Atsumi Ohara

H ∞ control for descriptor systems: a matrix inequalities approach

Abstract This paper considers the H ∞ control problem for descriptor systems that possibly have impulsive modes and/or jω-axis zeros. First, we propose matrix inequalities that give a generalized stability condition and an H ∞ norm condition for descriptor systems. Using these matrix inequalities, we show that the solvability of a set of matrix inequalities is necessary and sufficient to the existence of a proper controller that satisfies a prescribed H ∞ norm condition as well as stabilizing the closed-loop system and eliminating all impulsive modes. These inequalities are equivalent to certain linear matrix inequalities, to which we can get solutions whenever they exist using efficient polynomial-time algorithms.

LMI‐based controller synthesis: A unified formulation and solution

This paper proposes a unified approach to linear controller synthesis that employs various LMI conditions to represent control specifications. We define a comprehensive class of LMIs and consider a general synthesis problem described by any LMI of the class. We show a procedure that reduces the synthesis problem, which is a BMI problem, to solving a certain LMI. The derived LMI condition is equivalent to the original BMI condition and also gives a convex parametrization of all the controllers that solve the synthesis problem. The class contains many of widely-known LMIs (for H∞ norm, H2 norm, etc.), and hence the solution of this paper unifies design methods that have been proposed depending on each LMI. Further, the class also provides LMIs for multi-objective performance measures, which enable a new formulation of controller design through convex optimization.

Gain-scheduled controller design based on descriptor representation of LPV systems: application to flight vehicle control

This paper is concerned with a synthesis method of gain-scheduled controllers based on descriptor representations of LPV systems and its application to design of flight vehicle control. By representing a linear parameter-varying system in the descriptor form, parameter-dependent LMI to seek a gain-scheduled controller are formulated so that they have simple structure with respect to the parameter. A gain-scheduled controller is computed directly via an explicit formula in terms of the variable of LMI for LPV descriptor systems. This synthesis procedure is applied to design of flight vehicle control to satisfy multiple control specifications under large variation of the airspeed. Dynamics of the vehicle is modeled as an LPV descriptor system and a gain-scheduled controller is obtained by solving the proposed LMI. Control performance is improved by employing parameter-dependent Lyapunov matrices.

Dualistic differential geometry of positive definite matrices and its applications to related problems

Abstract Analysis on the set of positive definite matrices is involved in many engineering problems. We develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it. Consequently, various dualistic aspects of the set are elucidated. Next, using the theory, we derive new results and interesting geometrical interpretations for matrix approximation, positive definite matrix completion, and linear matrix inequality problems.

Geometry of q-Exponential Family of Probability Distributions

The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.

Information geometry of q-Gaussian densities and behaviors of solutions to related diffusion equations*

This paper presents new geometric aspects of the behaviors of solutions to the porous medium equation (PME) and its associated equation. First we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities. A dualistic relation between the two existing formalisms is elucidated. Next by equipping the manifold of q-Gaussian densities with such a structure, we derive several physically and geometrically interesting properties of the solutions. The manifold is proved invariant and attracting for the evolving solutions, which play crucial roles in our analysis. We demonstrate that the moment-conserving projection of a solution coincides with a geodesic curve on the manifold. Further, the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the Bregman divergence. Finally we show that the self-similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections.

LMI-based output feedback controller design-using a convex parametrization of full-order controllers

This paper proposes a unified output feedback controller synthesis method that employs LMI-represented specifications of H/sub 2/ norm control, H/sub /spl infin// norm control, root clustering and various multiobjective performances. For this purpose, we show a new parametrization of full-order controllers with a convex parameter set, and reduce synthesis problems to solving an LMI on the parameter space.

On solvability and numerical solutions of parameter-dependent differential matrix inequality

This paper considers the solvability condition and numerical algorithm for parameter-dependent differential affine matrix inequality. When the coefficient and solution matrices are assumed to be in a trigonometric polynomial form of the fixed order, the necessary and sufficient solvability condition is given in terms of linear matrix inequalities. The result is based on a simple idea making use of the positive real lemma to preserve positivity on an interval. Multidimensional parameter cases are also discussed.

A dually flat structure on the space of escort distributions

This note studies geometrical structure of the manifold of escort probability distributions and proves that the resultant geometry is dually flat in the sense of information geometry. We use a conformal transformation that flattens the alpha-geometry of the space of the discrete probability distributions in order to realize escort probabilities in the framework of affine differential geometry. Dual pairs of potential functions and affine coordinate systems on the manifold are derived, and the associated canonical divergence is shown to be conformal to the alpha-divergence.

Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

AbstractWe geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation.