On the realization of periodic functions

In this article, canonical and spectrally minimal infinite-dimensional state space realizations for periodic functions are considered. It is shown that the periodic functions having ℓ₁ Fourier coefficients are precisely those realizable by a Riesz spectral system (RSS) in which the system operator generates a periodic strongly continuous semigroup and the observation operator is bounded. This realization can easily be converted to a canonical and spectrally minimal form. It is shown how the use of RSS and Cesáro sums of Fourier series allows the construction of a state space realization for a given periodic function merely integrable over its period. Simple finite-dimensional approximations with error bounds are derived for the RSS realization. Regular well-posed linear systems (WPLS) are used to construct a Fuhrmann-type realization for a given periodic function integrable over its period. It is shown that the RSS realizations and WPLS realizations are precisely equally good at coping with the possible ill behavior of a given bounded periodic function integrable over its period, but the WPLS realization is not always spectrally minimal or canonical.     

State space approach to behavioral systems theory: the Dirac–Bergmann algorithm

This paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised. This representation is a first order hybrid representation of behaviors called pencil representation. An algorithm well known after Dirac and Bergmann (DB) is shown to be central in obtaining a constraint free and observable (CFO) state space representation of a behavior. Results and criteria for asymptotic stability, controllability, inclusions and Markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation.     

Solution modules and system equivalence

In this paper we consider a system as a relation between input, internal variables and output. This relation is given by the solution space of the system's equations. For time invariant linear systems in differential operator representation the solution space carries a K[s]-module structure defined by the ordinary differential operator. This algebraic structure is exploited systematically to develop a self-contained theory of strict system equivalence in time domain.

The module of free motions is considered as space of initial conditions. An algebraic characterization of systems having the same solution space is presented. System homomorphisms are defined as special K[s] homomorphisms between the solution modules. Two systems are called system-equivalent, if there exists a system-isomorphism between their solution spaces. It turns out that, this concept coincides with Rosenbrock's concept of strict. system equivalence. It. is shown that further concepts and results of linear system theory (construction of a state-space model in time domain, controllability and observability criteria, uniqueness theorem of linear realization theory) can be derived within this framework.     

Stochastic theory of minimal realization

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.     

Multivariate control theory in linear sequential machines

This paper presents a unified survey of some modern multivariate control theory aspects and techniques applied to linear sequential machines over a Galois field GF(p), Utilizing the concepts of controllability and observability for a Mealy-type machine the canonical decomposition problem, the state minimization problem, the identification problem, the transformation to canonical form problem, and the controllability/observability problem of combined machines are studied. Then the problem of designing a linear feedback controller for driving any state of a linear machine to the zero state after a minimum number of time steps A(clock periods), as well as the dual problem of designing a time-optimal state reconstructor for the same machine are solved. Finally, the problems of inverting a linear sequential machine and decoupling its inputs and outputs by using state feedforward and feedback are examined. Several examples illustrate the theoretical results.     

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 forms for the identification of multivariable linear systems

The advantage of using a unique parameterization in a numerical procedure for the identification of a system from operating records has been well established. In this paper several sets of canonical forms are described for state space models of deterministic multivariable linear systems; the members of these sets having therefore the required uniqueness property within the equivalence classes of minimal realizations of the system. In the identification of a stochastic system, it is shown how the problem depends also upon determining a unique factorization of the spectral density matrix of the system, and the sets of canonical forms obtained for the deterministic system are extended to this case.     

The input-output and controllable behavioural structures of a state space representation

Given the matrices of a linear state space representation, we find an expression for a universal left annihilator of the matrix zI - A -C and hence derive kernel representations for the input-output behaviour and by duality the controllable part. More generally in the discrete case we derive representations for the L -completion for different values of L, and the subbehaviours of trajectories reachable in a given time interval. The representations are in certain trim canonical forms, which are intimately connected with structure indices. As a by-product of the state elimination procedure, we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one.     

