The MIN PFS problem and piecewise linear model estimation

AbstractWe consider a new combinatorial optimization problem related to linear systems (MIN PFS) that consists, given an infeasible system, in finding a partition into a minimum number of feasible subsystems. MIN PFS allows formalization of the fundamental problem of piecewise linear model estimation, which is an attractive alternative when modeling a wide range of nonlinear phenomena. Since MIN PFS turns out to be NP-hard to approximate within every factor strictly smaller than 3/2 and we are mainly interested in real-time applications, we propose a greedy strategy based on randomized and thermal variants of the classical Agmon–Motzkin–Schoenberg relaxation method for solving systems of linear inequalities. Our method provides good approximate solutions in a short amount of time. The potential of our approach and the performance of our algorithm are demonstrated on two challenging problems from image and signal processing. The first one is that of detecting line segments in digital images and the second one that of modeling time-series using piecewise linear autoregressive models. In both cases the MIN PFS-based approach presents various advantages with respect to conventional alternatives, including wider range of applicability, lower computational requirements and no need for a priori assumptions regarding the underlying structure of the data

Global optimization for low-dimensional switching linear regression and bounded-error estimation

The paper provides global optimization algorithms for two particularly difficult nonconvex problems raised by hybrid system identification: switching linear regression and bounded-error estimation. While most works focus on local optimization heuristics without global optimality guarantees or with guarantees valid only under restrictive conditions, the proposed approach always yields a solution with a certificate of global optimality. This approach relies on a branch-and-bound strategy for which we devise lower bounds that can be efficiently computed. In order to obtain scalable algorithms with respect to the number of data, we directly optimize the model parameters in a continuous optimization setting without involving integer variables. Numerical experiments show that the proposed algorithms offer a higher accuracy than convex relaxations with a reasonable computational burden for hybrid system identification. In addition, we discuss how bounded-error estimation is related to robust estimation in the presence of outliers and exact recovery under sparse noise, for which we also obtain promising numerical results

Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

MULTI-MODAL BEHAVIOR AND CLUSTERING IN DYNAMICAL SYSTEM WITH APPLICATIONS TO WIND FARMS

he objective of this research is to develop a comprehensive model identification approach for complex multi-modal systems based on spectral theory for nonreversible Markov process that entails (i)model reduction techniques for a nonreversible Markov chain, (ii) the identification of the modal dynamics, and (iii) modeling and identification of local dynamics. This dissertation addresses the theoretical approach, algorithmic development, computational efficiency and numerical examples of the developed techniques.The dissertation then presents a novel methodology for clustering wind turbines of a wind farm into different groups. The method includes creation of a Markov transition matrix given the power output of each turbine, spectral analysis of the transition matrix and identification approach of each group. An application of the method is provided based on real data of a wind farms consisting of 25 turbines and 79 turbines, respectively. The application shows that those distinct wind farm groups with different dynamic output characteristics can be identified and the turbines in each group can also be determined

The MIN PFS problem and piecewise linear model estimation

We consider a new combinatorial optimization problem related to linear systems (MIN PFS) that consists, given an in-feasible system, in finding a partition into a minimum number of feasible subsystems. MIN PFS allows formalization of the fundamental problem of piecewise linear model estimation, which is an attractive alternative when modeling a wide range of nonlinear phenomena. Since MIN PFS turns out to be NP-hard to approximate within every factor strictly smaller than � and we are mainly interested in real-time applications, we propose a greedy strategy based on simple randomized and thermal variants of the classical Agmon-Motzkin-Schoenberg relaxation method for solving systems of linear inequalities. Our method provides good approximate solutions in a short amount of time. The potential of our approach and the per-formance of our algorithm are demonstrated on two challenging problems from image and signal processing. The first one is that of detecting line segments in digital images and the second one that of modeling time series using piecewise linear autoregressive models. In both cases the MIN PFS-based approach presents various advantages with respect to classical alternatives, including wider range of applicability, lower computational requirements and no need of a priori assumptions regarding the underlying structure of the data. USA