Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

Regimes of more and more frequent switchings in the optimal control problem of oscillations of <Emphasis Type="Italic">n</Emphasis> oscillators

This paper considers the control system of n oscillators executing forced oscillations under the action of a scalar-valued control force common for all oscillators whose module is bounded. The author proves the existence of an optimal singular regime and the assertion that the optimal control has at least countably many switchings that accumulate to a conjunction point of the singular and nonsingular parts of the trajectory. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Accumulation of switchings in distributed parameter problems

In recent times, optimal control theory for distributed parameter systems has been actively studied; among them, an important place is occupied by the class of systems describing oscillation processes. This work studies linear control distributed parameter systems of hyperbolic type. The minimization problem of a quadratic functional on the trajectories of the system is considered. By using the Fourier method, the problem is reduced to studying optimal solutions for a countable control system of ordinary differential equations. For Galerkin’s approximations of this system, it is proved that the optimal control is a chattering control, i.e., it has infinitely many switchings on a finite interval of time. The construction of the optimal synthesis uses the results of the theory of singular regimes and regimes with with more and more frequent switchings. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Time-optimal steering of an object moving in a viscous medium to a desired phase state

The two-point problem of the time-optimal attainment of a desired phase state by a multidimensional dynamic object is investigated. The motion occurs in a viscous medium by means of a limited force. The open-loop and/or feedback control laws constructed by numerical-analytical methods for arbitrary initial data. An asymptotically approximate solution of the maximum principle boundary-value problem is presented for short and long time intervals. The singularities of the optimal trajectory are established for the initial and final parts of the motion. The solution obtained of the two-point problem of the optimal control of the motion of a dynamic object in a homogeneous viscous medium by means of a force of bounded modulus is compared with the known solutions in special formulations.     

The existence of optimal control on the basis of Weierstrass’s theorem

The problem of existence of an optimal control is solved on the basis of Weierstrass’s classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass’s theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass’s theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems.     

On the terminal control problem

A minimum-time problem is considered, where the final point is locally controllable. It is shown that it is possible to construct a suboptimal control with a transfer time close to the optimal transfer time of the relaxed system. The resulting trajectory will satisfy initial and final conditions. Furthermore, it is shown that, if an optimal solution exists for the problem, then this optimal solution is also an optimal solution of the relaxed problem. In this case, the relaxed problem need not be solved.The authors wish to thank Dr. D. Hazan, Scientific Department, Ministry of Defense, Israel, for a fruitful discussion of this problem.     

Optimal synthesis in a two-dimensional problem with asymmetric constraints on the control

A two-dimensional optimal control problem is considered on the assumption that the terminal time of the process is not fixed and the integral objective functional depends on a parameter. Asymmetric constraints are imposed on the control parameter. Two cases are considered: constraints of the same sign and constraints of different signs. In the case of constraints of different signs, if the parameters of the problem satisfy certain relations, one obtains chattering control, alternating with a control with two switchings and a first-order singular are when these relations are violated. In the case of sign-definite control the controllability domain is part of the plane bounded by two semiparabolas. Three types of control law are then possible, in two of which the system will hit the boundary of the controllability domain and move along it, while the third features a first-order singular are. As the parameter of the problem is varied, the phase portrait undergoes evolution and one of these three types is interchanged with another. The optimality of these control laws is rigorously established using a dynamic programming method.     

Modes with switchings of increasing frequency in the problem of controlling a robot

Trajectories that are optimal with respect to high-speed response are constructed for a system for controlling a two-component manipulator (a robot). It is shown that when the initial conditions lie within a certain open region of the phase space, all optimal trajectories will have a segment of switchings of increasing frequency (SIF), i.e. a segment in which the control will undergo an infinite number of switchings in a finite time interval.

The synthesis of the optimal control in the R² plane containing the mode of SIF was first constructed by Fuller /1/. It was shown in /2/ that the synthesis is structurally stable in the sense that adding terms of higher order of smallness to the integrand and to the right-hand sides of the system of differential constraints does not affect the qualitative pattern of the optimal synthesis in the neighbourhood of the origin of coordinates.

The present paper explains that the synthesis in the problem of optimal control (relative to the high speed response) of the motion of the robot appears, in a certain sense, a direct product of the synthesis appearing in the Fuller problem and of the synthesis in the simplest problem of high-speed response (/3/, pp.38–47). The special aspect of the present paper consists of the proof of the proposition that switching surface is a piecewise-smooth manifold. The presence of the SIF mode is connected only with the fact that every trajectory intersects this surface an infinite number of times. In existing papers, the piecewise smoothness of the switching curve was proved for the two-dimensional problems using the SIF mode only for problems admitting of a one-parameter group of symmetries /1, 4–6/. A proof of the presence of SIF was given in /7, 8/.     

Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.     

Singular Arcs in the Optimal Evasion Against a Proportional Navigation Vehicle

The two-dimensional optimal evasion problem against a proportional navigation pursuer is analyzed using a nonlinear model. The velocities of both players have constant modulus, but change in direction. The problem is to determine the time-minimum trajectory (disengagement) or time-maximum trajectory (evasion) of the evader while moving from the assigned initial conditions to the final conditions. A maximum principle procedure allows one to reduce the optimal control problem to the phase portrait analysis of a system of two differential equations. The qualitative features of the optimal process are determined.     

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.     

Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance

This paper is concerned with the optimal transition and the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The abort landing problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered.It is assumed that a transition from descending flight to ascending flight is desired; that the initial state corresponds to quasi-steady flight with absolute path inclination of –3.0 deg; and that the final path inclination corresponds to quasi-steady steepest climb. Also, it is assumed that, as soon as the shear is detected, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the instantaneous altitude and a reference value, assumed constant. By suitable transformations, the Chebyshev problem is converted into a Bolza problem. Then, the Bolza problem is solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.Two types of optimal trajectories are studied, depending on the conditions desired at the final point. Type 1 is concerned with gamma recovery (recovery of the value of the relative path inclination corresponding to quasi-steady steepest climb). Type 2 is concerned with quasi-steady flight recovery (recovery of the values of the relative path inclination, the relative velocity, and the relative angle of attack corresponding to quasi-steady steepest climb). Both the Type 1 trajectory and the Type 2 trajectory include three branches: descending flight, nearly horizontal flight, and ascending flight. Also, for both the Type 1 trajectory and the Type 2 trajectory, descending flight takes place in the shear portion of the trajectory; horizontal flight takes place partly in the shear portion and partly in the aftershear portion of the trajectory; and ascending flight takes place in the aftershear portion of the trajectory. While the Type 1 trajectory and the Type 2 trajectory are nearly the same in the shear portion, they diverge to a considerable degree in the aftershear portion of the trajectory.Next, trajectory guidance is considered. Two guidance schemes are developed so as to achieve near-optimum transition from quasi-steady descending flight to quasi-steady ascending flight: acceleration guidance (based on the relative acceleration) and gamma guidance (based on the absolute path inclination).The guidance schemes for quasi-steady flight recovery in abort landing include two parts in sequence: shear guidance and aftershear guidance. The shear guidance is based on the result that the shear portion of the trajectory depends only mildly on the boundary conditions. Therefore, any of the guidance schemes already developed for Type 1 trajectories can be employed for Type 2 trajectories (descent guidance followed by recovery guidance). The aftershear guidance is based on the result that the aftershear portion of the trajectory depends strongly on the boundary conditions; therefore, the guidance schemes developed for Type 1 trajectories cannot be employed for Type 2 trajectories. For Type 2 trajectories, the aftershear guidance includes level flight guidance followed by ascent guidance. The level flight guidance is designed to achieve almost complete velocity recovery; the ascent guidance is designed to achieve the desired final quasi-steady state.The numerical results show that the guidance schemes for quasi-steady flight recovery yield a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, allows the aircraft to achieve the final quasi-steady state, and has good stability properties.This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company, and by Air Line Pilots Association.The authors are indebted to Dr. R. L. Bowles (NASA-LRC) and Dr. G. R. Hennig (BCAC) for helpful discussions.     

General theory of optimal trajectory for rocket flight in a resisting medium

This paper considers the problem of optimizing the flight trajectory of a rocket vehicle moving in a resisting medium and in a general gravitational force field. General control laws for the lift, the bank angle, and the thrusting program are obtained in terms of the primer vector, the adjoint vector associated to the velocity vector. Additional relations for the case of variable thrusting and integrals of motion for flight at maximum lift-to-drag ratio and flight in a constant gravitational field are obtained.This work was supported by Air Force Grant No. AFOSR-71-2129.     

Solutions for a class of optimal control problems with time delay,part 1

The optimal control of a system whose state is governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson (Ref. 1) to a situation where the trajectories are not necessary bounded. Also, we study the structure of approximate solutions for the problem on a finite interval.The author thanks A. Leizarowitz for fruitful discussions.     

