Suboptimal robust synthesis for MIMO plant under coprime factor perturbations

Suboptimal robust synthesis for MIMO nominal system under coprime factor perturbations is considered in classical and non-classical statements. In the classical statement, weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer. Suboptimal synthesis within ε tolerance is reduced to the solution of log₂(1/ε) standard mixed sensitivity problems of ℓ₁ optimization. In the non-classical statement, the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1/ε mixed sensitivity problems.     

具有未知干扰的互质因子摄动不确定性系统最优鲁棒调节: 优化设计及连续性

研究了含未知干扰的互质因子摄动离散时间系统达到最优性能鲁棒性的控制器设计及这种设计的连续性. 给出了含未知干扰的互质因子摄动离散时间系统最优鲁棒调节器的一种设计方法. 分析了含未知干扰互质因子摄动系统的性能鲁棒性优化设计的连续性, 证明了互质因子摄动系统性能鲁棒性的优化设计作为从设计模型到鲁棒控制器的映射是连续的, 并且, 若设计模型的参数集是紧的, 这种映射是一致连续的.     

互质因子摄动系统最优l1鲁棒控制:连续性与自适应控制

针对具有互质因子摄动和未知干扰的离散时间系统研究了一种自适应鲁棒控制策略.本文的主要工作包括三个方面.首先建立了互质因子摄动系统最优l1鲁棒控制设计的连续性.然后,提出了一种带变死区的参数鲁棒估计投影算法.最后,结合所提出的参数估计算法和最优l1鲁棒控制,利用确定性等价原理提出了互质因子摄动系统的一种新的自适应鲁棒控制方法.基于本文建立的l1优化设计的连续性,证明了自适应鲁棒控制的全局稳定性,给出了自适应控制系统稳定性的后验可计算条件.     

互质因子摄动系统最优е1鲁棒控制: 连续性与自适应控制

针对具有互质因子摄动和未知干扰的离散时间系统研究了一种自适应鲁棒控制策略. 本文的主要工作包括三个方面. 首先建立了互质因子摄动系统最优е1鲁棒控制设计的连续性. 然后,提出了一种带变死区的参数鲁棒估计投影算法. 最后, 结合所提出的参数估计算法和最优е1鲁棒控制,利用确定性等价原理提出了互质因子摄动系统的一种新的自适应鲁棒控制方法. 基于本文建立的е1优化设计的连续性, 证明了自适应鲁棒控制的全局稳定性,给出了自适应控制系统稳定性的后验可计算条件.     

Coprime factors reduction methods for linear parameter varying and uncertain systems

We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured ℓ₂-induced stability constraint required in the standard nonfactored case. Reduction error bounds in the ℓ₂-induced norm of the factorized mapping are given.     

On continuity and consistency of ℓ∞ optimal models

This paper is concerned with the question of continuity of the mapping from observed time series to models. The behavioral framework is adopted to formalize a model identification problem in which the observed time series is decomposed into a part explained by a model and a remaining part which is ascribed to noise. The misfit between data and model is defined symmetrically in the system variables and measured in the ℓ_∞ or amplitude norm. With the introduction of proper notions of convergence, it is shown that the misfit function continuously depends on both the data and the model. Two notions of consistency are formalized and it is shown that the continuity of the misfit function implies a consistent identification of optimal and suboptimal models.     

Permutation routing in double-loop networks: design and empirical evaluation

A double-loop network is an undirected graph whose nodes are integers 0,1,…,n−1 and each node u is adjacent to four nodes u±h₁(mod>n), u±h₂(mod>n), where 0<h₁<h₂<n/2. There are initially n packets, one at each of the n nodes. The packet at node u is destined to node π(u), where the mapping uπ(u) is a permutation. The aim is to minimize the number of routing steps to route all the packets to their destinations. If ℓ is the tight lower bound for this number, then the best known permutation routing algorithm takes, on average, 1.98ℓ routing steps (and 2ℓ routing steps in the worst-case).Because the worst-case complexity cannot be improved, we design four new static permutation routing algorithms with gradually improved average-case performances, which are 1.37ℓ, 1.35ℓ, 1.18ℓ, and 1.12ℓ. Thus, the best of these algorithms exceeds the optimal routing by at most 12% on average.To support our algorithm design we develop a program which simulates permutation routing in a network according to the given topology, routing model as well as communication pattern and measure several quality criteria. We have tested our algorithms on a large number of double-loop networks and permutations (randomly generated and standard).     

