Optimal Feedback Control Research Articles

Direct integral pseudospectral and integral spectral methods for solving a class of infinite horizon optimal output feedback control problems using rational and exponential Gegenbauer polynomials

This study is concerned with the numerical solution of a class of infinite-horizon linear regulation problems with state equality constraints and output feedback control. We propose two numerical methods to convert the optimal control problem into nonlinear programming problems (NLPs) using collocations in a semi-infinite domain based on rational Gegenbauer (RG) and exponential Gegenbauer (EG) basis functions. We introduce new properties of these basis functions and derive their quadratures and associated truncation errors. A rigorous stability analysis of the RG and EG interpolations is also presented. The effects of various parameters on the accuracy and efficiency of the proposed methods are investigated. The performance of the developed integral spectral method is demonstrated using two benchmark test problems related to a simple model of a divert control system and the lateral dynamics of an F-16 aircraft. Comparisons of the results of the current study with available numerical solutions show that the developed numerical scheme is efficient and exhibits faster convergence rates and higher accuracy.

Bioprocess Intensification of a Continuous-Flow Enzymatic Bioreactor via Productivity Dynamic Optimization under Modeling Uncertainty

In this contribution, a class of observer-based optimal feedback control is designed. The proposed feedback control is based on the Euler–Lagrange theoretical framework, and it is motivated by the productivity intensification from the chemical reactors, which is optimally increased. A Lagrangian is computed by employing the corresponding mass balance equation of a specifically selected biochemical compound. The resulting optimal controller is coupled with a novel uncertainty estimator with bounded feedback to derive an accurate estimation of the unknown terms and functions, mostly related to the reaction rate. Via Lyapunov analysis, it was shown that the proposed observer is asymptotically stable. The estimation of the unknown terms and functions is used by the proposed controller. The proposed methodology is applied to a generic model of an enzymatic biochemical continuous reactor with complex oscillatory dynamic behavior described by mass balance equations, so, in general, the proposed controller may be applied to any continuous stirred tank bioreactor; that is, the controller is independent of the specific kinetic functions. Numerical simulations show a satisfactory performance of the proposed control strategy.

Optimal Feedback in a Linear–Quadratic Optimal Control Problem for a Fractional-Order System

For a dynamical system described by a linear differential equation with a Caputo fractional derivative, we consider an optimal control problem of minimizing a quadratic terminal–integral performance functional. We propose and justify the construction of optimal feedback (optimal control synthesis) that generates the corresponding optimal control for any initial state of the system.

Online accelerated data‐driven learning for optimal feedback control of discrete‐time partially uncertain systems

SummaryIn this paper, we develop an online learning algorithm for solving the Bellman equation for affine in the control discrete‐time nonlinear uncertain dynamical systems. To ensure accelerated learning of our algorithm in generating optimal control policies, we use an actor‐critic structure predicated on higher‐order tuner laws. More specifically, we construct a Nesterov‐like architecture involving momentum‐based learning laws leading to an accelerated convergence of the optimal control policy. The proposed online learning‐based optimal control framework guarantees uniform ultimate boundedness of the closed‐loop system under the assumption that the system is persistently excited. Finally, two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

Model based optimal magnetic feedback control of multiple resistive wall modes with discrete coil and sensor arrays

A model-based optimal control method for multiple resistive wall mode (RWM) feedback stabilization has been developed and tested in plasma experiments at the EXTRAP T2R reversed field pinch (RFP) device. The controller is designed to target issues that arise in connection with RWM magnetic feedback stabilization systems based on discrete control coil and sensor arrays in tokamak and reversed field pinch devices. Multi-mode control capabilities in these systems is limited due to coupling of modes induced by the control system. The coupling originates from the generation of side-band control field harmonics and from aliasing of multiple harmonics in the sensor measurements. These couplings naturally leads to a multiple-input, multiple-output (MIMO) control problem. A model based state space controller has been designed based on a relatively simple physics model of the RWM plasma response. The physics model, which is applicable for the high aspect ratio RFP, is the basis for Fourier decoupling of the MIMO control problem into a set of single-input, multiple-output (SIMO) systems using the discrete Fourier transform (DFT). The linear, time-invariant physics model allows for the design of a state space model with states representing physical quantities; in this case the Fourier harmonics of the radial field at the resistive wall. Since the states cannot be directly measured, a Kalman filter is used for estimation of the states from the aliased sensor array measurements. A linear–quadratic (LQ) optimal state controller has been implemented. Design parameters, such as the LQ control cost function state weights and the Kalman filter input error covariances have been used to optimize the control operation in various ways. The controller has enhanced multi-mode control capabilities compared to earlier designs. For example it allows the prioritizing of suppression of one of the multiple magnetic field Fourier harmonics produced by a given control current DFT component. The controller has been tested in plasma experiments at EXTRAP T2R device, utilizing a newly installed extended sensor array, and the enhanced capabilities for multiple RWM feedback stabilization has been demonstrated.

