State-Space Model for Control Design

Since the dynamics of a batch reactor is characterized by a unitary relative order, the GMC law can be adopted [6, 14, 22, 40, 42, 65], In order to cope with model uncertainties, adaptive GMC approaches have been developed [56, 60, 62] in [27] some unknown quantities—namely, the effect of the heat released by the reaction and the heat transfer coefficient—are estimated by adopting the nonlinear adaptive observer proposed in [24] in [63], an ANN-based GMC approach is presented for semi-batch processes with relative order higher than one. 

Most of the above reviewed approaches, including those based on the mathematical model of the system, are developed and tested for simplified reaction networks and/or for particular reactive systems. Nonetheless, it is worthwhile to focus on more general reaction schemes and on the rigorous analysis of the main structural properties (i.e., stability and robustness) of the closed-loop system. 

In the following, the model-based controller-observer adaptive scheme in [15] is presented. Namely, an observer is designed to estimate the effect of the heat released by the reaction on the reactor temperature dynamics then, this estimate is used by a cascade temperature control scheme, based on the closure of two temperature feedback loops, where the output of the reactor temperature controller becomes the setpoint of the cooling jacket temperature controller. Model-free variants of this control scheme are developed as well. The convergence of the overall controller-observer scheme, in terms of observer estimation errors and controller tracking errors, is proven via a Lyapunov-like argument. Noticeably, the scheme is developed for the general class of irreversible nonchain reactions presented in Sect. 2.5. 

In order to design the controller, the model developed in Sect. 2.5 is conveniently reformulated in a state-space form. The following assumptions on the reactive process and on the reactor are used to derive the state-space form of the model. 

Assumption 5.1 The reactions involved in the process are exothermic and characterized by first-order kinetics therefore, the mass balances of the Nq compounds involved in the reaction have the form (2.27). 

---