Process Parameters F0 10 ft3min F1 20 ft3min T020 C T1 35 C

Process Parameters: F0= 10 ft3/min, F1: 20 ft3/min, T0-20 C, T1= 35 C, Vo= 20 ft3,Va= 20 ft3, pCp= 1.0 cal/K/ft, Actuator delays 2.0 min.

Solution

to design p controller that achieves a 50% disturbance attenuation level in the closed-loop system.then follow given preperties

We start with a simple controller of the form C(s) = KP , where KP is a constant that we will choose. In this case, the input to the plant is simply u(t) = KP e(t), which is proportional to the error. Thus, this type of controller is called a proportional controller. With this controller, the transfer function from r to y in the feedback control system becomes Try(s) = KP b0 s 2 + a1s + (a0 + KP b0) . Recall that the poles of this transfer function dictate how the system behaves to inputs. In particular, we would like to ensure that the system is stable (i.e., all poles are in the OLHP). Since the gain KP affects one of the coefficients in the denominator polynomial, it can potentially be used to obtain stability

to obtain pi controller

To obtain perfect tracking for step inputs, we will introduce an integrator into the controller (i.e., we will add a pole at the origin) in order to ensure that the system will be of type 1. The controller thus becomes C(s) = KP + KI s . In the time-domain, this corresponds to the input to the plant being chosen as u(t) = Kpe(t) + Z t 0 e( )d , and thus this is called a proportional-integral controller. With this controller, the transfer function from r to y is Try(s) = b0 s 2+a1s+a0 KP s+KI s 1 + b0 s 2+a1s+a0 KP s+KI s = b0(KP s + KI ) s 3 + a1s 2 + (a0 + KP b0)s + KI b0 . Note that we now have a third order system. Two of the coefficients of the denominator polynomial can be arbitrarily set by choosing KP and KI appropriately. Unfortunately, we still have no way to stabilize the 2 system if a1 < 0 (recall that for stability, all coefficients must be positive). Even if the system is stable with the given value of a1, we might want to be able to choose better pole locations for the transfer function in order to obtain better performance. To do this, we add one final term to the controll