Decoupling Zeros Research Articles

A Behavioral Approach to the Pole Structure of One-Dimensional and Multidimensional Linear Systems

We use the tools of behavioral theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional (nD) system. We make a natural division of the poles into controllable and uncontrollable poles. When the behavior in question has latent variables, we make a further division into observable and unobservable poles. In the case of a one-dimensional (1D) state-space model, the uncontrollable and unobservable poles correspond, respectively, to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices, and their left and right matrix fraction descriptions (MFDs). We find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behavior as the sum of subbehaviors associated with various poles. This is related to the integral representation theorem, which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

Decentralized control and periodic feedback

The decentralized stabilization problem for linear, discrete-time, periodically time-varying plants using periodic controllers is considered. The main tool used is the technique of lifting a periodic system to a time-invariant one via extensions of the input and output spaces. It is shown that a periodically time-varying system of fundamental period N can be stabilized by a decentralized periodic controller if and only if: 1) the system is stabilizable and detectable, and 2) the N-lifting of each complementary subsystem of identically zero input-output map is free of unstable input-output decoupling zeros. In the special case of N=1, this yields and clarifies all the major existing results on decentralized stabilization of time-invariant plants by periodically time-varying controllers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Zeros and controllability of distributed parameter systems

Decoupling zeros and invariant zeros of a distributed parameter system are investigated. The decoupling zeros are sub-divided into the set of input-decoupling zeros, output-decoupling zeros and input-output decoupling zeros. We make clear the relations between decoupling zeros and controllability andsol;or observability. We also consider invariant zeros and investigate the invariance of zeros and the preservation of controllability and observability by state feedback or output feedback.

Finite zero structure of linear periodic discrete-time systems

Periodic ordered sets of structural indices and a new and simpler characterization of the notions of invariant zero, transmission zero, pole and eigenvalue are introduced for a linear periodic discrete-time system. Also the notions of input and output decoupling zeros (together with their ordered sets of structural indices) are extended to this type of system and are shown to have a meaning and relations with the structural properties of the system, wholly similar to the time-invariant ones. The ordered sets of structural indices of non-zero poles, eigenvalues and zeros of any type are independent of time, while those (and even the existence) of the null invariant zero, transmission zero, input and output decoupling zero and pole can depend on time, as well as those of the null eigenvalue (although its algebraic multiplicity is independent of time). The ordered sets of structural indices of invariant zeros are not altered by a linear periodic state feedback (as well as those of input decoupling zeros) and...

Generalized proper inverse of polynomial matrices and the existence of infinite decoupling zeros

A necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices. The condition is then proved to be equivalent to the absence of infinite decoupling zeros. This condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form. It can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system, and to provide a new look of the unimodular matrix operation effect at infinity, for the left and right polynomial fraction forms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On Grassmann Invariants and Almost Zeros of Linear Systems

The canonical-IR [s]- Grassmann representative, g(X), of a rational vector space is defined as a new complete invariant of X. For the rational vector spaces Xc, Xb associated with a linear system, g(Xc), g(Xb) are used to define the new concepts of Almost Zeros (AZ) and Almost Decoupling Zeros (ADZ). The AZs and ADZs are natural extensions of the concepts of multivariable zeros and de coupling zeros respectively, and they are defi ned as local minima of a norm function defined on g(Xc), g(Xb) correspondingly. The dis tribution in the complex plane, the invariance properties and the significance for control problems of AZs and ADZs is discussed. The families of Strongly Zero Nonass ignable (SZNA) and Strongly Pole Nonassignable (SPNA) systems are defined. It is shown, that for SPNA (SZNA) systems the ADZs (AZs) define the centres of disks “trapping” closed-loop poles (zeros) of systems under any constant output feedback (squaring down). Criteria for determining upper bounds for the radii of the “trappining” disks are given. These results show that the AZs act as “nearly” fixed zeros under constant squaring down, and that the ADZs act as “nearly” fixed poles under constant output feedback.

Realizations in generalized state-space form for polynomial system matrices and the definitions of poles, zeros and decoupling zeros at infinity†

This paper contains three main results. Firstly, a realization procedure is presented that brings a polynomial system matrix of a non-proper multivariable system to generalized state-space form, such that all relevant properties including phenomena of redundancy associated with finite and infinite decoupling zeros are retained. Secondly, new definitions are proposed for poles, zeros and decoupling zeros at infinity of a general polynomial system matrix. As a third result, a theorem of Rosenbrock (1974 e) on the redundancy of LCR multiports is generalized to include the decoupling zeros at infinity.

Redundancy in Linear, Time-Invariant, Finite-Dimensional Systems

For systems in state-space form, redundancy can be analysed in terms of uncontrollability or unobservability. Alternatively we may use the concept of (finite) decoupling zeros, and this can readily be applied to more general descriptions of the system. For these more general descriptions other types of redundancy are also possible: in particular the idea of infinite decoupling zeros is described and applied.

Structural properties of linear dynamical systems

Abstract The system considered is Ex=Ax + Bu, y= Cx, with E singular. It is shown that infinite decoupling zeros can be defined, and these induce a fourfold decomposition, of that part of the system giving rise to the polynomial port of the transfer function matrix, analogous to Kalman'B canonical decomposition. Minimal indices are also defined, and are shown to be equal to dynamical indices based on the work of Forney and studied recently by Rosenbrock and Hayton.