Renjith R. Kumar

Genetic Algorithm Approach for Optimal Control Problems with Linearly Appearing Controls

For optimal control problems in Mayer form with all controls appearing only linearly in the equations of motion, this paper presents a method for calculating the optimal solution without user-specified initial guesses and without a priori knowledge of the optimal switching structure. The solution is generated in a sequence of steps involving a genetic algorithm (GA), nonlinear programming, and (multiple) shooting. The centerpiece of this method is a variant of the GA that provides reliable initial guesses for the nonlinear programming method, even for large numbers of parameters. As a numerical example, minimum-time spacecraft reorientation trajectories are generated. The described procedure never failed to correctly determine the optimal solution. INDING the solution to an optimal control problem is a dif- ficult and time-consuming task. By employing Pontryagins minimum principle in conjunction with simple or multiple shoot- ing to solve the resulting boundary-value problem (BVP), this task becomes equivalent to finding the numerical values of the costates (Lagrange multipliers) associated with the physical states of the underlying dynamic system at discrete times. Thus, the problem of solving an optimal control problem can be reduced to solving a non- linear system of equations. Usually, Newton-Raphson methods are well suited for this type of problem. However, due to the sensitiv- ity of the state-costate dynamical system, the task of finding initial guesses that lie within the domain of convergence can become arbi- trarily difficult. In addition, if a control appears only linearly in the equations of motion, the optimal solution is known to consist of a sequence of bang-bang and, possibly, singular subarcs. The switch- ing structure, however, is not known in advance and has to be found by trial and error. The present paper introduces a method for generating the opti- mal control solution for problems in which all controls appear only linearly in the equations of motion. In this method, the user need not provide initial guesses for the state history, the control history, the costate history, or the switching structure. Initial guesses that lie within the domain of convergence of a gradient search method are generated with a genetic algorithm (GA) using substring length 1 for each individual control parameter. A theoretical justification of the approach is given through hodograph analysis and convexity arguments. General convergence arguments pertaining to the GA are mainly heuristic and based on practical experience. Because of the probabilistic nature of GAs, this seems to be unavoidable.

Method for automatic costate calculation

A method for the automatic calculation of costates using only the results obtained from direct optimization techniques is presented. The approach exploits the relation between the time-varying costates and certain sensitivities of the variational cost function, a relation that also exists between the Lagrangian multipliers obtained from a direct optimization approach and the sensitivities of the associated nonlinear-programming cost function. The complete theory for treating free, control-constrained, interior-point-constrained, and state-constrained optimal control problems is presented. As a numerical example, a state-constrained version of the brachistochrone problem is solved and the results are compared to the optimal solution obtained from Pontryagins minimum principle. The agreement is found to be excellent. Nomenclature / = right-hand side of state equations ge = control equality constraints gi = control inequality constraints he = state equality constraints hi = state inequality constraints J = cost function M = interior-point constraints m = dimension of control vector u N = total number of nodes minus 1 = total number of subintervals n — dimension of state vector x PWC = set of piecewise continuous functions t = time tf = final time ti = nodes along the time axis to = initial time u = control vector x = state vector Xf = final state Xi = state vector at node £/ Xo = initial state \(t) = costate A/ = Lagrangian multiplier associated with differential constraints along subinterval / Hi - Lagrangian multiplier associated with state constraints at node i (Ti = Lagrangian multiplier associated with control constraints along subinterval / <£ = cost function if} j. = boundary conditions at final time •00 = boundary conditions at initial time

Simulation and Shuttle Hitchhiker validation of passive satellite aerostabilization

The Passive Aerodynamically Stabilized Magnetically Damped Satellite experiment will characterize and demonstrate passive aerodynamic stabilization and passive magnetic hysteresis damping of attitude rates. It is currently scheduled to be deployed on a Shuttle Hitchhiker flight. Although theoretically feasible, aerodynamically induced passive attitude stability represents a technology that has never been substantiated through actual flight experience. The two-week experiment will serve to validate overall performance predictions by the high-fidelity free-molecularflow simulation code developed at the Langley Research Center of NASA. The code can simulate with high fidelity the flight characteristics of a satellite in low Earth orbit. Aerostabilization, if proved, is highly desirable for future missions such as the Gravity and Magnetic Earth Surveyor. This paper describes the simulator, simulation results, and the Hitchhiker experiment in the context of the Gravity and Magnetic Earth Surveyor subsatellite aerostabilization requirements.

