In this study, a novel optimal tracker is developed for the linear quadratic tracking (LQT) problem using an output–feedback adaptive dynamic programming (ADP) framework. By leveraging the minimal polynomial of the exosystem matrix, we parameterize the steady-state input, state, and output, and incorporate them into the performance index of the LQT formulation. Unlike existing studies on LQT, in this study, we introduce a high-order filter to make the signals related to the high-order derivatives of the system output and input observable. Subsequently, we develop an incremental data ADP algorithm to learn the optimal dynamic output–feedback tracker, eliminate the impact of high-order filtering error and state reconstruction error on the iterative learning equation (ILE) within a specified time. Finally, the comprehensive simulation results show the effectiveness of the proposed output–feedback tracker.

When it comes to integrating inverter-based resources (IBRs) into modern grids with varying characteristics like unbalanced systems, low-inertia networks, or very weak grids, synthesizing the synchronization control method (SCM) of the IBR’s phase-locked loop can be a challenging task. This paper provides a unique solution to enhance the three-phase IBR’s SCM using the adaptive dynamic programming (ADP) method based on reinforcement learning. By making the SCM more intelligent and self-learning, IBRs can be easily integrated into diverse grids. To this end, this article investigates the synchronization process’s detailed dynamics, including all incorporating disturbances and parameters required for the first step in designing the ADP method. Afterward, this research synthesizes an optimal controller using an ADP method. It is a data-driven and practically sound approach to the problem under investigation. The new methodology is based on the adaptive optimal control employing measurement feedback to control the output regulation problem of uncertain synchronization process dynamics via the internal model principle. The proposed SCM design deploys an ADP learning methodology to tackle uncertain parameters and unknown disturbance signals to synchronize IBRs during transients, thereby enhancing IBRs’ synchronization in challenging conditions of modern power systems with unbalanced, low-inertia, and very weak grids. For comparison purposes, this paper applies a robust controller based on the well-established$\mu$synthesis approach (benefiting from the well-known$D\text{-}K$iteration process). Comparative simulations are performed; experiments are conducted to reveal the effectiveness and practicality of the ADP-based optimal SCM proposed in this paper.Note to Practitioners—As different nations strive to combat global warming and accelerate decarbonization, power and energy systems are undergoing a significant shift. Inverter-based resources are being used as an essential component to achieve these goals. However, studies have revealed that designing synchronization control methods of the inverter-based resources’ phase-locked loop in unbalanced, low-inertia, and very weak grids is challenging due to the need for accurate dynamic models and other factors. This study revisits the synchronization process’s detailed dynamics. It also proposes a novel adaptive dynamic programming strategy using intelligent self-learning approaches to the synchronization control method associated with inverter-based resources. This method utilizes an optimal control to synthesize the adaptive dynamic programming control strategy for the inverter-based resources’ synchronization process. Besides, it employs measurement feedback to control the output regulation problem of uncertain dynamics of inverter-based resources’ synchronization process via the internal model principle. As a result, this paper makes this process data-driven. It utilizes a learning methodology using adaptive dynamic programming to address uncertain parameters and unknown disturbance signals associated with the dynamics derived and formulated for the problem under investigation. Thus, the proposed method applies to controlling inverter-based resources’ synchronization process even in cases with slow parameter variations caused by different factors. It can compensate for all functional disturbance signals affecting the dynamics of the systems. In fact, unlike traditional methods that need an exact dynamic model of the inverter-based resources’ synchronization process to design and tune the controller to achieve a proper transient response, the proposed control system trains itself and does so. This study’s simulations and experiments reveal that the above points give the proposed approach a competitive edge over the existing methodologies.

This article addresses the stabilization challenges of nonholonomic systems under the threat of false data injection (FDI) attacks, which compromise the integrity of state information. A novel adaptive control strategy using Nussbaum-type gains is proposed to ensure the asymptotic stability of the closed-loop system while maintaining signal boundedness. The approach extends conventional Nussbaum designs to handle multiple unknown control directions. It integrates online learning mechanisms to mitigate the impact of FDI attacks. Additionally, adaptive backstepping and fuzzy-logic systems are utilized to approximate and compensate for unknown nonlinear dynamics. The methodology transforms nonholonomic systems into equivalent cascade structures to address inherent constraints and enable secure control input design. Simulation studies validate the effectiveness and resilience of the proposed control strategy, demonstrating significant improvements in stability and robustness in the presence of FDI attacks.

