Output Tracking Error Research Articles

Robust bilinear tracking control of a parabolic trough solar collector via saturation

This paper investigates the problem of robust tracking control of heat transport in a Parabolic Trough Solar Collector (PTSC), where the output has to track a desired reference trajectory. In this work, the PTSC is modeled by state-space bilinear dynamics. The manipulated variable is the pump volumetric flow rate, and the source term, i.e., solar irradiance, is assumed to be unmeasured. In addition, the actuator’s physical constraints induce saturation bounds on the manipulated variable and need to be considered explicitly in the controller design. To deal with these challenges, we first propose a saturated state-feedback law that meets the control objectives. Then, we reconstruct the unknown time-varying source term using an adaptive estimator. Later, through Lyapunov stability analysis, we prove that the closed-loop system and the output tracking error are uniformly ultimately stable. Numerical simulations attest to the performance of the proposed control strategy.

Semi‐global output regulation of continuous‐time linear systems subject to input saturation via output feedback: A model‐free method

SummaryThis article considers the output regulation problem of continuous‐time linear systems subject to input saturation in the scenario that the system dynamics are unknown. Based on the low‐gain technique and the output feedback technique, we construct an output feedback control law to solve the output regulation problem with input saturation. Since the system dynamics are unknown, we estimate the feedback and feedforward terms in the control law with a two‐step data‐driven method. An output feedback adaptive dynamic programming algorithm is proposed to estimate the feedback term. An online updating algorithm based on output tracking error is proposed to estimate the feedforward term directly without solving the output regulation equations. Both algorithms don't rely on the measurement of the internal states of the exosystem and the original system. We show that the input saturation caused by the iterative feedforward gain matrix in the online updating algorithm doesn't affect the convergence of the algorithm. With these two data‐driven algorithms, the control law doesn't rely on the dynamics anymore. The low‐gain control law obtained by the model‐free method prevents input saturation from occurring and hence solves the output regulation problem. Finally, a numerical example is provided to validate the theoretical results.

Output tracking of coupled stochastic strict-feedback higher-order nonlinear systems under pinning control

This paper focuses on the tracking problem for a class of coupled stochastic strict-feedback higher-order nonlinear systems under pinning control. Compared with existing stochastic nonlinear tracking designs, our approach stands out by simultaneously considering coupling and higher-order characteristics for the first time. By utilizing the backstepping technique and designing virtual controllers, we obtain actual controllers that ensure the closed-loop system has an almost surely unique solution and is bounded in probability. Additionally, the fourth-moment of the output tracking error can be rendered arbitrarily small in the long run. Notably, to address the cost implications of imposing a controller on each node, we employ a pinning control strategy to selectively control a portion of the nodes. Moreover, a global Lyapunov function is derived combining Lyapunov method with graph theory. Eventually, a numerical simulation is shown to verify the theoretical results.

Robust dynamic output feedback predictive fault‐tolerant control for discrete uncertain systems

AbstractA robust predictive fault‐tolerance control method integrating dynamic output feedback is proposed for discrete industrial processes with unmeasurable states, actuator faults, uncertainties, and external disturbances. First, a new iterative error model is developed, which extends the output tracking error to increase the freedom of controller regulation and ensure the tracking performance of the system. Second, a robust dynamic output feedback predictive fault‐tolerant controller is designed based on the preceding model. According to the Lyapunov stability theory, the robust stability criterion of the system is performed in the case of actuator faults, and the stability conditions of the system in the form of linear matrix inequality are given. The real‐time optimal control law gains are obtained by solving the stability conditions online. This method can effectively suppress the effects of actuator faults, uncertainties and external disturbances on the system with unmeasurable states, which can guarantee the tracking performance and stability of the system. The effectiveness of the proposed method is finally verified by using an injection molding process as an example.

Performance-Based Distributed Control of Multiagent Systems: A Dual Phase Approach.

