Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

Koskie, S.; Coumarbatch, C.; Gajić, Zoran

doi:10.1016/j.amc.2010.02.040

Cited by 14 publications

(10 citation statements)

References 14 publications

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“…Remark The conventional singularly perturbed approaches, for example , decomposed the original linear systems into the slow and fast subsystems and designed a composite controllers for its relative slow and fast subsystems. However, the aforementioned approaches cannot be employed to nonstandard SPSs (a class of singular perturbed systems with a singular matrix A 22 ) because A 22 is assumed as a nonsingular matrix during design controller:

\begin{matrix} aligned \\ right ẋ_{1} MathClass-open(t MathClass-close) & left = A_{11} x_{1} MathClass-open(t MathClass-close) + A_{12} x_{2} MathClass-open(t MathClass-close) + B_{1} u MathClass-open(t MathClass-close) & right \\ right ϵ ẋ_{2} MathClass-open(t MathClass-close) & left = A_{21} x_{1} MathClass-open(t MathClass-close) + A_{22} x_{2} MathClass-open(t MathClass-close) + B_{2} u MathClass-open(t MathClass-close), \end{matrix}

where x 1 ( t ) are slow state vectors, x 2 ( t ) are fast state vectors, and other parameters are described in .…”

Section: Resultsmentioning

confidence: 99%

“…The conventional singularly perturbed approaches, for example [34,35], decomposed the original linear systems into the slow and fast subsystems (37) and designed a composite controllers for its relative slow and fast subsystems. However, the aforementioned approaches cannot be employed to nonstandard SPSs (a class of singular perturbed systems with a singular matrix A 22 ) because A 22 is assumed as a nonsingular matrix during design controller:…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

Chen¹,

Sun

Min

et al. 2012

Intl J Robust & Nonlinear

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SUMMARYThis paper presents the novel approaches of designing robust fuzzy static output feedback H ∞ controller for a class of nonlinear singularly perturbed systems. Specifically, the considered system is approximated by a fuzzy singularly perturbed model. With the use of linear matrix inequality (LMI) methods, two methods are provided to design fuzzy static output feedback H ∞ controllers. The resulted controllers can guarantee that the closed‐loop systems are asymptotically stable and satisfy H ∞ performances for sufficiently small ϵ. In contrast to the existing results, the proposed approaches have two advantages: (i) the gains of controller are solved directly by a set of ϵ‐independent LMIs, and therefore, the problem of selecting the initial values in iterative LMIs algorithm can be avoided, and (ii) the smaller control input efforts are needed. The given methods are easy to implement and can be applied to both standard and nonstandard nonlinear singularly perturbed systems. Two numerical examples are provided to illustrate the effectiveness of the developed methods. Copyright © 2012 John Wiley & Sons, Ltd.

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\begin{matrix} aligned \\ right ẋ_{1} MathClass-open(t MathClass-close) & left = A_{11} x_{1} MathClass-open(t MathClass-close) + A_{12} x_{2} MathClass-open(t MathClass-close) + B_{1} u MathClass-open(t MathClass-close) & right \\ right ϵ ẋ_{2} MathClass-open(t MathClass-close) & left = A_{21} x_{1} MathClass-open(t MathClass-close) + A_{22} x_{2} MathClass-open(t MathClass-close) + B_{2} u MathClass-open(t MathClass-close), \end{matrix}

where x 1 ( t ) are slow state vectors, x 2 ( t ) are fast state vectors, and other parameters are described in .…”

Section: Resultsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

Chen¹,

Sun

Min

et al. 2012

Intl J Robust & Nonlinear

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“…where Q f and R f are symmetric positive definite matrices; P is the only solution for the Ricatti equation [63]:…”

Section: The Flexible-link Fast Subsystemmentioning

confidence: 99%

Robust fuzzy sliding mode control and vibration suppression of free-floating flexible-link and flexible-joints space manipulator with external interference and uncertain parameter

Xie

Chen

2021

Robotica

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The flexibility of the free-floating flexible space manipulator system’s link and joint may affect the control accuracy and cause vibrations. We studied the dynamics modeling, joint trajectory tracking control, and vibration suppressing problem of free-floating flexible-link and flexible-joints space manipulator system with external interference and uncertain parameter. The system’s dynamic equations are established using linear momentum conservation, angular momentum conservation, assumed mode method, and Lagrange equation. Then, the system’s singular perturbation model is established, and a hybrid control is presented. For the slow subsystem, a robust fuzzy sliding mode control is proposed to realize the joint desired trajectory tracking. For the fast subsystem, a speed difference feedback control and a linear-quadratic optimal control are designed to suppress the vibration caused by the flexible joints and the flexible link separately. The simulation comparison experiments under different conditions are taken. The simulate results demonstrate the proposed hybrid control’s validity.

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“…To remove the numerical stiffness in the Lyapunov equation given by expression (50), the latter will be decomposed in slow and fast parts [34]. The structure of L(ε) is assumed to be of the form 12 can be approximated, respectively, by the slow controller gain and the fast controller gain [35].…”

Section: Simplifying Of the Lyapunov Equationmentioning

confidence: 99%

Reconfigurable Control of Two-Time Scale Systems in Presence of Additive Faults

Tellili

Abdelkrim

Challouf

et al. 2015

ITC

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This work presents an adaptive approach for fault tolerant control of singularly perturbed systems, where both actuator and sensor faults are examined in presence of external disturbances. For sensor faults, an adaptive controller is designed based on an output-feedback control scheme. The feedback controller gain is determined in order to stabilize the closed-loop system in the fault free case and vanishing disturbance, while the additive gain is updated using an adaptive law to compensate for the sensor faults and the external disturbances. To correct the actuator faults, a state-feedback control method based on adaptive mechanism is considered. The both proposed controllers depend on the singular perturbation parameter ε leading to ill-conditioned problems. A well-posed problem is obtained by simplifying the Lyapunov equations and subsequently the controllers using the singular perturbation method and the reduced subsystems yielding to an ε-independent controller. The control scheme, designed based on the Lyapunov stability theory, guarantees asymptotic stability in presence of additive faults and external disturbances provided the singular perturbation parameter is sufficiently small. Finally, a numerical example is presented to demonstrate the effectiveness of the obtained results.

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Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

Cited by 14 publications

References 14 publications

New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

Robust fuzzy sliding mode control and vibration suppression of free-floating flexible-link and flexible-joints space manipulator with external interference and uncertain parameter

Reconfigurable Control of Two-Time Scale Systems in Presence of Additive Faults

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Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

Cited by 14 publications

References 14 publications

New results on static output feedback H ∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

New results on static output feedback H ∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

Robust fuzzy sliding mode control and vibration suppression of free-floating flexible-link and flexible-joints space manipulator with external interference and uncertain parameter

Reconfigurable Control of Two-Time Scale Systems in Presence of Additive Faults

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New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach

New results on static output feedback H_∞ control for fuzzy singularly perturbed systems: a linear matrix inequality approach