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Additional resources for Basic Process Engineering Control
N. The properties of the one-sided Laplace transform may be extended to the case of the vector function. In particular, the linearity property makes that the one-sided or two-sided Laplace transform of a linear combination of vectors A ⋅ x(t) + B ⋅ y(t) (with A and B constant matrices) is A⋅X(s)+B⋅Y(s), in the region of existence Ex ∩Ey . The onesided Laplace transform of the time differentiated vector signal x???? (t) is sX + (s) − x(0− ), in the region of existence s ∈ E+ . In the following, the notation “+” associated to the one-sided Laplace transform will be dropped and the MIMO case (k inputs, m outputs and n states) will be mainly considered.
24 Definition of the one-sided and two-sided Laplace transform Although the one-sided Laplace transform is predominantly used for linear timeinvariant systems description, the two-sided Laplace transform will be introduced first in order to get a comprehensive overview of this mathematical instrument : Definition. The two-sided Laplace transform is the application L that transforms the continuous signal x(t), defined on the entire time axis t ∈ R, into a complex function X(s) = L(x(t)) having the complex independent variable s, and being defined by the integral ∞ X(s) = ∫ x(t) ⋅ e−st dt, s ∈ E ⊂ C.
Additionally, the implied operation of integration is a relatively complicated and resource-consuming operation even for numerical computation. These considerations led to the development of other methods for investigating and describing the system’s behavior, based on the frequency response function, methods that are usually named the system analysis in the frequency domain. Frequency domain analysis is intuitive with respect to reflecting the system behavior and it is used for control system design .