Open-loop Solution Research Articles

Analytical Solution of Optimal Feedback Control for Radially Accelerated Orbits

The optimal feedback control problem for low-thrust trajectories with modulated inverse-square-distance radial thrust is studied in this paper. The problem is tackled by applying a generating-function method devised for linear systems. Instead of deriving open-loop solutions, arising from the two-point boundary-value problems in which the classical optimal control is stated, this technique allows us to obtain analytical closed-loop control laws. The idea behind this work consists of applying a globally diffeomorphic linearizing transformation that rearranges the original nonlinear dynamic system into a linear system of ordinary differential equations written in new variables. The generating-function technique is then applied to this new dynamic system, the optimal feedback control problem is solved, and the variables are transformed back into the original. Thus, we avoid the problem of expanding the vector field and truncating higher-order terms, because no remainders are lost in the approach undertaken. Practical examples are used to show the usefulness of the derived solution for modulated, inverse-square-distance, radially accelerated orbits.

Optimal discrete-pulse control of linear systems

Consideration was given to the linear problems of optimal control in the class of discrete-pulse and discrete-quasipulse control actions. Proposed was a dual method for construction of the open-loop solutions underlying the algorithms of the optimal controllers generating in real time the current values of the optimal feedbacks. The results obtained were illustrated by examples.

Near-Optimal Missile Avoidance Trajectories via Receding Horizon Control

The paper introduces a receding horizon control scheme for obtaining near-optimal controls in a feedback form for an aircraft trying to avoid a closing air-to-air missile. The vehicles are modeled as point-masses. Rotation kinematics of the aircraft are taken into account by limiting the pitch and roll rates as well as the angular accelerations of the angle of attack and the bank angle. The missile utilizes proportional navigation and it has a boost-sustain propulsion system. In the proposed scheme, the optimal controls of the aircraft over a short planning horizon are solved on-line by the direct shooting method at each decision instant. Thereafter, the state of the system is updated by using only the first controls in the sequence, and the process is repeated. The performance measure defining the objective of the aircraft can be chosen freely. In this paper, six performance measures consisting of the capture time, closing velocity, miss distance, gimbal angle, tracking rate, and control effort of the missile are considered. The quality of the receding horizon solutions computed by the scheme is validated by comparing them to the off-line computed optimal open-loop solutions.

Feedback Nash equilibria for non-linear differential games in pollution control

Dynamic problems of pollution and resource management with stock externalities often require a differential games framework of analysis. In addition they are represented realistically by non-linear transition equations. However, feedback Nash equilibrium (FBNE) solutions, which are the desired ones in this case, are difficult to obtain in problems with non-linear-quadratic structure. We develop a method to obtain numerically non-linear FBNE for a class of such problems, with a specific example for shallow lake pollution control. We compare FBNE solutions, by considering the entire equilibrium trajectories, with optimal management and open-loop solutions, and we show that the value of the best FBNE is in general worse than the open-loop and optimal management solutions.

The open-loop solution of the Uzawa–Lucas model of endogenous growth with N agents

We solve an N ∈ N player general-sum differential game. The optimization problem considered here is based on the Uzawa–Lucas model of endogenous growth. Agents have logarithmic preferences and own two capital stocks. Since the number of players is an arbitrary fixed number N ∈ N , the model’s solution is more general than the idealized concepts of the social planer’s solution with one player or the competitive equilibrium with infinitely many players. We show that the symmetric Nash equilibrium is completely described by the solution to a single ordinary differential equation. The numerical results imply that the influence of the externality along the balanced growth path decreases rapidly as the number of players increases. Off the steady state, the externality is of great importance, even for a large number of players.

Workers’ enterprises are not perverse: differential oligopoly games with sticky price

We take a differential game approach to study the dynamic behaviour of labour managed (LM) firms, in the presence of price stickiness. We find that the oligopoly market populated by LM firms reaches the same steady state equilibrium allocation as the oligopoly populated by profit-maximising (PM) firms, provided that the LM membership and the PM labour force are set before the market game starts. The conclusion holds under both the open-loop solution and the closed-loop solution. The result confirms the point made by Sertel (Eur Econ Rev 31:1619–1625, 1987) in a static framework.

