Indirect Optimization Method Research Articles

Improved Edelbaum's Approach to Optimize Low Earth/Geostationary Orbits Low-Thrust Transfers

Edelbaum's approach to the optimization of low-thrust transfers is revisited and some simplifications are removed. The variation of the spacecraft mass due to the propellant consumption is considered in the case of constant thrust, and the corresponding numerical result is compared with Edelbaum's solution. The approach is then extended to consider variable specific impulse and thrust magnitude with constant power level. The payload increment is first computed maintaining Edelbaum's suboptimal control strategy (i.e., constant-thrust direction during each half-revolution). An analytical solution of the quasi-circular one-revolution transfer is then found using the optimal control of both the thrust direction and magnitude. The very-low-thrust multirevolution problem is easily solved by assembling many one-revolution basic trajectories; in particular, the transfer from a 28.5 deg inclined low Earth orbit to the equatorial geostationary orbit is considered. Exact numerical solutions for both constant and variable specific impulse have also been obtained using an indirect optimization method: the accuracy of the solution based on the quasi-circular approximation has been verified

A modified iterative IOM approach for optimization of biochemical systems

The presented previously indirect optimization method (IOM) developed within biochemical systems theory (BST) provides a versatile and mathematically tractable optimization strategy for biochemical systems. However, due to the local approximations nature of the BST formalism, the iterative version of this technique possibly does not yield the true optimum solution. In this work, an algorithm is proposed to obtain the correct and consistent optimum steady-state operating point of biochemical systems. The existing linear optimization problem of the direct IOM approach is modified by adding an equality constraint of describing the consistency of solutions between the S-system and the original model. Lagrangian analysis is employed to derive the first order necessary optimality conditions for the above modified optimization problem. This leads to a procedure that may be regarded as a modified iterative IOM approach in which the optimization objective function includes an extra linear term. The extra term contains a comparison of metabolite concentration derivatives with respect to the enzyme activities between the S-system and the original model and ensures that the new algorithm is still carried out within linear programming techniques. The presented framework is applied to several biochemical systems and shown to the tractability and effectiveness of the method. The simulation is also studied to investigate the convergence properties of the algorithm and to give a performance comparison of standard and modified iterative IOM approach.

Indirect optimal control of a double quantum dot

An indirect optimization method for the dynamic control of non-Markovian open quantum systems is presented. It is demonstrated at the example of a spin-boson model for an electron in a double dot which couples to a bath of phonons. This study reveals strategies for controlling the effective system-bath coupling which become available when the system is addressed in its quantum regime. We give examples for population transfer, trapping and quantum operations.

PI design based on DMC strategy

Proportional and integral (PI) control parameters are determined by approximating a dynamic matrix control (DMC). Since accuracy of the approximation depends on DMC parameters in a complicated manner, a combination of direct and indirect optimization methods are employed in this paper. Quadratic programming (QP) is used to determine PI parameters, and a simple grid method is used to find optimal DMC parameters. A tuning formula for the PI controller based on a curve-fitting technique, is suggested. Computer simulations are performed to compare performance of the new PI design method with those achievable using some existing PI controllers.

Use of Analytical Gradients to Calculate Optimal Gravity-Assist Trajectories

A procedure for calculating the analytical derivatives required for an indirect optimization method of combined, long-duration finite burn and multiple gravity-assist trajectories is presented. The analytical derivatives are calculated using the state transition matrix associated with the complete set of the Euler-Lagrange equations of the optimal control problem on each trajectory segment as well as a state transition matrix that maps perturbations across any discontinuities in the state as a result of a zero-sphere-of-influence patched conic flyby or impulsive maneuver. As applications, the method is used to find Earth-to-Saturn trajectories that use combined low thrust and gravity assists and Earth-to-Saturn trajectories that use impulsive thrust and gravity assists. The state transition matrix derivatives are able to satisfy the trajectory constraints to orders of magnitude greater accuracy than the central difference derivatives. The state transition matrix method also requires less computational time to find an optimal trajectory.

Optimization of Variable-Specific-Impulse Interplanetary Trajectories

An indirect optimization method is used to find optimal trajectories that use solar electric propulsion. Missions that exploit variable specific impulse are compared to constant-specific-impulse trajectories, and the benefit, in terms of payload, is highlighted. The presence of limits on the attainable values of the specific impulse and the use of dual-mode thrusters, which operate only at two discrete values of specific impulse, are also considered. The analytical formulation of the problem is presented, and the necessary conditions for an optimal solution are discussed. Numerical examples deal with planet rendezvous missions; a Mercury mission would largely benefit from the management of the available power, which exhibits a tenfold variation caused by the vehicle's varying distance from the sun. The optimization method can also deal with planetary gravity assists; Mercury missions, which exploit Venus flybys, are optimized and their characteristics discussed. Results show that the dual-mode thrusters provide almost the same propellant savings as the more complex variable-specific-impulse thrusters.

