Minimal State Space Research Articles

Minimal realization of two-dimensional systems via a state space cyclic model

Abstract The problem of minimal state space realization of two-dimensional systems is considered. The approach followed is, initially, to derive a circuit realization of the given transfer function involving a minimum number of delay elements. To facilitate this realization, the transfer function is expanded into a continued fraction. Using this circuit realization, an algorithm is proposed which readily provides the matrices of the state space model. The simplicity in deriving this model is due to a novel state space model introduced, which is of cyclic structure. Several examples are presented to illustrate ihe proposed algorithm.

2-D discrete time lossless bounded real functions: minimal state space realisation

A circuit realisation for two-dimensional (2-D) nth order discretetime lossless bounded real functions, with minimum number of delay elements, is presented. Based on this circuit realisation the corresponding matrix vectors A, b, c' of the Givone-Roesser 2-D state space model, are derived. The dimension of the 2-D state space model is minimal.

Algorithm for reducing the minimal realization problem of two-dimensional systems to a system of bilinear equations

Abstract The problem of the minimal state space realization of two-dimensional transfer functions which are not of any special form such as separable, all pole, all zero, continued fraction expandable, etc. is considered. For this general type of transfer function, an algorithm is presented for the minimal state space realization which is computationally superior over known techniques. The proposed algorithm starts by deriving, prior to and independently of the state space vectors b and c and the scalar d, the matrix A of the space model, nearly by inspection. Subsequently, the vectors b and c and the scalar d are determined on the basis of a bilinear algebraic system of equations.

Minimal circuit and state space realization of three-term separable denominator 2-D filters

A simple method is presented for the minimal state space realization of two dimensional (2-D) filters, with a three-term separable denominator (3TSD) made up of two 1-D and one 2-D terms. The matrix A and the vectors b and c of the state space model are derived explicitly from the coefficients of the initial transfer function. The circuit implementation for the 2-D 3TSD filters is provided, as well. The number of delay elements employed represents a significant reduction compared to earlier non-minimal realizations.

A directional partial realization problem

This paper introduces and solves a new version of the minimal partial realization problem concerning a finite sequence of Markow parameters M 1,…, M r for the case when each M j is only specified on a subspace L J such that L 1⊃ … ⊃ L r . The results include a state space criterion for minimality, a procedure for constructing a minimal realization and a formula for the minimal state space dimension in terms of the original data.

A note on the state space realization of 2D FIR transfer functions

Some properties of minimal state space realizations of 2D FIR transfer functions are investigated. It is shown that hidden modes are allowed in minimal realization and, consequently, there exist internally unstable minimal realizations of FIR transfer functions.

Erratum: Minimal state space realisation of product factorable 2-D systems

Erratum: Minimal state space realisation of product factorable 2-D systems

Minimal state space search in parallel production systems

A methodology for transforming the original search space into an equivalent but minimal search space is proposed. First, the concept of dependences leads to a procedure for reduction of the search space. The search procedure using this method can produce a minimal and complete search space. It is shown that this method is applicable to parallel search as well. An added advantage of this method is that it does not exclude the use of heuristics. pi - lambda transformation is introduced to reduce the parallel search space.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Forward and backward Markovian state space models of second order process

This paper considers the realization of discrete time multivariable second order processes in forward and backward state space form. Using the forward and backward factorization of the spectral density and the associated infinite moving average representation of the process, the construction of minimal order state space models in canonic forms are discussed, and the relationships between forward and backward canonic forms are shown. The extremal points of the set of all minimal order state-space realizations are also derived.

Minimal state space realisation of product factorable 2-D systems

A simple technique is presented for the minimal state space realisation of product factorable two dimensional systems. The application of this method yields the matrices A, b, cT and the scalar d of the two dimensional Givone-Roesser state space model, having minimal dimension.

Two-dimensional modified Cauer form: circuit and state space realisation

A new two-dimensional (2-D) continued fraction expansion, based on the modified Cauer form, has been formulated. Moreover, a minimal state space model realisation has been derived, based on its signal flow graph, using the proposed 2-D continued fraction expansion.

