A new methodology for obtaining piecewise affine models using a set of linearisation points and voronoi partitions

To understand complex dynamical systems, approximations are often made by a linearisation about an operating point of interest. The drawback of this linear approximation is that it only describes the system locally, around the operating point. One possible solution to overcome this drawback is to approximate the complex nonlinear dynamical system with a piecewise affine (PWA) system. Approximating nonlinear dynamical systems is very important in system theory where one is interested in simplifying its analysis and numerical simulation. PWA modelling is a very powerful tool to represent nonlinear systems as a collection of a finite number of linear systems. In the literature of PWA, a uniform grid (UG) approximation is the current method being used to approximate a nonlinear system as a PWA system. The drawback of this method is the potential large amount of regions required to obtain a desired accuracy, which is most evident for systems with more than one variable in the domain of the nonlinearity. In order to reduce the number of regions, the proposed research will develop a new methodology for obtaining PWA models using a set of linearisation points (SLP) and the Voronoi partition. First, in order to generate a partition based on a SLP, the curvature of the nonlinearity is used as a tool for selecting appropriate locations for the linearisation points. Next, an algorithm is proposed to automate the SLP approximation for both curves and surfaces. The SLP and UG approximation methods are then compared over several simple examples. Finally, the newly proposed approximation methodology is applied to three case studies: modelling of a nonlinear mechanical system and modelling and control of an unmanned aerial vehicle (UAV) and a micro air vehicle (MAV

Piecewise Affine System Identification of a Hydraulic Wind Power Transfer System

Hydraulic wind power transfer systems exhibit a highly nonlinear dynamic influenced by system actuator hysteresis and disturbances from wind speed and load torque. This paper presents a system identification approach to approximate such a nonlinear dynamic. Piecewise affine (PWA) models are obtained utilizing the averaged nonlinear models of hysteresis in a confined space. State-space representation of PWA models is obtained over the allocated operating point clusters. The experimental results demonstrate a close agreement with that of the simulated. The experimental results and simulation show more than 91% match

Intersection-based Piecewise Affine Approximation of Nonlinear Systems

This paper presents a new algorithm for PWA approximation of nonlinear systems. Such an approximation is very important to enable a reduction in the complexity of models of nonlinear systems while keeping the global validity of the models. The paper builds on previous work on piecewise affine (PWA) approximation methods, in particular on the work done by Casselman and Rodrigues, known as the Set of Linearization Points (SLP) PWA approximation. The proposed extension method can be used to approximate any continuous function of one variable by a PWA function. The algorithm is based on the points at which the linearization lines intersect with each other. The method assumes that a desired approximation error and one linearization point are given. The algorithm, then performs several linearizations. It is shown that the new linearization points are optimal in the sense of decreasing the error between the exact function and the approximation. The main advantages of this methodology compared to previous approaches are the reduction of the number of pieces of the PWA function, the guarantee that the approximation is continuous, and that the derivative of the approximation and the derivative of the exact function are equal at all linearization points. A detailed collection of examples from different fields of study highlight the effectiveness and the flexibility of the proposed method. It is shown that the proposed method compares favorably with other methods

Pseudo Euler-Lagrange and Piecewise Affine Control Applied to Surge and Stall in Axial Compressors

This thesis addresses the control of the axial compressor surge and stall phenomena using Pseudo Euler-Lagrange and Piecewise Affine (PWA) controller synthesis techniques. These phenomena are considered as major gas turbine compressor instabilities that may result in failures such as the engine flame-out or severe mechanical damages caused by high blade vibration. The common approach towards the detection of the rotating stall and surge is to install various types of pressure sensors, hot wires and velocity probes. The inception of the rotating stall and surge is recognized by the presence of pressure fluctuation and velocity disturbances in the gas stream that are obtained through sensors. The necessary measure is then taken by applying proper stall and surge stabilizing control actions. The Lyapunov stability of pseudo Euler-Lagrange systems in the literature is extended to include additional nonlinear terms. Although Lyapunov stability theory is considered as the cornerstone of analysis of nonlinear systems, the generalization of this energy-based method poses a drawback that makes obtaining a Lyapunov function a difficult task. Therefore, proposing a method for generating a Lyapunov function for the control synthesis problem of a class of nonlinear systems is of potential importance. A systematic Lyapunov-based controller synthesis technique for a class of second order systems is addressed in this thesis. It is shown, in terms of stability characteristics, that the proposed technique provides a more robust solution to the compressor surge suppression problem as compared to the feedback linearization and the backstepping methods. The second contribution is a proposed new PWA approximation algorithm. Such an approximation is very important in reducing the complexity of nonlinear systems models while keeping the global validity of the models. The proposed method builds upon previous work on piecewise affine (PWA) approximation methods, which can be used to approximate continuous functions of n-variables by a PWA function. Having computed the PWA model of the stall and surge equations, the suppression problem is then solved by using PWA synthesis techniques. The proposed solution is shown to have higher damping characteristics as compared to the backstepping nonlinear method

