Final-value theorems

When r t) is a unit step, and riit) is zero, the final value theorem (equation (3.10)) gives the steady-state response... 

The denominator is now in the standard second-order system form of equation (3.42). The steady-state response may be obtained using the final value theorem given in equation (3.10). 

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. 

We arrive at the final value theorem after we cancel the f(0) terms on both sides. 

Here, f(t) = e2t. There is no upper bound for this function, which is in violation of the existence of a final value. The final value theorem does not apply. If we insist on applying the theorem, we will get a value of zero, which is meaningless. 

Yes, another trick question. If we apply the final value theorem without thinking, we would get a value of 0, but this is meaningless. With MATLAB, we can use... 

We can check the result with the final value theorem... 

We now take a formal look at the steady state error (offset). Let s consider a more general step change in set point, R = M/s. The eventual change in the controlled variable, via the final value theorem, is... 

Now if we have a unit step change R = 1/s, the steady state error via the final value theorem is (recall that e = e )... 

There are two noteworthy items. First, the closed-loop system is now second order. The integral action adds another order. Second, the system steady state gain is unity and it will not have an offset. This is a general property of using PI control. (If this is not immediately obvious, try take R = 1/s and apply the final value theorem. We should find the eventual change in the controlled variable to be c (°°) =1.)... 

To find the steadystate value of the error, we will use the final-value theorem from Chap. 9. 

When diffusion is not fast enough to avoid some spatial inhomogeneity, the value of the kernel must be considered. In general, eqn. (224) is too awkward to invert, so that the limiting value for long times can be determined using the final value theorem of Laplace, transforms, wherein g(s) is the Laplace transform of f(f)... 

The final value theorem gives the value of the response at t =... 

Using the final value theorem (equation 7.86) (sT does not become infinite for Re(s) > 0) ... 

Here we use the final value theorem whence we can replace the right hand side above by... 

This is the equation describing the dynamics of the longitudinal component of the magnetization obtained by the same method of truncation of the continued fraction as that employed in Section IV. This method, also used by Morita for dielectric relaxation [56], is a consequence of the final value theorem for Laplace transforms, which is... 

The shape of the response y(i) depends on the value of (overdamped, critically damped, or underdamped), but the ultimate value of the response can be found from the final-value theorem (Section 7.5) ... 

This is a very useful theorem because it allows us to compute easily the final value of a function from its z-transform. The final-value theorem states that... 

If y(nT) is the sampled response of a process, the final-value theorem yields the steady-state value of the process s reponse. 

The sampled-value response y(nT) is shown in Figure 29.7b and follows the same pattern as for the continuous case (see Figure 10.4). To find the steady-state value of the process output, invoke the final-value theorem. Thus... 

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