188 research outputs found
Finite dimensional backstepping controller design
We introduce a finite dimensional version of backstepping controller design
for stabilizing solutions of PDEs from boundary. Our controller uses only a
finite number of Fourier modes of the state of solution, as opposed to the
classical backstepping controller which uses all (infinitely many) modes. We
apply our method to the reaction-diffusion equation, which serves only as a
canonical example but the method is applicable also to other PDEs whose
solutions can be decomposed into a slow finite-dimensional part and a fast
tail, where the former dominates the evolution in large time. One of the main
goals is to estimate the sufficient number of modes needed to stabilize the
plant at a prescribed rate. In addition, we find the minimal number of modes
that guarantee the stabilization at a certain (unprescribed) decay rate.
Theoretical findings are supported with numerical solutions.Comment: 28 pages, 2 figure
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Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
Guaranteed cost boundary control of the semilinear heat equation
We consider a 1D semilinear reaction-diffusion system with controlled heat flux at one of the boundaries. We design a finite-dimensional state-feedback controller guaranteeing that a given quadratic cost does not exceed a prescribed value for all nonlinearities with a predefined Lipschitz constant. To this end, we perform modal decomposition and truncate the highly damped (residue) modes. To deal with the nonlinearity that couples the residue and dominating modes, we combine the direct Lyapunov approach with the S-procedure and Parseval’s identity. The truncation may lead to spillover: the ignored modes can deteriorate the overall system performance. Our main contribution is spillover avoidance via the L2 separation of the residue. Namely, we calculate the L2 input-to-state gains for the residue modes and add them to the control weight in the quadratic cost used to design a controller for the dominating modes. A numerical example demonstrates that the proposed idea drastically improves both the admissible Lipschitz constants and guaranteed cost bound compared to the recently introduced direct Lyapunov method
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Dynamics of Patterns
This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects
Full State Estimation of Continuum Robots From Tip Velocities: A Cosserat-Theoretic Boundary Observer
State estimation of robotic systems is essential to implementing feedback
controllers which usually provide better robustness to modeling uncertainties
than open-loop controllers. However, state estimation of soft robots is very
challenging because soft robots have theoretically infinite degrees of freedom
while existing sensors only provide a limited number of discrete measurements.
In this paper, we design an observer for soft continuum robotic arms based on
the well-known Cosserat rod theory which models continuum robotic arms by
nonlinear partial differential equations (PDEs). The observer is able to
estimate all the continuum (infinite-dimensional) robot states (poses, strains,
and velocities) by only sensing the tip velocity of the continuum robot (and
hence it is called a ``boundary'' observer). More importantly, the estimation
error dynamics is formally proven to be locally input-to-state stable. The key
idea is to inject sequential tip velocity measurements into the observer in a
way that dissipates the energy of the estimation errors through the boundary.
Furthermore, this boundary observer can be implemented by simply changing a
boundary condition in any numerical solvers of Cosserat rod models. Extensive
numerical studies are included and suggest that the domain of attraction is
large and the observer is robust to uncertainties of tip velocity measurements
and model parameters
Distributed Saturated Control for a Class of Semilinear PDE Systems: A SOS Approach
ProducciĂłn CientĂficaThis paper presents a systematic approach to deal with the saturated control of a class of distributed parameter systems which can be modeled by first-order hyperbolic partial differential equations (PDE). The approach extends (also improves over) the existing fuzzy Takagi-Sugeno (TS) state feedback designs for such systems by applying the concepts of the polynomial sum-of-squares (SOS) techniques. Firstly, a fuzzy-polynomial model via Taylor series is used to model the semilinear hyperbolic PDE system. Secondly, the closed-loop exponential stability of the fuzzy-PDE system is studied through the Lyapunov theory. This allows to derive a design methodology in which a more complex fuzzy state-feedback control is designed in terms of a set of SOS constraints, able to be numerically computed via semidefinite programming. Finally, the proposed approach is tested in simulation with the standard example of a nonisothermal plug-flow reactor (PFR).The research leading to these results has received funding from the European Union and from the Spanish Government (MINECO/FEDER DPI2015-70975-P)
Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion
Negative imaginary (NI) systems play an important role in the robust control
of highly resonant flexible structures. In this paper, a generalized NI system
framework is presented. A new NI system definition is given, which allows for
flexible structure systems with colocated force actuators and position sensors,
and with free body motion. This definition extends the existing definitions of
NI systems. Also, necessary and sufficient conditions are provided for the
stability of positive feedback control systems where the plant is NI according
to the new definition and the controller is strictly negative imaginary. The
stability conditions in this paper are given purely in terms of properties of
the plant and controller transfer function matrices, although the proofs rely
on state space techniques. Furthermore, the stability conditions given are
independent of the plant and controller system order. As an application of
these results, a case study involving the control of a flexible robotic arm
with a piezo-electric actuator and sensor is presented
Parameter identification problems in the modelling of cell motility
We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg–Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data. Our numerical tests allow us to compare the method with the two different formulations of the objective functional and we conclude that the results with both objective functionals seem to agree
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