Jaime A. Moreno

A Lyapunov approach to second-order sliding mode controllers and observers

In this paper a strong Lyapunov function is obtained, for the first time, for the super twisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic super twisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the higher order sliding modes research area.

Strict Lyapunov Functions for the Super-Twisting Algorithm

A method to construct a family of strict Lyapunov functions, i.e., with negative definite derivative, for the super-twisting algorithm, without or with perturbations, is provided. This second order sliding modes algorithm is widely used to design controllers, observers and exact differentiators. The proposed Lyapunov functions ascertain finite time convergence, provide an estimate of the convergence time, and ensure the robustness of the finite-time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm. Since the Lyapunov functions and their derivatives are quadratic forms, the operation with them is as simple as for linear time invariant systems.

Uniform Robust Exact Differentiator

The differentiators based on the Super-Twisting Algorithm (STA) yield finite-time and theoretically exact convergence to the derivative of the input signal, whenever this derivative is Lipschitz. However, the convergence time grows unboundedly when the initial conditions of the differentiation error grow. In this technical note a Uniform Robust Exact Differentiator (URED) is introduced. The URED is based on a STA modification and includes high-degree terms providing finite-time, and exact convergence to the derivative of the input signal, with a convergence time that is bounded by some constant independent of the initial conditions of the differentiation error. Strong Lyapunov functions are used to prove the convergence of the URED.

Variable Gain Super-Twisting Sliding Mode Control

In this note, a novel, Lyapunov-based, variable-gain super-twisting algorithm (STA) is proposed. It ensures for linear time invariant systems the global, finite-time convergence to the desired sliding surface, when the matched perturbations/uncertainties are Lipschitz-continuous functions of time, that are bounded, together with their derivatives, by known functions. The proposed algorithm has similar properties to the variable-gain first-order sliding mode control, but it provides alleviation to the chattering phenomenon. The results are verified experimentally.

Global observability analysis of sensorless induction motors

The current problems to successfully apply sensorless controllers for induction motors are the existence of operation regimes for which the performance is remarkably deteriorated, due to the difficulties in estimating correctly motor speed and flux, and the lack of a theoretical explanation for this kind of behavior. In this paper a global observability analysis for these machines is carried out. It is first shown that all indistinguishable trajectories of the system, i.e. pairs of state trajectories with the same input/output behavior, can be described by a differential equation on a manifold, named here the indistinguishable dynamics. Studying the stability properties of this latter system it can be shown that the induction motor is not completely observable nor detectable in a local or in a global sense, and for every set of parameters. This implies that it is impossible to construct a state observer for the motor that converges for every trajectory of the system. Moreover, the indistinguishable dynamics provides a systematic method to study, understand and explain particular operation regimes, and this is illustrated by some case studies of practical relevant operating conditions.

Static output feedback stabilization: necessary conditions for multiple delay controllers

This note focuses on the static output feedback stabilization problem for a class of single-input-single-output systems when the control law includes multiple (distinct) delays. We are interested in giving necessary conditions for the existence of such stabilizing controllers. Illustrative examples (second-order system, chain of integrators, or chain of oscillators) are presented, and discussed.

Super-twisting adaptive sliding mode control: A Lyapunov design

A novel super-twisting adaptive sliding mode controller is proposed. A drift uncertain term is assumed to be bounded with unknown boundary. The proposed Lyapunov-based approach consists in using dynamically adapted control gains that ensure the establishment, in a finite time, of a second order sliding mode. Finite convergence time is estimated. A numerical example confirms the efficacy of the proposed adaptive super-twisting control.

Lyapunov Approach for Analysis and Design of Second Order Sliding Mode Algorithms

Lyapunov functions are a basic tool for analysis and design in the modern control theory, and there are many different design methodologies based on Lyapunov theory. Second Order Sliding Modes, and in particular, the Super-Twisting Algorithm (STA), are a powerful tool for the design of controllers, observers and differentiators having very attractive dynamic features: they converge in finite time, even in presence of persistently acting bounded perturbations. This property, that we will call exactness, can be achieved because of the discontinuous nature of the STA. The design of control or observation algorithms based on Second Order Sliding Modes has been performed until now using either geometric or homogeneous approaches, but not Lyapunov methods. The reason for this situation is simple: only recently has been possible to find adequate Lyapunov functions for some of these algorithms. In this paper some recent advances in this direction will be presented and extended.

A linear framework for the robust stability analysis of a Generalized Super-Twisting Algorithm

In this paper a linear framework is proposed for the analysis and design of stable and robust stable Generalized Super-Twisting Algorithms (GSTA). The GSTA includes a linear version of the algorithm, the standard STA and a STA with extra linear correction terms, that provide more robustness and convergence velocity. This linear framework allows to construct strong Lyapunov functions of quadratic-like type for the GSTA by means of Algebraic Lyapunov Equations (ALE), in exactly the same form for the linear STA and for the GSTA. When nonlinear perturbations are present this framework leads to the construction of robust Lyapunov functions by solving Algebraic Riccati Inequalities (ARI) or Linear Matrix Inequalities (LMI), that are identical for the linear and the nonlinear versions of the GSTA. The corresponding frequency domain interpretations are also identical for the whole class of GSTA.

Optimal time control of bioreactors for the wastewater treatment

In this work the time optimal control problem for a biological sequencing batch reactor, used for wastewater treatment, is solved. This operation strategy increases greatly the efficiency of these plants. New is the consideration of the substrate concentration in the input flow, the main disturbance for these systems, as a variable signal and not as a constant parameter. The problem will be solved using a known method based on Greens theorem that allows one to obtain analytically the unique global solution. Furthermore, an optimal feedback control law can be derived that can be made robust against parameter uncertainties and the input disturbance. Simulations of a realistic model of an industrial wastewater treatment plant show the advantages of using an optimal strategy in the control of the plant, since this reduces, among other aspects, the costs of operation and the size of the plant. Copyright