Reliable state-space upper bounds for the peak structured singular value

A state-space method for computing upper bounds for the peak of the structured singular value over frequency for both real and complex uncertainties is presented. These bounds are based on the positivity and Popov criteria for one-sided, sector-bounded and for norm-bounded, block-structured linear uncertainty. These criteria are restated and used to derive upper bounds for the peak structured singular value by equating the feasibility of a linear matrix inequality which involves a plant state-space realization to the strict positive realness of a transfer function. Numerical examples are given to illustrate these upper bounds. © 1998 John Wiley & Sons, Ltd.     

Robustness analysis of uncertain linear singular systems withoutput feedback control

The robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered. The problem is transformed into a robust nonsingularity problem. Based on the linear fractional transformation (LFT) approach, the robustness bounds to preserve regularity, impulse immunity, and stability are found in terms of the structured singular value μ with respect to parametric uncertainties. The LFT approach provides a unified framework for robustness analysis of both uncertain linear continuous/discrete-time singular systems     

Robustness under bounded uncertainty with phase information

The authors consider uncertain linear systems where the uncertainties, in addition to being bounded, also satisfy constraints on their phase. In this context, the authors define the “phase-sensitive structured singular value” (PS-SSV) of a matrix and show that sufficient (and sometimes necessary) conditions for stability of such uncertain linear systems can be rewritten as conditions involving PS-SSV. They then derive upper bounds for PS-SSV, computable via convex optimization. They extend these results to the case where the uncertainties are structured (diagonal or block-diagonal, for instance)     

Robust stability of state-space models with structureduncertainties

A method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters. A method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed. The method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when cases with one or two uncertain parameters are considered. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems     

System zeros analysis via the Moore-Penrose pseudoinverse and SVDof the first nonzero Markov parameter

A new characterization of system zeros of an arbitrary linear system described by a state-space model S(A, B, C, D) is presented. The transmission zeros are characterized as invariant zeros of an appropriate strictly proper system with a smaller number of inputs and outputs than the original system. The approach is based on singular value decomposition (SVD) of the first nonzero Markov parameter. This result together with characterization of invariant and decoupling zeros, based on the Moore-Penrose inverse of the first nonzero Markov parameter and the Kalman canonical decomposition theorem, provided in the first part of the paper yield a complete characterization of system zeros of an arbitrary multi-input/multi-output system     

Time response bounds for linear parametric uncertain systems

Parametric uncertainties in linear time-invariant state-space models introduce perturbations in the eigenvalues and eigenvectors (eigenstructure), which modify the properties of a nominal system. Besides stability by the perturbed eigenvalues, controllability, observability, transient and steady-state values, etc., are some basic properties of a linear system significantly influenced by the perturbed eigenvectors. In this paper, time response analysis of parametric uncertain systems is considered. First the bounds on eigenvectors and parametric uncertainties are established for the perturbed eigenvalues to reside in a set of distinct circular regions. Secondly, analysis of such parametric uncertain systems is discussed by presenting bounds on the perturbed state variables in time coordinate. These bounds are then investigated for norm-bounded and structured uncertainty classifications. Input-output analysis of uncertain systems is illustrated using the initial conditions and input functions.     

On-line structured subspace identification with application to switched linear systems

This article is concerned with the identification of switched linear multiple-inputs–multiple-outputs state-space systems in a recursive way. First, a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features. More precisely, it does not require any singular value decomposition but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input–output models identification. Second, it is shown that this method can be implemented on-line to track both the range space of the extended observability matrix and its dimension and thereby, the system matrices. Third, by making use of an on-line switching times detection strategy, this method is applied to blindly identify switched systems and to label the obtained submodels. Simulation results on noisy data illustrate the abilities and the benefits of the proposed approach.     

Algebraic approach to controller design using structured singular value

The aim of this paper is to show an algebraic approach to controller design using the structured singular value (denoted μ) as a robust stability and performance indicator. The algebraic μ-synthesis is applied to three different problems–time-delay systems, the HIMAT vehicle model and the two-tank system. A way of treating general delayed systems with uncertain time delays via the linear fractional transformation is shown. A simple controller is derived, which handles uncertain time delay in both the numerator and denominator of an anisochronic system. The overall performance is verified by simulations for all systems and compared with the D–K iteration.     

Lipschitz广义非线性系统观测器设计

研究一类广义非线性系统的观测器设计问题．首先讨论了半正定Lyapunov函数下指数1广义非线性系统稳定及渐近稳定性,然后对一类由线性和Lipschitz非线性项组成的广义非线性系统,给出了渐近稳定观测器存在的条件,并把观测器反馈增益矩阵的设计归结为广义线性系统容许控制以及奇异值计算问题,证明了若容许广义线性系统矩阵的最小奇异值大于系统的Lipschitz常数,容许控制器增益矩阵就是待求的观测器反馈增益矩阵。     

时滞相关型离散时变时滞奇异系统的鲁棒镇定

讨论含参数不确定的离散时变时滞奇异系统的时滞相关的鲁棒状态反馈稳定化问题. 在一系列等价变换下, 阐述了其和一个不确定正常线性离散时变时滞系统的鲁棒状态反馈稳定化问题的等价关系;利用矩阵不等式方法, 给出一个对所有容许的不确定, 使得闭环系统正则、因果且稳定的时滞相关鲁棒状态反馈稳定化控制器存在的充分条件以及无记忆状态反馈控制器的一个解.     

