A New Solution to the Nonlinear Optimal Control for Lunar Landing Mission
- Paper number
IAC-08.C1.5.10
- Author
Prof. Jafar Roshanian, K.N.Toosi university of technology, Iran
- Year
2008
- Abstract
Optimal trajectory of a module for soft landing on the moon by minimizing the control effort expenditure is obtained. The problem is formulated as an optimal control problem with the thrust direction being the control variable. Applying the trigonometric series for approximating the performance measure and using the calculus of variations theory, can determine the optimal thrust angle from optimality conditions. In this way, for computing the optimal states, the governing state-space equations is transformed into the new differential equations expressed with respect to the control variable as an independent variable. Due to the simpler form of the state equations, it is integrated to yield the optimal state-space histories. Researcher and engineer have not been as successful in dealing with nonlinear optimal control problems as they have been in solving linear optimization problems in control. In general, optimal formulation of nonlinear dynamic systems either through dynamic programming or variational approach led to nonlinear partial differential equations. Numerical solution of these equations when dealing with complex nonlinear systems is always difficult and has certain difficulties, such as slow convergence rate and high sensitivity to initial guesstimates. Besides, if one manages to overcome these inherent difficulties, the determined optimal control strategy will be in an open-loop form, and thus, fully dependent on the initial condition. There are such problems arise frequently in aerospace applications and often there are limited control resources available for achieving desired objectives, so that finding an optimal control law which minimizes the control effort expenditure are very important. Optimal control solution for nonlinear lunar landing mission was obtained either by dynamic programming approach or through a variational formulation of the optimal control problem by Bryson. Time optimal control solution of the nonlinear lunching in the gravity of moon was obtained in the polar coordination system by a numerical technique named linear programming by Kirk. Furthermore, two performance measures for minimizing the control effort expenditure are discussed perfectly. Also, Souza obtained an optimal guidance law that minimized the commanded acceleration in three dimensions. This law is an exact solution to the two-point boundary value problem associated with the first variation necessary conditions. Pourtakdoust and Novinzadeh proposed a new technique to synthesize the optimal feedback law for nonlinear injection and nonlinear landing by using a fuzzy system in the process of developing the closed-loop fuzzy logic guidance. For finding an analytical solution to the lunar landing problem, at first, it should be assumed that an idealized spacecraft exists at the lunar parking orbit and moves under the action of a constant propulsive force making a control angle with the horizon. The state-space vector includes height, horizontal and vertical velocity of the module in each instant of time. Obviously, the position and velocity vector of the vehicle will change due to the action of forces acting on it. The lunar landing problem has three initial conditions at the lunar orbit and three terminal constraints at the end of mission. Indeed, lunar landing mission starts from an assumed initial state in the lunar orbit and finishes at the specific target on the surface of moon. The objective is to determine the analytical optimal control policy of this system for soft landing on the moon by minimizing the control effort expenditure. For finding an exact solution and avoiding from the unsolvable integrals which appear due to integrate from transformed differential equation, it is necessary to approximate the performance measure by using the trigonometric series. It should be considered that the time derivatives appearing in the state-space equations can be formulated with respect to the thrust angle. In this way, now the thrust angle becomes an independent variable and also, boundary conditions are expressed with respect to it. Due to the simpler form of obtained new differential equations, it can be possible to integrate from them to yield the result as a function of the control variable by using backward sweep method. Furthermore, there are six required unknown parameters include initial thrust angle, final thrust angle, final time and three constant parameters related to the co-state equations. These parameters are determined by solving a set of six nonlinear algebraic equations obtained from boundary conditions and optimality relations. After computing these parameters and substituting them, the optimal thrust angle and optimal state trajectories are determined. The results are compared with the numerical results computed by using steepest descend method as a validation work. The advantage of this guidance law is that it is simple, easily mechanized, and is the exact solution of the two-point boundary value problem. In fact, this research proposes a new exact solution for nonlinear two-point boundary value problem which can be utilized for many real world applications instantaneously.
- Abstract document
- Manuscript document
IAC-08.C1.5.10.pdf (🔒 authorized access only).
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