Non-quadratic Function Research Articles

A generalized, computationally versatile plasticity model framework - Part II: Theory and verification focusing on shear anisotropy

Shear-dominated deformation (SHDD) is pivotal in sheet metal forming; however, comprehensive modeling of plastic anisotropy in SHDD, specifically shear anisotropy considering both yield stress and plastic flow, has been inadequately addressed in existing literature. In this work, a generalized constitutive framework is introduced on the basis of stress triaxiality-dependent state variable to accurately emulate plastic anisotropy and the physics-based shear constraint in SHDD. The framework is capable to seamlessly integrate with existing yield criteria, preserving computational efficiency and versatility. Notably, the yield function, anisotropic parameters, and their optimization or analytical determination for the non-shear deformation state remain unaltered. When integrated with the Hill48 yield function, featuring either one or two anisotropic parameters within the generalized constitutive framework, precise characterization of yield strength and plastic flow in SHDD is achieved. The applicability of the framework extends to various anisotropic yield functions such as the widely employed Yld2k-2d and the sixth-order polynomial (Poly6) function as a class of associated flow rule-based yield functions, and one non-quadratic yield function for non-associated flow rule scenarios. Experimental validation with two engineering sheet metals, high-strength dual-phase steel DP980 and high-strength aluminum alloy AA7075-T6, was conducted. Comparative analyses with several recently proposed yield criteria, especially Poly6-18p, highlighted the efficiency of the proposed constitutive framework. Furthermore, this study explores intrinsic shear constraints, particularly the absence of through-thickness strains under in-plane shear stress. Additionally, it offers an enhanced description of plastic anisotropy in shear yield stress within the general framework, providing valuable insights into the complexities of SHDD.

Adaptive critic design for safety-optimal FTC of unknown nonlinear systems with asymmetric constrained-input

Safe fault tolerant control is one of the key technologies to improve the reliability of dynamic complex nonlinear systems with limited inputs, which is hard to solve and definitely a great challenge to tackle. Thus the paper presents a novel safety-optimal FTC (Fault Tolerant Control) approach for a category of completely unknown nonlinear systems incorporating actuator fault and asymmetric constrained-input, which can guarantee the system’s operation within a safe range while showcasing optimal performance. Firstly, a CBF (Control Barrier Function) is incorporated into the cost function to penalize unsafe behaviors, and then we translate the intractable safety-optimal FTC problem into a differential ZSG (Zero-Sum Game) problem by defining the control input and the actuator fault as two opposing sides. Secondly, a neural-network-based identifier is employed to reconstruct system dynamics using system data, and the resolution of handling asymmetric constrained-input with the introduced non-quadratic cost function is achieved through the design of an adaptive critic scheme, aiming to reduce computational expenses accordingly. Finally, through the theoretical stability analysis, it is demonstrated that all signals in the closed-loop system are consistently UUB (Uniformly Ultimately Bounded). Furthermore, the proposed method’s effectiveness is also verified in the simulation experiments conducted on a model of a single-link robotic arm system with actuator failure. The result shows that the algorithm can fulfill the safety-optimal demand of fault tolerant control in fault system with asymmetric constrained-input.

Dynamic Event-Triggered Safe Control for Nonlinear Game Systems With Asymmetric Input Saturation.

This article focuses on the Pareto optimal issues of nonlinear game systems with asymmetric input saturation under dynamic event-triggered mechanism (DETM). First, the safe control is guaranteed by transforming the system with safety constraints into the one without state constraints utilizing barrier function. The united cost function integrating nonquadratic utility function is constructed to provide the foundation to achieve the Pareto optimal solutions. Then, the adaptive dynamic programming method with concurrent learning is proposed to approximate the Pareto optimal strategies wherein both current and historical data are utilized. To further lessen the consumptions of computation/communication resources, the DETM is integrated into the adaptive algorithm framework which can avoid Zeno phenomena. All the signals of the closed-loop system are proved to be uniformly ultimately bounded. Finally, the simulation results are given to validate the effectiveness of the proposed method from several aspects.

Adaptive Dynamic Programming for Robust Event-Driven Tracking Control of Nonlinear Systems With Asymmetric Input Constraints.