Should controls be eliminated while solving optimal control problems via direct methods

Direct methods of solving optimal control problems include techniques based on control discretization, where the control function of time is parameterized, and collocation, where both the control and state functions of time are parameterized. A recently introduced direct approach of solving optimal control problems via differential inclusions parameterizes only the state, and constrains the state rates to lie in a feasible hodograph space. In this method, the controls, which are just artifacts used to parameterize the feasible hodograph space, are completely eliminated from the optimization process. Explicit and implicit schemes of control elimination are discussed. Comparison of the differential inclusions method is made to collocation in terms of number of parameters, number of constraints, CPU time required for solution, and ease of calculation of analytical gradients. A minimum time-to-climb problem for an F-15 aircraft is used as an example for comparison. For a special class of optimal control problems with linearly appearing bounded controls, it is observed that the differential inclusion scheme is better in terms of number of parameters and constraints. Increased robustness of the differential inclusion methodology over collocation for the Goddard problem with singular control as part of the optimal solutions is also observed. Background T HE most precise approach to solve optimal control problems is the variational1 approach based on Pontryagins minimum principle.2 This is an indirect approach as it involves solving the necessary conditions of optimality associated with the infinite dimensional optimal control problem rather than optimizing the cost of a finite dimensional discretization of the original problem directly. This method requires advanced analytical skills and to generate numerical solutions of the resulting two-point boundary-value problem is highly nontrivial. The controls are eliminated in the indirect method using the minimum principle. Thus, the optimal control is, in general, a nonlinear function of the state and costate variables. The most important application of the indirect method is the generation of benchmark solutions. Usually, good convergence is achieved only with excellent initial guesses for the nonintuitive costates. Additionally, the switching structure has to be guessed correctly in advance. For rapid trajectory prototyping, the safest approaches are the direct methods.3 These methods rely on a finite dimensional discretization of the optimal control problem to a nonlinear programming problem. Even though these methods do not enjoy the high precision and resolution of indirect methods, their convergence robustness makes them the method of choice of most practical applications. Moreover, these methods do not require the advanced mathematical skills necessary to pose and solve the variational problem.

Near-optimal three-dimensional air-to-air missile guidance against maneuvering target

Closed-loop guidance of a medium-range air-to-air missile (AAM) against a maneuvering target is synthesized using a three-phase, near-optimal guidance scheme. A large period of the closed-loop guidance is performed using neighboring optimal control techniques about open-loop optimal solutions obtained by solving the associated minimum time to intercept trajectories. The first phase is the boost phase guidance where the normal acceleration limit may be active due to high lofting of the boost-sustain-coast AAM. The guidance in this phase accommodates only errors in the missiles state variables, the target maneuvering being neglected. The boost phase guidance involves guidance in the presence of active control constraints. The second phase is the midcourse guidance where both state perturbations and target maneuvers are considered. Comparisons are made between guidance with gain indexing performed with clock time and with performance index to go. Models of aggressive target and run-away targets were used and the guidance scheme performance is excellent. Three methods of optimal gain evaluation are also discussed. Performance augmentation is obtained by using a center of attainability as a pseudotarget that fairs into the actual target as time to go becomes zero. The final phase is the terminal guidance, which employs proportional navigation and its variants.

Fuel-optimal stationkeeping via differential inclusions

Stationkeeping of one spacecraft in low Earth orbit with respect to another, or with respect to a reference point in space, is a common orbit maintenance and guidance requirement. This paper deals with formulating a infinite time fuel-optimal control problem using the Hill equations (also known as the Clohessy-Wiltshire equations) and solving it via a direct approach using concepts of hodograph space and differential inclusions. The differential inclusion based direct method has been selected due to its excellent convergence robustness. Using this methodology, numerous optimal solutions corresponding to various differential drag profiles and Stationkeeping error tolerances were easily obtained from trivial initial guesses. The major contribution of this paper is the interesting observation made regarding the structure of the fuel-optimal solutions as a function of the differential drag profiles and Stationkeeping error tolerances. Results from this study can be used for estimating fuel budgets and developing fuel-optimal Stationkeeping guidance laws.