Integrated nonlinear optimization of bioprocesses via linear programming

AbstractThe problem of integrated design and control of bioprocess plants is considered. A previously presented optimization approach for biochemical systems based on linear programming and modeling using the power law formalism (the Indirect Optimization Method, IOM) is extended. This method is enhanced in order to take into account both static and dynamic measures, and its use for the optimization of the integrated design of a bioprocess is illustrated. The chosen case study is a wastewater treatment plant, a bioprocess which typically presents controllability problems in real practice due to bad design methodologies. After defining an objective function reflecting both investment costs and “paracosts” (such as stability, flexibility, and controllability), a set of constraints determined by the system components and technical and economical factors is defined. A comparison of the results obtained with this new method and with a global optimization method reveals that, in both cases, significant improvements in both controllability and cost reduction are achieved, although the global method yields somewhat better improvements. The advantages and limitations of both methods are evaluated, concluding that the IOM, through its incorporation to a dynamic process simulator, can be successfully used to obtain, in a quick, inexpensive and interactive way, near‐optimal integrated designs for bioprocess plants.

Optimal Control of a Grey Solar Sail for Interplanetary Missions

An indirect optimization method is applied to minimize the trip time of interplanetary and escape missions using a grey (i.e., non-perfectly reflective) solar sail. The sail is assumed to be flat and the optical properties of its material (transmittance, absorptance, and reflectance) are taken into account; the reflection is not perfectly specular, but diffuse and back reflectance are also considered. The theory of optimal control and, in particular, Pontryagin's maximum principle provide the optimal orientation of the sail at each point of the trajectory. A real sail with non-unit reflectivity is feathered during some particular phases of the mission. A sail, which reemits the absorbed energy mainly from the rear side, may present jumps in the sail orientation for maximum performance. Numerical examples, concerning transfers to Mars and escape from the solar system, are presented to highlight the differences in control law and performance of ideal and real sails.

Mars gravity assist to improve missions towards main-belt asteroids

An indirect optimization method is applied to find rendezvous trajectories to reach asteroids of the main belt using solar electric propulsion. Direct trajectories and missions that receive gravity assist from Mars en route to the target are presented and compared in terms of final mass. Numerical results prove that missions with gravity assist surely perform better, if the trip time exceeds five years. Faster flyby missions are superior to direct trajectories if simple guidelines concerning the asteroid's orbital parameters are used to select the target and to take the greatest benefit from the Mars gravity assist. A direct mission can be carried out with a similar performance at time intervals which correspond to the Earth/asteroid synodic period. Gravity assist missions require the correct angular phasing of the three relevant bodies and mission opportunities are less frequent, a shortcoming which is overwhelmed by the large number of potential targets. Actual mission opportunities for a rendezvous in the next decade are found and presented for a small number of interesting asteroids.

Optimization of biochemical systems by linear programming and general mass action model representations

A new method is proposed for the optimization of biochemical systems. The method, based on the separation of the stoichiometric and kinetic aspects of the system, follows the general approach used in the previously presented indirect optimization method (IOM) developed within biochemical systems theory. It is called GMA-IOM because it makes use of the generalized mass action (GMA) as the model system representation form. The GMA representation avoids flux aggregation and thus prevents possible stoichiometric errors. The optimization of a system is used to illustrate and compare the features, advantages and shortcomings of both versions of the IOM method as a general strategy for designing improved microbial strains of biotechnological interest. Special attention has been paid to practical problems for the actual implementation of the new proposed strategy, such as the total protein content of the engineered strain or the deviation from the original steady state and its influence on cell viability.

Modeling, optimization and experimental assessment of continuous L‐(−)‐carnitine production by Escherichia coli cultures

In a previous paper Cánovas et al. (Biotechnol Bioeng 2002;77:764-775) presented a model for L-(-)-carnitine production using Escherichia coli O44 K74, in a cell-recycle bioreactor for the biotransformation of crotonobetaine into L-carnitine. In this work we optimize this biotechnological setup and experimentally verify the predicted optimal parameter profiles. Provided with a reliable and robust S-system description of the cell-bioreactor combined system, we applied the Indirect Optimization Method described by Torres et al. (Biotechnol Bioeng 1997;55(5):758-772; Food Technol Biotechnol 1998;36(3):177-184). This optimization approach provides different parameter value profiles, all of which are compatible with the cell physiology and the bioreactor operating conditions, that yield increased rates of L-(-)-carnitine production. Three parameters were seen to be of critical importance for maximizing L-(-)-carnitine production: the dilution rate, the initial crotonobetaine concentration, and the carnitine dehydratase activity. When the first two were changed in the experimental setup, there was a 74% increase in the L-(-)-carnitine production rate, performance that was in close agreement with the predictions of the model. In accordance with the optimized solution, a further improvement (90% increase in the L-(-)-carnitine production rate) could be attained by over-expressing up to 5 times the carnitine dehydratase basal activity. Thus the optimization approach shown herein provides experimental evidence of a new strategy which demonstrates the possible variables that can be subjected to modifications compatible with the cell physiology and bioreactor operating conditions, and which are able to yield increased rates of L-(-)-carnitine production.

Optimization of tryptophan production in bacteria. Design of a strategy for genetic manipulation of the tryptophan operon for tryptophan flux maximization.

