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81) −∞ where Eh is the region of existence of the two-sided Laplace transform. 82) 0− where Eh+ is the region of existence of the one-sided Laplace transform. If the impulse response function h(t) is zero for t < 0, then H(s) = H+ (s). A set of very important consequences for the analysis and synthesis of the systems are brought about by the transfer function concept. They will be presented in the following. (1) A first consequence of the new revealed concept of the transfer function is the fact that, in Laplace transform, there is a simple relationship between the Laplace transform of the output function Y(s) and the Laplace transform of the input function U(s).
27) for t ∈ T, T = [t0 , ∞], with n and m positive integer values and with f an application, f : Cn+m+2 × T → C. 10 Time response of the linear time-invariant systems Linear time-invariant (LTI) systems are described by linear differential equations having constant coefficients. 28) in which the qn and pm are nonzero coefficients. The domain of definition of the function u(t) and y(t) is the half-infinite T+ = [t0 , ∞) or infinite T∞ = R axis. The halfinfinite axis T+ is considered to include the moment t0− , leading to the interval T+ = [t0 −ε , ∞) with ε → 0.
If a small and bounded input or initial condition determines an unlimited increasing output, the system is denoted unstable. This mode of expressing stability leads to the concept of Bounded-Input BoundedOutput Stability (BIBO stability). Definition. A time-continuous system is denoted as stable in the bounded-input bounded-output sense (BIBO stable) if its response y to any finite amplitude input u remains of finite amplitude. A typical example of chemical systems instability is that of the reactor in which an exothermic reaction is taking place and for which the heat flow produced by the chemical reaction is superior to the evacuated heat flow of the cooling agent.