Decentralized Fixed Modes Research Articles

A Characterization of approximate decentralized fixed modes of time‐delay systems

AbstractThe concept of decentralized fixed mode (DFM) is crucial in determining the output feedback stabilizability by linear time‐invariant (LTI) decentralized controllers. However, in some cases, although a system has no DFMs, some of its modes may be very close to being fixed. It is thus important to characterize these modes, which are called approximate DFMs (ADFMs). In this paper, the characterization of ADFMs is discussed for LTI neutral (possibly descriptor‐type) time‐delay systems under static and dynamic decentralized output feedback control laws, and a mode mobility measure is proposed to provide quantitative information about ADFMs. To analytically compare the proposed measure, some of the commonly used quantitative measures for the characterization of ADFMs of finite‐dimensional systems have also been extended to the considered type of systems. The herein proposed approach allows multiple time delays and does not require additional expansions or preliminaries for multi‐input multi‐output systems, like some other commonly used methods. Thus, characterization with this measure, in general, requires less computational load, while it can provide more explicit information about the controller effort than the other considered measures.

An algebraic characterization of quotient decentralized fixed modes

In the control of linear time-invariant (LTI) decentralized systems, a decentralized fixed mode (DFM) is one which is immovable using an LTI decentralized controller. However, some DFMs can be moved using more complicated decentralized controllers; the ones which cannot are labelled quotient DFMs (QDFMs), since they arise from a related quotient system. The classical algorithm used to compute the QDFMs requires two steps: a partitioning of the sub-systems, using graph theory, followed by the application of standard tools from decentralized control. Here the goal is to provide a more direct approach to computing them.

On Computing Quotient Decentralized Fixed Modes

AbstractIn the decentralized control of linear time-invariant (LTI) systems, the notion of decentralized fixed modes (DFMs) was introduced to capture those modes which are immovable using LTI controllers. However, some DFMs can be moved using more complicated controllers. Recently a so-called “quotient system” associated with the original plant was investigated and it was proven that its DFMs, which we label quotient DFMs (QDFMs), are exactly those DFMs of the original system which cannot be moved using any form of nonlinear time-varying (NLTV) controller. The algorithm provided there to compute the QDFMs requires two steps: first, a partitioning of the sub-systems, using graph theory, and then application of standard tools from decentralized control. However, the first step is fragile, so the goal here is to provide a more robust approach to computing the QDFMs.

Decentralized Control of Systems with Approximate Decentralized Fixed Modes

The focus of this paper is directed towards the control of decentralized continuous systems using sampling and digital control methods. A digital control law together with a zero order hold forms a time-varying controller for continous-time systems, and a condition under which the resulted decentralized digital controller improves the performance of the continuous system is discussed. Different methods for implementing the digital controller are examined and compared.

On the quantitative characterization of approximate decentralized fixed modes using transmission zeros

In [16], it is shown that the spectrum of a linear time-invariant multivariable system, using decentralized linear time-invariant controllers, can only be assigned to a symmetric set of complex numbers that include the decentralized fixed modes (DFM). This is of fundamental importance to the decentralized stabilization problem. A measure of how close a mode is from being a DFM is introduced in [14]; this measure is used to quantify the difficulty of shifting such a mode. Following [4], modes which are "hard" to shift are considered to be approximate decentralized fixed modes. In this paper, we generalize the results of [5] to derive two new assignability measures. The first is coordinate invariant and is based on the distance between a mode and a set of transmission zeros. The second is coordinate dependent and is based on the minimum singular value of a set of transmission zero matrices.

Algebraic properties of decentralized control systems

The new concepts of the decentralized output feedback cycle index, the geometric multiplicities of decentralized fixed modes (DFM), and the decentralized output feedback variable polynomial (DVP) of a decentralized control system (DCS) are introduced. Their computational methods and some properties are given. It is shown that the decentralized output feedback cycle index is equal to one when the DCS has no decentralized fixed modes (DFM), or equal to the maximum of the geometric multiplicities of DFM otherwise. Almost every decentralized output feedback law can be used to make the zeros of the DVP distinct. It is also shown that almost any decentralized output feedback law will make the zeros of the DVP disjoint from any given finite set of points on the complex plane. It is then shown that if a DCS has no DFM then it can be made both controllable and observable with respect to any non-trivial input and output by applying almost any decentralized output feedback law.