A constructive algorithm for the complete set of compensators for two-dimensional feedback system design

In the recent factorization approach to linear feedback system design, it has been found that in order to completely solve a generalized Ω-stabilization problem, a coprime condition in the ring of Ω-stable transfer functions must be solved. In this note, two-dimensional (2- D) systems are considered, and this Ω-stable coprime condition is solved for Ω-stable rings. A well-known 2-D zero-coprime condition is a special case of this condition. This is the first constructive method for obtaining the complete set of solutions, or compensators, for 2-D feedback system design. Some examples of systems treated as 2-D systems are: temporal-spatial, spatial-spatial, delay-differential, multipass, etc.     

The disturbance decoupling problem in decentralized linear multi-variable systems by local dynamic feedback

Following the works of Hu and Zheng [7] on the problem of disturbance decoupling in decentralized systems with non-dynamic feedback (DDPP), this paper studies the problem of disturbance decoupling in decentralized linear multi-variable systems with one or two local dynamic feedbacks (abbreviated as DDDPDC1 or DDDPDC2 respectively). A counter-example is given to indicate that the main results of Cury [1] are incorrect. Developing the concepts such as structure subspace pair and structure subspace vector, the necessary and sufficient conditions of DDDPDC1 and DDDPDC2 using low order compensators is derived.     

Compensator synthesis using (C,A,B)-pairs

This paper gives a general framework (in the geometric style) for the design of compensators for linear multivariable systems. Basic is the notion of a "(C,A,B)-pair of subspaces." This concept is newly defined here, and we give its simplest application (to the problem of disturbance decoupling by observation feedback). We then formulate a general compensator synthesis principle and show how several well-known synthesis techniques can be derived from it. We also show that "almost all" compensators can be interpreted as observer-based compensators, and discuss the relation between the compensator problem and the stable cover problem.     

The disturbance decoupling problem for time-delay nonlinear systems

In this paper, the disturbance decoupling problem (DDP) for a class of SISO nonlinear systems with multiple delays in the input and the state is studied. A pioneering mathematical approach is introduced for this class of systems and is claimed to be the cornerstone of the problem. Necessary and sufficient conditions are given for the existence of a bicausal feedback that solves the DDP. Sufficient conditions for the existence of a solution within other classes of compensators are included as well     

Decoupling by dynamic measurement feedback with stability:necessary and sufficient conditions

The noninteracting control problem of linear time-invariant multivariable systems by dynamic measurement feedback is solved in the state-space geometric approach. The generalization of W.M. Wonham's EDP (1979) is considered. The problem of the reduction of compensator's order is not treated, but the solution derived is able to solve simultaneous problems like the decoupling and the disturbance decoupling problems     

Optimal control of 2-D systems

Two-dimensional (2-D) optimal control theory that parallels one-dimensional (1-D) optimal control is developed. A generalized performance measure suited to 2-D systems is introduced. The canonical equations associated with this performance measure and a general nonlinear model are obtained. The 2-D linear quadratic regulator problem is formulated, and its canonical equations are derived for the Roesser model. An earlier result by T. Kaczorek and J. Klamka (1986) for the solution of the minimum-energy problem with fixed-final local state is rederived using this approach. A new problem, minimum-energy with fixed-final-pass local states is formulated and solved, and a numerical example is given     

Status of noninteracting control

The current status of decoupling theory for linear constant multivariable systems is described. The subject is treated in vector space terms and appropriate background concepts including invariant and controllability subspaces are discussed. Suggestions are given for translating vector space operations into matrix operations suitable for computation. The controllability subspace is used to formulate the restricted (static compensation) decoupling problem. Although the most general version of this problem is unsolved, there are known solutions for three special cases. A complete solution to the extended (dynamic compensation) decoupling problem is known. If a linear constant multivariable system can be decoupled at all, by any means whatever, then it can always be decoupled using linear dynamic compensation. The internal structure of a decoupled system is described in simple matrix terms. Using this representation, it is possible to characterize the system pole distributions which may be achieved while preserving a decoupled structure. A procedure is outlined for synthesizing a dynamic compensator of low order which will decouple a system. The procedure actually provides minimal order decoupling compensators for systems in which the number of open-loop inputs equal the number of outputs to be controlled.     

General response formula and minimum energy control for the generalsingular model of 2-D systems

The general response formula for the general singular model of 2-D linear systems is derived. The concepts of local reachability and local controllability are extended to the singular model. Necessary and sufficient conditions for local reachability and local controllability are established. The minimum energy control problem for the singular model is formulated and solved     

A parametric frequency domain approach to decoupling with coupled rows

This contribution considers the incomplete decoupling of non-minimum phase and non-decouplable systems by static state feedback, where a stable diagonal decoupling is not possible. By introducing a coupled row in the reference transfer matrix, so that one output is affected by several reference inputs, an incompletely decoupled but internally stable closed loop system is obtained. This decoupling problem is solved using a parametric approach to the design of state feedback controllers in the frequency domain. Explicit expressions are derived for the design parameters of the state feedback controller achieving the decoupled reference transfer matrix. A simple example demonstrates the proposed design procedure.     

