Robust Limits of Risk Sensitive Nonlinear Filters

Deterministic filter models are considered, and a criterion for a deterministic filter to be robust is introduced. Among the candidates for robust deterministic filters are so-called minimax estimators. In the second part of the paper, a risk sensitive stochastic approach to nonlinear filtering is considered, in which the traditional expected mean squared error criterion is replaced by an expected exponential-of-mean squared error. Minimax filters are obtained as totally risk averse limits of risk sensitive filters. Date received: July 22, 1999. Date revised: June 19, 2000.     

Robustness of Nonlinear Delay Equations with Respect to Input Perturbations: a Trajectory-Based Approach

 The effect of input perturbations on the stability properties of nonlinear time-varying delay differential equations is studied from a trajectory-based point of view. Thereby the semi-global stability results from [MSR2], where a Lyapunov approach is taken, are broadened to a much larger class of delay differential equations. Applications to the stabilization of partially linear cascade systems with delay using partial state feedback are briefly outlined. Date received: January 9, 2001. Date revised: April 15, 2002. RID="*" ID="*"This paper presents research results of the Belgian programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture (IUAP P4/02). The scientific responsibility rests with its authors. Luc Moreau is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium) (F.W.O.-Vlaanderen) and a recipient of an Honorary Fellowship of the Belgian American Educational Foundation. Part of this work was done while Luc Moreau was supported by a BOF grant of the Ghent University.     

Input-to-State Stability for Nonlinear Time-Varying Systems via Averaging

 We introduce two definitions of an averaged system for a time-varying ordinary differential equation with exogenous disturbances (“strong average” and “weak average”). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the weak average exists. It is shown that input-to-state stability (ISS) of the strong average of a system implies uniform semi-global practical ISS of the actual system. This result generalizes the result of [TPA] which states that global asymptotic stability of the averaged system implies uniform semi-global practical stability of the actual system. On the other hand, we illustrate by an example that ISS of the weak average of a system does not necessarily imply uniform semi-global practical ISS of the actual system. However, ISS of the weak average of a system does imply a weaker semi-global practical “ISS-like” property for the actual system when the disturbances w are absolutely continuous and . ISS of the weak average of a system is shown to be useful in a stability analysis of time-varying cascaded systems. Date received: April 6, 1999. Date revised: April 11, 2000.     

Formal Elimination for Multidimensional Systems and Applications to Control Theory

Following Douglas's ideas on the inverse problem of the calculus of variations, the purpose of this article is to show that one can use formal integrability theory to develop a theory of elimination for systems of partial differential equations and apply it to control theory. In particular, we consider linear systems of partial differential equations with variable coefficients and we show that we can organize the integrability conditions on the coefficients to build an “intrinsic tree”. Trees of integrability conditions naturally appear when we test the structural properties of linear multidimensional control systems with some variable or unknown coefficients (controllability, observability, invertibility, …) or for generic linearization of nonlinear systems. Date received: June 29, 1998. Date revised: January 29, 2000.     

On Local Structural Stability of Differential 1-Forms and Nonlinear Hypersurface Systems on a Manifold with Boundary

In this paper we consider smooth differential 1-forms and smooth nonlinear control-affine systems with (n−1)-inputs evolving on an n-dimensional manifold with boundary. These systems are called hypersurface systems under the additional assumption that the drift vector field and control vector fields span the tangent space to the manifold. We locally classify all structurally stable differential 1-forms on a manifold with boundary. We give complete local classification of structurally stable hypersurface systems on a manifold with boundary under static state feedback defined by diffeomorphisms, which preserve the manifold together with its boundary. Date received: March 30, 2000. Date revised: October 30, 2000.     

非线性系统的实现与最小阶问题

在系统的微分域及相应的微分向量空间上定义了一个非交换的多项式环(算子环), 并利用这个环定义非线性系统的传递函数.用微分向量空间为工具,讨论单输入/单输出非线 性系统的实现问题.主要结果回答了:1)在什么条件下,不同的输入/输出微分方程有相同(等 价)的实现;2)在未知实现的条件下,如何确定输入/输出微分方程最小实现的阶数.覆盖了线 性系统理论的相关结果.     

