Input- and output-decoupling structural zeros of linear systems described by Rosenbrock's polynomial matrices

This paper introduces the notion of input- (output-) decoupling structural zeros for linear time-invariant systems, described by Rosenbrock's system polynomial matrices. This notion is based on the definition of polynomial system matrices having the same structure that is given in this paper. Namely the input- (output-) decoupling structural zeros are decoupling zeros that are present for every choice of the system parameters. The main results are derived using digraph theory. An example illustrates the procedure for detecting structural zeros.     

Multidimensional structured matrices and polynomial systems

We apply and extend some well-known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing, that is, computations with structured matrices and solving a system of polynomial equations. In the first part of our paper, we extend the Toeplitz and Hankel structures of matrices and some of their known properties to some new classes of structured (quasi-Hankel and quasi-Toeplitz) matrices, naturally associated to systems of multivariate polynomial equations. In the second part of the paper, we prove some relations between these structured matrices, which extend the classical relations of the univariate case. Supported by NSF Grant CCR 9625344 and PSC CUNY Awards Nos. 667340     

Decentralised fixed modes of networked MIMO systems

In this paper, decentralised fixed modes (DFMs) of a networked system are studied. The network topology is directed and weighted and the nodes are higher-dimensional linear time-invariant (LTI) dynamical systems. The effects of the network topology, the node-system dynamics, the external control inputs, and the inner interactions on the existence of DFMs for the whole networked system are investigated. A necessary and sufficient condition for networked multi-input/multi-output (MIMO) systems in a general topology to possess no DFMs is derived. For networked single-input/single-output (SISO) LTI systems in general as well as some typical topologies, some specific conditions for having no DFMs are established. It is shown that the existence of DFMs is an integrated result of the aforementioned relevant factors which cannot be decoupled into individual DFMs of the node-systems and the properties solely determined by the network topology.     

Flat output characterization for linear systems using polynomial matrices

This paper is devoted to the study of linear flat outputs for linear controllable time-invariant systems in polynomial matrix form. We characterize the transformations expressing the system variables in terms of a linear flat output and derivatives, called defining matrices, as the kernel of a polynomial matrix. An application to trajectory planning is presented, showing the usefulness of the present characterization.     

Inversion of polynomial matrices by interpolation

Known polynomial interpolation methods to polynomial matrices are generalized to obtain new algorithms for the computation of the inverse of such matrices. The algorithms use numerically stabilizable manipulations of constant matrices. Among the three methods investigated Lagrange's interpolation seems especially suitable for the purpose     

Elimination of fixed modes in decentralized systems by means of sampling

In this contribution it is shown that fixed modes in continuous-time decentralized control systems can be eliminated by the use of sampling. Criteria are derived which distinguish the fixed modes which are still present in the sampled-data system from those which are eliminated.     

Characterizations of decentralized fixed modes for interconnected systems

A study of the characterization of decentralized fixed modes for large-scale interconnected systems is made. This is done by obtaining a recursive characterization of decentralized fixed modes, in which the existence of fixed modes of a ν-control agent system is expressed in terms of the existence of fixed modes of ν ? 1 control agent systems. An interpretation of these conditions is then made in the frequency domain. Simple conditions, in terms of the controllability of the system's subsystems, are then found for a composite system consisting of ν interconnected subsystems to have no decentralized fixed modes. This result clarifies some recent discussion on the fixed modes of interconnected systems; in particular, it proves that Fessas's conjecture is true for systems with two control agents, but false for systems with more than two control agents.     

Stabilization of fixed modes in expansions of LTI systems

The objective of this note is to propose a method for stabilization of structurally fixed modes in expansions of LTI dynamic systems in the scope of overlapping decentralized control design based on the expansion/contraction framework, enabling successful design for a broader class of problems than considered so far. The method is based on a judicious choice of complementary matrices in the expanded space. Numerical examples are provided to illustrate simplicity and efficiency of the proposed approach.     

