**PID controller** =>

1) The proportional control gives a control signal proportional to the error signal e(t).

2) The integral control calculates the integral of error thereby giving out a control signal as long as there is a error.

3) The derivative action computes the rule of change of error and gives out a control signal which increase the speed of response.

4) Hence PID controller, which combines all the three mode is the most superior and it is given by

m(t) = $K_p e(t) + K_i \int_{0}^{t} e(t)dt + K_d \frac{d}{dt}e(t) + m_o$

computing the laplace transform we get,

$M(s) = K_p E(s) + \frac{K_i}{s} E(s) + K_ds E(s) + M_o$

Block diagram =>

using block reduction, we get

we know, $\frac{c(s)}{k(s)} = \frac{G(s)}{1+G(s)H(s)}$

$\frac{C(S)}{K(S)} = \frac{(K_p + K_d s + \frac{K_i}{s}) (\frac{{w_n}^2}{s(s+2 \epsilon w_n)})}{1+(K_p + K_d s + \frac{K_i}{s}) (\frac{{w_n}^2}{s(s+2 \epsilon w_n)})}$

$\frac{C(s)}{K(s)} = \frac{(K_p + K_d s + \frac{K_i}{s}) {w_n}^2}{s(s+2 \epsilon w_n)+(K_p + K_d s + \frac{K_i}{s}) {w_n}^2}$

$\frac{C(s)}{K(s)} = \frac{(\frac{K_p s + K_d s^2 + K_i}{s}){w_n}^2}{(s^2 + 2 \epsilon w_n s)+(\frac{K_p s + K_d s^2 + K_i}{s}){w_n}^2}$

$\frac{C(s)}{K(s)} = \frac{(K_p s + K_d s^2 + K_i)({w_n}^2)}{s(s^2 + 2 \epsilon w_n s) + (K_p s + K_d s^2 + K_i)({w_n}^2)}$

$\frac{C(s)}{K(s)} = \frac{(K_p s + K_d s^2 + K_i)({w_n}^2)}{s^3 + 2 \epsilon w_n s^2 + K_p s {w_n}^2 + K_d s^2 {w_n}^2 + K_i {w_n}^2}$

$\frac{C(s)}{K(s)} = \frac{(K_ps +K^2_{ds} + K_i)w^2_n}{s^3 + (2\epsilon w_n+K_dw^2_n)s^2 + K_pw^2_ns + K_iw^2_n}$

This is the Transform function of PID controller.

**Advantages of PID controller**,

1) Simple

2) Easiest of the continuous controller's tune

3) Provides good stability, rapid response and relatively stable.

**Model predictive control system**

1) Model predictive control system (MPC) refers to a class of computer control algorithms that utilize an explict process model to predict the future response of a plant.

2) At each control interval an MPC algorithm atempts to optimize future plant behaviour by computing a sequence of future manipulated variables.

3) The first input in optimal sequence is then sent into the plant and the entire calculations in repeated at subsequent control intervals.

4) MPC technology can now be found in wide variety of application area includes chemical, food processing, automotive and aeroplane application.

**Advantages of MPC system**

1) Relatively easy tuning

2) Requires little computation

3) control a great variety of processes

4) Very useful when future reference