Grid modernization and large-scale integration of inverter-based resources (IBRs) into distribution systems have resulted in the development of new control strategies relying on information and communication technologies. To this end, this article proposes a distributed secondary control algorithm using an event-triggered mechanism for exchanging information among IBRs over digital communication channels in islanded modern microgrids. Unlike the existing event-triggered studies, the proposed method is based on a nonlinear mapping technique for encoding shared data over digital communication channels, making it suitable for real-world applications. It enables the control system to use digitized and encoded data instead of typical continuous analog information, resulting in the more efficient usage of communication infrastructures. As a result, it can be regarded as a practical algorithm for stabilizing voltage and frequency during the transient and steady-state response of autonomous modern microgrids considering computational constraints and the limited bandwidth of communication systems. Finally, comparative simulation studies and experimental results validate the performance and effectiveness of the proposed algorithm.

This article investigates the event-triggered cooperative adaptive optimal output regulation problem for unknown multiagent systems (MASs) under switching network. To address communication disruptions between subsystems and the leader, a distributed observer is provided to estimate the reference signals. Without using system dynamics, an event-triggered mechanism is established to reduce computation and communication costs. Then, event-triggered adaptive optimal controllers are developed by using the available input/state data. By exploiting the Lyapunov stability theory and the method of input-to-state stability (ISS), rigorous stability analysis is conducted, and conditions for MASs to achieve the leader-to-formation stability (LFS) are provided. Additionally, the sensitivity of the suboptimality index to system parameters is analyzed. Finally, an application to cooperative adaptive cruise control (CACC) is presented to validate the proposed approach.

This paper proposes a multivariable, adaptive, robust (MAR) control strategy for islanded inverter-based resources (IBRs) operating as grid-forming inverters. The proposed method is employed in the inner control loop of the primary layer in the hierarchical or decentralized structures for the islanded operation of microgrids. The MAR control scheme is responsible for stabilizing IBRs’ output voltage in autonomous operations of microgrids, considering mismatched input voltage disturbances from the grid side and a large amount of system uncertainty. The control methodology introduced in this paper does not rely on the system’s physical parameters, such as microgrid topology, load dynamics, LCL filters, and output connectors. As a result, there is no need to know the nominal values or the bounds of uncertainties in system dynamics. The MAR control method uses online adaptation rules first to identify and then adjust the control parameters of the closed-loop system based on an arbitrary dynamic model. In other words, the MAR method replaces the actual dynamics of IBRs with predetermined dynamics of interest. Simulation results in the MATLAB/Simulink environment confirm the capability of the scheme introduced for the closed-loop stabilization and voltage regulation in the presence of disturbances and a significant amount of uncertainty under various case studies; moreover, comparative simulations by comparing the presented method with other studies using sliding mode control are provided. Finally, experiments verify the effectiveness and practicality of the proposed MAR control scheme.Note to Practitioners—Inverter-based resources are integral parts of current and especially future power and energy systems; with increasing concerns about carbon footprints, the tendency to substitute traditional synchronous generators with inverter-based resources increases. This transition towards the widespread use of power electronics devices needs careful studies regarding the stability and control of power converters. Although existing studies are addressing potential control system challenges, they suffer from complex mathematical computations and the need for the system’s preliminary information. With this in mind, this study proposes a multivariable, adaptive, robust control strategy for the inner voltage control loop of grid-forming inverters. This method utilizes online estimation algorithms to identify inverter-based resources’ parameters and tune control system parameters simultaneously, making it applicable even to cases with slow parameter variations caused by aging or environmental changes. It can compensate for potential voltage disturbances from the grid side and enable the designer to replace undesirable dynamics of inverter-based resource units with arbitrary and stable dynamics of interest. In fact, unlike traditional methods that need control parameters and gains to be tuned to achieve a proper dynamic response, the control system designer can choose reference dynamics and enforce the closed-loop system to imitate the dynamical model selected. This model is usually chosen based on established priorities, such as response time and other transient behaviors. Moreover, this method does not require complex mathematical and algebraic calculations to design and implement. It can be easily applied to inverter-based resource units after selecting the desired reference dynamics, as shown through this study’s experimental result. The above points give this method a competitive edge over the existing algorithms, especially in practical applications.