Worst Case Power Generating Capabilities of Nonlinear Systems

In this paper we analyze the worst case power generating capabilities of a class of nonlinear systems which exhibit a power gain property. This class of systems includes systems which exhibit persistent excitation in the absence of inputs. Examples include limit cycle systems and chaotic systems. In order to capture the power generating capability of a nonlinear system, we define a worst case average cost per unit time performance index. This performance index, called the available power, is in effect the most power that can be generated by a system via the application of any input. The main result of the paper is that the input which achieves this worst case performance is typically a persistent input whose power is given explicitly by a function of the derivative of the available power with respect to the power gain of the system. A natural corollary of this result is that the available power may be recast as an optimization over power inputs. Date received: 24 February 2000. Date revised: 11 May 2001.     

A Global Asymptotic Stability Result for a Class of Totally Asynchronous Discrete Nonlinear Systems

This paper proves a global stability result for a class of nonlinear discrete-time systems that are subject to regular desynchronization, also known as total asynchronism. The class of systems studied has its origins in a discrete-time neural net model. The techniques used are of interest in terms of the use of a Lyapunov function for the study of convergence of asynchronous nonlinear dynamical systems and also in terms of applications to neural networks. In the latter context, the main result of this paper strengthens a result of an earlier paper on neural networks, and shows that a class of discrete-time continuous-valued neural nets of the Hopfield type displays global convergence properties even when there exists total asynchronism in the updating of neuron states. Date received: May 2, 1995. Date revised: October 19, 1998.     

A Trajectory-Based Approach for the Stability Robustness of Nonlinear Systems with Inputs

 We show, for two different definitions of semiglobal practical external stability, that the stability property holds on semi-infinite time intervals if and only if it holds on arbitrarily long but finite time intervals. These results have immediate applications in the analysis of the stability properties of highly oscillatory systems with inputs using averaging or for systems with inputs that are slowly varying. Results are stated for general flows and the stability is given with respect to arbitrary (not necessarily compact) sets. Date received: May 11, 2001. Date revised: March 14, 2002.     

航天器姿轨耦合非线性同步控制

研究航天器飞行稳定控制建模问题。航天器动力学模型的精确建立,要求采用单独建立轨道或姿态的模型无法满足任务高精度要求,从非线性相对轨道动力学方程和修正罗德里格斯参数(MRP)表示的姿态运动学方程出发,建立了航天器六自由度的相对耦合动力学方程。为了给出姿轨运动的基准,分别设计了航天器理想姿态和椭圆加指数接近轨道。针对耦合非线性动力学方程设计了非线性同步控制律,并通过Lyapunov证明闭环系统的全局渐近稳定性。通过仿真结果可以看出,非线性同步控制算法能使轨道和姿态误差逐步趋于零,为航天器姿轨耦合设计提供了依据。     

Observability and Forward–Backward Observability of Discrete-Time Nonlinear Systems

 In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward–backward observability is equivalent to the stronger notion of (forward) observability. Date received: January 19, 2001. Date revised: May 14, 2002.     

Nonlinear equation approach for inequality elastostatics: a two-dimensional BEM implementation

The numerical solution of variational inequality problems in elastostatics is investigated by means of recently proposed equivalent nonlinear equations. Symmetric and nonsymmetric variational inequalities and linear or nonlinear, but monotone, complementarity problems can be solved this way without explicit use of nonsmooth (nondifferentiable) solvers. As a model application, two-dimentional unilateral contact problems with and without friction effects approximated by the boundary element method are formulated as nonsymmetric variational inequalities, or, for the two-dimensional case as linear complementarity problems, and are numerically solved. Performance comparisons using two standard, smooth, general purpose nonlinear equation solvers are included.     

