Optimal Periodic Control Research Articles

Optimal periodic control for scalar dynamics under integral constraint on the input

This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control \begin{document}$ u $\end{document} , under an integral constraint on \begin{document}$ u $\end{document} . We give conditions for which the value of the cost function at steady state with a constant control \begin{document}$ \bar u $\end{document} can be improved by considering periodic control \begin{document}$ u $\end{document} with average value equal to \begin{document}$ \bar u $\end{document} . This leads to the so-called over-yielding met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.

Optimal periodic control of spin systems: Application to the maximization of the signal-to-noise ratio per unit time

We propose an optimal control algorithm for periodic spin dynamics. This non-trivial optimization problem involves the design of a control field maximizing a figure of merit, while finding the initial and final states of the dynamics, which are not known but are subjected to specific periodic conditions. As an illustrative example, we consider the maximization of the signal to noise ratio per unit time of spin systems. In the case of a homogeneous spin ensemble and a very short control duration, we show numerically that the optimal field corresponds to the Ernst angle solution. We investigate the optimal control process for longer control durations and their sensitivity to offset inhomogeneities.

Optimal Periodic Control of Hypersonic Cruise Vehicle: Trajectory Features

Optimal periodic control is an effective technique to reduce aerodynamic heating and fuel consumption of hypersonic cruise vehicles. Herein, this optimal control problem has been solved by nonlinear programming and passed the posteriori check of optimality. The results indicate that the lift coefficient and velocity in dynamical model could be recognized as constants. Accordingly, the original 3D system consists of velocity, flight path angle, and altitude, which could be decoupled into a 2D subsystem and a 1D one. The 2D subsystem could describe the correlation between the flight path angle and altitude. And the period length is mainly dependent on the maximum altitude difference. The 1D subsystem has been proved to be feasible to describe the variation of kinetic energy or velocity. On basis of the subsystems, the heating and fuel performances of periodic cruise have been studied. The heating performance is mainly dependent on the maximum altitude difference, and the fuel consumption is mainly dependent on the drag coefficient. The features concluded in this paper can support rapid trajectory planning or suboptimal feedback design of hypersonic cruise vehicles.

On periodic optimal solutions of persistent sensor planning for continuous-time linear systems

This work investigates informative planning of sensing agents over infinite time horizon when the system of interest is expressed as a continuous-time linear system. The objective of this planning problem, termed persistent monitoring problem, is to maintain the monitoring uncertainty at the minimum. The method reduces the persistent planning problem into a periodic planning problem; it is formulated as a periodic optimal control or optimization problem to determine the optimal periodic sensor plan as well as the period. The plan induces the periodic Riccati equation and is proven to lead an arbitrary initial uncertainty state to the optimized periodic trajectory. It is also proven that any infinite-horizon (non-periodic) sensor plan is able to be approximated arbitrary well by a periodic sensor plan. A suboptimal filtering mechanism is proposed by using the resulting optimal periodic solution. Two numerical examples on (a) a relaxed periodic sensor scheduling for a two dimensional linear system, and (b) persistent monitoring by a mobile sensor of two-dimensional diffusion dynamics show the validity of the proposed approach.

Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems

In this paper, a control mechanism is presented for optimal synchronization of two non-smooth fractional order chaotic systems with parametric uncertainty based on nonlinear fractional order proportional derivative (NLFPD) controller combined with optimal periodic control signals. Unlike synchronization methods based on FPID controllers, in this approach, optimal tuning of the NLFPD controller and determination of optimal periodic control signals for synchronization process are presented in the form of non-smooth fractional order optimal control (FOC) problems. Optimal periodic control signals are demonstrated as a generalized expansion in the sense of Fourier expansion that is used to accelerate the synchronization process. Using the generalization of a numerical method in nonlinear optimal control problems and the Grunwald-Letnikov(GL) fractional derivative definition, the synchronization problem based on the suggested mechanism is transformed into the form of a smooth FOC problem. By defining a base-time soft switch, a supervisory approach is added to the proposed control mechanism for desired and robust performance against parametric uncertainty in the slave system. Finally, to illustrate the proposed control mechanism, the synchronization of two identical fractional order Chua systems with simulation results is presented. The Results show that using the suggested control mechanism, the synchronization is fast and robust against parametric uncertainty in Chua slave system.

Fuel-Optimal Periodic Control of Passenger Cars in Cruise Based on Pontryagin’s Minimum Principle

This paper aims to exploit the fuel saving potential of a passenger car in cruising scenarios by deriving an optimal periodic control strategy based on Pontyagin’s Minimum Principle. Gear shifting and coupling between the engine and the driveline are explicitly included in the optimal control problem, and this leads to a hybrid system. By investigating the analytically derived optimal trajectory in the state space, we confirm the optimal periodic control. Considerable advantages in fuel economy up to 4.8% compared to conventional steady control are demonstrated.

