Linear Fractional Transformation Research Articles

Identification, Uncertain Modelling, and Robust Control of Embedded Systems

This paper presents a methodology embodying identification procedures, uncertain modeling, and robust control design of embedded multivariable control systems. Concerning the identification, this methodology involved the determination of probabilistic uncertainty bounds for multivariable plants based on the black box or gray box identification. The bounds obtained were used in the derivation of an uncertain model in the form of upper Linear Fractional Transformation (LFT). This model was used in the robust control design implementing μ-synthesis. The problems arising on the different design stages were illustrated by an example presenting the embedded robust control of a two-input two-output analog model. The plant was identified by using black box and gray box identification methods that produced the necessary information to develop the corresponding uncertainty models. Two discrete-time robust controllers relevant to the two types of identification were designed and embedded in the physical system. Simulation results for the embedded closed-loop system and experimental results obtained by using the robust controllers were compared.

Nonlinear H-Infinity Model Reference Controller Design

In this work, a nonlinear and robust controller is proposed for nonlinear uncertain systems by consolidating the H-infinity state feedback control with the model reference control approach. The proposed controller is motivated to achieve a satisfactory transient response by inserting a reference model. The upper and lower linear fractional transformations are constructed in order to extract the structured parametric uncertainties. The controller design procedure is performed based on the small gain theorem. Next, the proposed controller gain matrix is calculated by solving the H-infinity algebraic Riccati equation. The Lyapunov stability analysis is used to show the stability properties of the controlled system. To show the effectiveness of the proposed controller, two types of nonlinear uncertain systems are presented as case studies. Finally, the results demonstrate that the proposed controller is powerful and efficient in compensating the nonlinear uncertain systems and in imposing the asymptotic tracking between the system states and the reference model states.

N-Regular functions in quaternionic analysis

In this paper, we study left and right [Formula: see text]-regular functions that originally were introduced in [I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics II, accepted in Adv. Theor. Math. Phys]. When [Formula: see text], these functions are the usual quaternionic left and right regular functions. We show that [Formula: see text]-regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy–Fueter type reproducing formulas. Arguably, these Cauchy–Fueter type reproducing formulas for [Formula: see text]-regular functions are quaternionic analogues of Cauchy’s integral formula for the [Formula: see text]th-order pole [Formula: see text] We also find two expansions of the Cauchy–Fueter kernel for [Formula: see text]-regular functions in terms of certain basis functions, we give an analogue of Laurent series expansion for [Formula: see text]-regular functions, we construct an invariant pairing between left and right [Formula: see text]-regular functions and we describe the irreducible representations associated to the spaces of left and right [Formula: see text]-regular functions of the conformal group and its Lie algebra.

Optimized fault detection using bond graph in linear fractional transformation form

This article deals with the optimal robust fault detection problem using the bond graph in its linear fractional transformation form. Generally, this form of the bond graph allows the generation of two perfectly separate analytical redundancy relations, that are used as residual and threshold. However, the uncertainty calculation method gives overestimated thresholds. This may, for instance, lead to undetectable faults. Therefore, enhancing the robustness of fault detection and isolation algorithms is of utmost importance in designing a bond graph–based fault detection system. The main idea of this article is to develop optimized thresholds to ensure an optimal detection, otherwise this article proposes a method to detect tiny magnitude faults concerning parameter’s uncertainties. This work considers the issue of optimal fault detection as an optimization problem of the gap between the residuals and its threshold. New uncertainty values will be calculated in a way that these estimated parameters ensure the desired optimized gap between residuals and thresholds. These estimated uncertainty values will be used to generate optimized adaptive thresholds. Through these thresholds, we increase the sensitivity of the residuals to tiny magnitude faults, and we ensure an optimal and early detection.

A new approach based on the discriminant system of polynomial for robust stability and stabilization of two-dimensional systems

In this paper, we present necessary and sufficient stability and robust stability conditions for two-dimensional (2D) systems described by the Fornasini-Marchesini (FM) second model in terms of the discriminant systems of polynomial. This paper simplifies the traditional method of stability into a tractable method by the fractional linear transformation (FLT). More specifically, we reduce the stability analysis to a easy issue whether some polynomials are positive definite. Then we use the same idea to consider the stabilization and robust stabilization issues. Finally, the effectiveness of the proposed results is demonstrated by a practical example and two numerical examples.

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.

