Topic
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.
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01 Jun 1989TL;DR: In this article, the construction of tangent plane continuous piecewise quadric surfaces that interpolate finite sets of essentially arbitrary points in IR 3 according to a given topology is described in terms of a piecewise linear interpolant.
Abstract: This paper is concerned with the construction of tangent plane continuous piecewise quadric surfaces that interpolate finite sets of essentially arbitrary points in IR 3 according to a given ‘topology’ which is described in terms of a piecewise linear interpolant. Moreover, within certain ranges depending on the topology and the location of data points, given normal directions at the points are also matched by the interpolating piecewise quadric surface.
53 citations
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01 Jan 2005
TL;DR: This thesis addresses the problem of constrained optimal control of discrete-time linear hybrid systems by developing algorithmic implementations that give the exact solution to the optimal control problems and describes an efficient implementation of a general multi-parametric quadratic program.
Abstract: This thesis addresses the problem of constrained optimal control of discrete-time linear hybrid systems. In their most general form hybrid systems are characterized by the interaction of continuous valued components (governed by differential or difference equations) and logic rules. Among the variety of equivalent descriptions of discrete-time hybrid systems reported in the literature we focus on the dass of constrained piecewise affine (PvVA) systems. Discrete-time P\VA models can describe a large number of processes. Moreover, they can approximate nonlinear discrete-time dynamics via multiple linearizations at different operating points. Even though PWA systems are a special dass of nonlinear systems most of the nonlinear system and control theory does not apply because it requires certain smoothness assumptions. For the same reason we also cannot simply use linear control theory in some approximate manner to design controllers for P\VA systems. In the past most tools for the analysis and control of hybrid systems were ad hoc supported by extensive simulation. The aim of this thesis is to furt her advance systematic procedures and develop algorithmic implementations that give the exact solution to the optimal control problems. Arecurring theme in the thesis is the construction of efficient algorithms for solving various instances of optimal control problems. An instrumental tool in the development of such algorithms is the concept of multi-parametric programming, where a quadratic (or linear) optimization problem is solved off-line for a range of parameters. Specifically, an efficient implementation of a general multi-parametric quadratic program is described together with the in-depth analysis of the properties of the solution. The optimal control problem of a constrained linear discrete-time system can now be formulated as a multi-parametric quadratic program by treating the state vector as a parameter. The optimal solution is a piecewise affine state-feedback control law that is defined over a poly-hedral partition of the feasible state-space. This allows users to carry out most of the time-consuming/complex computation off-line, while on-Iine implementation (control action computation) reduces to a simple set-membership test. By exploiting the properties of the value function and the optimal controllaw, new algorithms are developed that avoid storing the polyhedral regions. The new algorithms significantly reduce the on-line storage demands and computational complexity during evaluation of the PWA feedback controllaw. Next, the finite time optimal control problem for constrained discrete-time linear hybrid systems based on quadratic or linear performance criteria is tackled. Basic theoretical results on the structure of the optimal …
53 citations
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01 Jan 2004TL;DR: In this article, the authors investigate some particular classes of hybrid systems subject to a class of time delays; the time delays can be constant or time varying, and they present the corresponding classes of piecewise continuous Lyapunov functions.
Abstract: We investigate some particular classes of hybrid systems subject to a class of time delays; the time delays can be constant or time varying. For such systems, we present the corresponding classes of piecewise continuous Lyapunov functions.
53 citations
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TL;DR: A new method is proposed, the main features of which are to treat the optimal dimensioning problem as a bilevel programming problem, then to simplify the lower level problem by conjugate duality theory and to handle the upperlevel problem by its piecewise linear and convex nature.
Abstract: This paper is concerned with the optimal dimensioning problem in which the layout of a pipe network is given but several available diameters can be selected for each pipe. The purpose is to minimize total cost while satisfying certain restrictions. A new method is proposed, the main features of which are to treat the problem as a bilevel programming problem, then to simplify the lower level problem by conjugate duality theory and to handle the upper level problem by its piecewise linear and convex nature. Some numerical testing results are also presented.
53 citations
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TL;DR: This method is applied to approximate the efficient point curve of the Bicriterial Minimum Cost Flow Problem, which is a piecewise linear convex curve that may have an exponential number of breakpoints in the worst case.
53 citations