Optimal Periodic Control Research Articles

Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Periodic Optimal Control of a Plug Flow Reactor Model with an Isoperimetric Constraint

We study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type “A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow $$\\end{document} product” carried out in a plug flow reactor (PFR) in the presence of an inert component. An isoperimetric optimal control problem with periodic boundary conditions and input constraints is formulated for the considered mathematical model in order to maximize the mean amount of product over the period. For the single-input system, the optimality of a bang-bang control strategy is proved in the class of bounded measurable inputs. The case of controlled flow rate input is also analyzed by exploiting the method of characteristics. A case study is performed to illustrate the performance of the reaction model under different control strategies.

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.

Bioprocess Modelling, Intensification, and Control


 
 
 Chemical bio-process intensification, driven by the need for sustainability and cost-effectiveness, has gained significant attention. This study explores dynamic intensification (DI), a strategy to enhance efficiency by modifying system dynamics, operation, and control. Two key approaches are investigated: intensification by design, involving deliberate changes to physical and chemical aspects, and intensification by control, optimizing process parameters and feedback loops.
 The research focuses on the Optimal Periodic Control (OPC) method, dating back to the 1960s but now revisited with modern technology. To investigate the feasibility and effectiveness of dynamic intensification, a comprehensive methodology encompassing modeling, optimization, linearization, dynamic intensification, and control was designed. This methodology ensures systematic and standardized testing of DI in various bioprocesses within the industry.
 Through rigorous case studies, such as Activated Sludge Model No. 1 (ASM1) and an antibiotic perfusion bioreactor, the study demonstrates the practical benefits of dynamic intensification, including enhanced process efficiency and control. Applying model predictive control (MPC) further improves process efficiency, and the intensified systems are thoroughly compared to their respective base cases.
 The insights gained from this research contribute to advancements in bioprocessing technologies across industries, paving the way for sustainable and efficient engineering practices. Engineers can leverage this knowledge and the developed methodology to optimize chemical bioprocesses through innovative intensification strategies, ultimately promoting more eco-friendly and economical bio-processes. The study's comprehensive methodology offers valuable insights and understanding for bioprocess engineers seeking to enhance system efficiency and contribute to a greener future.
 
 

Dynamic soaring in wind turbine wakes

Dynamic soaring for UAVs is a flight technique that enables continuous, powerless periodic flight patterns in the presence of a wind gradient. However, sufficiently large wind gradients are uncommon over land, while at offshore locations the largest wind gradients are located close to the ocean surface, thereby limiting the scope of practical application. An intrinsic feature of wind turbines is that they inherently produce very sharp wind gradients in the near wake. Therefore, in this paper, we propose and investigate periodic stationary dynamic soaring trajectories in the near wake of wind turbines. We additionally consider the potential of dynamic soaring for revitalizing the wind turbine wake. To this end, we apply periodic optimal control based on a simplified model for the glider dynamics and the wind profile in the wake. The cost function maximizes the revitalization of the wake. We compute optimal orbits for a range of different wing spans and different mass-scaling assumptions. The largest glider configuration, with a wingspan of 10 m and a mass of 222.6 kg, achieves a wake revitalization of about 0.94% of the total turbine thrust.

New optimal periodic control policy for the optimal periodic performance of a chemostat using a Fourier–Gegenbauer-based predictor-corrector method

In its simplest form, a chemostat consists of microorganisms or cells that grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions of the cell growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an “overyielding” state in which the performance becomes higher than that at steady-state operation. In this paper, we show that an optimal periodic control policy for maximizing chemostat performance can be accurately and efficiently derived numerically using a novel class of integral pseudospectral (IPS) methods and adaptive h-IPS methods composed through a predictor–corrector algorithm. New formulas for the construction of Fourier pseudospectral (PS) integration matrices and barycentric-shifted Gegenbauer (SG) quadratures are derived. A rigorous study of the errors and convergence rates of SG quadratures, as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions, is presented. We also introduce a novel adaptive scheme for detecting jump discontinuities and reconstructing a piecewise analytic function from PS data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.

Singular Arcs in Optimal Periodic Controls for Scalar Dynamics and Integral Input Constraint

We revisit recent results about optimal periodic control for scalar dynamics with input integral constraint, under lack of convexity and concavity. We show that in this more general framework, the optimal solutions are bang-singular-bang and generalize the bang-bang solutions for the convex case and purely singular for the concave one. We introduce a non-local slope condition to characterize the singular arcs. The results are illustrated on a class of bioprocesses models.

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.

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.

Speed‐loop frequency‐adaptive and current‐loop optimal harmonic periodic controllers for fault‐tolerant SPMSM drive systems

The surface permanent magnet synchronous motor (SPMSM) drive system has been widely used in the industry due to its high power density, high efficiency and easy to control. nature The author proposes a speed-loop frequency-adaptive periodic controller and a current-loop optimal harmonic periodic controller for a fault-tolerant SPMSM drive system, including normal operating conditions and faulty operating conditions. The faulty conditions consist of an insulated gate bipolar transistor (IGBT) open-circuit and an IGBT short-circuit. A digital signal processor, TMS-320F-2808, manufactured by Texas Instruments, is used as a control centre to execute the proposed fault-detection, fault-diagnosis, frequency-adaptive and optimal harmonic periodic control algorithms. Experimental results show the proposed advanced periodic controllers can provide better performance than the proportional integral controller and the classic periodic controller, including transient responses, load disturbance responses, and tracking responses under normal and faulty conditions.

