scholarly journals Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Carlos Aguilar-Ibañez ◽  
Miguel S. Suarez-Castanon ◽  
José de Jesús Rubio
Keyword(s):  
Lyapunov Function ◽  
Time Derivative ◽  
Dissipation Function ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Lyapunov Approach ◽  
Beam System ◽  
Inverse Stability ◽  
The Stability ◽  
Invariance Theorem

A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.

A multiple convex Lyapunov function for asynchronous control of discrete-time switched systems

2021 ◽  
pp. 014233122110265
Author(s):  
Jiahao Cui ◽  
Ruihua Wang ◽  
Shumin Fei
Keyword(s):  
Lyapunov Function ◽  
Convex Function ◽  
Discrete Time ◽  
Switched Systems ◽  
Closed Loop System ◽  
Asynchronous Control ◽  
Control Scheme ◽  
Function Combination ◽  
The Stability ◽  
Stability Results

In this paper, the problem of asynchronous control for a class of discrete-time switched systems is investigated under mode-dependent integrated dwell time (MDIDT) switching. By constructing a time-dependent convex function, a multiple convex Lyapunov function (MCLF) is firstly proposed for the asynchronous control of the switched systems. Under the MDIDT switching strategy, the matching interval is divided reasonably, and the convex function combination is constructed on the partitioned interval. Under asynchronous switching, the Lyapunov function is continuous when the subsystem mode is switched, but discrete when the controller mode is changed. Then, the increase of the Lyapunov function in the mismatched interval will be offset by the attenuation in the matched interval. In light of these merits, the stability results of the system are deduced, and the asynchronous controller is devised to guarantee the globally uniformly exponentially stability of the closed-loop system. Comparing with the traditional asynchronous control methods, the proposed method has less conservative results and larger stability regions. Finally, a numerical example and an application example are demonstrated to verify the validity and superiority of the asynchronous control scheme.

Refined Stability Conditions for Linear Uncertain Systems With Multiple Time-Varying Delays

2006 ◽  
Vol 129 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Chih-Peng Huang
Keyword(s):  
Uncertain Systems ◽  
State Feedback ◽  
Multiple Time ◽  
Time Varying ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Time Varying Delay ◽  
The Stability ◽  
Varying Delay ◽  
Explicit Criteria

This paper mainly proposes distinct criteria for the stability analysis and stabilization of linear uncertain systems with time-varying delays. Based on the Lyapunov theorem, a sufficient condition of the unforced systems with single time-varying delay is first derived. By involving a memoryless state feedback controller, the condition will be extended to treat with the resulting closed-loop system. These explicit criteria can be reformulated in LMIs forms, so we will readily verify the stability or design a stabilizing controller by the current LMI solver. Furthermore, the considered systems with multiple time-varying delays are similarly addressed. Numerical examples are given to demonstrate that the proposed approach is effective and valid.

On the Global Stability of a Class of Nonlinear Time-Varying Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 399-406
Author(s):  
N. N. Puri
Keyword(s):  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Equilibrium Solution ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
State Variables ◽  
Time Varying Systems ◽  
Negative Semidefinite ◽  
The Stability ◽  
Conditions Of Stability

In this paper the problem of the stability of motion of the equilibrium solution x1 = x2… = xn = 0 is studied, in the sense of Lyapunov, for a class of systems represented by a system of differential equations dxi/dt = Fi (x1, x2…xn, t), i = 1, 2…n or x˙ = A (x,t)x. Various x1 are known as state variables and Fi (0, 0…0, ∞) = 0. The various elements of square matrix A (x, t) are functions of time as well as functions of state variables x. Two different methods for generating Lyapunov functions are developed. In the first method the differential equation is multiplied by various state variables and integrated by parts to generate a proper Lyapunov function and a number of matrices α, α1…αn, S1, S2…Sn. The second method assumes a quadratic Lyapunov function V = [x′S(x,t)x], x′ being the transpose of x. The elements of S(x,t) may be functions of time and the state variables or constants. The time derivative V˙ is given by V˙ = x′[B′A + S˙]x = x′T(t,x)x where B x gives the gradient ∇V, and S˙ is defined as ∂S/∂t. For the equilibrium solution x1 = x2… = xn = 0 to be stable it is required that V˙ should be negative definite or negative semidefinite and V should be positive definite. These considerations determine the sufficient conditions of stability.

An Image-Based Visual Servoing for Manipulator

2011 ◽  
Vol 186 ◽  
pp. 277-280
Author(s):  
Shi Xiang Tian ◽  
Sheng Ze Wang
Keyword(s):  
Tracking Control ◽  
Visual Servoing ◽  
Closed Loop ◽  
Robot Manipulator ◽  
Single Camera ◽  
Closed Loop System ◽  
Simulation Experiments ◽  
Lyapunov Approach ◽  
The Stability ◽  
Three Degree Of Freedom

In this paper, an image-based controller for tracking control of robot manipulators using a single camera is proposed. The proposed controller has robustness to parametric uncertainties of the robot manipulator and compensation for uncertainties included in the image Jacobian. The stability of the closed-loop system is proved by Lyapunov approach. The performance of the proposed method is demonstrated by simulation experiments on a 3-link robot manipulator with three degree of freedom.

