The decoupling of multivariable systems using matrix polynomial equations

A matrix polynomial equation is derived which governs the behaviour of time-invariant multivariable systems under linear state variable feedback. This is used to derive the f-invariants of the system. The problem of decoupling is considered using this matrix polynomial equation and conditions for decoupling are easily derived from it. An algorithm for decoupling is then presented. Examples are given to illustrate the procedure.     

Design of multivariable systems via polynomial equations

Matrix polynomial equations are derived for linear time-invariant multivariable systems with linear state-variable feedback. The problems of exact model matching and decoupling are analysed using these equations. Examples are given to illustrate the approach.     

An expanded matrix representation for multivariable systems

The purpose of this paper is to develop a matrix representation for linear multivariable systems which describes bilateral properties like output interaction and can be used to simplify the analysis of systems composed of interconnected multivariable elements. This representation is analogous to the circuit parameters used to describe two-port communications networks. Examples of the application of the proposed matrix to a physical system are given and certain properties are discussed which can be described in terms of either the system matrix or the principle of invariance.     

A matrix nullspace approach for solving equality-constrained multivariable polynomial least-squares problems

We present an elimination theory-based method for solving equality-constrained multivariable polynomial least-squares problems in system identification. While most algorithms in elimination theory rely upon Groebner bases and symbolic multivariable polynomial division algorithms, we present an algorithm which is based on computing the nullspace of a large sparse matrix and the zeros of a scalar, univariate polynomial.     

On the solution and impulsive behaviour of polynomial matrix descriptions of free linear multivariable systems

In this note we examine the solution and the impulsive behaviour of autonomous linear multivariable systems whose pseudo-state beta(t) obeys a linear matrix differential equation A(rho)beta(t) = 0 where A(rho) is a polynomial matrix in the differential operator rho:=d/dt. We thus generalize to the general polynomial matrix case some results obtained by Verghese and colleagues which regard the impulsive behaviour of the generalized state vector x(t) of input free generalized state space systems.     

Deadbeat algorithms for output controllers in discrete-time multivariable systems

This paper studies the problem of designing output deadbeat controllers to force the state of discrete-time multivariable systems to zero in a finite number of samples. Two algorithms are considered. The first is based on the fact that the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the left and right eigenvectors must be in certain subspaces. In the second algorithm, the output gain matrix is computed through the optimization of certain parameters of the controller, while maintaining its structural constraints. Computer programs have been developed to realize the two algorithms and examples are given to illustrate the feasibility of the techniques.     

On the number of solutions of multivariable polynomial systems

Systems of multivariable polynomial equations play a key role in many important control and stability problems. In this paper we derive analytical expressions for the number of solutions as a function of the polynomial coefficients. The new results are obtained by combining standard results from elimination theory with the properties of inner determinants. For problems where one of the variables is constrained to a given interval, the number of solutions can be expressed in terms of Sturm's sequences. Tests for system solvability, and the uniqueness of a solution are also presented. A numerical example illustrates that the new results also provide a promising noniterative approach for solving systems of polynomial equations.     

Robust real-time algorithms for identification of linear multivariable time-varying systems

The paper presents a discussion on the problem of the robust real-time identification of linear multivariable time-varying dynamic systems working in a noisy environment. Two methodologically different approaches to the design of such algorithms are presented. The first is based on the one-step estimation, optimal in the sense of the minimal conditional mean-square error, combined with convenient approximations of the underlying error covariance matrix. The second is based on the general formulation of robustified stochastic approximation algorithms, characterized by a suitable non-linear transformation of normalized residuals. Particular algorithms are derived on the basis of step-by-step optimization with respect to the weighting matrix of the algorithm. Monte Carlo simulation results illustrate the discussion, and show the efficiency of the proposed robust algorithms in the presence of large disturbance realizations, the so-called outliers.     

Data filtering based least squares algorithms for multivariable CARAR-like systems

This paper focuses on the identification problem of multivariable controlled autoregressive autoregressive (CARAR-like) systems. The corresponding identification model contains a parameter vector and a parameter matrix, and thus the conventional least squares methods cannot be applied to directly estimate the parameters of the systems. By using the hierarchical identification principle, this paper presents a hierarchical generalized least squares algorithm and a filtering based hierarchical least squares algorithm for the multivariable CARAR-like systems. The simulation results show that the two hierarchical least squares algorithms are effective.     

New fast hybrid matrix multiplication algorithms

New hybrid algorithms are proposed for multiplying (n × n) matrices. They are based on Laderman’s algorithm for multiplying (3 × 3)-matrices. As compared with well-known hybrid matrix multiplication algorithms, the new algorithms are characterized by the minimum computational complexity. The multiplicative, additive, and overall complexities of the algorithms are estimated.     

