External estimates of the reachability sets of nonlinear controlled systems

Consideration was given to formation of the external estimates of the reachability sets of the nonlinear controlled systems in the form of the level sets of the functions satisfying the differential Hamilton-Jacobi inequalities. The trajectories of the nonlinear system are estimated by modifying the estimates of its linear part. The constructions proposed are based on comparing the systems of differential inequalities. Applications of the reachability sets of the multivariable controlled systems with nonlinear cross connections to the ellipsoidal estimates were examined.     

Robust optimization of nonlinear dynamic systems with application to a jacketed tubular reactor

We present robust optimization techniques for dynamic systems which are affected by time-varying uncertainties. After reviewing existing techniques from the field of reachability analysis and ellipsoidal calculus, we discuss how to over-estimate the influence of uncertainty in nonlinear dynamic systems. The corresponding strategies lead to a framework which can be used to solve min–max optimal control problems in a conservative approximation. The technique is illustrated by applying it to a robust optimal control problem for a nonlinear jacketed tubular reactor. Inside this reactor a highly nonlinear and exothermic chemical reaction takes place which is uncertain due to fouling at the reactor wall. We regard safety constraints on the temperature which must be satisfied for all possible scenarios.     

On Convergence of External Ellipsoidal Approximations of the Reachability Domains of Discrete Dynamic Linear Systems

The ellipsoid technique is widely used in the guaranteed estimation for approximation of the reachability domains of dynamic systems. The present paper considered the issues of external ellipsoidal estimation of the current and limiting reachability sets of a stable discrete dynamic linear system. Recurrent estimation algorithms using the criterion of minimum trace of the weighted ellipsoid matrix were developed for these systems, and their limiting properties were considered.     

Stability robustness bounds for linear state–space models with structured uncertainty based on ellipsoidal set-theoretic approach

This paper is concerned with the problem of robust stability of linear dynamic systems with structured uncertainty by means of ellipsoidal set-theoretic approach. In this paper, the uncertainty in the physical parameters is expressed in terms of an ellipsoidal set in appropriate vector space. Two ellipsoidal set-theoretic approaches are presented for giving sufficient conditions for robust stability property of the systems with structured uncertainty. The bound produced by the ellipsoidal extension function theorem is shown to be less conservative than the one predicted by the Lagrange multiplier method. In order to introduce the ellipsoidal extension function theorem, in Appendix A of this paper, we try to present the theory of ellipsoidal algebra, following the thought of interval analysis. First of all, we give the concept of ellipsoidal numbers and define their arithmetic operations. Based on them, we finally introduce ellipsoidal vectors and ellipsoidal functions. In terms of the inclusion monotonic property of ellipsoidal functions, we present and prove the ellipsoidal extension function theorem.     

Hamilton-Jacobi inequalities and the optimality conditions in the problems of control with common end constraints

The canonical theory of the necessary and sufficient conditions for global optimality based on the sets of nonsmooth solutions of the differential Hamilton-Jacobi inequalities of two classes of weakly and strongly monotone Lyapunov type functions was developed. These functions enable one to estimate from above and below the objective functional of the optimal control problem and determine the internal and external approximations of the reachability set of the controlled dynamic system.     

Reach set computation and control synthesis for discrete-time dynamical systems with disturbances

The paper is devoted to the formulation of the reachability problem for discrete-time dynamical systems with disturbances. The concept of maxmin and minmax forward and backward reach sets is addressed. Invariance of the backward reach set is discussed. The emphasis of the paper is on discrete-time linear systems, for which the ellipsoidal computational method is described. The synthesis of maxmin and minmax closed-loop control for steering the system to a given target set using ellipsoidal backward reach set approximations is explained. The ellipsoidal method covered in the paper is implemented in the Ellipsoidal Toolbox for MATLAB, a popular collection of ellipsoidal calculus routines freely available online.     

A differential inclusion approach for modeling and analysis of dynamical systems under uncertainty

In this paper we deal with the application of differential inclusions to modeling nonlinear dynamical systems under uncertainty in parameters. In this case, differential inclusions seem to be better suited to modeling practical situations under uncertainty and imprecision than formulations by means of fuzzy differential equations. We develop a practical algorithm to approximate the reachable sets of a class of nonlinear differential inclusion, which eludes the computational problems of a previous set-valued version of the Heun’s method. Our algorithm is based on a complete discretization (time and state space) of the differential inclusion and it suits hardware features, handling the memory used by the method in a controlled fashion during all iterations. As a case of study, we formulate a differential inclusion to model an epidemic outbreak of dengue fever under Cuban conditions. The model takes into account interaction of human and mosquito populations as well as vertical transmission in the mosquito population. It is studied from the theoretical point of view to apply the Practical Algorithm. Also, we estimate the temporal evolution of the different human and mosquito populations given by the model in the Dengue 3 epidemic in Havana 2001, through the computation of the reachable sets using the Practical Algorithm.     

The equality problem for vector addition systems is undecidable

We demonstrate the usefulness of Petri nets for treating problems about vector addition systems by giving a simple exposition of Rabin's proof of the undecidability of the inclusion problem for vector addition system reachability sets, and then proceed to show that the inclusion problem can be reduced to the equality problem for reachability sets.     

Observers in nonlinear model‐based predictive control

A closed‐loop reformulation of the dual‐mode paradigm affords significant advantages in model‐based predictive control of systems subject to uncertainty and/or disturbances and/or nonlinear dynamics. This paper considers earlier results based on ellipsoidal invariant sets and proposes extensions to the output feedback nonlinear case by defining invariant sets both for the system state and the error dynamics of a nonlinear observer. Particular attention is paid to the class of systems with separable nonlinearities but the results carry over to the general case. Copyright © 2000 John Wiley & Sons, Ltd.     

