Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints

Methods to construct the ellipsoidal estimates of the reachability sets of a nonlinear dynamic system with scalar pulse control and uncertainty in the initial data were proposed. The considered pulse system was rearranged in the ordinary differential inclusion already without any pulse components by means of a special discontinuous time substitution. The results of the theory of ellipsoidal estimation and the theory of evolutionary equations of the multivalued states of dynamic systems under uncertainty and ellipsoidal phase constraints were used to estimate the reachability sets of the resulting nonlinear differential inclusion.     

Robust MPC of constrained discrete-time nonlinear systems based on approximated reachable sets

A robust MPC for constrained nonlinear systems with uncertainties is presented. Outer bounds of the reachable sets of the system are used to predict the evolution of the system under uncertainty. A method that uses zonotopes to represent the approximated reachable sets is proposed. The closed-loop system is ultimately bounded thanks to a contractive constraint that drives the system to a robust invariant set.     

Impulse differential inclusions: a viability approach to hybridsystems

Impulse differential inclusions are introduced as a framework for modeling hybrid phenomena. Connections to standard problems in the area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples     

车辆跟随控制策略的状态可达集建模及验证方法

车辆动态行为的不确定性会造成跟随车辆控制的不确定性.传统的车辆跟随控制方法只是针对车辆的单一行为动态的不确定性进行建模,无法遍历车辆所有可能的控制输入,因而,既无法一次性提供跟随策略下完整的可行控制方案,也不足以在理论上保证对策略安全检测的可信性.为此,提出车辆跟随控制策略的状态可达集建模及验证方法.该方法将控制策略转换为能用可达集计算和表征的多级安全判定事件,利用随机可达集的状态遍历特征描述车辆控制输入的不确定性,通过对可达集交集的判断,辨识所有初始条件对应的危险与安全控制行为,为驾驶员提供完整的可选择控制方案;然后利用马尔科夫链逼近可达集,近似表达车辆行为的不确定性,依据驾驶员行为习惯统计验证策略的安全性,实现对控制策略的有效建模分析.实验结果表明,所提出的建模及验证方法不仅可以完备地表征车辆不确定行为,提供交通情形中跟随策略相应的完整控制方案,也可实现对策略安全性的精确验证.     

基于可达集的鲁棒模型预测控制

设计了一种基于可达集的鲁棒模型预测控制算法.首先确定了一个鲁棒不变集,并将此不变集用作模型预测控制的终端约束集;接着采用终端约束集对可达集的包含度作为优化指标;最后,采用预测时域逐渐减小的控制策略以保证在线优化存在可行解.从理论上证明了吸引域内的任意点在有限时域内都会被引导至终端约束集并始终停留在此集之内,并由仿真算例验证了本文所设计鲁棒模型预测控制算法的可行性.     

Stability and performance analysis of switched linear systems subject to time delays and actuator saturation

This paper is devoted to the stability and the performance for the switched linear systems with time delays and actuator saturation. Problems to be considered include the reachable set under a class of bounded energy disturbances and time delays. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Using the PDI and the NDI describing actuator saturation systems, sufficient conditions for the output feedback control laws are derived to stabilise the systems. The estimation of the attraction domain is to ensure that the state remains inside the level set of a certain Lyapunov function where the PDI and NDI are valid. The sufficient conditions for stability and performances are derived as linear matrix inequalities. A example based on jitterbug tools is used to illustrate the effectiveness of the proposed approach.     

Convergence and error bounds of adaptive filtering under model structure and regressor uncertainties

Adaptive filtering algorithms are investigated when system models are subject to model structure errors and regressor signal perturbations.System models for practical applications are often approximati...     

Nonlinear maximum likelihood estimation of electricity spot prices using recurrent neural networks

Electricity spot prices are complex processes characterized by nonlinearity and extreme volatility. Previous work on nonlinear modeling of electricity spot prices has shown encouraging results, and we build on this area by proposing an Expectation Maximization algorithm for maximum likelihood estimation of recurrent neural networks utilizing the Kalman filter and smoother. This involves inference of both parameters and hyper-parameters of the model which takes into account the model uncertainty and noise in the data. The Expectation Maximization algorithm uses a forward filtering and backward smoothing (Expectation) step, followed by a hyper-parameter estimation (Maximization) step. The model is validated across two data sets of different power exchanges. It is found that after learning a posteriori hyper-parameters, the proposed algorithm outperforms the real-time recurrent learning and the extended Kalman Filtering algorithm for recurrent networks, as well as other contemporary models that have been previously applied to the modeling of electricity spot prices.     

Using hybrid automata for set-membership state estimation with uncertain nonlinear continuous-time systems

This paper deals with set-membership state estimation for continuous-time systems from discrete-time measurements, in the unknown but bounded error framework. The classical predictor–corrector approach to state estimation uses interval Taylor methods for solving the prediction phase, which are known to have poor performance in presence of large model or input uncertainty. In this paper, we show how to derive more efficient predictors by using a nonlinear hybridization method which builds hybrid automata to characterize the boundaries of reachable sets. The derived continuous–discrete set-membership predictor–corrector estimator is then tested with simulated data from a bioreactor. Our method is compared to classical continuous-time interval observers and is shown to have promising performance.     

Reachability analysis of continuous-time piecewise affine systems

This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.     

