On-line optimization of batch processes with nonlinear manipulated input

The population model of bone remodelling employed the optimal control

“…The optimal control methodology, implemented in many population-like systems, [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] is based on the fundamental work of Pontryagin. 30 In a general case the optimal control problem can be written according to the Pontryagin maximum principle 30 .…”

“…One can see that the criterion ( 13) describes a metabolic penalty for alteration of bone material described by variable x 4 , from unit (first term in expression ( 13)) and the penalty for variation of mechanical stress from a standard one, recounting by the value s, second term in (13). Then the Hamiltonian will be…”

A feedback-based implementation scheme for batch process optimization

“…An optimal feedback control law for linear-quadratic problems with time delays in both the state as well as the input was derived in [Soliman72]. For end-point optimization problems, state feedback laws were synthesized for affine-in-input systems [Palanki93] and systems nonlinear both in the states and inputs [Rahman96a]. In [Rahman98], nonlinear state feedback laws are developed for on-line optimization of batch / semi-batch processes with multiple manipulated inputs.…”

“…The computation of the reoptimized input trajectories may become prohibitive when the sampling rate is high or fast system dynamics are encountered. A second approach is to develop analytical optimal feedback laws [Soliman72,Palanki93,Rahman96a,Rahman98]. In this approach, the manipulated input is derived as a function of the system parameters and states or costates.…”

Real-Time Dynamic Optimization of Non-Linear Batch Systems

“…To try and compensate for the amount of computing time required, the profiles may be parameterized to known trajectories that have met with success (see (Visser et al, 2000)). In a series of papers by Palanki and Rahman (Palanki et al, 1993), (Palanki and Rahman, 1994), (Rahman and Palanki, 1996), and (Rahman and Palanki, 1998) a method is introduced that provides a geometric approach to handling batch optimization. They show how to develop feedback laws for end-point optimization problems under a variety of state space variations.…”