Some studies on control system using lie groups

This project related to control systems, Lie groups and Lie algebra. Firstly, we have dis- cussed about control systems with its examples. Then various types of control systems and its applications. Control theory has continued to advance with advancing technology and has emerged in modern times as a highly developed discipline. Lie theory, the theory of Lie groups, Lie algebras and their applications is a fundamental part of mathematics. On this project review Lie groups controllability for left-invariant control systems on Lie groups are addressed

Motion on lie groups and its applications in control theory

The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group G to that on a subgroup H, provided a particular solution of an associated problem in G/H is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.http://www.sciencedirect.com/science/article/B6VN0-49F836Y-3/1/3e135eb33cca05a026b85455b554430

Relative CC"-Numerical Ranges for Applications in Quantum Control and Quantum Information

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.Comment: accompanying paper to math-ph/070103

Geometric methods for designing optimal filters on Lie groups

In control theory, the problem of having available good measurements is of primary importance in order to perform good tracking and control. Unfortunately, in real-life applications, sensing systems do not provide direct measurements about the pose (and its rate) of mechanical systems, while, in other situations, measurements are so noisy that require pre-processing to filter out disturbances and biases. These problems could be faced by using filters and observers. In this thesis, we apply a second-order optimal minimum-energy filter constructed on Lie groups to several planar bodies. We start by studying the application of the filter to the matrix Lie group TSE(2), i.e. the tangent bundle of the Special Euclidean group SE(2); moreover, a comparison with the extended Kalman filter is presented. After that, we consider the Chaplygin sleigh case, that is a mechanical system with a nonholonomic constraint. Then, we move our attention to the case of an articulated convoy with hooking constraints. Finally, we apply the filter to a real case scenario consisting of a scaled model representing a parking truck semi-trailer system. Particular attention is posed to the description of the geometric structure that underlies the dynamics and to the choice of the measurement equation, the affine connection, and the other parameters that define the filters. Simulations show the effectiveness of the proposed filters. The use of Lie groups theory for designing the filters is challenging, but the accuracy of the results, obtained considering the geometric structure and the symmetries of the system justifies the effort

Generalized eigenvalue problems with specified eigenvalues

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithm

Non-Equilibrium Structural and Dynamic Behaviors of Polar Active Polymer Controlled by Head Activity

Thermodynamic behavior of polymer chains out of equilibrium is a fundamental problem in both polymer physics and biological physics. By using molecular dynamics simulation, we discover a general non-equilibrium mechanism that controls the conformation and dynamics of polar active polymer, i.e., head activity commands the overall chain activity, resulting in re-entrant swelling of active chains and non-monotonic variation of Flory exponent $\nu$. These intriguing phenomena lie in the head-controlled railway motion of polar active polymer, from which two oppose non-equilibrium effects emerge, i.e., dynamic chain rigidity and the involution of chain conformation characterized by the negative bond vector correlation. The competition between these two effects determines the polymer configuration. Moreover, we identify several generic dynamic features of polar active polymers, i.e., linear decay of the end-to-end vector correlation function, polymer-size dependent crossover from ballistic to diffusive dynamics, and a polymer-length independent diffusion coefficient that is sensitive to head activity. A simple dynamic theory is proposed to faithfully explain these interesting dynamic phenomena. This sensitive structural and dynamical response of active polymer to its head activity provides us a practical way to control active-agents with applications in biomedical engineering.Comment: 9 pages, 5 figure