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In the present technical note we study the fundamental limitation on stability that arise when an additive coloured Gaussian noise (ACGN) channel is explicitly considered over either the control or measurement paths of a linear time invariant (LTI) feedback loop. By considering a linear setting we can naturally express the fundamental limitation as a lower bound on the channel signal-to-noise ratio (SNR) required for stabilisability. We start by first obtaining a closed-form expression for the squared L 2 norm of a partial fraction expansion with repeated poles in the Laplace domain. We then use the squared L 2 norm result to obtain the closed-form expression for the infimal SNR required for stabilisability. The proposed closed-form includes the case of repeated unstable plant poles and non minimum phase (NMP) zeros.

In this technical note, we present a new frequency weighted balancing related technique which is based on a parametrized combination of the unweighted balanced truncation technique [9] ,[8] and the partial fraction expansion technique [11] . The reduced-order models which are guaranteed to be stable in the case of double-sided weighting, are obtained by either direct truncation or singular perturbation approximation. Simple, elegant and easily computable a priori error bounds are also derived. Numerical examples and comparison with other well-known techniques show the effectiveness of the proposed technique.

In this paper, we present a semianalytical method for evaluating quadratic cost functionals for linear time-invariant systems with multiple time delays. The main computational tasks involved in the method are the solution of a symbolic Diophantine equation by the Euclid algorithm and the factorization of a finite-degree polynomial. To illustrate the numerical implementation of the proposed method, an example is provided.

A full state vector observer is derived for a class of linear autonomous time lag systems represented by differential equations. The major features of the approach are: (1) an observer is constructed just for the set of unstable or poorly damped modes of the delayed system once they have been determined, and (2) the design is accomplished by using a partial-fraction expansion separating the unstable part from the system and then applying well-known finite dimensional state vector techniques. A procedure for the construction is given along with some numerical results to illustrate the simplicity of the observer. >

Finite spectrum assignment for time-delay systems is the elimination of delay operators from the characteristic function of the closed-loop system and the arbitrary assignment of poles. The control consists of polynomials in the delay operator and finite Laplace transforms. An algorithm for computing the control matrix is presented. In particular, the control matrix over rational functions of a delay operator is computed and expanded to partial fractions. Partial fractions are systematically transformed to finite Laplace transforms. >

A state-space technique is developed for finding the block partial fraction expansion of an irreducible right matrix fraction description with distinct and/or repeated block poles. The developed method is the generalized version of the existing methods which consider only distinct block poles.

An algorithm is developed to derive the partial fraction expansion of the ratio of two polynomials. A unified approach using tables is presented in order to solve both single and multiple pole cases.

This paper represents a first attempt to derive a closed-form (Hankel-norm) optimal solution for multivariable system reduction problems. The basic idea is to extend the scalar ease approach in [5] to deal with the multivariable systems. The major contribution lies in the development of a minimal degree approximation (MDA) theorem and a computation algorithm. The main theorem describes a closed-form formulation for the optimal approximants, with the optimality verified by a complete error analysis. In deriving the main theorem, some useful singular value/vector properties associated with block-Hankel matrices are explored and a key extension theorem is also developed. Imbedded in the polynomial-theoretic derivation of the extension theorem is an efficient approximation algorithm. This algorithm consists of three steps: i) compute the minimal basis solution of a polynomial matrix equation; ii) solve an algebraic Riccati equation; and iii) find the partial fraction expansion of a rational matrix.

This paper presents a complete solution for the optimum linear system which operates on n stationary and correlated random processes so as to minimize error variance in filtering or prediction. A simple closed-form answer results if the matrix \Phi(s) of spectra of the input signals can be factored such that \Phi(s) = G(-s)G^{T}(s) where G(s) and G^{1}(s) represent matrices of stable transforms in the Laplace variables. A general factoring procedure for rational matrices is presented. G(s) can be viewed as the system which would reproduce signals with the spectrum of \Phi(s) when excited by n uncorrelated unit-density white-noise sources. In the case of a multidimensional filter, when G(s) is separated by partial fractions into two terms, S(s) + N(s) , having 1hp poles from the signal and noise spectra, respectively, the optimum unity-feedback filter is shown to have a forward-loop transference of S(s)N^{-1}(s) .

The controllability and observability of linear stationary multivariable control systems is investigated in the frequency domain. If the minimal polynomial of the system is of maximum degree, then a simple test involving mode cancellations determines controllability (observability). For the general case, a simple test based on a partial fraction expansion yields the necessary and sufficient conditions for controllability (observability).

A computer program has been developed to find the partial fraction expansion of a ratio of two polynomials with real coefficients. The program is based on a method suggested by P. Henrici (Eidenossische Technische Hochschule, Zurich, Switzerland) and utilizes real numbers only, thus dispensing with complex arithmetic. The program assumes nonrepeated roots for the denominator polynomial. Henrici's technique may be extended to ether situations.

For the linear time invariant system \bigl[\matrix {y_{c}(s)\cr y_{m}(s)}\bigr] = \bigl[\matrix{G(s)&H(s)\cr M(s)&N(s)} \bigr] \bigl[\matrix {u_{c}(s)\cr u_{r}(s)}\bigr] a necessary and sufficient condition based on the proper rational transfer function matrices \{G, H, M, N\} is given for the existence of a proper stabilizing controller u_{c}(s) = C(s)y_{m}(s) . The condition states that the McMillan degrees of M^{+}(s) and \bigl[\matrix{G^{+}(s)&H^{+}(s)\cr M{+}(s)&N{+}(s)}\bigr] must be equal, where \{G^{+}(s), H^{+}(s), M^{+}(s), N^{+}(s)\} represents the unstable components of the partial fraction expansions of \{G(s), H(s), M(s), N(s)\} .

In the above paper, the author presents a recursive algorithm for determination of the terms in a partial fraction expansion associated with multiple poles. However, this recursive algorithm has been derived previously in a different publication [1].

A simple method for finding the inverse Vandermonde matrix is presented. The technique involves determination of the partial fraction expansion of rational functions. Only synthetic division and longhand division are required, which make the process very suitable for hand calculation. Alternatively, simple recurrent relations are also developed for computer use.