Minimal realization of transfer function matrices via oneorthogonal transformation

The minimal realization of a given arbitrary transfer function matrix G(s) is obtained by applying one orthogonal similarity transformation to the controllable realization of G( s). The similarity transformation is derived by computing the QR or the singular value decomposition of a matrix constructed from the coefficients of G(s). It is emphasized that the procedure has not been proved to be numerically stable. Moreover, the matrix to be decomposed is larger than the matrices factorized during the step-by-step procedures given     

A computer program for the minimal realization of transfer function matrices

A computer program for finding the minimal realizations of transfer function matrices is described. An example is included to illustrate the procedure.     

Minimal realization of transfer function matrices A comparative study of different methods

A comparative study is made of three basically different methods for obtaining minimal-order realizations of linear multivariable systems in the form of state equations for specified rational transfer function matrices. Computational efficiency and suitability for practical implementation are the main criteria used for the comparison. The possibilities of direct realization in canonical forms suitable for special applications are also examined.     

Approximated modeling and minimal realization of transfer function matrices with multiple time delays

The modeling and minimal realization techniques for a specific multiple time-delay continuous-time transfer function matrix with a delay-free denominator and a multiple (integer/fractional) time-delay numerator matrix have been developed in the literature. However, this is not the case for a general multiple time-delay continuous-time transfer function matrix with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. This paper presents a new approximated modeling and minimal realization technique for the general multiple time-delay transfer function matrices. According to the proposed technique, an approximated discrete-time state-space model and its corresponding discrete-time transfer function matrix are first determined, by utilizing the balanced realization and model reduction methods with the sampled unit-step response data of the afore-mentioned multiple time-delay (known/unknown) continuous-time systems. Then, the modified Z-transform method is applied to the obtained discrete-time transfer function matrix to find an equivalent specific multiple time-delay continuous-time transfer function matrix with multiple time delays in only the inputs and outputs, for which the existing control and design methodologies and minimal realization techniques can be effectively applied. Illustrative examples are given to demonstrate the effectiveness of the proposed method.     

Linear mechanical systems and dyadic transfer function matrices

For some linear mechanical systems, which can be written as a dyadic transfer function matrix (DTM), MIMO controller design can be reduced to a number of SISO designs. Internal stability of the SISO designs guarantees stability of the resulting MIMO controller. By relating the insights from modal analysis to system theory, necessary and sufficient conditions are derived for linear mechanical systems to have the DTM property. The main theorems are illustrated on an active suspension design of an agricultural spray boom.     

Detecting stable matrices

Stable matrices are related to the values of their entries by the concept of an all negative quasi-dominant diagonal. Stable matrices are related to the signs of their entries by the signs of cycles in their corresponding signed directed graphs. This paper establishes relationships between stable matrices and conditions on the values of cycles. For some cases the conditions on the values of cycles are satisfied if and only if the matrix has a quasi-dominant diagonal.     

New algorithm for minimal balanced realization of transfer function matrix

A simple method for calculating the gramians of any stable realization using algebraic computation is described. This allows the development of a new algorithm to obtain a minimal balanced realization. The algorithm presented is shown to be computationally simpler and more efficient than previous methods and is illustrated by an example.     

Minimal stable partial realization

In this paper two equivalent sets of necessary and sufficient conditions for the existence of an asymptotically stable partial realization are developed. Both sets are expressed as multivariable polynomial equations which may be tested for the existence of a solution in a finite number of rational steps via decision methods. Should a solution exist, it may be evaluated with the aid of polynomial factorization. The first set of conditions are based on results due to Ho and Kalman, and are useful for the case where the number of specified Markov parameters is greater than the order of the realization. For other cases, the second set of conditions which include results from a companion paper on minimal observers, require less computational effort to be tested.     

Decentralized control using quasi-block diagonal dominance of transfer function matrices

The purpose of this paper is to present a generalization of the Nyquist array method to blockwise decompositions, which is based upon a new version of block diagonal dominance. An additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices, which increases considerably the class of control systems which can be designed via block diagonal dominance. Within this framework, the controllers for individual subsystems can be designed independently of each other, so that their union represents a decentralized controller for the overall system.     

Approximation and realization of Hankel matrices

The approximation of a Hankel matrix by finite rank Hankel matrices is considered. A constructive procedure is proposed for the approximate realization of a linear analytic system by means of bilinear systems.     

Balanced realizations near stable invariant manifolds

It is shown that the notion of balancing a state space realization about a stable, isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension, for example, a stable limit cycle.     

Stochastic approximation and realization of Hankel matrices

Dynamic systems are considered whose outputs can be represented either by a deterministic series of the input variables or by a stochastic series of brownian motion processes. Approximate representations of such series are proposed. The representations are based on the Hankel matrix representation of the series. First, it is shown that any Hankel matrix, belonging to suitable classes, can be approximated in the deterministic and stochastic sense by finite rank Hankel matrices. Then, a method is proposed to design bilinear realizations of finite rank Hankel matrices, effective in the deterministic and stochastic sense.     

A generalized-Jordan-form-approach one-step irreducible realization of matrices

This note presents a one-step irreducible realization (A,B,C,D) from a rational matrixG(s)with multiple poles. The introduction of a generalized-Jordan-formAmakes it possible to consider additional network synthesis constraints in determiningCcolumn by column andBrow by row. Only Gauss-Jordan reduction is needed. One example is given for illustration.     

Digital decoupling controller design for multiple time-delay continuous-time transfer function matrices

This paper presents an extended adjoint decoupling method to conduct the digital decoupling controller design for the continuous-time transfer function matrices with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. First, based on the sampled unit-step response data of the afore-mentioned multiple time-delay system, the conventional balanced model-reduction method is utilised to construct an approximated discrete-time model of the original (known/unknown) multiple time-delay continuous-time transfer function matrix. Then, a digital decoupling controller is designed by utilising the extended adjoint decoupling method together with the conventional discrete-time root-locus method. An illustrative example is given to demonstrate the effectiveness of the proposed method.     

The minimal order of rational transfer function realization from its z-domain samples

We present here a new method for realization of minimal-order proper rational transfer functions from z-domain samples. A criterion for determining the minimal order of the transfer function is given. The theory is developed first for strictly proper rational transfer functions and then extended to the case of transfer functions with a direct-coupling term. The theory is valid for arbitrary z-domain samples. The special case of uniform frequency response samples is shown to allow a computationally efficient derivation of the realization. Examples are given to illustrate the theory. The theory developed here has many applications, such as modelling of communication channels, identification of systems from frequency measurements and infinite impulse response filter design.     

On implementable transfer matrices

A straightforward proof is given of the sufficient condition for implementable transfer matrices. A similar result is obtained also when the controlled output is not accessible for feedback     

A new algorithm for the minimal balanced realization of a transfer function matrix

A new algorithm is proposed here to obtain a minimal balanced realization directly from the transfer function matrix (TFM). This method which employs the singular-value decomposition (SVD) of an infinite block-Hankel matrix requires neither an initial minimal realization nor the solution of a Lyapunov matrix equation. The formulation is solely in terms of the coefficients of the transfer function matrix.     

Minimal periodic realizations of transfer matrices

Periodic controllers designed based on the so-called lifting technique are usually represented by transfer matrices. Real operations require that the controllers be implemented as periodic systems. The problem of realizing an Nn_o×Nn_i proper rational transfer matrix as an n_i-input n_o-output N-periodic discrete-time system is studied. An algorithm for obtaining periodic realizations which have a minimal number of states is proposed. The result can also be used to remove any redundant states that exist in a periodic system