Suboptimality and stability of linear distributed-parameter systems with finite-dimensional controllers

In order to implement feedback control for practical distributed-parameter systems (DPS), the resulting controllers must be finite-dimensional. The most natural approach to obtain such controllers is to make a finite-dimensional approximation, i.e., a reduced-order model, of the DPS and design the controller from this. In past work using perturbation theory, we have analyzed the stability of controllers synthesized this way, but used in the actual DPS; however, such techniques do not yield suboptimal performance results easily. In this paper, we present a modification of the above controller which allows us to more properly imbed the controller as part of the DPS. Using these modified controllers, we are able to show a bound on the suboptimality for an optimal quadratic DPS regulator implemented with a finite-dimensional control, as well as stability bounds. The suboptimality result may be regarded as the distributed-parameter version of the 1968 results of Bongiorno and Youla.This research was supported by the National Science Foundation under Grant No. ECS-80-16173 and by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0124. The author would like to thank the reviewer for many helpful suggestions.     

Stability of interconnected impulsive switched systems subject to state dimension variation

The contribution of this article is twofold. First we deal with the stability of continuous-time interconnected impulsive switched systems for which the dimension of the system’s state vector may change at different modes. A dwell time condition is derived to ensure the exponential convergence of the state trajectories, and when the system is subject to a non-vanishing perturbation, a superior bound of the state trajectories is provided as well. The main advantage of the solution we propose is its direct application to practical fields such as metallurgy or traffic control through the use of semi-definite programming solvers. Second, we introduce in the framework a platoon of vehicles in automated highway systems where vehicles may join or leave the platoon. Then, we detail the longitudinal stability problem of such a system and illustrate the features of the proposed stability conditions through numerical simulations.     

Exponentially stabilizing finite-dimensional controllers for linear distributed parameter systems: Galerkin approximation of infinite dimensional controllers

The Galerkin method is presented as a way to develop finite-dimensional controllers for linear distributed parameter systems (DPS). The direct approach approximates the open-loop DPS and then generates the controller from this approximation; the indirect approach approximates the infinite-dimensional stabilizing controller. The indirect approach is shown to converge to the stable closed-loop system consisting of DPS and infinite-dimensional controller; conditions are presented on the behavior of the Galerkin method for the open-loop DPS which guarantee closed-loop stability for large enough finite-dimensional approximations.     

Perturbation of the trigonometric system in L 1(0, π)

In this paper, we study a special perturbation of the function system obtained from the Fejér kernel. It is shown how this relates to the stability of bases and complete systems as well to the stability of the trigonometric system. An approximation algorithm in systems resulting from a perturbation of the original system is given.     

随机选择修理工的可修复系统指数稳定性分析

讨论了一个随机选择修理工的可修复系统解的指数稳定性,首先通过对积分微分方程组描述的可修复系统生成的系统算子的本质谱的增长性约束和扰动后本质谱半径的变化情况进行分析,进而得到了可修复系统解的指数稳定性.     

鲁棒稳定多项式的摄动界

鲁棒稳定多项式的摄动界安森建,王恩平（中国科学院系统科学研究所,北京１０００８０）基金项目：国家自然科学基金资助项目．１）作者现在通讯地址：北京大学力学系,邮政编码：１００８７１．１９９２年１月００４日收到,１９９２年４月７日收到第一次修改稿,１９９...     

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

We present a rigorous derivation of the Ericksen‐Leslie equation starting from the Doi‐Onsager equation by the Hilbert expansion method. The existence of the Hilbert expansion is related to an open question of whether the energy of the Ericksen‐Leslie equation is dissipated. On this point, we show that the energy is dissipated for the Ericksen‐Leslie equation derived from the Doi‐Onsager equation. The most difficult step is to prove a uniform bound for the remainder of the Hilbert expansion. This step is connected to the spectral stability of the linearized Doi‐Onsager operator around a critical point and the lower bound estimate for a bilinear form associated with the linearized operator. By introducing two important auxiliary operators, we can obtain the detailed spectral information for the linearized operator around all the critical points. We establish a precise lower bound of the bilinear form by introducing a five‐dimensional space called the Maier‐Saupe space.© 2015 Wiley Periodicals, Inc.     

Decentralized event-based control: Stability analysis and experimental evaluation

Event-based control aims at reducing the amount of information which is communicated between sensors, actuators and controllers in a networked control system. The feedback link is only closed at times at which an event indicates the need for an information update to retain a desired performance. Between consecutive event times the control loop acts as a continuous system, whereas at the event times it performs a state jump. Thus, the event-based control loop belongs to the class of hybrid dynamical systems. In this paper a new method for decentralized event-based control is proposed. Two methods are presented for the stability analysis of the decentralized event-based state feedback control of physically interconnected systems. The comparison principle leads to a stability criterion that provides an upper bound for the coupling strength for which the stability of the uncoupled event-based control loops implies ultimate boundedness of the interconnected event-based system. It is shown that ultimate boundedness of the event-based state-feedback loop is implied by the asymptotic stability of the continuous state-feedback system. Furthermore, it is explained how the number of events can be reduced by estimating the interconnection signals between the subsystems and two different estimation methods are proposed. The derived methods are demonstrated for a thermofluid process by simulation and experiments.     