Optimal Finite-Time Feedback Controllers for Nonlinear Systems with Terminal Constraints

This paper presents a dynamic-programming approach for determining finite-time optimal feedback controllers for nonlinear systems with nonlinear terminal constraints. The method utilizes a polynomial series expansion of the cost-to-go function with time-dependent gains that operate on the state variables and constraint Lagrange multipliers. These gains are computed from backward integration of differential equations with terminal boundary conditions, derived from the constraint specifications. The differential equations for the gains are independent of the states. The Lagrange multipliers at any particular time are evaluated from the knowledge of the current state and the gain values. Several numerical examples are considered to demonstrate the applications of this methodology. The accuracy of the method is ascertained by comparing the results with those obtained by using open-loop solutions to the respective problems. Finally, results of the application of the developed methodology to a spacecraft detumbling problem are presented.

Range Maximization for Emergency Landing After Engine Cutoff

A pilot who experiences engine cutoff when flying at an envelope point of high velocity and low altitude needs as much range and altitude as possible for safe landing. An algorithm is presented that converts the velocity excess into altitude and range in an optimal way. Such an algorithm can be implemented after engine cutoff as the first stage of an emergency landing. The optimization problem is identified as a two-timescale problem composed of a short-duration initial boundary layer, followed by a long segment dominated by slow behavior. The slow segment represents the classical quasi-steady-state glide, whereas the boundary layer determines the fast dynamic phase in which the excess of the dynamic pressure is converted into the steady-state dynamic pressure while reaching the highest altitude. The boundary-layer section is solved by the dichotomic basis method and applied to the F-16, and the ensuing solution is compared to the exact one. The match between the two is very good. with timescale separation based on three-timescale separation: The velocity and path angle were assumed to be faster than the altitude and range, and the path angle was assumed to be faster than the ve- locity. Such a separation produces a main boundary layer that keeps the altitude on hold while considering the dynamic fast change of the velocity and path angle. The main boundary layer has an internal sublayer that keeps the velocity on hold, changing the path angle first. This formulation defines boundary layers such that each one has only one active variable, thus, enabling an analytical solution of the boundary layers that produces an open-loop solution of the trajectory and a feedback law for the control. The ensuing solu- tions were compared to the exact ones, and the matching between the approximated solutions and the exact numerical solutions were studied. The open-loop solutions showed good agreement for those cases where the initial dynamic pressure was much higher than the stabilized one. The feedback solutions were reasonable except for induced oscillations that appeared when the boundary layer inter- cepted the steady-state segment. The oscillations result because the path-angle time constant is not faster than that of the velocity when approaching the stabilized section. In this paper, the same maximum range problem is treated, but as a two-time-constant problem instead of a three-time-constant problem. The path angle and the velocity time constants are con- sidered comparable, but faster than the altitude and range. Under this assumption, numerical solutions based on the dichotomic basis method 1 are derived. The dichotomic basis approach's objective is to develop an indirect solution method for nonlinear two-timescale optimal control problems, a solution method based on underlying geometric structure of the trajectories of the Hamiltonian system but not requiring a priori knowledge of this structure. Two capabilities are required: splitting the Hamiltonian system into slow and fast parts and splitting the fast part into contracting and expanding parts. The expanding part is suppressed in the initial boundary layer, thus, enabling an easier convergence of the solution.

SYNTHESIS OF OPTIMAL FEEDBACKS FOR LINEAR SYSTEMS UNDER STATE CONSTRAINTS

A method of on-line constructing an optimal feedback control for linear systems under state and control constraints is suggested. At first, a fast algorithm of constructing an open-loop control is worked out. It is based on successive correction of the solution to the optimal control problem with intermediate state constraints by changing the location of constraints and introducing additional constraints for identifying contact points and boundary arcs. Then, the scheme of constructing open-loop solutions is modified in order to generate a realization of optimal closed-loop control in any control process.

