This article first proposes a discrete-time sliding flexible performance preset boundary (DT-SFPPB)-based control algorithm for input saturated discrete-time nonlinear systems (IS-DTNSs). Compared to the existing discrete-time prescribed performance control (DT-PPC) algorithms, the PPB of them present a “trumpet” shape, resulting in fundamental conservation of the transient performance, and whenever the initial error is altered, it is essential to recheck whether the new error meets the original constraint condition, if not, a new PPB with a larger measure has to be reselected. By designing a novel DT-SFPPB associated with the initial error, which can always envelope the initial error with an arbitrarily preset initial measure, indicating that the proposed approach can be utilized for IS-DTNSs with arbitrary initial error without compromising the initial transient performance. Furthermore, the coupling effect between performance preset and input saturation is also considered, by designing a novel equilibrium boundary related to saturation, so that the proposed approach can achieve the synergy between performance preset and input security, i.e., the designed DT-SFPPB can flexibly expand when input saturation occurs to avoid vulnerability, and when the control input is within the safe boundary, it rapidly reverts to the original PPB to guarantee the specified performance metrics. The findings demonstrate that the developed approach guarantees that the system output tracks the desired signal with the specified performance metrics, and all of the tracking errors are always enveloped within their corresponding DT-SFPPBs. The devised approach is exemplified by means of simulation examples.

This article first proposes a sliding flexible prescribed performance boundary-guided reinforcement learning (SFPPB-RL) control approach for input-constrained nonlinear systems (ICNSs). By designing a sliding flexible prescribed performance boundary, which not only can adaptively adjust the initial boundary according to the initial error, but also dynamically adjust the constraint relaxation according to the coupling correlation between the input constraint and the performance constraint, a novel prescribed performance control (PPC) approach is proposed. Compared with the existing "horn" shape performance boundary-based PPC methods, the limitation of having to repeatedly debug design parameters or sacrifice initial transient performance to meet different initial error requirements is eliminated. Meanwhile, the coupling effect between the input constraint and the performance constraint is also considered, and the balance between input safety and control performance is achieved by constructing an auxiliary system. Furthermore, combining identifier-critic-actor structure-based RL strategy and backstepping technique, a sliding flexible PPB-guided reinforcement learning (SFPPB-RL) optimal control algorithm is developed, which minimizes the cost function while ensuring input safety and prescribed performance indicators. The validity of the proposed algorithm is demonstrated via simulations.

This article first presents a dual flexible prescribed performance control (DFPPC) approach of input saturated high-order nonlinear systems (IS-HONSs). Compared to the existing PPC approaches of IS-HONSs, under which the performance constraint boundaries (PCBs) are usually fixed and bounded, resulting in a restriction of the initial error in the algorithm implementation; in addition, the coupling relationship between performance constraints and input saturation is usually ignored, resulting in the methods are very fragile when input saturation occurs. By designing the novel tensile model-based PCBs that depend on output and input constraints, the proposed DFPPC method provides sufficient resilience for both the initial conditions and the input saturation, so that the proposed DFPPC method can not only be suitable for multiple types of initial errors by adjusting the parameters, including , , and , where , and denote the initial PCBs; but also can achieve a good balance between input saturation and performance constraints, i.e., when the control input reaches or exceeds the saturation threshold, the PCBs can adaptively extend to avoid the singularity, and when the control input returns to the saturation threshold range, the PCBs are then adaptively restored to the original PCBs. The results show that the proposed DFPPC algorithm guarantees semi-global boundedness for all closed-loop signals, while ensuring that the system output accurately tracks the desired signal, and it consistently maintains the tracking error within the PCBs. The developed algorithm is illustrated by means of simulation instances.