Adaptive attitude-tracking control of spacecraft considering on-orbit refuelling

2020 ◽  
pp. 014233122097313
Author(s):  
Yiqi Xu
Keyword(s):  
Tracking Control ◽  
Closed Loop ◽  
Closed Loop System ◽  
Tracking Errors ◽  
Attitude Tracking ◽  
Control Scheme ◽  
Inertia Model ◽  
And Performance ◽  
Attitude Tracking Control ◽  
The Stability

This paper studies the attitude-tracking control problem of spacecraft considering on-orbit refuelling. A time-varying inertia model is developed for spacecraft on-orbit refuelling, which actually includes two processes: fuel in the transfer pipe and fuel in the tank. Based upon the inertia model, an adaptive attitude-tracking controller is derived to guarantee the stability of the resulted closed-loop system, as well as asymptotic convergence of the attitude-tracking errors, despite performing refuelling operations. Finally, numerical simulations illustrate the effectiveness and performance of the proposed control scheme.

scholarly journals Iterative stability analysis for general polynomial control systems

2021 ◽  
Author(s):  
Bo Xiao ◽  
Hak-Keung Lam ◽  
Zhixiong Zhong
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Control Systems ◽  
Polynomial System ◽  
Stability Conditions ◽  
General Polynomial ◽  
Main Challenge ◽  
Detailed Simulation ◽  
Feasible Solutions ◽  
The Stability

AbstractThe main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Simultaneous Optimal Design of the Lyapunov-Based Semi-Active Control and the Semi-Active Vibration Control Device: Inverse Lyapunov Approach

2010 ◽  
Author(s):  
Kazuhiko Hiramoto ◽  
Taichi Matsuoka ◽  
Akira Fukukita ◽  
Katsuaki Sunakoda
Keyword(s):  
Optimal Design ◽  
Lyapunov Function ◽  
Vibration Control ◽  
Active Control ◽  
Control Device ◽  
Ball Screw ◽  
Design Parameters ◽  
Resistance Force ◽  
Control Law ◽  
Lyapunov Approach

We address a simultaneous optimal design problem of a semi-active control law and design parameters in a vibration control device for civil structures. The Vibration Control Device (VCD) that is being developed by authors is used as the semi-active control device in the present paper. The VCD is composed of a mechanism of a ball screw with a flywheel for the inertial resistance force and an electric motor with an electric circuit for the damping resistance force. A new bang-bang type semi-active control law referred to as Inverse Lyapunov Approach is proposed as the semi-active control law. In the Inverse Lyapunov Approach the Lyapunov function is searched so that performance measures in structural vibration control are optimized in the premise of the bang-bang type semi-active control based on the Lyapunov function. The design parameters to determine the Lyapunov function and the design parameters of the VCD are optimized for the good performance of the semi-active control system. The Genetic Algorithm is employed for the optimal design.

LPV Based Sliding Mode Control for Limit Protection of Aircraft Engines

2018 ◽  
Author(s):  
Shubo Yang ◽  
Xi Wang
Keyword(s):  
Closed Loop ◽  
Sliding Mode ◽  
Aircraft Engine ◽  
Gain Scheduling ◽  
Engine Control ◽  
Aircraft Engines ◽  
Sliding Surface ◽  
Closed Loop System ◽  
Limit Protection ◽  
The Stability

Limit protection, which frequently exists as an auxiliary part in control systems, is not the primary motive of control but is a necessary guarantee of safety. As in the case of aircraft engine control, the main objective is to provide the desired thrust based on the position of the throttle; nevertheless, limit protection is indispensable to keep the engine operating within limits. There are plenty of candidates that can be applied to design the regulators for limit protection. PID control with gain-scheduling technique has been used for decades in the aerospace industry. This classic approach suggests linearizing the original nonlinear model at different power-level points, developing PID controllers correspondingly, and then scheduling the linear time-invariant (LTI) controllers according to system states. Sliding mode control (SMC) is well-known with mature theories and numerous successful applications. With the one-sided convergence property, SMC is especially suitable for limit protection tasks. In the case of aircraft engine control, SMC regulators have been developed to supplant traditional linear regulators, where SMC can strictly keep relevant outputs within their limits and improve the control performance. In aircraft engine control field, we all know that the plant is a nonlinear system. However, the present design of the sliding controller is carried out with linear models, which severely restricts the valid scope of the controller. Even if the gain scheduling technique is adopted, the stability of the whole systems cannot be theoretically proved. Research of linear parameter varying (LPV) system throws light on a class of nonlinear control problems. In present works, we propose a controller design method based on the LPV model to solve the engines control problem and achieve considerable effectiveness. In this paper, we discuss the design of a sliding controller for limit protection task of aircraft engines, the plant of which is described as an LPV system instead of LTI models. We define the sliding surface as tracking errors and, with the aid of vertex property, present the stability analysis of the closed-loop system on the sliding surface. An SMC law is designed to guarantee that the closed-loop system is globally attracted to the sliding surface. Hot day (ISA+30° C) takeoff simulations based on a reliable turbofan model are presented, which test the proposed method for temperature protection and verify its stability and effectiveness.

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