A new sliding surface for discrete second order sliding mode control of time delay systems

Nizar, Abdennebi; Said, Nouri Ahmed

doi:10.1109/ssd.2012.6198049

Cited by 7 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%

“…More recently, Kim et al used LMI numerical tools for solving this optimization problem. Indeed, LMI is currently a research area where a wide number of proposals for specifying sliding surfaces have been continuously given …”

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 99%

“…The solution of this problem given in the works of Nizar et al in terms of LMI will be detailed in the LMI solution of the numerical example 2. Yan et al introduced one time‐dependent sufficient condition for the existence of the sliding surface in LMI terms, giving one solution for specifying

C \in {double-struckR}^{m \times false(n - m false)}

and proving that the following sliding surface definition:

s false(k false) = K L_{1}^{- 1} x_{1} false(k false) + x_{2} false(k false) = 0

guarantees that the sliding dynamic is asymptotically stable if, for certain positive number κ , there exists some positive scalars δ 1 , δ 2 > 0 and the ( n − m ) × ( n − m ) matrices

L_{1} > 0, L_{2}, L_{3}, U_{1} > 0, U_{2} > 0, {\overset{W}{true}}_{1}, {\overset{W}{true}}_{2}, {\overset{W}{true}}_{3}

and the matrix

K \in {double-struckR}^{m \times false(n - m false)}

such that the following LMIs is fulfilled:

([], \begin{matrix} center \\ array Θ_{1} & array Θ_{2} & array 0 & array 0 & array L_{1} & array ϑ & array L_{2}^{T} & array L_{2}^{T} \\ array * & array Θ_{3} & array ς & array 0 & array 0 & array 0 & array L_{3}^{T} & array L_{3}^{T} \\ array * & array * & array - U_{1} & array U_{1} H_{} \end{matrix})

…”

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 99%

“…Solution Finally, for illustrating an LMI approach for the design of SM switching surface, we have chosen the proposition given in the work of Nizar and Saïd, where the system dynamics is described as follows:

x false(k + 1 false) = normalΦ x false(k false) + {normalΦ}_{d} x false(k - h false) + normalΓ u false(k false),

where h is delay parameter and

{normalΦ}_{d} \in {double-struckR}^{n \times n}

Section: Illustrative Examplesmentioning

confidence: 99%

x false(k + 1 false) = normalΦ x false(k false) + {normalΦ}_{d} x false(k - h false) + normalΓ u false(k false),

where h is delay parameter and

{normalΦ}_{d} \in {double-struckR}^{n \times n}

. For this example, Φ d = 0 because the time delay is not considered because we have selected the nominal system obtained in the work of Xie et al Following the LMI designing proposed in the works of Nizar et al, the system must be transformed to the regular form, then the system representation given in is needed. Then, using theorem 1 in the work of Nizar et al, where it is proven that a system given in is asymptotically stable via DVSC if there exists symmetrical matrices

P, Q \in {double-struckR}^{n \times n}

such that the following LMI holds:

([], \begin{matrix} center \\ array W - L & array * & array * \\ array 0 & array - L & array * \\ array Φ_{11} L + Φ_{12} w_{1} & array Φ_{d 11} L + Φ_{d 12} w_{1} & array - L \end{matrix}) < 0,

where W = P −1 QP −1 , L = P −1 , w 1 = − CL , and where (∗) denotes the transposed elements in the symmetric location.…”

Section: Illustrative Examplesmentioning

confidence: 99%

See 4 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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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%

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 99%

C \in {double-struckR}^{m \times false(n - m false)}

and proving that the following sliding surface definition:

s false(k false) = K L_{1}^{- 1} x_{1} false(k false) + x_{2} false(k false) = 0

guarantees that the sliding dynamic is asymptotically stable if, for certain positive number κ , there exists some positive scalars δ 1 , δ 2 > 0 and the ( n − m ) × ( n − m ) matrices

L_{1} > 0, L_{2}, L_{3}, U_{1} > 0, U_{2} > 0, {\overset{W}{true}}_{1}, {\overset{W}{true}}_{2}, {\overset{W}{true}}_{3}

and the matrix

K \in {double-struckR}^{m \times false(n - m false)}

such that the following LMIs is fulfilled:

([], \begin{matrix} center \\ array Θ_{1} & array Θ_{2} & array 0 & array 0 & array L_{1} & array ϑ & array L_{2}^{T} & array L_{2}^{T} \\ array * & array Θ_{3} & array ς & array 0 & array 0 & array 0 & array L_{3}^{T} & array L_{3}^{T} \\ array * & array * & array - U_{1} & array U_{1} H_{} \end{matrix})

…”

Section: Discrete‐time Lti Systems Sliding Surface Designmentioning

confidence: 99%

x false(k + 1 false) = normalΦ x false(k false) + {normalΦ}_{d} x false(k - h false) + normalΓ u false(k false),

where h is delay parameter and

{normalΦ}_{d} \in {double-struckR}^{n \times n}

Section: Illustrative Examplesmentioning

confidence: 99%

x false(k + 1 false) = normalΦ x false(k false) + {normalΦ}_{d} x false(k - h false) + normalΓ u false(k false),

where h is delay parameter and

{normalΦ}_{d} \in {double-struckR}^{n \times n}

P, Q \in {double-struckR}^{n \times n}

such that the following LMI holds:

([], \begin{matrix} center \\ array W - L & array * & array * \\ array 0 & array - L & array * \\ array Φ_{11} L + Φ_{12} w_{1} & array Φ_{d 11} L + Φ_{d 12} w_{1} & array - L \end{matrix}) < 0,

where W = P −1 QP −1 , L = P −1 , w 1 = − CL , and where (∗) denotes the transposed elements in the symmetric location.…”

Section: Illustrative Examplesmentioning

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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A New Sliding Function for Discrete Predictive Sliding Mode Control of Time Delay Systems

Nizar

Houda

Said

2013

Int. J. Autom. Comput.

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The control of time delay systems is still an open area for research. This paper proposes an enhanced model predictive discrete-time sliding mode control with a new sliding function for a linear system with state delay. Firstly, a new sliding function including a present value and a past value of the state, called dynamic surface, is designed by means of linear matrix inequalities (LMIs). Then, using this dynamic function and the rolling optimization method in the predictive control strategy, a discrete predictive sliding mode controller is synthesized. This new strategy is proposed to eliminate the undesirable effect of the delay term in the closed loop system. Also, the designed control strategy is more robust, and has a chattering reduction property and a faster convergence of the system s state. Finally, a numerical example is given to illustrate the effectiveness of the proposed control.

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