On polyhedral estimates for reachable sets of discrete-time systems with bilinear uncertainty

We describe techniques for constructing external and internal polyhedral (parallelepiped-valued) estimates for reachability sets of discrete-time systems with bilinear uncertainty, i.e., systems with originally linear but incompletely specified dynamics, when system coefficients exhibit uncertainty of the interval type. Our primary attention is focused on internal estimates. The evolution of reachability set estimates is given by recursive relations. We give numerical modeling results.     

External ellipsoidal estimates of the attainability sets of differential impulse systems

The problems of estimating a tube of solutions of dynamic systems described by differential equations containing generalized (impulse) control functions are considered. It is supposed that a special constraint of an ellipsoidal type is imposed on the control, and, in particular, that the vectors of jumps of generalized equations are to belong to a specified ellipsoid of the appropriate finite-dimensional space. To obtain external estimates of the attainability sets of the system, the results of ellipsoidal calculus, which is well developed nowadays, are applied. A new method for constructing estimates based on the external ellipsoidal approximation of the convex set of a union of ellipsoids is also proposed. Particular examples of constructing estimates of the attainability set are considered based on the proposed methods.     

不确定规划中循环可达状态信息的维护

动态环境下,动作执行的不确定性会因外部因素存在变动,因此将导致不确定系统中的状态可达关系可能发生改变.为解答这一问题,论文对信息传递法中状态之间可达关系的更新方式进行改进,提出一种新的状态可达关系的维护算法.该算法将变更的状态之间可达关系与原可达矩阵对比,利用邻接矩阵中对应可达信息对变更后状态的可达信息进行修改,然后通...     

基于多面体包含的非线性混成系统可达性分析

针对一类非线性混成系统的可达性问题,提出了一种基于多面体包含的分析方法。首先介绍了混成系统及其可达性,讨论了如何应用多面体包含对多项式混成系统进行线性近似,并采用量词消去和非线性优化方法来构造相应的线性混成系统,然后运用验证工具SpaceEx求得原非线性混成系统的过近似可达集,并应用于验证系统的安全性。     

一类非仿射系统的鲁棒控制

考虑汽车转向控制中出现的微分包含系统的控制问题。首先用一非仿射不确定系统来描述这一微分包含系统，基于耗散理论分析了这一类非仿射不确定系统的鲁棒控制问题。并针对该非线性系统的有界性，给定系统鲁棒稳定的充分条件是满足一线性矩阵不等式，最后给出了一个仿真计算实例以验证算法的有效性。     

非线性DEDS的标准结构

非线性DEDS是指由极大极小函数描述的系统, 常见于计算机科学、控制论、运筹学等领域, 考虑非自治非线性DEDS的结构问题, 通过引入白色图和凝白色图, 得到了系统能达和能观的两个充要条件以及系统的标准结构, 同时还给出了它们的矩阵表示.     

Some decision problems related to the reachability problem for Petri nets

In this paper, we show that (1) the question to decide whether a given Petri net is consistent, M_o-reversible or live is reduced to the reachability problem in a unified manner, (2) the reachability problem for Petri nets is equivalent to the equality problem and the inclusion problem for the sets of all firing sequences of two Petri nets, (3) the equality problem for the sets of firing sequences of two Petri nets with only two unbounded places under homomorphism is undecidable, (4) the coverability and reachability problems are undecidable for generalized Petri nets in which a distinguished transition has priority over the other transitions, and (5) the reachability problem is undecidable for generalized Petri nets in which some transitions can reset a certain place to zero marking.     

Control design for stochastic one-sided Lipschitz differential inclusion system with time delay

This paper is concerned with the control design problem for stochastic one-sided Lipschitz differential inclusion system with time delay. Different from the current works on the differential inclusion system, the nonlinear function with time delay is assumed to be one-sided Lipschitz. First, the theory of stochastic differential equation is extended to stochastic differential inclusion and the stability criterion is given. Next, convex hull Lyapunov function is used to construct the nonlinear controllers, and the sufficient conditions for stability are derived by a set of bilinear matrix inequalities. Finally, three simulation examples are provided to verify the validity of the designed controllers.     

Vulnerability of dynamic systems†

The purpose of this paper is to formulate and partially resolve the problem of vulnerability of dynamic systems in the context of the theory of directed graphs. A system is considered vulnerable if its structural properties such as input (output) reachability or structural controllability (observability) is destroyed by a perturbation characterized as removal of a line, or set of lines, from the corresponding graph. Graph-theoretic procedures are developed to identify the minimal sets of lines which are essential for preserving the structural properties of the system.     

On the construction of resolving controls in control problems with phase constraints

Problems associated with the construction of resolving controls in problems of dynamics of control systems with phase constraints are considered. Here, it is topical to construct and estimate the reachability sets of control systems and differential inclusions. Moreover, the problem of steering the phase vector of the controlled system into a given point of the reachability set is very important. This problem is naturally connected with various problems of optimal control, in particular, with the time optimal control problem. In the general case, this problem cannot be solved explicitly. In this paper, a method for constructing a control that steers the phase vector into a specified neighborhood of a point in the reachability set is proposed. The method is based on the so-called guide constructions.     

On the analysis of nonlinear nilpotent switched systems using the Hall–Sussmann system

We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that a nice reachability analysis may be reduced to the reachability analysis of a specific canonical system, the nilpotent Hall–Sussmann system. Furthermore, local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems. We derive several new results revealing the elegant Lie-algebraic structure of the nilpotent Hall–Sussmann system.