Large-scale Stochastic Systems Research Articles

SUBSPACE METHOD FOR ANALYZING LARGE-SCALE NONLINEAR STOCHASTIC DYNAMIC SYSTEMS WITH POLYNOMIAL TYPE OF NONLINEARITES

In this paper, the probabilistic solutions of the multi-degree-of-freedom (MDOF) or large-scale stochastic dynamic systems with polynomial type of nonlinearity and excited by Gaussian white noise excitations are obtained and investigated with the subspace method proposed recently by the authors. The space of the state variables of large-scale nonlinear stochastic dynamic (NSD) system excited by white noises is separated into two subspaces. Both sides of the Fokker–Planck–Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure (EPC) method. A simple flexural beam on nonlinear elastic springs is analyzed with the subspace method to show the effectiveness of the subspace-EPC method in this case.

Decentralized Output-feedback Stabilization of Large-scale Stochastic Nonlinear Systems with Polynomial Nonlinearity

摘要: 基于齐次占优方法研究 具有多项式型非线性项的大规模随机非线性系统的分散输出反馈镇定问题. 本文的主要贡献在于利用高增益齐次占优方法来解决大规模随机非线性系统的分散控制问题. 这种方法能彻底地去掉传统结果中所要求的线性增长条件, 不仅推广了以前的结果还得到了一些新的结果. 仿真验证了输出反馈控制器的有效性. 关键词: 大规模随机非线性系统 / 输出反馈镇定 / 分散 / 齐次占优

Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems

In this paper, an adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large-scale stochastic nonlinear systems without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. On the basis of the fuzzy state observer, and by combining the adaptive backstepping technique with decentralized control design, an adaptive fuzzy decentralized output feedback control approach is developed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. A simulation example is provided to show the effectiveness of the proposed approach.

Delay-dependent generalized H2 control for discrete T-S fuzzy large-scale stochastic systems with mixed delays

Delay-dependent generalized H2 control for discrete T-S fuzzy large-scale stochastic systems with mixed delays This paper is concerned with the problem of stochastic stability and generalized H2 control for discrete-time fuzzy large-scale stochastic systems with time-varying and infinite-distributed delays. Large-scale interconnected systems consist of a number of discrete-time interconnected Takagi-Sugeno (T-S) subsystems. First, a novel Delay-Dependent Piecewise Lyapunov-Krasovskii Functional (DDPLKF) is proposed, in which both the upper and the lower bound of delays are considered. Then, two improved delay-dependent stability conditions are established based on this DDPLKF in terms of Linear Matrix Inequalities (LMIs). The merit of the proposed conditions lies in its reduced conservatism, which is achieved by circumventing the utilization of some bounding inequalities for cross products of two vectors and by considering the interactions among the fuzzy subsystems in each subregion. A decentralized generalized H2 state feedback fuzzy controller is designed for each subsystem. It is shown that the mean-square stability for discrete T-S fuzzy large-scale stochastic systems can be established if a DDPLKF can be constructed and a decentralized controller can be obtained by solving a set of LMIs. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed method.

Local feedback Pareto strategy for weakly coupled large-scale discrete-time stochastic systems

In this study, the author discusses a Pareto strategy implemented via state and static output feedback for a class of weakly coupled large-scale discrete-time stochastic systems with state- and control-dependent noise. The asymptotic structure along with the uniqueness and positive semi-definiteness of the solutions of cross-coupled non-linear matrix equations (CNMEs) is newly established via the implicit function theorem. The main contribution of this study is the proposal of a parameter-independent local state and static output feedback Pareto strategy. Moreover, a computational approach for solving the CNMEs is also considered if the information about the small parameter is available. Particularly, a new iterative algorithm based on the linear matrix inequality is established to design a Pareto strategy. Finally, in order to demonstrate the effectiveness of the proposed design method, a numerical example is provided for practical aircraft control problems.

Pth Mean Practical Stability for Large‐Scale Itô Stochastic Systems with Markovian Switching

Motivated by the study of a class of large‐scale stochastic systems with Markovian switching, this correspondence paper is concerned with the practical stability in the pth mean. By investigating Lyapunov‐like functions and the basic comparison principle, some criteria are derived for various types of practical stability in the pth mean of nonlinear stochastic systems. The main contribution of these results is to convert the problem of practical stability in the pth mean of stochastic systems into the one of practical stability of the comparative deterministic systems.

Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.

