Grid- and transformation-free model order reduction for linear parameter varying systems
Final Report Abstract
Over the last three decades, linear parameter-varying (LPV) systems have developed into a powerful framework that enables the extension of efficient analysis and synthesis tools for linear timeinvariant (LTI) systems, to a wide class of time-varying and nonlinear systems. However, since analysis and synthesis conditions have to be evaluated on a grid over the admissible parameter range, computational complexity and thus model reduction becomes an issue. The most powerful model reduction techniques for LTI systems, which are based on balancing a state space model, have therefore also been extended to LPV models. The methods for LPV model reduction that are available so far, are based on solving the model reduction problem on a grid over the parameter space, which for large systems might render the reduction problem itself intractable. The idea for this project was inspired by the fact that gridding the parameter space for LPV systems can under certain circumstances be avoided, in particular when the LPV system is expressed in a special form (as a linear fractional representation), when a particular way of solving the problem is employed, and when certain constraints are imposed on the solution. One contribution of this project is to show how the LPV model reduction problem can be expressed in a form that lends itself to applying the above idea. The benefits of this approach have been demonstrated in numerical studies, using a multiple spring-mass-damper system of tuneable complexity as a benchmark. In these studies, it is shown that the proposed grid-free approach drastically extends the size of LPV systems that can be reduced within given limits of computation time, compared with existing grid-based methods. Whereas the first contribution of the project aims at avoiding the gridding of the parameter space, a second contribution is aiming at avoiding the balancing transformation on which standard linear reduction techniques are based. This can be achieved by formulating the model reduction problem as a fictitious optimal control problem, and by employing fixed-structure synthesis tools for LPV systems. The benefits of doing this are twofold: By solving a fictitious L2 optimal control problem, an upper bound on the induced L2 norm of the modelling error can be minimised directly, instead of having to be assessed a posteriori. And secondly, formulating the reduction problem as a fixedstructure controller synthesis problem, makes it possible to impose in addition to a low order a desired structure on the reduced model. This can be exploited in cases where the original model possesses a particular structure in its input-output behaviour, which is desired to be preserved in the reduced model. Or it may be desired to impose a particular structure on the reduced model, for example in application areas such as active aeroservoelastic control, where it is desired to obtain reduced models in a canonical modal form. Thus, a second outcome of this project is a method for LPV model order reduction, where not only the order but also any desired structural features can be imposed on the reduced state space model. The practicality of this approach is again demonstrated in numerical studies on a benchmark problem.
Publications
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Grid-free model order reduction for linear parameter-varying systems via full-block S procedure, Proceedings of the European Control Conference, 1579- 1584, 2020
L. Heeren & H. Werner
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Grid-free constraints for parameter-dependent generalised Gramians via full-block S procedure, Proceedings of the European Control Conference, 1073- 1078, 2022
L. Heeren & H. Werner
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Transformation-free fixed-structure model reduction for LPV systems
L. Heeren, A. Datar, A.M. Gonzalez & H. Werner
