A new discrete sliding mode control for systems with time varying delay: Multi-Structure approach

Nizar, Abdennebi; Ltaïef, Majda; Said, Nouri Ahmed

doi:10.1109/ssd.2013.6564098

Cited by 3 publications

(13 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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}

. 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%

“…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. The matrices Φ 11 , Φ 12 , Φ d 11 , Φ d 12 are given from the regular representation of .Therefore, the switching surface is achieved as follows:

C = ([], - w_{1} L^{- 1} I_{m}) M,

where M is given in .…”

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

View full text Add to dashboard Cite

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.

show abstract

“…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}

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%

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,

C = ([], - w_{1} L^{- 1} I_{m}) M,

where M is given in .…”

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

View full text Add to dashboard Cite

show abstract

Delayed sliding surface for time delay system

Yassine

Mehdi

Nouri

2017

2017 18th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA)

View full text Add to dashboard Cite

Análisis y Diseño de Algoritmos de Control Discreto de Sistemas MIMO Lineales y No Lineales Aplicando Técnicas de Control de Estructura Variable

Delgado¹

View full text Add to dashboard Cite

Over the last four decades, variable structure controllers with sliding mode (VSC-SM ) have been extensively studied, demonstrating to be a robust solution among robust nonlinear processes control techniques. Since the late 80s, several research works have been focused on the application of VSC controllers applied to discrete time or sampled data systems, which are known as DVSC-SM, where the most extensive source of analysis has been devoted to the robustness and invariance properties of VSC-SM controllers when applied to discrete systems. The main aim of this doctoral thesis work is to study, analyze and propose a design scheme of DVSC-SM controllers for lineal and nonlinear multivariable discrete time processes. For this purpose, a new design philosophy is proposed, where various design features have been considered that have not been analyzed in DVSC design approaches. Among them, the physical limitations and the nonideal dynamic sliding mode dynamics. The most innovative aspect is the inclusion of a new design methodology applied to lineal sliding surfaces MIMO systems and the extension to nonlinear multivariable systems, in addition to a new robust control law applied to lineal and nonlinear MIMO systems, including a chattering reduction scheme. Finally, to illustrate the e ciency of the proposed schemes, several numerical examples applied to lineal and nonlinear systems are included.

show abstract

A new discrete sliding mode control for systems with time varying delay: Multi-Structure approach

Cited by 3 publications

References 21 publications

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

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

Delayed sliding surface for time delay system

Análisis y Diseño de Algoritmos de Control Discreto de Sistemas MIMO Lineales y No Lineales Aplicando Técnicas de Control de Estructura Variable

Contact Info

Product

Resources

About