Linear Fractional Representation Research Articles

H? Control of Discrete-time Uncertain Periodic Systems with Delays

This paper deals with the problem of H1 control for a class of linear discrete-time periodic system with delays. The obtained results are then extended for the time-delay periodic system with Linear Fractional Representation (LFR) uncertainty. Furthermore, linear matrix inequality (LMI)-based su cient conditions for H1 control are established. Two numerical examples are given to illustrate the applicability of the proposed approach.

Robust Stability Analysis of a Linear Time-PeriodicActive Helicopter Rotor

Robust stability of a helicopter hingeless rotormodel used for air resonance alleviation is investigated in this paper. This aeroelastic phenomenon can be described naturally considering the rotor as a whole through a time-periodic change of coordinates called multiblade coordinates transformation. To assess robust stability of the resulting continuous linear time-periodic system, this paper defines a set of uncertainties specific to the helicopter rotor problem. The uncertain system is written in the form of a linear fractional representation and transformed using a frequency-lifting technique. This leads to a time-invariant representation of the uncertain system, the so-called harmonic transfer function. The extension of the standard -analysis to such lifted systems is proposed in the paper. Compared with a similar approach from literature based on an elementary example, the technique requires a moderate increase of the number of inputs and outputs, thus reducing the numerical effort involved in the computation of the -bounds. Themethodology, applied to the uncertain closed-loop helicopter rotor, shows robust stability for the defined set of uncertainty. The results are compared to an analysis performed on the base of a linear time-invariant system to quantify how strongly robust stability bounds are underestimated when the periodicity is not accounted for.

Initial Estimates for the LFR Nonlinear Model Structure via the Best Linear Approximation

In this paper, a novel method is proposed for the identification of a fairly general nonlinear structure called Linear Fractional Representation (LFR), also related to Lur'e type systems. It consists of one static nonlinearity (SNL) connected to the input and the output via a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) block. The nonlinear LFR structure encompasses, e.g., Wiener-Hammerstein and nonlinear feedback models. The procedure starts from 2 state-space models corresponding to the best linear approximation at 2 input variance levels. The MIMO LTI block is estimated by exploiting the approximate structural relationships, taking the state transformations carefully into account. Using the measured input and output, the input and the output of the SNL block are then reconstructed, yielding a nonparametric estimate of the SNL, which is finally converted into a parametric estimate. In the whole procedure, the internal signals, the linear and the nonlinear part need not be known. No stability requirement was imposed on the linear models used. The functional form of the SNL is not needed to find the MIMO LTI block and the nonparametric SNL estimate.The simulation results, supporting the theory, show the superior quality of the obtained initial estimate of the LFR nonlinear model compared to both linear models. This shows that the method is a very promising approach in the field of block-oriented nonlinear modelling.

Identification of a Block-Structured Model with Localised Nonlinearity

This paper considers the identification of a rather general nonlinear time-invariant system, consisting of a Multiple-Input Multiple-Output (MIMO) linear dynamic part and one static nonlinear part. It is sometimes referred to as Linear Fractional Transformation (LFT) or Linear Fractional Representation (LFR). The structure will be called nonlinear LFR and includes many standard block-structured models, such as Wiener, Hammerstein, Wiener-Hammerstein and nonlinear feedback. The identification does assume neither the states, nor the internal signals over the static nonlinearity to be measured. The static nonlinearity (SNL) is assumed to be polynomial. After estimation of a nonlinear state-space model with certain structural properties, the SNL can be separated from the MIMO linear part. Next, the linear system is represented by a combination of four linear dynamic blocks, yielding extra insight. The method is illustrated via an experimental-data example.

