• Title, Summary, Keyword: Lyapunov-like function

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Weak attractors and Lyapunov-like functions

  • Kim, Jong-Myung;Kye, Young-Hee;Lee, Keon-Hee
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.457-462
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    • 1996
  • Recently Hurley [3] proved that if A is a weak attractor of a discrete dyanamical system f then there exists a Lyapunov-like function for A. The purpose of this note is to study whether the converse of the above result does hold or not.

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Uniform ultimate boundedness of control systems with matched and mismatched uncertainties by Lyapunov-like method

  • Sung, Yulwan;Shibata, Hiroshi;Park, Chang-Young;Kwo, Oh-Kyu
    • 제어로봇시스템학회:학술대회논문집
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    • pp.119-122
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    • 1996
  • The recently proposed control method using a Lyapunov-like function can give global asymptotic stability to a system with mismatched uncertainties if the uncertainties are bounded by a known function and the uncontrolled system is locally and asymptotically stable. In this paper, we modify the method so that it can be applied to a system not satisfying the latter condition without deteriorating qualitative performance. The assured stability in this case is uniform ultimate boundedness which is as useful as global asymptotic stability in the sense that it is global and the bound can be taken arbitrarily small. By the proposed control law we can deal with both matched and mismatched uncertain systems. The above facts conclude that Lyapunov-like control method is superior to any other Lyapunov direct methods in its applicability to uncertain systems.

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ON STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS BY LYAPUNOV'S DIRECT METHOD

  • Park, Jong-Yeoul;Phat, Vu-Ngoc;Jung, Il-Hyo
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.805-821
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    • 2000
  • The paper deals with asymtotic stabillity of nonlinear nonautinomous systems by Lyapunov's direct method. The proposed Lyapunov-like function V(t, x) needs not be continuous in t and Lipschitz in x in a Banach space. The class of systems considered is allowed to be nonautonomous and infinite-dimensional and we relax the boundedness, the Lipschitz assumption on the system and the definite decrescent condition on the Lyapunov function.

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CRITERIA FOR A NEW CPNTEPT OF STABILITY

  • Lakshmikanthan, V.
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.657-664
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    • 2000
  • A new concept of stability that includes Lyapunov and orbital stabilities and leads to concepts in between them is discussed in terms of a given topology of the function space. The criteria for such new concepts to hold are investigted employing suitably Lyapunov-like functions and the comparison principle.

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Uniform ultimate boundedness and global asympotic stabilization for systems with mis-matched uncertainties (비 매칭 불확실성이 있는 비선형시스템의 균일 종국적 유계성 및 대역적 점근 안정성)

  • 장충환;성열완;이건일
    • Journal of the Korean Institute of Telematics and Electronics S
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    • v.35S no.7
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    • pp.29-36
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    • 1998
  • In this paper we propose a control law using a Lyapunov-like function that makes stable the systems which have mis-matched uncertainties. The existing control law using a Lyapunov-like function, which gives global saymptotic stability, is designed under the assumption of a targetsystem to be stable locally. But we broaden here the class of target systems by designing the control law which can give uniform ultimate boundedness to even the systems not satisfing the locally asymptotic stability. And we also show that the control law giving global asymptotic stability can be designed more systematically through using the uniform ultimate boundedness.

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An Adaptive Control Method of Robot Manipulators using RBFN (RBFN을 이용한 로봇 매니퓰레이터의 적응제어 방법)

  • 이민중;최영규;박진현
    • 제어로봇시스템학회:학술대회논문집
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    • pp.420-420
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    • 2000
  • In this paper, we propose an adaptive controller using RBFN(radial basis function network) for robot manipulators The structure of the proposed controller consists of a RBFN and VSC-1 ike control. RBFN is used in order to approximate かon system, and VSC-like control to guarantee robustness On the basis of the Lyapunov stability theorem, we guarantee the stability for the total system. And the learning law of RBFN is established by the Lyapunov method, Finally, we apply the proposed controller to tracking control for a 2 link SCARA type robot manipulator.