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.

Asymptotic Tracking for Nonminimum Phase Linear Systems via Steady-State Compensation

The problem of output tracking is addressed in this article by a reformulation in terms of a two-point boundary value problem for the plant's zero dynamics, which is allowed to be unstable. On this basis, a hybrid feedback regulator ensuring output tracking is then designed. The robustness properties of the proposed regulator as well as connections with periodic (cheap) optimal control, output regulation, and possible extensions to the nonlinear setting are also discussed.

Improvement of performances of the chemostat used for continuous biological water treatment with periodic controls

We investigate the benefit of operating the chemostat model with periodic controls for biological water decontamination. The first problem that we consider is to minimize the average output concentration of pollution under a constraint on the total quantity of water treated over the period. The second one consists in maximizing this quantity under a constraint on the average output concentration. We first give conditions on the growth characteristics of micro-organisms for which an improvement of the objective function (for the two problems) is possible, compared to the value at steady-state. We then provide the (global) optimal periodic control strategies for the first problem, and we then show a duality property both problems, which allows us to deduce the solutions of the second problem from the solutions of the first one. Results are illustrated on Monod’s, Haldane’s, Hill’s, and Contois’s kinetics.

Simplifying Complex Network Stability Analysis via Hierarchical Node Aggregation and Optimal Periodic Control.

In this study, the stability of a hierarchical network with delayed output is discussed by applying a kind of optimal periodic control. To reduce the number of the nodes of the original hierarchical network, an aggregation algorithm is first presented to take some nodes with the same information as an aggregated node. Furthermore, the stability of the original hierarchical network can be guaranteed by the optimal periodic control of the aggregated hierarchical network. Then, an optimal control scheme is proposed to reduce the bandwidth waste in information transmission. In the control scheme, the time sequence is separated into two parts: the deterministic segment and the dynamic segment. With the optimal control scheme, two targets are achieved: 1) the outputs of the original and aggregated hierarchical system are both asymptotically stable and 2) the nodes with slow convergent rate can catch up with the convergence speeds of other nodes.

Periodic optimal control of nonlinear constrained systems using economic model predictive control

In this paper, we consider the problem of periodic optimal control of nonlinear systems subject to online changing and periodically time-varying economic performance measures using model predictive control (MPC). The proposed economic MPC scheme uses an online optimized artificial periodic orbit to ensure recursive feasibility and constraint satisfaction despite unpredictable changes in the economic performance index. We demonstrate that the direct extension of existing methods to periodic orbits does not necessarily yield the desirable closed-loop economic performance. Instead, we carefully revise the constraints on the artificial trajectory, which ensures that the closed-loop average performance is no worse than a locally optimal periodic orbit. In the special case that the prediction horizon is set to zero, the proposed scheme is a modified version of recent publications using periodicity constraints, with the important difference that the resulting closed loop has more degrees of freedom which are vital to ensure convergence to an optimal periodic orbit. In addition, we detail a tailored offline computation of suitable terminal ingredients, which are both theoretically and practically beneficial for closed-loop performance improvement. Finally, we demonstrate the practicality and performance improvements of the proposed approach on benchmark examples.

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.

Optimal Periodic Control of Connected Multiple Vehicles With Heterogeneous Dynamics and Guaranteed Bounded Stability

The platooning of connected and automated vehicles has the potential to significantly improve the fuel efficiency of road transportation. Shortening the carfollowing distance to reduce aerodynamic drag is often used to improve fuel economy in today's platoons, but suffers high risk of rear-end collision. This paper presents an alternative solution to reduce platoon fuel consumption, i.e., periodic longitudinal control, which remains effective under the condition of middle or long following distance. The periodic controller can be applied to heterogeneous platoons, in which the optimal switching rule is designed for each vehicle to maintain a safe following distance while guaranteeing a collective bounded stability. Sectionalized switching maps based on state trajectory analysis are designed to choose the appropriate driving mode. The bounded stability for heterogeneous platoons is proved through the set-invariance theory, which ensures that the inter-vehicle distances are confined in a desirable range. Simulation results show that the proposed method can save up to 18% fuel compared to linear quadratic (LQ) controller, and all vehicles achieve bounded range error and velocity fluctuation.

Frequency-domain Methods and Polynomial Optimization for Optimal Periodic Control of Linear Plants

We consider the problem of periodic trajectory design for single-output systems which may be subject to periodic external disturbances. We show how trajectories optimizing a possibly nonquadratic and nonconvex polynomial performance objective can be found by using the frequency-domain description of the plant by converting the problem to a polynomial optimization problem (POP) in the Fourier coefficients of the external input signals. The method is suited for distributed-parameter systems, since the system transfer functions are not required to be rational; the computational complexity of the method depends on the order of the polynomial nonlinearities in the performance objective as well as the number of required harmonics, but is independent of the underlying system dimension.

Optimal periodic control of nonlinear chemical reactions with a time‐varying flow rate

AbstractThis paper deals with a mathematical model of a non‐isothermal chemical reaction described by nonlinear ordinary differential equations with two‐dimensional control. The control variables correspond to the possibility of manipulating the concentration of the input reactant and the total flow rate. We consider the task of maximizing the reactor performance as an isoperimetric optimal control problem with periodic boundary conditions. Necessary optimality conditions are derived by the Pontryagin maximum principle with Lagrange multipliers corresponding to the isoperimetric constraints.