Model of the hysteresis loop of soft-magnetic amorphous alloys with the usage of a modified linear fractional function

Model of the hysteresis loop of soft-magnetic amorphous alloys with the usage of a modified linear fractional function

Robust Distributed Filters Design for Discrete-Time Spatially Interconnected Systems with LFT Uncertainties

In this paper, we investigate discrete-time uncertain spatially interconnected systems (USISs), where uncertainties are modeled by linear fractional transformation (LFT). First, the well-posedness, quadratic stability and contractiveness of discrete-time USISs are introduced. Second, a sufficient condition is proposed to guarantee that discrete-time USISs are well-posed, quadratically stable and contractive. Then, a more tractable condition is derived to check the well-posedness, quadratic stability and contractiveness of discrete-time USISs via a modified bilinear transformation. Besides, the robust distributed filters which inherit the structure of the plants are designed. A sufficient and necessary condition is presented to guarantee the existence of the robust distributed filters. Finally, a vehicle platoon model demonstrates the effectiveness of the proposed scheme.

Compound Fault Diagnosis and Sequential Prognosis for Electric Scooter with Uncertainties

This paper addresses diagnosis and prognosis problems for an electric scooter subjected to parameter uncertainties and compound faults (i.e., permanent fault and intermittent fault with non-monotonic degradation). First, the diagnostic bond graph in linear fractional transformation form is used to model the uncertain electric scooter and derive the analytical redundancy relations incorporating the nominal part and uncertain part, based on which the adaptive thresholds for robust fault detection and the fault signature matrix for fault isolation can be obtained. Second, an adaptive enhanced unscented Kalman filter is proposed to identify the fault magnitudes and distinguish the fault types where an auxiliary detector is introduced to capture the appearing and disappearing moments of intermittent fault. Third, a dynamic model with usage dependent degradation coefficient is developed to describe the degradation process of intermittent fault under various usage conditions. Due to the variation of degradation coefficient and the presence of non-monotonic degradation characteristic under some usage conditions, a sequential prognosis method is proposed where the reactivation of the prognoser is governed by the reactivation events. Finally, the proposed methods are validated by experiment results.

New family of triangular norms for decreasing generators in the form of a logarithm of a linear fractional function

The purpose of this paper is to establish restrictions on the parameters under which a function in the form of a natural logarithm of a linear fractional function is a decreasing generator. In addition to the general case, all special cases of generators of this type are considered. A triangular norm is constructed for each generator. In fact, a new parametric class of decreasing generators that generalizes and systematizes known generators is proposed. Such generators generate triangular norms represented by rational functions. The results of this paper can be used to construct aggregation operations.

Virtual refrigerant charge sensor for booster refrigeration systems

Abstract In this paper a virtual refrigerant charge sensor (VRCS) for a booster refrigeration system with a receiver unit is developed and proposed. The VRCS estimates the liquid charge in the receiver unit by utilizing a combined mass and energy balance equation for mapping the change in the receiver unit’s time constant to the liquid volume in the receiver unit. The time constant for the receiver unit is found by utilizing the commonly available data in an online subspace identification method. The parameters estimated by the online subspace identification method are recast in a linear fractional transformation structure to identify parameter changes due to liquid volume change. Hence, the VRCS functions in a plug and play framework where only the initial refrigerant charge in percentage is required to be known. Results from utilizing the VRCS on experimental data from a charge test conducted on a test setup located at Danfoss A/S are shown, where the VRCS shows an accurate estimation of the liquid charge in the receiver. The results from the VRCS can be employed for: early prediction of low charge in the system, early detection of a too large leakage rate, estimation of the current charge on a daily basis, and estimation of the amount of charge lost.

Invariant densities for composed piecewise fractional linear maps

The starting point for this paper was the following problem: If we know the invariant densities for two maps S and T can we say something about the invariant density of the map $$U = T \circ S$$ ? This problem seems not to be touched on within ergodic theory. In this note we look at the inverse problem. Let the invariant density for $$U = T \circ S$$ be known. What can we say about invariant densities for S and T? We discuss a simple model, namely the class of all piecewise fractional linear maps with two branches on the unit interval [0, 1].

Affine Dependence of Network Observability/Controllability on Its Subsystem Parameters and Connections

This paper investigates observability/controllability of a networked dynamic system (NDS) in which system matrices of its subsystems are expressed through linear fractional transformations (LFT). Some relations have been obtained between this NDS and descriptor systems about their observability/controllability. A necessary and sufficient condition is established with the associated matrices depending affinely on subsystem parameters/connections. An attractive property of this condition is that all the required calculations are performed independently on each individual subsystem. Except well-posedness, not any other conditions are asked for subsystem parameters/connections. This is in sharp contrast to recent results on structural observability/controllability which is proven to be NP hard. Some characteristics are established for a subsystem which are helpful in constructing an observable/controllable NDS. It has been made clear that subsystems with an input matrix of full column rank are helpful in constructing an observable NDS, while subsystems with an output matrix of full row rank are helpful in constructing a controllable NDS. These results are extended to an NDS with descriptor form subsystems. As a byproduct, the full normal rank condition of previous works on network observability/controllability has been completely removed. On the other hand, satisfaction of this condition is shown to be appreciative in building an observable/controllability NDS.