Novel extended Kalman filter + H∞$$ {\mathbf{H}}_{\infty } $$ optimal output feedback control configuration for small satellites with high pointing stability requirements

SummaryOne of the most challenging subsystems for the CubeSats is the attitude determination and control system (ADCS) because it involves hardware and software integration, advanced strategies and algorithms to determine and control the attitude that impacts on space maneuvering and well working of a large set of payload (e.g., the optical payload). Although ADCS is largely studied, it still requires further investigations and analysis in order to achieve high‐performance and a reduced efforts. This paper presents an effective and reliable solution for the ADCS of CubeSats involved in Earth observation missions. The solution is based on an innovative framework that leverages the strengths of two distinct methodologies—H‐infinity optimal output feedback control and the extended Kalman filter (EKF)‐to significantly enhance the pointing stability of satellite systems. While H‐infinity optimal output feedback control is traditionally associated with partial state knowledge, we intentionally extend its application to scenarios with complete state information. The study includes a comparative performance analysis involving three algorithm configurations: the classic combination of EKF and model predictive control (MPC), the utilization of H‐infinity optimal output feedback control without EKF, and our proposed integrated approach. Performance evaluations are based on extrinsic indicators, demonstrating the advantages of our method in terms of pointing stability and overall control system performance. This research underscore the significance of combining innovative thinking with well‐established methodologies, unlocking new possibilities for stability enhancement in a range of engineering applications.

New discussion on optimal feedback control for Caputo fractional neutral evolution systems governed by hemivariational inequalities

The main focus of this article is to investigate the existence of feedback optimal control for the neutral fractional evolution systems in Hilbert spaces in the sense of the Caputo fractional derivatives. In order to establish the necessary conditions for the proposed problem, we apply the semigroup property, the fixed point theorem of multivalued maps, and the properties of generalized Clarke's subdifferentials. Then, by using the Filippov theorem and the Cesari property, a set of sufficient conditions is formulated to ensure the existence of a feasible pair for the feedback control systems. Finally, we apply our main results to obtain the optimal feedback control pair. In the end, an example is given to illustrate our theory.

AN Optimization control Techniques for Aircraft Yaw Control Lateral Dynamics

Presently, advanced control theory plays a major role in the aircraft system for the control of Yaw, Roll and Pitch angles, which was very much necessary for the stabilization of aircraft system. Following the works of the control system community for aircraft applications, this paper concentrates on the control of Yaw angle by designing various optimal Linear Quadratic Regulator (LQR) controllers using standard existing tools. The optimal controllers are designed for the aircraft, tested and validated using MATLAB/SIMULINK environment, and the results are compared with different system conditions to select the best optimal LQR feedback controllers in future aircraft control systems.

Mechanisms for handling uncertainty in sensorimotor control in sports: a scoping review

ABSTRACT In complex naturalistic sensorimotor behaviour, uncertainty arises from ambiguities and delays in sensory inputs as well as noise in sensory detection and motor execution. In sports, where human capacity reaches its limits, handling uncertainty is crucial. In fundamental motor-control research, five mechanisms for handling uncertainty – multisensory integration, prior-knowledge integration, risk optimisation, redundancy exploitation, and impedance control – have been proposed based on a rich body of evidence, mostly investigating simple arm and hand movement tasks. Here we review the literature investigating more complex tasks and examine to what extent these mechanisms explain handling uncertainty in sensorimotor control in sports. A systematic search following the PRISMA guidelines resulted in the consideration of 82 studies. These studies provide robust empirical evidence for the mechanisms of multisensory integration, prior-knowledge integration, and redundancy exploitation in complex naturalistic behaviour, whilst only a few publications focused on the other two mechanisms. Furthermore, only a few studies test model-based predictions that can be derived from the theoretical frameworks to a satisfactory extent. Finally, beyond discussing these explanatory mechanisms in isolation, we propose a unifying model that builds upon the theory of optimal feedback control, in which the mechanisms can be related to each other coherently.

