Periodic Control Problem Research Articles

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Heuristics for perishable inventory systems under mixture issuance policies

We consider a periodic review inventory control problem of perishable goods with a fixed lifetime. Instead of considering either FIFO or LIFO issuance policy, we incorporate customers’ behavior in the retail setting using a mixture of FIFO and LIFO issuance policy. First, we characterize properties of inventory transitions under a mixture of FIFO and LIFO policies and show that the sequence of demand arrival does not influence the inventory state at the end of the period. Second, we study the properties of optimal solutions under different issuance policies to gain insights for heuristic designs under mixture issuance policies. Third, we propose a two-stage heuristic for the LIFO policy, demonstrating its approximation accuracy through comparison with optimal solutions. Building on this, we design an effective heuristic for the mixture issuance policy by combining heuristics for both FIFO and LIFO policies. The numerical experiments with stationary and nonstationary demand validate the performance of our proposed heuristics against alternative heuristics across various issuance policies.

Periodic event-triggered bipartite containment control for nonlinear multi-agent systems with input delay

ABSTRACT This paper studies the periodic event-triggered bipartite containment control problem of multi-agent systems with unmeasurable states and input delays. To deal with the system input delay problem, a Pade approximation technique is introduced. By utilising the sampled output of the system, a fuzzy state observer is constructed to estimate the unknown state. A periodic event-triggered mechanism is designed, which can effectively reduce the usage of communication resources. Compared with the traditional event-triggered control manner, a periodic event-triggered control manner only monitors the preset threshold condition at the sampling instant to determine whether to update the current control signal, which can automatically guarantee the minimum lower bound of the execution interval, avoiding the occurrence of Zeno behaviour. Stability analysis verifies that all signals in the closed-loop system are bounded, and the bipartite tracking error can converge to a small neighbourhood of the origin. Finally, a practical example further verifies the effectiveness of the designed control scheme.

A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control problems

A new diagonalization technique for the parallel-in-time solution of linear-quadratic optimal control problems with time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization efficiently transforms the discretized optimality system into nt (=number of time steps) decoupled complex valued 2ny × 2ny systems, where ny is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.

Optimal Control of the Initial Condition in the Problem of Gas Lifting

A partially periodic control problem is considered, where the control parameter enters the initial condition. We study the formalization related to the calculus of variations. The necessary conditions are written out in the form of Euler–Lagrange equations, with the help of which an algorithm for finding the optimal program trajectories is developed. The results are illustrated by an example when the motion is described by a time-averaged hyperbolic equation at a sufficiently large well depth during a gas lift.

Periodic LQG Wind Turbine Control with Adaptive Load Reduction

A periodic linear quadratic Gaussian (LQG) control law augmented with a reference point adaption to enable adequate rotor speed tracking and sufficient load reductions for a wind turbine is presented. The solution of the periodic LQG control problem is based on solving two periodic Ricatti differential equations in continuous time with a multiple shooting integration technique. For this, the available gridded linear time-variant description of the turbine is converted to a harmonic representation using harmonic Fourier approximation. While the periodic LQG controller provides rotor speed tracking and effective damping of the aeroelastic blade modes, the reference point adaption explicitly reduces the loads resulting from the periodic operation of the turbine rotor at the rotor rotational frequency. The performance of the proposed control system is compared against a baseline controller in realistic wind scenarios using a high fidelity nonlinear simulator. The results show a significant damage equivalent load reduction while maintaining adequate rotor speed tracking.

Inventory control with modulated demand and a partially observed modulation process

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $$K=0$$ and the existence of an optimal (s, S) policy if $$K>0$$ , where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $$K=0$$ , the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $$K>0$$ , the values of s and S and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $$K=0$$ case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

Linear programming based optimality conditions and approximate solution of a deterministic periodic optimal control problem in discrete time

This paper is focused on a discrete time periodic optimal control problem. We establish linear programming based optimality conditions and present a method for the construction of near optimal controls. A numerical example is included to demonstrate the results.

