3,562 research outputs found
Complex order control for improved loop-shaping in precision positioning
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
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
MasterÂŽs thesis in Mechatronics (MAS500
Digital waveguide simulation of convex acoustic pipes
This work deals with the physical modelling of acoustic pipes for real-time simulation, using the âDigital Waveguide Networkâ 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 âKelly-Lochbaumâ 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 âWaveguidesâ, we meet some substates of the system which do not take effect on the outputs.
But, using the âKelly-Lochbaumâ 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 âWaveguideâ 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
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 ââ Optimality Criterion
This expository paper sets out the principal results in ââ 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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