On the Rod Heating Problem by Sources on the Class of Zonal Controls Using the Current and Past State at Measurement Points

We study an optimal feedback control problem for the rod heating process by means of lumped sources. The control actions are the powers of the sources, the values of which are defined on the class of zonal controls. The values of the parameters of zonal control actions are determined by subsets of the state space, to which belong the values of the process state at the measurement points at the current and past time moments. The posed problem is reduced to a parametric optimal control problem on determining a finite-dimensional vector of values of the parameters of zonal control actions. We have obtained optimality conditions for the values of the parameters of zonal control actions. These conditions contain formulas for the gradient of the objective functional with respect to the optimizable parameters. They make it possible to solve the reduced problem numerically with the application of efficient first-order optimization methods.

Dynamic optimal control of flow front position in injection molding process: A control parameterization-based method

High dynamic and precise control of process variables within permissible limits is the key to high product quality in the injection molding industry. Among these variables, the injection flow rate of molten polymer is particularly significant. In this paper, an effective optimal open-loop and state feedback controller design method is designed to realize the optimal tracking control of the melt polymer flow front position flow of melt polymer fluid during injection molding. To this end, the dynamic model of the filling process is firstly established, taking into account the nonlinearity of hydraulic control system and the effect of molten polymer flow on the dynamics of the injection ram, and then the dynamic optimal tracking control problem of flow front position of molten polymer in injection molding machine is proposed. Then, an open-loop controller and a multi-level feedback control law are well designed, wherein the control signal and the control feedback kernels are parameterized by control parameterization. The controller design problems are transformed into sequential optimal parameter decision problems. The explicit expressions of the gradient information of the objective function and constraints on the parameters of the decision variables are solved by using the state sensitivity equation analysis method, and the gradient information is combined with the nonlinear programming algorithm to solve the optimization problems efficiently. Finally, the viability and effectiveness of the proposed design method are verified by numerical simulations.

A weather optimal heading-based incremental feedback control for dynamic positioning of ships

In this brief, the weather optimal heading-based incremental feedback control algorithm is built for dynamic positioning (DP) of ships in the presence of model uncertainties and unknown environmental disturbances induced by wind, waves and currents. The proposed control algorithm has two parts, i.e., the weather optimal heading portion and the control portion. An improved ZPC-W approach with the deviation of the lateral control force is designed in order to enhance the convergence rate of the desired weather optimal heading and avoid overshoot as well. In addition, based on the iterative sliding mode function, a concise incremental feedback control scheme is designed with the pitch ratio of the thruster as the control input. It is measurable in the practical plant and has nothing to do with ship models with good stability and strong robustness. Lastly, simulation studies are performed to illustrate the performance and the superiorities of the proposed control algorithm.

A generalization of the Riccati recursion for equality‐constrained linear quadratic optimal control

AbstractThis paper introduces a generalization of the well‐known Riccati recursion for solving the discrete‐time equality‐constrained linear quadratic optimal control problem. The recursion can be used to compute problem solutions as well as optimal feedback control policies. Unlike other tailored approaches for this problem class, the proposed method does not require restrictive regularity conditions on the problem. This allows its use in nonlinear optimal control problem solvers that use exact Lagrangian Hessian information. We demonstrate that our approach can be implemented in a highly efficient algorithm that scales linearly with the horizon length. Numerical tests show a significant speed‐up of about one order of magnitude with respect to state‐of‐the‐art general‐purpose sparse linear solvers. Based on the proposed approach, faster nonlinear optimal control problem solvers can be developed that are suitable for more complex applications or for implementations on low‐cost or low‐power computational platforms. The implementation of the proposed algorithm is made available as open‐source software.

Integral sliding mode‐based event‐triggered nearly optimal tracking control for uncertain nonlinear systems

SummaryIn this paper, an event‐triggered nearly optimal tracking control method is investigated for a class of uncertain nonlinear systems by integrating adaptive dynamic programming (ADP) and integral sliding mode (ISM) control techniques. An ISM‐based discontinuous control law with a neural network (NN) adaptive term is designed to eliminate the influence of the uncertainties and obtain the sliding mode dynamics which is equivalent to the tracking error dynamics without uncertainties, and relax the known upper‐bounded condition of uncertainties. In order to guarantee the stability of tracking error system and the considerable optimality, under the ADP technique, a critic NN is applied to approximate the optimal value function for solving the event‐triggered Hamilton‐Jacobi‐Bellman equation and the event‐triggered nearly optimal feedback control is obtained. The feedback control law is updated and transmitted to plant only when events occur, thus both the communication and the computational resources can be saved. Furthermore, the stability of tracking error is proven thanks to Lyapunov's direct method. Finally, we provide two simulation examples to validate the developed control scheme.