Parametric and classical resonance in passive satellite aerostabilization

Purely passive aerostabilization of satellites has never been flight demonstrated. The Shuttle hitchhiker passive aerodynamically stabilized magnetically damped satellite experiment would be the first flight experiment of its kind that, in conjunction with results from a high-fidelity computer simulator, would corroborate attitude stability. Pure aerostabilization, with no gravity gradient restoring torques, if proved, is highly desirable for future missions such as the gravity and magnetic Earth surveyor. High-fidelity nonlinear simulation results indicate interesting attitude behavior, such as cone angle transients that provoke the need for sound theoretical justification. A wind vane in a wind field model is used to derive simple analogous Mathieu-Hill equations of motion, the stability properties of which are predicted via Floquet theory. Parametric resonance caused by higher order of the once per orbit density harmonics, varying natural frequency of oscillation as a result of altitude decay, and varying wind magnitude due to global winds are studied in detail. The simple time-varying linear wind vane analogy captures the essence of the observations made with the complex nonlinear simulation. Classical resonance because of step changes in the solar torques as a result of Earth occultation is also discussed. Based on insight obtained from the stability properties observed for the wind vane analogy, an optimal satellite is designed that provides best attitude performance while maintaining sufficient lifetime and other mission constraints. Nomenclature : area of analogous wind vane, m2 = average ballistic coefficient of subsat, kg/m2 = moment of inertia about an axis of rotation, kg-m2 : aerodynamic moment stiffness, N-m/rad = length of back shell, cm : length of front shell, cm : mass of subsat, kg = center-of-pres sure to center-of-mas s offset, m = inner radius of front shell, cm : aerodynamic torque, N-m : thickness of back shell, cm : thickness of front shell, cm = velocity magnitude of wind, m/s = cone angle of the satellite, rad : damping ratio = coefficient of damping, N-m-s/rad = angle of rotation of wind vane from reference wind direction, rad = density of air, kg/m3 = amplitude of once/orbit harmonic of Q, kg/m3 = density of back shell, kg/m3 = bias component of density of air profile over one orbit, kg/m3 = density of front shell, kg/m3 = accommodation coefficients = natural frequency of oscillation of wind vane, rad/s = once per orbit frequency, rad/s

FINITE DIFFERENCE SCHEME FOR AUTOMATIC COSTATE CALCULATION

A method for the automatic calculation of costates using only results obtained from direct optimization techniques is presented. The approach is based on finite differences and exploits the relation between the costates and certain sensitivities of the cost function. The complete theory for treating free, control constrained, interior-point constrained, and state constrained optimal control problems is presented. An important advantage of the method presented here is that it does not require a priori identification of the optimal switching structure. As a numerical example, a state constrained version of the Brachistochrone problem is solved, and the results are compared to the optimal solution obtained from Pon try a gins minimum principle. The agreement is found to be excellent.

Air data system optimization using a genetic algorithm

An optimization method for flush-orifice air data system design has been developed using the Genetic Algorithm approach. The optimization of the orifice array minimizes the effect of normally distributed random noise in the pressure readings on the calculation of air data parameters, namely, angle of attack, sideslip angle and freestream dynamic pressure. The optimization method is applied to the design of Pressure Distribution/Air Data System experiment (PD/ADS) proposed for inclusion in the Aeroassist Flight Experiment (AFE). Results obtained by the Genetic Algorithm method are compared to the results obtained by conventional gradient search method.

Concatenated Approach to Trajectory Optimization

the covariance simulationincorporatesthe equations developedpreviously.Inasmuchasthe Tomahawkmissileis being considered,the terrain scene is a terminal area digital scene matching and area correlation(DSMAC) scene. Shown in Fig. 2 is a typical CEP plot from the covariance simulation, which illustrates that the highly accurate positional solution obtained from the relative GPS correction vector scheme degrades only minimally between the DSMAC scene and the target. For target penetration considerations, the results presented here incorporate a terminal trajectory with a high-angle dive to impact the target. At the target, the navigational error contributes 2.6%, the steering error 10.2%, the scene center location error 20.9%, and the relative target location error 66.3%, which is consistent with the earlier discussion regarding relative error magnitudes. An examination of Fig. 2 also reveals that the correction vector scheme affords a substantial improvement over the situation where the Kalman e lter accepts a high-quality DSMAC positional update and, subsequently, the missile e ies in a free inertial manner to the target (the mode of operation for the current Block III Tomahawk missile ). Because improvements in satellite imaging technology are expected to permit larger DSMAC scene to target separations while maintaining the same scene/target relative error, free inertial navigation to the target will clearly become less desirable, and relative GPS schemes such as the one proposed herein more desirable. Summary