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GLOBAL ROBUST STABILITY OF TIME-DELAY SYSTEMS WITH DISCONTINUOUS ACTIVATION FUNCTIONS UNDER POLYTOPIC PARAMETER UNCERTAINTIES

  • Wang, Zengyun;Huang, Lihong;Zuo, Yi;Zhang, Lingling
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.89-102
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    • 2010
  • This paper concerns the problem of global robust stability of a time-delay discontinuous system with a positive-defined connection matrix under polytopic-type uncertainty. In order to give the stability condition, we firstly address the existence of solution and equilibrium point based on the properties of M-matrix, Lyapunov-like approach and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. Second, we give the delay-independent and delay-dependent stability condition in terms of linear matrix inequalities (LMIs), and based on Lyapunov function and the properties of the convex sets. One numerical example demonstrate the validity of the proposed criteria.

The Sliding Controller designed by the Indirect Adaptive Fuzzy Control Method (간접 적응 퍼지 제어기법에 의한 슬라이딩 제어기 설계)

  • Choi, Chang-Ho;Yim, Wha-Yeong
    • Proceedings of the KIEE Conference
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    • pp.2283-2286
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    • 2000
  • Sliding control is a powerful approach to controlling nonlinear and uncertain systems. Conventional sliding mode control suffer' from high control gain and chattering problem. also it needs mathematic! modeling equations for control systems. A Fuzzy controller is endowed with control rules and membership function that are constructed on the knowledge of expert, as like intuition and experience. but It is very difficult to obtain the exact values which are the membership function and consequent parameters. In this paper, without mathematical modeling equations, the plant parameters in sliding mode are estimated by the indirect adaptive fuzzy method. the proposed algorithm could analyze the system's stability and convergence behavior using Lyapunov theory. so sliding modes are reconstructed and decreased tracking error. moreover convergence time took a short. An example of inverted pendulum is given for demonstration of the robustness of proposed methodology.

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Exponential Stabilization of a Class of Underactuated Mechanical Systems using Dynamic Surface Control

  • Qaiser, Nadeem;Iqbal, Naeem;Hussain, Amir;Qaiser, Naeem
    • International Journal of Control, Automation, and Systems
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    • v.5 no.5
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    • pp.547-558
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    • 2007
  • This paper proposes a simpler solution to the stabilization problem of a special class of nonlinear underactuated mechanical systems which includes widely studied benchmark systems like Inertia Wheel Pendulum, TORA and Acrobot. Complex internal dynamics and lack of exact feedback linearizibility of these systems makes design of control law a challenging task. Stabilization of these systems has been achieved using Energy Shaping and damping injection and Backstepping technique. Former results in hybrid or switching architectures that make stability analysis complicated whereas use of backstepping some times requires closed form explicit solutions of highly nonlinear equations resulting from partial feedback linearization. It also exhibits the phenomenon of explosions of terms resulting in a highly complicated control law. Exploiting recently introduced Dynamic Surface Control technique and using control Lyapunov function method, a novel nonlinear controller design is presented as a solution to these problems. The stability of the closed loop system is analyzed by exploiting its two-time scale nature and applying concepts from Singular Perturbation Theory. The design procedure is shown to be simpler and more intuitive than existing designs. Design has been applied to important benchmark systems belonging to the class demonstrating controller design simplicity. Advantages over conventional Energy Shaping and Backstepping controllers are analyzed theoretically and performance is verified using numerical simulations.

A Hybrid Guidance Law for a Strapdown Seeker to Maintain Lock-on Conditions against High Speed Targets

  • Lee, Chae Heun;Hyun, Chul;Lee, Jang Gyu;Choi, Jin Yung;Sung, Sangkyung
    • Journal of Electrical Engineering and Technology
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    • v.8 no.1
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    • pp.190-196
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    • 2013
  • This paper proposes a new guidance law, which considers the Field of View (FOV) of the seeker when a missile has a strapdown seeker mounted instead of a gimbal seeker. When a strapdown seeker, which has a narrow FOV, is used for tracking a target, the FOV of the seeker is an important consideration for guidance performance metrics such as miss distance. We propose a new guidance law called hybrid guidance (HG) to address the shortcomings of conventional guidance laws such as proportional navigation guidance (PNG), which cannot maintain lock-on conditions against high speed targets due to the narrow FOV of the strapdown seeker. The aim of the HG law is to null miss distance and to maintain the look angle within the FOV of the strapdown seeker. In order to achieve this goal, we combine two guidance laws in the HG law. One is a PNG law to null the LOS rate, and the other is a sliding mode guidance (SMG) law derived to keep the look angle within the FOV by employing a Lyapunov-like function with a sliding mode control methodology. We also propose a method to switch these two guidance laws at certain look angles for better guidance performance.