Singular Arcs Research Articles

Approximate Finite-Horizon Optimal Control for Input-Affine Nonlinear Systems with Input Constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optimal input is in general not known. The control input can be found solving an Hamilton-Jacobi-Bellman (HJB) partial differential equation (pde): it typically consists of a combination of bang-bang arcs and singular arcs. In the paper a methodology to approximate the solution of the HJB pde arising in the finite-horizon optimal control problem with input constraints is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of an example: the minimum time optimal control problem for an industrial wastewater treatment plant.

A simple trolley-like model in the presence of a nonlinear friction and a bounded fuel expenditure

Lomonosov Moscow State UniversityFaculty of Computational Mathematics and CyberneticsRussia, Moscow, Leninskiye Gory, MSUe-mail: barbudo.sam@cs.msu.suAbstractWe consider a problem of maximization of the distance traveled by amaterial point in the presence of a nonlinear friction under a bounded thrustand fuel expenditure. Using the maximum principle we obtain the form ofoptimal control and establish conditions under which it contains a singularsubarc. This problem seems to be the simplest one having a mechanicalsense in which singular subarcs appear in a nontrivial way.Keywords: optimal control problem, Pontryagin Maximum Principle,extremals, singular arcs.2010 Mathematics Subject Classification: 49K15, 49N90.

Structural Stability for Bang-Singular-Bang Extremals in the Minimum Time Problem

We study the structural stability of a bang-singular-bang extremal in the minimum time problem between fixed points. The dynamics is single-input and control-affine with bounded control. On the nominal problem ($r=0$), we assume the coercivity of a suitable second variation along the singular arc and regularity both of the bang arcs and of the junction points, thus obtaining the strict strong local optimality for the given bang-singular-bang extremal trajectory. Moreover, as in the classically studied regular cases, we assume a suitable controllability property, which grants the uniqueness of the adjoint covector. Under these assumptions we prove that for any sufficiently small $r$, there is a bang-singular-bang extremal trajectory which is a strict strong local optimizer for the $r$-problem. A uniqueness result in a neighborhood of the graph of the nominal extremal pair is also obtained. The results are proved via the Hamiltonian approach to optimal control and by taking advantage of the implicit functio...

Junction point on partially singular trajectories

In recent times, several works have been performed on the design of fuel optimal trajectories for space navigation. These works show the possibility of the existence of partially singular trajectories for systems that are linear analytic (Park et al. 2010). Linear analytic systems may show the existence of partially singular subarcs, and the point where these subarcs meet is called a junction point. Thus, knowledge about junction conditions became necessary when solving the optimal control problem for such systems. This led to the development of two ‘theorems’ on junction conditions, given by McDanell and Powers (McDanell, J.P. and Powers W.F. (1971), ‘Necessary Conditions for Joining Optimal Singular and Nonsingular Sub Arcs’, SIAM Journal of Control, 9, 161–173). However, the second ‘theorem’, which is now known as a conjecture, could not satisfy all classes of linear analytic system. Therefore, the aim of this study was to detect and correct the errors in the derivation of the McDanell and Powers conjecture. The error in their derivations was corrected and then tested on two newly mathematically constructed systems. The results of these tests were found to be satisfactory. This implies that by making the necessary corrections, the conjecture can still be useful in generating a general theorem for all classes of systems.

Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima

We address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate from a given initial condition to a given target value in a minimal amount of time. Recently, the optimal synthesis (optimal feeding strategy) has been obtained in systems in which the microorganisms involved are represented by increasing growth functions or growth functions with one maxima, with either Monod or Haldane functions, respectively (widely used in bioprocesses modeling). In the present work, we allow impulsive controls corresponding to instantaneous dilutions, and we assume that the growth function of the microorganism present in the process has exactly two local maxima. This problem has been tackled from a numerical point of view without impulsive controls. In this article, we introduce two singular arc feeding strategies, and we define explicit regions of initial conditions in which the optimal strategy is either the first singular arc strategy or the second strategy.