Optimal purchase and advertising for a product with immediate sale start

Summary We consider the problem of maximizing the discounted net profit of a firm which purchases a quantity of some product at a given time and afterwards advertises and sells the product progressively. We distinguish among the three possibilities of assuming the final time to be either fixed, or bounded, or free. In all cases, after stating the problem in the optimal control theory framework, we prove the existence of an optimal solution and characterize it using the Maximum Principle necessary conditions. Furthermore, we prove that the convexity of the purchase cost function is a sufficient condition for the uniqueness of the optimal solution. Partially supported by MURST.     

Maximum Divert for Planetary Landing Using Convex Optimization

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.     

Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching times. The control-affine system under consideration is transformed into a driftless system by assuming that the controls possess properties of a partition of unity. Then the problem of constructing periodic trajectories is studied analytically by applying the Fliess series expansion over a small time horizon. We propose analytical results concerning the relation between the boundary conditions and switching parameters for an arbitrary number of switchings. These analytical results are applied to a mathematical model of non-isothermal chemical reactions. It is shown that the proposed control strategies can be exploited to improve the reaction performance in comparison to the steady-state operation mode.     

Optimal abort landing trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. The abort landing problem is considered with reference to flight in a vertical plane. It is assumed that, upon sensing that the airplane is in a windshear, the pilot increases the power setting at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.The performance index being minimized is the peak value of the altitude drop. The resulting optimization problem is a minimax problem or Chebyshev problem of optimal control, which can be converted into a Bolza problem through suitable transformations. The Bolza problem is then solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Numerical results are obtained for several combinations of windshear intensities, initial altitudes, and power setting rates.For strong-to-severe windshears, the following conclusions are reached: (i) the optimal trajectory includes three branches: a descending flight branch, followed by a nearly horizontal flight branch, followed by an ascending flight branch after the aircraft has passed through the shear region; (ii) along an optimal trajectory, the point of minimum velocity is reached at about the time when the shear ends; (iii) the peak altitude drop depends on the windshear intensity, the initial altitude, and the power setting rate; it increases as the windshear intensity increases and the initial altitude increases; and it decreases as the power setting rate increases; (iv) the peak altitude drop of the optimal abort landing trajectory is less than the peak altitude drop of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory; (v) the survival capability of the optimal abort landing trajectory in a severe windshear is superior to that of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory.Portions of this paper were presented at the IFAC 10th World Congress, Munich, Germany, July 27–31, 1987 (Paper No. IFAC-87-9221).This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company (BCAC), and by Air Line Pilots Association (ALPA). Discussions with Dr. R. L. Bowles (NASA-LRC) and Mr. C. R. Higgins (BCAC) are acknowledged.     

Informational Sets in a Model Problem of Homing

A linear pursuit problem in the plane under incomplete pursuer information about the evader is investigated. At discrete time instants, the pursuer measures with errors the angle of vision to the evader, the angular velocity of the line of sight, and the relative distance. Other combinations of measurable parameters are possible (for example, angle of vision and relative distance or angle of vision only). The measurements errors obey certain geometric constraints. The initial uncertainties on the evader coordinates and velocities are given in advance. Having a resource of impulse control, the pursuer tries to minimize the miss distance. The evader control is bounded in modulus.The problem is formulated as an auxiliary differential game. Here, the notion of informational set is central. The informational set is the totality of pointwise phase states consistent with the history of the observation-control process. The informational set depends on the current measurements; it changes in time and plays the role of a generalized state, which is used for constructing the pursuer control.A control method designed for the linear pursuit problem is used in the planar problem of a vehicle homing toward a dangerous space object. The nonlinear dynamics is described by the Kepler equations. Nonlinear terms of the equations in relative coordinates are small and are replaced by an uncertain vector parameter, which is bounded in modulus and is regarded as an evader control. As a result, we obtain the mentioned control problem in the plane.The final part of the paper is devoted to the simulation of a space vehicle homing toward a dangerous space object. In testing the control method developed, two variants are considered: random measurement errors and game method of constructing the measurements; the latter is also described in the paper.     

Application of relaxed solutions to minimum sensitivity optimal control

Relaxed variational techniques are applied to a minimum sensitivity control problem. Sensitivity of a trajectory is minimized to perturbations in initial conditions. Rather than using the optimal control that does indeed exist and that satisfies the final conditions exactly, a suboptimal control is used that transfers the system from the given initial state to an arbitrary small neighborhood of the given final state, and that results in a considerably better performance than the optimal solution. The suboptimal control is constructed using the optimal controls of the relaxed problem.This paper is based upon the Ph.D. dissertation by the author at Purdue University, Lafayette, Indiana. The author wishes to thank Professor Violet B. Haas, School of Electrical Engineering, Purdue University, for introducing him to relaxed variational problems and for many very helpful suggestions through the course of this work.