On parametrizations for the minimal partial realization problem

The uniqueness of minimal partial realizations of input-output observations with elements in a Hilbert space is investigated. A sufficient condition and a necessary condition based on input-output data, without any assumptions on the state sequences, are derived. Under these conditions, models obtained by a class of identification schemes, known as "subspace methods", are the unique minimal partial realizations. Markov parameter identification from input-output observations is studied. Balanced approximations for rapidly decaying systems can be obtained from this approach.

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

The objective of the present paper is the study of models which view the given 1-D data as the impulse response of a minimal order discrete linear filter with data and coefficients confined to a finite field. This representation is known as “minimal partial realization” problem in linear system theory and has been tackled successfully by Ho and Kalman. Interestingly, the same problem is addressed and solved in a coding theory context by the well-known Berlekamp-Massey (bm) algorithm. We present a modified recursive version of thebm algorithm specially suited to our task of signal modelling. By introducing an absolute error bound, one can further solve the approximation problem computationally. The paper includes a number of examples and a mention of the software for this purpose.

In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

Based on a generalization of the Markov parameters, a generalized method for the construction of minimal realizations is presented. The method can give a family of minimal realizations, which may be useful, for example, in the problem of minimal partial realization. In contrast to the Markov parameters, special forms of the generalized Markov parameters may readily be obtained from experimental data. In particular, it is shown that the Laguerre expansion coefficients of a system provide a special form of the generalized Markov parameters of an associated system, and by using the proposed method, an exact minimal state variable model of the system can be restored from its Laguerre truncation.

The problem of finding minimal realizations of linear constant systems from finite order input-output Markov matrix sequences is considered. The paper identifies from the sequences sets of independent structural and numerical quantities which are invariants of equivalent state space representations and completely characterize any minimal realization of the sequence. These sets, termed bases of invariants, acquire a “nesting” property by which a subsequent basis of a higher order finite sequence is obtained from the previous basis by addition of some new invariants. Two canonical state space representations of special forms that reflect the input and output structural properties of the underlying systems are presented and readily derived from these bases by a simple algorithm which is provided. Necessary and sufficient conditions for the existence of a unique minimal partial realization to a given finite Markov sequence are given in terms of the invariants of its nested basis. The set of all minimal partial realizations $S_{n_r }^r $ that, in the case of existence of more than one solution, corresponds to many distinct systems, is thoroughly investigated. A minimal set of undetermined quantities that parametrize $S_{n_r }^r $ is obtained. These parameters are used to characterize $S_{n_r }^r $ either in the form of bases of invariants or in the form of the canonical representations, and it is also shown that an arbitrary assignment of values to these parameters leads to a minimal realization of the given finite sequence. Additional properties of these parameters that may be desirable in certain identification problems are also discussed.

The partial realization problem for 2D discrete linear shift-invariant systems is discussed. The authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2D linear system characterized by a given 2D impulse response array. The method used is based on the 2D Berlekamp-Massey algorithm which has a close connection with the 2D Hankel matrix derived from the 2D array. While it compares in efficiency with other methods for identification and approximation of 2D linear systems, it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree. >

A directional partial realization problem

On minimality in the partial realization problem

: This paper is concerned with the development of a complete set of invariants and canonical forms under feedback for linear systems characterized by proper rational transfer matrices. The invariants are determined in the frequency domain and consist of the Kronecker set of controllability indices together with a canonical form for the numerator of the transfer matrix under the action of a stabilizer subgroup of the unidoular group of polynomial matrices. Techniques used are algebro-geometric in nature. (Author)

The method of approximating the desired variable characteristics by a transfer function containing one kind of variable parameter in terms of minimal realization is described. Two models of the approximating variable transfer function that is minimally realizable are proposed. In the conventional model considering minimal realization, the poles are independent of the variable parameter lambda , while in the proposed models, both the poles and zeros depend on lambda . Consequently, they can represent a wider range of the variable characteristics. Making use of the property that the approximating transfer function is minimally realizable, the method of improvement of the approximation accuracy for the specified value of lambda by use of Markov parameters is also presented. >

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

In the previous chapter we derived the two most important models for the purpose of identification of their dynamic parameters. Both models are shown in their canonical form and are represented by a set of dynamic parameters which are usually linear combinations of individual link dynamic parameters. This simplifies the task of identifying these parameters. This chapter is dedicated to various identification schemes which can be used to identify the set of canonical parameters. The techniques proposed here apply to both geared and direct drive robots. From a practical point of view, in order to carry out experimental identification it is necessary to measure joint positions, velocities, accelerations, and joint torques and/or forces. For the purpose of load identification the robot has to be equipped with force/torque sensors and any experimental set-up in a laboratory environment should satisfy these requirements. However, since these sensors are quite expensive, robots are seldom equipped with them in practice.

Power electronic converters due to their inherent switching nature introduce nonlinearity in an electric drive system. That is sometimes exhibited in the form of the coexistence of several stable regimes (or attractors) for the same set of parameters and inputs, referred to as multistability. We study the behavior of a chopper-fed DC drive for the several sets of parameters and inputs, when the system exhibits multistability, in the presence of noise. In particular, we qualitatively characterize the coexisting attractors by what we call their robustness to noise, by comparing the level of noise required to observe the noise-induced tipping from one attractor into another. We depict that the “main” attractor (related to the nominal operation of the drive) is the most robust-to-noise one among the coexisting for all cases being analyzed. In those terms we explain the drive's behavior in the noisy environment―whether the effect of multistability takes the form of tipping or intermittency.

In 1969, J. Massey published a now-famous paper showing, among other things, that an iterative algorithm introduced by Berlekamp for decoding BCH codes also solved the problem of finding a shortest-length feedback shift register circuit for generating a given finite sequence of digits. This nice physical interpretation opened the door to connections with many other problems, including the minimal partial realization problems of linear system theory, Pade approximations and continued fractions, the fast algorithms of Levinson and Schur for Toeplitz matrices, inverse scattering, VLSI implementations, etc. This paper is an informal account of some of the different contexts in which the Berlekamp-Massey algorithm have been encountered in the work of the author and his students.