Robust stability and performance of stochastic uncertain systems on an infinite time interval

In this paper, we consider a robust stability problem for continuous time stochastic uncertain systems. The uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input. This uncertainty is assumed to satisfy a certain relative entropy constraint. The solution to a specially parametrized risk-sensitive performance analysis problem is used to estimate the level of guaranteed performance for the stochastic uncertain system under consideration. This solution is obtained by solving an algebraic Riccati equation. The corresponding performance bound holds for all admissible uncertainties and is nonconservative.     

Robust filtering of stochastic uncertain systems on an infinite time horizon

In this paper, we consider a robust filtering problem for continuous time stochastic uncertain systems.The uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input. This uncertainty is assumed to satisfy a certain relative entropy constraint. The solution to a specially parameterized risk-sensitive stochastic filtering problem is used to construct a filter for the original uncertain system which guarantees an optimal worst-case filtering error. The corresponding minimax optimal filter is obtained by solving a pair of algebraic Riccati equations.     

Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances

In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure.     

Stability of stochastic interval systems with time delays

Consider a given exponentially stable system undergoing a random perturbation which is dependent on a past state of the solution of the system. Suppose this stochastically perturbed system is described by a stochastic differential-functional equation. In this paper, we establish a sufficient condition that the perturbed system remains exponentially stable. Using a specific example, we show how this condition may be used, and we extend it to deal with multiple time delays.     

Optimal and stable controllable stochastic systems synthesis with incomplete state information on an unbounded time interval

We propose a method of control strategy synthesis for a stochastic dynamical system on an unbounded time interval that provides for the stability of the system and optimality of stabilization cost per unit of time with respect to a given criterion. We assume that a control strategy may depend only on a part of the state vector components subject to measurement. We consider both the general nonlinear and special linear cases.     

Stability radii of infinite dimensional systems with stochastic uncertainty and their optimization

In this paper we consider infinite dimensional systems which are subjected to stochastic structured multiperturbations. We first characterize the stability radii of these systems in terms of a Lyapunov equation and the corresponding Lyapunov inequalities. Then we investigate the problem of maximizing the stability radius by linear state feedback. We show that the supremal achievable stability radius can be determined via the resolution of a parametrized Riccati equation. Illustrative examples are included. Copyright © 2006 John Wiley & Sons, Ltd.     

Robust stabilization of stochastic differential inclusion systems with time delay

This paper is concerned with robust stabilization of stochastic differential inclusion systems with time delay.A nonlinear feedback law is established by using convex hull quadratic Lyapunov function s...     

Stability and stabilization of nonlinear discrete‐time stochastic systems

This paper mainly studies the locally/globally asymptotic stability and stabilization in probability for nonlinear discrete‐time stochastic systems. Firstly, for more general stochastic difference systems, two very useful results on locally and globally asymptotic stability in probability are obtained, which can be viewed as the discrete versions of continuous‐time Itô systems. Then, for a class of quasi‐linear discrete‐time stochastic control systems, both state‐ and output‐feedback asymptotic stabilization are studied, for which, sufficient conditions are presented in terms of linear matrix inequalities. Two simulation examples are given to illustrate the effectiveness of our main results.     

Robust aerodynamic shape design based on an adaptive stochastic optimization framework

Optimization techniques combined with uncertainty quantification are computationally expensive for robust aerodynamic optimization due to expensive CFD costs. Surrogate model technology can be used to improve the efficiency of robust optimization. In this paper, non-intrusive polynomial chaos method and Kriging model are used to construct a surrogate model that associate stochastic aerodynamic statistics with airfoil shapes. Then, global search algorithm is used to optimize the model to obtain optimal airfoil fast. However, optimization results always depend on the approximation accuracy of the surrogate model. Actually, it is difficult to achieve a high accuracy of the model in the whole design space. Therefore, we introduce the idea of adaptive strategy to robust aerodynamic optimization and propose an adaptive stochastic optimization framework. The surrogate model is updated adaptively by increasing training airfoils according to historical optimization results to guarantee the accuracy near the optimal design point, which can greatly reduce the number of training airfoils. The proposed method is applied to a robust aerodynamic shape optimization for drag minimization considering uncertainty of Mach number in transonic region. It can be concluded that the proposed method can obtain better optimal results more efficiently than the traditional robust optimization method and global surrogate model method.     

Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems

In this work, probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs. A suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations, leading to dual algorithms for reachability computations. In particular, the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired ‘safe’ region of the state space is characterized in terms of a value function, and ‘maximally safe’ Markov policies are determined via dynamic programming. These results are of interest not only for safety analysis and design, but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems. The temperature regulation problem presented in the paper as a case study is one such case.     

Asymptotic stability and discretization on an infinite interval

By imposing some conditions on the discrete dynamical systems generated by one step discretization methods, we construct some non linear one step methods of the first and second order, which turn out to beA-stable orL-stable according to the current definitions. Numerical experiments are carried out on some common stiff test problems, confirming the validity of the methods.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

Finite‐time stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper investigates the finite‐time control problem for a class of stochastic nonlinear systems with stochastic integral input‐to‐state stablility (SiISS) inverse dynamics. Motivated by finite‐time stochastic input‐to‐state stability and the concept of SiISS using Lyapunov functions, a novel finite‐time SiISS using Lyapunov functions is introduced firstly. Then, by adopting this novel finite‐time SiISS small‐gain arguments, using the backstepping technique and stochastic finite‐time stability theory, a systematic design and analysis algorithm is proposed. Given the control laws that guarantee global stability in probability or asymptotic stability in probability, our design algorithm presents a state‐feedback controller that can ensure the solution of the closed‐loop system to be finite‐time stable in probability. Finally, a simulation example is given to demonstrate the effectiveness of the proposed control scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

Quadratic stability and stabilization of dynamic interval systems

This note presents necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. The results are obtained in terms of linear matrix inequalities (LMIs) and extended to the quadratic stability and stabilization of linear systems with uncertain parameters. With the powerful LMI toolbox, it is very convenient to solve these problems. The illustrative examples show that this method is effective and less conservative to check the robust stability and to design the stabilizing controller for dynamic interval systems.     

广义区间系统的稳定性与二次能稳分析

对于一类广义区间系统,利用Lyapunov方法,研究了该系统的稳定性.通过广义Lyapunov不等式的建立,得到了此系统结构稳定的充分必要条件是广义Lyapunov不等式有解.在此基础上,建立了广义Riccati不等式,由此不等式得到了此系统二次能稳的充要条件是广义Riccati不等式有解.这些结果将对进一步研究此系统有着重要的基础作用.最后,举例说明了主要结果.     

随机状态空间系统的梯度优化辨识

提出了随机状态空间系统参数的梯度优化辨识方法。通过极小化输出预报误差而获得系统的参数估计。提出了动态选择雅可比矩阵奇异值比率确定参数搜索方向的方法,用以解决因雅可比矩阵的线性相关性引起的算法失效问题。给出了融合参数局部逼近性能信息的辨识算法,并得到了算法收敛速度的解析表达式。数值仿真实验的结果说明了所提出方法的有效性。     

Stability and stabilization of stochastic systems with multiplicative noise

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L _A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.