An idea for Lyapunov function design for arbitrary order continuous twisting algorithms

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Summary Professor Utkin proposed an example showing that the amplitude of chattering caused by the presence of parasitic dynamics (stable actuators) in some systems governed by the First‐Order Sliding‐Mode Controller is lower than that produced by the Super‐Twisting Algorithm. This example served to motivate this paper reconsidering the problem of comparison of chattering in systems with stable actuators, and driven by Discontinuous Sliding‐Mode Controllers (DSMCs) and Continuous Sliding‐Mode Controllers (CSMCs). Comparison of chattering produced by DSMC and CSMC taking into account their amplitudes, frequencies, and average power (AP) needed to maintain the system into real‐sliding modes, allowing to conclude the following: (i) for systems with slow actuators, the amplitude of oscillations and AP produced by DSMC be smaller than those caused by CSMC; (ii) for bounded disturbances with fixed Lipschitz constant, there exist sufficiently fast actuators for which the amplitude of oscillations and AP produced by CSMC be smaller than those caused by DSMC.

\begin{matrix} right u & left = - λ_{1} ⌊ σ ⌉^{\frac{1}{r + 1}} - ⋯ - λ_{r} ⌊ σ^{(r - 1)} ⌉^{\frac{1}{2}} + v, & right \\ right \dot{v} & left = - λ_{r + 1} ⌊ σ ⌉^{0} - ⋯ - λ_{2 r} ⌊ σ^{(r - 1)} ⌉^{0}, \end{matrix}

Section: Construction Of Describing Functions For Csmcmentioning

confidence: 99%

“…Consider, for instance, c = 1.1 , then the chattering parameters predicted by HB, amplitude (24), frequency (25) and AP (26), take the form…”

Section: Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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When is it reasonable to implement the discontinuous sliding‐mode controllers instead of the continuous ones? Frequency domain criteria

Pérez‐Ventura

Fridman

2018

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“…finite-time theoretically exact compensation of Lipschitz P/U; 2. sliding accuracy of order (r+1) with respect to the output ((r+1)th order of precision) in the face of time discretizations of the control input and actuator time constant; 3. robust convergence of the output and its first r time derivatives to the origin in finite time ((r + 1)-sliding motion) while assuming only the information of the output and its first (r − 1) time derivatives.Examples of CSM controllers for systems of relative degree two are: continuous twisting (CT) algorithm, 17 continuous terminal SM, 18 and discontinuous integral controller. 19 For an arbitrary relative degree, continuous algorithms have also been derived (eg, see other works [19][20][21][22] ).…”

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confidence: 99%

“…Examples of CSM controllers for systems of relative degree two are: continuous twisting (CT) algorithm, 17 continuous terminal SM, 18 and discontinuous integral controller. 19 For an arbitrary relative degree, continuous algorithms have also been derived (eg, see other works [19][20][21][22] ).…”

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confidence: 99%

Continuous sliding‐mode control for singular systems

Hernández

Golkani

2018

Singular systems with matched Lipschitz perturbations and uncertainties are considered in this paper. Since continuous solutions of an impulse-free singular system require continuous input signals, a two-step continuous sliding-mode control strategy to compensate matched Lipschitz perturbations and uncertainties in singular systems is proposed. Our suggested methodology is tested in a singular representation of a DC motor pendulum of relative degree two. The performance of the proposed strategy is assessed by comparing the accuracy, in both cases, with and without considering small noise in the output, obtained through other continuous sliding-mode control, and reconstruction/compensation of perturbations and uncertainties techniques.Recently, CSM controllers for sliding outputs of relative degree r have been developed. Such controllers provide:1. finite-time theoretically exact compensation of Lipschitz P/U; 2. sliding accuracy of order (r+1) with respect to the output ((r+1)th order of precision) in the face of time discretizations of the control input and actuator time constant; 3. robust convergence of the output and its first r time derivatives to the origin in finite time ((r + 1)-sliding motion) while assuming only the information of the output and its first (r − 1) time derivatives.Examples of CSM controllers for systems of relative degree two are: continuous twisting (CT) algorithm, 17 continuous terminal SM, 18 and discontinuous integral controller. 19 For an arbitrary relative degree, continuous algorithms have also been derived (eg, see other works [19][20][21][22] ).

Reaction wheel pendulum control using fourth‐order discontinuous integral algorithm

Gutiérrez‐Oribio

Mercado‐Uribe

Moreno

2020

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The fourth-order model of the reaction wheel pendulum is considered and a fourth-order discontinuous integral algorithm is used for stabilization and tracking of the system, using a continuous control signal. The states reach the origin or a reference signal in finite-time, even in presence of uncertain control coefficient and a kind of matched and unmatched uncertainties/disturbances. A homogeneous Lyapunov function is designed to ensure local finite-time stability of the system, which can be used for designing the controller gains. Simulations and experimental results illustrate the performance and advantages of the presented algorithm. K E Y W O R D S finite-time convergence, higher-order sliding-mode control, nonlinear control, tracking control, wheel pendulum Abbreviations: 3-DIA, third order discontinuous integral algorithm; 4-DIA, fourth order discontinuous integral algorithm; RWP, reaction wheel pendulum.