The issue of sliding flexible prescribed performance control (SFPPC) of input saturated nonlinear systems (ISNSs) is first studied in this article. Compared to the traditional PPC and the finite-time PPC algorithms for ISNSs, under which the performance constraint boundaries (PCBs) present the symmetrical or asymmetric “horn” shape, which leads to a large jitter in the tracking error before the system reaches steady state; and once the parameters are selected, the PCBs are fixed, when the initial state (or reference signal) changes, it is necessary to reverify whether the initial error still satisfies the initial constraint condition. By designing a new pair of sliding flexible PCBs (SFPCBs) associated with the initial error, a novel SFPPC algorithm is presented in this article, which presents two main advantages: 1) the SFPCBs can slide adaptively with the initial tracking error without increasing the measure of the initial PCBs, implying that the proposed SFPPC algorithm can be applied to ISNSs with arbitrary initial errors without sacrificing the initial control performance; 2) the proposed SFPPC algorithm achieves a tradeoff between performance constraint and input saturation, i.e., the SFPCBs can adaptively increase when the control input exceeds the maximum allowable threshold, effectively avoiding singularity, and when the control input is within the saturation threshold range, the SFPCBs can adaptively revert back to the original PCBs. The results demonstrate that the proposed SFPPC approach can guarantee that the system output tracks the desired signal, and the tracking error always kept within the SFPCBs that depend on initial error, input, and output constraints. The developed algorithm is exemplified by means of simulation instances.

This article presents a unified adaptive fuzzy control approach for high-order nonlinear systems (HONSs) with multitype state constraints. Existing methods always require the upper and lower constraint boundaries are strictly positive and negative functions (or constants), respectively, which is often inconsistent with the actual constraints. In this article, "multitype state constraint" means that the upper and lower constraint boundaries include multiple types, such as both being strictly positive (or negative), sometime be positive or negative, and so on (cases ①-⑥). By designing a unified mapping function (UMF), the multitype state constraints are processed under removal the feasibility conditions (FCs). Furthermore, a technical design makes the proposed method also applicable to unconstrained HONSs without changing the control structure. By means of a fuzzy-logic system (FLS) and fixed-time stability theory (FTST), the proposed algorithm can ensure that the tracking error converges to a zero-centered neighborhood within a fixed time, and the singularity which often appears in the existing fixed-time control (FTC) methods of HONSs is effectively avoided. Simulation results demonstrate the scheme developed.

This article presents a prescribed-time output feedback control (PTOFC) algorithm for cyber-physical systems (CPSs) under output constraint occurring in any finite time interval (OC-AFT) and malicious attacks. The OC-AFT meaning that the output constraint only occurs during a finite number of time periods while being absent in others, which is more general and complex than traditional infinite-time/deferred output constraints. A stretch model-based nonlinear mapping function is constructed to handle the OC-AFT, and a salient advantage is that the proposed algorithm is also suit for CPSs with infinite-time/deferred output (or funnel) constraints, as well as those that are constraint-free, without necessitating changes to the control structure. The uncertain terms (including system model uncertainties, malicious attacks, and external disturbances) are compensated by fuzzy logic systems. Furthermore, a novel practical prescribed-time stability criterion is proposed, under which a novel PTOFC scheme is given. The results demonstrate that the proposed scheme can ensure that both tracking error and observation error converge to a neighborhood centered on zero within a prescribed time, while accommodating the OC-AFT and malicious attacks. Additionally, the settling time remains unaffected by control parameters and initial states, and the limitations of excessive initial control inputs and singularity problems in existing prescribed-time control algorithms are eliminated. The developed algorithm is exemplified through simulation instances.