Structure and parametrization of periodic linear systems

In this paper, the beginnings of a structure theory for discrete-time periodic linear systems are developed. Canonical forms for state space realizations are described, structural invariants and their combinatorics are studied and geometric aspects of the parametrization of periodic systems are treated.     

ARMA canonical forms obtained from constructibility invariants

In recent approaches, multivariable ARMA models have been derived from observable canonical MFDs, but it was realized that these models have a number of serious limitations and disadvantages. This paper presents a new approach based on the idea of observing the state from past outputs, which leads to the construction of monic ARMA models defined by the constructibility invariants. The derivation and structural properties of these constructibility forms are investigated.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Canonical forms for bilinear input/output maps

In an earlier paper [1], necessary and sufficient conditions were obtained for a state-space realization of a bilinear inout/output map to be quasi-reachable and observable, and procedures were introduced for reducing any realization not of this type to one which is. In rids paper, it is shown that any two such quasi-reachable and observable (or canonical) realizations are isomorphic, and using this result, it is possible to generate canonical forms for these realizations by means of complete sets of independent invariants.     

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.     

On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so-called singular value functions of a system describe the relative importance of each state component from an input–output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, distinct sets of singular value functions and balanced realizations are possible.     

Invariants and canonical forms for systems structural and parametric identification

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled.With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set.Then the set of polynomial input-output models of the type P(z)y(t)=Q(z)u(t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced.Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced.The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system.The results obtained in the identification of a real process are then reported.     

Minimal state space realisation of continuous-time linear time-variant input–output models

In the linear time-invariant (LTI) framework, the transformation from an input–output equation into state space representation is well understood. Several canonical forms exist that realise the same dynamic behaviour. If the coefficients become time-varying however, the LTI transformation no longer holds. We prove by induction that there exists a closed-form expression for the observability canonical state space model, using binomial coefficients.     

Moduli and canonical forms for linear dynamical systems II: The topological case

In this paper we study real linear dynamical systems \(\dot x = Fx + Gu,y = Hx,x \in R^n \) = state space,u ∈ R ^m = input space,y ∈ R ^p = output space, under the equivalence relation induced by base change in state space; or in other words we study triples of matrices with real coefficients (F, G, H) of sizesn × n, n × m, p × n respectively, under the action(F, G, H.) →(TFT ^?1,TG, HT ^?1) ofGL _n (R), the group of invertible realn × n matrices. One of the central questions studied is: “do there exist continuous canonical forms for this equivalence relation?”. After various trivial obstructions to the existence of such forms have been removed the answer is very roughly: no ifm ≥ 2, p ≥ 2, yes ifm = 1, orp = 1. For a precise statement cf. theorem 1.7. Existence or nonexistence of continuous canonical forms is related to the existence of a universal family of real linear dynamical systems. More precisely continuous canonical forms exist if such a universal family exists and if the underlying vector bundle of this family is the trivial vector bundle. In the case studied we show that a universal family in the appropriate sense does exist. The methods used are purely (differential) topological and in particular do not involve any algebraic geometry. There is a corresponding algebraic theory over any fieldk instead ofR which is the subject of part III of this series of papers.     

Observable state space realizations for multivariable systems

This paper derives two canonical state space forms (i.e., the observer canonical form and the observability canonical form) from multiple-input multiple-output systems described by difference equations. The state space model is expressed by the first-order difference equation and is equivalent to the input–output representation. More specifically, by setting the different state variables, the difference equations or the input–output representations can be transformed into two observable canonical forms and the canonical state space model can be also transformed into the difference equations. Finally, two examples are given.     

Computational algorithms for linear control systems: a brief survey

This paper presents a brief survey of computational algorithms for the analysis and synthesis of linear control systems described in the state space. An attempt is made to select the most efficient methods for analysis of the stability, controllability and observability, the reduction into canonical forms, the pole assignment synthesis and the synthesis of optimal systems with quadratic cost. Some aspects of the development of mathematical software for solving these problems are also discussed.