A Second Order Sliding Mode Controller with Predefined-Time Convergence

This paper presents the basis to design a well-suited control law which guarantees predefined-time convergence for a class of second-order systems. In contrast to the case of finite-time and fixed-time controllers, a predefined-time controller allows to set the bound of the convergence time, explicitly during the control design. Furthermore, in the case of no disturbance, the least upper bound of the convergence time can be predefined directly from the control definition. A Lyapunov-like characterization for predefined-time stability is performed. Numerical results are discussed to show the reliability of the proposed method.Consejo Nacional de Ciencia y Tecnologí

A novel predefined time PD-type ILC paradigm for nonlinear systems

Intelligent robotics has drawn a great deal of attention due to its high precision, stability, and reliability, which are the basic key factors for industrial automation. This paper proposes an iterative learning control (ILC) technique with predefined-time convergence as a solution to an applied engineering problem, namely, that local time cannot be preset when a second-order nonlinear system undertakes control of the accurate tracking of local time under any initial iterative value. A time-varying sliding surface with an initial value of zero was designed, and it was theoretically proven that the trajectory tracking error in the sliding surface could converge to zero within a predefined time. The iterative control problem of trajectory tracking was thus changed to an iterative control problem of time-varying sliding-mode surface tracing with a starting value of zero. A PD-type closed-loop ILC with a time-varying sliding mode surface was designed such that the trajectory tracking error converged and stabilized on the sliding mode surface after a finite number of learning iterations. The control goal for the system's output was the ability to track the desired trajectory accurately within a predefined time interval, and it was achieved by combining this with the predefined time convergence characteristics of the time-varying sliding mode surface. Numerical simulation of trajectory tracking control of a repetitive motion manipulator was used to verify the effectiveness of the proposed controller and its robustness in the face of external disturbances.Web of Science111art. no. 5

A class of predefined-time stable dynamical systems

This article introduces predefined-time stable dynamical systems which are a class of fixed-time stable dynamical systems with settling time as an explicit parameter that can be defined in advance. This concept allows for the design of observers and controllers for problems that require to fulfil hard time constraints. An example is encountered in the fault detection and isolation problem, where mode detection in a timely manner needs to be guaranteed in order to apply a recovery action. Furthermore, through the notion of strong predefined-time stability, the approach hereinafter presented permits to overcome the problem of overestimation of the convergence time bound encountered in previous methods for the analysis of finite-time stable systems, where the stabilization time is often an unbounded function of the initial conditions of the system. A Lyapunov analysis is provided together with a detailed discussion of the applications to consensus and first order sliding mode controller design

On optimal predefined-time stabilization

This paper addresses the problem of optimal predefined-time stability. Predefined-time stable systems are a class of fixed-time stable dynamical systems for which the minimum bound of the settling-time function can be defined a priori as an explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined-time stabilization problem for a given nonlinear system are provided. These conditions involve a Lyapunov function that satisfies a certain differential inequality for guaranteeing predefined-time stability. It also satisfies the steady-state Hamilton–Jacobi–Bellman equation for ensuring optimality. Furthermore, for nonlinear affine systems and a certain class of performance index, a family of optimal predefined-time stabilizing controllers is derived. This class of controllers is applied to optimize the sliding manifold reaching phase in predefined time, considering both the unperturbed and perturbed cases. For the perturbed case, the idea of integral sliding mode control is jointly used to ensure robustness. Finally, as a study case, the predefined-time optimization of the sliding manifold reaching phase in a pendulum system is performed using the developed methods, and numerical simulations are carried out to show their behavior