A Stability Result With Application to Nonlinear Regulation1

2002 ◽  
Vol 124 (3) ◽  
pp. 452-456 ◽  
Author(s):  
Wilbur Langson ◽  
Andrew Alleyne
Keyword(s):  
Nonlinear Systems ◽  
Stability Result ◽  
Linear Representation ◽  
State Regulation ◽  
System Stability ◽  
Stability And Convergence ◽  
Region Of Attraction ◽  
Linear Quadratic ◽  
Local Asymptotic Stability ◽  
Lq Optimal Control

This work considers a class of nonlinear systems whose feedback controller is generated via the solution of a State Dependent Riccati Equation (SDRE) as proposed in Banks and Manha and Cloutier. A pseudo-linear representation of the class of nonlinear systems is described and a stability analysis is performed. This analysis leads to sufficiency conditions under which local asymptotic stability is present. These conditions allow for the computation of a Region of Attraction estimate for system stability. These results are then applied to study stability and convergence properties of closed loop systems that arise when the SDRE technique is used. Many of the benefits of Linear Quadratic (LQ) Optimal Control, such as a tradeoff between state regulation and input effort, are readily transparent in the nonlinear scheme. The tradeoff ability is the major advantage of the SDRE over several other nonlinear control schemes. The computed Region of Attraction, while sufficient, is demonstrated to also be quite conservative. An example is used to examine the SDRE approach.

LS-SVM Adaptive Back-Stepping Control for a Class of Uncertain Nonlinear Systems

2012 ◽  
Vol 588-589 ◽  
pp. 1409-1413
Author(s):  
Guo Dong Zhu ◽  
Hui Lin ◽  
Chen Wang
Keyword(s):  
Support Vector Machine ◽  
Nonlinear Systems ◽  
Least Squares ◽  
System Stability ◽  
Stability And Convergence ◽  
Support Vector ◽  
Control Law ◽  
Uncertain Nonlinear Systems ◽  
Machine Theory ◽  
Back Stepping Control

Based on back-stepping control design, adaptive control and least squares support vector machine theory, a new least squares support vector machine adaptive back-stepping control law was designed for strictly block type of feedback nonlinear systems control with uncertainties. Least squares support vector machine theory method to approximate a nonlinear function of uncertain nonlinear systems by analyzing the disadvantage of common back-stepping. New control law of the nonlinear systems is achieved without accurate mathematical model. The method overcomes the impact of the bounded uncertainties on the control system. On this basis, the system stability and convergence are proved by Lyapunov method. Simulation results indicate that the designed control law has strong robustness and adaptability, uncertainties that exist in the strict block feedback nonlinear systems.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

scholarly journals Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kai Du ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Differential System ◽  
Optimization Problems ◽  
Stackelberg Game ◽  
Mean Field ◽  
Open Loop ◽  
Stackelberg Equilibrium ◽  
Linear Quadratic ◽  
Stackelberg Differential Game ◽  
Lq Optimal Control

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

scholarly journals Studying State Convergence of Input-to-State Stable Systems with Applications to Power System Analysis

Energies ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 92 ◽  
Author(s):  
Antonio T. Alexandridis
Keyword(s):  
Nonlinear Systems ◽  
Power Systems ◽  
Power System ◽  
System Analysis ◽  
Nonlinear Models ◽  
Sufficient Conditions ◽  
Stability And Convergence ◽  
Convergence To Equilibrium ◽  
Analysis Tool ◽  
Wide Range

In stability studies, the response of a system enforced by external, known or unknown, inputs is of great importance. Although such an analysis is quite easy for linear systems, it becomes a cumbersome task when nonlinearities exist in the system model. Nevertheless, most of the real-world systems are externally enforced nonlinear systems with nonzero equilibriums. Representative examples in this category include power systems, where studies on stability and convergence to equilibrium constitute crucial objectives. Driven by this need, the aim of the present work is twofold: First, to substantially complete the theoretical infrastructure by establishing globally valid sufficient conditions for externally enforced nonlinear systems that converge to nonzero equilibriums and, second, to deploy an efficient method easily applicable on practical problems as it is analyzed in detail on a typical power system example. To that end, in the theoretical first part of the paper, a rigorous nonlinear analysis is developed. Particularly, starting from the well-established nonlinear systems theory based on Lyapunov techniques and on the input-to-state stability (ISS) notion, it is proven after a systematic and lengthy analysis that ISS can also guarantee convergence to nonzero equilibrium. Two theorems and two corollaries are established to provide the sufficient conditions. As shown in the paper, the main stability and convergence objectives for externally enforced systems are fulfilled if simple exponential or asymptotic converging conditions can be proven for the unforced system. Then, global or local convergence is established, respectively, while for the latter case, a novel method based on a distance-like measure for determining the region of attraction (RoA) is proposed. The theoretical results are examined on classic power system generation nonlinear models. The power system examples are suitably selected in order to effectively demonstrate the proposed method as a stability analysis tool and to validate all the particular steps, especially that of evaluating the RoA. The examined system results clearly verify the theoretical part, indicating a rather wide range of applications in power systems.

STABILITY OF SISO NONLINEAR SYSTEMS WITH PARAMETERS DISTURBANCES

2012 ◽  
Vol 26 (25) ◽  
pp. 1246008
Author(s):  
OLGA SHPILEVAYA
Keyword(s):  
Nonlinear Systems ◽  
System Dynamics ◽  
Control Systems ◽  
Lyapunov Method ◽  
Switched System ◽  
System Stability ◽  
Single Input Single Output ◽  
Single Output ◽  
System Output ◽  
Single Input

We study single-input single-output (SISO) control systems with the rapid piecewise-smooth parameters disturbances. The system dynamics are described by switched system models. The system output is regulated with the help of the nonlinear astatic controller with parameters which depend on some disturbance properties. The system stability is studied by second Lyapunov method.

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