The microelectromechanical system (MEMS) gyroscope is a complex nonlinear system with multiple variables, strong coupling, and susceptibility to stochastic disturbances. This paper presents an adaptive fuzzy control scheme for stochastic MEMS gyroscopes, with the primary objectives of reducing control vibration and achieving high precision prescribed performance tracking with low communication resources within a fixed-time backstepping framework. To address the stochastic disturbances and unknown nonlinear system dynamics, the interval type-3 fuzzy logic system (IT3FLS) is introduced. Additionally, a novel quadratic prescribed performance function (QPPF) is proposed to ensure satisfactory transient and steady-state performance of the system while mitigating initial control vibrations during fast error convergence. Furthermore, an event-triggered mechanism (ETM) is developed using a switching threshold strategy to minimize the communication load without compromising control accuracy. By utilizing the fixedtime command-filtered backstepping design method and newly introduced error-compensating signals, the issue of “explosion of complexity” is effectively resolved, and filtering errors are adequately compensated. The proposed control scheme guarantees that the tracking errors converge to a predefined set of arbitrarily small residuals in probability. In addition, all the closed-loop signals are within a fixed time bounded in probability (FTBIP). The simulation results validate the effectiveness and superiority of the proposed scheme.

A flexible prescribed performance control (FPPC) approach for input saturated nonlinear systems (ISNSs) with unmeasurable states is first presented in this article. Compared to the standard prescribed performance control (SPPC) or funnel control methods for ISNSs, the “flexibility” of the proposed FPPC algorithm is reflected in two aspects: 1) the proposed FPPC algorithm simultaneously considers multiple key indicators (including the steady state accuracy, convergence time, and overshoot), which are widely demanded in industrial production; 2) the proposed FPPC algorithm achieves a tradeoff between performance constraint and input saturation, i.e., the performance boundary can adaptively increase when the control input exceeds the saturation threshold, effectively avoiding singularity; conversely, when the control input is within the saturation threshold range, the performance constraint boundary can adaptively revert back to the original performance boundary. In addition, the unmeasured states are observed by the state observer, and the unknown nonlinear functions are approximated by fuzzy logic systems. The results demonstrate that the proposed output feedback control algorithm can ensure that all closed-loop signals are semiglobally bounded, the system output can track the desired signal within a prescribed time, and the tracking error is consistently maintained within flexible performance boundaries that depend on input and output constraints. The developed algorithm is exemplified through simulation instances.

A novel practical prescribed-time control (PPTC) approach for high-order nonlinear systems (HONSs) subject to state constraints is studied in this article. Different from the existing methods which always require the constraint boundaries to be continuous functions, the state constraints considered in this article are discontinuous (i.e., the state constraints occur only in some time periods and not in others), which can be found in many practical systems. By designing a novel stretch model-based nonlinear mapping function (NMF), the state constraints are dealt with directly, and the limitations that the virtual control function depends upon the feasibility condition (FC) and the tracking error depends upon the constraint boundaries in the conventional schemes are removed. Meanwhile, the proposed method is a unified one, which is also effective for HONSs with conventional continuous state constraints/ deferred state constraints/ funnel constraints or constraints-free without altering the control structure. Furthermore, by designing a newly time-varying scaling transformation function (STF), a more relaxed criterion for practical prescribed-time stable (PPTS) is given, based on which a newly PPTC algorithm is designed. The result shows that the proposed algorithm can preset the upper bound of the settling time, which does not depend upon the initial state of the system and control parameters, the limitations of singularity problem and excessive initial control input in existing methods are removed. Simulation examples verify the algorithm developed.

A unified approach to fixed-time tracking control of stochastic high-order nonlinear systems (HONSs) with or without full-state constraints is studied in this article. By introducing two important universal-constrained functions and using coordinate transformation technology, the stochastic HONS with full-state constraints is transformed into an equivalent one without state constraints. Compared with the existing control schemes, the proposed scheme not only eliminates the feasibility conditions, but also accurately analyzes the scope of tracking error. Moreover, the limitation that the constraint functions need to be bounded is also removed; thus, the proposed scheme is a unified approach that can be applied to stochastic HONSs with or without state constraints. With the help of fixed-time stability theory and adaptive fuzzy control technology, a novel fixed-time fuzzy controller is designed. In addition, an adaptive event-triggering mechanism that the threshold parameters can be adjusted adaptively according to the tracking performance is introduced to reduce the communication burden. Simulation results verify the scheme developed.