Harald Pfifer

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.

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.

Modal Matching for LPV Model Reduction of Aeroservoelastic Vehicles

A model order reduction method is proposed for models of aeroservoelastic vehicles in the linear parameter-varying (LPV) systems framework, based on state space interpolation of modal forms. The dynamic order of such models is usually too large for control synthesis and implementation since they combine rigid body dynamics, structural dynamics and unsteady aerodynamics. Thus, model order reduction is necessary. For linear timeinvariant (LTI) models, order reduction is often based on balanced realizations. For LPV models, this requires the solution of a large set of linear matrix inequalities (LMIs), leading to numerical issues and high computational cost. The proposed approach is to use well developed and numerically stable LTI techniques for reducing the LPV model locally and then to transform the resulting collection of systems into a consistent modal representation suitable for interpolation. The method is demonstrated on an LPV model of the body freedom flutter vehicle, reducing the number of states from 148 to 15. The accuracy of the reduced order model (ROM) is confirmed by evaluating the ν-gap metric with respect to the full order model and by comparison to another ROM obtained by state-of-the-art LPV balanced truncation techniques.

Robust Control Design for Active Flutter Suppression

Flutter is an unstable oscillation caused by the interaction of aerodynamics and structural dynamics. It can lead to catastrophic failure and therefore must be strictly avoided. Weight reduction and aerodynamically efficient high aspect ratio wing design reduce structural stiffness and thus reduce flutter speed. Consequently, the use of active control systems to counter these adverse aeroservoelastic effects becomes an increasingly important aspect for future flight control systems. The paper describes the process of designing a controller for active flutter suppression on a small, flexible unmanned aircraft. It starts from a greybox model and highlights the importance of individual components such as actuators and computation devices. A systematic design procedure for an H∞-norm optimal controller that increases structural damping and suppresses flutter is then developed. A second key contribution is the development of thorough robustness tests for clearance in the absence of a high-fidelity nonlinear model.

Structural model identification of a small flexible aircraft

The system identification of a light-weight, high-aspect ratio wing is presented. Experimental data is obtained from a ground vibration test. The input signals are sine sweep wave forces and the outputs are the corresponding acceleration responses of the aircraft. Subspace algorithms are used to estimate a state-space model of the aircraft. Minimization of the model prediction error is performed to fit the frequency response data. As result, the estimated model identifies six structural modes between 5 Hz and 30 Hz.

Robust performance analysis: A review of techniques for dealing with infinite dimensional LMIs

This paper compares three techniques for dealing with infinite dimensional linear matrix inequalities (LMIs) for robust performance analysis: the gridding based approximation, the polytopic relaxation and the linear fractional representation based relaxation. The latter draws on the Full Block S-Procedure with different types of multipliers. All three techniques are applied in two benchmark studies at the example of an aeroelastic system. The studies are backed up by results from the Robust Control Toolbox for Matlab.

Updating a finite element based structural model of a small flexible aircraft

The generation of a finite element based structural model of a small, flexible unmanned aircraft is presented. The paper focuses on obtaining a simple model suitable for control design based on a two step procedure. In an initial step, static and dynamic tests of the wings are conducted. These experiments give first estimates of the material properties (e.g., stiffness) of the aircraft. A finite element model consisting of simple beam elements is constructed based on these first estimates. In the next step, the modal data of the aircraft is extracted from a ground vibration test. The initial finite element model is then updated using this modal data. An optimization problem is proposed to minimize the difference in the modal properties, i.e. the modal frequencies and mode shapes, between the model and the experimental data. The free parameters of the optimization are chosen based on physical insights of the system. The resulting finite element model closely matches the experimental data both in terms of modal properties as well as input/output behavior.

Unsteady Aerodynamics Modeling for a Flexible Unmanned Air Vehicle

An unsteady aerodynamics model of a flexible unmanned air vehicle is presented in this paper. The unsteady aerodynamics results from structural vibrations of the flexible airframe which affect the airflow around it. The doublet lattice method, which is a potential flow based panel method, is used for obtaining the basic unsteady aerodynamics model. It gives the pressure distribution on a lifting surface harmonically oscillating in steady flow. The basic concepts, underlying assumptions and approximations of the method are discussed. The extensive post processing which is done for the model obtained from the doublet lattice method is also described. The aerodynamic model is first transformed into suitable coordinates to account for structural vibrations effects. A rational function fitting is then carried out to obtain the final model which is suitable for time domain analysis. We use these methods to obtain an unsteady aerodynamics model of the Body Freedom Flutter vehicle, a flexible test bed aircraft. The results of this application are discussed. The final model is used for developing a nonlinear simulation for the flexible aircraft which is capable of simulating phenomena such as flutter and is important for integrated control law synthesis for the aircraft. The software for DLM as well as the post processing tools are made available as an open source research tool for aeroelastic systems research.

System Identification of a Small Flexible Aircraft - Invited

This paper describes system identification of a light weight, high aspect-ratio flying-wing type aircraft. For this aircraft, complex flexible structure poses challenges in correctly identifying and validating the aircraft structural modes. Also, structural deformation coupled with unsteady aerodynamics in flight makes aerodynamic characterization difficult. The aircraft is identified in flight with inputs carefully selected to excite the dynamic modes of interest. In this study, the lightly damped structural modes are identified based on spectral analysis. The aerodynamic parameters are estimated using prediction error methods. The results of the estimation are validated on data at different flight conditions, as well as by comparison to subspace based black box models. A focus of the paper lies on providing useful insights and lessons learned during the flight test campaign.

Applied LPV control exploiting the separation principle for the single axis positioning of an industrial manipulator

This paper describes the application of linear parameter varying (LPV) control design to the positioning control of the first axis of an industrial manipulator. The manipulator is modeled as a single flexible joint robot. The parameter dependency of the manipulator dynamics results from movements of the other axes. They determine the inertia that has to be moved by the first axis. The controller is designed via a mixed sensitivity weighting scheme for LPV control. Structural properties of the weighting scheme allow the application of the separation principle. By splitting the synthesis problem into an observer and a state feedback problem, the complexity of the convex optimization problem which has to be solved is greatly reduced. A multistage parameter optimization is employed to tune the control system. Finally, the design is compared to an output feedback PID and an ad-hoc gain-scheduled observer based PI state feedback controller on a real-time testbed with a KUKA standard industrial manipulator.