Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

Closed-form solutions of singular KYP lemma: strongly passive systems,and fast lossless trajectories

In this paper, we deal with a special class of passive systems, which possess the characteristic property of having no finite spectral zeros. We call these systems strongly passive. It is well known that, for these systems, storage functions, i.e. solutions to the linear matrix inequality (LMI) arising from the Kalman–Yakubovich–Popov (KYP) lemma, cannot be obtained by the conventional approach of algebraic Riccati equations (AREs) and Hamiltonian matrices. In this paper, we first show that a strongly passive system always admits a unique storage function. We then provide a closed-form expression for this unique storage function. Using the closed-form formula of the unique storage function we characterise the ‘lossless’ trajectories of strongly passive systems and show that such systems admit impulsive lossless trajectories on the half-line; we call them fast lossless trajectories. This adds to the existing notion that such systems do not admit any ‘slow’ lossless trajectories.     

On the C^r-Closing Lemma and the Koebe–Morse Coding of Geodesics on Surfaces

We consider the -closing lemma for vector fields with finitely many singularities on an orientable closed surface M of genus g2. Given a nontrivially recurrent trajectory, there is the corresponding geodesic having the same asymptotic directions (both negative and positive). Using the Koebe–Morse coding for the corresponding geodesic, we introduce the notion of p-expansions in the form of two sequences of nonnegative integers. The main result is the following. Suppose a vector field , r, has a nontrivially recurrent trajectory l through a point m; then there exists arbitrarily close to X (in the -topology) having a periodic trajectory through m provided that the Koebe–Morse coding of the corresponding geodesic g(l) has p-expansions of unrestricted type.     

Parametric macromodeling for S‐parameter data based on internal nonexpansivity

We propose a novel parametric macromodeling method for systems described by scattering parameters, which depend on multiple design variables such as geometrical layout or substrate features. The new concept of internal nonexpansivity for bounded real systems is introduced. It is used in combination with suitable interpolation schemes to interpolate a set of state‐space matrices, and hence poles and residues indirectly, to build accurate parametric macromodels. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed parametric macromodeling method. Copyright © 2012 John Wiley & Sons, Ltd.     

基于LMI方法的离散分段线性系统静态输出反馈控制

研究了离散分段线性系统静态输出反馈控制问题. 基于分段二次李亚普诺夫函数, 给出了静态输出反馈镇定控制器综合的新的充分的线性矩阵不等式条件. 同时, 通过使用Finsler引理, 引入一组具有特殊结构的松弛变量以减少设计保守性. 与以后方法相比, 所提出的方法具有好的性能, 并且在已有方法失效的情况下仍可用. 文中还将此方法推广到H无穷控制. 最后给出三个例子说明方法的有效性.     

干扰环境下的挠性航天器姿态跟踪控制

挠性航天器在姿态机动过程会受环境干扰力矩的作用,研究有扰情况下挠性航天器的姿态跟踪控制问题.首先,设计了干扰观测器在线估计频率已知的外部干扰信号,然后,基于干扰观测器设计了非线性反馈姿态跟踪控制器以跟踪目标姿态,并且该控制器只需要误差四元数和角速度的反馈信息.其次,采用Lyapunov方程和Barbalat引理证明了控...     

The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

Embedding of time-varying contractive systems in lossless realizations

The lossless embedding problem, also known as the Darlington synthesis or unitary extension problem, considers the extension of a given contractive system to become the partial input-output operator of a lossless system. In this paper the embedding problem is solved for discrete-time time-varying systems with finite but possibly time-varying state dimensions, for the strictly contractive as well as the boundary case. The construction is done in a state space context and gives rise to a time-varying Riccati difference equation which is shown to have a closed-form solution. As a corollary, a discrete-time Bounded Real Lemma is formulated, linking contractiveness of an input-output operator to conditions on its state realization.This research was supported in part by the commission of the EC under the ESPRITBRA program 6632 (NANA2).     

Periodic value problems of differential systemson infinite-dimensional spaces and applications to differential geometry

Periodicity questions of differential systems on infinite-dimensional Hilbert spaces arestudied via a new methodology which is based on Fan-Knaster-Kuratowski-Mazurkiewicz theorem. We obtain in this way an alternative, to the classical fixed point theory, approach to the study of such type of problems with various applications on issues of mathematical analysis and differential geometry. Two examples of such applications are included.