This paper investigates the robust optimal control problem of a class of continuous-time, partially linear, interconnected systems. In addition to the dynamic uncertainties resulted from the interconnected dynamic system, unknown bounded disturbances are taken into account throughout the learning process, wherein the system’s dynamics and the disturbances are assumed unknown. These challenges lead the collected online data to be imperfect. In this scenario, traditional data-driven control techniques, such as adaptive dynamic programming (ADP) and robust ADP, encounter a challenge in approximating the optimal control policy precisely due to imperfect data and computational errors. In this paper, a novel data-driven robust policy iteration method is proposed to simultaneously solve the robust optimal control problems. Without relying on the knowledge of the system’s dynamics, the external disturbances or the complete state, the implementation of the proposed method only needs to access the input and partial state information. Based on the small-gain theorem, the notions of strong unboundedness observability and input-to-output stability, it is guaranteed that the learned robust optimal control gain is stabilizing and that the solution of the closed-loop system is uniformly ultimately bounded despite the existence of dynamic uncertainties and unknown external disturbances. The simulation results reveal the efficiency and practicality of the proposed data-driven control method.Note to Practitioners—This work is motivated by the use of reinforcement learning to improve the quality of designing adaptive optimal controllers for engineering applications. Adaptive dynamic programming methods, in particular policy iteration (PI), are widely used in solving optimal control problems. However, due to the iterative nature of PI, the approximated optimal control policy may be inaccurate and imprecise. Especially, when using imperfect system’s measurements instead of the modelling information. This can result in causing the learned control policy to deviate from the actual optimal policy. This becomes more challenging in the existence of dynamic uncertainties and unknown external disturbances which corrupt the measurements and result in imperfect data. This work investigates the conditions on the uncertainties such that the proposed novel data-driven PI algorithm is robust to system’s uncertainties, unknown external disturbances and imperfect measurements. The approximated robust optimal control policy performs robustly in the existences of imperfect data and uncertainties, and at the same time is close enough to the optimal control policy.

This paper studies the problem in transportation networks and introduces a novel reinforcement learning-based algorithm, namely. Different from almost all canonical sota solutions, which are usually computationally expensive and lack generalizability to unforeseen destination nodes, segac offers the following appealing characteristics. segac updates the ego vehicle’s navigation policy in a sample efficient manner, reduces the variance of both value network and policy network during training, and is automatically adaptive to new destinations. Furthermore, the pre-trained segac policy network enables its real-time decision-making ability within seconds, outperforming state-of-the-art sota algorithms in simulations across various transportation networks. We also successfully deploy segac to two real metropolitan transportation networks, namely Chengdu and Beijing, using real traffic data, with satisfying results.

As an important branch of machine learning, reinforcement learning (RL) has proved its efficiency in many emerging applications in science and engineering. A remarkable advantage of RL is that it enables agents to maximize their cumulative rewards through online exploration and interactions with unknown (or partially unknown) and uncertain environments, which is regarded as a variant of data-driven adaptive optimal control methods. However, the successful implementation of RL-based control systems usually relies on a good quantity of online data due to its data-driven nature. Therefore, it is imperative to develop data-efficient RL methods for control systems to reduce the required number of interactions with the external environment. Moreover, network-aware issues, such as cyberattacks, dropout packet and communication latency, and actuator and sensor faults, are challenging conundrums that threaten the safety, security, stability, and reliability of network control systems. Consequently, it is significant to develop safe and resilient RL mechanisms.

In this paper, a data-driven framework was designed to predict manufacturing failure. The framework includes an autoregression model with the least mean square algorithm, a linear regression model with prediction intervals for short-term and long-term failure detection, and a feature extraction model with empirical mode decomposition. The analytical results validate that the designed data-driven model is a good candidate for failure predictions in smart manufacturing processes.

This article addresses the resilient practical cooperative output regulation problem (RPCORP) for multiagent systems subjected to both denial-of-service (DoS) attacks and actuator faults. Fundamentally different from the existing solutions to RPCORPs, the system parameters considered in this article are unknown to each agent, and a novel data-driven control approach is introduced to handle such an issue. The solution starts with developing resilient distributed observers for each follower in the presence of DoS attacks. Then, a resilient communication mechanism and a time-varying sampling period are introduced to, respectively, ensure the neighbor state is available as soon as attacks disappear and to avoid targeted attacks launched by intelligent attackers. Furthermore, a model-based fault-tolerant and resilient controller is designed based on the Lyapunov approach and the output regulation theory. In order to remove the reliance on system parameters, we leverage a new data-driven algorithm to learn controller parameters via the collected data. Rigorous analysis shows that the closed-loop system can resiliently achieve practical cooperative output regulation. Finally, a simulation example is given to illustrate the effectiveness of the achieved results.

