Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Adaptive Control of Piecewise Linear Systems: the State Tracking Case

Abstract: Nonlinear controlled systems at multiple operating points are modeled as piecewise linear systems, where changes in operating points are modeled as switches between constituent linearized systems. This note studies the adaptive state feedback for state tracking control problem for such systems. Piecewise linear reference model systems are used for generating desired state trajectories and their stability properties are studied. Adaptive state feedback control schemes are developed, and their stability and tracking performance are analyzed and evaluated by simulation examples. It is shown that exponential tracking performance can be achieved if the reference input is sufficiently rich and the switches are sufficiently slow.

Stabilizing Model Predictive Control of Stochastic Constrained Linear Systems

Abstract: This paper investigates stochastic stabilization procedures based on quadratic and piecewise linear Lyapunov functions for discrete-time linear systems affected by multiplicative disturbances and subject to linear constraints on inputs and states. A stochastic model predictive control (SMPC) design approach is proposed to optimize closed-loop performance while enforcing constraints. Conditions for stochastic convergence and robust constraints fulfillment of the closed-loop system are enforced by solving linear matrix inequality problems off line. Performance is optimized on line using multistage stochastic optimization based on enumeration of scenarios, that amounts to solving a quadratic program subject to either quadratic or linear constraints. In the latter case, an explicit form is computable to ease the implementation of the proposed SMPC law. The approach can deal with a very general class of stochastic disturbance processes with discrete probability distribution. The effectiveness of the proposed SMPC formulation is shown on a numerical example and compared to traditional MPC schemes.

Global optimization of a combinatorially complex generalized pooling problem

Abstract: Global optimization strategies are described for a generalization of the pooling problem that is important to the petrochemical, chemical, and wastewater treatment industries. The problem involves both discrete variables, modeling the structure of a flow network, and continuous variables, modeling flow rates, and stream attributes. The continuous relaxation of this mixed integer nonlinear programming problem is nonconvex because of the presence of bilinear terms in the constraint functions. We propose an algorithm to find the global solution using the principles of the reformulation-linearization technique (RLT). A novel piecewise linear RLT formulation is proposed and applied to the class of generalized pooling problems. Using this approach we verify the global solution of a combinatorially complex industrial problem containing 156 bilinear terms and 55 binary variables, reducing the gap between upper and lower bounds to within 1.2%. © 2005 American Institute of Chemical Engineers AIChE J, 2006

Sensitivity Analysis for Mean-Variance Portfolio Problems

Abstract: This paper shows how to perform sensitivity analysis for Mean-Variance MV portfolio problems using a general form of parametric quadratic programming. The analysis allows an investor to examine how parametric changes in either the means or the right-hand side of the constraints affect the composition, mean, and variance of the optimal portfolio. The optimal portfolio and associated multipliers are piecewise linear functions of the changes in either the means or the right-hand side of the constraints. The parametric parts of the solution show the rates of substitution of securities in the optimal portfolio, while the parametric parts of the multipliers show the rates at which constraints are either tightening or loosening. Furthermore, the parametric parts of the solution and multipliers change in different intervals when constraints become active or inactive. The optimal MV paths for sensitivity analyses are piecewise parabolic, as in traditional MV analysis. However, the optimal paths may contain negatively sloping segments and are characterized by types of kinks, i.e., points of nondifferentiability, not found in MV analysis.

Visualizing nonlinear vector field topology

Abstract: We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.