H 2 optimal control of a chemical reactor J´ an Mikleˇ s, L’uboˇ s ˇ Cirka, and Miroslav Fikar Abstract —This paper deals with H 2 control of a continuous stirred tank reactor (CSTR). The coolant temperature and the temperature in the reactor are respectively considered as control and controlled variables. A connection between H 2 polynomial approach and the mixed sensitivity optimiza- tion for feedback control of CSTR is presented. The control algorithm has been implemented with the help of the Poly- nomial Toolbox for MATLAB. Simulation results demon- strate the robustness properties of H 2 control of CSTR. Keywords —H 2 optimal control, controller with integral ac- tion, chemical reactor. I. Introduction T RADITIONALLY linear methods have been used to design controllers for CSTRs. Control schemes, that have been proposed, include conservative linear controllers and linear controllers with gain scheduling. These methods are based on the first-order approximation of the actual system at a single point and a discrete set of operating points, respectively. Consequently, these techniques cannot account for large perturbations or operation away from the steady-state operating curve. There are various design methods, which utilize more accurate nonlinear models. A precise treatment of model nonlinearities has emerged for example with the differential geometric techniques of linearization [1]. These methods have been applied to the aforementioned CSTR problem to obtain exact state linearization, exact input-output lin- earization and state linearization with disturbance rejec- tion. The major shortcoming of these techniques is their lack of robustness guaranties. The problem of robust stability of closed loop systems has received a considerable amount of attention in recent years. In particular, H 2 and H ∞ approaches to optimal control design and loop shaping has provided some promis- ing results in the area of robust stabilization of plants. The modern control area is usually believed to span the period from 1960 to 1980. This coincided with the intro- duction of state-space-based synthesis techniques. These state-space approaches were thought as being time-domain methods, although Parseval’s theorem enabled the control problem to be posed equally well in frequency domain (clas- sical approach). At the present time, H ∞ optimal control is employed both in the state-space [2] as well as with the polynomial methods [3]. In H ∞ loop shaping design it is necessary to model plant uncertainty of the nominal plant. Chemical reactors (CSTR) are commonly used in the process industries. The reactor is a nonlinear process with time-varying parameters. It is necessary to control these Slovak University of Technology in Bratislava, Radlinsk´ eho 9, 81237 Bratislava, Slovakia, Tel: +421 2 524952 69, Fax: +421 2 52496469, e-mail: mikles@cvt.stuba.sk reactors to improve their behavior. Usually, input-output models are used for control purposes. These models only represent an approximation of the considered CSTR dy- namics. In fact, the CSTR parameters can be varying. Then it is necessary to implement robust control strategies for improving the behavior of the CSTR. H ∞ optimization of CSTR control deals with the minimization of the peak value of certain closed-loop frequency response functions. It can be seen, that H 2 optimization with shaping is the H 2 version of the well-known mixed sensitivity problem of H ∞ optimization [4]. Therefore, robust control algorithms can be based on H 2 optimal control. In this paper control of a chemical reactor is studied. Control design is based on H 2 optimal control design. The resulting controller has integral action. The rest of the paper is organized as follows. Section 2 is devoted to the formulation of H 2 optimal control. In Sec- tion 3, modeling issues relevant to the CSTR H 2 control problem is first addressed. Then the H 2 controller with in- tegral action is designed for CSTR model. The conclusions will be drawn in Section 4. II. H 2 Optimal Control The standard H 2 optimal control problem consists of sta- bilizing a linear system in such a way that its transfer ma- trix attains a minimum norm in the Hardy space H 2 . The plant is modeled by the transfer matrix G(s)=  G 11 (s) G 12 (s) G 21 (s) G 22 (s)  (1) with G 11 (s) = C 1 (sI − A) −1 B 1 , G 12 (s) = C 1 (sI − A) −1 B 2 + D 12 , G 21 (s) = C 2 (sI − A) −1 B 1 + D 21 , G 22 (s) = C 2 (sI − A) −1 B 2 , where matrices A, C 1 ,B 1 ,B 2 ,D 12 ,C 2 ,D 21 are given by the input-state-output model of the plant ˙ x(t) = Ax(t)+ B 1 v(t)+ B 2 u(t), z (t) = C 1 x(t)+ D 11 v(t)+ D 12 u(t), (2) y(t) = C 2 x(t)+ D 21 v(t)+ D 22 u(t). It is assumed that: (A, B 1 ) stabilizable, (A, C 1 ) detectable, (A, B 2 ) stabilizable, (A, C 2 ) detectable D T 12 C 1 =0,B 1 D T 21 =0,D T 12 D 12 = I, D 21 D T 21 = I, D 11 = D 22 =0.