Constructing maximum likelihood estimates for statistically uncertain linear systems

We consider the parameters estimation problem for a statistically uncertain linear model, i.e., a model whose observations contain both random perturbations with known distributions and uncertain perturbations for which we only know the domain of their possible values. To solve this problem, we use an approach related to the maximum likelihood method for statistically uncertain systems. We show that as the variances of random perturbations tend to zero, maximum likelihood estimates converge to the information set of the system without random perturbations.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

具有传感器增益退化的不确定系统融合估计器

研究具有传感器增益退化、模型不确定性的多传感器融合估计问题, 其中传感器增益退化现象描述为统计特性已知的随机变量, 模型的不确定性描述为系统矩阵受到随机扰动. 设计一种局部无偏估计器结构, 并建立以局部估计器增益为决策变量、以有限时域下融合估计误差为代价函数的优化问题. 在给出标量融合权重时, 考虑到求得最优的局部估计器增益的解析形式较为困难, 通过最小化代价函数的上界得到一组次优的局部估计器增益. 最后通过算例仿真表明了所设计融合估计器的有效性.

     

Optimal and suboptimal solutions to stochastically uncertain problems of quintile optimization

In this paper, problems of stochastic optimization under incomplete information on distribution of random perturbations with the quintile and probability criteria are considered. The minimax approach is used when optimal solutions are chosen. Conditions for equivalency of direct and inverse problems of stochastic optimization under incomplete statistical information are studied. The solution method for statistically uncertain problems of optimization with the quintile criterion basing on the use of generalized confidence sets for statistically uncertain random quantities is proposed. The use of confidence sets for finding suboptimal solutions to the problem of stochastic optimization under incomplete information is considered. Examples of the application of obtained relations are represented.     

Estimation of random information sets of multistep systems

The estimation problems for the linear multistep controlled system under mixed perturbations are considered. It is assumed that deterministic perturbations are bounded by convex and compact constraints and random perturbations are Gaussian. Random information sets, called multiestimes for brevity, are defined. In the absence of random components, the introduced multiestimates coincide with information sets in the theory of guaranteed estimation. The multiestimate structure is considered. It is shown that multiestimates can be represented as the sum of a random vector and a deterministic set, which depend on several parameters. In turn, this set of parameters determines uniquely both conditional and unconditional probability of inclusion of the multiestimate in the covering set. Several special cases are analyzed, and the form of the covering set is discussed. A modification of the problem under communication constraints is proposed. This modification takes into account the limited capability of the digital data transfer channel. Relations between the accuracy of multiestimate parameters identification and the length of the transmitted word are obtained. A number of the obtained results are illustrated by examples.     

区域极点约束下线性离散系统的Riccati鲁棒控制

讨论区域极点约束下,含结构参数扰动的不确定线性离散系统的鲁棒控制问题,即设计一鲁棒状态反馈控制器,使线性离散系统在可允许的参数扰动下,闭环矩阵极点始终位于一预先给定的圆形区域中,从而闭环系统具有期望的动态性能.上述控制目的可通过求解一含参数的代数离散Riccati方程达到.     

Suboptimal estimation of systems with large parameter uncertainties

The design of a filter that must estimate the state of a system when large parameter uncertainties in the plant dynamics or plant and measurement noise covariance matrices are present is considered. Filter designs are evaluated relative to the trade off between filter sensitivity to the uncertain parameters and minimum mean-square estimation error. This is accomplished by reformulating the estimation problem as an optimization problem with vector-valued performance index. The dominant design has form similar to the Kalman-Bucy filter, and employ open-loop compensation for the uncertain parameters.     

A Nonlinear Filtering Scheme for Multistage Statistically Uncertain Systems

A filtering scheme for multistage systems under uncertain probability distributions of perturbations is studied. A recurrent scheme for constructing point state estimates is designed and its relation with guaranteed filtering theory is determined. This scheme covers the case in which both uncertain and random perturbations are present.     

A Multi-step Output Feedback Robust MPC Approach for LPV Systems with Bounded Parameter Changes and Disturbance

This paper considers a multi-step output feedback robust model predictive control (OFRMPC) approach for the linear parameter varying (LPV) systems with bounded changes of scheduling parameters and bounded disturbance. Less conservative bounds of future estimation error sets and system parametric uncertain sets are predicted by considering bounded changes of scheduling parameters in LPV systems. In the multi-step OFRMPC approach, an optimization problem is solved to obtain a sequence of controller gains, which considers predictions of future bounds of estimation error sets and system parametric uncertain sets. The optimized sequence of controller gains corresponding to a sequence of Lyaponov matrices have less constraint conditions and also introduce more degree of freedom for the optimization. The proposed multi-step OFRMPC guarantees robust uniform ultimately bounded of the estimation error and robust stability of the observer system. A numerical example is given to demonstrate the effectiveness of the approach.     

具有测量数据丢失的离散不确定时滞系统鲁棒Kalman滤波

研究了具有测量数据丢失的离散不确定时滞系统鲁棒Kalman滤波问题, 其中时延存在于系统状态和观测值中. 模型的不确定性通过在系统矩阵中引入随机参数扰动来表示, 测量数据丢失现象则通过一个满足Bernoulli分布且统计特性已知的随机变量来描述. 基于最小方差估计准则, 利用射影性质和递归射影公式得到一个新的滤波器设计方法, 并且保证了滤波器的维数与原系统相等. 与传统的状态增广方法相比, 当时延比较大时, 该方法可以有效降低计算量. 最后, 给出一个仿真例子说明所提方法的有效性.     