Optimization method for solving bang-bang and singular control problems

In this paper we study optimal control problems with the control variable appearing linearly. A novel method for optimization with respect to the switching times of controls containing both bang-bang and singular arcs is presented. This method transforms the control problem into a finite-dimensional optimization problem by reformulating the control problem as a multi-stage optimization problem. The optimal control problem is partitioned as several stages, with each stage corresponding to a particular control arc. A control vector parameterization approach is applied to convert the control problem to a static nonlinear programming (NLP) problem. The control profiles and stage lengths act as decision variables. Based on the Pontryagin maximal principle, a multi-stage adjoint system is constructed to calculate the gradients required by the NLP solvers. Two examples are studied to demonstrate the effectiveness of this strategy.

Optimization of Unpowered Descents of Commercial Aircraft in Altitude-Dependent Winds

Unpowered descents of commercial transport aircraft are optimized in the presence of altitude-dependent winds with the objective of maximizing range. The optimal problem and an optimized constant calibrated-airspeed procedure are analyzed. The optimal problem is studied using the theory of singular optimal control. The optimal control is of the bang–singular–bang type, and the optimal trajectories are formed by a singular arc and two minimum/maximum-path-angle arcs joining the singular arc with the given initial and final points. The constant calibrated-airspeed procedure isoptimizedusing parametricoptimization theory. Results are presented fora model of a Boeing 767-300ER and for linear wind profiles. The effects of both the average wind and the wind shear on the optimalresults,aswellastheeffectsoftheaircraftweight,areanalyzed.Thewindshearisshowntohaveacleareffect onthemaximumrange.Thecomparisonbetweenthetwosetsofresultsshowsthattheoptimizedconstantcalibratedairspeeddescentisveryclosetooptimal.Fromtheoperationalpointofview,theoptimumdescentcalibratedairspeed is defined in terms of the wind profile and the aircraft weight.

Minimal Time Control of Fed-Batch Processes With Growth Functions Having Several Maxima

We address the issue of minimal time optimal control of fed-batch reactor in presence of complex non monotonic kinetics. Several extremal paths with singular arcs can be locally optimal. We show how a regularization technique can help determining the optimal synthesis, based on a numerical approach.

Structural Stability Investigation of Bang-Singular-Bang Optimal Controls

The paper is devoted to parametric optimal control problems with a scalar, partially singular optimal control function. In contrast to the case of pure bang-bang behavior, the investigation of structural stability properties for partially singular controls so far has been rarely addressed in literature. The central result of the paper deals with the case of one first order singular arc under regular concatenation to bang-arcs. Conditions will be provided which ensure the Lipschitz stability of bang-singular junction times positions with respect to small parameter changes. Three examples illustrate the main theorem.

Time-optimal trajectory planning of robot manipulators in point-to-point motion using an indirect method

In this study, an iterative method for computing the time-optimal point-to-point control of robot manipulators is studied, subject to limits on the actuator torques and actuator jerks. This method uses the indirect solution of an open-loop optimal control problem so that the optimal control problem is translated to a non-linear two-point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and solving the TPBVP, a simple iterative method is proposed. This method is based on solving the minimum-energy problem without the necessity of finding the switching point. So, the resultant TPBVP can be solved using the usual algorithms. In addition, since the solution of the TPBVPs is sensitive to the initial guess, a simple procedure for making proper initial guesses is introduced. Furthermore, since the inverse kinematic of redundant manipulators is an ill-posed problem, the general form for boundary conditions corresponding to the final time is considered. In addition, it is shown that the proposed method has no sensitivity for specifying the time-optimal trajectory of manipulators that may contain singular arcs.

Bang-singular-bang extremals: sufficient optimality conditions

In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.

Minimum-Cost Cruise at Constant Altitude of Commercial Aircraft Including Wind Effects

A IRCRAFT trajectory optimization is a subject of great importance in air traffic management from the point of view of defining optimal flight procedures that lead to energy-efficient flights. In practice, the airlines consider a cost index (CI) and define the direct operating cost (DOC) as the combined cost of fuel consumed and flight time weighted by the CI. Their goal is to minimize the DOC. However, in the presence of unexpected winds, the flight timemay differ considerably from the scheduled time, which leads to an arrival-error cost that can be added to the DOC to obtain the total cost (TC). Minimum-DOC trajectories have been studied by different authors [1–6]. The related problem of minimum fuel with fixed final time has been analyzed as a minimum-DOC problem with free final time in [3,5,7] (the problem is to find the time cost for which the corresponding free final time DOC-optimal trajectory arrives at the assigned time); this same problem is addressed in [8], analyzing the effects of mismodeled winds in a scenario formed by the final cruise and descent segments. The problem of minimum-cost flight, considering not only the DOC but also the arrival-error cost, is analyzed in [9,10], taking into account factors such as crew overtime cost, passenger dissatisfaction cost, and losses due to missed connections. In this Note, the problem of minimum-cost cruise at constant altitude in the presence of strong winds, including the arrival-error cost, is analyzed, considering the general unsteady problem, with variable aircraft mass, and without any restriction on cruise altitude. The main objective is to analyze the optimal trajectories that lead to minimum cost, defined as optimal speed laws (speed as a function of aircraft mass). The analysis is made using the theory of singular optimal control (see [11]), which has the great advantage of providing feedback control laws (control variables as functions of the state variables) that can be directly used to guide the aircraft along the optimal path. These optimal control laws are analyzed as well. In this work, the initial and final speeds are given, so that the optimal control is of the bang-singular-bang type, and the optimal paths are formed by a singular arc and two minimum/maximumthrust arcs joining the singular arc with the given initial and final points (see [6,12]). In previous work related to optimum cruise at constant altitude [13,14], only the singular arc was studied; hence, a more general formulation of the optimal problem is addressed now, apart from considering the arrival-error cost and including wind effects (average horizontal winds). In this analysis of the minimum-TC problem, the arrival-error cost depends on the difference between the actual and the scheduledflight times, and it is defined to be positive, so that both late and early arrivals are penalized (the objective is to achieve high arrival-time accuracy). It will be shown that, for some values of the parameters of the problem, minimum cost is obtained when the final time coincides with the scheduled time of arrival; that is, when the arrival-error cost is zero. This critical case is in fact a problem with fixed final time. Results are presented for a model of a Boeing 767-300ER.

High-Accuracy Trajectory Optimization for a Trans-Earth Lunar Mission

The trajectory optimization of a spacecraft subject to the gravitational effects of the moon, Earth, and sun are considered. The problem is how to achieve Earth-interface conditions from a low lunar orbit. Practical constraints of maximum thrust, fuel budget, and flight time generates a constrained, nonautonomous, nonlinear optimal control problem. Severe constraints on the fuel budget combined with high-accuracy demands on the endpoint conditions necessitate a high-accuracy solution to the trajectory optimization problem. The problem is first solved using the standard Legendre pseudospectral method. The optimality of the solution is verified by an application of the covector mapping principle. It is shown that the thrust structure consists of three finite burns with nearly linear steering-angle time histories. A singular arc is detected and is interpreted as a singular plane change maneuver. The Bellman pseudospectral method is then employed for mesh refinement to improve the accuracy of the solution.

Tracking the singular arc of a continuous bioreactor using sliding mode control

A sliding mode controller is developed for optimal transient operation of a continuous bioreactor. The sliding mode is the singular arc from the solution of an optimal control problem. The proposed controller is applied through simulations to an anaerobic digester and its performance is evaluated in terms of optimality and robustness properties.

Optimal Rendezvous Trajectories of a Controlled Spacecraft and a Tumbling Object

This paper formulates and solves the problem of minimum-time and minimum-energy optimal trajectories of rendezvous of a powered chaser and a passive tumbling target, in a circular orbit. Both translational and rotational dynamics are considered. In particular, ending conditions are imposed of matching the positions and velocities of two points of interest onboard the vehicles. A collision-avoidance condition is imposed as well. The optimal control problems are analytically formulated through the use of the Pontryagin minimum principle. The problems are then solved numerically, by using a direct collocation method based on the Gauss pseudospectral approach. Finally, the obtained solutions are verified through the minimum principle, solved by a shooting method. The simulation results show that the pseudospectral solver provides solutions very close to the optimal ones, except in the case of presence of singular arcs when it may not provide a feasible solution. The computational time needed by the pseudospectral solver is a small fraction of the one needed by the indirect approach, but it is still considerably too large to allow for its use in real-time onboard guidance.

Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis

We describe optimal protocols for a class of mathematical models for tumor anti-angiogenesis for the problem of minimizing the tumor volume with an a priori given amount of vessel disruptive agents. The family of models is based on a biologically validated model by Hahnfeldt et al. and includes a modification by Ergun et al, but also provides two new variations that interpolate the dynamics for the vascular support between these existing models. The biological reasoning for the modifications of the models will be presented and we will show that despite quite different modeling assumptions, the qualitative structure of optimal controls is robust. For all the systems in the class of models considered here, an optimal singular arc is the defining element and all the syntheses of optimal controlled trajectories are qualitatively equivalent with quantitative differences easily computed.

Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach

The aim of this paper is to investigate from the numerical point of view the coupling of the Hamilton-Jacobi-Bellman (HJB) equation and the Pontryagin minimum principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered.

Minimum-Fuel Cruise at Constant Altitude with Fixed Arrival Time

A N IMPORTANT problem in air traffic management (ATM) is the design of aircraft trajectories that meet certain arrival time constraints at given waypoints, for instance, at the top of descent (TOD), at the initial approach fix (IAF), or at the runway threshold. These are called 4-D trajectories, which are a key element in the trajectory-based-operations (TBO) concept proposed by NextGen and SESAR for the future ATM system. The time constraints must also be met in certain cases in which the nominal trajectories have to be modified to resolve detected conflicts (e.g., lost of separation minima) between aircraft; for example, Bilimoria and Lee [1] analyze aircraft conflict resolution with an arrival time constraint at a downstream waypoint. On the other hand, the design of fuel-optimal trajectories that lead to energy-efficient flights is another important problem which has been treated extensively in the literature, see, for instance, Burrows [2], Neuman andKreindler [3],Menon [4] and the references therein. Fuel-optimal trajectories with fixed arrival times are studied by Sorensen and Waters [5], Burrows [6] and Chakravarty [7], who analyze the 4-D fuel-optimization problem as a minimum directoperating-cost (DOC) problem with free final time, that is, the problem is to find the time cost for which the corresponding free final-time DOC-optimal trajectory arrives at the assigned time. In this work we analyze the problem of minimum-fuel cruise at constant altitudewith afixed arrival time as a singular optimal control problem, building upon the works of Pargett and Ardema [8] and Rivas and Valenzuela [9], who analyze the problem of maximumrange cruise at constant altitude also as a singular optimal control problem; the case of unsteady cruise is considered. The singular arcs and the corresponding optimal control are obtained as a function of the final time. The optimal paths are obtained as well, which define a variable-Mach cruise at constant altitude. The influence of cruise altitude on the optimal paths is analyzed, and the minimum fuel is calculated. The final-time constraint may be defined, for example, by a flight delay imposed on the nominal (preferred) cruise trajectory (which in our case is the minimum-fuel cruise trajectory with free final time); comparison with a standard constant-Mach procedure to absorb delays is made. Results are presented for a model of a Boeing 767-300ER. Problem Formulation

Compressibility Effects on Maximum Range Cruise at Constant Altitude

I NA recent paper, Pargett and Ardema [1] analyze the problem of range maximization in cruise at constant altitude as a singular optimal control problem. The same problem is also analyzed using different approaches by Miele [2] and Torenbeek [3], who consider the case of quasi-steady flight. In [1], the singular arc that defines the optimal path is studied; in this study, a simple aircraft model defined by a parabolic drag polar of constant coefficients is considered and only one value of cruise altitude. Many other analyses of cruise optimization can be found in the literature, with different flight constraints and performance indices; see, for instance, the study of Menon [4], in which both speed and altitude are allowed to vary, and the many references therein. In this work, we generalize the analysis of [1] considering a general drag polar, so that compressibility effects can be taken into account. The singular arc in this general case is obtained and is particularized for both asymmetric and symmetric compressible parabolic drag polars. The results show that compressibility effects are very important; the differences with the incompressible case are shown to be not only quantitative, but also qualitative. We also analyze the influence of cruise altitude on the optimal paths, showing that it can be important from a qualitative point of view. Themaximum range is calculated as a function of cruise altitude; in particular, the best altitude that leads to the largest maximum range is obtained. Results are presented for a model of a Boeing 767-300ER.