Generation of a reduced-order LPV/LFT model from a set of large-scale MIMO LTI flexible aircraft models

In the civilian aviation industry, the aeroelastic behavior of an aircraft is often modeled at frozen flight and mass configurations using high fidelity numerical tools. Unfortunately, the resulting large-scale models cannot be handled in such form by modern analysis and control techniques, which generally require the considered models to be written as low-order Linear Fractional Representations (LFR). In this context, a methodology is described to derive a reduced-order Linear Parameter Varying (LPV) model from a reference set of large-scale Multiple Input Multiple Output (MIMO) Linear Time Invariant (LTI) models describing a given system at frozen configurations. The proposed approach is in two steps. The reference models are first reduced using recent advances in Krylov methods, leading to a set of low-order state-space representations with consistent state vectors. An LPV model is then obtained by polynomial approximation and converted into an LFR of reasonable size. A special effort is made to avoid data overfitting by using as simple as possible approximation formulas. The method is applied to a long-range commercial aircraft model developed in an industrial context: a set of large-scale flexible models linearized at different mass configurations is converted into a single low-order LPV model. More generally, any kind of purely numerical models for which the analytical structure is unknown can be considered.

A New Elementary Operation Approach to Multidimensional Realization and LFR Uncertainty Modeling: The MIMO Case

This paper proposes a new elementary operation approach (EOA) to multidimensional (-D) realization and linear fractional representation (LFR) modeling for multi-input and multi-output (MIMO) -D systems, as an extension of the new EOA proposed for the single-input and single-output (SISO) case by the authors recently. It is shown that, due to the substantial differences between the SISO and MIMO systems, the extension is not straightforward and further significant development is necessary. A matrix relation property among the associated matrices under the augmenting and admissible elementary operations is first revealed. Based on this matrix relation property, the realization problem for the MIMO -D case is formulated as an elementary operation problem of a certain -D polynomial matrix, which makes the extension possible. General constructive procedures are then established for the regular realizations based on the right and left matrix fraction descriptions (MFDs) of a given transfer matrix, respectively, such that one can easily implement this approach by a computer program in, e.g., MATLAB or Maple. Numerical and symbolic examples are provided to illustrate the main ideas and the effectiveness of the proposed approach.

A LPV approach for early fault detection in aircraft control surfaces servo-loops

The work presented in this paper is undertaken within the European FP7 funded ADDSAFE* project. The goal is to propose new fault diagnosis techniques that could significantly help developing environmentally-friendlier aircraft, by optimizing structural load design objectives. In this paper, a Linear Parameter Varying (LPV) model-based fault detection scheme is proposed for robust and early detection of faults in aircraft control surfaces servo-loop. The proposed methodology is based on the H∞/H_ LPV technique proposed in (Henry, 2012), for LPV systems modeled in a linear fractional representation (LFR) fashion. Simulation results using a “high-fidelity” simulator provided by AIRBUS, are presented to demonstrate the potential of the proposed technique.

PWM modeling for attitude control of a launcher during ballistic phase and comparative stability analysis

The present paper exposes a novel description of the effect of a pulse generator on a double integrator representing one axis of rotation for a space vehicle. By considering the difference between the angular velocity obtained with zero-order-hold (ZOH) actuator and the one with a PWM actuator, a perturbation operator expressing the influence of the PWM on the angular speed is defined. Using this configuration in a linear fractional representation (LFR), a stability analysis allows comparing LMI-based and IQC-based stability results for a one degree of freedom rotating body with uncertain PWM and saturation and permits dwelling on improvements to make for further analyses.

Equilibrium Transition Study for a Hybrid MAV

Wind tunnel testing was performed on a VTOL aircraft in order to characterize longitudinal flight behavior during an equilibrium transition between vertical and horizontal flight modes. Trim values for airspeed, pitch, motor speed and elevator position were determined. Data was collected by independently varying the trim parameters, and stability and control derivatives were identified as functions of the trim pitch angle. A linear fractional representation model was then proposed, along with several methods to improve longitudinal control of the aircraft.