An Unscented Kalman Filter–based iterative learning controller for multibody rotating scan optical spacecraft

In this paper, to address the periodic control problem facing high-precision observation for multibody rotating scan optical spacecraft connected with active magnetic bearing (AMB), the Unscented Kalman Filter (UKF)–based iterative learning controller (UILC) is proposed. In the UILC, an adaptive backstepping controller (ABC) is deployed in the main loop to make the system state converge, and an UKF module is substituted for the memory module to use feedback information in real time and eliminate error propagation between adjacent periods. In the paper, the 9 degrees of freedom (DOF) error discrete dynamics are first derived to facilitate the convergence proved in the discrete domain. Then, the convergence analysis of the closed-loop system is investigated, and the state error supremum of closed-loop system in the mean square is indicated. Finally, for a comparison with the classical iterative learning controller (ILC), a series of numerical simulations are performed. As indicated by the simulation results, the UILC is capable of eliminating periodic deviation and error propagation during the adjacent periods.

Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory Management

A fundamental yet notoriously difficult problem in operations management is the periodic inventory control problem under positive lead time and lost sales. More recently, there has been interest in the problem setting where the demand distribution is not known a priori and must be learned from the observations made during the decision-making process. In “Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory Management,” Agrawal and Jia present a reinforcement learning algorithm that uses the observed outcomes of past decisions to implicitly learn the underlying dynamics and adaptively improve the decision-making strategy over time. They show that, compared with the best base-stock policy, their algorithm achieves an optimal regret bound in terms of the time horizon and scales linearly with the lead time of the inventory ordering process. Furthermore, they demonstrate that their approach is not restricted to the inventory problem and can be applied in an almost black box manner to more general reinforcement learning problems with convex cost functions.

Periodic Event-Triggered Control for a Class of Nonminimum-Phase Nonlinear Systems Using Dynamic Triggering Mechanism

In this paper, the output feedback based periodic event-triggered control problem is considered for a class of nonminimum-phase nonlinear systems via dynamic event-triggering mechanism. When only the sampled-data output is known, a new periodic event-triggered control method is proposed via output feedback and the control input updates based on a discrete-time dynamic event-triggering condition. Despite the unstable zero dynamics, the proposed control method can asymptotically stabilize the hybrid control systems by closing the loop only when it is necessary. In contrast to the continuous-time static one, the proposed dynamic triggering condition has several advantages including the easier digital implementation and the larger average inter-event time interval. The delicate analysis gives the explicit expression of the maximum allowable sampling period, and the global asymptotic stability can be achieved for the hybrid control systems. Finally, the effectiveness of the proposed periodic event-triggered control method is verified by a numerical simulation.

Periodic event-triggered formation control for multi-UAV systems with collision avoidance

In this paper, periodic event-triggered formation control problems with collision avoidance are studied for leader–follower multiple Unmanned Aerial Vehicles (UAVs). Firstly, based on the Artificial Potential Field (APF) method, a novel sliding manifold is proposed for controller design, which can solve the problem of collision avoidance. Then, the event-triggered strategy is applied to the distributed formation control of multi-UAV systems, where the evaluation of the event condition is continuous. In addition, the exclusion of Zeno behavior can be guaranteed by the inter-event time between two successive trigger events have a positive lower bound. Next, a periodic event-triggered mechanism is developed for formation control based on the continuous event-triggered mechanism. The periodic trigger mechanism does not need additional hardware circuits and sophisticated sensors, which can reduce the control cost. The stability of the control system is proved by the Lyapunov function method. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed control protocol.

Maximizing average throughput in oscillatory biochemical synthesis systems: an optimal control approach.

A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input, the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the system’s output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem, which can be analysed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of nonlinear occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.

Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations

In this paper, we study second-order optimality conditions for some optimal control problems governed by some semi-linear parabolic equations with periodic state constraint in time. We obtain a necessary condition and a sufficient condition in terms of the second order derivative of the associated Lagrangian. These two conditions correspond to the positive definite and the nonnegativity of the second order derivative of the Lagrangian on the same cone, respectively.

A Variational Approach to the Problem of Continuous Dependence of Solutions to Second Order Periodic System

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.

Neural-Network Approximation-Based Adaptive Periodic Event-Triggered Output-Feedback Control of Switched Nonlinear Systems.

