**2. Application of Tabu search for optimal control of a polymerization reactor**

The determination of open-loop time varying control policies that maximize or minimize a given performance index is referred as optimal control. The optimal control policies that guarantee the product property requirements and the operational constraints can be computed off-line, and are executed on-line in such a way that the process is operated in accordance with these control policies. Achievement of product quality is a major issue in polymerization processes since the molecular or morphological properties of a polymer product majorly influences its physical, chemical, thermal, rheological and mechanical properties as well as polymer applications. In a free radical copolymerization, composition drifts caused by variations in reactivities of comonomers can be mitigated by continuous addition of more reactive monomer while maintaining a constant mole ratio. The end use properties

of the polymer product such as flexibility, strength and glass transition temperature are affected by the copolymer composition. Further, the molecular weight (*MW*) and molecular weight distribution (*MWD*) affects the important end use properties such as viscosity, elasticity, strength, toughness and solvent resistance. Hence, the determination of optimal control trajectories is important for the operation of polymerization reactors in order to produce a polymer with the desired product characteristics.

In the past, various methods have been reported for optimal control of polymerization reactors [12–16]. Most of the studies are based on classical methods of optimization such as Pontryagin's maximum principle [17], which has been applied to solve the optimal control problems of different polymerization reactors [18–23]. However, the classical methods have certain limitations for optimal control of polymerization reactors and these drawbacks have been discussed in literature [24]. The stochastic and evolutionary optimization methods are found beneficial over the conventional gradient-based search techniques because of their ability in locating the global optimum of multi-modal functions and searching design spaces with disjoint feasible regions.

### **2.1 Optimal control problem**

The general open-loop optimal control problem of a lumped parameter batch/semi-batch process with fixed terminal time can be stated as follows. Find a control vector *u*(t) over *tf* [*to, tf*] to maximize (minimize) a performance index *J*(*x*,*u*):

$$J(\mathbf{x}, u) = \int\_{t\_0}^{t\_f} \rho[\mathbf{x}(t), u(t), t] \, dt \tag{4}$$

Subject to

$$\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{\kappa}(t), \ \boldsymbol{u}(t), t), \quad \boldsymbol{\kappa}(t\_0) = \boldsymbol{\kappa}\_0 \tag{5}$$

$$h[\mathfrak{x}(t), \mathfrak{u}(t)] = \mathbf{0} \tag{6}$$

$$\mathbb{E}\left[\mathbf{x}(t),\boldsymbol{\mu}(t)\right] \le \mathbf{0} \tag{7}$$

$$\mathbf{x}^{L} \le \mathbf{x}(t) \le \mathbf{x}^{U} \tag{8}$$

$$u^L \le u(t) \le u^U \tag{9}$$

In the above, *J* refers the performance index, *x* is the state variable vector, *u* is the control variable vector. Eq. (5) represents the system of ordinary differential equations with their initial conditions, Eqs. (6) and (7) specify the equality and inequality algebraic constraints and Eqs. (8) and (9) are the upper and lower bounds for the state and control variables.

#### **2.2 Multistage dynamic optimization strategy**

A multistage optimization results due to the natural extension of a single stage optimization. In a system involving multiple stages, the output from one stage becomes the input to the subsequent stage. The multistage dynamic optimization procedure can be referred elsewhere [24]. This type of optimization needs special techniques to split the problem into computationally manageable units. In multistage dynamic optimization, the optimal control problem of the entire batch duration is divided into finite number of time instants referred to as discrete stages. The *A Metaheuristic Tabu Search Optimization Algorithm: Applications to Chemical… DOI: http://dx.doi.org/10.5772/intechopen.98240*

control variables and the corresponding state variables that satisfy the objective function are evaluated in stage wise manner.

In this work, a multistage dynamic optimization strategy with sequential implementation procedure is presented for optimal control of polymerization reactors. The procedure for solving the optimal control problem is similar to that of the dynamic programming based on the principle of optimality [25]. The meta heuristic features of tabu search (TS) are exploited by implementing this strategy on a semibatch styrene–acrylonitrile (SAN) copolymerization reactor. The procedure involves in discretizing the process into *N* stages, defining the objective function, *f <sup>i</sup>* , the control vector, *ui* and the state vector, *x<sup>i</sup>* for stage *i*. This procedure is briefed in the following steps:

1. The optimum value of the objective function, *f* <sup>1</sup> [*x*<sup>1</sup> ] for stage 1 driven by the best control vector *u*<sup>1</sup> along with the state vector *x*<sup>1</sup> is represented as

$$f^1[\mathbf{x}^1] = \min \quad f^1\_o[\mathbf{x}^1, \mathbf{u}^1] \tag{10}$$

2. The value of the objective function, *f* <sup>2</sup> [*x*<sup>2</sup> ] for stage 2 is determined based on the best control vector *u*<sup>2</sup> along with the state vector *x*<sup>2</sup> as given by

$$f^2[\mathbf{x}^2] = f^1[\mathbf{x}^1] + \begin{array}{c} \min \quad f^2\_o[\mathbf{x}^2, \mathbf{u}^2] \\ \mathbf{u}^2 \end{array} \tag{11}$$

3. Recursive generalization of the above procedure for the *kth* stage is represented by

$$f^{k}\left[\mathbf{x}^{k}\right] = f^{k-1}\left[\mathbf{x}^{k-1}\right] + \min \quad f^{k}\_{o}\left[\mathbf{x}^{k}, \mathbf{u}^{k}\right] \tag{12}$$

In this procedure, *f k <sup>o</sup>* represents the performance index of stage *k*.
