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Digital Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
The Final Value Theorem: The final value theorem may be viewed as the converse of the initial value theorem. It also applies only with the one-sided Z-transform. This theorem enables us to determine the behavior of r(k) as k → ∞ from its Z-transform. The final value theorem can be derived from the time shift theorem as follows:
Tuning rules for feedforward control from measurable disturbances combined with PID control: a review
Published in International Journal of Control, 2021
An interesting analysis can be obtained by using the final value theorem and the initial value theorem. From (3), the transfer function between d and y becomes Using from the Lambda tuning rule, the expression can be simplified to If the load disturbance is a unit step, the process output becomes Now, the final value theorem can be used to calculate the integral of y after the step load disturbance, resulting in Here, fore simplicity, it is assumed that the setpoint is zero. Determining controller gain K from the Lambda tuning rule (5) finally gives Since the controller is tuned using the Lambda tuning rule, the response is overdamped as long as λ is not too short. In this case, IE is equal to IAE which is one of the most common measures of process control performance. Equation (6) shows that the IE value is proportional to the gain of and that it increases when the delay of the process and the desired time constant of the closed-loop system, λ, increases.
Why do nonlinearities matter? The repercussions of linear assumptions on the dynamic behaviour of assemble-to-order systems
Published in International Journal of Production Research, 2019
We exploit the Initial Value Theorem (IVT) and Final Value Theorem (FVT) to mathematically crosscheck the correctness of the transfer function, guide the appropriate initial condition required by a simulation and to understand the final steady-state value of the dynamic response so as to help verify any simulation. Hence, the initial and final values of AINVAS, AINVSA, and ORATESA in responding to a unit step input are obtained.
Dynamic analysis and design of a semiconductor supply chain: a control engineering approach
Published in International Journal of Production Research, 2018
Junyi Lin, Virginia L.M. Spiegler, M.M. Naim
We now turn to the analysis of Initial Value Theorem (IVT) and Final Value Theorem (FVT). The IVT is a useful tool to cross-check mathematically the correctness of a transfer function and guide the appropriate initial condition required by a simulation. The FVT is useful to understand the steady state value of the dynamic response of a transfer function and can help verify the simulation. Equation (16) presents the initial and final values of FGI, , WS and AWIP in responding to a unit step input for the semiconductor hybrid MTS-MTO system.