Physical models of generic aircraft-dynamics are usually represented as parametric, nonlinear-differential equations. Proofing robust stability and performance for such models for the whole flight envelope and for all possible parameter variations is a very time consuming task. In industry this is usually done by exhaustive gridding approaches, where it is still not guaranteed to find a global worst case parameter combination. One way to avoid this problem is to apply Linear Fractional Transformation (LFT) based analysis and controller synthesis techniques. The basic steps required for the application of these methods are sketched in the figure above and are briefly described in the following.

**Generation of a linear parametric state-space system**

A prerequisite for the application of LFT-based methods is that the nonlinear model is approximated by a linear parametric state-space system usually denoted as linear parameter varying (LPV) model. At our institute we developed two different approaches to generate LPV models from a nonlinear model:

1) One way requires aircraft model implementations in MODELICA and the usage of the object-oriented modelling environment Dymola. The key aspect of modelling with Dymola/MODELICA is the possibility to obtain nonlinear dynamic models which exhibit explicit parametric dependencies. For this approach we developed several tools [7,8,9]:

- the DLR Flight Dynamics Library for object-oriented modelling to generate particular aircraft models (within Dymola)
- tools for numerical computations to determine equilibrium points (efficient trimming within SIMULINK)
- tools for symbolic computations to perform system linearizations (within MAPLE)

2) Our second approach is based on a very efficient approximation of a grid of linear state-space systems [1,2]. In many cases the aircraft models are only given in linearized form at a specified set of grid points (e.g., discrete set of mass cases for an aeroelastic aircraft model or aerodynamic data given in large tables or as neural networks). For these models we developed tools that generate highly accurate LPV-models such that the transformation to LFT-form can be conducted in an optimal way, i.e. the order/complexity of the resulting LFT-models is low [3].

# Generation of LFT-representations using the LFR-Toolbox

The LFR-toolbox is a MATLAB toolbox to build LFT-representations of uncertain system models. LFT-representations can be obtained either directly from symbolic models or via an object oriented manipulation of LFT-objects (overloaded MATLAB functions for addition/subtraction, multiplication, inversion, column/row concatenation). The new version of the LFR-toolbox [5] includes recent developments and enhancements, which are mainly focused to improve the capabilities for low order LFT-modelling.

## Symbolic pre-processing

The role of symbolic pre-processing [4] of multivariate rational models is to convert individual elements, entire rows/columns or even the whole symbolic model/matrix to special decomposed forms, which immediately allow to obtain low order LFT-representations. For the decomposition of a single rational matrix element the Horner evaluation scheme, the “optimal operation count” evaluation and conversion to partial fraction or continued-fraction forms are supported by the LFR-toolbox. One of the most promising new techniques is the variable splitting based factorization technique [4], which can be applied in combination with an enhanced implementation of the structured tree-decomposition technique [4]. For efficiency, most of these methods are directly implemented in MAPLE and called from the user friendly environment of MATLAB via the Extended Symbolic Toolbox. A conversion of these tools to the new symbolic MuPad kernel of Matlab will be finished soon.

## Object-Oriented LFT-Realization

The new version of the LFR-toolbox relies on a special form of the generalized LFT-representation [6] supporting also constant blocks in the feedback matrix Delta. This allows realizing arbitrary rational expressions in LFT-form and circumvents the problem that for the object-oriented approach rational expressions like 1/p had to be symbolically normalized before realizing the LFT-representation. This improvement generally leads to LFT-representations of lower order. Furthermore, the new LFT-object definition in the LFR-toolbox is transparent, user friendly and supports several types of uncertainties ensuring full compatibility to the Robust Control Toolbox of MATLAB.

## Numerical Order Reduction

The LFR-toolbox provides several order reduction tools for exact and approximate order reduction. To achieve efficiency of computation, numerical robustness and a high accuracy of results, the order reduction computational kernel of the LFR-Toolbox is formed by mex-functions based on powerful Fortran routines from the LAPACK-based public domain control library SLICOT. For exact order reduction, one may choose between repeated 1-d and n-d Kalman decomposition based order reduction.A collection of model reduction tools, covering the balanced truncation, singular perturbation approximation and Hankel-norm approximation approaches, can be employed for approximate 1-d order reduction.

The combination of all these features – symbolic preprocessing, generalized LFT-representation, numerical efficient order reduction – allows to obtain low order LFT-representations for parametric uncertain models.

## Related Publications

[1] Pfifer H., Hecker S., Michalka G.

LFT-based stability analysis of a generic guided missile. Proc. of AIAA Guidance, Navigation and Control Conference, Chicago, Illinois, USA, 2009, submitted

[2] Pfifer H., Hecker S.

Generation of optimal linear parametric models for LFT-based robust stability analysis and control design. Proc. of IEEE Conference on Decision and Control, Cancun, Mexico, 2009

[3] Hecker S.

Improved mu-analysis results by using low order uncertainty modelling techniques. AIAA Journal of Guidance, Control and Dynamics, Vol. 31, No. 4, 2008

[4] Hecker, S., Varga, A,

Symbolic manipulation techniques for low order LFT-based parametric uncertainty modelling. International Journal of Control, Vol. 79, No. 11, pp. 1485-1494, 2006

[5] Hecker, S., Varga, A., Magni, J. F.

Enhanced LFR-Toolbox for MATLAB. Aerospace Science and Technology (AST), 8, 2004.

[6] Hecker, S., Varga, A.

Generalized LFT-based Representation of Parametric Uncertain Models. European Journal of Control, Vol. 10, No. 4, 2004.

[7] Varga, A., Looye, G., Moormann, D., Grübel, G.:

Automated generation of LFT-based parametric uncertainty descriptions from generic aircraft models. Mathematical and Computer Modelling of Dynamical Systems, vol. 4, pp. 249-274, 1998.

[8] Varga, A., Looye, G.:

Symbolic and numerical software tools for LFT-based low order uncertainty modeling. Proc. of IEEE International Symposium on Computer Aided Control System Design, CACSD'99, Kohala Coast, Hawaii, pp. 1-6, 1999.

[9] Varga, A.:

Computational challenges in flight control design. Proc. of 3rd IEEE Workshop on Computer-Intensive Methods in Control and Data Processing, Prague, pp. 175-186, 1998.