Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

Symmetricity of the solution of semidefinite programming

Symmetricity of an optimal solution of Semi-Definite Programming (SDP) is discussed based on the symmetry property of the central path that is traced by a primal-dual interior-point method. A symmetric SDP is defined by operators for rearranging elements of matrices and vectors, and the solution on the central path is proved to be symmetric. Therefore, it is theoretically guaranteed that a symmetric optimal solution is always obtained by using a primal-dual interior-point method even if there exist other asymmetric optimal solutions. The optimization problem of symmetric trusses under eigenvalue constraints is shown to be formulated as a symmetric SDP. Numerical experiments illustrate convergence to strictly symmetric optimal solutions.     

Intelligent robust control for uncertain nonlinear time-varying systems and its application to robotic systems.

This paper addresses the problem of designing adaptive fuzzy-based (or neural network-based) robust controls for a large class of uncertain nonlinear time-varying systems. This class of systems can be perturbed by plant uncertainties, unmodeled perturbations, and external disturbances. Nonlinear H(infinity) control technique incorporated with adaptive control technique and VSC technique is employed to construct the intelligent robust stabilization controller such that an H(infinity) control is achieved. The problem of the robust tracking control design for uncertain robotic systems is employed to demonstrate the effectiveness of the developed robust stabilization control scheme. Therefore, an intelligent robust tracking controller for uncertain robotic systems in the presence of high-degree uncertainties can easily be implemented. Its solution requires only to solve a linear algebraic matrix inequality and a satisfactorily transient and asymptotical tracking performance is guaranteed. A simulation example is made to confirm the performance of the developed control algorithms.     

An algorithm for robust explicit/multi-parametric model predictive control

A new algorithm for robust explicit/multi-parametric Model Predictive Control (MPC) for uncertain, linear discrete-time systems is proposed. Based on previous work on Dynamic Programming (DP), multi-parametric Programming and Robust Optimization, the proposed algorithm features, (i) a DP reformulations of the MPC optimization problem, (ii) a robust reformulation of the constraints, and (iii) a multi-parametric programming step, where the control variables are obtained as explicit functions of the state variable, such that the state and input constraints are satisfied for all admissible values of the uncertainty. A key feature of the proposed procedure is that, as opposed to previous methods, it only solves a convex multi-parametric programming problem for each stage of the DP procedure.     

Robust one-step receding horizon control for constrained systems

This paper proposes a robust receding horizon control scheme for discrete-time uncertain linear systems with input and state constraints. The control scheme is based on the minimization of the worst-case one-step finite horizon cost with a finite terminal weighting matrix. It is shown that the proposed receding horizon control robustly asymptotically stabilizes uncertain constrained systems under some matrix inequality conditions on the terminal weighting matrices. This robust receding horizon control scheme has a larger feasible initial-state set and a more general structure than existing robust receding horizon controls for uncertain constrained systems under the same design parameters. The proposed controller is obtained using semidefinite programming. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic stabilization of a class of uncertain composite systems

A class of stabilizing, generalized feedback controls is presented for a class of uncertain dynamical systems. The uncertain systems are based on a composite prototype system consisting of two nonlinearly coupled subsystems, with a non-asymptotically-stabilizable linearization. To encompass all possible realizations of uncertainty, a problem formulation based on differential inclusions is adopted. The generalized feedback controls, described in terms of set-valued maps, have practical analogues in the form of discontinuous feedback controls. An intrinsic element of the control strategy is designed to render a prescribed nonlinear manifold, containing the state origin, invariant and globally finite-time attractive. For each uncertain system, the generalized feedback controls guarantee global asymptotic stability of the zero state.     

Efficient semidefinite solutions for TDOA-based source localization under unknown PS

Two efficient solutions via Semi-Definite Programming (SDP) are proposed for source localization problems using time difference of arrival (TDOA)-based ranging measurements when the propagation speed (PS) is unavailable and considered as a variable. For this problem, we propose a relaxed SDP (RSDP) solution, the performance of which is suboptimal. Accordingly, we propose a two-stage SDP method to improve the performance by applying the rank-reduction method. Besides, we also propose a penalty function-based SDP (PF-SDP) by introducing the penalty term. By doing so, the cost function becomes tighter so that the solution performs better. The simulated results show that the performance of two-stage SDP is sufficiently close to the Cramér-Rao Lower Bound (CRLB) accuracy at high noise levels. The PF-SDP outperforms the two-stage SDP in the presence of low noise levels.     

不确定时滞系统基于观测器的鲁棒控制器设计

研究了不确定线性时滞系统的状态观测器和基于观测器的鲁棒控制器设计问题，其中不确定性是时变的，通过构造增广系统，利用线性矩阵不等式方法，获得了该不确定系统存在状态观测器和基于观测器的鲁棒控制器的充分条件，同时给出了相应的状态观测器和基于观测器的鲁棒控制器，所给示例说明了本文方法的设计步骤和有效性。     

Forwarding, backstepping, and finite spectrum assignment for time delay systems

The nonlinear control ideas and results of forwarding and backstepping have been recently extended to systems with time delay. In this paper we discuss how to apply these tools to design robust controls for linear delay systems. We first show that the integral cross-term forwarding produces a feedback law that achieves finite spectrum of the closed loop system. Combining it with backstepping provides a structured design method for a large class of systems with time delay. Robustness of the controller to uncertainties, including uncertain time delay, is illustrated by an example.     

