Stability analysis of discrete input output second order sliding mode control

Houda, Romdhane; Dehri, Khadija; Nouri, Ahmed Saïd

doi:10.1504/ijmic.2014.064291

Cited by 13 publications

(8 citation statements)

References 18 publications

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“…The second‐order sliding mode control is a particular case of the high‐ order sliding mode control, which consists in forcing the sliding surface and their first derivatives to zero. In order to obtain a discrete second‐order sliding mode control, the following condition must be satisfied :

S (k + d + 1) = S (k + d) = 0_{(p \times 1)}

with S ( k + d ) as the sliding function vector in the case of classical sliding mode control. It is given by

\begin{matrix} S (k + d) & = C (q^{- 1}) ξ (q) (Y (k) - Y_{r} (k)) \\ = C (q^{- 1}) ξ (q) E (k) \end{matrix}

where C ( q −1 ) is a polynomial matrix defined as

C (q^{- 1}) = I_{p} + C_{1} q^{- 1} + \dots + C_{n_{C}} q^{- n_{C}}; \dim C_{τ_{3}} = (p, p); τ_{3} \in [1, n_{C}]

E ( k ) is the error vector defined as

E (k) = Y (k) - Y_{r} (k)

Y r ( k ) is the ...…”

Section: Discrete Second‐order Sliding Mode Control For Multivariablementioning

confidence: 99%

“…For decouplable multivariable systems, the interactor matrix is expressed by

ξ (q) = q^{d} I_{p}

Therefore, the sliding function vector becomes

\begin{matrix} S (k) & = C (q^{- 1}) (Y (k) - Y_{r} (k)) \\ = C (q^{- 1}) E (k) \end{matrix}

In the case of second‐order sliding mode control, the new sliding function vector σ ( k ) is selected by the following expression :

σ (k) = S ((), k) + β S ((), k - 1)

We noted that β was chosen in the interval ]0,1[ in order to ensure the convergence of the sliding function vector σ ( k ). To synthesize the equivalent control law, it is essential to solve the following diophantine polynomial matrix equation:

C ((), q^{- 1}) = \overset{F}{true} ((), q^{- 1}) A ((), q^{- 1}) normalΔ ((), q^{- 1}) + q^{- 1} \overset{G}{true} ((), q^{- 1})

where

\overset{F}{true} ((), q^{- 1}) = I_{p}

\overset{G}{true} ((), q^{- 1}) = {\overset{G}{true}}_{0} + {\overset{G}{true}}_{1} q^{-<...>}

…”

Section: Discrete Second‐order Sliding Mode Control For Multivariablementioning

confidence: 99%

“…In the case of second-order sliding mode control, the new sliding function vector .k/ is selected by the following expression [23,25]:…”

Section: Discrete Second-order Sliding Mode Control For Decouplable Mmentioning

confidence: 99%

“…In the case of discrete second-order sliding mode control, the discontinuous control vector is given by [23][24][25] 3810 H. ROMDHANE, K. DEHRI AND A. S. NOURI Therefore, the discrete second-order sliding mode control law can be expressed as follows:…”

Section: Discrete Second-order Sliding Mode Control For Decouplable Mmentioning

confidence: 99%

“…In all of the aforementioned works, the sliding mode control based on LMIs is developed with a state space model.In [22], we proposed synthesizing a discrete second-order sliding mode control for single-input single-output systems via input-output representation. This proposed strategy was applied to a real discrete second-order system in [23]. Because many industrial processes are multivariable, we synthesized a discrete second-order sliding mode control for decouplable multivariable systems [24] and for nondecouplable multivariable systems [25].…”

mentioning

confidence: 99%

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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.

show abstract

S (k + d + 1) = S (k + d) = 0_{(p \times 1)}

with S ( k + d ) as the sliding function vector in the case of classical sliding mode control. It is given by

\begin{matrix} S (k + d) & = C (q^{- 1}) ξ (q) (Y (k) - Y_{r} (k)) \\ = C (q^{- 1}) ξ (q) E (k) \end{matrix}

where C ( q −1 ) is a polynomial matrix defined as

C (q^{- 1}) = I_{p} + C_{1} q^{- 1} + \dots + C_{n_{C}} q^{- n_{C}}; \dim C_{τ_{3}} = (p, p); τ_{3} \in [1, n_{C}]

E ( k ) is the error vector defined as

E (k) = Y (k) - Y_{r} (k)

Y r ( k ) is the ...…”

Section: Discrete Second‐order Sliding Mode Control For Multivariablementioning

confidence: 99%

“…For decouplable multivariable systems, the interactor matrix is expressed by

ξ (q) = q^{d} I_{p}

Therefore, the sliding function vector becomes

\begin{matrix} S (k) & = C (q^{- 1}) (Y (k) - Y_{r} (k)) \\ = C (q^{- 1}) E (k) \end{matrix}

In the case of second‐order sliding mode control, the new sliding function vector σ ( k ) is selected by the following expression :

σ (k) = S ((), k) + β S ((), k - 1)

C ((), q^{- 1}) = \overset{F}{true} ((), q^{- 1}) A ((), q^{- 1}) normalΔ ((), q^{- 1}) + q^{- 1} \overset{G}{true} ((), q^{- 1})

where

\overset{F}{true} ((), q^{- 1}) = I_{p}

\overset{G}{true} ((), q^{- 1}) = {\overset{G}{true}}_{0} + {\overset{G}{true}}_{1} q^{-<...>}

…”

Section: Discrete Second‐order Sliding Mode Control For Multivariablementioning

confidence: 99%

“…In the case of second-order sliding mode control, the new sliding function vector .k/ is selected by the following expression [23,25]:…”

Section: Discrete Second-order Sliding Mode Control For Decouplable Mmentioning

confidence: 99%

Section: Discrete Second-order Sliding Mode Control For Decouplable Mmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

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

Self Cite

View full text Add to dashboard Cite

show abstract

Preliminaries

Sharma¹,

Janardhanan²

2018

Discrete-Time Higher Order Sliding Mode

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Synthesis of an Optimal Sliding Function Using LMIs Approach for Time Delay Systems

Houda

Dehri

Nouri

2016

Applications of Sliding Mode Control

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Stability analysis of discrete input output second order sliding mode control

Cited by 13 publications

References 18 publications

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

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

Preliminaries

Synthesis of an Optimal Sliding Function Using LMIs Approach for Time Delay Systems

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