Simon Hecker

Generalized LFT-Based Representation of Parametric Uncertain Models

In this paper we introduce a general descriptor-type LFT representation of rational parametric matrices. This generalized representation allows to represent arbitrary rationally dependent multivariate functions in LFT-form. As applications, we develop explicit LFT realizations of the transfer-function matrix of a linear descriptor system whose state space matrices depend rationally on a set of uncertain parameters. The resulting descriptor LFT-based uncertainty models generally have smaller orders than those obtained by using the standard LFT-based modelling approach.

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.

Symbolic manipulation techniques for low order LFT-based parametric uncertainty modelling

Symbolic techniques are very useful in obtaining low order LFT-representations of linear parametric models. The main role of symbolic manipulations is to find, via suitable pre-processing steps, equivalent representations of rationally dependent parametric matrices, which automatically lead to lower order LFT-representations. In this paper we give an overview of symbolic processing methods and we propose some new techniques and several enhancements of existing methods. All proposed methods are implemented in the latest version of the LFR-toolbox and served to illustrate the strengths of symbolic processing in obtaining low order LFT-representations of two challenging parametric model examples.

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.

Enhanced LFR-toolbox for MATLAB

We describe recent developments and enhancements of the LFR-toolbox for MATLAB for building LFT-based uncertainty models. A major development is the new LFT-object definition supporting a large class of uncertainty descriptions: continuous- and discrete-time uncertain models, regular and singular parametric expressions, more general uncertainty blocks (nonlinear, time-varying, etc.). By associating names to uncertainty blocks the reusability of generated LFT-models and the user friendliness of manipulation of LFR-descriptions have been highly increased. Significant enhancements of the computational efficiency and of numerical accuracy have been achieved by employing efficient and numerically robust FORTRAN implementations of order reduction tools via Mex-function interfaces. The new enhancements in conjunction with improved symbolical preprocessing lead generally to a faster generation of LFT-models with significantly lower orders

LPV controller synthesis for a generic missile model

The paper presents an example of a linear, parameter varying (LPV) controller synthesis for a generic nonlinear missile. It is based on obtaining a parameter dependent Ljapunov function, in order to ensure stability of the system. The LPV model of the missile is constructed by means of Jacobian linearization at fixed parameter values. Performance requirements of the control system are specified in an induced L2-norm framework, which is similar to the H∞-synthesis in the linear time invariant (LTI) case. The suitable weighting functions are found using a multi-objective optimization approach. Finally, the LPV controller is assessed by a comparison to a classical PI controller.

SYMBOLIC TECHNIQUES FOR LOW ORDER LFT-MODELLING

Symbolic preprocessing techniques are very useful to obtain low order LFT-representations for parametric models. In this paper we give an overview about existing preprocessing methods and we present new techniques and enhancements of existing methods. All methods are implemented in the new version 2 of the LFR-toolbox and their capabilities are illustrated by a challenging aircraft parametric uncertainty modelling example.

A desktop environment for assessment of fault diagnosis based fault tolerant flight control laws

We present a simulation based software environment conceived to allow an easy assessment of fault diagnosis based fault tolerant control techniques. The new tool is primary intended for the development of advanced flight control applications with fault accommodation abilities, where the requirements for increased autonomy and safety play a premier role.

Robust H ∞ -based vehicle steering control design

Two vehicle steering controllers to improve the yaw dynamics of a mid-size passenger car are designed based on robust H∞ synthesis techniques. The controllers fulfill the desired mixed-sensitivity performance specifications robustly with respect to large uncertainties in the vehicle model parameters for longitudinal speed, road adhesion, mass and moment of inertia. The parametric uncertainties are non-conservatively considered using a minimal-order linear fractional representation (LFR) for the uncertain vehicle model during control design. One approach is based on μ-synthesis, which guarantees robust performance assuming that parameters are uncertain but constant. The second design is based on linear parameter varying (LPV) control design techniques, guaranteeing robust performance also in case of bounded variation rates of the longitudinal speed, which is an important property in real-life situations like μ-split braking. To allow real-time implementation, frequency weighted controller reduction techniques are applied to reduce the order and to eliminate high frequency dynamics of the controllers.

LFT Model Generation via ℓ1-Regularized Least Squares

The paper presents a general approach to approximate a nonlinear system by a linear fractional representation (LFR), which is suitable for LFT-based robust stability analysis and control design. In a first step, the nonlinear system will be transformed into a quasi linear parameter varying (LPV) system. In the second step, the nonlinear dependencies in the quasi-LPV, which are not rational in the parameters, are approximated using polynomial fitting based on l1-regularized least squares. Using this approach an almost Pareto front between the accuracy and complexity of the resulting LFR can be efficiently obtained. The effectiveness of the proposed method is demonstrated by applying it to a nonlinear missile model of industrial complexity.