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The relationship between the unimodular matrices relating coprime and column reduced matrix fraction descriptions (MFD) of a nonproper transfer function, and the restricted system equivalence (r.s.e.) transformations relating the corresponding generalised state space realisations is considered. It is shown that the r.s.e and unimodular transformations can be directly obtained from each other by inspection. The r.s.e. transformations leading to the canonical form are derived from the unimodular transformations leading to the echelon canonical form of the composite matrix of the MFD of the system.

This chapter focuses on the equivalence of non-nested models. The concept of equivalence is a fundamental one in stochastic modeling. Equivalence of submodels need not be the result of underidentification of the original model, which also need not be equivalent to the submodels. In all cases of equivalence, the data cannot be used to distinguish between the equivalent models. The chapter formulates conditions for equivalence and thus, tries to answer the basic question as to when differently formulated models describe reality in the same way. It is shown that equivalent models need not be submodels of a common underidentified, that is, overparametrized model. The chapter gives conditions for equivalence that can also be applied in case of an identified common model. However, the regularity assumptions made in the chapter cannot be considered mild. There are relevant cases of equivalence where the regularity assumptions are not satisfied. The chapter considers, as a special case, systems of simultaneous equations; here the conditions that characterize equivalence allow for a computerized evaluation.

Publisher Summary The goal of an observation method is to determine the observable behaviors of a system under various conditions. The conditions might involve different observers, different stimuli to the system, or simply range over all possible executions of a nondeterministic system. Observable behaviors can be useful in expressing the requirement that a system satisfies a specification, which might require or exclude certain observable behaviors. An observation method can be used for defining an equivalence notion between systems: two systems are equivalent whenever no observation can distinguish them. Many different observation methods and observation equivalences have been proposed for nondeterministic systems and especially for nondeterministic concurrent systems. This chapter discusses observational equivalence. Proving the properties of observational equivalence requires transfinite induction. However, it is easy to prove that two trees are observationally equivalent as it is enough to exhibit a bisimulation relating them. The chapter discusses, under very general assumptions, the existence of canonical representatives for observational equivalence classes.

Publisher Summary The linear, time invariant compartmental models are of primary interest in the pharmacokinetics. The goal of the compartmental analysis is usually to find the simplest model that can fit the experimental data within the limits of measurement errors. This chapter describes the use of a new tool for tackling the a priori model selection and model simplification problems in compartmental analysis. This new tool is the concept of structural equivalence that is a generalization of the conventional concept of system equivalence and is closely related to the notion of structural identiflability. The results are based on the rigorous mathematical formulation. The chapter presents the mathematical results with two examples to show possible applications of the concept with respect to some practical problems of pharmacokinetics.