Finite settling time stabilisation for multivariable discrete-time systems: a polynomial equation approach

The multivariable case of Finite Settling Time Stabilisation (FSTS) of linear discrete-time systems is considered in this paper. An algebraic approach is adopted which leads to the solution of a polynomial matrix Diophantine equation. This gives rise to the parametrisation of all FSTS controllers in a Ku?era–Youla–Bonjiorno sense and the FSTS problem is further reduced to a linear algebra problem over the real numbers. Subsequently, the family of all causal FSTS controllers is parametrised, and necessary and sufficient conditions for strong FSTS (stable controllers) are derived. The minimal McMillan degree solution and minimal complexity controllers are examined and new bounds are given. The analysis provides the means for the parametrisation of families of FSTS controllers with certain complexity. Finally the problems of tracking, disturbance rejection and partially assigned dynamics in FST sense are considered and conditions for their solvability are given.     

Asymptotically optimal algorithms for approximate agreement

This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases. Alan Fekete was born in Sydney Australia in 1959. He studied Pure Mathematics and Computer Science at the University of Sydney, obtaining a B.Sc.(Hons) in 1982. He then moved to Cambridge, Massachusetts, where he obtained a distributed Ph.D. degree, awarded by Harvard University's Mathematics department for work supervised by Nancy Lynch in M.I.T.'s Laboratory for Computer Science. He spend the year 1987–1988 at M.I.T. as a postdoctoral Research Associate, and is now Lecturer in Computer Science at the University of Sydney. His research concentrates on understanding the modularity in distributed algorithms, especially those used for concurrency control in distributed databases.A preliminary version of this paper has appeared in the Proceedings of the 5th ACM Symposium on Principles of Distributed Computing (August 1986). This work was begun in the Department of Mathematics, Harvard University, and completed at the Laboratory for Computer Science at Massachusetts Institute of Technology. The work was supported in part (through Professor N. Lynch) by the Office of Naval Research under Contract N00014-85-K-0168, by the Office of Army Research under contract DAAG29-84-K-0058, by the National Science Foundation under Grants MCS-8306854, DCR-83-02391, and CCR-8611442, and by the Defense Advanced Research Projects Agency (DARPA) under Contract N00014-83-K-0125     

Achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters

The notion of interactor matrix or equivalently the Hermite normal form, is a generalization of relative degree to multivariable systems, and is crucial in problems such as decoupling, inverse dynamics, and adaptive control. In order for a system to be input-output decoupled using static state feedback, the existence of a diagonal interactor matrix must first be established. For a multivariable linear system which does not have a diagonal interactor matrix, dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system. Such precompensation often depends on the parameters of system, and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty. In this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters. More precisely, we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix. These conditions are associated with the so-called (non)generic singularity of certain matrix related to the system structure but independent of the system parameters. The result of this paper is expected to be useful in robust and adaptive designs.     

Compensatory neurofuzzy systems with fast learning algorithms

In this paper, a new adaptive fuzzy reasoning method using compensatory fuzzy operators is proposed to make a fuzzy logic system more adaptive and more effective. Such a compensatory fuzzy logic system is proved to be a universal approximator. The compensatory neural fuzzy networks built by both control-oriented fuzzy neurons and decision-oriented fuzzy neurons cannot only adaptively adjust fuzzy membership functions but also dynamically optimize the adaptive fuzzy reasoning by using a compensatory learning algorithm. The simulation results of a cart-pole balancing system and nonlinear system modeling have shown that: 1) the compensatory neurofuzzy system can effectively learn commonly used fuzzy IF-THEN rules from either well-defined initial data or ill-defined data; 2) the convergence speed of the compensatory learning algorithm is faster than that of the conventional backpropagation algorithm; and 3) the efficiency of the compensatory learning algorithm can be improved by choosing an appropriate compensatory degree.     

Stability of a matrix polynomial in continuous systems

Two sufficient conditions under which the roots of the determinant of a given (m times m) matrix polynomial ofnth order lie in the open left-half plane have been obtained. The first condition is given in terms of the positive definiteness of an (mn times mn) symmetric matrix, while the second condition is given in terms of the positive definiteness of an (m times m) matrix that is a function ofs, Res leq 0. These conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial. Therefore, the explicit computation of the determinant polynomial is not required.     

Stability of a matrix polynomial in discrete systems

Two sufficient conditions that the determinant of a nonsingular real (m times m) matrix polynomial ofnth order has all its roots inside the unit circle have been obtained. These conditions are represented in terms of rational functions of the coefficient matrices. Therefore, these conditions do not require the computation of the determinant polynomial. The first condition is given in terms of the positive definiteness of an (mn times mn) symmetric matrix, while the second condition is expressed by the positive definiteness of an (m times m) Hermitian matrix which is a function ofz, |z| leq 1. The first condition implies the second, and hence is more restrictive than the second.