OPEN-LOOP AND CLOSED-LOOP MODELS OF DYNAMIC OLIGOPOLY IN THE CRUISE LINE INDUSTRY

In this paper, the competition of dynamic oligopoly in the cruise line industry is modeled as an N-person nonzero-sum noncooperative dynamic game where a finite number of cruise lines compete to maximize their profits over a fixed planning horizon. The noncooperative Nash equilibrium capacity investment strategies of cruise lines are theoretically analyzed under the open-loop and closed-loop information structures. The optimality conditions for open-loop and closed-loop Nash equilibrium solutions are derived using a Pontryagin-type maximum principle and given economic interpretations so as to demonstrate the differences between the open-loop and closed-loop Nash equilibrium solutions. The dynamic oligopolistic competition of three cruise lines in a hypothetical setting is numerically analyzed by using the iterative algorithms for open-loop and closed-loop models. Numerical results provide a number of important managerial guidelines for cruise capacity investment decisions. The paper concludes with a discussion on future research directions.

Dynamic Oligopoly with Sticky Prices: Closed-Loop, Feedback, and Open-Loop Solutions

We investigate a dynamic oligopoly game with price adjustments. We show that the subgame perfect equilibria are characterized by larger output and lower price levels than the open-loop solution. The individual (and industry) output at the closed-loop equilibrium is larger than its counterpart at the feedback equilibrium. Therefore, firms prefer the open-loop equilibrium to the feedback equilibrium, and the latter to the closed-loop equilibrium. The opposite applies to consumers.

A co-evolutionary method for pursuit-evasion games with non-zero lethal radii

This study suggests a co-evolutionary method for solving pursuit-evasion games with consideration of non-zero lethal radii. The proposed method has three key features. First, it can handle both the final time problem and the miss distance problem simultaneously, by adopting a separated payoff function. Second, the Stackelberg equilibrium instead of the security strategy solution is employed to consider the maximin characteristics of an open-loop solution. Finally, an additional evolving group is introduced to treat an unprescribed final time. Numerical simulations are performed to verify the proposed method by comparing it with the gradient-based method. In addition, the effect of lethal radius is discussed based on the numerical results.

Real option theory from finance to batch distillation

Batch distillation processes have gained renewed interest because of the recent development in small-scale industries producing high-value-added, low-volume specialty chemicals. The flexibility and unsteady state nature of batch distillation constitute a challenge for the designer. A particularly difficult problem is the optimal control problem involving open loop solution for the reflux ratio profile. This is because of the complexity of the formulation and the large computational effort associated to its solution. The mathematical and numerical complexities of the optimal control problem get worse when uncertainty is present in the formulation. In this work, by applying the optimality conditions from the real option theory based on the Ito's Lemma [Investment under uncertainity (1994); Memoirs Am. Math. Soc. 4 (1951) 1; Appl. Math. Opt. 4 (1974) 374], the mathematical tools needed to solve optimal control problems in batch distillation columns when uncertainties in the state variables are present have been developed. Furthermore, the coupled maximum principle and NLP approach developed by Diwekar [Am. Inst. Chem. Eng. J. 38 (1992) 1551] has been extended for solving the optimal control problem in the uncertain case. This new algorithm has been implemented in the MultiBatchDS batch distillation process simulator. Finally, a numerical case-study is presented to show the scope and application of the proposed approach.

Optimization of Economic Growth Model

An optimal control problem for the economic growth model is under consideration. Optimal open-loop solutions are constructed on the base of the sufficient optimality conditions. Final result is presented as optimal feedback controls.

Design of Optimal Feedbacks in the Class of Inertial Controls

Two optimal control problems in the class of inertial controls (continuous controls of bounded values and rates) are investigated: the first is the optimal excitation of dynamic systems and the related design of attainability sets and the second is a linear optimal control problem with terminal constraints. They are solved by a dynamic adaptive linear programming method. First, open-loop solutions are constructed and then an algorithm for the optimal controller for the linear optimal control problem is designed, which in real time generates current optimal feedback. Examples are given to illustrate the results.

