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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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TL;DR: In this paper, a linear complementary problem formulation is proposed for post-buckling analysis of geometrically nonlinear structures with nonfrictional contact constraints, and the arc-length method with updated normal plane constraint is combined to generate a path-following algorithm as a predictor-corrector procedure.
7 citations
01 Jan 2008
TL;DR: In this paper, the authors present a modeling framework based on the measure-differential inclusion representation of the Lagrangian dynamics, which enables the incorporation of set-valued force laws and control laws on acceleration and velocity level.
Abstract: The scope of this dissertation addresses numerical and theoretical issues in the impulsive control of hybrid finite-dimensional Lagrangian systems. In order to treat these aspects, a modeling framework is presented based on the measure-differential inclusion representation of the Lagrangian dynamics. The main advantage of this representation is that it enables the incorporation of set-valued force laws and control laws on acceleration and velocity level concisely. This property of the measure-differential inclusion representation renders the description of the hybrid behaviour of Lagrangian systems in the framework of set-valued control and force laws possible. Based on the MDI representation of Lagrangian dynamics the impactive blocking is analysed as a set-valued impulsive unbounded control law. The application to mechanical systems with impulsively-blockable degrees of freedom is presented. The numerical application of this set-valued control law is the formulation of a nonlinear programming (NLP) algorithm for underactuated mechanical systems with impulsively-blockable DOF. The natural numerical treatment of the measure-differential inclusion representation is based on Moreau’s sweeping process. By applying this discretisation scheme together with an augmented Lagrangian based NLP method that performs the minimisations with a modified conjugate gradients method an optimisation scheme is presented. A numerical example is applied to the impulsive optimal control of a manipulator with one impulsively-blockable degrees of freedom. A further numerical method is introduced for the class of switching Lagrangian systems. This numerical method is a shooting method that performs the numerical integrations based on the sweeping process and the minimisations by making use of the augmented Lagrangian concept. The augmented Lagrangian is minimised by an optimisation method that relies on function value comparisons. This relatively easily implemented numerical method is applied to a wheeled robot which is a tenth-order dynamical system. This system has four different operating modes and time and control effort (quasi-) optimal trajectories are presented. The theoretical results of the dissertation include the statement and the derivation of necessary conditions for the impulsive optimal control of finite-dimensional Lagrangian systems. In this analysis, Lagrangian systems are considered on which the impulses are induced solely by the impulsive control action. The challenge in the derivation of these necessary conditions has been the concurrent discontinuity of state and costate on a Lebesgue negligible time instant. In order to tackle this problem, the instances of impulsive control action are considered as an internal boundaries on the time domain. By the introduction of the concepts of internal boundary variations and discontinuous transversality conditions by the author this problem is resolved and necessary conditions for mechanical systems in the first-order and second-order representations are derived. The discontinuous transversality conditions that result from the consideration of the internal boundary variations in the time domain are discovered and analysed by the author of the dissertation and are applied to the impulsive optimal control of Lagrangian systems.
7 citations
Cites background from "Linear complementarity, linear and ..."
...6 -The Mixed Linear Complementarity Problem [70] Given A ∈ Rn×n, B ∈ Rm×m,C ∈ Rn×m, D ∈ Rm×n, a ∈ R, b ∈ R, find v ∈ R,u ∈ R satisfying a + Au + Cv = 0, b + Du + Bv ≥ 0, v ≥ 0, 〈v,b + Du + Bv〉 = 0....
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...and [70] by Murty, detailed treatment of complementarities and optimisation can be found....
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...24: The Variational Inequality Problem [70] Given a mapping F(z) = {Fi(z)} : R → R, and K ⊂ R, K 6= ∅, find a z∗ ∈ K satisfying 〈y − z∗, F(z∗)〉 ≥ 0, ∀y ∈ K....
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...and [70] by Murty detailed treatment of complementarities and optimisation can be found....
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...5 -The Linear Complementarity Problem [70] Given A ∈ Rn×n, b ∈ R find w = (wj) ∈ R, z = (zj) ∈ R satisfying w −Az = b; w, z o 0; 〈w, z〉 = 0....
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01 Dec 2011
TL;DR: A new receding-horizon approach for performing constraint-based control using a WSN using a conservative abstraction-relaxation approach that simplifies the original mixed-integer programming problem into a convex quadratic programming problem.
Abstract: Constraint-based control over wireless sensor networks (WSNs) require control strategies that achieve a desired closed-loop system performance while using minimal network resources. In addition to constraints associated with distributed control, WSNs have limitations on bandwidth, energy consumption, and transmission range. This paper introduces and experimentally evaluates a new receding-horizon approach for performing constraint-based control using a WSN. By leveraging the system controllability, the receding-horizon controller is formulated as a mixed-integer programming problem which, at each time step, simultaneously generates a control sequence and sensor selection schedule such that the desired performance is achieved while minimizing the energy required to perform data acquisition and control. For systems containing many sensors, a multi-step state estimator is employed to implement the receding-horizon controller using a conservative abstraction-relaxation approach that simplifies the original mixed-integer programming problem into a convex quadratic programming problem. A wireless process control test bed consisting of 8 coupled water tanks and 16 wireless sensors are used to experimentally evaluate the receding-horizon controller.
7 citations
Cites methods from "Linear complementarity, linear and ..."
...The equations describing the dynamics of the interconnected water tank system are nonlinear Thus, in order to apply the proposed techniques, we linearized the dynamics around quiescent points L10, L20 for the upper and lower tanks, respectively....
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...This protocol is the standard for low-power wireless communications and is the base of the wireless industrial protocols WirelessHART and ISA100 [31], [32]....
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15 Dec 1998
TL;DR: In this paper, a continuum model of large discrete networks in planar domains is discussed, where the Kirchhoff law, boundary conditions and capacity constraints lead in a system optimization approach to a infinite dimensional constrained optimisation problem and to mixed variational inequalities.
Abstract: This contribution discusses a continuum model of large discrete networks in planar domains. For this model, the Kirchhoff law, boundary conditions and capacity constraints lead in a system optimisation approach to a infinite dimensional constrained optimisation problem and to “mixed” variational inequalities. Mixed finite element methods can be formulated for these variational inequalities such that computable discretizations of the continuum problem are obtained.
7 citations
TL;DR: The main aim of this paper is to present an application of mathematical modeling and optimal control theory to an ecological engineering problem related to preserve and enhance natural stocks of fish migrating between saltwater and freshwater.
Abstract: A river fishway is a hydraulic structure enabling fish to overcome stream obstructions such as dams in hydroelectric power plants. The main aim of this paper is to present an application of mathematical modeling and optimal control theory to an ecological engineering problem related to preserve and enhance natural stocks of fish migrating between saltwater and freshwater. Particularly, we improve the optimal shape design of a fishway. This problem is formulated within the framework of the optimal control of partial differential equations, approximated by a discrete optimization problem, and solved by using both a gradient method (a Spectral Projected-Gradient algorithm) and a derivative-free method (the Nelder-Mead algorithm). Finally, numerical results are compared and analyzed for a standard real-world situation.
7 citations
Cites methods from "Linear complementarity, linear and ..."
...Since M is a Z-matrix (all off-diagonal elements are non-positive), then this LCP(q,M) can be solved by the so-called Chandrasekaran’s algorithm [21]: Step 0....
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