Confidence Analysis of Linear Unbiased Estimates under Uncertain Unimodal Noise Distributions

Journal of Computer and Systems Sciences International - The estimation problem for a linear parametric function in a linear regression model with uncertain symmetric unimodal noise distributions...     

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.     

Efficient robust predictive control

Predictive constrained control of time-varying and/or uncertain linear systems has been effected through the use of ellipsoidal invariant sets (Kothare et al., 1996). Linear matrix inequalities (LMIs) have been used to design a state-dependent state-feedback law that maintains the state vector inside invariant feasible sets. For the purposes of prediction however, at each time instant, the state feedback law is assumed constant. In addition, due to the large number of LMIs involved, online computation becomes intractable for anything other than small dimensional systems. Here we propose an approach that deploys a fixed state-feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state-feedback law. The problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online. Over and above the very significant reduction in computational cost, the extra degrees of freedom allow for better performance and wider applicability     

Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or perturbations

This paper investigates the problem of lag synchronization for a kind of chaotic neural networks with discrete and distributed delays (mixed delays). The driver system has uncertain parameters and uncertain nonlinear external perturbations, while the response system has channel noises. A simple but all-powerful robust adaptive controller is designed to circumvent the effects of uncertain external perturbations such that the response system synchronize with the driver system. Based on the invariance principle of stochastic differential equations and some suitable Lyapunov functions, several sufficient conditions are developed to solve this problem. Moreover, under certain conditions, parameters of the uncertain master system can be estimated. Numerical simulations are exploited to show the effectiveness of the theoretical results.     

A double-distribution statistical algorithm for composite laminate optimization

The paper proposes a new evolutionary algorithm termed Double-Distribution Optimization Algorithm (DDOA). DDOA belongs to the family of estimation of distribution algorithms (EDA) that build a statistical model of promising regions of the design space based on sets of good points and use it to guide the search. The efficiency of these algorithms is heavily dependent on the model accuracy. In this work, a generic framework for enhancing the model accuracy by incorporating statistical variable dependencies is presented. The proposed algorithm uses two distributions simultaneously: the marginal distributions of the design variables, complemented by the distribution of physically meaningful auxiliary variables. The combination of the two generates more accurate distributions of promising regions at a low computational cost. The paper demonstrates the efficiency of DDOA for three laminate optimization problems where the design variables are the fiber angles, and the auxiliary variables are integral quantities called lamination parameters. The results show that the reliability of DDOA in finding the optima is greater than that of simple EDA and a standard genetic algorithm, and that its advantage increases with the problem dimension.     

Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach

In this paper the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance‐observer‐based control (DOBC) framework. The unknown external disturbances are supposed to be generated by an exogenous system, where some classic assumptions on disturbances can be removed. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. Design schemes are presented for both the full‐order and reduced‐order disturbance observers via LMI‐based algorithms. For the plants with known nonlinearity, it is shown that the full‐order observer can be constructed by augmenting the estimation of disturbances into the full‐state estimation, and the reduced‐order ones can be designed by using of the separation principle. For the uncertain nonlinearity, the problem can be reduced to a robust observer design problem. By integrating the disturbance observers with conventional control laws, the disturbances can be rejected and the desired dynamic performances can be guaranteed. If the disturbance also has perturbations, it is shown that the proposed approaches are infeasible and further research is required in the future. Finally, simulations for a flight control system is given to demonstrate the effectiveness of the results. Copyright © 2005 John Wiley & Sons, Ltd.     

Design and analysis of robust residual generators for systems under feedback control

In this paper, a general approach is presented to design robust fault detection and isolation (FDI) filters for LTI uncertain systems under feedback control. A two-step design procedure is proposed to derive non-conservative robust FDI filters. The first step consists of designing an optimal FDI filter that maximizes fault sensitivity performance and simultaneously minimizes the influence of unknown inputs, for a large class of model perturbations. The design problem is formulated so that all free parameters are optimized via Linear Matrix Inequality techniques. The second step consists of a systematic post-design analysis procedure. A test is proposed to check if all FDI objectives are achieved for all specified model perturbations. The problem is formulated using an appropriate performance index over a specified frequency range. The solution is based on the generalized structured singular value. Finally, experimental results from a pilot laboratory system are presented to demonstrate the effectiveness of the proposed method.     

A robust estimation approach for uncertain systems with perturbed measurements

This paper deals with state estimation problem for uncertain continuous‐time systems. A numerical treatment is proposed for designing interval observers that ensures guaranteed upper and lower bounds on the estimated states. In order to take into account possible perturbations on the system and its outputs, a new type of interval observers is introduced. Such interval observers consist of two coupled general Luenberger‐type observers that involve dilatation functions. In addition, we provide an optimality criterion in order to find optimal interval observers that lead to tight interval error estimation. The proposed existence and optimality conditions are expressed in terms of linear programming. Also, some illustrative examples are given. Copyright © 2015 John Wiley & Sons, Ltd.