Reevaluation and renegotiation of climate change coalitions—a sequential closed-loop game approach

Major international environmental problems can be characterized as stock externality phenomena. Long-term international environmental coalitions are subject to unexpected shocks. Certain shocks might have strong impacts on regional environmental policies and their commitments to the coalition. The open-loop solutions in optimal control problems and differential games cannot capture regions’ reactions to those shocks. In this paper, we develop a feasible modeling approach that uses the closed-loop strategies to study the issues of reevaluation and renegotiation of climate change coalitions. For the purpose, we design an algorithm that decomposes a closed-loop strategy into a sequence of open-loop equilibria and implement the algorithm in the RICE model (Am. Econ. Rev. 86 (1996) 741), an integrated assessment model of climate change and economies. We analyze policy and methodological implications of the closed-loop strategies through simulations of the RICE model.

A Leader-Follower Stochastic Linear Quadratic Differential Game

A leader-follower stochastic differential game is considered with the state equation being a linear Ito-type stochastic differential equation and the cost functionals being quadratic. We allow that the coefficients of the system and those of the cost functionals are random, the controls enter the diffusion of the state equation, and the weight matrices for the controls in the cost functionals are not necessarily positive definite. The so-called open-loop strategies are considered only. Thus, the follower first solves a stochastic linear quadratic (LQ) optimal control problem with the aid of a stochastic Riccati equation. Then the leader turns to solve a stochastic LQ problem for a forward-backward stochastic differential equation. If such an LQ problem is solvable, one obtains an open-loop solution to the two-person leader-follower stochastic differential game. Moreover, it is shown that the open-loop solution admits a state feedback representation if a new stochastic Riccati equation is solvable.

A model of optimal advertising expenditures in a dynamic duopoly

This paper develops a dynamic model of oligopolistic advertising competition. The model is general enough to include predatory advertising and informative advertising as particular cases. The analysis is conducted in a differential game framework and compares the open-loop and feedback equilibria to the efficient outcome. It is found that for the informative advertising competition game, advertising levels are closer to the collusive outcomes in a feedback equilibrium. In the case of predatory advertising, expenditures are inefficiently high in a feedback equilibrium and the open-loop solution is more efficient.

Optimal and asymptotically optimal decision rules for sequential screening and resource allocation

We consider the problem of maximizing the expected sum of n variables X/sub k/ chosen sequentially in an i.i.d. sequence of length N. It is equivalent to the following resource allocation problem: n machines have to be allocated to N jobs of value X/sub k/ (k=1,...,N) arriving sequentially, the ith machine has a (known) probability p/sub i/ to perform the job successfully and the total expected reward must be maximized. The optimal solution of this stochastic dynamic-programming problem is derived when the distribution of the X/sub k/s is known: the optimal threshold for accepting X/sub k/, or allocating the ith machine to job k, is given by a backward recurrence equation. This optimal solution is compared to the simpler (but suboptimal) open-loop feedback-optimal solution for which the threshold is constant, and their asymptotic behaviors are investigated. The asymptotic behavior of the optimal threshold is used to derive a simple open-loop solution, which is proved to be asymptotically optimal (N/spl rarr//spl infin/ with n fixed) for a large class of distributions for X/sub k/.

Optimal and near-optimal take-off manoeuvres in the presence of windshear

This paper is concerned with the development of short-range forward-look guidance strategies for take-off manoeuvres in the presence of windshear, with reference to flight in the vertical and horizontal planes. In addition to the horizontal shear, the presence of a downdraught is assumed. First, two open-loop optimal escape strategies are considered, i.e. a performance maintaining strategy and a survival strategy, where it is assumed that the windshear is detected in an early stage of the encounter. Next, two closed-loop guidance strategies featuring short-range forward-look detection, which closely approximate the open-loop optimal solutions, are developed. Numerical results show that lateral manoeuvring, in which an aircraft is turned away from the microburst, improves the flight performance significantly in both optimal escape strategies. The simulation results demonstrate that the employed closed-loop guidance strategies are able to approximate closely the open-loop optimal trajectories, in both longitudinal and lateral escape manoeuvres.