Hybrid multilevel programming with uncertain random parameters

Multilevel programming is developed for modeling decentralized decision-making processes. For different management requirements and risk tolerances of different-level decision-makers, the decision-making criteria applied in different levels cannot be always the same. In this paper, a hybrid multilevel programming model with uncertain random parameters based on expected value model (EVM) and dependent-chance programming (DCP), named as EVM–DCP hybrid multilevel programming, is proposed. The corresponding concepts of Nash equilibrium and Stackelberg–Nash equilibrium are given. For some special case, an equivalent crisp mathematical programming is proposed. An approach integrating uncertain random simulations, Nash equilibrium searching approach and genetic algorithm is designed. Finally, a numerical experiment of uncertain random supply chain pricing decision problem is given.     

Robustness of uncertain systems in the absence of matching assumptions

We consider a class of uncertain dynamical systems containing uncertain elements. These may be due to model-parameter uncertainty, input disturbance and measurement noise. We require no a priori information on the uncertain elements except that their restraint sets are assumed to be known and compact. This paper describes a general approach for designing controls to assure practical stability in the absence of matching assumptions (which most previous results have needed). The result is applicable to non-linear systems. A specialization to linear systems is then given and a comparison is made with previous work.     

A Frequency Condition for the Generalized Recurrent Difference in the Synthesis of H∞-Suboptimal, Decentralized, and Robust Regulators

A new approach to designing control laws under uncertainty produces easily verifiable tests for their generalized recurrent differences and does not require the solution of matrix equations or inequalities. This test is expressed as a frequency inequality, which is verified by verifying whether a polynomial has a real positive root of odd multiplicity. The underlying principles of the method are the necessary and sufficient conditions for a given stabilizing linear feedback to be the local minimax control satisfying the Bellman–Isaac inequality in a differential game with a general quadratic functional. The design of controls for Lur'e systems and systems with uncertain bounded parameters as well as the design of H -suboptimal regulators and decentralized controls for multiconnected systems with unknown cross-coupling are shown to involve a local minimax control problem.     

Asymptotic stability of controlling uncertain dynamical systems

Asymptotic stabilization of general uncertain dynamical systems is investigated. A new class of continuous feedback controls is proposed to guarantee asymptotic stability for any uncertain systems whose nominal system is uniformly asymptotically stable. The analysis is based on a new theoretical result on asymptotical stability. The required information about uncertain dynamics in the system is merely that the uncertainties are bounded in euclidean norm by a known function of the system state.     

IGP/MPLS混合的IP网络不确定流量规划方法

随着IP网络的迅速发展,网络的业务量越来越呈不确定性和不可预测性。针对此问题,提出了以软管模型为基础的不确定业务量模型下的IGP/MPLS混合IP流量规划问题的混合整数规划(MIP)模型,并将该模型分解成权重设计子问题和流量分配子问题进行求解。实验结果表明,该方法能在标签交换路径(LSP)较少的情况下达到较理想的优化效果。     

不确定系统鲁棒LQ设计的稳定裕度分析和工程应用研究

针对不确定线性系统, 研究鲁棒LQ设计的稳定裕度分析问题. 首先提出不确定系统α_0裕度稳定鲁棒界, 针对结构和非结构不确定系统, 分别给出了鲁棒LQ控制系统的鲁棒稳定裕度性分析方法, 进一步给出对结构不确定系统优化鲁棒界的算法. 最后是造纸打浆过程实例的鲁棒稳定裕度性分析以及相应仿真结果.     

A reduction paradigm for output regulation

The goal of this paper is to provide a reduction paradigm for the design of output regulators which can be of interest for nonlinear as well as linear uncertain systems. The main motivation of the work is to provide a systematic design tool to deal with non‐minimum‐phase uncertain systems for which conventional high‐gain stabilization methods are not effective. The contribution of the work is two‐fold. First, this work extends a previous reduction paradigm for output regulation of nonlinear systems. Furthermore, in the case of the uncertain controlled dynamics being linear, we show how the proposed framework leads to a number of systematic design tools of interest for non‐minimum‐phase linear systems affected by severe uncertainties. A numerical control example of a linearized model of an inverted pendulum on a cart is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Recursive scaling design for robust global nonlinear stabilization via output feedback

This paper proposes a novel approach to robust backstepping for global stabilization of uncertain nonlinear systems via output feedback. The design procedure developed in this paper is based on the concept of state‐dependent scaling, which handles output‐feedback stabilization problems of strict‐feedback systems with various structures of uncertainties in a unified way. The proposed method is suitable for numerical computation. The theory of the method employs the Schur complements formula instead of Young's inequality and completing the squares. This paper shows a condition of allowable uncertainty size under which an uncertain system is globally stabilized by output feedback. A class of systems is shown to be always globally stabilizable for arbitrarily large nonlinear size of uncertainties. A recursive procedure of robust observer design for such a class of uncertain systems is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

不确定性时滞系统的鲁棒镇定控制器设计——LMI 方法*

采用线性矩阵不等式方法,研究一类不确定性动态系统,这类系统具有多重时变状态时滞多重时变控制输入时滞,其不确定性满足范数有界条件。我们得到一了类不确定性时滞系统的可状态反馈镇定的充分条件。     

基于非线性观测器设计的混沌同步控制

提出一种基于非线性观测器设计的混沌同步控制方法,给出了一类含有不确定参数混沌系统的观测器实现条件.该控制策略可满足混沌系统中存在一些不确定参数的要求.通过对Lorenz混沌系统的理论分析和数字仿真,证明了其有效性.     

不确定线性系统LQ 设计的鲁棒性分析

针对不确定线性系统,研究LQ设计的鲁棒最优性分析问题。在建立Riccati方程和频域传递子数矩阵关系的基础上,给出了不确定闭环系统鲁棒最优的一个充分条件。对结构和非结构不确定系统,分别给出了LQ控制系统的鲁棒最优性分析方法,进一步给出了改进结构不确定系统鲁棒最优性的一种优化方法。实例分析及仿真结果表明了所给结论的正确性和有效性。