Structured fault detection filters for LPV systems modeled in an LFR manner

SUMMARYThis paper investigates the design of robust‡ fault detection and isolation filters for linear parameter‐varying systems modeled in a linear fractional representation fashion. The goal is to obtain structured fault detection filters with enhanced fault transmission H− gain and large H∞ nuisance attenuation. It is shown by means of the scaling matrices technique and the projection lemma that the synthesis of the residual structuring and the filter state‐space matrices can be performed simultaneously using linear matrix inequality optimization techniques. Computational aspects are discussed and it is shown that the proposed solution is structurally well defined. Closed‐loop time simulations demonstrate the efficiency of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.

Generation of Optimal Linear Parametric Models for LFT-Based Robust Stability Analysis and Control Design

We present a general approach to generate a linear parametric state-space model, which approximates a nonlinear system with high accuracy and is optimally suited for linear fractional transformation (LFT) based robust stability analysis and control design. At the beginning a Jacobian-based linearization is applied to generate a set of linearized state-space systems describing the local behavior of the nonlinear plant about the corresponding equilibrium points. These models are then approximated using multivariable polynomial fitting techniques in combination with global optimization. The objective is to find a linear parametric model, which allows the transformation into a linear fractional representation (LFR) of least possible order. A gap metric constraint is included during the optimization in order to guarantee a specified accuracy of the transfer function of the linear parametric model. The effectiveness of the proposed method is demonstrated by applying it to a simple benchmark problem as well as to two industrial applications, one being a nonlinear missile model the other a nonlinear transport aircraft model.

State feedback stabilization of a class of uncertain periodic systems

AbstractThis paper deals with the robust stability and stabilization problem for a class of uncertain linear periodic discrete time system. This class is defined as the class of systems subject, at first, to LFR (Linear fractional representation)-based uncertainty and then to LFR and polytopic uncertainties together. The LFR-based uncertainty encompasses the case of norm bounded uncertainty as a special case. The use of a full block S-procedure allows to reformulate the stability LMI conditions of an uncertain system through a dual form of quadratic Lyapunov inequality in equivalent efficient LMI conditions that can be easily checked by any LMI solver.The obtained conditions are then used to propose a periodic state feedback controller ensuring the robust stabilizability of the obtained closed loop system. The use of the PDLF (parameter-dependent Lyapunov functions) technique enables one to take account in the same time of polytopic uncertainties. Numerical examples are provided to illustrate the proposed results.

Identification of Piecewise Affine LFR Models of Interconnected Systems

Identification of interconnected systems is a challenging problem in which it is crucial to exploit the available knowledge about the interconnection structure. This paper addresses the identification of discrete-time dynamical models in linear fractional representation form, composed by the interconnection of a linear time-invariant block and a static nonlinearity. An iterative identification approach is adopted, which alternates the estimation of the linear and the nonlinear components. Standard identification techniques are applied to the linear part, whereas recently developed piecewise affine identification techniques are employed for modeling the static nonlinearity. The proposed method takes advantage of the interconnection structure to identify models which are more accurate and often much simpler than those obtained when applying black-box piecewise affine identification techniques to the overall system. This is demonstrated through the application of the adopted identification algorithm to the silverbox problem, a popular real-life benchmark in nonlinear system identification.

Robust Stability of Time-Varying Uncertain Systems With Rational Dependence on the Uncertainty

Robust stability of time-varying uncertain systems is a key problem in automatic control. This note considers the case of linear systems with rational dependence on an uncertain time-varying vector constrained in a polytope, which is typically addressed in the literature by using the linear fractional representation (LFR). A novel sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI) feasibility test by exploiting homogeneous polynomial Lyapunov functions, the square matrix representation and an extended version of Polya's theorem which considers structured matrix polynomials. It is shown that this condition is also necessary for second-order systems, and that this condition is less conservative than existing LMI conditions based on the LFR for any order.