多变量自校正控制器的状态空间设计

本文针对输入、输出维数相同的多变量随机系统提出了两种状态空间自校正控制器-- 多变量状态空间极点配置自校正控制器和多变量状态空间自校正跟踪控制器.文中给出两种 控制器结构和理论分析,前者适用于稳定/不稳定、最小相位/非最小相位线性时不变多变量系 统,后者适用于稳定/不稳定、最小相位线性时不变多变量系统,简单的结构和优良的性能使两 种算法能较好地应用于实时系统.     

A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process

This paper presents a methodology for system identification of continuous-time state-space models from finite sampled input-output signals. The estimation problem of the consecutive time-derivatives and integrals of the input-output signals is considered. The appropriate frequency characteristcs of a linear filtering based on the Poisson moment functionals in regards to the derivative or integral estimation problem is shown. The proposed method combines therefore the Poisson moment functionals technique with subspace based state-space system identification methods. The developed algorithm is based on a generalized singular value decomposition to compensate the noise colouring caused by the linear prefiltering of the input-output data. Rules of thumb are presented to choose the design parameters and new regards to the selection of the Poisson filter cut-off frequency are introduced. Finally, the proposed method is applied to a multivariable winding processes. The experimental results emphasize the applicability of the developed methodology.     

On the robustness of uncertain time-delay systems with structured uncertainties

This paper considers the robust stability and robust performance of uncertain delay systems subject to possibly structured uncertainties using structured singular values with phase information. Computationally efficient sufficient conditions for robust stability and performance problem are derived.     

Explicit solution to the singular discrete-time stationary linear filtering problem

A closed form solution to the stationary discrete-time linear filtering problem is obtained explicitly in terms of the system state-space matrices in the limiting singular case where the measurement noise tends to zero. Simple expressions, in closed form, are obtained for the Kalman gain matrix both for uniform and nonuniform rank systems and the explicit eigenstructure of the Kalman filter closed loop matrix is derived. The minimum error covariance matrices of the a priori and a posteriori filtered estimates are obtained using this special eigenstructure, and a remarkably different behavior of the solution in the minimum- and nonminimum-phase cases is found.     

一类不确定离散奇异系统的鲁棒稳定化

讨论了离散奇异系统矩阵E中含时不变参数不确定的鲁棒状态反馈稳定化问题.首先,在一系列等价变换下,阐述了其和一个不确定正常线性离散系统的鲁棒状态反馈稳定化问题的等价关系;然后,利用线性矩阵不等式(LMI)方法,给出了鲁棒状态反馈稳定化控制器存在的一个充分必要条件,控制器的设计方法及控制器的一个解;最后,通过一个数值算例验证了本设计方法的有效性.     

Reduced explicit constrained linear quadratic regulators

In this note, it is studied how structural properties of certain linear systems can be exploited to derive reduced dimension multiparametric quadratic programs that lead to explicit piecewise linear feedback solutions to the state and input constrained linear quadratic regulation problem. The reduced dimensionality typically results in suboptimal controllers of lower complexity, with associated computational advantages in the online implementation. At the heart of the methods are state-space projections based on the singular value decomposition.     

Singular linear-quadratic optimal control problem for a class of discrete singular systems with multiple time-delays

The singular linear-quadratic optimal control problem for a class of discrete-time linear singular systems with multiple time-delays is investigated. The equivalent relationship between this problem and the non-singular linear-quadratic problem for standard state-space discrete-time linear systems without time-delay is derived, and the necessary and sufficient condition guaranteeing this equivalent relationship is also given. The optimal control is synthesized as state feedback, and the closed-loop system is regular, impulse-free and without time-delay. An example is given to show the validity of the proposed method.     

The structured phase margin for robust stability analysis of linear systems with phase and time delay uncertainties

In this paper, we introduce the notion of the structured phase margin for characterising stability margins for a dynamical system with block-structured phase uncertainty. Since the structured phase margin may in general be difficult to compute, we derive an easily computable lower bound in terms of a generalised eigenvalue problem. This bound is constructed by choosing stability multipliers that are tailored to the structure of the phase uncertainty. The results in this paper complement and generalise phase information results presented in the literature involving principal phases, multivariable phase margin, phase spread, phase envelope, phase matching, and phase-sensitive structured singular value. Finally, using the structured phase margin, we present new and improved delay-dependent stability criteria for stability analysis of time-delay systems.     

Decoupling of structured systems by parameter-independentprecompensation and state feedback

In this paper, structured systems described by state-space models are considered. For these systems, the entries of the state-space model matrices are assumed to be either fixed zeros or free independent parameters. We study the dynamic decoupling of structured linear systems. It turns out that a large class of systems which is not state feedback decouplable can be made decouplable by the addition of a simple structured precompensator. Such a precompensator then has the property to be independent of the system parameters. We give a necessary and sufficient condition for a structured system to be made feedback decouplable using a parameter independent precompensator. Furthermore, we prove that when such a solution exists, the precompensator can be chosen to be diagonal. In this case the precompensator is constructed via a simple combinatorial algorithm. Our approach is based on a graph representation of the structured system. All the results of this paper are related to some particular shortest sets of input-output paths in the graph associated with the structured system