Decentralized control of uncertain nonlinear stochastic systems based on DSC

In this paper, the decentralized stabilization control approach based on the dynamic surface control (DSC) is proposed for a class of large-scale interconnected stochastic nonlinear systems. The proposed approach combined the existing dynamic surface control (DSC) with back-stepping technique. This approach can overcome the problem of “explosion of complexity” inherent in the back-stepping method. Thus, the proposed control approach is simpler than the traditional back-stepping control method for the large-scale interconnected stochastic nonlinear systems. The stability analysis shows that all the signals in the closed-loop system are uniformly ultimately bounded (UUB). Finally, an example is provided to illustrate the effectiveness of the proposed control system.

Guaranteed cost PID control for uncertain discrete-time stochastic systems with additive gain perturbations

This paper investigates a novel design method for robust nonfragile proportional-integral-derivative (PID) control that is based on the guaranteed cost control (GCC) problem for a class of uncertain discrete-time stochastic systems with additive gain perturbations. On the basis of linear matrix inequality (LMI), a class of fixed PID controller parameters is obtained, and some sufficient conditions for the existence of the GCC are derived. Although the additive gain perturbations are included in the feedback systems, both the stability of closed-loop systems and adequate cost bound are attained. As a sequel, decentralized GCC PID for a class of discrete-time uncertain large-scale stochastic systems is also considered. Finally, the numerical results demonstrate the efficiency of the proposed controller synthesis. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Multi-objective controller design for uncertain large-scale stochastic systems with time delays via linear matrix inequality optimizations

The present paper investigates the problem of robustly decentralized asymptotical stabilization with an

Feature of Dynamic Games for Discrete-time Weakly Coupled Large-scale Stochastic Systems

Feature of Dynamic Games for Discrete-time Weakly Coupled Large-scale Stochastic Systems

Local Mode Dependent Decentralized Stabilization of Uncertain Markovian Jump Large-Scale Systems

This technical note is concerned with the robust stabilization of a class of stochastic large-scale systems governed by a finite state Markov process. Using the method of integral quadratic constraints, a sufficient condition is developed to design decentralized stabilizing controllers which use local system states and local system operation modes to produce local control inputs. The condition is given in terms of a set of rank constrained linear matrix inequalities. The theory is illustrated by an example.

The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback

The guaranteed cost control (GCC) problem involved in decentralized robust control of a class of uncertain nonlinear large-scale stochastic systems with high-order interconnections is considered. After determining the appropriate conditions for the stochastic GCC controller, a class of decentralized local state feedback controllers is derived using the linear matrix inequality (LMI). The extension of the result of the study to the static output feedback control problem is discussed by considering the Karush–Kuhn–Tucker (KKT) conditions. The efficiency of the proposed design method is demonstrated on the basis of simulation results.

Adaptive Stabilization Control of a Class of Large-scale Stochastic Nonlinear Systems with Time-delays and Unknown Virtual Control Coefficients

In this paper,the adaptive stabilization problem is investigated for a class of large-scale stochastic time-delay nonlinear systems with unknown virtual control coefficients and unknown noise covariances.First,corresponding to the unknown virtual control coefficients and unknown noise covariances,some parameters to be estimated are properly chosen. Then,in view of the effect of the time-varying delay on the stability of the closed-loop system,an suitable Lyapunov- Krasovskii functional is constructed,based on which and the well-known integrator backstepping method,a memoryless state-feedback control law is systematically designed.Under some mild conditions,it is shown that the equilibrium of the closed-loop system is globally stable in probability,and all the signals of the closed-loop system except for parameter estimates converge to zero almost surely.A simulation example is given to illustrate the effectiveness of the proposed approach.

弱結合大規模確率システムのためのNash均衡戦略

弱結合大規模確率システムのためのNash均衡戦略

Robust decentralized controller with multi-objective performance design for stochastic large-scale systems via linear matrix inequalities approach

The present paper focuses on a class of stochastic large-scale interconnected systems with norm-bounded and time-varying parameter uncertainties. The problem addressed is the design of a robust decentralized controller such that the corresponding interconnected closedloop system satisfies certain multi-objective performance requirements for all admissible uncertainties. The objectives of this robust design consist of individual state variance limitation, white noise attenuation constraints on H∞ norm, and pole placement specification. Suffcient conditions, based on a quadratic Lyapunov function, for multi-objective robustness are derived in terms of linear matrix inequalities (LMIs). It is shown that the problem of multi-objective robust decentralized control can be solved and the effectiveness of the proposed approach is illustrated by a numerical example.