A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: the SISO case

In this paper, a new elementary operation approach is proposed to the problem of multidimensional (n-D) Roesser state-space model realization or linear fractional representation (LFR) uncertainty modeling. The new approach can overcome the main difficulties encountered by Galkowski's approach and always generates a standard or regular realization for a given causal n-D transfer function. In particular, the n-D realization problem is formulated as an elementary operation problem of a certain n-D polynomial matrix in a way totally different from Galkowski's approach so that the singularity problem can be completely avoided. A general constructive realization procedure and an algorithm for evaluating the realization order associated with this procedure will be proposed, which can be easily implemented by symbolic software in, e.g., MATLAB or Maple. Some further techniques for a realization with lower order will also be shown. Symbolic and numerical examples will be presented throughout the paper to illustrate the basic ideas as well as the effectiveness of the proposed approach.

On-ground aircraft control design using a parameter-varying anti-windup approach

As an original alternative to dynamic inversion techniques, a non-standard anti-windup control strategy is developed in this paper in order to improve the on-ground control system of a civilian aircraft. Using a linear fractional representation (LFR) of the aircraft in combination with an original approximation of the nonlinear ground forces by saturation-type nonlinearities, the proposed design method delivers low-order and robust controllers, which are automatically adapted to the runway state and to the aircraft longitudinal velocity. The efficiency of the design scheme is assessed by several nonlinear simulations.

Redheffer Representations and Relaxed Commutant Lifting

It is well known that the solutions of a (relaxed) commutant lifting problem can be described via a linear fractional representation of the Redheffer type. The coefficients of such Redheffer representations are analytic operator-valued functions defined on the unit disc D of the complex plane. In this paper we consider the converse question. Given a Redheffer representation, necessary and sufficient conditions on the coefficients are obtained guaranteeing the representation to appear in the description of the solutions to some relaxed commutant lifting problem. In addition, a result concerning a form of non-uniqueness appearing in the Redheffer representations under consideration and an harmonic maximal principle, generalizing a result of A. Biswas, are proved. The latter two results can be stated both on the relaxed commutant lifting as well as on on the Redheffer representation level.

Robust ℋ2 filtering for LTI systems with linear fractional representation

This article introduces a new approach to ℋ2 robust filtering design for continuous and discrete-time LTI systems subjected to linear fractional parameter uncertainty representation. The novelty consists on the determination of a performance certificate in terms of the gap between lower and upper bounds of a minimax programming problem which defines the optimal robust filter and the associated equilibrium cost. The calculations are performed through convex programming methods, applying slack variables, known as multipliers, to handle the fractional dependence of the plant transfer function with respect to the parameter uncertainty. The theory is illustrated by means of an example borrowed from the literature and a practical application involving the design of a robust filter for the load voltage estimation on a transmission line with a stub feeding an unknown resistive load.

Bitangential Interpolation in Generalized Schur Classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in Derkach and Dym (On linear fractional transformations associated with generalized J-inner matrix functions. Integ Eq Oper Th (2009, in press) arXiv:0901.0193).

Generation of flexible aircraft LFT models for robustness analysis

In the aeronautical industry, the robustness properties of an aircraft are usually assessed using intensive and time-consuming simulations. Fortunately, several optimization techniques can be implemented and applied for clearance of flight control laws, in order to improve the efficiency and reliability of the certification process. Some of them, such as Lyapunov-based analysis and μ-analysis, require the considered models to be written as Linear Fractional Representations (LFR), which is unfortunately not the case of the aeroelastic models available in an industrial context. A whole methodology is thus proposed in this paper to convert a set of numerical flexible aircraft models into a suitable LFR which depends on the aircraft configuration and the flight conditions. This is a challenging issue, since the size of the initial models is very large and the state vector does not have the same physical meaning for the whole model set. Nevertheless, an LFR is obtained, which is representative of the aircraft behavior in the sense that its eigenvalues and frequency responses almost exactly match those of the initial flexible models. Moreover, its complexity proves compatible with the use of robustness analysis tools. Numerical results indeed show that the considered clearance problem is computationally tractable., clearance of flight control laws, LFT modeling and reduction techniques, aeroelastic models, robustness analysis.