The emergence of feedforward periodicity for the fed-batch penicillin fermentation process

Abstract In this paper, focus is on identifying the feedforward structure for the emergence of input periodicity at the optimal situation, and the fed-batch penicillin fermentation process is applied for case study. Through information of optimal control, the reversed system analysis methods are constructed, and the criterion for the emergency of optimal periodic control (OPC) is built, and possibly, analytical method to compute out the OPC problem would be proposed.

Numerical Optimal Control With Periodicity Constraints in the Presence of Invariants

Periodic optimal control problems (POCPs) based on dynamic models holding invariants are often problematic to treat using standard numerical methods. The difficulty stems from a failure of standard constraint qualifications and typically hinders the convergence of the numerical solver, or even defeats it. Optimization problems having weak constraint qualifications can be treated using dedicated solvers, at the price of a more involved algorithmic. In this paper, we analyze the constraint qualification of POCPs holding invariants, and propose three simple and computationally inexpensive modifications of the formulation that allow for a recovery of linear independence constraint qualification, while not affecting the second-order sufficient conditions for optimality. Hence, the resulting POCP can be tackled via standard solvers, without special treatment. The application of these approaches is detailed for the case of POCPs holding index-reduced differential-algebraic equations and representations of the $\mathrm{SO}\left(3\right)$ Lie group.

On the Structure of the Optimal Solution to a Periodic Drug-Delivery Problem

This paper examines the shaping of a drug's delivery—in this case, nicotine—to maximize its efficacy. Previous research: (i) furnishes a pharmacokinetic–pharmacodynamic (PKPD) model of this drug's metabolism; (ii) shows that the drug-delivery problem is proper, meaning that its optimal solution is periodic; (iii) shows that the underlying PKPD model is differentially flat; and (iv) exploits differential flatness to solve the problem by optimizing the coefficients of a truncated Fourier expansion of the flat output trajectory. In contrast, the work in this article provides insight into the structure of the theoretical solution to this optimal periodic control (OPC) problem. First, we argue for the existence of a bijection between feasible periodic input and state trajectories of the problem. Second, we exploit Pontryagin's maximum principle to show that the optimal periodic solution has a bang–singular–bang structure. Building on these insights, this article proposes two different numerical methods for solving this OPC problem. One method uses nonlinear programming (NLP) to optimize the states at which the optimal solution transitions between the different solution arcs. The second method approximates the control input trajectory as piecewise constant and optimizes the discrete values of the input sequence. The paper concludes by discussing the computational costs of these two algorithms as well as the importance of the associated insights into the structure of the optimal solution trajectory.

Optimal stationary control of diesel engines using periodic control

Measurements and optimal control are used to study whether the fuel economy of a diesel engine can be improved through periodic control of the wastegate, illustrating how modern optimal control tools can be used to identify non-trivial solutions that can improve performance. The measurements show that the pumping torque of the engine is changed when the wastegate is controlled in a periodic manner versus stationary even if the mean position is the same. If this decreases the fuel consumption or not is seen to be frequency and operating point dependent. The measurements indicate that the phenomenon occurs in the time scales capturable by mean value engine models (MVEM). The operating points are further analyzed using a MVEM and optimal control. It is shown that whether the optimal solution exhibits periodic oscillations or not is operating point dependent, but is not due to the instantaneous nature of the controls. Even if an actuator model is added the oscillations persist for reasonable time constants, the frequency of the oscillations is however affected. Further it is shown that the periodic control can be predicted by optimal periodic control theory and that the frequency of the control affects the resulting efficiency.

Periodic optimal control problems governed by semilinear parabolic equations with impulse control

This paper is concerned with periodic optimal control problems governed by semilinear parabolic differential equations with impulse control. Pontryagin's maximum principle is derived. The proofs rely on a unique continuation estimate at one time for a linear parabolic equation.

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.

Optimal Periodic Control of an Ideal Stirling Engine Model

We consider an optimal control problem for a model of a Stirling engine that is actively controlled through its displacer piston motion. The framework of optimal periodic control (OPC) is used as the setting for this active control problem. We use the idealized isothermal Schmidt model for the system dynamics and formulate the control problem so as to maximize mechanical power output while trading off a penalty on control (displacer motion) effort. An iterative first-order algorithm is used to obtain the optimal periodic motion of the engine and control input. We show that optimal motion is typically nonsinusoidal with significant higher harmonic content, and that a significant increase in the power output of the engine is possible through the optimal scheduling of the displacer motion. These results indicate that OPC may provide a framework for a large class of energy conversion and harvesting problems in which active actuation is available.

Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion

In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks.

