30,967 research outputs found
An analytic center cutting plane method to determine complete positivity of a matrix
We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic 1. Linear Algebra 29(1):46-58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput 32(1):40-561 Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like O(d(2)) for d x d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl. Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided
Optimum Weight Selection Based LQR Formulation for the Design of Fractional Order PI{\lambda}D{\mu} Controllers to Handle a Class of Fractional Order Systems
A weighted summation of Integral of Time Multiplied Absolute Error (ITAE) and
Integral of Squared Controller Output (ISCO) minimization based time domain
optimal tuning of fractional-order (FO) PID or PI{\lambda}D{\mu} controller is
proposed in this paper with a Linear Quadratic Regulator (LQR) based technique
that minimizes the change in trajectories of the state variables and the
control signal. A class of fractional order systems having single non-integer
order element which show highly sluggish and oscillatory open loop responses
have been tuned with an LQR based FOPID controller. The proposed controller
design methodology is compared with the existing time domain optimal tuning
techniques with respect to change in the trajectory of state variables,
tracking performance for change in set-point, magnitude of control signal and
also the capability of load disturbance suppression. A real coded genetic
algorithm (GA) has been used for the optimal choice of weighting matrices while
designing the quadratic regulator by minimizing the time domain integral
performance index. Credible simulation studies have been presented to justify
the proposition.Comment: 6 pages, 5 figure
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
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