In this article, we investigate the distributed tracking control problem for networked uncertain nonlinear strict-feedback systems with unknown time-varying gains under a directed interaction topology. A dual phase performance-guaranteed approach is established. In the first phase, a fully distributed robust filter is constructed for each agent to estimate the desired trajectory with prescribed performance such that the control directions of all agents are allowed to be nonidentical. In the second phase, by establishing a novel lemma regarding Nussbaum function, a new adaptive control protocol is developed for each agent based on backstepping technique, which not only steers the output to track the corresponding estimated signal asymptotically with arbitrarily prescribed transient response but also extends the application scope of the proposed control scheme largely since the unknown control gains are allowed to be time-varying and even state-dependent. In such a way, the underlying problem is tackled with the output tracking error converging into an arbitrarily preassigned residual set exhibiting an arbitrarily predefined convergence rate. Besides, all the internal signals are ensured to be semi-globally ultimately uniformly bounded (SGUUB). Finally, two examples are provided to illustrate the effectiveness of the co-designed scheme.

Robust predictive fault-tolerant control under actuator random failures

A robust predictive fault-tolerant switching control method is presented for a class of industrial processes that are prone to random actuator failures. The primary contribution is the combination of stochastic control theory with robust predictive control to represent the occurrence of system failure in the form of probability, thus significantly enhancing the conventional fault-tolerant control approach. First, a state-space model of the system is established, which is then transformed into an enhanced form that includes state deviations and output tracking errors. This augmented model enables the formulation of fault-tolerant switching control laws with fault probabilities, ensuring system convergence and tracking capabilities. Subsequently, linear matrix inequality constraints are given to find the switching controller gains under recovery and fault probabilities. Under certain probabilities, failure triggers fault-tolerant control, while recovery prompts a shift back to conventional control. Final, we validate the feasibility of our approach with a case study involving pressure regulation.

Adaptive fuzzy tracking control for multiple‐input multiple‐output nonlinear systems with sensor faults and experiment in the flexible‐joint robot

AbstractThis paper is centered on the adaptive asymptotic tracking control of multi‐input multi‐output uncertain strict‐feedback nonlinear systems in the presence of both multiplicative and additive sensor faults. Through adaptive accommodation techniques, the proposed control scheme can dynamically compensate for sensor faults, thereby ensuring precise and stable control. The control scheme is devised using the backstepping design that incorporates positive integrable time‐varying functions, thereby ensuring the boundedness of all signals in the closed‐loop system and guaranteeing that the output tracking error asymptotically converges to zero. Additionally, extensive simulations and experimental evaluations conducted on the Quanser platform with 2‐link flexible‐joint robots demonstrate the efficacy of the proposed control scheme in mitigating the impact of sensor faults and achieving accurate tracking performance.

Adaptive predefined-time tracking control of uncertain permanent-magnet synchronous motor drive chaotic system

A predefined-time tracking control algorithm is formulated for the Permanent Magnet Synchronous Motor(PMSM) driver system subjected to chaotic oscillations and undeterministic parameters, which can determine the upper bound of settling-time by controller gains. Radial Basis Function Neural Networks(RBFNNs) are employed to approximate the unknown nonlinear functions as universal approximators. The innovative predefined-time tracking differentiators are constructed to estimate the derivatives of the virtual inputs, resolving “the explosion of complexity” in traditional backstepping method. Then, an adaptive predefined-time controller is proposed in the backstepping framework. Lyapunov’s theoretical analysis proves that all signals in the closed-loop systems are uniformly ultimately bounded and the output tracking errors can converge to a small neighborhood of the origin within the predefined-time. Finally, simulation studies are performed to demonstrate that the proposed control scheme can quash the chaotic behaviors of the PMSM drive system and ensure fast-tracking performance even under unknown parameters.