Nonlinear stabilization in infinite dimension

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infinite-dimensional setting. First we present designs for nonlinear ODEs with constant, time-varying or state-dependent input delays, which arise in numerous applications of control over networks. Second, we present a design for nonlinear ODEs with a wave (string) PDE at its input, which is motivated by the drilling dynamics in petroleum engineering. Third, we present a design for systems of (two) coupled nonlinear first-order hyperbolic PDEs, which is motivated by slugging flow dynamics in petroleum production in off-shore facilities. Our design and analysis methodologies are based on the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping. We present several simulation examples that illustrate the design methodology.     

Nonlinear robust performance analysis using complex-step gradient approximation

In this paper, the complex-step method is applied in the setting of numerical optimisation problems involving dynamical systems modelled as nonlinear differential equations. The main advantage of the complex-step method for gradient approximation is that it entails no subtractive cancellation error, and therefore the truncation error can be made arbitrarily (to machine precision) small. The method is applied to two robust performance analysis problems. The accuracy and convergence rate of the solutions computed using the proposed approach are seen to be significantly better than those achieved using standard gradient approximation methods.     

The complexity of matrix rank and feasible systems of linear equations

We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix (as well as other problems) are complete under logspace reductions.?As an important part of presenting this classification, we show that the "exact counting logspace hierarchy" collapses to near the bottom level. We review the definition of this hierarchy below. We further show that this class is closed under NC¹-reducibility, and that it consists of exactly those languages that have logspace uniform span programs (introduced by Karchmer and Wigderson) over the rationals.?In addition, we contrast the complexity of these problems with the complexity of determining if a system of linear equations has an integer solution. Received: June 9, 1997.     

On the Learnability of Recursive Data

We establish some general results concerning PAC learning: We find a characterization of the property that any consistent algorithm is PAC. It is shown that the shrinking width property is equivalent to PUAC learnability. By counterexample, PAC and PUAC learning are shown to be different concepts. We find conditions ensuring that any nearly consistent algorithm is PAC or PUAC, respectively.?The VC dimension of recurrent neural networks and folding networks is infinite. For restricted inputs, however, bounds exist. The bounds for restricted inputs are transferred to folding networks.?We find conditions on the probability of the input space ensuring polynomial learnability: the probability of sequences or trees has to converge to zero sufficiently fast with increasing length or height.?Finally, we find an example for a concept class that requires exponentially growing sample sizes for accurate generalization. Date received: September 5, 1997. Date revised: May 29, 1998.     

Nonlinear viscoelastic membranes

The subject of nonlinear viscoelastic membranes is an important class of problems within nonlinear viscoelasticity that involves interesting and important applications, computational issues and applied mathematics.The viscoelastic materials discussed in this paper are described by nonlinear single integral constitutive equations. After presenting the general constitutive framework, two fundamental membrane problems are formulated: the inflation of a circular membrane and the extension and inflation of a circular tube. Both problems involve large axially symmetric deformations and lead to a system of nonlinear partial differential–Volterra integral equations. A numerical method of solution is presented that combines methods for solving nonlinear Volterra integral equations and nonlinear ordinary differential equations. Finally, some properties of the equations are discussed that are related to the possibility that there may exist a critical time when the solution develops multiple branches.     

Small Parameter Limit for Ergodic, Discrete-Time, Partially Observed, Risk-Sensitive Control Problems

We show that discrete-time, partially observed, risk-sensitive control problems over an infinite time horizon converge, in the small noise limit, to deterministic dynamic games, in the sense of uniform convergence of the value function on compact subsets of its domain. We make use of new results concerning large deviations and existence of value functions. Date received: May 21, 1999. Date revised: April 7, 2000.     

Nonlinear and dynamic structural analysis using combined approximations

It is shown how the combined approximations (CA) approach, developed originally for linear reanalysis, can improve the solution efficiency of nonlinear and dynamic analysis problems. In such problems the analysis equations are modified repeatedly during the solution process. The CA approach is based on the integration of several concepts and methods. The advantage is that efficient local approximations and accurate global approximations are combined to achieve an effective solution procedure. Some numerical examples demonstrate how the CA method can be used as a general tool in various structural analysis problems.