Time-optimal control of linear systems

A useful procedure for the solution of time-optimal control problems for linear systems is presented. The procedure consists of a direct search method for the determination of the best diagonal weighting matrix used in a Liapunov function for the system. The control policy chosen to minimize the forward difference of the Liapunov function yields very rapid approach to the origin. Numerical results are presented for a linear six-plate gas absorber to show the feasibility of the proposed method. Even in the presence of disturbances the system can be taken to and kept at the vicinity of the origin.     

Global optimization of the bifunctional catalyst problem

As shown by Luus et al. (1992), determination of the optimum catalyst blend along a tubular reactor to maximize the yield of a particular component is a difficult problem due to the large number of local optima. By using iterative dynamic programming, the effects of the region reduction factor and the choice of the initial region size are analyzed in detail for this problem. Even if a single grid-point for the state is used, the likelihood of obtaining the global optimum is very high if the initial region size is chosen to be sufficiently large.     

Bioequivalence of controlled-release calcium antagonists

In this review, several deficiencies of published bioequivalence studies for controlled-release calcium antagonists have become apparent. As a consequence, some of the published conclusions based on such studies must be viewed with care. A proper statistical analysis of bioequivalence is not frequently reported. A proper statistical analysis of the pharmacokinetic variables involves the calculation of 90% confidence intervals (CI) for the test: reference ratio of the means of the pharmacokinetic variables of the test and reference product. The CI must fall completely within the predetermined bioequivalence range (usually 0.8 to 1.25) for the products to be declared bioequivalent. Serious methodological errors, such as a conclusion of bioequivalence based on a lack of statistically significant difference between products, and conversely, a conclusion of bioequivalence because of a statistically significant difference, or because of a mere failure to show bioequivalence, are still made. With calcium antagonists in particular, an assessment of the rate of absorption and of the maximum concentration is important, as those characteristics may have implications for the safety profile with this class of drugs. As a minimum, in single doses studies the maximum concentration (Cmax), and the time to the maximum concentration (tmax), and in multiple-dose studies the Cmax, and the peak-trough fluctuation (%PTF) must be considered. Some bioequivalence studies of calcium antagonists are deficient in this respect. To show bioequivalence for controlled-release formulations, multiple-dose studies are required but some published bioequivalence studies contain only single-dose assessments. Similarly, bioequivalence studies under fed conditions are rarely published, although food may have a significant effect on the absorption rate of these drugs. Some calcium antagonists, such as verpamil, show stereo-selective pharmacokinetics, so that enantiomers may have to be investigated. Unfortunately, few of the published studies of controlled-release calcium antagonists satisfy all requirements. One would expect that data submitted to regulatory authorities for approval of generic formulations are more complete; published data are in many cases not satisfactory.     

Model reduction of high-order multistage systems by the method of orthogonal collocation

A straightforward model reduction procedure is developed for high-order multistage systems. It is proposed to generate a partial differential equation from the original system and then to use the orthogonal collocation procedure to establish a low-order reduced model. This low-order model can then be used for time-optimal control with the assurance that the optimal control, as determined for the reduced system, can be applied to the original system. The procedure is illustrated with a 20-plate gas absorber which is eventually reduced to 5 state equations.     

Optimization of a model IV fluidized catalytic cracking unit

Maximization of a profit function related to a fluidized catalytic cracking unit model was carried out by Luus‐jaakola optimization procedure. A 7‐dimensional search is carried out on a FCC unit described by 113 nonlinear algebraic equations and 9 differential equations. Despite the low sensitivity and the existence of several local optima, the global optimum was obtained with reasonable amount of computational effort. At the optimum, the profit function is 1% higher than when the air blowers are constrained to operate at their maximum capacity.     

Further developments in the new approach to boundary condition iteration in optimal control

In solving the boundary value problem resulting from the use of Pontryagin's maximum principle, a transformation matrix is used to relate the sensitivity of the final state to the initial state. This avoids the need to solve the (n × n) differential equation to give the transition matrix, and yields very rapid convergence to the optimum. To ensure convergence, iterative dynamic programming (IDP) is used for a number of passes to yield good starting conditions for this boundary condition iteration procedure. Clipping technique is used to handle constraints on control. Five optimal control problems are used to illustrate and to test the procedure.     

A mathematical model of dehydroisomerization of methylcyclopentane using a bifunctional catalyst blend

A mathematical model of dehydroisomerization of methylcyclopentane in presence of bifunctional catalysts under industrial operating conditions is presented. The model consists of six differential equations. The rate constants are expressed as polynomial functions of the catalyst blend, and coefficients of these polynomials were determined from numerous measurements, taken along the axis of a tubular reactor. The resulting polynomial coefficients led to reaction rate constants which yielded results close to the actual process.     

Evaluation and improvement of control vector iteration procedures for optimal control

To improve the convergence rate to the optimal policy, a simple but effective way of determining the stepping parameter for the control policy adjustment is developed. The application of the proposed approach to four systems described by ordinary differential equations shows that the existing methods are improved considerably when the proposed method is incorporated.