Moving Polygon Methods for Incompressible Fluid Dynamics

Hybrid particle-mesh numerical approaches are proposed to solve incompressible fluid flows. The methods discussed in this work consist of a collection of particles each wrapped in their own polygon mesh cell, which then move through the domain as the flow evolves. Variables such as pressure, velocity, mass, and momentum are located either on the mesh or on the particles themselves, depending on the specific algorithm described, and each will be shown to have its own advantages and disadvantages. This work explores what is required to obtain local conservation of mass, momentum, and convergence for the velocity and pressure in a particle-mesh CFD simulation method. Current particle methods are explored and analyzed for their benefits and deficiencies, and newly developed methods are described with results and analysis. A new method for generating locally orthogonal polygonal meshes from a set of generator points is presented in which polygon areas are a constraint. The area constraint property is particularly useful for particle methods where moving polygons track a discrete portion of material. Voronoi polygon meshes have some very attractive mathematical and numerical properties for numerical computation, so a generalization of Voronoi polygon meshes is formulated that enforces a polygon area constraint. Area constrained moving polygonal meshes allow one to develop hybrid particle-mesh numerical methods that display some of the most attractive features of each approach. It is shown that this mesh construction method can continuously reconnect a moving, unstructured polygonal mesh in a pseudo-Lagrangian fashion without change in cell area/volume, and the method\u27s ability to simulate various physical scenarios is shown. The advantages are identified for incompressible fluid flow calculations, with demonstration cases that include material discontinuities of all three phases of matter and large density jumps

Decentralized Resource Allocation through Constrained Centroidal Voronoi Tessellations

The advancements in the fields of microelectronics facilitate incorporating team elements like coordination into engineering systems through advanced computing power. Such incorporation is useful since many engineering systems can be characterized as a collection of interacting subsystems each having access to local information, making local decisions, interacting with neighbors, and seeking to optimize local objectives that may well conflict with other subsystems, while also trying to optimize certain global objective. In this dissertation, we take advantage of such technological advancements to explore the problem of resource allocation through different aspects of the decentralized architecture like information structure in a team. Introduced in 1968 as a toy example in the field of team decision theory to demonstrate the significance of information structure within a team, the Witsenhausen counterexample remained unsolved until the analytical person-by-person optimal solution was developed within the past decade. We develop a numerical method to implement the optimal laws and show that our laws coincide with the optimal affine laws. For the region where the optimal laws are non-linear, we show that our laws result in the lowest costs when compared with previously reported costs. Recognizing that, in the framework of team decision theory, the difficulties arising from the non-classical information structure within a team currently limit its applicability in real-world applications, we move on to investigating Centroidal Voronoi Tessellations (CVTs) to solve the resource allocation problem. In one-dimensional spaces, a line communication network is sufficient to obtain CVTs in a decentralized manner, while being scalable to any number of agents in the team. We first solve the static resource allocation problem where the amount of resource is fixed. Using such static allocation solution as an initialization step, we solve the dynamic resource allocation problem in a truly decentralized manner. Furthermore, we allow for flexibility in agents\u27 embedding their local preferences through what we call a civility model. We end the dissertation by revisiting the application of Demand-response in smart grids and demonstrate the developed decentralized dynamic resource allocation method to solve the problem of power allocation in a group of building loads

A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme