Singular Optimal Control Problems Research Articles

Solution to Singular Optimal Control by Backward Differential Flow

This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

Stochastic Singular Optimal Control Problem with State Delays: Regularization, Singular Perturbation, and Minimizing Sequence

An optimal control problem with quadratic cost functional for a linear stochastic system with pointwise and distributed time delays in the state variable is considered. The case where the cost functional does not contain a control cost is treated. The latter means that the problem under consideration is a singular optimal control problem. This control problem is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an integral of the square of the control with a small positive weighting coefficient. Thus, the new problem is a regular optimal control problem. Moreover, it is a cheap control problem. By using the control optimality conditions, the solution of this problem is reduced to solution of the set of four equations: one Riccati-type matrix ordinary differential equation, two Riccati-type matrix partial differential equations of the first order, and one trivial scalar ordinary differential equation. This set of equations is with deviating arguments, and it is singularly perturbed. An asymptotic solution of this set of equations is constructed and justified. Based on this asymptotic solution, an asymptotically suboptimal state-feedback control is constructed for the cheap control problem. Finally, it is shown that this control constitutes a minimizing sequence for the original problem. The infimum of the cost functional in the original optimal control problem also is obtained. An illustrative example is presented.

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

A second-order maximum principle for singular optimal stochastic controls

A singular optimal stochastic control problem is studied. Asecond-order maximum principle is presented. The second-orderadjoint processes are involved, though the diffusion of the controlsystem is control independent. The range theorem of vector-valuedmeasures is used to prove the maximum principle. Examples are givento illustrate the applications.

The solution of singular optimal control problems using the modified line-up competition algorithm with region-relaxing strategy

In this study, singular optimal control problems are solved by line-up competition algorithm (LCA) under the framework of control parametrization. Two steps that promote the convergence quality of LCA are taken. One is to use normal (Gaussian) sampling policy to replace uniform sampling policy to accelerate initial convergence, while the other is to introduce region-relaxing strategy to enhance the refinement of solutions in final convergence. Four typical examples are given to illustrate the proposed algorithm. The results show that such modifications make LCA more robust and efficient in the solutions of singular optimal control problems.

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.

Generalized solutions for singular optimal control problems

This text presents a complete theory of existence/uniqueness and the structure of generalized solutions for singular linear-quadratic optimal control problems. Generalized optimal controls are distributions of order r and the corresponding generalized trajectories are distributions of order (r − 1). r is the “order of singularity” of the problem, an integer no greater than the dimension of the state space. Its value is obtained through a certain reduction procedure. In the final section, some perspectives and partial results concerning the extension of these results to nonlinear problems are briefly discussed.

Generalized synthesis for singular nonlinear optimal control problems

We consider the problem of minimizing a quadratic noncoercive functional along the trajectories of a control-affine system. Due to lack of coercivity, existence of "classical" minimizers cannot, in general, be guaranteed. Under appropriate commutativity assumptions the problem can be extended into the space of generalized controls of class W ?1,? and reduced into a new problem which is generically coercive but nonconvex. We show how to extend further the problem in order to include generalized controls which are "generalized derivatives of one-parameter families of regular probability measures," thus achieving convexification. Generalized trajectories for this type of controls exist only in a weak sense. We discuss a version of the maximum principle suitable to this class of problems and show how a generalized synthesis can be obtained.

Numerical solution methods for singular control with multiple state dependent forms

We use several recently developed techniques along with two established optimal control softwares to solve a variation of a widely studied singular optimal control problem. The problem displays some interesting features from both the analytical and numerical point of view. In particular, we show that, during singular arcs, the optimal control takes on one of two distinct state dependent forms. We also demonstrate how the numerical solution methods can easily get trapped in what may appear to be local optima when applied to this problem.

Numerical procedure to approximate a singular optimal control problem

In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution – the optimal cost of the original optimal control problem – we present a complete discrete method based on the use of some finite elements and penalization techniques.

Singular linear-quadratic optimal control problem for a class of discrete singular systems with multiple time-delays

The singular linear-quadratic optimal control problem for a class of discrete-time linear singular systems with multiple time-delays is investigated. The equivalent relationship between this problem and the non-singular linear-quadratic problem for standard state-space discrete-time linear systems without time-delay is derived, and the necessary and sufficient condition guaranteeing this equivalent relationship is also given. The optimal control is synthesized as state feedback, and the closed-loop system is regular, impulse-free and without time-delay. An example is given to show the validity of the proposed method.

