how to calculate Plant transfer function in PID control system?

I make simple PID control system example.

in continuous sys

Pid contorl system : Gc=tf([1 1],[1 0]);

Plant system : Pc=tf([1 1],[1 1 1]);

and make simple feed back system As below

All system : Ac=(Gc*Pc)/(1+Gc*Pc);

And I make discrete systme like this

Discrete Control system : Gd=c2d(Gc,0.1);

All System : Ad=c2d(Ac,0.1);

As u know that i can get very easely to use c2d funcion to verigy Plant system in discrete system like to use Pd=Ad=c2d(Pc,0.1);

But i try to use solve the equation of feed back system like this

Pd=Ad/(Gd-Gd*Ad);

after that i test the step response test

figure(1)

step(Ac,Ad,100)

figure(2)

step(Gc,Gd,100)

Pd=Ad/(Gd-Gd*Ad);

figure(3)

step(Pc,Pd,100)

figure 1 and 2 are good but figure 3 Plant is different with continuous ans discrete

is there some mistake ? i Dont konw i Just follow the thory

can you help me??

[Prasanna Gude]

Can you attach the figures that you are expecting?

Answer 1

[Prasanna Gude]

As of my understanding, your problem arises due to the following command:

Pd = Ad/(Gd-Gd*Ad);

The above mathematical equation and the c2d function follow different approaches to get the output.

In the mathematical equation, infinite series associated with each transfer function will be computed to get the Discrete Plant transfer function. Whereas in c2d function, ‘zoh’(Zero-order hold) Discretization method is followed to discretize the continuous signal.

Instead use the following commands to obtain the output that you are expecting:

Pd = c2d(Pc,0.1);
step(Pc,Pd,100)