engleski

Modelling Error Estimation for Reduced Order Coupled Dynamical Systems

Introduction Modelling of coupled dynamical systems represents a cornerstone in many applied technical sciences. Demands for control of spatially distributed dynamical systems is also increasing. Examples include flexible multibody systems (FMBS) such unmanned aerial vehicles (UAV), satellites and aircrafts, micro electro-mechanical systems (MEMS), very large-scale integrated (VLSI) circuit design and similar. Choosing the appropriate discretization mesh (grid) for the coupled dynamical system to be controlled is not trivial: spatial discretization error increases with the reduction of mesh resolution and the order of the resulting coupled dynamical system dramatically increases with mesh refinement. Model order reduction (MOR) methods plays an essential role here - at the cost of introducing the MOR error. Methods There exist many techniques that deal with the error estimation of a specific spatial discretization technique and/or MOR method. However, it is was shown that those are usually hard to be combined and implemented for efficient controller synthesis. To deal with this, author proposes to model the coupled system affected by discretization and MOR errors as an uncertain dynamical system – where each error (from both spatial discretization and MOR) is modeled as a full-block unstructured additive uncertainty model. With the novelty of the procedure being the modelling of the uncertainty at the level of a subsystem – i.e structure preservation. Robustness analysis of the resulting uncertain system can be carried out using the classical tools, such as $mu$-tools or using the integral quadratic constraints (IQCs). Preliminary results For a special class of spatially distributed dissipative dynamical system, modelling of the uncertainty at a subsystem level, was proven to be practically useful: (i) uncertainty modelling at the level of a subsystem preserves the structure of the coupled system, (ii) for coupled systems with many interconnected dissipative subsystems, uncertainty at the level of a subsystem, can often be reduced – taking into account dissipative properties of the subsystems’ surroundings – resulting in a less conservative uncertainty models, (iii) it is possible to fine tune the required discretization level of each subsystem as well as to choose appropriate (or even different) MOR method per subsystem, and (iv) using the IQCs it is possible to take into consideration the different types of uncertainties into the overall robustness analysis – such as parametric uncertainties and nonlinearities. Discussion It is shown possible to find a low-order approximation of the coupled uncertain dynamical system. The obtained low-order coupled system is robustly stable and the overall dynamics are well correlated in comparison to the original high-order (nominal) coupled system – and as such suitable for modern decentralized or distributed controller synthesis.

spatial discretization error, model order reduction error, robustness analysis, integral quadratic constrains

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