An event-triggered output feedback control approach is proposed via a disturbance observer and adaptive dynamic programming (ADP). The solution starts by constructing a nonlinear disturbance observer, which only depends on the measurement of system output. A state observer is then developed based on approximation information of system dynamics via neural networks. In order to avoid continuous transmission and reduce the communication burden in the closed-loop system, an event-triggered mechanism is introduced such that the control signal is updated only at a specific instant when a triggered condition is violated. By virtue of the disturbance observer and state observer, an output-feedback ADP control approach then is developed, where only a critic network is employed to estimate the value function. Based on the Lyapunov stability theory, the stability of the closed-loop system is rigorously analyzed, and the effectiveness of the proposed control approach is verified by two simulation examples.

This article considers the robust optimal control problem for a class of nonlinear systems in the presence of unmodeled dynamics. An adaptive optimal controller is designed using the online actor–critic learning and is robustified against unmodeled dynamics. To deal with unmodeled dynamics, an auxiliary signal with the system state as its input signal is designed to capture the input-to-state stability. In addition to the critic network for value function approximation, a novel robustifying term is developed and introduced into the actor network to ensure robustness during the learning process. It is shown that both the actor and the critic weights learning converge to their optimal values while guaranteeing the boundedness of all the signals in the closed loop. Simulation examples are conducted to verify the efficacy of the presented scheme.

This article proposes an adaptive, optimal, data-driven control approach based on reinforcement learning and adaptive dynamic programming to the three-phase grid-connected inverter employed in virtual synchronous generators (VSGs). This article takes into account unknown system dynamics and different grid conditions, including balanced/unbalanced grids, voltage drop/sag, and weak grids. The proposed method is based on value iteration, which does not rely on an initial admissible control policy for learning. Considering the premise that the VSG control should stabilize the closed-loop dynamics, the VSG outputs are optimally regulated through the adaptive, optimal control strategy proposed in this article. Comparative simulations and experimental results validate the proposed method's effectiveness and reveal its practicality and implementation.

This article investigates the optimal operational control (OOC) problem for a class of industrial systems consisting of multiple unit devices with fast dynamics and an unknown operational process with slow dynamics. First, the OOC problem is formulated as a noncascade optimal control problem of two-time-scale systems with a novel performance function. Second, using singular perturbation theory, a decentralized composite control scheme is proposed by decomposing the original optimal problem into reduced-order fast and slow subsystem problems. Then, in the framework of reinforcement learning, an online controller design method for the slow subsystem is proposed by using the online measurement, and an offline controller design for the fast subsystem is proposed by using the unit device models. The obtained decentralized composite optimal controller achieves both the desired operational index tracking and disturbance rejection without requiring the dynamics of the operational process. Different from the existing cascade design methods, the proposed approach regulates the unit devices and operational process simultaneously, as well as overcomes the potential high dimensionality and ill-conditioned numerical issues. Finally, a mixed separation thickening process and a numerical example are given to illustrate the presented results.

In order to deal with optimal setpoint tracking (OST) problems, a discounted cost function has been introduced in the existing work. However, the optimal tracking controllers developed according to the discounted cost function may not ensure asymptotic tracking and the stability of the closed-loop systems. To overcome these limitations, in this article, we propose a novel adaptive optimal control method to minimize a cost function without a discount factor. The proposed method starts from a reformulation of the infinite-horizon OST problem for linear discrete-time systems with external disturbances. We derive an algebraic Riccati equation for solving the OST problem, whose solution is uniquely determined under mild conditions. It is proved that the obtained controller accommodates the disturbance and realizes the output tracking with zero steady-state error. In the framework of reinforcement learning, a$Q$-learning algorithm is devised to learn the suboptimal control policy by using measured data. The present learning algorithm does not require that the disturbance is measurable and can be implemented completely model-free. Finally, two examples on dc motor system and F-16 aircraft plant are provided to corroborate our design methodology.

