SIMULATION OF OPTIMAL CONTROL OF ORBITAL VEHICLE THRUST DURING LAUNCH

In 21st century economically feasible and less expensive ways of space exploration and industrialization become the great challenge for mankind. One of the promising approaches is development of modern SSTO (single-stage-to-orbit) and VTOL (vertical-take-off-landing) technologies, leveraging latest achievements in new materials and propulsion systems. The paper presents results of simulation of dynamic systems, moving in central gravitational field, in an environment, where atmospheric drag force has a serious impact on system’s dynamics. Model, proposed in paper, has been analyzed and build, based on Pontrjagin’s principle of optimality. Optimal regular and singular thrust control of engine is analyzed. Research has been conducted on relations between optimal control feasibility and SSTO parameters, including initial acceleration and average (along trajectory) drag coefficient. Impact of drag dependence on Mach number on accuracy of computed optimal trajectory has been considered. One approach to solve the problem has been developed, where impact of proposed optimal control implementation on problem’s optimality criteria is evaluated. The derived formula for calculating optimal control is invariant to conditions on right end of ascent trajectory

Planar tilting maneuver of a spacecraft: singular arcs in the minimum time problem and chattering

In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the initial values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.Comment: 43 pages, 18 figure

WELL-POSEDNESS OF THE SHOOTING ALGORITHM FOR STATE CONSTRAINED OPTIMAL CONTROL PROBLEMS WITH A SINGLE CONSTRAINT AND CONTROL ∗

Abstract. This paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint. Additional conditions are displayed, under which the so-called alternative formulation is equivalent to Pontryagin’s minimum principle. The shooting algorithm appears to be well-posed (invertible Jacobian), iff (i) the no-gap second order sufficient optimality condition holds, and (ii) when the constraint is of order q ≥ 3, there is no boundary arc. Stability and sensitivity results without strict complementarity at touch points are derived using Robinson’s strong regularity theory, under a minimal second-order sufficient condition. The directional derivatives of the control and state are obtained as solutions of a linear quadratic problem. Key words. Optimal control, Pontryagin’s principle, state constraints, junction conditions, shooting algorithm, no-gap second-order optimality conditions, strong regularity, sensitivity analysis, directional derivatives. AMS subject classifications. 49M05, 49K40, 34B15, 34E10. 1. Introduction. For optimal control problems satisfying the strengthened Legendre-Clebsch condition, Pontryagin’s principle allows to express the control as

Some results on optimal control with unilateral state constraints

International audienceIn this paper, we study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau's sweeping process, that is, a specific distributional differential inclusion, and from ten Dam's geometric theory [A.A. ten Dam, K.F. Dwarshuis, J.C. Willems, The contact problem for linear continuous-time dynamical systems: A geometric approach, IEEE Trans. Automat. Control 42 (4) (1997) 458–472; A.A. ten Dam, Unilaterally Constrained Dynamical Systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869, 1997] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau's sweeping process (developed in Acary et al. [V. Acary, B. Brogliato, D. Goeleven, Higher order Moreau's sweeping process: Mathematical formulation and numerical simulation, Math. Programm. A 113 (2008) 133–217]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem

Constrained Buckling of Variable Length Elastica

University of Minnesota Ph.D. dissertation. 2017. Major: Civil Engineering. Advisor: Emmanuel Detournay. 1 computer file (PDF); 236 pages.The physical understanding of the response of slender elastic bodies restrained inside constraints under various loading and boundary conditions is of a great importance in engineering and medical applications. The research work presented in this thesis is especially concerned with the buckling response of an elastic rod (the elastica) subjected to unilateral constraints under axial compression. It seeks to address two main issues: (i) the conditions that lead to the onset of instability, and (ii) the factors that define the bifurcation diagram. Two distinct classes of problems are analyzed; (i) the classical buckling problem of a constant length elastica and (ii) the insertion buckling problem of a variable length elastica. Their main difference is the generation of a configurational force at the insertion point of the sliding sleeve in the insertion problem, which is not present in the classical problem. The thesis describes two distinct methodologies that can solve these constrained buckling problems; (1) a geometry-based method, and (2) an optimal control method. The geometry-based method is used to analyze the post-buckling response of a weightless planar elastica subjected to unilateral constraints. The method rests on assuming a deformed shape of the elastica and on uniquely segmenting the elastica consistent with a single canonical segment (clamped-pinned). An asymptotic solution of the canonical problem is then derived and the complete solution of the constrained elastica is constructed by assembling the solution for each segment. Nevertheless, the application of the optimal control method is more generic. It can be used to solve any constrained buckling problem under general boundary and loading conditions. Based on Hamiltonian mechanics, the optimality conditions, which constitute the Pontryagin’s minimum principle, involve the minimization of the Hamiltonian with respect to the control variables, the canonical equations and the transversality conditions. The main advantage of the optimal control method is the assumption of strong rather than weak variation of the involved variables, which leads to the additional Weierstrass necessary condition (“optimal” equilibrium state). Based on it, several factors such as the effect of the self-weight of the elastica and the clearance of the walls are investigated