Adaptive Two-Loop Control Scheme

The controller scheme developed in the following is based on the well-known GMC paradigm [22, 27] reviewed in Sect. 5.4.2. The key idea of this technique is that of globally linearizing the reactor dynamics by acting on the jacket temperature 7], which is, in turn, controlled by a standard linear (e.g., PID) controller. Since 7] does not play the role of the input manipulated variable, the only way to impose an assigned behavior to the jacket temperature is that of computing a suitable setpoint 7j,des, to be passed by a control loop closed around 7], Both in [22] and [27], the mathematical relationship between the jacket temperature and the setpoint is assumed to be a known linear first-order differential equation, from which Tj es is 

The whole control scheme is represented in Fig. 5.2. The first control loop (inner loop) is closed around the jacket temperature in such a way to track a desired temperature, 7j,des(0 = J2,des(0 to be determined then, an outer loop is closed around the reactor temperature so as to track the desired reactor temperature profile, 7r,des(0 = yi,des(0- The outer controller computes the desired jacket temperature on the basis of the reactor tracking error e = ypdes - yi and of the estimate of aq, while the inner controller receives y2,des as input and computes the temperature of the fluid entering the jacket, i.e., the manipulated input u. 

Since the control goal is the tracking of a temperature profile for the reactor, according to the GMC method, a desired profile for y must be chosen. By considering (5.12) and the expression of y = xatc+i given by (5.15), the following equality can be imposed  

An alternative version of the control law (5.38), (5.39) can be obtained by setting the gains gi,r and gis to zero. Therefore, the following control law can be used  

The convergence properties of the error variables (including the parameter estimation errors 60 = 6 - 90 and 9C = 9 - 9C) for the overall controller-observer scheme defined by (5.23)-(5.25) and (5.38)-(5.40) are established by the following theorem  

---