Initial value theorem

We now present two theorems which can be used to find the values of the time-domain function at two extremes, t = 0 and t = °°, without having to do the inverse transform. In control, we use the final value theorem quite often. The initial value theorem is less useful. As we have seen from our very first example in Section 2.1, the problems that we solve are defined to have exclusively zero initial conditions. 

The final value theorem is valid provided that a final value exists. The proofs of these theorems are straightforward. We will do the one for the final value theorem. The proof of the initial value theorem is in the Review Problems. 

The value of the response at t = 0 is given by the initial value theorem which states that ... 

Given the z-transform of a function, the initial-value theorem allows us to compute the initial value of the function as follows ... 

Initialization of control algorithms, 637 Initial value theorems ... 

Initial-Value Theorem 141 Things to Think About 141... 

Find fit = 0). Using the initial-value theorem, we have... 

Back transformation from the transfer function to a differential equation in the time-domain, can be achieved by substitution of 5 -x(5 ) = dx(t)/dt. The variable s is independent of the time and indicates more or less the rate of change. This can be explained by the final-value theorem and the initial value theorem apphed to a variation in the input variable of a system. 

The parameters explained in Sections 1.1 through 1.5.2.5.4 are in the frequency domain. The parameters in the time domain are the same as those in the frequency domain in the perfect conductor case because they are frequency independent and, thus, time independent. In the case of the imperfecdy conducting earth and conductor, only the parameters at infinite frequency are known anal5 cally. These parameters should correspond to the parameters at t = 0 in the time domain from the initial value theorem of the Laplace transform, that is... 

Analogous to the final value theorem, the initial value theorem can be stated as... 

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