模糊系统的串联补偿解耦

本文提出并解决模糊关系系统的串联补偿解耦问题,给出了串联解耦补偿器的结构、实现 解耦的一个充分条件的串联补偿解耦问题的解存在的一个充要条件.     

分散化系统中的结构关联集及其应用

本文提出了分散化结构关联集(DSIS)的概念.它揭示了具有二个控制站的分散化系 统中所具有不同类型的不变子空间之间的关系,并可用来确定分散化动态补偿器的结构.最 后讨论了它在分散化动态补偿器干扰解耦问题中的应用.     

Linear feedback decoupling--Transfer function analysis

The problem of linear system decoupling is examined based on recent results on linear feedback. New insight is obtained, through which resolution of the decoupliug problem is accomplished by calculations, performed directly on the given transfer matrix. Computation of the decoupling compensators follows by easy constructions. The problem of feedback block decoupling with internal stability, is also formulated and resolved.     

2－D系统的干扰解耦

本文讨论了２－Ｄ系统的干扰解耦问题（ＤＤＰ），将１－Ｄ系统理论中的有关结果推广到了２－Ｄ线性离散常系数一般模型（２－ＤＧＭ），得到了问题可解的充分条件和必要条件以及相应的算法．这些结果较好地改进了现有结果     

离散时间奇异系统的可测扰动解耦

为离散时间广义非线性控制系统的可测扰动提供一种反演算法．运用一类正则动态补偿器解决了系统的解耦问题，并且证明系统既可利用动态反馈进行解耦，也可利用拟表态反馈进行解耦．     

Squaring down by static and dynamic compensators

Given a system, which is not necessarily invertible and which has an unequal number of inputs, a method of `squaring down', that is a method of designing pre-compensators and postcompensators such that the resulting system has an equal number of inputs and outputs and is invertible, is presented. The compensators are, in general, dynamic and have the property that the additional finite zeros induced by them are assignable to the open left-half complex plan. Furthermore, the compensators are asymptotically stable, and hence do not deteriorate the robustness and performance of an eventual feedback design. Also, the compensator design preserves the stabilizability, detectability, and minimum-phase properties and the infinite zero structure of the original system. Thus, a method of designing nonsquare systems by converting them to square invertible systems is introduced. Two applications of such a design philosophy, (1) diagonal decoupling with state feedback; and (2) almost disturbance decoupling with output feedback, are pointed out     

Feedback decoupling-controller design of 3-D systems in state space

This paper solves the input-output decoupling problem of three-dimensional (3-D) systems formulated in state-space representation. The control policy adopted is of the static state feedback type u=Kx+Nw where K, N are appropriate matrices to be determined, x is the system state vector, and w is the new input vector assumed equidimensional to the actual input vector u. The procedure derived determines K and N such that the resulting closed-loop system has a diagonal and nonsingular transfer-function matrix. The case, where only partial input-output decoupling is possible, is also considered, and the corresponding state-feedback matrices K and N are determined. The results are illustrated by simple numerical examples. The required 3-D generalization of the well known Cayley-Hamilton theorem is provided.     

Near-decoupling of linear multivariable systems

Available exact-decoupling conditions by constant compensation do not address the important class of ‘nearly decouplable’ systems. For practical systems, parameter perturbations, measurement errors and system-modelling inaccuracies may conceal the system decouplability by constant- or low-order compensation, thus calling for unnecessarily high-order dynamics. In this paper, state-space decoupling conditions formulated as rank-degeneracy conditions are used to detect how close a system is to one that is exactly decouplable. The procedure simultaneously provides the compensators necessary to achieve the indicated level of decoupling. The conditions are extended to cases where it is required to use a fixed constant compensator for decoupling under different sets of perturbations     

同时不变子空间与鲁棒干扰解耦

本文主要讨论了同时不变子空间的三种特征:即时域特征、频域特征和几何特征.特别对 包含在给定子空间H中的最大同时不变子空间给出了新的描述.利用这些结果,文中首先给 出了离散扰动系统鲁棒干扰解耦问题的充要条件,然后推广到连续扰动系统,获得了一个 Kharitonov型的结果,使得两种扰动下的鲁棒干扰解耦问题在理论上得到了完整的解决.     

Two-input one-output compensators

Two-input one-output compensators can be used for decoupling and exact model-matching design. They are more general than the well-known output feedback and observer-controller-type feedback. However, the design of two-input one-output compensators is also difficult. In this paper we present a design procedure for a two-input one-output compensator for multivariable systems. In the design, we only require an added dynamic to ensure the solution-existence condition. Thus the design of two-input one-output compensators is put on the same level as the familiar output feedback and observer-controller-type feedback designs.