In this article, a data-driven distributed control method is proposed to solve the cooperative optimal output regulation problem of leader-follower multiagent systems. Different from traditional studies on cooperative output regulation, a distributed adaptive internal model is originally developed, which includes a distributed internal model and a distributed observer to estimate the leader's dynamics. Without relying on the dynamics of multiagent systems, we have proposed two reinforcement learning algorithms, policy iteration and value iteration, to learn the optimal controller through online input and state data, and estimated values of the leader's state. By combining these methods, we have established a basis for connecting data-distributed control methods with adaptive dynamic programming approaches in general since these are the theoretical foundation from which they are built.

In many pilot research and development (R&D) microgrid projects, engine-based generators are employed in their power systems, either generating electrical energy or being mixed with the heat and power technology. One of the critical tasks of such engine-based generation units is the frequency regulation in the islanded mode of modernized microgrid (MMG) operation; MMGs are microgrids equipped with advanced controls to address more emerging scenarios in smart grids. For having a stable and reliable MMG, we need to synthesize an optimal, robust, primary frequency controller for the islanded mode of MMG of the future. This task is challenging because of unknown mechanical parameters, occurrence of uncertain disturbances, uncertainty of loads, operating point variations, and the appearance of engine delays, and hence nonminimum phase dynamics. This article presents an innovative primary frequency control for the engine generators regulating the frequency of an islanded MMG in the context of smart grids. The proposed approach is based on an adaptive optimal output-feedback control algorithm using adaptive dynamic programming (ADP). The convergence of algorithms, along with the stability analysis of the closed-loop system, is also shown in this article. Finally, as experimental validation, hardware-in-the-loop (HIL) test results are provided in order to examine the effectiveness of the proposed methodology practically.Note to Practitioners—This article was motivated by the problem of primary frequency controls in modernized microgrids (MMGs) using engine generators, which are still one of the prime sources of regulating frequency in pilot research and development (R&D) microgrid projects. Although MMGs will be integral parts of the smart grid of the future, their primary controls in the islanded mode are not advanced enough and not considering existing theoretical challenges scientifically. Existing approaches to regulate frequency using industrially accepted methods are highly model-based and not optimal. Besides, they are not considering the nonminimum phase dynamics. These dynamics are mainly associated with the engine delays—an inherent issue of mechanical parts—for islanded microgrids. This article suggests a new adaptive optimal output-feedback control approach based on the adaptive dynamic programming (ADP) to the abovementioned problem under consideration. By using the proposed methodology, MMGs can deal with the issues mentioned earlier, which are challenging. The proposed approach is optimally rejecting uncertain disturbances (considering the load uncertainty and operating point variations) and reducing the impacts of nonminimum phase dynamics caused by the engine delay. Based on our currently available hardware-in-the-loop (HIL) device’s capability of modeling power systems’ components in real time, our HIL-based experiments demonstrate that this approach is feasible.

This article presents an event-triggered output-feedback adaptive optimal control method for continuous-time linear systems. First, it is shown that the unmeasurable states can be reconstructed by using the measured input and output data. An event-based feedback strategy is then proposed to reduce the number of controller updates and save communication resources. The discrete-time algebraic Riccati equation is iteratively solved through event-triggered adaptive dynamic programming based on both policy iteration (PI) and value iteration (VI) methods. The convergence of the proposed algorithm and the closed-loop stability is carried out by using the Lyapunov techniques. Two numerical examples are employed to verify the effectiveness of the design methodology.

This paper presents a cooperative/shared framework of the driver and his/her semi-autonomous vehicle in order to achieve desired steering performance. In particular, a copilot controller and the driver together operate and control the vehicle. Exploiting the classical small-gain theory, our proposed shared steering controller is developed independent of the unmeasurable internal states of the human driver, and only relies on his/her steering torque. Furthermore, by adopting data-driven adaptive dynamic programming and an iterative learning scheme, the shared steering controller is studied from the measurable data of the driver and the vehicle. Meanwhile, the accurate knowledge of the driver and the vehicle dynamics is unnecessary, which settles the problem of their potential parametric variations/uncertainties in practice. The effectiveness of the proposed method is validated by rigorous analysis and demonstrated by numerical simulations.

