LPV/LFT Modeling and its Application in Aerospace

Kurzfassung

In this thesis, methods for rapidly developing LPV/LFT models for aerospace applications are proposed. The capabilities of these methods are demonstrated on the example of two different nonlinear missile models of industrial complexity. In addition, possible applications for both LPV models are shown. For one model a modern LPV controller is synthesized based on the induced L2-norm framework. For the other model the robust stability of the closed loop with a given nonlinear control system is assessed in face of time varying parameters. Two general algorithms for generating LFT models have been developed, which can be applied to arbitrary nonlinear systems, as long as the system behavior can be accurately described/approximated with polynomial or rational parametric state-space systems. The bases for efficient LFT based methods like μ-analysis are lean and accurate models in LFR form. Hence, an optimization problem is derived to find an accurate parametric approximation of a nonlinear system, which provides an optimal structure in terms of least order LFR generation. In addition to this complex optimization problem, a convex relaxation is proposed based on l1-regularized least squares. It is very time efficient and can compute an almost Pareto front between the LFR order and accuracy. Furthermore, state-of-the-art algorithms proposed in [Hec07] are used for the transformation of the LPV models into LFRs. In addition, a general procedure to approximate a parametric LFR with a reduced order one including unstructured uncertainty is developed. A numerical multidimensional order reduction method is applied to a parametric LFR and the reduction error is overbounded by an unstructured uncertainty. It is shown that a tradeoff between the complexity and the quality of an LFR model can be achieved by using such a mixed parametric/unstructured modeling approach. The algorithm is applied to a benchmark model of the longitudinal motion of a missile. For this benchmark model a μ-controller synthesis problem is considered. The quality of different mixed models is studied and compared to a fully unstructured approximation method. Finally, it is possible to design a μ-controller with almost the same performance as the full order LFR model with a significantly simpler model. The capabilities of the developed LPV/LFT modeling procedure are demonstrated on the example of two highly nonlinear missile models of industrial complexity. The nonlinear dynamics of both missiles are cast into a quasi-LPV models making use of the function substitution technique proposed in [Tan97]. An enhancement of the approach in [Tan97] is used which is ideally suited for aerospace application. For the first example the LFT generation method described in Chapter 3 is applied. The closed loop LFT model of this missile in conjunction with a given nonlinear dynamic inversion based controller is later used in Chapter 4 to show the applicability of modern robust stability analysis tools. In the second example a grid-based LPV system of a missile model is obtained, which serves as the basis for the controller design in in Chapter 5. Both quasi-LPV models are validated by comparison with the full nonlinear systems using Monte-Carlo simulations. The advantage of the function substitution technique over a more traditional Jacobian linearization based LPV model is shown. For one of the missile benchmarks an LPV controller is designed, which is based on induced L2-norm performance specifications and on obtaining a parameter dependent Lyapunov function [Wu95]. A special focus is set on developing numerical reliable synthesis functions, e.g. by optimizing the condition number of the Lyapunov matrix. A multi-objective parameter optimization is employed to improve the performance of the control system. Finally, the resulting controller is compared to a classical controller based on eigenstructure assignment. The results show that the LPV controller provides better robustness, while achieving the same level of performance. A sophisticated robust stability analysis framework based on the Full Block S-Procedure, as introduced in [Det01], has been applied to a highly complex closed loop LFT model. Unlike previous work in this area, see for instance [GMPT09], robust stability in face of time varying parameters with bounded parameter variation rate is studied. In addition, a new approach on computing a stability radius of the LFR is proposed, providing a better idea on how close to instability the system is. The used model contains a large number of uncertain parameters, allowing judgment of the applicability of the method on realistic industrial applications. The conservatism of the approach is estimated by a complimentary nonconvex worst case optimization. The novel stability radius computation performs well in comparison to established methods like the worst case optimization or μ-analysis. References [Det01] Marco Dettori. LMI techniques for control with application to a Compact Disc player mechanism. PhD thesis, Technische Universiteit Delft, 2001. [GMPT09] A. Garulli, A. Masi, S. Paoletti, and E. Türkoglu. Clearance of flight control laws via parameter-dependent Lyapunov functions. In Proceedings of IFAC Symposium on Robust Control Design, Haifa, Israel, 2009. [Hec07] S. Hecker. Generation of low order LFT Representations for Robust Control Applications. VDI Verlag, 2007. [Tan97] W. Tan. Applications of linear parameter varying control theory. Master’s thesis, University of California at Berkeley, 1997. [Wu95] F. Wu. Control of Linear Parameter Varying Systems. PhD thesis, University of California, 1995.