This article considers the robust dynamic event-driven tracking control problem of nonlinear systems having mismatched disturbances and asymmetric input constraints. Initially, to tackle the asymmetric constraints, a novel nonquadratic value function is constructed for the original system. This makes the asymmetrically constrained tracking control problem transformed into an unconstrained optimal regulation problem. Then, a dynamic event-driven mechanism is proposed. Meanwhile, the event-driven Hamilton-Jacobi-Bellman equation (ED-HJBE) is developed for the optimal regulation problem in order to acquire the optimal control with distinctly decreased computational burden. To solve the ED-HJBE, a single critic neural network (CNN) is designed in the adaptive dynamic programming framework. Meanwhile, the gradient descent method is employed to update the CNN's weights. After that, both the weight estimation error and the tracking error are proved to be uniformly ultimately bounded via Lyapunov's direct method. Finally, simulations of the spring-mass-damper system and the pendulum plant are separately utilized to validate the established theoretical claims.

Distributed Optimal Attitude Synchronization Control of Multiple QUAVs via Adaptive Dynamic Programming.

This article proposes a distributed optimal attitude synchronization control strategy for multiple quadrotor unmanned aerial vehicles (QUAVs) through the adaptive dynamic programming (ADP) algorithm. The attitude systems of QUAVs are modeled as affine nominal systems subject to parameter uncertainties and external disturbances. Considering attitude constraints in complex flying environments, a one-to-one mapping technique is utilized to transform the constrained systems into equivalent unconstrained systems. An improved nonquadratic cost function is constructed for each QUAV, which reflects the requirements of robustness and the constraints of control input simultaneously. To overcome the issue that the persistence of excitation (PE) condition is difficult to meet, a novel tuning rule of critic neural network (NN) weights is developed via the concurrent learning (CL) technique. In terms of the Lyapunov stability theorem, the stability of the closed-loop system and the convergence of critic NN weights are proved. Finally, simulation results on multiple QUAVs show the effectiveness of the proposed control strategy.

Integral reinforcement learning‐based optimal tracking control for uncertain nonlinear systems under input constraint and specified performance constraints

AbstractThis article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

Adaptive sampling artificial-actual control for non-zero-sum games of constrained systems

Considering physical constraints encountered by actuators, this paper addresses the non-zero-sum game of continuous nonlinear systems with symmetric and asymmetric input constraints through aperiodic sampling artificial-actual control. Initially, the artificial system built by the improved Elman dynamic neural networks (EDNNs) has artificial-actual interaction with the physical system, which provides a new perspective for predicting the system state. By constantly learning and adjusting parameters, EDNNs can gradually approximate the dynamic behavior of the real system to achieve more effective control. Aiming at accommodating diverse input constraints, the non-quadratic value function constructed from a smoothly bounded function is devised. Then, the polynomial parameterized adaptive dynamic programming (ADP) is employed to approximate the solution of the coupled Hamilton–Jacobi equation (HJE), deriving optimal control laws for two players. To improve the efficiency of data communication, three adaptive sampling mechanisms including event-triggered mechanism (ETM) with relative threshold, dynamic ETM (DETM) and self-triggered mechanism (STM) are introduced in turn during the iterative learning process of control sequences. DETM further extends sampling intervals by incorporating internal dynamic variables, while STM determines the next trigger time through soft calculation without hardware monitoring. All three trigger modes can ensure the system stability while avoiding the Zeno phenomenon, and relevant proofs are given. Finally, the simulation validates the effectiveness of the designed algorithm and highlights the unique characteristics of each trigger mode.

Optimal Spin Polarization Control for the Spin-Exchange Relaxation-Free System Using Adaptive Dynamic Programming.

This work is the first to solve the 3-D spin polarization control (3DSPC) problem of atomic ensembles, which controls the spin polarization to achieve arbitrary states with the cooperation of multiphysics fields. First, a novel adaptive dynamic programming (ADP) structure is proposed based on the developed multicritic multiaction neural network (MCMANN) structure with nonquadratic performance functions, as a way to solve the multiplayer nonzero-sum game (MP-NZSG) problem in 3DSPC under the constraints of asymmetric saturation inputs. Then, we utilize the MCMANNs to implement the multicritic multiaction ADP (MCMA-ADP) algorithm, whose convergence is proven by the compression mapping principle. Finally, the MCMA-ADP is deployed in the spin-exchange relaxation-free (SERF) system to provide a set of control laws in 3DSPC that fully exploits the multiphysics fields to achieve arbitrary spin polarization states. Numerical simulations support the theoretical results.