Simple solutions of the Burgers and Hopf equations

We describe all solutions of the Burgers equation of analytic complexity not exceeding . It turns out that all such solutions fall into four families of dimensions not exceeding that are represented by elementary functions. An example of a family of solutions of the Burgers equation of complexity is given. A similar problem is also solved for the Hopf equation. It turns out that all solutions to the Hopf equation of complexity form a two-parameter family of fractional-linear functions which coincides with one of the families of solutions of the Burgers equation.

Symmetries of slice monogenic functions

In this paper we consider the symmetry behavior of slice monogenic functions under Möbius transformations. We describe the group under which slice monogenic functions are taken into slice monogenic functions. We prove a transformation formula for composing slice monogenic functions with Möbius transformations and describe their conformal invariance. Finally, we explain two construction methods to obtain automorphic forms in the framework of this function class. We round off by presenting a precise algebraic characterization of the subset of slice monogenic linear fractional transformations within the set of general Möbius transformations.

Advanced aeroelastic robust stability analysis with structural uncertainties

Robust flutter analysis described in this paper is based on the robust control theory framework. Therefore, a time-domain linear fractional transformation representation of the perturbed aeroelastic system is modeled. Then, the robust stability is analyzed by means of the structured singular value mu, which is defined as an alternative measure of robustness. Robust flutter analysis deals with aeroelastic (or aeroservoelastic) stability analysis taking structural dynamics, aerodynamics and/or unmodeled system dynamics uncertainties into account. Flutter is a well-known dynamic aeroelastic instability phenomenon caused by an interaction between structural vibrations and unsteady aerodynamic forces, whereby the level of vibration may trigger large amplitudes, eventually leading to catastrophic failure of the structure. The primary motivation of the robust flutter analysis is that this method allows the computation of the worst-case flutter velocity which can support, for example, the flight test program by a valuable robust flutter boundary. This paper addresses the issue of an approach for aeroelastic robust stability analysis with structural uncertainties with respect to physical symmetric and asymmetric stiffness perturbations on the wing structure by means of tuning beams.

A note on N-polynomials over finite fields

A simple type of Cohen's transformation consists of a polynomial and a linear fractional transformation. We study the effectiveness of Cohen transformation to find N-polynomials over finite fields.

Functional models for entire symmetric operators in rigged de Branges Pontryagin spaces

The theory of operator extensions in rigged Pontryagin spaces is used to develop two functional models for closed symmetric entire operators S with finite deficiency indices (p,p) acting in a separable Pontryagin space K. In the first functional model it is shown that every such operator S is unitarily equivalent to the multiplication operator in a de Branges-Pontryagin space B(E) of p×1 vector valued entire functions. The second functional model is used to parametrize a class of compressed resolvents of extensions S˜ of S in terms of the range of a linear fractional transformation that is associated with the model. This approach is independent of the methods used by Krein and Langer to parameterize a related class of extensions.

Nonconservative LMI techniques for robust stabilisation of spatially interconnected systems

This paper is concerned with the robust stabilisation problem of spatially interconnected systems (SISs) with linear fractional transformation (LFT) representation of uncertainties. A robust stabilisability function for SISs is built with the aid of Routh–Hurwitz criterion. By solving two semidefinite programs (SDPs) with sums-of-squares (SOS) polynomial constraints, necessary and sufficient conditions for establishing the existence of robust stabilising controllers are derived, implying that the derived robust stabilisation results are nonconservative. Moreover, a numerically tractable algorithm is proposed to obtain square matrix representation (SMR) of real polynomials, which enables the SOS constraints to be equivalently checked via linear matrix inequalities (LMIs). A simulation example is finally included to demonstrate the efficiency of the proposed method.

An extension of the structured singular value to nonlinear systems with application to robust flutter analysis

The paper discusses an extension of mu (or structured singular value), a well-established technique from robust control for the study of linear systems subject to structured uncertainty, to nonlinear polynomial problems. Robustness is a multifaceted concept in the nonlinear context, and in this work the point of view of bifurcation theory is assumed. The latter is concerned with the study of qualitative changes of the steady-state solutions of a nonlinear system, so-called bifurcations. The practical goal motivating the work is to assess the effect of modeling uncertainties on flutter, a dynamic instability prompted by an adverse coupling between aerodynamic, elastic, and inertial forces, when considering the system as nonlinear. Specifically, the onset of flutter in nonlinear systems is generally associated with limit cycle oscillations emanating from a Hopf bifurcation point. Leveraging mu and its complementary modeling paradigm, namely linear fractional transformation, this work proposes an approach to compute margins to the occurrence of Hopf bifurcations for uncertain nonlinear systems. An application to the typical section case study with linear unsteady aerodynamic and hardening nonlinearities in the structural parameters will be presented to demonstrate the applicability of the approach.