The exclusive bus lane (XBL) is one of the most popular bus transit systems in the U.S. The Lincoln Tunnel utilizes an XBL through the tunnel in the AM peak period. This paper proposes a novel data-driven cooperative adaptive cruise control (CACC) algorithm that aims to minimize a cost function for connected and autonomous buses along the XBL. Different from existing model-based CACC algorithms, the proposed approach employs the idea of reinforcement learning, which does not rely on accurate knowledge of bus dynamics. Considering a time-varying topology, where each autonomous vehicle can only receive information from preceding vehicles that are within its communication range, a distributed controller is learned real-time by online headway, velocity, and acceleration data collected from the system trajectories. The convergence of the proposed algorithm and the stability of the closed-loop system are rigorously analyzed. The effectiveness of the proposed approach is demonstrated using a well-calibrated Paramics microscopic traffic simulation model of the XBL corridor. The simulation results show that the travel time in the autonomous version of the XBL are close to the present day travel time even when the bus volume is increased by 30%.

This brief proposes a novel solution to problems related to the measurement feedback adaptive optimal output regulation of discrete-time linear systems with input time-delay. Based on reinforcement learning and adaptive dynamic programming, an approximate optimal control policy is obtained via recursive numerical algorithms using online information. Convergence proofs for the proposed algorithms are given. Notably, the exact knowledge of the plant and the exosystem is not needed. The learned control policy is only a function of retrospective input and measurement output data. Theoretical analysis and an application to a grid-connected inverter show that the proposed methodologies serve as effective tools for solving adaptive and optimal output regulation problems.

This paper proposes a novel data-driven control approach to address the problem of adaptive optimal tracking for a class of nonlinear systems taking the strict-feedback form. Adaptive dynamic programming (ADP) and nonlinear output regulation theories are integrated for the first time to compute an adaptive near-optimal tracker without any a priori knowledge of the system dynamics. Fundamentally different from adaptive optimal stabilization problems, the solution to a Hamilton-Jacobi-Bellman (HJB) equation, not necessarily a positive definite function, cannot be approximated through the existing iterative methods. This paper proposes a novel policy iteration technique for solving positive semidefinite HJB equations with rigorous convergence analysis. A two-phase data-driven learning method is developed and implemented online by ADP. The efficacy of the proposed adaptive optimal tracking control methodology is demonstrated via a Van der Pol oscillator with time-varying exogenous signals.

In this paper, a data-driven non-model-based approach is proposed for the adaptive optimal control of a class of connected vehicles that is composed of n human-driven vehicles only transmitting motional data and an autonomous vehicle in the tail receiving the broadcasted data from preceding vehicles by wireless vehicle-to-vehicle (V2V) communication devices. Considering the cases of range-limited V2V communication and input saturation, several optimal control problems are formulated to minimize the errors of distance and velocity and to optimize the fuel usage. By employing an adaptive dynamic programming technique, the optimal controllers are obtained without relying on the knowledge of system dynamics. The effectiveness of the proposed approaches is demonstrated via the online learning control of the connected vehicles in Paramics' traffic microsimulation.

Due to the popularity of social networks and human-carried/human-affiliated devices with sensing abilities, like smartphones and smart wearable devices, a novel application was necessitated recently to organize group activities by learning historical data gathered from smart devices and choosing invitees carefully based on their personal interests. We proposed a private and efficient social activity invitation framework. Our main contributions are ( 1 ) defining a novel friendship to reduce the communication/update cost within the social network and enhance the privacy guarantee at the same time; ( 2 ) designing a strong privacy-preserving algorithm for graph publication, which addresses an open concern proposed recently; ( 3 ) presenting an efficient invitee-selection algorithm, which outperforms the existing ones. Our simulation results show that the proposed framework has good performance. In our framework, the server is assumed to be untrustworthy but can nonetheless help users organize group activities intelligently and efficiently. Moreover, the new definition of the friendship allows the social network to be described by a directed graph. To the best of our knowledge, it is the first work to publish a directed graph in a differentially private manner with an untrustworthy server.

A floating gate charge-sharing circuit that can be electrically programmed for precise positive and negative voltage changes, and can be implemented in a standard CMOS VLSI process is presented. With the advantage of its long-term charge-retention, the floating gate charge-sharing circuit is suitable for providing a compact, non-volatile and high-precision analog trimming method to trim the offset voltage resulting from unavoidable mismatches in analog circuits such as op-amps and comparators.>