Discrete-time control of continuous systems with approximate decentralized fixed modes

In this paper, the discrete-time control of decentralized continuous-time systems, which have approximate decentralized fixed modes, is studied. It is shown that under certain conditions, discrete-time controllers can improve the overall performance of the decentralized control system, when a linear time-invariant continuous-time controller is ineffective. In order to obtain these conditions, a quantitative measure for different types of approximate fixed modes in a decentralized system is given. In this case, it is shown that discrete-time zero-order hold (ZOH) controllers, and in particular, that generalized sampled-data hold functions (GSHF), can significantly improve the overall performance of the resultant closed-loop system. The proposed sampled-data controller is, in fact, a linear time-varying controller for the continuous-time system.     

Generalized inverse of polynomial matrices

Investigations are made into conditions that preserve the algebraic structure of a matrix in its generalized inverse, with particular reference to polynomial matrices. The procedure also constructs such a type of inverse, when it is found to exist.     

A graph-theoretic method to find decentralized fixed modes of LTI systems

This paper deals with the decentralized pole assignability of interconnected systems by means of linear time-invariant (LTI) controllers. A simple graph-theoretic approach is proposed to identify the distinct decentralized fixed modes (DFMs) of the system, i.e., the unrepeated modes which cannot be moved by means of a LTI decentralized controller. The state-space representation of the system is transformed to the decoupled form using a proper change of coordinates. For any unrepeated mode, a matrix is then computed which resembles the transfer function matrix of the system at some point in the complex plane. A bipartite graph is constructed accordingly in terms of the computed matrix. Now, the problem of verifying if this mode is a DFM of the system reduces to checking if the constructed graph has a complete bipartite subgraph with a certain property. The sole restriction of this work is that it is only capable of identifying the distinct DFMs of a system. However, it is axiomatic that most of the modes of the real-world systems are normally distinct. The primary advantage of the present paper is its simplicity, compared to the existing ones which often require evaluating the rank of several matrices.     

Periodic compensation of a class of decentralized systems with fixed modes

This paper first finds that the high frequency (ω) periodic controller suggested in Lee, Meerkov, and Runolfsson (1987) for fixed mode removal, and pole and zero placement, of a class of decentralized systems (A,B_i,C_i) all the strongly interconnected channels of which satisfy C_iB_i=0, causes large, O(ω), oscillations in the outputs. Next it proposes an alternative controller, a dual of the above, that achieves the compensation with oscillations of O(1) only.     

Observers for systems described by transport equations

The design of observers for processes involving transport phenomena with time-varying transportation velocity is considered. The observed systems are modelled by first-order bilinear partial differential equations of the hyperbolic type. Using geometric techniques, the asymptotic reconstruction of sets of linear functionals of the state is examined, yielding existence criteria and design procedures for finite-dimensional exact and approximate observers. Attention is paid to both cases of distributed measurements and of point measurements.     

Identifiabilty of systems described by convolution equations

Parameter identifiability is studied for a class of finite- and infinite-dimensional systems described by convolution equations. The notion of sufficiently rich input which enforces identifiability is also addressed and the results are obtained assuming knowledge of solutions on a bounded time interval.     

New coprimeness over multivariable polynomial matrices and its application to control delay systems

This paper presents a new notion of coprimeness over multivariable polynomial matrices, where a single variable is given priority over the remaining variables. From a characterization through a set of common zeros of the minors, it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable. An application of the presented coprimeness to control systems with non-commensurate delays and finite spectrum assignment is also presented. Because the presented coprimeness is stronger than minor coprimeness, non-commensurate delays are difficult to deal with in algebraic control theory. The “rational ratio condition” between delays, which can reduce non-commensurate delays to commensurate delays, proves to be both powerful and practical concept in algebraic control theory for delay systems.     

Operational matrices of piecewise linear polynomial functions with application to linear time-varying systems

Application of the piecewise linear polynomial functions expansion is extended to linear time-varying systems. With the treatment of the product of two time functions, two types of operational matrix are developed. By applying these operational matrices, the dynamic equations are transformed into a set of algebraic equations. A recursive algorithm is derived and the system equations can be solved with very low dimensional matrix inversions. This represents a considerable saving of computer memory capacity and computing time.     

On the sufficient conditions for the equality of the unassignable polynomial and Davison's fixed polynomial of strongly connected systems

In this technical note, it is established that the unassignable polynomial defined for a not strongly connected decentralized control system is not equal to Davison's fixed polynomial. This leads to a "sufficient condition" for the equality of the unassignable polynomial and Davison's fixed polynomial for strongly connected systems.