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# Journal of Professor & SPaC member

Last Updated : 03/2016
• ## An improved fragmentation approach to sampled-data synchronization of chaotic Lur’e systems

JunMin Park, Seok Young Lee, PooGyeon Park

Nonlinear Analysis: Hybrid Systems

Abstract This paper considers the sampled-data synchronization problem for two identical chaotic Lur’e systems. Under aperiodic samplings, the output of the chaotic Lur’e systems is available only at sampling instants. An improved fragmentation approach addresses a system description of the fragmented state located between the last sampling instant and present time. The sampled output of the fragmented state system is updated only at sampling instants, like the identical chaotic Lur’e systems. Thus, utilizing the additional information on the fragmented state system helps in handling the sampling interval. In the looped-functional framework, a sampled-data synchronization criterion is proposed for stability analysis and control synthesis about stabilization in terms of linear matrix inequalities. Simulation results show the effectiveness of the proposed criterion.
• ## A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input

In Seok Park, Nam Kyu Kwon, PooGyeon Park

Nonlinear Analysis: Hybrid Systems

Abstract This paper proposes a linear programming approach for stabilization of positive Markovian jump systems (PMJSs) with a saturated single input. The proposed approach first derives a sufficient condition for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. By introducing an intermediate scalar whose absolute value is less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input matrix and constructing a special form of the controller gains, this approach obtains a modified condition applicable for the linear programming. Finally, four numerical examples show that the proposed approach gives the larger domain of attraction than the existing approach based on the quadratic Lyapunov function.
• ## An improved stability criteria for neutral-type Lur’e systems with time-varying delays

JunMin Park, Seok Young Lee, PooGyeon Park

Journal of the Franklin Institute

Abstract This paper improves stability criteria for neutral-type Lur’e systems with time-varying delays, where the nonlinearity satisfies sector and slope restrictions. A proposed Lyapunov-Krasovskii functional consisting of a quadratic term and integral terms for the time-varying delays and the nonlinearities, has four different characteristics. First, the quadratic term utilizes not only the current and delayed states but also the nonlinear vectors. Second, the integral terms for nonlinearities fully exploit the characteristics of sector and slope restrictions. Third, the integral terms for nonlinearities also exploit the characteristic of incremental restriction induced from the slope restriction. Fourth, this paper utilizes a vector related to the time derivative of the neutral delayed state to handle the neutral delay. Based on the proposed Lyapunov-Krasovskii functional, the improved stability criteria are derived in terms of linear matrix inequalities. Numerical examples show that the proposed criteria present less conservative results than the previous criteria.
• ## Affine Bessel–Legendre inequality: Application to stability analysis for systems with time-varying delays

Won Il Lee, Seok Young Lee, PooGyeon Park

Automatica

Abstract Recently, some novel inequalities have been proposed such as the auxiliary function-based integral inequality and the Bessel–Legendre inequality which can be obtained from the former by choosing Legendre polynomials as auxiliary functions. These inequalities have been successfully applied to systems with constant delays but there have been some difficulties in application to systems with time-varying delays since the resulting bounds contain the reciprocal convexity which may not be tractable as it is. This paper proposes an equivalent form of the Bessel–Legendre inequality, which has the advantage of being easily applied to systems with time-varying delays without the reciprocal convexity.
• ## H∞control for Markovian jump fuzzy systems with partly unknown transition rates and input saturation

In Seok Park, Nam Kyu Kwon, PooGyeon Park

Journal of the Franklin Institute

Abstract This paper considers less conservative conditions of H∞ control for Markovian jump fuzzy systems (MJFSs) with partly unknown transition rates and input saturation. To find H∞control for level γ, a set invariance condition and the stabilization conditions are first formulated in terms of parameterized linear matrix inequalities (PLMIs) by considering the properties of transition rates. Then, to derive the less conservative stabilization conditions, all possible slack variables are incorporated into the relaxation process with fully considering the property of the fuzzy weights. Two numerical examples are provided to illustrate the effectiveness of the proposed method.
• ## Output-feedback control for singular Markovian jump systems with input saturation

