights.

LFT Modeling and Robust Stability Analysis of Missiles with Uncertain Parameters

The structured singular value (μ) analysis based method has many advantages for the robust stability analysis of missiles with uncertain parameters. Nevertheless, the present linear fractional transformation (LFT) modeling process, which is the basis of μ analysis, is complex, and not suitable for automatic implementation; on the other hand, μ analysis requires a large amount of computation, which is a burden for large-scale application. A constructive procedure, which is computationally more efficient, and which may lead to a lower order realization than existing algorithms, is proposed for LFT modeling. To reduce the calculation burden, an analysis method is developed, based on skew μ. On this basis, calculation of the supremum of μ over a fixed frequency range converts into a single skew μ value calculation. Two algorithms are given, to calculate the upper and lower bounds of skew μ, respectively. The validity of the proposed method is verified through robust stability analysis of a missile with real uncertain parameters.

LFT modelling and H&lt;SUB align="right"&gt;&amp;infin; control of a non-linear MIMO system with cross coupled dynamics

This paper presents a systematic approach towards the control-oriented linear fractional transformation (LFT) modelling for the H∞ controller design of a non-linear multi-input multi-output (MIMO) system. A very interesting mechatronic platform, namely, twin-rotor laboratory scale helicopter, which is a highly non-linear and cross coupled dynamic system and pose challenges for many classical linear control techniques, has been considered as a candidate system to illustrate the proposed LFT modelling and the design of H∞ controller for non-linear MIMO system. A novel methodology has been presented to consider the model uncertainties arising due to the difference between the linearised mathematical model and the actual system, presence of disturbance signal, and the model order reduction during the derivation of the LFT model. A two-degrees-of-freedom (2-DOF) framework has been adopted for the design of the H∞ controller. Robust stability and performance of the resulting controller have been verified in simulation studies.

Bridging the gap between linear and nonlinear worst-case analysis: an application case to the atmospheric phase of the VEGA launcher

This article presents the application of the structured singular value worst-case approach to the VEGA launcher during the atmospheric ascent phase. The analysis uses a linear fractional transformation model, formed from a subset of the uncertainty and dispersion parameters defined for the nonlinear VEGA system, to identify physically feasible worst-cases. This analysis is complementary to traditional Monte Carlo campaigns in that it provides an analytical guaranteed existence of worst-case conditions. To exemplify the difference, a dense (100,000 runs) Monte Carlo campaign in a high-fidelity VEGA nonlinear simulator yielded no performance-violating cases but, the results presented show that the application of mu-analysis is capable of identifying numerous parametric uncertainty combinations away from the extremes resulting in performance violations in the same VEGA nonlinear simulator at a fraction of the computational time. Thus, the present application shows that linear analytical approaches such as mu-analysis (when supported by proper development of a linear fractional transformation model) have capabilities that complement the traditional design verification and validation process for launchers.

Stochastic Intensity Propagation via LFT Modelling

AbstractRecently linear fractional transformation (LFT) modelling has demonstrated considerable potential for the filtering problem of highly nonlinear state space systems. A characteristic of the LFT model is the feedback loop which encapsulates the nonlinearity of the state-space model in a structure that is both simple and sparse. For a broad class of practical problems the LFT gives an equivalent representation of the nonlinear state space models which is more efficient due to its structure. With an approximation localized to a simple nonlinearity in the feedback path, it has been shown that this approach works reasonably well where conventional linearization techniques fail. In this paper, we propose nonlinear filtering via LFT modelling for propagating the stochastic conditional intensity function for problems in multi-target filtering. Simulation results are shown to demonstrate the performance of the proposed filter.

Development of an Integrated LPV/LFT Framework: Modeling and Data-Based Validation Tool

In this paper, the development of a software tool for linear parameter varying (LPV)/linear fractional transformation (LFT) modeling and data-based validation techniques is described. This tool represents a step towards industrialization of these techniques, which although still suffering from a certain theoretical complexity are increasingly shown to help deal effectively with important real world applications. The tool is a key component for the development of an integrated LPV/LFT framework for control modeling, analysis, and design which is also a necessity for successful transfer of LPV/LFT techniques to industry. The focus of this paper is on describing the development of the integrated modeling/validation software and examining the applicability problems associated with these techniques. The technical readiness of the tool is exemplified through the development of validated LFT models used for LPV control design and analysis of a nonlinear longitudinal model of the NASA HL-20 re-entry vehicle.

Nonlinear Bayesian Filtering Using the Unscented Linear Fractional Transformation Model

For nonlinear state space model involving random variables with arbitrary probability distributions, the state estimation given a sequence of observations is based on an appropriate criterion such as the minimum mean square error (MMSE). This leads to linear approximation in the state space of the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), which work reasonably well only for mildly nonlinear systems. We propose a Bayesian filtering technique based on the MMSE criterion in the framework of the virtual linear fractional transformation (LFT) model, which is characterized by a linear part and a simple nonlinear structure in the feedback loop. LFT is an exact representation for any differentiable nonlinear mapping, so the virtual LFT model is amenable to a wide range of nonlinear systems. Simulation results demonstrate that the proposed filtering technique gives better approximation and tracking performance than standard methods like the UKF. Furthermore, for highly nonlinear systems where UKF diverges, the LFT model estimates the conditional mean with reasonable accuracy.

Closed form Filtering for Linear Fractional Transformation Models

The nonlinear optimal Bayes filter is computationally intractable and at present there exists no analytical method for the nonlinear filtering problem. The standard approaches for linear approximation in the literature work reasonably well for the class of systems that are only mildly nonlinear. In this paper, we introduce a filtering approach for the most general class of nonlinear models by transforming the state space model to an equivalent representation given by the linear fractional transformation (LFT) model which is nonlinear in the feedback loop only. Based on an approximation localized to the feedback path only, we derive a closed form solution to Bayes recursion for the LFT model. We give simulation results to demonstrate the potential of the proposed filtering approach for applications where conventional methods fail.

A symbolic matrix decomposition algorithm for reduced order linear fractional transformation modelling

In this paper an algorithm that provides an equivalent, but of reduced order, representation for multivariate polynomial matrices is given. It combines ideas from computational symbolic algebra, polynomial/matrix algebraic manipulations and information logic. The algorithm is applied to the problem of finding minimal linear fractional transformation models. Statistical performance analysis of the algorithm reveals that it consistently outperforms currently available algorithms.

Validation of Flight Control Law Based on LFT and Structured Singular Value

AbstractThe main goal of flight clearance is to use the structured singular value (μ) analysis of the flight control system when some parameters in the system vary in a certain range. As theμ-analysis can only be done on a linear fractional transformation (LFT) model, the first step of flight clearance is to generate the LFT model of the flight control system. In this paper, based on the introduction of basic theory of LFT andμ-analysis, an X-fighter is chosen as an example to utilize the proposed methods. In order to realize the flight clearance process automatically, a novel software package based on MATLAB programming language is developed. The results of simulation experiments validate the feasibility and effectiveness of the novel methods proposed in this paper.