Novel quantized iterative learning control based on spherical polar coordinates

AbstractThis study investigates the iterative learning control method for discrete‐time systems with data quantization, and it employs a quantizer based on spherical polar coordinates. The quantizer uses spherical polar coordinates to transform an unquantized signal into one with a predetermined quantization level. The quantization capability is dependent on the size of the support sphere, which is dynamically adjustable. In the scenarios of system output quantization, general quantization, and tracking error quantization, the learning control schemes are developed with consideration of the smallest sphere quantization features. The radii of the support spheres are designed independently based on the characteristics of the quantized signals (system output, control input, tracking error). The findings suggest that the system can achieve a bounded error convergence in the scenarios of system output and general quantization. In the context of quantization in tracking error, it is achievable to attain tracking performance with zero error. The example of a permanent magnet synchronous motor is provided to illustrate the findings of the study.

Command filter based input quantized adaptive tracking control for multi‐input and multi‐output non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints

SummaryThis paper addresses the problem of adaptive tracking control for multi‐input and multi‐output (MIMO) non‐strict feedback systems with unmodeled dynamics and full state time‐varying constraints. To tackle the interference of unmodeled dynamics, the dynamic signal generated by the auxiliary system is used. Hyperbolic tangent function is used as a nonlinear mapping tool to transform the constrained system into an unconstrained one. Hysteresis quantizer is introduced to mitigate the chattering phenomenon and quantization error in the quantization signal. The derivative of virtual signal can be approximated more efficiently by command filter. Furthermore, an error compensation mechanism is established to mitigate the error introduced by the command filter. Unknown nonlinear functions are approximated by radial basis function neural networks (RBFNNs). Stability analysis of the proposed controller is performed through the Lyapunov stability theory and the output tracking error can be constrained within a specified range. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method.

Research on Maximum Power Control of Direct-Drive Wave Power Generation Device Based on BP Neural Network PID Method

Ocean wave energy is a new type of clean energy. To improve the power generation and wave energy conversion efficiency of the direct-drive wave power generation system, by addressing the issue of large output errors and poor system stability commonly associated with the currently used PID (proportional, integral, and derivative) control methods, this paper proposes a maximum power control method based on BP (back propagation) neural network PID control. Combined with Kalman filtering, this method not only achieves maximum power tracking but also reduces output ripple and tracking error, thereby enhancing the system’s control quality. This study uses a permanent magnet linear generator as the power generation device, establishes a system dynamics model, and predicts the main frequency of irregular waves through the Fast Fourier Transform method. It designs a desired current tracking curve that meets the maximum power strategy. On this basis, a comparative analysis of the control accuracy and stability of three control methods is conducted. The simulation results show that the BP neural network PID control method improves power generation and exhibits better accuracy and stability.

Almost sure and mean square convergence of iterative learning control system with state feedback under random packet dropouts

This paper investigates the almost sure and mean square convergence of a state feedback-based iterative learning control scheme for a class of discrete-time multi-input–multi-output linear systems with a direct feedback term in the presence of random data dropouts. Firstly, the realisability of the control system is considered. It is shown that the realisability of the control system is only determined by the direct feedback matrix, which has nothing to do with the input–output coupling matrix, and the direct feedback matrix to be full-row rank is enough to guarantee the system realisable. Secondly, the convergence of output tracking errors is analysed in both the almost sure and mean square senses, respectively. By the contraction mapping method combined with matrix decomposition and transformation techniques, it is strictly proved that the output sequence can converge to any given desired trajectory in the sense of almost sure and mean square, respectively, if and only if the general spectral radius condition is satisfied. Finally, an example is given to show the correctness of the theoretical analysis.

Robust predictive minmax fault‐tolerant control for multi‐phase batch processes under partial actuator faults and asynchronous switching

AbstractIn view of the uncertainties, partial actuator faults, asynchronous switching as well as unknown external interference in multi‐phase batch processes, a robust predictive minmax fault‐tolerant control approach is presented. First, considering that a controller and a system state cannot switch synchronously during switching, a match‐and‐mismatch switching model is constructed. Then, an output tracking error is introduced to establish an equivalent extended switching model. In this extended model, the robust control issue is transformed into the minmax optimization issue by establishing the minmax performance index. On this basis, the control laws with control input and interference input are designed. Second, sufficient conditions are given to assure the system separately has asymptotic and exponential stabilities in each phase and batch. By calculating the given sufficient conditions online, the gains of the control laws, and the minimum and maximum operating time under match and mismatch cases are obtained. With the maximum operating time, the controller receives the switching signal beforehand to eliminate the mismatch case of switching. Additionally, a robust criterion is supplied to confirm the system's robustness. Finally, a simulation study of an injection molding process shows that the approach is valid and feasible.