A Multiresolution Analysis-Assisted Reinforcement Learning Approach to Run-by-Run Control

In recent years, the run-by run (RbR) control mechanism has emerged as a useful tool for keeping complex semiconductor manufacturing processes on target during repeated short production runs. Many types of RbR controllers exist in the literature of which the exponentially weighted moving average (EWMA) controller is widely used in the industry. However, EWMA controllers are known to have several limitations. For example, in the presence of multiscale disturbances and lack of accurate process models, the performance of EWMA controller deteriorates and often fails to control the process. Also, the control of complex manufacturing processes requires sensing of multiple parameters that may be spatially distributed. New control strategies that can successfully use spatially distributed sensor data are required. This paper presents a new multiresolution analysis (wavelet) assisted reinforcement learning (RL)-based control strategy that can effectively deal with both multiscale disturbances in processes and the lack of process models. The novel idea of a wavelet-aided RL-based controller represents a paradigm shift in the control of large-scale stochastic dynamic systems of which the control problem is a subset. Henceforth, we refer our new control strategy as a WRL-RbR controller. The WRL-RbR controller is tested on a multiple-input-multiple-output chemical mechanical planarization process of wafer fabrication for which the process model is available. Results show that the RL controller outperforms EWMA-based controllers for low autocorrelation. The new controller also performs quite well for strongly autocorrelated processes for which the EWMA controllers are known to fail. Convergence analysis of the new breed of the WRL-RbR controller is presented. Further enhancement of the controller to deal with model-free processes and for inputs coming from spatially distributed environments are also discussed. Note to Practitioners-This work was motivated by the need to develop an intelligent and efficient RbR process controller, especially for the control of processes with short production runs as in the case of the semiconductor manufacturing industry. A novel controller that is presented here is capable of generating optimal control actions in the presence of multiple time-frequency disturbances, and allows the use of realistic (often complex) process models without sacrificing robustness and speed of execution. Performance measures, such as reduction of variability in process output and control recipe, minimization of initial bias, and ability to control processes with high autocorrelations are shown to be superior in comparison to the commercially available exponentially weighted moving average controllers. The WRL-RbR controller is very generic, and can be easily extended to processes with drifts and sudden shifts in the mean and variance. The viability of extending the controller to distributed input parameter sensing environments, including those for which process models are not available, is also discussed

Stabilization for a Class of Large-Scale Stochastic Nonlinear Systems with Decentralized Controller Design

In this paper, a state feedback adaptive stabilization for a class of large-scale stochastic nonlinear systems is designed with Lyapunov and Backstepping method. In the systems there are uncertain terms, whose bounds are governed by a set of unknown parameters. The designed controllers would make the close-loop systems asymptotically stable and adaptive for the unknown parameters. As an application, a second order example is delivered to illustrate the approach.

Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems

In this paper, the problem of decentralized adaptive output-feedback stabilization is investigated for large-scale stochastic nonlinear systems with three types of uncertainties, including parametric uncertainties, nonlinear uncertain interactions and stochastic inverse dynamics. Under the assumption that the inverse dynamics of the subsystems are stochastic input-to-state stable, an adaptive output-feedback controller is constructively designed by the backstepping method. It is shown that under some general conditions, the closed-loop system trajectories are bounded in probability and the outputs can be regulated into a small neighborhood of the origin in probability. In addition, the equilibrium of interest is globally stable in probability and the outputs can be regulated to the origin almost surely when the drift and diffusion vector fields vanish at the origin. The contributions of the work are characterized by the following novel features: (1) even for centralized single-input single-output systems, this paper presents a first result in stochastic, nonlinear, adaptive, output-feedback asymptotic stabilization; (2) the methodology previously developed for deterministic large-scale systems is generalized to stochastic ones. At the same time, novel small-gain conditions for small signals are identified in the setting of stochastic systems design; (3) both drift and diffusion vector fields are allowed to be dependent not only on the measurable outputs but some unmeasurable states; (4) parameter update laws are used to counteract the parametric uncertainty existing in both drift and diffusion vector fields, which may appear nonlinearly; (5) the concept of stochastic input-to-state stability and the method of changing supply functions are adapted, for the first time, to deal with stochastic and nonlinear inverse dynamics in the context of decentralized control.

Large-scale stochastic hereditary systems under Markovian structural perturbations. Part III. Qualitative analysis

In this final part of the work, the convergence and stability analysis of large-scale stochastic hereditary systems under random structural perturbations is investigated. This is achieved through the development and the utilization of comparison theorems in the context of vector Lyapunov-like functions and decomposition-aggregation method. The byproduct of the investigation suggests that the qualitative properties of decoupled stochastic hereditary subsystems under random structural perturbations are preserved, as long as the self-inhibitory effects of subsystems are larger than cross-interaction effects of the subsystems. Again, it is shown that these properties are affected by hereditary and random structural perturbations effects. It is further shown that the mathematical conditions are algebraically simple, and are robust to the parametric changes. Moreover, the work generates a concept of block quasimonotone nondecreasing property that is useful for the investigation of hierarchic systems. These results are further extended to the integrodifferential equations of Fredholm type.