Minimum ℓ1, ℓ2, and ℓ∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations

Cadzow, J. A., Minimum ℓ₁, ℓ₂, and ℓ_∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations, Digital Signal Processing12 (2002) 524–560Many practical problems encountered in digital signal processing and other quantitative oriented disciplines entail finding a best approximate solution to an overdetermined system of linear equations. Invariably, the least squares error approximate solution (i.e., minimum ℓ₂ norm) is chosen for this task due primarily to the existence of a convenient closed expression for its determination. It should be noted, however, that in many applications a minimum ℓ₁ or ℓ_∞ norm approximate solution is preferable. For example, in cases where the data being analyzed contain a few data outliers a minimum ℓ₁ approximate solution is preferable since it tends to ignore bad data points. In other applications one may wish to determine an approximate solution whose largest error magnitude is the smallest possible (i.e., a minimum ℓ_∞ norm approximate solution). Unfortunately, there do not exist convenient closed form expressions for either the minimum ℓ₁ or the minimum ℓ_∞ norm approximate solution and one must resort to nonlinear programming methods for their determination. Effective algorithms for determining these two solutions are herein presented (see Cadzow, J. A., Data Analysis and Signal Processing: Theory and Applications).     

Nonlinear state feedback for ℓ optimal control

This paper considers ℓ¹ optimal control problems with full state feedback. In contrast to ^∞ optimal control, previous work has shown that linear ℓ¹ optimal controllers can be dynamic and arbitrarily high order. However, this paper shows that continuous memoryless nonlinear state feedback perform as well as dynamic linear state feedback. The derivation, which is nonconstructive, relies on concepts from viability theory.     

On the pole location of a system designed by robust stability-degree assignment

In this paper, we examine the pole location of the feedback system composed of the nominal plant and the H_∞ central controller designed by the robust stability-degree assignment. Namely, the exact pole location at γ=∞ and the behavior near the infimum of γ are clarified where γ is the upper bound of the H_∞ norm constraint. The original design goal is to stabilize the plant against additive perturbations with the regional pole placement condition Re s<−α, and the design problem is reduced to the one-block H_∞ control problem.     

On system gains for linear and nonlinear systems

System gains, and bounds for system gains, are determined for stable linear and nonlinear systems in different signal setups which include ℓ_p signal setups and certain persistent signal generalizations of the ℓ₂ and ℓ₁ signal setups. These results show that robust H_∞ and ℓ₁ control generalize to very versatile persistent signal settings. Relationships between different system gains are also derived. Finally, an application of nonlinear system gain bounds is given by establishing induced ℓ_∞ modelling error bounds for a class of (generalized) piecewise linear systems approximated with simpler linear time-invariant models.     

Robust stabilization of nonlinear systems via stable kernel representations with L2-gain bounded uncertainty

The approach to robust stabilization of linear systems using normalized left coprime factorizations with _∞ bounded uncertainty is generalized to nonlinear systems. A nonlinear perturbation model is derived, based on the concept of a stable kernel representation of nonlinear systems. The robust stabilization problem is then translated into a nonlinear disturbance feedforward _∞ optimal control problem, whose solution depends on the solvability of a single Hamilton-Jacobi equation.     

Continuity properties of LQG optimal controllers

It is shown that the LQG optimal controller is a continuous function of the plant. The result is proved for a class of plants which contains the class of strictly proper finite-dimensional plants. The topology employed is the one generated by convergence of the closed-loop transfer functions in an induced _∞ sense. This topology is slightly stronger than the usual ₂ gap metric convergence on transfer functions.     

Tight error bounds for projection algorithms in conditional set membership estimation

In set membership estimation, conditional problems arise when the estimate must belong to a given set of assigned structure. Conditional projection algorithms provide estimates that are suboptimal in terms of the worst-case estimation error. In order to precisely evaluate the suboptimality level of these estimators, tight upper bounds on the estimation errors must be computed as a function of the conditional radius of information, which represents the minimum achievable error. In this paper, tight bounds are derived for ℓ_∞ and ℓ₁ estimation errors, in a general setting which allows to consider any compact set of feasible problem elements and linearly parameterized estimates.     

Modeling the system of suboptimal robust tracking under unknown upper bounds on the uncertainties and external disturbances

This paper addresses a problem of suboptimal robust tracking for a discrete-time plant under unknown upper bounds on external disturbances. The transfer function of the nominal plant model is assumed to be known, whereas the upper bounds on the external disturbance, measurement noise, and coprime factor perturbations are assumed to be unknown. The results of numerical modeling of the algorithm of suboptimal robust tracking based on relaxed verification of the estimates of upper bound in the closed loop and the use of control criterion associated with such verification as the identification criterion are presented and analyzed. The results of numerical modeling illustrate efficiency of the proposed method of control design.     

On robust stabilization regions of unstructured uncertainties

Relations on the robust stabilization regions of four main forms of unstructured uncertainties are investigated. The robust stabilization regions represented by three other perturbations are derived from coprime factor perturbation and additive perturbation, respectively. Furthermore, it is shown that the normalized coprime factor H^∞ robust controller can also be explained in additive and multiplicative perturbations.     

H∞回路成形设计的鲁棒性

回路成形法设计中是用互质因子摄动来表示系统不确定性的. 文中对这种互质因子摄动进行了较详尽的分析, 指出系统中的弱阻尼模态会增大互质因子摄动的范数, 因而降低了允许的摄动值, 使系统实际上失去了鲁棒性. 所以H∞回路成形设计并不一定像所期望的那样具有鲁棒性. 文中并用一个参数摄动下的鲁棒性为例来进行说明.