Open-loop Optimal Solutions Research Articles

Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems

<p style='text-indent:20px;'>In this paper, we consider linear-quadratic (LQ) leader-follower Stackelberg differential games for mean-field type stochastic systems with jump diffusions, where the system includes mean-field variables, i.e., the expected value of state and control variables. We first solve the LQ mean-field type control problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the coupled integro-Riccati differential equations (CIRDEs) via the Four-Step Scheme. Next, we solve the problem of the leader, which is the LQ control problem subject to the mean-field type forward-backward stochastic system with jump diffusions, where the constraint characterizes the rational behavior of the follower. Using the variational approach, we obtain the (mean-field type) stochastic maximum principle. However, to obtain the state-feedback representation of the open-loop optimal solution of the leader, there is a technical challenge due to the jump process. We consider two different cases, in which the state-feedback type control in terms of the CIRDEs can be characterized by generalizing the Four-Step Scheme. We finally show that the state-feedback type controls of the open-loop optimal solutions for the leader and the follower constitute the Stackelberg equilibrium.</p>

Linear-Quadratic Stochastic Stackelberg Differential Games for Jump-Diffusion Systems

This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem of the leader using the variational method. However, to obtain the state-feedback representation of the open-loop solution for the leader, there is a technical challenge due to the jump process. To overcome this limitation, we consider two different cases, in which the state-feedback type optimal control for the leader in terms of the ISRDE can be characterized by generalizing the Four-Step Scheme. Finally, in these two cases, we show that the state-feedback representation of the open-loop optimal solutions for the leader and the follower constitutes the Stackelberg equilibrium. Note that the (indefinite) LQ control problem of the leader is new and nontrivial due to the coupled FBSDE constraint induced by the rational behavior of the follower.

Time optimal entrainment control for circadian rhythm.

The circadian rhythm functions as a master clock that regulates many physiological processes in humans including sleep, metabolism, hormone secretion, and neurobehavioral processes. Disruption of the circadian rhythm is known to have negative impacts on health. Light is the strongest circadian stimulus that can be used to regulate the circadian phase. In this paper, we consider the mathematical problem of time-optimal circadian (re)entrainment, i.e., computing the lighting schedule to drive a misaligned circadian phase to a reference circadian phase as quickly as possible. We represent the dynamics of the circadian rhythm using the Jewett-Forger-Kronauer (JFK) model, which is a third-order nonlinear differential equation. The time-optimal circadian entrainment problem has been previously solved in settings that involve either a reduced-order JFK model or open-loop optimal solutions. In this paper, we present (1) a general solution for the time-optimal control problem of fastest entrainment that can be applied to the full order JFK model (2) an evaluation of the impacts of model reduction on the solutions of the time-optimal control problem, and (3) optimal feedback control laws for fastest entrainment for the full order Kronauer model and evaluate their robustness under some modeling errors.

Orbital rendezvous performance comparison of differential geometric and ZEM/ZEV feedback guidance algorithms

In this paper, the performance of two distinct classes of feedback guidance algorithms is evaluated for a spacecraft rendezvous problem utilizing a continuous low-thrust propulsion system. They are the DG (Differential Geometric) and ZEM/ZEV (Zero-Effort-Miss/Zero- Effort-Velocity) feedback guidance algorithms. Even though these two guidance algorithms do not attempt to minimize the onboard fuel consumption or ΔV directly, the ΔV requirement is used as a measure of their orbital rendezvous performance for various initial conditions and a wide range of the rendezvous time (within less than one orbital period of the target vehicle). For the DG guidance, the effects of its guidance parameter and terminal time on the closed-loop performance are evaluated by numerical simulations. For the ZEM/ZEV guidance, its nearfuel- optimality is further demonstrated for a rapid, short-range orbital rendezvous, in comparison with the corresponding open-loop optimal solutions. Furthermore, the poor ΔV performance of the ZEM/ZEV guidance for a slow, long-range orbital rendezvous is remedied by simply adding an initial drift phase. The ZEM/ZEV feedback guidance algorithm and its appropriate variants are then shown to be a simple practical solution to a non-impulsive rendezvous problem, in comparison with the DG guidance as well as the open-loop optimal guidance.

