Input-output Equivalence Research Articles

Equivalent 2-D nonsingular Roesser models for discrete linear repetitive processes

ABSTRACTThe elementary operations algorithm is used to establish that a system matrix describing a discrete linear repetitive process can be transformed to that of a 2-D nonsingular Roesser model where all the input–output properties are preserved. Moreover, the connection between these system matrices is shown to be input–output equivalence. The exact forms of the resulting system matrix and the transformation involved are established. Some areas for possible future use/application of the developed results are also briefly discussed.

Linear Input-Output Equivalence and Row Reducedness of Discrete-Time Nonlinear Systems

The problem of linear input-output (i/o) equivalence of meromorphic nonlinear control systems, described by implicit higher order difference equations, is studied. It is proved that any system is linearly i/o equivalent to a row-reduced form. The constructive algorithm is given for finding the required transformation. The latter amounts to 1) multiply the set of i/o equations <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\varphi=0$</tex></formula> from left by a unimodular matrix <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$A(\delta)$</tex> </formula> , whose entries are non-commutative polynomials in the forward-shift operator <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\delta$</tex> </formula> , and 2) define certain multiplicative subset of the difference ring of analytic functions which introduces some inequations that should be satisfied.

Input-Output Model Equivalence of Spin Systems: A Characterization Using Lie Algebra Homomorphisms

In this paper, we consider the problem of model equivalence for quantum systems. Two models are said to be (input-output) equivalent if they give the same output for every admissible input. In the case of quantum systems, we take as output the expectation value of a given observable or, more generally, a probability distribution for the result of a quantum measurement. We link the input-output equivalence of two models to the existence of a homomorphism of the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie algebra $su(N)$ is useful to find such a homomorphism and to determine the classes of equivalent models. We consider in detail the important cases of two level systems with a Cartan structure and of spin networks. In the latter case, complete results are given generalizing previous results to networks of spin particles with arbitrary values of the spins. In treating this problem, we give independent proofs of some instrumental results on the subalgebras of $su(N)$.

Equivalence of linear control systems on time scales

The notions of transfer matrix, transfer equivalence, and i nput-output equivalence for linear control systems on time scales are introduced. These concepts generalize the cor- responding continuous- and discrete-time versions. Necessary and sufficient conditions for transfer and input-output equivalence are presented. As the main tool, an extension of the Laplace transform for functions defined on a time scale is use d.

Input-Output Equivalence of Spin Networks under Multiple Measurements

Two quantum control systems that are driven by an external field are said to be input–output equivalent if, for any control field, the measured value of a given observable is the same. Equivalent models cannot be distinguished by experiments involving state evolutions and measurements. In this paper, we characterize the equivalent models of networks of spin Open image in new window driven by electro-magnetic fields for which the expectation value of the total magnetization is measured. Extending previous results and definitions that only dealt with the case of a single measurement, we describe the class of equivalent models under a sequence of Von Neumann measurements. The results are motivated by the problem of parameter identification for Heisenberg spin systems modeling molecular magnets.

Input-Output Equivalence Transformations for Discrete-Time Nonlinear Systems

The paper characterizes the input-output (i/o) equivalence transformations for discrete-time nonlinear system described by the set of nonlinear higher order difference equations in the inputs and the outputs. It has been demonstrated that applying the i/o equivalence transformation to the set of equations ϕ = 0 amounts to multiplying ϕ from left by the unimodular matrix U(δ), whose entries are non-commutative polynomials in the forward-shift operator δ. Then it has been proved that using the i/o equivalence transformations, the set of equations can be transformed into the row-reduced form, whenever the set of equations is well-defined. Finally, the constructive algorithm is provided for calculation of the transformation, which extends the corresponding transformation for linear systems and is directly implementable via the computer algebra systems like Mathematica or Maple.

Model matching under external and input-output equivalence

The model-matching problem for systems described by external models is considered in frameworks of both external and input-output equivalence. Necessary conditions for the solvability of the problem are produced, and it is shown that in certain cases these conditions are also sufficient. In the case where necessary and sufficient conditions exist, the solutions of the problem are obtained in a constructive way and a parametrization of solutions is given.

Generalized transfer functions and input-output equivalence

The concept of a transfer function is extended to general dynamic systems described by non-regular linear algebraic and integro-differential equations. The input consistency and uniqueness of the system output for the given consistent input are discussed. These two properties are closely related to the existence of a transfer function which itself does not need to be unique. Input-output equivalence in the sense that two systems have the same output for the given consistent input is proved to be necessary and sufficient for the existence of an input-output isomorphic map between the solution spaces of the systems under zero initial conditions.

Some New Aspects on Descriptor form Representation of Systems

Properties of the descriptor form representation of linear time-invariant multivariable systems have been studied using polynomial system methodology. A new normal form for the input-output equivalence on descriptor form representations has been presented. This normal form makes the complicated and partly misleading controllability and observability considerations presented in the literature clear and easy to carry out.

System matrix characterization of input-output equivalence

The notion of input-output equivalence is defined and its characterization in system matrix terms is given.

Generalized linear systems with lumped or distributed parameters and differential vector spaces

Abstract By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

Generalized linear systems with lumped or distributed parameters and differential vector spaces

Abstract By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

A Criterion Based Approach to Parametrization and Identification of Multivariable Systems

The problem of system identification is considered in terms of (approximate) modelling of a given sequence of measurement data. Given a data sequence of a multivariable process, there arc two aspects that play a central role in determining the properties of the identified models: the set of models that is taken into account, and the identification criterion that selects the identified models from this set. When parametrizing a set of models ℳ very often use is made of a notion of model equivalence in order to arrive at a uniquely identifiable model set ℳ* ⊂ ℳ It is undesirable that the dynamical properties of the identified model are dependent on the specific choice of ℳ*, i.e. the specific parametrization chosen; consequently it is shown that the applied model equivalence, which conventionally refers to input–output equivalence, has to be in accordance with the identification criterion. Therefore a criterion based model equivalence is introduced which for equation error identification methods leads to a non conventional set of canonical forms.

Program equivalence and context-free grammars

Tree equivalence is a relation among polyadic recursion schemes. This relation is broad enough to be interesting: equivalent schemes may not be obviously equivalent and may still differ in computationally important ways. We show that this relation is also narrow enough to imply input-output equivalence. Is tree equivalence decidable? We assign context-free grammars to recursion schemes in such a way that schemes are tree equivalent iff their grammars generate the same language. Known results on LL(k) grammars then imply that tree equivalence is decidable for a class of schemes which includes the monadic recursion schemes without constants. Some important nonmonadic schemes are also included.