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This article investigates the input–output finite-time stability (IO-FTS) for nonlinear hidden semi-Markov switching systems (S-MSSs) via the asynchronous sliding mode control (SMC) approach. Under the assumption that a detector is set up to estimate the mode value, the mode of the original system is not directly accessible. The asynchrony between the system and controller is described as a hidden semi-Markov model (HSMM). Many practical factors, such as semi-Markov switching parameters, finite-time interval, asynchronous phenomenon, uncertain parameters, and nonlinearity, are taking into account during the SMC design process. We aim to design an efficient finite-time asynchronous SMC scheme under a hidden semi-Markov switching effect. A novel sliding switching surface (SSS) is constructed, in which the SMC law rises asynchronously with original S-MSSs. Then, by means of finite-time stability, an asynchronous SMC law is synthesized to guarantee that the associated hidden S-MSSs fulfill the reaching condition within a finite time. Furthermore, sufficient conditions are derived in view of the IO-FTS of sliding mode dynamics under the framework of a novel inequality lemma. Results are given for the application of this control design method to a single-link robot arm model (SLRAM). [ABSTRACT FROM AUTHOR]

In this article, the issue of positive $\mathcal {L}_{1}$ filter design is investigated for positive nonlinear stochastic switching systems subject to the phase-type semi-Markov jump process. Many complicated factors, such as semi-Markov jump parameters, positivity, T–S fuzzy strategy, and external disturbance, are taken into consideration. Practical systems under positivity constraint conditions and unpredictable structural changes are characterized by positive semi-Markov jump systems (S-MJSs). First, by the key properties of the supplementary variable and the plant transformation technique, phase-type S-MJSs are transformed into Markov jump systems (MJSs), which means that, to an extent, these two kinds of stochastic switching systems are mutually represented. Second, with the help of the normalized membership function, the associated nonlinear MJSs are transformed into the local linear MJSs with specific T–S fuzzy rules. Third, by choosing the linear copositive Lyapunov function (LCLF), stochastic stability (SSY) criteria are given for the corresponding system with $\mathcal {L}_{1}$ performance. Some solvability conditions for positive $\mathcal {L}_{1}$ filter are constructed under a linear programming framework. Finally, an epidemiological model illustrates the effectiveness of the theoretical findings. [ABSTRACT FROM AUTHOR]

The issue of sliding-mode control (SMC) design is studied for a class of nonlinear semi-Markovian switching systems (S-MSSs) via T–S fuzzy approach. Compared with previous literature, novel integral sliding-mode surfaces (ISMSs) are developed to depend on the designed controller gains and projection matrices. The aim of this work is to design an appropriate fuzzy SMC law under complex stochastic semi-Markovian switching process. For this purpose, the novel ISMSs are developed under the T–S fuzzy modeling framework. Then, based on the weak infinitesimal operator theory, sufficient conditions are given for stochastic stability criteria relying on the sojourn-time. Furthermore, an appropriate fuzzy SMC law is proposed to drive the state signals onto the predefined fuzzy manifold. Finally, an electric circuit model illustrates the effectiveness of the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper concerns a positive $\mathcal {L}_{1}$ observer for positive nonlinear semi-Markovian switching systems (MSSs) via the expansion of Taylor formula and the fuzzy Lyapunov function approach, in which semi-Markovian switching parameters, positivity, Takagi–Sugeno (T–S) fuzzy, and external disturbance are all considered in a unified framework. A fuzzy Lyapunov function approach with less conservativeness is introduced into the research of positive systems. In the system under consideration, positive S-MSSs with the semi-Markovian process can describe more complex systems in a practical control process. The main motivation of this paper is that the practical system subject to positivity and abrupt changes can be described by positive nonlinear S-MSSs, which always needs to consider the external disturbance. First, by using the normalized membership function approach, positive nonlinear S-MSSs can be represented by local positive T–S fuzzy S-MSSs. Second, by constructing the fuzzy Lyapunov function, some sufficient conditions are proposed for stochastic stability and $\mathcal {L}_{1}$ -gain performance analysis, respectively. Then, a positive $\mathcal {L}_{1}$ observer in a novel standard linear programming condition is designed to guarantee the resulting closed–loop augmented system is positive and stochastically stable with a required $\mathcal {L}_{1}$ -gain performance. Finally, a practical example about the epidemiological model is introduced to show the effectiveness of the main theory. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive fixed-time convergent continuous controller designed to solve a stock management problem with the objective to drive stock and supply chain levels at the reference values, subject to loss rate disturbances whose bounds are unknown. The only measurable state of the supply chain is the inventory retailer stock level, whereas the supply line inventory level should be estimated. The designed controller includes a fixed-time convergent differentiator, an adaptive fixed-time convergent disturbance observer, and a fixed-time convergent regulator. The adaptive fixed-time convergent observer is used to estimate a disturbance without excessively increasing the controller gains. The controller design is validated in a case study of stock management. The calculated upper estimate for the total settling (convergence) time and the obtained simulation results confirm the fixed-time convergence and the robustness of the designed controller. [ABSTRACT FROM AUTHOR]

In order to understand how geometric errors propagate and how deviations accumulate in a six-axis motion platform (SMP), a new geometric error model based on the stream of variation (SOV) theory is presented in this paper. SOV theory is widely used in industrial engineering with several steps or phases. The conventional geometric error model only calculates the initial and final orientation, yet the deviations after each step in the whole process are still unknown, which are critical parameters in measuring the product quality. In this new error modeling method, each step in the alignment and welding process in SMP can be considered as a station. Thus, the deviations can also be derived after each station, which is beneficial for error identification. Based on the new error modeling approach involving SOV theory, the validation of an optimized configuration is developed by a series of calculations results. By observing the deviations after each station, the optimized configuration can improve accuracy and reduce power loss compared to a traditional configuration. The new error modeling approach based on SOV theory is systematic and comprehensive, and can be applied in other similar environments. [ABSTRACT FROM AUTHOR]