This paper uses the semi-tensor product (STP) of matrices and adopts algebraic methods to study the controllability, reachability, and stabilizability of extended finite state machines (EFSMs). First, we construct the bilinear dynamic system model of the EFSM, laying the foundation for further research. Second, combined with this bilinear dynamic system model, we propose theorems for the controllability, reachability, and stabilizability of the bilinear dynamic system model of the EFSM. Finally, we design an algorithm to determine the controllability and stabilizability of the EFSM. The correctness of the main results is verified through examples.

This brief aims to introduce the concept of robust control from continuous-state systems to the field of finite state machines (FSMs). It expands on the concept of observability by proposing current state observability and current state set observability for FSMs. The concept of robust controllability is introduced for FSMs, including robust controllability between states and robust controllability between states and state sets. An existence condition for robust controllers of FSMs is established. Consequently, a robust controller is designed to enable FSMs with uncertainty to possess robust controllability, ensuring that FSMs exhibit the desired state evolution behavior under uncertain disturbances. Unlike the robust control of continuous-state systems, the built robust control systems of FSMs use reference states that represent the desired state transitions as the reference inputs of the closed-loop systems. Additionally, a brief discussion is conducted on how to further investigate robust control problems of FSMs within the STP framework. Note to Practitioners—The motivation of this paper is to address the control of uncertainty in Finite State Machines (FSMs), but it also has significant implications for other logic dynamic systems, such as Boolean networks, game systems, and finite-value systems. Existing control methods for FSMs are typically based on supervised control, which limits control over system uncertainty. Drawing inspiration from robust control in continuous state systems (CSS), this paper presents a robust control approach for FSMs. This approach enables the application of robust control methods from CSS to control the uncertainty in FSMs. The usage of this approach closely resembles the well-known and widely utilized robust control methods employed in practical industrial domains involving CSS. Several new concepts, ideas, and methods are proposed in this paper, followed by the design of a corresponding controller. Numerical examples demonstrate the feasibility of this approach, although it has not been validated in actual industrial systems. Nevertheless, the findings can undoubtedly be applied to many industrial systems in theory, such as communication and networks, control systems, embedded systems, and natural language processing. In future research, we will explore the application of the proposed robust control for FSMs in specific real-world industrial systems.

Traditional matrix-based approaches in the field of finite state machines construct state transition matrices, and then use the powers of the state transition matrices to represent corresponding dynamic transition processes, which are cornerstones of system analysis. In this study, we propose a static matrix-based approach that revisits a finite state machine from its structure rather than its dynamic transition process, thus avoiding the “explosion of complexity” problem inherent in the existing approaches. Based on the static approach, we reexamine the issues of closed-loop detection and controllability for deterministic finite state machines. In addition, we propose controllable equivalent form and minimal controllable equivalent form concepts and give corresponding algorithms.

Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines (FSMs) in most existing results, we consider the problem by viewing FSMs as logical dynamic systems. Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory, the concepts of t -equivalent states and t -source equivalent states are introduced. Based on the state transition dynamic equations of FSMs proposed in recent years, several mathematical formulations of t -equivalent states and t -source equivalent states are proposed. These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem. Using these mathematical formulations, two methods are designed to find all the t -equivalent states and t -source equivalent states of FSMs. Further, two ways of reducing the state space of FSMs are found. These can be implemented without computers but with only pen and paper in a mathematical manner. In addition, an open question is raised which can further improve these methods into unattended ones. Finally, the correctness and effectiveness of the proposed methods are verified by a practical language model.

Inertia load change, torque disturbance, and sophisticated mechanical nonlinearity impose great challenge on the high-precision control design of motor servo systems. To achieve better tracking and disturbance rejection performance, a practical control scheme, which combines a fixed-time-convergent generalized extended state observer (FGESO) with a fixed-time-convergent state feedback control law, is proposed in this article. First, an FGESO is designed to estimate the lumped disturbance and the fixed-time convergence of the estimation errors can be achieved. In the FGESO, there are three parameters and the correction term is selected as a continuous switching function of the output estimation error, which can facilitate the tuning process and greatly alleviate the chattering problem. Second, a control law based on the bilimit homogeneity technique is provided to regulate the tracking errors within a finite time independent of the initial condition. The fixed-time stability of the closed-loop system can be rigorously proved based on the Lyapunov theory. To facilitate the application of the proposed method, a practical two-step tuning procedure is established. Finally, extensive simulations and experiments are performed on a brushless direct current motor to validate the effectiveness and robustness of the proposed method.