Basis of invariants and canonical forms for linear dynamic systems

This paper is a study of the problem of how to parametrize the set of all finite order constant linear systems. The parameters are interpreted as independent invariants for the equivalence relation which defines two systems to be equivalent when they have the same impulse response. Two kinds of canonical representations of the systems are constructed from the invariants, one of the state-space equations type and the other of the transfer function type.     

Computing the regularization of a linear differential–algebraic system

We study the regularization problem for linear differential–algebraic systems. As an improvement of former results we show that any system can be regularized by a combination of state-space and input-space transformations, behavioral equivalence transformations and a reorganization of variables. The additional state feedback which is needed in earlier publications is shown to be superfluous. We provide an algorithmic procedure for the construction of the regularization and discuss computational aspects.     

Computation of transfer function matrices of linear multivariable systems

This paper is concerned with the computation of transfer function matrices of linear multivariable systems described by their state-space equations. The algorithm proposed here performs orthogonal similarity transformations to find the minimal order subsystem corresponding to each input-output pair and uses some determinant identities to determine the elements of the transfer function matrices. Numerical examples are included to illustrate the performance of the proposed algorithm.     

A multipoint continued-fraction expansion for linear system reduction

This note presents a multipoint continued-fraction expansion (MCFE) to obtain reduced models for linear time-invariant systems. Algorithms for the expansion and inversion of the MCFE are outlined. A realization for the MCFE of a transfer function is suggested, and the corresponding MCFE canonical state-space model is established. A new similarity transformation matrix is constructed to transform a state-space model in the phase-variable canonical form to one in the MCFE canonical form. The attractive feature of the present MCFE is that it is general in form and the resulting reduced models approximate well the original system in both the frequency and time responses.     

Fundamental equivalence of generalized state-space systems

This paper defines a fundamental form of equivalence between two generalized state-space systems in terms of mappings of the solution sets of the describing differential equations together with mappings of the sets of restricted initial conditions. Properties of this equivalence relation are discussed and it is shown to be an alternative characterization of the system matrix relationship of complete system equivalence.     

Equivalence of dynamical systems by bisimulation

A general notion of bisimulation is defined for linear input-state-output systems, using analogies with the theory of concurrent processes. A characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory. Bisimulation is shown to be a notion which unifies the concepts of state-space equivalence and state-space reduction, and which allows to study equivalence of systems with nonminimal state-space dimension. The notion of bisimulation is especially powerful for "nondeterministic" dynamical systems, and leads in this case to a notion of equivalence which is finer than equality of external behavior. For abstractions of systems it is shown how the results specialize to previously obtained results by other authors. Extensions of the main results to the nonlinear case are provided.     

Reduced-order modelling of linear discrete-time systems via continued-fraction expansion of a z transfer function about z = 1 and z = a alternately

A new time-domain procedure is suggested for obtaining reduced-order models of linear time-invariant discrete-time systems. The procedure is based on presenting a new form of continued-fraction expansion (CFE) about z = 1 and z = a alternately, and deriving a realization form for the CFE. An algorithm is presented for obtaining the new CFE of the z transfer function of a linear discrete-time system from its state-space model directly, without having to determine the corresponding rational z transfer function. Also presented is a systematic approach to deriving two similarity transformation matrices: one is used to transform a state-space model from a general form to the CFE canonical form, and the other is used to transform a state-space model from the phase-variable canonical form to the CFE canonical form. Finally, an approximate aggregation matrix is constructed to relate the state vector of the original system to that of a reduced model obtained by the present method. The proposed procedure is illustrated with examples.     

Cauer continued-fraction expansion about s = 0 and s = a for biased reduced-order state-space models

A method of linear-system reduction is presented which produces biased state-space models such that combinations of retained time-moments and weighted time-moments may be varied. The method is based on the concept of representing the system transfer function in a biased Cauer form of continued-fraction expansion (CFE) about s = 0 and s = a. A new algorithm for obtaining the continued-fraction expansion of the transfer function of a linear time-invariant system from its state-space model directly, without first determining the corresponding transfer function, is derived. A realization which uses only integrators for the biased Cauer continued-fraction expansion about s = 0 and s = a is presented. The corresponding canonical state-space model is established, which allows the reduced-order state-space model to be formed by directly partitioning the system matrices. Also presented are two new similarity transformation matrices; one is used to transform a state-space model from a general form to a state-space model in the biased CFE canonical form, and the other is used to transform a state-space model from phase-variable canonical form to the biased CFE canonical form.     

