When considering modeling and control problems often one must study functionals of the system variables and their derivatives: this happens for example in optimal control, in the theory of Lagrangian or Hamiltonian mechanics, in Lyapunov stability theory, etc. When studying linear systems, such functionals are usually taken to be bilinear or quadratic.
Given the predominance of the state-space paradigm in system and control theory, usually only (integrals of) functionals of the state and of the input of the system are considered. However, in reality linear systems are hardly ever described by state-space models: the result of a first principles modeling procedure is usualy a set of higher-order differential equations possibly involving algebraic relations among the variables. Consequently, the need arises to develop a theory of bilinear- and quadratic functionals of the variables of a linear system described by such a model, without transforming it first in state-space form. These functionals must necessarily involve, besides the variables of the system, also their higher-order derivatives.
An efficient representation of such functionals by means of two-variable polynomial matrices has been proposed in [1], where the concepts of bilinear- and quadratic differential form have been introduced. The association of BDFs and QDFs with two-variable polynomials allows to develop a calculus that has applications in various areas of systems and control theory. In such calculus, operations and properties of the bilinear- or quadratic functionals correspond to algebraic operations and properties of the two-variable polynomial matrices representing the functionals. In the first part of this talk some of the basic concepts of bilinear- and quadratic differential forms are introduced; the second part deals with various applications to system- and control theory problems such as the modelling of linear Hamiltonian systems, an equipartition of energy principle, and J-spectral factorization.
References
[1] Willems, J.C. and Trentelman, H.L., “On quadratic differential forms”, SIAM J. Control Opt., vol. 36, no. 5, pp. 1703-1749, 1998.
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