On Average Performance and Stability of Economic Model Predictive Control

Control performance and cost optimization can be conflicting goals in the management of industrial processes. Even when optimal or optimization-based control synthesis tools are applied, the economic cost associated with plant operation is often only optimized according to static criteria that pick, among all feasible steady states, those with minimal cost. In mathematical terms, an economic cost functional differs from stage costs commonly adopted in model predictive control (MPC) as it need not be minimal at its best equilibrium. This note collects and illustrates some recent advances in receding horizon optimization of nonlinear systems that allow the control designer to simultaneously and dynamically optimize transient and steady-state economic performance. In particular, we show that average performance of economic MPC is never worse than the optimal steady-state operation. We introduce a dissipation inequality and supply function that extend previous sufficient conditions for asymptotic stability of economic MPC. Dissipativity is also shown to be a sufficient condition for concluding that steady-state operation is optimal. We show how to modify an economic cost function so that steady-state operation is asymptotically stable when that feature is deemed desirable. Finally, for the case when steady-state operation is not optimal, we develop two modified MPC controllers that asymptotically guarantee 1) improved performance compared to optimal periodic control and 2) satisfaction of constraints on average values of states and inputs.

Approximate Robust Optimal Control of Periodic Systems with Invariants and High-Index Differential Algebraic Systems

In this paper we present solution approaches for uncertain periodic optimal control problems with invariants that arise e.g., after index reduction of high-index differential algebraic systems. There are two difficulties to be addressed: first, we encounter a redundancy in the periodic boundary constraints which is due to the presence of invariants. And second, we have to deal with the presence of uncertainties. To address the first problem we discuss both a projection and a null-space based reformulation approaches that avoid the redundancies in the constraints. Concerning the uncertainties, we discuss an approximate robust optimal control formulation based on Lyapunov differential equations. Here, the invariants and periodic boundary constraints have to be taken into account, too. We illustrate our techniques by optimizing an open-loop controlled inverted pendulum which is described by index three differential algebraic equations and is affected by unknown forces.

Determining the optimum cyclic operation of adsorption chillers by a direct method for periodic optimal control

Adsorption refrigeration systems provide a sustainable possibility to reduce the environmental impact of refrigeration and air-conditioning as they allow for sources of otherwise unused excess waste heat to be reused for cooling purposes. Adsorptive cooling is a discontinuously operated cycling process, and it is well known that the determination of an optimal cycling time yielding maximum cooling power is a key to the design of an efficient mode of operation. The optimal cycle time however strongly depends on operating conditions such as ambient air temperature, available heat source temperature, desired target cooling temperature, and achievable volume flow rates of the secondary heat transfer circuits. In this contribution, we apply a direct method for periodic optimal control to optimize two-bed adsorption chillers. We present a first principles dynamic model of the underlying thermal process. We show that direct methods for periodic optimal control allow for quick and reliable computation of optimal cycle times for a given set of parameters. Contrary to pre-existing methods, fast computation times and guaranteed optimality of the solutions computed by our approach makes it viable to extensively study the simulated optimal cyclic operation of two-bed adsorption chillers under a wide range of varying conditions.

Optimal Unmanned Aerial Vehicle Flights for Seeability and Endurance in Winds

Video-equippedunmanned aerial vehicles are highly useful inmissions of surveillance,monitoring, and sensing. In these flights, the unmanned aerial vehicles must maintain the ability to see the targets as well as to save energy for prolonged endurance. This paper examines basic patterns of unmanned aerial vehicle flights that can maximize the ability to see targets, or “seeability,” and/orminimize power consumptions to assist unmanned aerial vehiclemission planning. In this paper, a point-massmodel is used to describe unmanned aerial vehiclemotions. A seeabilitymodel is established that peaks when the unmanned aerial vehicle is flying at a certain angle from the normal vector perpendicular to the surface of the target. Unmanned aerial vehicle flights are formulated as nonlinear periodic optimal control problems. The performance indices are selected to maximize the average seeability, to minimize the average power consumption, or to achieve a balance of the two. Motion constraints due to unmanned aerial vehicle performance capabilities and safety are imposed. The effects of different levels of constant wind velocities are considered. These nonlinear optimal control problems are converted into parameter optimization for numerical solutions. Extensive numerical solutions are obtained for unmanned aerial vehicle level flights with constant airspeeds and variable airspeeds, as well as three-dimensional flights. Clear tradeoffs between maximum seeability and minimum power are established.

Optimal Discrete-Time Design of Three-Axis Magnetic Attitude Control Laws

The problem of designing discrete-time attitude controllers for three-axis stabilization of magnetically actuated spacecraft is considered. Several methods are discussed and an approach to the tuning of various classes of projection-based controllers is proposed relying on periodic optimal output feedback control techniques. The main advantages of the proposed methods are discussed and illustrated in a simulation study.

Optimal periodic output feedback control: a continuous-time approach and a case study

This article deals with the problem of optimal static output feedback control of linear periodic systems in continuous time, for which a continuous-time approach, which allows to deal with both stable and unstable open loop systems, is presented. The proposed approach is tested on the problem of designing attitude control laws for a Low-Earth Orbit (LEO) satellite on the basis of feedback from a triaxial magnetometer and a set of high-precision gyros. Simulation results are used to demonstrate the feasibility of the proposed strategy and to evaluate its performance.