A continuum approach to multiaxial high-cycle fatigue modeling for ductile metallic materials

In this work we propose a generalization of the endurance function employed in the continuum approach to high-cycle fatigue modeling by Ottosen et al. (2008). The von Mises-type backstress-modified effective stress term of the original formulation is replaced by the non-quadratic Hershey-Hosford function, which introduces flexibility with respect to the predicted fatigue limit in pure shear. While the original model assumes a fixed ratio between the fatigue limits in fully reversed torsion and fully reversed bending of 0.5774 for all materials, the modification enables the ratio to take on values on the interval from 0.5 to 0.5852 by calibration to experimental data that is readily available in the literature. Like the original formulation simplifies to the invariant-based Mises-Sines fatigue limit criterion for proportional stress histories, the current proposal can be written as a Hershey-Hosford-Sines criterion. A systematic validation to experimental multiaxial infinite fatigue life data from eight different mild and semi-hard metals with fully reversed torsion and bending fatigue limit ratios lower than 0.6 demonstrates that the proposed modification increases the accuracy and reduces the number of non-conservative predictions.

Adaptive Virotherapy Strategy for Organism With Constrained Input Using Medicine Dosage Regulation Mechanism.

In this article, the constrained adaptive control strategy based on virotherapy is investigated for organism using the medicine dosage regulation mechanism (MDRM). First, the tumor-virus-immune interaction dynamics is established to model the relations among the tumor cells (TCs), virus particles, and the immune response. The adaptive dynamic programming (ADP) method is extended to approximately obtain the optimal strategy for the interaction system to reduce the populations of TCs. Due to the consideration of asymmetric control constraints, the nonquadratic functions are proposed to formulate the value function such that the corresponding Hamilton-Jacobi-Bellman equation (HJBE) is derived which can be deemed as the cornerstone of ADP algorithms. Then, the ADP method of a single-critic network architecture which integrates MDRM is proposed to obtain the approximate solutions of HJBE and eventually derive the optimal strategy. The design of MDRM makes it possible for the dosage of the agentia containing oncolytic virus particles to be regulated timely and necessarily. Furthermore, the uniform ultimate boundedness of the system states and critic weight estimation errors is validated by Lyapunov stability analysis. Finally, simulation results are given to show the effectiveness of the derived therapeutic strategy.

Asymmetric Constrained Optimal Tracking Control With Critic Learning of Nonlinear Multiplayer Zero-Sum Games.

By utilizing a neural-network-based adaptive critic mechanism, the optimal tracking control problem is investigated for nonlinear continuous-time (CT) multiplayer zero-sum games (ZSGs) with asymmetric constraints. Initially, we build an augmented system with the tracking error system and the reference system. Moreover, a novel nonquadratic function is introduced to address asymmetric constraints. Then, we derive the tracking Hamilton-Jacobi-Isaacs (HJI) equation of the constrained nonlinear multiplayer ZSG. However, it is extremely hard to get the analytical solution to the HJI equation. Hence, an adaptive critic mechanism based on neural networks is established to estimate the optimal cost function, so as to obtain the near-optimal control policy set and the near worst disturbance policy set. In the process of neural critic learning, we only utilize one critic neural network and develop a new weight updating rule. After that, by using the Lyapunov approach, the uniform ultimate boundedness stability of the tracking error in the augmented system and the weight estimation error of the critic network is verified. Finally, two simulation examples are provided to demonstrate the efficacy of the established mechanism.

Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty

Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty

Adaptive optimal control of switched linear systems with input constraints

In this paper, an adaptive optimal control scheme is proposed for a class of switched linear systems with input constraints and unknown dynamics. Firstly, in the case that the system dynamics is known, the optimal controller and optimal switching condition are determined by introducing a non-quadratic cost function and employing the embedding transformation technique. After that, an adaptive algorithm is established for estimating the unknown dynamics of the switched linear systems. Then, when the system dynamics is unknown, the adaptive optimal controller and optimal switching condition are deduced by resorting to the certainty equivalent principle. A common Lyapunov function is constructed, by using which, it is proved that the closed-loop system is globally uniformly asymptotically stable. Simulations are finally conducted to further illustrate the effectiveness of the proposed method.

ADP-Based Event-Triggered Constrained Optimal Control on Spatiotemporal Process: Application to Temperature Field in Roller Kiln.

The precise control of the spatiotemporal process in a roller kiln is crucial in the production of Ni-Co-Mn layered cathode material of lithium-ion batteries. Since the product is extremely sensitive to temperature distribution, temperature field control is of great significance. In this article, an event-triggered optimal control (ETOC) method with input constraints for the temperature field is proposed, which takes up an important position in reducing the communication and computation costs. A nonquadratic cost function is adopted to describe the system performance with input constraints. First, we present the problem description of the temperature field event-triggered control, where this field is described by a partial differential equation (PDE). Then, the event-triggered condition is designed according to the information of system states and control inputs. On this basis, a framework of the event-triggered adaptive dynamic programming (ETADP) method that is based on the model reduction technology is proposed for the PDE system. A critic network is used to approach the optimal performance index by a neural network (NN) together with that an actor network is used to optimize the control strategy. Furthermore, an upper bound of the performance index and a lower bound of interexecution times, as well as the stabilities of the impulsive dynamic system and the closed-loop PDE system, are also proved. Simulation verification demonstrates the effectiveness of the proposed method.

A new Q‐function structure for model‐free adaptive optimal tracking control with asymmetric constrained inputs

SummaryThis article aims to design a model‐free adaptive tracking controller for discrete‐time nonlinear systems with unknown dynamics and asymmetric control constraints. First, a new Q‐function structure is designed by introducing the control input into the tracking error of the next moment, in order to eliminate the final tracking error, avoid the steady control, and ignore the discount factor. Second, via system transformation, a general performance index is developed to overcome the challenge caused by asymmetric constraints of implicit control inputs. By this operation, the constrained tracking problem is converted to an unconstrained optimal tracking problem without the traditional nonquadratic performance function that is only applicable to explicit control inputs. Then, a value‐iteration‐based Q‐learning (VIQL) algorithm is derived to seek the optimal Q‐function and the optimal control policy by using offline data rather than the mathematical model. Next, the convergence, monotonicity, and stability properties of VIQL are investigated to demonstrate that the iterative Q‐function sequence can converge to the optimal Q‐function under ideal conditions. To realize the VIQL algorithm, the critic neural network is employed to approximate the Q‐function. Finally, simulation results and comparative experiments are conducted to demonstrate the validity and effectiveness of the present VIQL scheme.

Optimized convergence of stochastic gradient descent by weighted averaging

Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the arithmetic mean of all iterates converges considerably slower to the optimal solution than the iterates themselves. And also in the presence of noise, when a termination of the stochastic gradient method after a finite number of steps is considered, the arithmetic mean is not necessarily the best possible approximation to the unknown optimal solution. This paper aims at identifying optimal strategies in a particularly simple case, the minimization of a strongly convex function with i. i. d. noise terms and termination after a finite number of steps. Explicit formulas for the stochastic error and the optimality error are derived in dependence of certain parameters of the SGD method. The aim was to choose parameters such that both stochastic error and optimality error are reduced compared to arithmetic averaging. This aim could not be achieved; however, by allowing a slight increase of the stochastic error it was possible to select the parameters such that a significant reduction of the optimality error could be achieved. This reduction of the optimality error has a strong effect on the approximate solution generated by the stochastic gradient method in case that only a moderate number of iterations is used or when the initial error is large. The numerical examples confirm the theoretical results and suggest that a generalization to non-quadratic objective functions may be possible.

Decentralized Event-Triggered Tracking Control for Unmatched Interconnected Systems via Particle Swarm Optimization-Based Adaptive Dynamic Programming.