Philosophy, structure, and examples of relegated control

With the availability of fast and economical microprocessors, effective design of systems that are immune to failure of individual or groups of sensors, actuators, or computational units is feasible. The system can be made tolerant of the failure of individual subsystems, but functions with reduced efficiency. One way to realize this design is to formulate the dynamics of the systems in larger than minimal state spaces, to process the large number of sensory inputs needed, to relegate control functions to different inputs and to provide reliable communication among the subsystems. The imbedding of the state of the system in a larger state space allows the system to have direct access to its minimal state and indirect access by computing it, hence the need for many sensors. The sensors themselves can then measure directly physical parameters of interest or indirectly by providing a processor with measurements from which the processor computes the needed parameter. This paper deals with the concept of relegation of control as a special kind of generalized nonlinear decoupling control. A structure is proposed that relegates control of specific functions to subsets of inputs. The concept is illustrated by a nonlinear robotic example where the control of constraint forces (due to contact, grip, hold, touch, etc.), control of trajectory of motion, control of stability, and control of collision avoidance are relegated to different inputs. The inputs can be the actuator outputs of force and torque applied to the mechanical system or alternatively the inputs to the actuators themselves. Any conflict in fulfilling the four functions is arbitrated at a higher level. Compromises among the functions, priorities of functions over each other and assignments of inputs in primary, secondary, or as lower contributors to function are elaborated, programmed, and stored. This structure allows integration of a certain amount of intelligence in a robotic system at the lowest level.

A simplified realization algorithm for singular systems

A procedure for realizing a generalized transfer function matrix into a minimal state space singular system is presented. The procedure is a simplification of the methods based on Hankel matrices. Compared to existing methods, the proposed one reduces the computing time and the memory storage required, since it uses Hankel matrices of smaller order. An example of the technique is given.

Identification of Minimal Order State Space Models from Stochastic Input-Output Data

This paper discusses the problem of identifying a minimal order state space representation of a multivariable linear time invariant system from Gaussian stationary input-output measurements. A procedure for identifying the system’s order is proposed, based on an approximate probability distribution of the squared singular values of the Hankel matrix built from the sample cross-covariances. The approximate distribution converges to the true one as the number of measurements becomes large. The order determination procedure also identifies sets of linearly independent rows and linearly independent columns of the Hankel correlation matrix which form a basis for a minimal order representation of the system.

Minimal state space realisation of 3-D systems

A computational simple technique for minimal state-space realisation of 3-D systems is presented. The method starts with the idea of expanding the transfer function into continued fraction and subsequently represent the system in block diagram form. This block diagram allows the determination of minimal state-space by inspection. The structure of the 3-D model is of cyclic form with respect to the state variables and it is easy to implement. The efficiency of the method is illustrated by an example.

On minimality in the partial realization problem

A compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices M 1,…, M r to a minimal partial realization of M 1,…, M r . Furthermore, a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained.

On dual zero spaces and inverse systems

Control system response possibilities for a finite-dimensional, linear, time-invariant plant are governed by an interaction between the possible plant transfer function inverses, when they exist, and the closed-loop response maps themselves. Possible minimal state spaces for such inverses are comprised algebraically of a space of plant zeros and a variable space of poles, with the latter having a characterization dependent upon whether the inverse is left or right. In this note, we unify the study of state spaces for left and right inverse systems by defining a dual zero space. The dualization is natural and extends to both the state space and the variable pole space.

From time series to linear system—Part II. Exact modelling

In the second part of this paper the problem of finding an exact model for a q-dimensional infinite time series is considered. First a mathematical vocabulary for discussing exact modelling is developed. It is then shown how the results of Part I guarantee the existence of a most powerful (AR) model for an observed time series. Two algorithms for obtaining such an (AR) model are subsequently derived. One of these algorithms gives a shortest lag input/output model. The problem of obtaining a minimal state space realization of the observed time series is also considered. In order to do that, realization theory based on the truncated behaviour is developed. As an extensive example, the classical situation with impulse response measurements is discussed.

On the factorization of discrete-time rational spectral density matrices

For a discrete-time rational spectral density matrix \Phi(z) , the relationship between the factorizations \Phi(z) = Z(z) + Z^{T} (z^{-1}) and \Phi(z) = W(z)QW^{T}(z) , in terms of minimal state space realizations of Z(z) and W(z) , are derived in a straightforward way, without resorting to the use of the bilinear transformation. The obvious application of this result to performing a spectral factorizafion of \Phi(z) is discussed.

Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime factorization of a rational matrix

The series connection \Sigma_{12} linear time-invariant systems \Sigma_1 and \Sigma_2 that have minimal state space system descriptions is considered. From these descriptions strict system equivalent polynomial matrix system descriptions in the manner of Rosenbrock are derived. They are based on the factorization of the transfer matrix of the subsystems as a ratio of two right or left coprime polynomial matrices. They give rise to a simple polynomial matrix system description of the tandem connection \Sima_{12} . Theorem 1 states that for the complete controllability and observability of the state space system description of \Sigma_12 it is necessary and sufficient that certain denominator and numerator groups are coprime. Consequences for feedback systems are drawn in Corollary 1. The role of pole-zero cancellations is explained by Lemma 3 and Corollaries 2 and 3.