Parameterization and identification of multivariable state-space systems: A canonical approach

In this paper, the problem of determining a canonical state-space representation for multivariable systems is revisited. A method is derived to build a canonical state-space representation directly from data generated by a linear time-invariant system. Contrary to the classic construction methods of canonical parameterizations, the technique developed in this paper does not assume the availability of any observability or controllability indices. However, it requires the -matrix of any minimal realization of the system to be non-derogatory. A subspace-based identification algorithm is also introduced to estimate such a canonical state-space parameterization directly from input–output data.     

Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time

In the present paper the problem of equivalence under coordinate changes and output transformations to observer canonical forms is addressed in discrete time for multi-output systems. Necessary and sufficient conditions are given for local equivalence to this form which yields a straightforward observer design with linear error dynamics.     

Generalization of Transfer Equivalence for Discrete-Time Non-Linear Systems: Comparison of Two Definitions

The problem of system reduction is studied for discrete-time non-linear single-input single-output systems described by high-order input–output (i/o) difference equations, that is, given the i/o equation, we can find a minimal representation which is equivalent to the original system with the order being as small as possible. Comparison of two definitions of reducibility is provided. The reducibility concepts addressed are both generalizations of the well-known notion of transfer equivalence to the case of non-linear control systems. Two roles of transfer equivalence are covered by these extensions. The first is that of identity of the outputs for any fixed control sequence under zero initial conditions. The second property is that of pole/zero cancellation that may occur in the transfer function of equivalent systems. The relationship between two reducibility definitions related to two equivalence notions is examined and the reducibility definition which extends pole/zero cancellation property is shown to be stronger. Finally, the computational aspects of system reduction are discussed.     

Note on ‘ On the definition of transfer function ’

An algorithm for directly obtaining the canonical state-space model corresponding to the Cauer third form of continued fraction expansion (CFE) from a given general state-space model is presented. The algorithm can be used to determine the transfer function of linear time-invariant system from its state-space model as well as to obtain the reduced order models..Two new similarity matrices, one transforms a state-space equation from a general form to a Cauer third CFE canonical form, and the other transforms a state-space model in phase-variable form to a state-space model in a Cauer third CFE canonical form, are derived. Using these matrices an approximate relationship between the original state vector and the state-vector of reduced model obtained by the method of Cauer third CFE is established     

Model reduction for state-space symmetric systems

In this paper, the model reduction problem for state-space symmetric systems is investigated. First, it is shown that several model reduction methods, such as balanced truncation, balanced truncation which preserves the DC gain, optimal and suboptimal Hankel norm approximations, inherit the state-space symmetric property. Furthermore, for single input and single output (SISO) state-space symmetric systems, we prove that the H_∞ norm of its transfer functions can be calculated via two simple formulas. Moreover, the SISO state-space symmetric systems are equivalent to systems with zeros interlacing the poles (ZIP) under mild conditions.     

A global parametrization of asymptotically stable linear systems

In this paper we construct a homeomorphism from the set of p × m transfer functions of McMillan degree n onto the open subspace of asymptotically stable linear systems. This homeomorphism yields a one-to-one correspondence between

1. (i) canonical forms for state space equivalence of minimal systems and

2. (ii) canonical forms for state space equivalence of asymptotically stable minimal systems.

Implications for the topology of various sets of asymptotically stable systems are given.     

Model reduction for state-space symmetric systems

In this paper, the model reduction problem for state-space symmetric systems is investigated. First, it is shown that several model reduction methods, such as balanced truncation, balanced truncation which preserves the DC gain, optimal and suboptimal Hankel norm approximations, inherit the state-space symmetric property. Furthermore, for single input and single output (SISO) state-space symmetric systems, we prove that the H_∞ norm of its transfer functions can be calculated via two simple formulas. Moreover, the SISO state-space symmetric systems are equivalent to systems with zeros interlacing the poles (ZIP) under mild conditions.