The Columbia History of Jews and Judaism in America (review)

Reviewed by: The Columbia History of Jews and Judaism in America Marjorie N. Feld The Columbia History of Jews and Judaism in America. Edited by Marc Lee Raphael. New York: Columbia University Press. 2008. Lee Raphael has ambitious goals for this collection. He aims to capture the tension between religious and secular identities in the Jewish community (5), as well as the Marc centrality of the relationships among Jewishness, Judaism, and Americanness for Jewish individuals (6). Is the history of American Jews to be told as a narrative of assimilation and declining religiosity? Of diminishing anti-Semitism and rapid social mobility? Of political consensus or increasing internal division? Raphael's contributors engage each of these themes. Because the book is organized first by chronology (the book's first six essays) and then by topic (the final twelve essays), it attempts to avoid one of the most troubling tendencies of U.S. Jewish history: to provide a singular narrative arc to the American Jewish experience that follows religious Jewish life and Zionism while marginalizing all other expressions of Jewishness. The chronological essays offer useful synthetic introductions. Brief biographies of leaders and laypeople alike appear in these pages, along with discussions of suburbanization, the Nazi Holocaust, synagogue and Jewish organizational growth and decline. Even for well-read scholars of this history, there are surprise facts: Eli Faber, for example, tells us that despite much anti-Semitic exclusion, Jews could be city constables in eighteenth-century New York City (36). There are also missed opportunities where the collection does not break out of the paradigm of a singular narrative arc, presenting a narrow reading of Jewishness. Riv-Ellen Prell, for example, proposes that the attacks on Israel in 1967 brought about a "new American [End Page 146] Jewish consensus... that placed Israel and Jerusalem at the center of American Jewish life" (137). Though there is ample material here on Jewish contributions to American Communism and other radical movements, Prell fails to mention Jewish anti-Zionism as existing outside of this so-called "consensus." This is a glaring omission, especially since the work of Michael Staub (some of which thankfully appears here as a topical essay) has carefully documented crucial "dissenting perspectives" in the cold war era (325). A few pages later Stephen Whitfield discusses the topic of postwar anti-Semitism among African Americans, calling it the "only—and most glaring—exception to the trend (of declining American anti-Semitism)" (144). Again, the glaring absence is of a richer contextual discussion: of Jewish racism and contributions to white supremacist oppression, and too the impact of Third World Solidarity with Palestinians among African Americans. These omissions carry with them their own conservative agenda: in reading back into history a consensus on complicated issues, one stemming from a religious and Zionist brand of Jewishness, those forwarding this agenda see only one set of Jewish values and identity worth perpetuating in contemporary American Jewish life. The invisibility of dissenting perspectives continues to shape a false sense of consensus among American Jews. In documenting some of the internal debates of American Jews, the topical essays—especially Staub's—offer other narratives and perspectives. Melissa Klapper writes of the vociferous postwar debates over the Orthodox Jewish day school movement and offers an interesting window onto internal divisions among American Jews: non-Orthodox Jews pledged loyalty to public education while "indictments of ghettoization and assimilation" were offered by both sides (209). William Toll and Mark Bauman provide two welcome essays that depart the East Coast and look at varieties of Jewish life along the Pacific and in the American South. American Studies scholars may lament the general lack of material on American Jews' interactions with other cultural groups, and also on cultural representations of Jewishness. Jeffrey Shandler's promising title "What is American Jewish Culture?" yields a brief, disappointing study of how this culture has been studied. Linda Raphael's concluding essay is the only one dedicated to analyzing representations of Jewishness, yet she offers only a recitation of well-trod readings of the conservative canon of American Jewish literature: Cahan, Yezierska, Roth and Ozick. Ultimately, despite its ambitious title and introduction, Raphael's collection rarely strays from the old...