鲁棒镇定受到干扰的非线性微分包含系统

研究带有干扰的非线性微分包含系统的镇定问题. 基于凸锥Lyapunov函数方法, 首先, 对于无干扰的这类系统, 设计可使闭环系统全局镇定的连续反馈控制律. 其次, 通过状态反馈, 对于受到两类有界干扰的这类微分包含系统的可达集进行了估计. 最后, 通过一个仿真例子说明所提出的设计方法的有效性.     

Set-valued maps for image segmentation

The goal of this paper is an analytical basis of a region growing method using set-valued maps. They are to provide an approach to image segmentation that does not use a priori conditions on the final segment. Seizing a suggestion of Demongeot and Leitner ([8]), we start with a compact subset of the grey-valued image, and the region growing method is based on the continuous deformation of sets for decreasing some error functional. Avoiding any further restrictions on these sets leads to describing the process as a set-valued map. The ansatz of the deforming sets utilizes reachable sets of differential inclusions that admit more than one velocity of propagation at each point. So set-valued maps underlie a mathematical segmentation problem posed and solved in the first part. Then we present a suggestion how to apply the analytical results to a very simple computer implementation. Received: 2 March 2000 / Accepted: 27 May 2001     

Stabilization of a class of uncertain nonlinear affine systems subject to control constraints

Practical and asymptotic stabilization is investigated for a class of uncertain nonlinear affine control systems subject to constraints on the control inputs. The uncertain systems are modelled as non‐linear perturbations to a known non‐linear idealized system and a problem formulation based on differential inclusions is adopted. A class of constrained generalized state‐feedback controls (containing both continuous and discontinuous selections) is developed, which guarantees stabilization with a specified region of attraction. In the presence of residual uncertainty, sufficient conditions for global stabilizability are presented, whilst local results are obtained by relaxing these conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

Recurrent neural network modeling combined with bilinear model structure

In this paper, we propose a neural network modeling scheme for nonlinear systems. The proposed architecture is a new combination of neural network and bilinear system model in which the terms of cross-products of input and output signals within the bilinear model are taken as the inputs into the neural network. Compared with the original bilinear system, this kind of network model possesses much more adjustable parameters to fulfill the system identification. Moreover, instead of the general back-propagation method an evolutionary computation called the differential evolution algorithm is presented to update the network parameters. This algorithm is with multiple direction searches toward the global optimal solution for given optimization problem. To show the feasibility of the proposed scheme, a nonlinear chemical process system of continuously stirred tank reactor is illustrated. Many simulations and examinations are considered to verify the robustness of the proposed neural network structure on the modeling performance, including different sets of initial conditions of the algorithm and model orders.     

Hybrid control synthesis for eventuality specifications using level set methods

This paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications. Given a hybrid system modelled by a hybrid automaton and a target set of states, the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time. Due to the existence of set-valued disturbance inputs, the problem is studied in a game-theoretic framework. Having shown that a least restrictive solution does not exist, we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property. To implement the algorithm, reachable sets of pursuit-evasion differential games need to be computed. For that reason level set methods are employed, where the boundary of the reachable set is characterized as the zero level set of a Hamilton–Jacobi equation. The procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology. Finally, to demonstrate the practical character of our results, a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology.     

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.     

A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games

We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.     

Web服务自动化测试技术

赋时Petri网为装配序列规划提供了有效的建模方法,但其在求解最优装配序列时受到组合复杂性的严重制约。零压缩二叉决策图(ZBDD)是处理大规模组合集合和0-1稀疏向量的一种有效符号技术,能够有效缓解组合爆炸问题。将赋时Petri网与ZBDD结合起来,给出了一种求解装配序列最优解的有效方法。首先通过转换算法将赋时Petri网转换为等价的普通Petri网,接下来给出普通Petri网可达状态及迁移引发函数的ZBDD表示方法,最后基于ZBDD给出最优装配序列求解算法。实例验证表明,该算法在求解过程中通过隐式符号操作实现了Petri网的可达状态搜索,有效缓解了计算过程中的组合复杂性。     

Adaptive stabilization of multi-input nonlinear systems

Universal stabilizers are presented for two classes of nonlinear systems, linear in their multiple control inputs: (I) a class of pth order controlled differential inclusions on R″ with full state available for feedback, and (II) a class of nonlinearly perturbed linear systems with restricted state availability. The stabilizers are of a discontinuous feedback form (embedded in a set-valued map), and incorporate adaptive matrix-valued gain functions which exploit the existence of finite spectrum-unmixing sets associated with the systems under consideration. The analysis draws on an extension, to differential inclusions, of LaSalle's invariance principle for ordinary differential equations.     

Discrete Interval Type 2 Fuzzy System Models Using Uncertainty in Learning Parameters

Fuzzy system modeling (FSM) is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty. Conventional FSM techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system. However, since type 1 fuzzy sets express the belongingness of a crisp value x' of a base variable x in a fuzzy set A by a crisp membership value mu_A(x'), they cannot fully capture the uncertainties due to imprecision in identifying membership functions. Higher types of fuzzy sets can be a remedy to address this issue. Since, the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set, the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades. In this paper, we propose a black-box methodology that can identify robust type 2 Takagi-Sugeno, Mizumoto and Linguistic fuzzy system models with high predictive power. One of the essential problems of type 2 fuzzy system models is computational complexity. In order to remedy this problem, discrete interval valued type 2 fuzzy system models are proposed with type reduction. In the proposed fuzzy system modeling methods, fuzzy C-means (FCM) clustering algorithm is used in order to identify the system structure. The proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of FCM, known as the level of membership, and its variation over a specific set of values which generate the uncertainty associated with the system structure