Segmented hybrid event‐triggered control for underactuated autonomous underwater vehicles with an asymmetrical prescribed performance constraint

AbstractIn this technical article, the trajectory‐tracking problem of event‐based adaptive prescribed performance control for underactuated autonomous underwater vehicles (AUVs) is considered. The primary innovation of this article is the proposal of a segmented event‐triggered mechanism (SETM) that incorporates preselected convergence time. This mechanism allows for the regulation of communication frequency in the control channel (controller‐to‐actuator) according to the dynamic characteristics of the AUV system at different stages. In addition to presenting new design scheme based on SETM, an enhanced vision known as segmented hybrid event‐triggered mechanism (SHETM) is introduced. Notably, the minimum and maximum triggering intervals (MITI and MATI) for these two ETMs can be calculated from the corresponding resettable dynamic variables presented in the trigger conditions. Subsequently, by utilizing the proposed asymmetrical prescribed performance control (APPC) strategy, the system's output tracking error behaviors in both transient and steady‐state stages can be qualitatively predetermined through design parameters within the boundary function. Moreover, the minimum learning parameter (MLP) based RBFNN adaptive control law is developed to counteract the effects of model uncertainties and ocean current perturbations. Finally, rigorous theoretical analysis and simulation results confirm the viability of the proposed scheme.

Prescribed performance under input saturation for uncertain strict-feedback systems: A switching control approach

We consider the problem of achieving prescribed performance metrics on the output tracking error, for uncertain strict-feedback systems, while satisfying inelastic control input constraints in the form of magnitude saturation. Inspired by the prescribed performance control (PPC) methodology, we propose a switching control strategy comprised of two operational units: a traditional PPC and a Safety Controller. The former enforces the output error to evolve strictly inside a user-defined performance envelope that incorporates all the desired performance characteristics. However, in the presence of input saturation this envelope can be easily violated, leading to PPC instability. On the other hand, the Safety Controller enables the safe escaping, thus avoiding the appearance of this internal instability phenomenon. We prove that the closed-loop system solution is complete and non-zeno. Furthermore, we show that the tracking error evolves inside a modified performance envelope, guaranteeing a prescribed minimum convergence rate and a controlled maximum steady-state error. Simulation studies clarify and verify the theoretical findings.

Adaptive tracking control of nonlinear systems with unknown time‐varying parameters via a monotonically increasing function

AbstractThis article considers the adaptive tracking control problem of a class of strict feedback nonlinear systems with unknown time‐varying parameters. Note that the recent adaptive method, that is, congelation of variables, is built upon the assumption that the known basis functions are zeros as zero states. To remove this assumption, a novel monotonically increasing function is designed to transform the output tracking error and the backstepping tracking errors into increasing forms, which is integrated into the congelation of variables to design the ‐modification adaptive tracking control law. It is noted that the proposed form of monotonically increasing function is general, which includes positive proportional functions, exponential functions, and logarithmic functions and so on. To yield the boundedness of increasing tracking errors, a barrier Lyapunov function is designed. The asymptotic convergence of the resulting adaptive tracking system is analyzed in detail. Simulations demonstrate the effectiveness of the proposed method.

