New Control Theory and Its Application

Abstract

The PID control system design method based on transfer functions was established in the 1950s and has since made contributions to automation processes in various industrial fields. Transfer functions have features that make you aware of rough characteristics of input–output responses based on the positioning of the poles and zero point and also discuss frequency responses and the stability of the control system using their Bode diagram. Now we will once again review the restrictions required to express dynamic characteristics of the control target with transfer functions.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-4-431-54195-0_12

Chapter 11 Exercises

• 1. Express the Eq. (5.1) shown in Chap. 5 with a state equation; however, the output equation is for current i(t).

• 2. Prove that the closed loop control system is stabilized by the control input shown in Eq. (11.13).

• 3. Prove that the Eq. (11.12) of the evaluation function is minimized by feeding back the control input shown in Eq. (11.13).

• 4. Design an optimal regulator based on the following evaluation function “J” as the minimum, assuming that the control target is

  $\frac{{\mathrm{ d}x(t)}}{{\mathrm{d}t}}=\left[ {\begin{array}{llll} 0 & \ \ 1 \\0 &{ -1} \\ \end{array}} \right] x(t)+ \left[ \begin{array}{llll} 0 \\ 1 \end{array}\right] u(t)$

  $J= \int\limits_0^{\infty } {\left\{ {{x^{\mathrm{ T}}}(\tau )\left[ {\begin{array}{llll} 1 & 0 \\0 & 1 \\ \end{array}} \right]x(\tau )+{u^2}(\tau )} \right\}} \mathrm{d}\tau$

• 5. Derive Eq. (11.23).

• 6. Explain that how to correct the weight by learning is expressed in a simple format like Eq. (11.33) if the sigmoid function such as Eq. (11.30) is used as an input–output function for neural network.