Hierarchical Sliding-Mode Surface-Based Adaptive Critic Tracking Control for Nonlinear Multiplayer Zero-Sum Games via Generalized Fuzzy Hyperbolic Models

This paper investigates the hierarchical sliding-mode surface (HSMS)-based adaptive critic tracking control problem for nonlinear multiplayer zero-sum games (ZSGs). First, a generalized fuzzy hyperbolic models-based identifier is employed to approximate the unknown nonlinear functions. According to the derived data-driven model, a static control policy is proposed to transform the considered optimal tracking control problem into an optimal regulation control problem of an induced error system. Subsequently, based on sliding mode control technology and the concept of hierarchical design, a novel optimal feedback control policy is developed to regulate the tracking error by minimizing a performance index function related to the HSMS. The solution to the Hamilton-Jacobi-Isaacs equation of nonlinear multiplayer ZSGs is obtained via the HSMS-based critic neural network. Furthermore, the historical stored data is utilized to conquer the difficulty associated with the persistence of excitation conditions. Based on Lyapunov stability theory, it is strictly proven that the tracking error and the critic neural network weight are uniformly ultimately bounded. Finally, two examples are presented to demonstrate the effectiveness of the proposed control scheme.

Calibration of Viscous Damping–Stiffness Control Force in Active and Semi-Active Tuned Mass Dampers for Reduction of Harmonic Vibrations

Tuned mass dampers (TMDs) are commonly used to mitigate vibrations in civil structures. There is a growing demand for new solutions that offer similar effectiveness as TMDs but with reduced mass. In this context, this paper investigates active (ATMD) and semi-active (STMD) tuned mass dampers with relative displacement and velocity feedback. The control force of the ATMD is assumed to be the sum of viscous damping and either positive or negative stiffness forces. This control force is calibrated for a specific parameter K such that the effectiveness of the ATMD in reducing harmonic vibrations matches that of the TMD with K times larger mass. The optimal calibration is derived based on the mathematical reformulation of an existing optimal acceleration feedback control algorithm. The control approach for the ATMD is then applied to the STMD. Subsequently, the sub-optimal STMD is analyzed, with a focus on its limitations arising from the clipping of active forces. Finally, the paper presents a calibration of the STMD using a numerical optimization method. It is demonstrated that the maximum achievable performance of the numerically optimized STMD matches that of the TMD with three times larger mass.

An optimal control problem for the continuity equation arising in smart charging

This paper is focused on the mathematical modeling and solution of the optimal charging of a large population of identical plug-in electric vehicles (PEVs) with mixed state variables (continuous and discrete). A mean field assumption is formulated to describe the evolution interaction of the PEVs population. The optimal control of the resulting continuity equation of the mixed system under state constraints is investigated. We prove the existence of a minimizer. We then characterize the solution as the weak solution of a system of two coupled PDEs: a continuity equation and of a Hamilton-Jacobi equation. We provide regularity results of the optimal feedback control.

An Investigation on Existence and Optimal Feedback Control for Fractional Neutral Stochastic Evolution Hemivariational Inequalities

An Investigation on Existence and Optimal Feedback Control for Fractional Neutral Stochastic Evolution Hemivariational Inequalities

Optimal Feedback Control for the Proportion of Energy Cost in the Upper-Arm Reaching Movement.

The minimum expected energy cost model, which has been proposed as one of the optimization principles for movement planning, can reproduce many characteristics of the human upper-arm reaching movement when signal-dependent noise and the co-contraction of the antagonist's muscles are considered. Regarding the optimization principles, discussion has been mainly based on feedforward control; however, there is debate as to whether the central nervous system uses a feedforward or feedback control process. Previous studies have shown that feedback control based on the modified linear-quadratic gaussian (LQG) control, including multiplicative noise, can reproduce many characteristics of the reaching movement. Although the cost of the LQG control consists of state and energy costs, the relationship between the energy cost and the characteristics of the reaching movement in the LQG control has not been studied. In this work, I investigated how the optimal movement based on the LQG control varied with the proportion of energy cost, assuming that the central nervous system used feedback control. When the cost contained specific proportions of energy cost, the optimal movement reproduced the characteristics of the reaching movement. This result shows that energy cost is essential in both feedforward and feedback control for reproducing the characteristics of the upper-arm reaching movement.

Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.