Adaptive sensor fault tolerant control with prescribed performance for unmanned autonomous helicopter based on neural networks

PurposeThis paper aims to study a neural network-based fault-tolerant controller to improve the tracking control performance of an unmanned autonomous helicopter with system uncertainty, external disturbances and sensor faults, using the prescribed performance method.Design/methodology/approachTo ensure that the tracking error satisfies the prescribed performance, the authors adopt an error transformation function method. A control scheme based on the neural network and high-order disturbance observer is designed to guarantee the boundedness of the closed-loop system. A simulation is performed to prove the validity of the control scheme.FindingsThe developed adaptive fault-tolerant control method makes the system with sensor fault realize tracking control. The error transformation function method can effectively handle the prescribed performance requirements. Sensor fault can be regarded as a type of system uncertainty. The uncertainty can be approximated accurately using neural networks. A high-order disturbance observer can effectively suppress compound disturbances.Originality/valueThe tracking performance requirements of unmanned autonomous helicopter system are considered in the design of sensor fault-tolerant control. The inequality constraint that the output tracking error must satisfy is transformed into an unconstrained problem by introducing an error transformation function. The fault state of the velocity sensor is considered as the system uncertainty, and a neural network is used to approach the total uncertainty. Neural network estimation errors and external disturbances are treated as compound disturbances, and a high-order disturbance observer is constructed to compensate for them.

Command filter-based event-triggered control for stochastic MEMS gyroscopes with finite-time prescribed performance

This paper proposes an adaptive neural control strategy for stochastic microelectromechanical system (MEMS) gyroscopes, aiming to achieve a prescribed performance in a finite time. The radial basis function neural network is introduced to address the system’s unknown nonlinear dynamics and stochastic disturbances. Then, the technology of finite-time prescribed performance function, along with the method of command-filtered backstepping design, is utilized to ensure both transient and steady-state performance and simultaneously solve the problem of "explosion of complexity." Moreover, a switching threshold event-triggered control law is proposed to cut down on communication resources and eliminate corresponding parametric inequality restrictions. The proposed adaptive state feedback control strategy is able to guarantee that the output tracking error converges to a prescribed, arbitrarily small residual set. Additionally, the closed-loop system’s signals can be semi-globally ultimately uniformly bounded in probability. Finally, numerical simulations demonstrate the effectiveness and superiority of the proposed strategy.

Funnel-Based Adaptive Neural Fault-Tolerant Control for Nonlinear Systems with Dead-Zone and Actuator Faults: Application to Rigid Robot Manipulator and Inverted Pendulum Systems

This study addresses an adaptive neural funnel fault-tolerant control problem for a class of strict-feedback nonlinear systems with actuator faults and input dead zone. To guarantee the boundedness of the tracking error, a modified transformation for funnel error is devised and incorporated into the control design process. To manage unknown nonlinear functions, radial basis function neural networks (RBFNN) are employed in designing an adaptive neural funnel fault-tolerant controller through the backstepping technique. The proposed controller guarantees the output tracking error stays within a predefined funnel, and all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, simulations of a rigid robot manipulator system and an inverted pendulum system are conducted to validate the practicality and effectiveness of the proposed control method.

Event-driven prescribed performance distributed formation control for MSVs based on second-order step response with actuator input saturation

This study aims to investigate the problem of event-driven distributed formation control for marine surface vehicles (MSVs) system, including prescribed performance constraint and actuator input saturation. By constructing a novel prescribed performance function based on second-order step response, the intuitive physical significance is realized, and the performance specifications imposed on the output tracking errors are achieved. Subsequently, a double-layer adaptive sliding-mode disturbance observer (ASMDO) is presented to compensate for lumped uncertain terms. Within the controller design stage, an auxiliary dynamic system is built to introduce the auxiliary intermediate control law, thus avoiding the need to directly design anti-saturation generalized control input for the MSVs system. Then, by incorporating external dynamic variable, a dynamic event-triggered mechanism (DETM) is proposed to regulate the transmission between the controller and actuators. Under the DETM, the designed controller only transmits the control force and torque into actuators at the specified triggering moment, and the trigger moment is independent of the internal state of the system. After that, the minimum inter-event time (MIET) of the proposed DETM is proved to be strictly positive, which ensures Zeno-free behavior. Lastly, the efficiency and advantages of the proposed event-driven distributed formation controller are verified through numerical simulations.