Chan-eun Park, Nam Kyu Kwon, PooGyeon Park

Nonlinear Dynamics

Abstract This paper considers the problem of dynamic output-feedback stabilization for singular Markovian jump systems with input saturation. The stabilization and set invariance conditions are first formulated in terms of non-convex matrix inequalities which is not linear matrix inequalities (LMIs). This paper, however, successfully derives the necessary and sufficient conditions for the non-convex inequalities in terms of LMIs. Also, an optimization problem is formulated to find the largest contractively invariant set in mean square sense of the closed-loop systems. Two numerical examples show the validity of the derived results.
• ## Orthogonal-polynomials-based integral inequality and its applications to systems with additive time-varying delays

Seok Young Lee, Won Il Lee, PooGyeon Park

Journal of the Franklin Institute

Abstract Recently, a polynomials-based integral inequality was proposed by extending the Moon’s inequality into a generic formulation. By imposing certain structures on the slack matrices of this integral inequality, this paper proposes an orthogonal-polynomials-based integral inequality which has lower computational burden than the polynomials-based integral inequality while maintaining the same conservatism. Further, this paper provides notes on relations among recent general integral inequalities constructed with arbitrary degree polynomials. In these notes, it is shown that the proposed integral inequality is superior to the Bessel–Legendre (B–L) inequality and the polynomials-based integral inequality in terms of the conservatism and computational burden, respectively. Moreover, the effectiveness of the proposed method is demonstrated by an illustrative example of stability analysis for systems with additive time-varying delays.
• ## Delays-Dependent region partitioning approach for stability criterion of linear systems with multiple time-varying delays

Kab Seok Ko, Won Il Lee, PooGyeon Park, Dan Keun Sung

Automatica

Abstract This paper considers a delay-dependent stability criterion for linear systems with multiple time-varying delays. To exploit all possible information for the relationships among the marginally delayed states (x(t−τiM)x(t−τi+1M)), the exactly delayed states (x(t−τi(t)),x(t−τi+1(t))), and the current state x(t) for each pair (i,i+1) of time-varying delays, a delays-dependent region partitioning approach in double integral terms is proposed. By applying the Wirtinger-based integral inequality and the reciprocally convex approach to terms resulted from the region partitioning, a stability criterion is derived in terms of linear matrix inequalities. Numerical examples show that the resulting criterion outperforms the existing one in literature.
• ## \mathcal {H} _\infty  H∞ state-feedback control for continuous-time Markovian jump fuzzy systems using a fuzzy weighting-dependent Lyapunov function

Nam Kyu Kwon, In Seok Park, PooGyeon Park

Nonlinear Dynamics

Abstract This paper proposes a method for designing an HH∞ state-feedback fuzzy controller for continuous-time Markovian jump fuzzy systems (MJFSs) with partly unknown transition rates. To find HH∞ control for level γγ, the stabilization conditions are first formulated in terms of parameterized linear matrix inequalities (PLMIs) for the MJFSs based on a fuzzy weighting-dependent Lyapunov function. Besides, to derive less conservative stabilization conditions, all possible slack variables are incorporated into the relaxation process with fully considering the property of the fuzzy weights. Finally, three examples are provided to verify the effectiveness of the proposed method.
• ## Stabilization condition of one-step receding horizon control for discrete-time linear systems with model uncertainties

Nam Kyu Kwon, In Seok Park, PooGyeon Park, Chaneun Park

IEEE Transactions on Automatic Control

Abstract For the dynamic output-feedback stabilization of continuous-time singular Markovian jump systems, this paper introduces the necessary and sufficient condition, whereas the previous research works suggested the sufficient conditions. A special choice of the block entries of Lyapunov matrices leads to derive the necessary and sufficient condition in terms of linear matrix inequalities. A numerical example shows the validity of the derived results.