Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part II: Application to Control System Design

1981 ◽  
Vol 103 (1) ◽  
pp. 22-27 ◽  
Author(s):  
J. J. Beaman ◽  
J. K. Hedrick
Keyword(s):  
Stochastic Systems ◽  
Pole Placement ◽  
Linear Feedback ◽  
Statistical Linearization ◽  
Feedback Controllers ◽  
Nonlinear Stochastic Systems ◽  
Input Noise ◽  
Gaussian Method ◽  
And Control ◽  
Digital Simulations

Using the techniques of Gaussian statistical linearization and the extension given in Part I, this paper describes the synthesis of linear feedback controllers for nonlinear stochastic systems. The method used is that of pole placement of the statistically linearized “eigenvalues”. The technique is described in terms of a design example, a position servomechanism with backlash, it is shown that for this type of system the standard Gaussian method works well for large input noise levels but can lead to an unstable design for low input levels. The extended fourth cumulant method is satisfactory for both cases studied. The results of the analysis are compared to Monte Carlo digital simulations to test their accuracy.

scholarly journals Hyperchaos Numerical Simulation and Control in a 4D Hyperchaotic System

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Junhai Ma ◽  
Yujing Yang
Keyword(s):  
Lyapunov Exponents ◽  
Equilibrium Points ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Bifurcation Diagrams ◽  
Theory Analysis ◽  
Feedback Controllers ◽  
Dynamic Simulations ◽  
Wide Range ◽  
And Control

A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.

Optimal Linear Feedback Control for a Class of Nonlinear Nonquadratic Non-Gaussian Problems

1991 ◽  
Vol 113 (4) ◽  
pp. 568-574 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Control System ◽  
Stochastic Systems ◽  
Feedback Controller ◽  
Performance Criteria ◽  
Linear Feedback ◽  
Feedback Gain ◽  
Linear Feedback Controller ◽  
Gaussian Method ◽  
Optimal Linear ◽  
Non Gaussian

An optimal linear feedback controller designed for a class of nonlinear stochastic systems with nonquadratic performance criteria by a non-Gaussian approach is presented. The non-Gaussian method is developed through expressing the unknown stationary output density function as a weighted sum of the Gaussian densities with undetermined parameters. With the aid of a Gaussian-sum density, the optimal feedback gain for a control system with complete state information is derived. By assuming that the separation principle is valid for the class of stochastic systems, a nonlinear precomputed-gain filter is then implemented. The method is illustrated by a Duffing-type control system and the performance of a linear feedback controller designed through both quadratic and nonquadratic performance indices is compared.

scholarly journals Feedback Stabilization for a Class of Nonlinear Stochastic Systems with State- and Control-Dependent Noise

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yu-Hong Wang ◽  
Tianliang Zhang ◽  
Weihai Zhang
Keyword(s):  
State Feedback ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Feedback Stabilization ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Global State ◽  
Nonlinear Stochastic Systems ◽  
Feedback Stabilizability ◽  
And Control

This paper mainly studies the state feedback stabilizability of a class of nonlinear stochastic systems with state- and control-dependent noise. Some sufficient conditions on local and global state feedback stabilizations are given in linear matrix inequalities (LMIs) and generalized algebraic Riccati equations (GAREs). Some obtained results improve the previous work.

scholarly journals A Review on Analysis and Synthesis of Nonlinear Stochastic Systems with Randomly Occurring Incomplete Information

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hongli Dong ◽  
Zidong Wang ◽  
Xuemin Chen ◽  
Huijun Gao
Keyword(s):  
Incomplete Information ◽  
Stochastic Systems ◽  
Future Research ◽  
Missing Measurements ◽  
Parameter Uncertainties ◽  
Nonlinear Stochastic Systems ◽  
Randomly Occurring Uncertainties ◽  
Analysis And Synthesis ◽  
System States ◽  
And Control

In the context of systems and control, incomplete information refers to a dynamical system in which knowledge about the system states is limited due to the difficulties in modeling complexity in a quantitative way. The well-known types of incomplete information include parameter uncertainties and norm-bounded nonlinearities. Recently, in response to the development of network technologies, the phenomenon of randomly occurring incomplete information has become more and more prevalent. Such a phenomenon typically appears in a networked environment. Examples include, but are not limited to, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring saturation, randomly missing measurements and randomly occurring quantization. Randomly occurring incomplete information, if not properly handled, would seriously deteriorate the performance of a control system. In this paper, we aim to survey some recent advances on the analysis and synthesis problems for nonlinear stochastic systems with randomly occurring incomplete information. The developments of the filtering, control and fault detection problems are systematically reviewed. Latest results on analysis and synthesis of nonlinear stochastic systems are discussed in great detail. In addition, various distributed filtering technologies over sensor networks are highlighted. Finally, some concluding remarks are given and some possible future research directions are pointed out.

Robust H∞ Control for Uncertain Nonlinear Stochastic Systems with Delayed State and Control

2011 ◽  
Vol 58-60 ◽  
pp. 685-690
Author(s):  
Cheng Wang ◽  
Yun Xu
Keyword(s):  
Linear Matrix Inequalities ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Feedback Controller ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Nonlinear Stochastic Systems ◽  
Stochastically Stable ◽  
Nonlinear Uncertain Systems ◽  
And Control

This paper considers the issue of robust H∞ control for a class of nonlinear uncertain systems with delayed states and control, and the feedback controller is designed. By constructing proper Lyapunov-krasovskii function, the resulting closed-loop system is stochastically stable for all admissible uncertainties, time-delays and nonlinearities, and satisfies a prescribed H∞ performance. Sufficient conditions for the system to be robustly stochastically asymptotically stable are derived, by using linear matrix inequalities and Lyapunov-krasovskii stability theory. The feedback controller is obtained by solving the linear matrix inequalities. Numerical example is provided to show the validity of the proposed approaches.

Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique

1981 ◽  
Vol 103 (1) ◽  
pp. 14-21 ◽  
Author(s):  
J. J. Beaman ◽  
J. Karl Hedrick
Keyword(s):  
Stochastic Systems ◽  
Practical Method ◽  
Order System ◽  
Statistical Linearization ◽  
Linearization Technique ◽  
Expansion Technique ◽  
Mean Square Response ◽  
Simplifying Assumptions ◽  
The Stability ◽  
And Control

A practical method of improving the accuracy of the Gaussian statistical linearization technique is presented. The method uses a series expansion of the unknown probability density function which includes up to fourth order terms. It is shown that by the use of the Gram-Charlier expansion a simple generating function can be derived to evaluate analytically the integrals required. It is also shown how simplifying assumptions can be used to substantially reduce the required extra equations, e.g. symmetric or assymetric and single input nonlinearities. It is also shown that the eigenvalues of the statistically linearized system can be used to estimate the stability and speed of response of the nonlinear system. The reduced expansion technique is applied to first and second order nonlinear systems and the predicted mean square response is compared to the Gaussian statistical linearization and the exact solution. The prediction of the time response of the mean of a nonlinear first order system by the use of the statistically linearized eigenvalues is compared to a 300 run Monte Carlo digital solution.

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