A New Sliding Surface for Discrete Second Order Sliding Mode Control for Time Delay Systems with Time Varying Uncertainties

Nizar, Abdennebi; Nouri, Ahmed Saïd

doi:10.5120/8334-1857

Cited by 10 publications

(16 citation statements)

References 12 publications

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“…Similar problems has been analyzed in other works using LMI‐based approaches. Consider the following regular representation:

\begin{matrix} alignleft \\ align-1 x_{1} (k + 1) & align-2 = [Φ_{11} + Δ Φ_{11} (k)] x_{1} (k) + Φ_{d 11} x_{1} (k - d (k)) \\ align-1 & align-2 + [Φ_{12} + Δ Φ_{12} (k)] x_{2} (k) \\ align-1 x_{2} (k + 1) & align-2 = [\begin{array}{c} {normalΦ}_{21} & {normalΦ}_{22} \\ {normalΦ}_{d 21} z^{- d false(k false)} & {normalΦ}_{d 22} z^{- d false(k false)} \end{array}] [\begin{array}{c} x_{1} false(k false) \\ x_{2} false(k false) \end{array}] + \\ align-1 & align-2 + Γ u (k) + f (k, x (k)), \end{matrix}

where z − d ( k ) is a time variant delay d ( k ) and

f : double-struckZ \times {double-struckR}^{n} \to {double-struckR}^{m}

is a nonlinear function representing the external disturbances and not modeled dynamics.…”

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 59%

“…Similar problems has been analyzed in other works [68][69][70]78,79 using LMI-based approaches. Consider the following regular representation:…”

Section: Lmi Approachesmentioning

confidence: 61%

“…The LMI is an efficient numerical tool that can be applied for obtaining the solution of optimization problems where some convex objective function must be minimized subjected to some constraint functions. 74 In related works, [23][24][25][26]44,67,[75][76][77] LMI-based approaches have been widely used for designing sliding surfaces for continuous time systems and also for discrete-time systems (see other works 22,27,[68][69][70][71]78,79 ). In these research works, the switching surface specification is obtained using LMI formulations, which allows that the sliding surface can be designed for perturbed systems with uncertainties, even with mismatched ones.…”

Section: Lmi Approachesmentioning

confidence: 99%

“…This control law guarantees the minimal u(k) for the maximal disturbance (k), hence, it is referred as Lyapunov min-max control law. 26,80 Using the Ricatti equation previously defined in (79), then…”

Section: Min-max Controller Approachmentioning

confidence: 99%

“…On the contrary to the previous eigenvalues assignment approaches, the following ones are based on optimal methods. Considering the Lyapunov function V(k) = x T (k)Px(k), then, solving the Ricatti equation given in (79) for Q = I n , the optimal matrix P ∈ ℝ n×n can be obtained using the dlqr MATLAB function. The P matrix can be used for obtaining an optimal sliding surface via Spurgeon 30 Computing the equivalent control, then the equivalent system matrix for the SM is as follows:…”

Section: Solution 4 (Cheng and Chang Approach)mentioning

confidence: 99%

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Discrete‐time systems sliding mode switching hyperplane design: A survey

Luis-Delgado

Al‐Hadithi

Jiménez

2019

Optim Control Appl Methods

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Summary As regards to variable structure control with sliding mode (VSC‐SM), it is known that designing sliding surface is a critical task because sliding surfaces must guarantee that the steady state of the controlled system achieves some desired performance including global (or global asymptotic) stability. This problem has been extensively analyzed for a wide class of systems: scalar, multivariable, linear, nonlinear, time‐variant, discrete‐time system, etc. The main purpose of this paper is to present a thorough survey about the main approaches related to the design of discrete‐time switching surfaces applied to VSC‐SM. Research works covering the design of sliding surfaces for the aforementioned systems are considered and some illustrative examples are also included to explain some of the design methodologies.

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“…Similar problems has been analyzed in other works using LMI‐based approaches. Consider the following regular representation:

\begin{matrix} alignleft \\ align-1 x_{1} (k + 1) & align-2 = [Φ_{11} + Δ Φ_{11} (k)] x_{1} (k) + Φ_{d 11} x_{1} (k - d (k)) \\ align-1 & align-2 + [Φ_{12} + Δ Φ_{12} (k)] x_{2} (k) \\ align-1 x_{2} (k + 1) & align-2 = [\begin{array}{c} {normalΦ}_{21} & {normalΦ}_{22} \\ {normalΦ}_{d 21} z^{- d false(k false)} & {normalΦ}_{d 22} z^{- d false(k false)} \end{array}] [\begin{array}{c} x_{1} false(k false) \\ x_{2} false(k false) \end{array}] + \\ align-1 & align-2 + Γ u (k) + f (k, x (k)), \end{matrix}

where z − d ( k ) is a time variant delay d ( k ) and

f : double-struckZ \times {double-struckR}^{n} \to {double-struckR}^{m}

is a nonlinear function representing the external disturbances and not modeled dynamics.…”

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 59%

“…Similar problems has been analyzed in other works [68][69][70]78,79 using LMI-based approaches. Consider the following regular representation:…”

Section: Lmi Approachesmentioning

confidence: 61%

Section: Lmi Approachesmentioning

confidence: 99%

Section: Min-max Controller Approachmentioning

confidence: 99%

Section: Solution 4 (Cheng and Chang Approach)mentioning

confidence: 99%

See 3 more Smart Citations

Discrete‐time systems sliding mode switching hyperplane design: A survey

Luis-Delgado

Al‐Hadithi

Jiménez

2019

Optim Control Appl Methods

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Discrete second-order sliding mode control based on optimal sliding function vector for multivariable systems with input-output representation

Houda

Dehri

Nouri

2016

Int. J. Robust. Nonlinear Control

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Summary The technique of linear matrix inequalities is a powerful method for solving optimization problems. In this paper, a sliding function vector was calculated using linear matrix inequalities approach. This technique provided optimal values of the coefficients of the sliding function vector, which led to the reduction of the reachability phase. Then, a discrete second‐order sliding mode control for multivariable systems was developed using this optimal sliding function vector. Two examples were used in order to illustrate the effectiveness of the proposed strategy. Simulation results prove good performances in terms of reduction of the reachability phase. Copyright © 2016 John Wiley & Sons, Ltd.

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