Close loop transfer function
The transfer function of the system is given by:
as is assumed to be a first-order system, its equation is given by:
where is the sampling time, and the time constant of the open loop system.
is the PI controller, its equation is given by:
The transfer function of the closed loop system is now given by:
Previous equation can be simplified:
The new transfer function is given by:
Static gain of the closed loop system
Let’s consider the response of the closed loop system when the input is a unity step ():
According to the final value theorem for Z-transforms, the static gain of the system is given by :
The static gain of the system is equal to 1, the static error will be equal to zero.
Time constant of the closed loop system
The closed loop system is also a first-order :
where is the sampling time, and the time constant of the closed loop system. Note that can be less than (the time constant of the open loop system). If the system is more reactive (but is also more energy consuming). In practice, is a good compromise. From the previous equation :
To get the same response time in closed and open loop, the previous equation becomes :
The system is stable if all the poles are located inside the unity circle. Here, as the system is a first-order, there is one pole : . The system is stable if:
The previous equation can be rewrite as:
Note that if is calculated from ( ), the term is included in and the system is necessarely stable because it leads to .
The transfert function of the controller expressed in the discrete-time domain is given by:
Let’s expressed the previous equation in the discrete-time domain:
I want to thank Laurent Hardouin from the University of Angers for his help and explainations.