In the present work we have applied the indirect optimization method (Torres, N. V. et al. Biotechnol. Bioeng. 1996, 49, 247-258) to the maximization of tryptophan biosynthesis in Escherichia coli. The optimization procedure is applied to an updated model of this biochemical system (Xiu, Z-L et al., J. Biotechnol. 1997, 58, 125-140) and thus extended to a problem that includes the processes of transcription and translation. The model representation used by these authors is first translated into the corresponding S-system version. Then, to guarantee cell viability, we impose a set of constraints on some variable and parameter values, all of which are able to be modulated by available techniques. Our results show that it is possible to attain a stable and robust steady state with a rate of tryptophan production increased more than 4 times. This is achieved by changing four key parameters related to the efflux of tryptophan, the growth rate, the inhibition constant, and the tryptophan repressor level. Moreover, it is demonstrated that we can reach this optimum state in a sequential manner, each step leading us to a better situation in relation to the previous one. Thus, only by doubling the tryptophan excretion we can triplicate the rate of tryptophan production. A further, although lesser, improvement can be attained by increasing 4-fold the rate of growth and subsequently by weakening the inhibitory feedback interaction of tryptophan on the enzymes leading to its synthesis. Finally, a significant jump in the rate of production can be obtained if the level of the trp operon could be decreased. When a second approach was considered, in which the growth rate is kept constant in the optimized profile, we found out that by modulation of the parameters it is possible to increase more than 2-fold the rate of tryptophan production.

Optimization of ? V Earth-Gravity-Assist Trajectories

An analysis of delta-velocity Earth-gravity-assist (D V-EGA) trajectories is carried out by using an indirect optimization method. In comparison to previous literature, the eccentricity of the Earth’ s orbit is taken into account, and simplifying assumptions, concerning the position and direction of velocity impulses, are removed. However, the focus is on the application of the theory of optimal control to complex interplanetary missions and D V-EGA trajectories are only illustrative examples. In particular, it is shown that the optimization of this kind of mission, according to the patched-conic approximation, can be obtained by considering only the two-body problem equations in the heliocentric frame. The necessary optimum conditions for the Earth e yby are extended to consider a minimum-height constraint, which is frequently required in this maneuver. The procedure is also capable of recognizing when a numerical solution is only suboptimal and a powered e yby could improve the mission.

An indirect optimization method for biochemical systems: Description of method and application to the maximization of the rate of ethanol, glycerol, and carbohydrate production inSaccharomyces cerevisiae

Three metabolic models for the production of ethanol, glycerol, and carbohydrates in yeast are optimized with respect to different production rates. While originally nonlinear, all three optimization problems are reduced in such a way that methods of linear programming can be used. The optimizations lead to profiles of enzyme activities that are compatible with the physiology of the cells, which guarantees their viability and fitness, and yield higher rates of the desired final end products than the original systems. In order to increase ethanol rate production at least three times, six enzymes must be modulated. By contrast, when the production of glycerol or carbohydrates is optimized, modulation of just one enzyme (in the case of glycerol) or two enzymes (in the case of carbohydrates) is necessary to yield significant increases in product flux rate. Comparisons of our results with those obtained from other methods show great similarities and demonstrate that both are valid methods. The choice of one or the other method depends on the question of interest. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 758-772, 1997.

An indirect optimization method for biochemical systems: Description of method and application to the maximization of the rate of ethanol, glycerol, and carbohydrate production in Saccharomyces cerevisiae

Three metabolic models for the production of ethanol, glycerol, and carbohydrates in yeast are optimized with respect to different production rates. While originally nonlinear, all three optimization problems are reduced in such a way that methods of linear programming can be used. The optimizations lead to profiles of enzyme activities that are compatible with the physiology of the cells, which guarantees their viability and fitness, and yield higher rates of the desired final end products than the original systems. In order to increase ethanol rate production at least three times, six enzymes must be modulated. By contrast, when the production of glycerol or carbohydrates is optimized, modulation of just one enzyme (in the case of glycerol) or two enzymes (in the case of carbohydrates) is necessary to yield significant increases in product flux rate. Comparisons of our results with those obtained from other methods show great similarities and demonstrate that both are valid methods. The choice of one or the other method depends on the question of interest. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 758–772, 1997.

Nonequilibrium simulated-annealing algorithm applied to reliability optimization of complex systems

This paper applies an improved nonequilibrium simulated-annealing (I-NESA) technique to find: (1) the global optimum of system cost of two kinds of complex systems subject to constraints on system reliability, and (2) the optimum number of redundancies which maximize the system reliability, subject to constraints on system cost, weight, and volume in a multistage mixed system. The efficacy of I-NESA in solving both varieties of problems is demonstrated by comparing its results with those of simulated annealing (SA). I-NESA, using the Glauber algorithm and an exponential cooling schedule, provides a stable global solution to all the problems considered. The essential features of I-NESA, (1) the nonequilibrium concept while coming out of an inner iteration, and (2) incorporation of the simplex-like heuristic, make it very fast and stable in obtaining the global solution when compared to the traditional SA. Fast convergence was observed in all the problems studied. I-NESA is a useful alternative to either indirect optimization methods or to some random search techniques, in solving problems like those in this paper.