Functional Analysis Approach to Minimum Energy Maneuver Problem for Flexible Space Structures

A singular optimal control problem with a performance index given by the total energy of the system is formulated, and functional analysis ideas are used to reduce the control problem to consider certain integral equations. In particular, the optimal control profile for minimum-energy control is obtained in an analytic manner. It is not necessary for the approach to take the second variation into consideration in spite of singular problems. The number of integral equations to be solved is equal to the number of control variables, which is usually much smaller than that of the state variables in the case of large space structures. The time-optimized control can be obtained numerically for the minimum-energy maneuver as the least-time maneuver that does not violate the constraints on the control inputs. The results of the present formulation are compared with that of multiple bang-bang time-optimal control using a simple model. The advantages of the present formulation are discussed on such control performance issues as the reduction of the control effort and the appropriate implementation of continuous controlled jets.

Linear operator inequalities for strongly stable weakly regular linear systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems.

Use of Luus–Jaakola optimization procedure for singular optimal control problems

Use of Luus–Jaakola optimization procedure for singular optimal control problems

On the Convergence of Gradient Methods Used in Solving Singular Optimal Control Problems

We study the convergence and stability of gradient methods for numerical solution of singular optimal control problems for systems with distributed parameters that satisfy a priori inequalities in negative norms.

Newton's Method for a Class of Weakly Singular Optimal Control Problems

Newton's method for optimal control of highly nonlinear partial differential equations is analyzed using a 2-norm technique. We consider the case where neither the linearization of the equality constraint e characterizing the differential equation is surjective nor a second order sufficient optimality condition holds for the topology on which e is well defined. Such problems occur, for instance, in optimal control of semilinear elliptic equations or for parameter estimation problems. Despite the above mentioned difficulties, sufficient conditions for second order convergence are obtained.

Singular optimal control problem of linear singular systems with linear-quadratic cost *

This paper investigates the singular linear-quadratic optimal control problem of a kind of linear time-in variant singular systems containing impulse mode. The relation between this problem and the linear-quadratic optimal control problem of standard state-space linear systems is established by using an equivalent transformation based on the Weierstrass canonical form of singular systems. The existence and uniqueness of the solution, the method to compute the optimal control and corresponding state trajectory, and the feedback synthesis of the optimal control of this problem is also given

An extended model for pairwise conflict resolution in air traffic management

A conflict resolution model for aircraft traversing planar intersecting trajectories is developed and solved. The model incorporates the flight dynamics of each aircraft. Three types of optimal conflict avoidance manoeuvres are examined. The first is the minimum-time safe deviation to a designated objective and the second and third are the minimum-time safe deviation to and from a designated ground track. These objectives lead to singular optimal control problems where the control variable constraint is defined in terms of the maximum allowable turn rate ∣u(t)∣⩽uM,0⩽t⩽tf, for the aircraft executing the avoidance manoeuvre and the non-autonomous state variable constraint is defined in terms of the radius rp of the protected zone about the potentially conflicting aircraft. A fast, accurate procedure is derived for computing the optimal steering program which requires the solution of a coupled differential-algebraic system of equations to determine the control law on any boundary arc of an extremal trajectory. An example pairwise trajectory conflict is analysed in detail. The incorporation of variable airspeed v(t) and an acceleration/deceleration control u2(t) is also discussed. Copyright © 1999 John Wiley & Sons Ltd.

The optimal control of collision avoidance trajectories in air traffic management

A collision avoidance/conflict resolution model for aircraft traversing planar intersecting trajectories is developed based on the actual flight dynamics of each aircraft and the criteria that a flight regime correction is optimal if it represents a minimum-time safe deviation from a preassigned flight program. The formulation of the model leads to a singular optimal control problem where the control variable constraint is the maximum acceptable aircraft turn rate for passenger comfort and safety and the state variable constraint is defined in terms of the radius of the protected zone about a potentially conflicting aircraft. A robust, accurate real-time solution procedure is derived for computing the steering program for the execution of the optimal safe avoidance maneuver. Several representative examples of pairwise trajectory conflicts are analysed in detail. A brief discussion of how the model might be generalized to incorporate changes in airspeed and altitude as well as more comprehensive dynamics is included.

On the optimal control of the Vidale-Wolfe advertising model

The Vidale-Wolfe advertising model is a singular optimal control problem with a non negative control. Sufficient conditions on a generalized time-varying market, for which a solution can be found, are given. Time is parametrized to describe impulsive optimal trajectories in a conventional manner. The solution is then found and verified by using the Hamilton-Jacobi-Bellman equation on the parametrized problem. © 1997 John Wiley & Sons, Ltd.