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

A discontinuous recurrent neural network with predefined time convergence for solution of linear programming

The aim of this paper is to introduce a new recurrent neural network to solve linear programming. The main characteristic of the proposed scheme is its design based on the predefined-time stability. The predefined-time stability is a stronger form of finite-time stability which allows the a priori definition of a convergence time that does not depend on the network initial state. The network structure is based on the Karush-Kuhn-Tucker (KKT) conditions and the KKT multipliers are proposed as sliding mode control inputs. This selection yields to an one-layer recurrent neural network in which the only parameter to be tuned is the desired convergence time. With this features, the network can be easily scaled from a small to a higher dimension problem. The simulation of a simple example shows the feasibility of the current approach.Consejo Nacional de Ciencia y Tecnologí

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

A Note on Predefined-Time Stability

This note presents a new characterization for predefined-time-stable systems based on Lyapunov stability. In contrast to the previous results in predefined-time stability, the proposed characterization allows the construction of predefined-time stabilizing controllers with polynomial terms instead of exponential ones, removing the exponential nature hypothesis of predefined-time-stable systems. Moreover, the existing Lyapunov characterization of predefined-time stability is shown to be a consequence of the new theorem presented in this paper. Finally, the proposed approach is used for the construction of robust predefined-time stabilizing controllers for first-order systems. A simulation example shows the feasibility of the proposed methods.Consejo Nacional de Ciencia y Tecnologí

Non-Singular Predefined-Time Stable Manifolds

In this paper it is introduced a class of non-singular manifolds with predefinedtime stability. That is, for a given dynamical system with its trajectories constrained to this manifold it can be shown predefined-time stability to the origin. In addition, the function that defines the manifold and its derivative along the system trajectories are continuous, therefore no singularities are presented for the system evolution once the constrained motion starts. The problem of reaching the proposed manifold is solved by means of a continuous predefined-time stable controller. The proposal is applied to the predefined-time exact tracking of fully actuated and unperturbed mechanical systems. It is assumed the availability of the state and the desired trajectory as well as its two first derivatives. As an example, the proposed solution is applied over a two-link planar manipulator and numerical simulations are conducted to show its performance.ITESO, A.C.CINVESTAV-IP

A Second Order Predefined-Time Control Algorithm

The predefined-time stabilization of second-order systems, i.e. the fixed-time stabilization with settling time as a function of the controller parameters, is revisited in this paper. The proposed controller is a time-based switched controller which first drive the system trajectories to a linear manifold in predefined time and then uses a nested second order controller. The application of the results is demonstrated for the trajectory tracking control in fully actuated mechanical systems. An illustrative example of the control of a two-link planar manipulator with predefined-time convergence shows the effectiveness of the proposed algorithm

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