The problem of the large-scale interconnected system (LSIS) control is prevalent in practical engineering and is becoming increasingly complex. In this article, we propose a novel decentralized event-triggered tracking control (ETTC) strategy for a class of continuous-time nonlinear LSIS with unmatched interconnected terms and asymmetric input constraints. First, auxiliary subsystems are established to address the unmatched cross-linking terms. Next, the dynamics states of the tracking error and the exosystem are combined to construct a nominal augmented subsystem. By employing a nonquadratic performance function, the input-constrained decentralized tracking control problem is transformed into an optimal control problem for the nominal augmented subsystem. A group of independent parameters and event-triggered conditions are designed to save communication bandwidth and computational resources. Subsequently, the critic-only adaptive dynamic programming (ADP) method is used to solve the Hamilton-Jacobi-Bellman equation (HJBE) associated with the optimal control problem. To improve training success rate, the weights of the critic neural network (NN) are updated by introducing a particle swarm optimization algorithm (PSOA). The tracking error and the NN weights are proved to be uniformly ultimately bounded (UUB) under the proposed ETTC by using the Lyapunov extension theorem. Finally, the simulation example of an unmatched interconnected system is provided to verify the validity of the proposed decentralized method.

Observer-Based Consensus Control for MASs With Prescribed Constraints via Reinforcement Learning Algorithm.

In this article, an adaptive optimal consensus control problem is studied for multiagent systems (MASs) with external disturbances, unmeasurable states, and prescribed constraints. First, by using neural networks (NNs), a composite observer is constructed to estimate the unmeasurable states and disturbances simultaneously. Then, the consensus error is guaranteed within a prescribed boundary by presenting an improved prescribed performance control (PPC) technique, and the initial conditions for the error are eliminated. In addition, the updating laws of actor-critic NNs are established by using a simplified reinforcement learning (RL) algorithm based on the uniqueness of optimal solution, and the asymmetric input saturation is resolved by designing auxiliary system instead of using nonquadratic cost functions in other optimal control methods. Finally, the boundedness of all signals in the closed-loop system is proved by using Lyapunov stability theory. The effectiveness of the proposed control method is verified by a simulation example.

Reinforcement Learning for Robust Dynamic Event-Driven Constrained Control.

We consider a robust dynamic event-driven control (EDC) problem of nonlinear systems having both unmatched perturbations and unknown styles of constraints. Specifically, the constraints imposed on the nonlinear systems' input could be symmetric or asymmetric. Initially, to tackle such constraints, we construct a novel nonquadratic cost function for the constrained auxiliary system. Then, we propose a dynamic event-triggering mechanism relied on the time-based variable and the system states simultaneously for cutting down the computational load. Meanwhile, we show that the robust dynamic EDC of original nonlinear-constrained systems could be acquired by solving the event-driven optimal control problem of the constrained auxiliary system. After that, we develop the corresponding event-driven Hamilton-Jacobi-Bellman equation, and then solve it through a unique critic neural network (CNN) in the reinforcement learning framework. To relax the persistence of excitation condition in tuning CNN's weights, we incorporate experience replay into the gradient descent method. With the aid of Lyapunov's approach, we prove that the closed-loop auxiliary system and the weight estimation error are uniformly ultimately bounded stable. Finally, two examples, including a nonlinear plant and the pendulum system, are utilized to validate the theoretical claims.

Fully Distributed Optimal Economic Dispatch for Microgrids under Directed Communication Networks Considering Time Delays

Distributed generation and demand-side management are expected to play a more prominent role in future power systems. However, the increased number of generations and load demands pose new challenges to optimal energy management in a microgrid. In this paper, an economic dispatch model for microgrids considering Traditional Generators (TGs), energy storage units, wind turbines (WTs), and flexible loads is established. To tackle the Economic Dispatch Problem (EDP) over directed communication networks, a fully distributed algorithm developed by leveraging a two-step state information exchange mechanism is proposed. In addition, by employing a fixed stepsize, the proposed algorithm demonstrates rapid convergence. Furthermore, our algorithm is well-suited for nonquadratic convex cost functions. Subsequently, we extend our algorithm to address imperfect communication scenarios. Even in the presence of arbitrarily large yet bounded time delays, our algorithm exhibits robustness. Finally, several numerical examples are given to verify the correctness and effectiveness of the developed results.