Lattice piecewise affine approximation of explicit nonlinear model predictive control with application to trajectory tracking of mobile robot

To promote the widespread use of mobile robots in diverse fields, the performance of trajectory tracking must be ensured. To address the constraints and nonlinear features associated with mobile robot systems, we apply nonlinear model predictive control (MPC) to realize the trajectory tracking of mobile robots. Specifically, to alleviate the online computational complexity of nonlinear MPC, this paper devises a lattice piecewise affine (PWA) approximation method that can approximate both the nonlinear system and control law of explicit nonlinear MPC. The kinematic model of the mobile robot is successively linearized along the trajectory to obtain a linear time-varying description of the system, which is then expressed using a lattice PWA model. Subsequently, the nonlinear MPC problem can be transformed into a series of linear MPC problems. Furthermore, to reduce the complexity of online calculation of multiple linear MPC problems, we approximate the optimal solution of the linear MPC by using the lattice PWA model. That is, for different sampling states, the optimal control inputs are obtained, and lattice PWA approximations are constructed for the state control pairs. Simulations are performed to evaluate the performance of our method in comparison with the linear MPC and explicit linear MPC frameworks. The results show that compared with the explicit linear MPC, our method has a higher online computing speed and can decrease the offline computing time without significantly increasing the tracking error

An Invitation to the Generalized Saturation Conjecture

We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the Littlewood--Richardson numbers.Comment: 79 pages, new sections, new results and example

Simplex basis function based sparse least squares support vector regression

In this paper, a novel sparse least squares support vector regression algorithm, referred to as LSSVR-SBF, is introduced which uses a new low rank kernel based on simplex basis function, which has a set of nonlinear parameters. It is shown that the proposed model can be represented as a sparse linear regression model based on simplex basis functions. We propose a fast algorithm for least squares support vector regression solution at the cost of O(N) by avoiding direct kernel matrix inversion. An iterative estimation algorithm has been proposed to optimize the nonlinear parameters associated with the simplex basis functions with the aim of minimizing model mean square errors using the gradient descent algorithm. The proposed fast least square solution and the gradient descent algorithm are alternatively applied. Finally it is shown that the model has a dual representation as a piecewise linear model with respect to the system input. Numerical experiments are carried out to demonstrate the effectiveness of the proposed approaches

Polyhedral Tools for Control

Polyhedral operations play a central role in constrained control. One of the most fundamental operations is that of projection, required both by addition and multiplication. This thesis investigates projection and its relation to multi-parametric linear optimisation for the types of problems that are of particular interest to the control community. The first part of the thesis introduces an algorithm for the projection of polytopes in halfspace form, called Equality Set Projection (ESP). ESP has the desirable property of output sensitivity for non-degenerate polytopes. That is, a linear number of linear programs are needed per output facet of the projection. It is demonstrated that ESP is particularly well suited to control problems and comparative simulations are given, which greatly favour ESP. Part two is an investigation into the multi-parametric linear program (mpLP). The mpLP has received a lot of attention in the control literature as certain model predictive control problems can be posed as mpLPs and thereby pre-solved, eliminating the need for online optimisation. The structure of the solution to the mpLP is studied and an approach is pre- sented that eliminates degeneracy. This approach causes the control input to be continuous, preventing chattering, which is a significant problem in control with a linear cost. Four new enumeration methods are presented that have benefits for various control problems and comparative simulations demonstrate that they outperform existing codes. The third part studies the relationship between projection and multi-parametric linear programs. It is shown that projections can be posed as mpLPs and mpLPs as projections, demonstrating the fundamental nature of both of these problems. The output of a multi-parametric linear program that has been solved for the MPC control inputs offline is a piecewise linear controller defined over a union of polyhedra. The online work is then to determine which region the current measured state is in and apply the appropriate linear control law. This final part introduces a new method of searching for the appropriate region by posing the problem as a nearest neighbour search. This search can be done in logarithmic time and we demonstrate speed increases from 20Hz to 20kHz for a large example system

Saturation problem for affine Kac-Moody algebras

This thesis is a study of the saturated tensor cones of the affine Kac-Moody algebras A1,1 and A2,2. In general, we show that the occurrence of certain components in the tensor product of two highest weight integrable representations implies the occurrence of other components. For A1,1 and A2,2, we are able to prove the occurrence of enough components to explicitly determine the saturated tensor cone and saturation factors. Moreover, in these two cases, we show that the saturated tensor cone is given by the inequalities conjectured in Brown-Kumar.Doctor of Philosoph

Multipartite Quantum States and their Marginals

Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is based on arXiv:1302.6990 and arXiv:1210.046

On dynamical properties of quasicrystals

In the first half of this thesis we study the properties of the dynamical hull associated with model sets arising from irregular Euclidean Cut and Project Schemes. We provide deterministic as well as probabilistic constructions of irregular windows whose associated Cut and Project Schemes yield Delone dynamical systems with positive topological entropy. Moreover, we provide a construction of an irregular window whose associated dynamical hull has zero topological entropy but admits a unique ergodic measure. Furthermore, we show that dynamical hulls of irregular model sets always admit an infinite independence set. Hence, the dynamics cannot be tame. We extend this proof to a more general setting and show that tame implies regular for almost automorphic group actions on compact spaces. In the second half of this thesis, we provide and discuss a generalization of the Cut and Project formalism. We show that this new formalism yields all sets generated by Euclidean Cut and Project Schemes as well as non-Meyer sets. Furthermore, we give a sufficient criterion to obtain Meyer sets by this formalism in Euclidean space

Landau-Ginzburg potentials via projective representations

We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.Comment: v2 27 pages, some editorial change