Continuous Nested Algorithms : The Fifth Generation of Sliding Mode Controllers

Fridman, Leonid; Moreno, Jaime A.; Bandyopadhyay, B.; Kamal, Shyam; Chalanga, Asif

doi:10.1007/978-3-319-18290-2_2

Cited by 74 publications

(88 citation statements)

References 31 publications

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“… Let the vector of states Σ with virtual state

normalΓ = f + \int_{t_{0}}^{t} U_{d} d τ

, then the extended system can be written as

σ^{false(r false)} = U_{c} + normalΓ, \overset{Γ}{true} \in false[- normalΔ, normalΔ false] + U_{d},

where a Lipschitz perturbation

false| \overset{f}{true} false| \leq normalΔ

is assumed. Some examples of these controllers are the Super‐Twisting, Higher‐Order Super‐Twisting, Continuous‐Terminal, Discontinuous Integral, and Continuous Twisting, where the gains of the algorithm are usually selected from Δ. The inclusion and the controller are homogeneous of degree q = −1 with the dilation

{\overset{d}{true}}_{κ} : (() t, σ, \dot{σ}, …, σ^{(r)}) \to (() κ t, κ^{m_{1}} σ, κ^{m_{2}} \dot{σ}, …, κ^{m_{r}} σ^{(r)}),

where m 1 = r + 1, m 2 = r , …, m r = 1 are the homogeneity weights and κ > 0. The ( r + 1)‐CSMC enforce the ( r + 1)th‐order sliding‐mode in a finite‐time, ie, there exist a time t r such that

\forall t \geq t_{r} : σ false(t false) = \overset{σ}{true} false(t false) = ⋯ = σ^{false(r false)} false(t false) = 0 .

Note that, in ( r + 1)‐CSMC, the vector field is tangent to e...…”

Section: Preliminariesmentioning

confidence: 99%

“…The ( r + 1)‐CSMC enforce the ( r + 1)th‐order sliding‐mode in a finite‐time, ie, there exist a time t r such that

\forall t \geq t_{r} : σ false(t false) = \overset{σ}{true} false(t false) = ⋯ = σ^{false(r false)} false(t false) = 0 .

Note that, in ( r + 1)‐CSMC, the vector field is tangent to each surface defined by σ ( i ) ( t ) = 0, i = 0,1,…, r , while it is directed toward the derivative of the virtual state

\overset{Γ}{true} = 0

.…”

Section: Preliminariesmentioning

confidence: 99%

“…where m 1 = r + 1, m 2 = r, … , m r = 1 are the homogeneity weights and > 0. (c) The (r + 1)-CSMC enforce the (r + 1)th-order sliding-mode 18,38 in a finite-time, ie, there exist a time t r such that ∀ t ≥ t r ∶ (t) =̇(t) = · · · = (r) (t) = 0.…”

Section: Ideal Sliding-modesmentioning

confidence: 99%

“…The output of the linear system (38) in closed loop with Twisting (40) has the following performance.…”

Section: Figurementioning

confidence: 99%

“…where the gains depend on the Lipschitz constant Δ of the disturbance, 17 k 1 = 2Δ 2/3 , k 2 = 5Δ 1/2 , and k 3 > Δ. Note that (47) enforces the third-order sliding-mode in system (38) when the actuator effects are instantaneous ( = 0). Figure 9 shows system (38) in closed loop with the nonlinearity (47).…”

Section: Parameters Of Chattering Predicted By Hb For the Discontinuomentioning

confidence: 99%

See 4 more Smart Citations

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

Pérez‐Ventura

Fridman

2018

Intl J Robust & Nonlinear

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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.

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“… Let the vector of states Σ with virtual state

normalΓ = f + \int_{t_{0}}^{t} U_{d} d τ

, then the extended system can be written as

σ^{false(r false)} = U_{c} + normalΓ, \overset{Γ}{true} \in false[- normalΔ, normalΔ false] + U_{d},

where a Lipschitz perturbation

false| \overset{f}{true} false| \leq normalΔ

{\overset{d}{true}}_{κ} : (() t, σ, \dot{σ}, …, σ^{(r)}) \to (() κ t, κ^{m_{1}} σ, κ^{m_{2}} \dot{σ}, …, κ^{m_{r}} σ^{(r)}),

\forall t \geq t_{r} : σ false(t false) = \overset{σ}{true} false(t false) = ⋯ = σ^{false(r false)} false(t false) = 0 .

Note that, in ( r + 1)‐CSMC, the vector field is tangent to e...…”

Section: Preliminariesmentioning

confidence: 99%

“…The ( r + 1)‐CSMC enforce the ( r + 1)th‐order sliding‐mode in a finite‐time, ie, there exist a time t r such that

\forall t \geq t_{r} : σ false(t false) = \overset{σ}{true} false(t false) = ⋯ = σ^{false(r false)} false(t false) = 0 .

Note that, in ( r + 1)‐CSMC, the vector field is tangent to each surface defined by σ ( i ) ( t ) = 0, i = 0,1,…, r , while it is directed toward the derivative of the virtual state

\overset{Γ}{true} = 0

.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Ideal Sliding-modesmentioning

confidence: 99%

“…The output of the linear system (38) in closed loop with Twisting (40) has the following performance.…”

Section: Figurementioning

confidence: 99%

Section: Parameters Of Chattering Predicted By Hb For the Discontinuomentioning

confidence: 99%

See 3 more Smart Citations

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

Pérez‐Ventura

Fridman

2018

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

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Adaptive high‐order sliding mode controller‐observer for MIMO uncertain nonlinear systems

Hendel

Khaber

Essounbouli

2019

Asian Journal of Control

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In this paper, an adaptive high-order sliding mode controller-observer is proposed for multi input-multi output uncertain systems with application to robotic manipulators. Firstly, a type-2 Takagi-Sugeno fuzzy system is used to model the original system. Next, with the unavailability of velocity measurement, a fuzzy high-order sliding mode observer is designed to estimate both the joint velocities and uncertainties. Moreover, based on a super twisting second-order sliding mode, the proposed robust controller generates smooth control and guarantees finite-time convergence to the third-order sliding set with respect to the sliding variable by keeping second-order sliding mode constraint. The controller gains are generated based on an adaptive estimation scheme without overestimation. The finite-time stability of the suggested controller is proved by using an homogeneous and strict Lyapunov function. Finally, the obtained simulation results for two-link robot manipulators demonstrate the effectiveness of the proposed controller.

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Digital sliding mode controllers for active control of ground vehicles

Lúa

Gennaro

Guzmán³

et al. 2021

Asian Journal of Control

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This paper aims at designing, in the discrete-time setting and using sliding mode techniques, some controllers ensuring the tracking of desired references of lateral and yaw velocities for a ground vehicle with active front steering and rear torque vectoring actuators. The vehicle is described by an enhanced discrete-time model, recently presented in the literature. These three controllers have been designed considering the modified equivalent control method for perturbation attenuation, the discrete-time-like super-twisting algorithm, and the discrete-time reaching law. They have been tested with a simulation study using CarSim, where the three controllers are compared under parametric variations, external perturbations, and different sampling periods.

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Continuous Nested Algorithms : The Fifth Generation of Sliding Mode Controllers

Cited by 74 publications

References 31 publications

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

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

Adaptive high‐order sliding mode controller‐observer for MIMO uncertain nonlinear systems

Digital sliding mode controllers for active control of ground vehicles

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