3,562 research outputs found

    Complex order control for improved loop-shaping in precision positioning

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    This paper presents a complex order filter developed and subsequently integrated into a PID-based controller design. The nonlinear filter is designed with reset elements to have describing function based frequency response similar to that of a linear (practically non-implementable) complex order filter. This allows for a design which has a negative gain slope and a corresponding positive phase slope as desired from a loop-shaping controller-design perspective. This approach enables improvement in precision tracking without compromising the bandwidth or stability requirements. The proposed designs are tested on a planar precision positioning stage and performance compared with PID and other state-of-the-art reset based controllers to showcase the advantages of this filter

    An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

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    Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

    Fractional-order controller design with partial pole-zero cancellation

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    MasterÂŽs thesis in Mechatronics (MAS500

    Digital waveguide simulation of convex acoustic pipes

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    This work deals with the physical modelling of acoustic pipes for real-time simulation, using the “Digital Waveguide Network” approach and the horn equation. With this approach, a piece of pipe is represented by a two-port system with a loop which involves two delays for wave propagation, and some subsystems without internal delay. A well-known form of this system is the “Kelly-Lochbaum” framework. It allows the reduction of the computation complexity and it gives a physically meaningful interpretation of the involving subsystems. In this paper, we focus this work on the simulation of pipes with a convex profile, for which a curvature coefficient is constant and negative. In the literature, it has been shown that such pipes have trapped modes. With the formalism of automatic control, adapted for “Waveguides”, we meet some substates of the system which do not take effect on the outputs. But, using the “Kelly-Lochbaum” framework with the horn equation, two problems occur: first, even if the outputs are bounded, some substates have their values which diverge; second, there is an infinite number of such substates. The system is then unstable and cannot be simulated as such. The solution of this problem is obtained with two steps. First, we show that there is a simple standard form compatible with the “Waveguide” approach, for which there is an infinite number of solutions which preserve the input/output relations. Second, we look for one solution which guarantees the stability of the system and which makes easier the approximation in order to get a low-cost simulation

    Dynamics with Infinitely Many Derivatives: The Initial Value Problem

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    Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off.Comment: 40 pages, no figures. Added comments concerning fractional operators and the implications of restricting the contour of integration. Typos correcte

    Linear Control Theory with an ℋ∞ Optimality Criterion

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    This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods
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