Consider a convex set S = {x is an element of D: G(x) >= 0}, where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D subset of R-n is a domain on which G(x) is defined, and G(x) &...
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Consider a convex set S = {x is an element of D: G(x) >= 0}, where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D subset of R-n is a domain on which G(x) is defined, and G(x) >= 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) D = R-n, G(x) is a matrix polynomial and matrix sos-concave;(ii) D is compact convex, G(x) is a matrix polynomial and strictly matrix concave on D;(iii) G(x) is a matrix rational function and q-module matrix concave on D. Explicit constructions of semidefinite representations are given. Some examples are illustrated.
The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This pap...
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The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations of the GPP: the k-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting *** are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.
This paper presents a proof that the use of polynomial Lyapunov functions is not conservative for studying exponential stability properties of nonlinear ordinary differential equations on bounded regions. The main res...
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This paper presents a proof that the use of polynomial Lyapunov functions is not conservative for studying exponential stability properties of nonlinear ordinary differential equations on bounded regions. The main result implies that if there exists an n-times continuously differentiable Lyapunov function which proves exponential decay on a bounded subset of ℝ n , then there exists a polynomial Lyapunov function which proves that same rate of decay on the same region. Our investigation is motivated by the use of semidefinite programming to construct polynomial Lyapunov functions for delayed and nonlinear systems of differential equations.
Abstract Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equatio...
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Abstract Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE), partial differential equations (PDE) and optimal control problems (OCP), and preliminary numerical results are reported.
We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additi...
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We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additional term that vanishes in a neighborhood of the origin. The augmented controller outperforms the original controller in terms of the estimate of the region of attraction. The paper translates the results discussed in Braun et al. (2022a,b), from the continuous-time setting to the discrete-time setting, and numerically verifies that the results derived for continuous-time systems are recovered if the discrete-time system is obtained through an Euler discretization of a continuous-time system with a sufficiently small sampling rate.
This paper presents a public domain toolbox built on top of the software LMITOOL and the semidefinite programming package. The Multiobjective Robust Control Toolbox is a collection of synthesis LMI-based tools for a l...
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This paper presents a public domain toolbox built on top of the software LMITOOL and the semidefinite programming package. The Multiobjective Robust Control Toolbox is a collection of synthesis LMI-based tools for a large class of nonlinear uncertain systems. It provides solutions for the robust state- and output-feedback controller synthesis under various specifications. These specifications include performance requirements via α-stability, L 2 -gain bounds and command input and outputs bounds.
We propose a new combinatorial optimization problem, the so-called Directed Circular Facility Layout Problem (DCFLP). The (DCFLP) seems to be quite interesting as it contains several other problems that have been disc...
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We propose a new combinatorial optimization problem, the so-called Directed Circular Facility Layout Problem (DCFLP). The (DCFLP) seems to be quite interesting as it contains several other problems that have been discussed extensively in literature as special cases. We show that the (DCFLP) is closely related to the Single-Row Facility Layout Problem (SRFLP) and hence we adapt the leading algorithmic method for the (SRFLP) by suggesting an appropriate modelling approach for the (DCFLP). Finally we show that this algorithmic approach yields promising computational results on a variety of benchmark instances.
作者：
Leonardo F. TosoGiorgio ValmorbidaCentraleSupélec
Université Paris-Saclay Laboratoire des signaux et systèmes 91190 Gif-sur-Yvette France CentraleSupélec
CNRS Université Paris-Saclay Inria Saclay - Projet DISCO Laboratoire des signaux et systèmes 91190 Gif-sur-Yvette France
We propose a method for the stability analysis of linear hybrid systems with periodic jumps. The method relies on the solution to polynomial inequalities based on the Handelman decomposition. Compared to existing appr...
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We propose a method for the stability analysis of linear hybrid systems with periodic jumps. The method relies on the solution to polynomial inequalities based on the Handelman decomposition. Compared to existing approaches, such as sum-of-squares (SoS) and Polya's theorem, the proposed method reduces the computation time to obtain stability certificates.
The paper presents in detail alternative robust model predictive control approach based on the optimization of convergence rate subject to nominal system, and additional saturation of control inputs. The approach is a...
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The paper presents in detail alternative robust model predictive control approach based on the optimization of convergence rate subject to nominal system, and additional saturation of control inputs. The approach is a compromise between guaranteed convergence rate and high computational complexity on the one hand, and larger set of feasible initial conditions and lower computational burden on the other hand. The applicability of the proposed strategy is verified using a case study of uncertain chemical reactor stabilization.
Abstract We consider the safety of a V-bar hopping manoeuvre which forms part of the final stage in autonomous space rendezvous. This manoeuvre is controlled by thrusters: we assume these inputs are perfect impulses a...
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Abstract We consider the safety of a V-bar hopping manoeuvre which forms part of the final stage in autonomous space rendezvous. This manoeuvre is controlled by thrusters: we assume these inputs are perfect impulses and model the system in a hybrid automaton framework. We consider the operation of the system under bounded parametric uncertainties, e.g. thruster misalignment, and use the concept of Barrier function certificates to assess system safety. This methodology provides an efficient tool for the systematic investigation of the safety property and does not rely on Monte-Carlo simulations. In particular, the existence of a Barrier function certificate guarantees that all trajectories of the system starting from a given initial set do not enter a predefined unsafe region under any possible combination of parameter deviations. Such a Barrier function certificate can be constructed efficiently using the Sum of Squares (SOS) decomposition and semi-definite programming (SDP).
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