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# Question 2 (Generalized Controller Design): Consider the open loop system GH (=)=- (=-0.75)(=-1)² a) Design a controlle...

Question 2 (Generalized Controller Design)

Consider the open loop system GH=

Transcribed: Question 2 (Generalized Controller Design): 1 Consider the open loop system GH (2) =- (z-0.75)(z-1)* a) Design a controller that assigns the closed-loop system poles to z,2 = 0.25± j0.25 and the rest poles to z =0.

Closed loop system:

It is a system in which the output is dependent on the input of the system. The output has a feedback system that ia connected with the input. Examples are AC, voltage stabilizer, electric iron, etc.

Open loop system has $GH\left(S\right)=\frac{1}{\left(Z-0.75\right){\left(Z-1\right)}^{2}}$

The general controller block diagram is shown below.

the transfer function of the block diagram is

Put C(Z) in the transfer function equation.

Poles are at

The polynomial is

Compare (1) with (2) and get

$\begin{array}{rcl}{K}_{i}& =& 0.125.......\left(3\right)\\ & & \\ 2.5+{K}_{d}& =& 1\\ {K}_{d}& =& -1.50......\left(4\right)\\ & & \\ {K}_{P}-0.75& =& -0.50\\ {K}_{P}& =& 0.25.........\left(5\right)\end{array}$

Put the above value in the C(Z) equation.

$\begin{array}{rcl}C\left(Z\right)& =& \frac{{K}_{d}.{Z}^{2}+{K}_{p}Z+{K}_{i}}{Z}\\ C\left(Z\right)& =& \frac{-1.50{Z}^{2}+0.25Z+0.125}{Z}\end{array}$

Put the value (3), (4), and (5) in the eq(1).

The transfer function with the controller is

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