Noncausal Continuous Time Systems

Abstract

AbstractIn this chapter we study various types of noncausal continuous time systems. Contrary to the usual continuous time systems obeying the equations $$ x\left( t \right) = - iAx\left( t \right) + Bu\left( t \right), $$ ((7.1a)) $$ y\left( t \right) = - iCx\left( t \right) + Du\left( t \right), $$ ((7.1b)) where t ∈ ℝ+ is time, u(t) is input, y(t) is output, x(t) is the state, and −iA generates a strongly continuous semigroup, we now consider t ∈ ℝ and require −iA to be exponentially dichotomous. This amounts to dropping the causality assumption on the linear system. Various theories can be developed, parallelling existing theories for causal systems. In Section 7.1 we require −iA to be exponentially dichotomous and B and C to be bounded. This includes the direct generalization of finite-dimensional linear systems theory, where A, B, C, and D are all matrices and A does not have real eigenvalues. In Section 7.2 we pass to a formalism with two state spaces (one densely and continously imbedded into the other), where the exponentially dichotomous operator −iA on the larger state space extends that on the smaller state space, the input operator B is bounded from the input space into the larger state space, and the output operator C is bounded from the smaller state space into the output space. Also adopting a complex Hilbert space setting, we thus obtain the so-called extended Pritchard-Salamon realizations. At the same time we discuss left and right Pritchard-Salamon realizations, where only one state space is used at the time.KeywordsTransfer FunctionOperator FunctionBounded Linear OperatorContinuous SemigroupComplex Hilbert SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.