Online optimization of LTI systems under persistent attacks: Stability,tracking, and robustness

We study the stability properties of a closed-loop system composed of a dynamical plant and a feedback controller, the latter generating control signals that can be compromised by a malicious attacker. We consider two classes of feedback controllers: a static output-feedback controller, and a dynamical gradient-flow controller that seeks to steer the output of the plant towards the solution of a convex optimization problem. In both cases, we analyze the stability properties of the closed-loop system under a class of switching attacks that persistently modify the control inputs generated by the controllers. Our stability analysis leverages the framework of hybrid dynamical systems, Lyapunov-based arguments for switching systems with unstable modes, and singular perturbation theory. Our results reveal that, under a suitable time-scale separation between plant and controllers, the stability of the interconnected system can be preserved when the attack occurs with “sufficiently low frequency” in any bounded time interval. We present simulation results in a power-grid example that corroborate the technical findings.     

Remarks on virtual bound states for semi—bounded operators

We calculate the number of bound states appearing below the spectrum of a semi—bounded operator in the case of a weak, indefinite perturbation. The abstract result generalizes the Birman—Schwinger principle to this case. We discuss a number of examples, in particular higher order differential operators, critical Schrodinger operators, systems of second order differential operators, Schrodinger type operators with magnetic fields and the Two—dimensional Pauli operator with a localized magnetic field.     

Clifford分析中一类高阶奇异积分算子关于区域摄动的稳定性

定义了Clifford分析中一类高阶奇异Teodorescu算子,利用几个重要的不等式研究了这类算子关于积分区域的边界摄动的稳定性.     

The galerkin method and feedback control of linear distributed parameter systems

In the development of feedback control theory for distributed parameter systems (DPS), i.e., systems described by partial differential equations, it is important to maintain the finite dimensionality of the controller even though the DPS is infinite dimensional. Since this dimension is directly related to the available on-line computer capacity, it must be finite (and not very large) in order to make the controller implementable from an engineering standpoint. In previous work, it has been our intention to investigate what can be accomplished by finite-dimensional control of infinite-dimensional systems; in particular, we have concentrated on controller design and closed-loop stability. The starting point for all of this is some means for producing a finite-dimensional approximation—a reduced-order model—of the actual DPS. When the “modes” of the DPS are known, the natural candidate for model reduction is projection onto the modal subspace spanned by a finite number of critical modes. Unfortunately, in real engineering systems, these modes are never known exactly and some other reasonable approximation must be used. In this paper, the model reduction is based on the well-known Galerkin procedure. We generate the Galerkin reduced-order model and develop a finite-dimensional controller from it; then we analyze the stability of this controller in closed loop with the actual DPS. Our results indicate conditions under which model reduction based on consistent Galerkin approximations will lead to stable finite-dimensional control.     

多组多滞后区间时变系数线性关联控制系统的结构与关联鲁棒镇定

本文建立了多组多滞后区间时变系数线性关联控制系统的结构概念,采用刘永清建立的鲁棒镇定等价法,给出了具有扰动结构参数的多组多滞后区间时变系数线性关联控制系统的结构与关联鲁棒镇定,同时给出了鲁棒镇定的扰动参数与滞后界限的估计公式。     

An Improved Eigenvalue Perturbation Bound on a Nonnormal Matrix and Its Application in Robust Stability Analysis

This paper presents an improved estimation of the eigenvalue perturbation bound developed by the author. The result is useful for robust stability analysis of linear control systems.     

Nash-Moser iteration and singular perturbations

We present a simple and easy-to-use Nash-Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter ε→0. The novel feature is to allow loss of powers of ε as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems in the degenerate case that the velocity of the profile is a characteristic mode of the hyperbolic operator.     

Stability of perturbed switched nonlinear systems with delays

This paper addresses the stability problems of perturbed switched nonlinear systems with time-varying delays. It is assumed that the nominal switched nonlinear system (perturbation-free system) is uniformly exponentially stable and that the perturbations satisfy a linear growth bound condition. It is revealed that there exists an upper bound of perturbation guaranteeing that the perturbed system preserves the stability property of the nominal system, locally or globally, depending on both perturbations and the nominal system itself. An example is provided to illustrate the proposed theoretical results.     

关于开或闭模型的指数稳定性

讨论了系统解的渐进稳定性和指数稳定性,证明了系统在Banach空间中生成正的C_0-半群以及系统算子0本征值的存在性,系统算子的谱点均为于复平面的左半平面且在虚轴上除0外无谱,并通过分析系统本质谱界经过扰动后的变化,进一步表明在一定条件下,系统的动态解以指数形式收敛于系统的稳态解.     

Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ^∞ bound on the ratio between their densities, Wasserstein distances, and Kullback–Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.     

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.     

Quantitative stability analysis of optimal solutions in PDE-constrained optimization

PDE-constrained optimization problems under the influence of perturbation parameters are considered. A quantitative stability analysis for local optimal solutions is performed. The perturbation directions of greatest impact on an observed quantity are characterized using the singular value decomposition of a certain linear operator. An efficient numerical method is proposed to compute a partial singular value decomposition for discretized problems, with an emphasis on infinite-dimensional parameter and observation spaces. Numerical examples are provided.