This study considers an adaptive neural-network (NN) periodic event-triggered control (PETC) problem for switched nonlinear systems (SNSs). In the system, only the system output is available at sampling instants. A novel adaptive law and a state observer are constructed by using only the sampled system output. A new output-feedback adaptive NN PETC strategy is developed to reduce the usage of communication resources; it includes a controller that only uses event-sampling information and an event-triggering mechanism (ETM) that is only intermittently monitored at sampling instants. The proposed adaptive NN PETC strategy does not need restrictions on nonlinear functions reported in some previous studies. It is proven that all states of the closed-loop system (CLS) are semiglobally uniformly ultimately bounded (SGUUB) under arbitrary switchings by choosing an allowable sampling period. Finally, the proposed scheme is applied to a continuous stirred tank reactor (CSTR) system and a numerical example to verify its effectiveness.

A nonlinear tracking model predictive control scheme for dynamic target signals

We present a nonlinear model predictive control (MPC) scheme for tracking of dynamic target signals. The scheme combines stabilization and dynamic trajectory planning in one layer, thus ensuring constraint satisfaction irrespective of changes in the dynamic target signal. For periodic target signals we ensure exponential stability of the optimal reachable periodic trajectory using suitable terminal ingredients and a convexity condition for the underlying periodic optimal control problem. Furthermore, we introduce an online optimization of the terminal set size to automate the trade-off between fast convergence and operation close to the constraints. In addition, we show how stabilization and dynamic trajectory planning can be formulated as partially decoupled optimization problems, which reduces the computational demand while ensuring recursive feasibility and convergence. The main tool to enable the proposed design is a novel reference generic offline computation that provides suitable terminal ingredients for tracking of dynamic reference trajectories. The practicality of this approach is demonstrated on benchmark examples, which demonstrates superior performance compared to state of the art approaches.

Comparative analysis of model reduction strategies for circuit based periodic control problems

AbstractThis paper is a comparative analysis of two prominent iterative algorithms for model order reduction of linear time‐varying (LTV) periodic systems where the system's matrices are singular. Our proposed method is based on a reformulation of the LTV model to an equivalent linear time‐invariant (LTI) model using a suitable discretization procedure. The resulting LTI model is reduced in two ways, once by applying a balanced truncation method and once by applying a Krylov‐based method known as iterative rational Krylov algorithm (IRKA). During the application of balanced truncation, the low‐rank Cholesky factorized alternating directions implicit (LRCF‐ADI) method is used to estimate the solutions of the corresponding LTI form of Lyapunov equations. Since the system's matrices are singular, the concept of pseudo‐inverse is adopted to compute the shift parameters needed in the LRCF‐ADI iterations. For the Krylov‐based IRKA, our work is twofold. We solve the time‐invariant Lyapunov equation for the observability Gramian and apply a moment‐matching Krylov technique. The accuracy and effectiveness of the two proposed techniques are demonstrated with the help of frequency response graphs, bode plots, and eigenstructure of the main and reduced models.

Periodic pulse control of Hopf bifurcation in a fractional-order delay predator\u2013prey model incorporating a prey refuge

This paper is concerned with periodic pulse control of Hopf bifurcation for a fractional-order delay predator–prey model incorporating a prey refuge. The existence and uniqueness of a solution for such system is studied. Taking the time delay as the bifurcation parameter, critical values of the time delay for the emergence of Hopf bifurcation are determined. A novel periodic pulse delay feedback controller is introduced into the first equation of an uncontrolled system to successfully control the delay-deduced Hopf bifurcation of such a system. Since the stability theory is not well-developed for nonlinear fractional-order non-autonomous systems with delays, we investigate the periodic pulse control problem of the original system by a semi-analytical and semi-numerical method. Specifically, the stability of the linearized averaging system of the controlled system is first investigated, and then it is shown by numerical simulations that the controlled system has the same stability characteristics as its linearized averaging system. The proposed periodic pulse delay feedback controller has more flexibility than a classical linear delay feedback controller guaranteeing the control effect, due to the fact that the pulse width in each control period can be flexibly selected.

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.