Optimal control for pointwise asymptotic stability in a hybrid control system

Pointwise asymptotic stability, or semistability, is a property of the set of equilibria of a dynamical system, where every equilibrium is Lyapunov stable and every solution is convergent to some equilibrium. Under an appropriate version of asymptotic controllability assumption, it is shown that the property can be achieved in a hybrid control system by open-loop optimal solutions of an infinite-horizon optimal control problem. For discrete-time systems, the optimal solutions can be generated by feedback. Regularity of the optimal value function and the existence of hybrid optimal controls are also studied.

Applications of Generalized Zero-Effort-Miss/Zero-Effort-Velocity Feedback Guidance Algorithm

The performance of the zero-effort-miss/zero-effort-velocity feedback guidance algorithm is evaluated through practical space application examples. The zero-effort-miss/zero-effort-velocity feedback guidance algorithm is, in general, not an optimal solution; however, it is an optimal solution in a uniform gravitational environment. It is also conceptually simple and easy to implement and, thus, has great potential for autonomous onboard implementation. It is shown that, for some classic ballistic missile intercept and asteroid intercept scenarios, the zero-effort-miss/zero-effort-velocity algorithm can even compete with corresponding open-loop optimal solutions, while its feedback characteristics make it more suitable to deal with uncertainties and perturbations. By employing the zero-effort-miss/zero-effort-velocity algorithm in the highly nonlinear orbital transfer and raising problems and comparing with corresponding open-loop optimal solutions, its simplicity and near-optimality are further verified.

3307 開ループ制御による回転型振子の最短時間運動(OS3-2 設計と制御,OS3 交通・物流システムの制御,オーガナイズド・セッション(OS))

This paper investigates the application of optimal control results to a rotary pendulum. The optimal control problem is prescribed as a minimum-time maneuver of the rotary pendulum, and it is solved using pseudospectral (PS) optimal control theory. The computed optimal results including a constraint of pendulum angle are applied as open-loop commands to the real system. Preliminary experimental results demonstrate the feasibility of commanding the rotary pendulum with open-loop optimal solutions generated offline.

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.

Decreasing the sensitivity of open-loop optimal solutions in decision making under uncertainty

We consider a framework for approximating the effects of uncertainty within the mean–variance optimization model for nonlinear systems. We present formulations for controlling the effects of parameter uncertainty characterized by continuous probability density functions (PDFs). Robust strategies, capable of capturing up to second-order effects of the uncertainty, are derived and tested on a dynamic optimisation problem arising from a chemical engineering application. An associated sensitivity-robustness concept is discussed and shown to be an instance of the general approximation framework developed. The accuracy of the approximations (or the local nature of the robust solutions) is studied using Monte-Carlo (MC) simulations.

Lateral Escape Guidance Strategies for Microburst Windshear Encounters

We present the outcome of a preliminary research effort involving the development of a near-optimal lateral escape guidance technique for aircraft encountering a microburst on final approach. The proposed guidance technique relies either on in situ or on relatively short-range forward-look windshear detection. Simulated guidance solutions are evaluated in terms of recovery-altitude performance and robustness to uncertainty in microburst size and strength. Based on a comparison with exact open-loop optimal solutions, the employed technique of directing the aircraft toward a target recovery altitude constitutes itself as a very promising longitudinal guidance strategy. The lateral escape strategy, though very simple, also proved to be very effective, reconfirming the overall benefits of lateral escape vis-a-vis nonturning escape. Before a practical implementation of the guidance strategies can take place, several issues still need be addressed. This paper briefly outlines the missing elements to provide a framework for future research and development.

Near-optimal three-dimensional air-to-air missile guidance against maneuvering target

Closed-loop guidance of a medium-range air-to-air missile (AAM) against a maneuvering target is synthesized using a three-phase, near-optimal guidance scheme. A large period of the closed-loop guidance is performed using neighboring optimal control techniques about open-loop optimal solutions obtained by solving the associated minimum time to intercept trajectories. The first phase is the boost phase guidance where the normal acceleration limit may be active due to high lofting of the boost-sustain-coast AAM. The guidance in this phase accommodates only errors in the missile's state variables, the target maneuvering being neglected. The boost phase guidance involves guidance in the presence of active control constraints. The second phase is the midcourse guidance where both state perturbations and target maneuvers are considered. Comparisons are made between guidance with gain indexing performed with clock time and with performance index to go. Models of aggressive target and run-away targets were used and the guidance scheme performance is excellent. Three methods of optimal gain evaluation are also discussed. Performance augmentation is obtained by using a center of attainability as a pseudotarget that fairs into the actual target as time to go becomes zero. The final phase is the terminal guidance, which employs proportional navigation and its variants.