Consistency tests in guaranteed simulation of nonlinear uncertain systems with application to an activated sludge process

In this paper, interval arithmetic simulation techniques are presented to determine guaranteed enclosures of the state variables of both continuous and discrete-time systems with uncertain but bounded parameters. In nonlinear uncertain systems axis-parallel interval boxes are mapped to complexly shaped regions in the state space that represent sets of possible combinations of state variables. The approximation of each region by a single interval box causes an accumulating overestimation from time-step to time-step, usually called the wrapping effect. The algorithm presented in this paper minimizes the wrapping effect by applying consistency techniques based on interval Newton methods. Subintervals that do not belong to the exact solution at a given time can be eliminated in order to give a tighter but still conservative approximation of the exact solution. Additionally, efficient splitting and merging strategies are employed to limit the number of subintervals. The proposed algorithm is applied to the simulation of an activated sludge process in biological wastewater treatment.     

Extensions of ValEncIA-IVP for reduction of overestimation,for simulation of differential algebraic systems,and for dynamical optimization

Simulation techniques are commonly used to analyze the influence of uncertainties of initial conditions and systemparameters on the trajectories of the state variables of dynamical systems. In this context, interval arithmetic approaches are of interest. They are capable of determining guaranteed bounds of all reachable states if worst-case bounds of the above-mentioned uncertainties are known. Furthermore, interval algorithms ensure the correctness of numerical results in spite of rounding errors which inevitably arise if floating point operations are carried out on a computer. However, naive implementations of interval algorithms often lead to overestimation, i.e., too conservative enclosures which can make the results meaningless. In this contribution, we summarize the basic routines of ValEncIA-IVP which computes interval enclosures of all reachable states of dynamical systems described by ordinary differential equations ODEs. ValEncIA-IVP , VAL idation of state ENC losures using I nterval A rithmetic for I nitial V alue P roblems, can be applied to the simulation of systems with both uncertain parameters and uncertain initial conditions. Advanced techniques for reduction of overestimation are demonstrated for a simplified catalytic reactor. Afirst approach to using VanEncIA-IVP for the simulation of sets of differential algebraic equations is outlined. Finally, an outlook on the integration of ValEncIA-IVP in an interval arithmetic framework for computation of optimal and robust control strategies for continuous-time processes is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interval methods as a simulation tool for the dynamics of biological wastewater treatment processes with parameter uncertainties

This paper presents sophisticated interval algorithms for the simulation of discrete-time dynamical systems with bounded uncertainties of both initial conditions and system parameters. Since naive implementations of interval algorithms might lead to guaranteed enclosures of all system states which are too conservative to be practically useful, we present algorithmic extensions of classical approaches which are applicable to the simulation of non-cooperative systems with time-varying uncertain parameters. Overestimation arising in the interval evaluation of dynamical system models due to the wrapping effect is reduced by an exact pseudo-linear transformation of nonlinear state equations and by new heuristics for the subdivision of interval enclosures which especially prefer splitting of unstable intervals. To highlight the typical procedure for parameterization of interval-based simulation routines and to demonstrate their efficiency, a nonlinear model of biological wastewater treatment processes is discussed. For this application, we consider the maximum specific growth rate of substrate consuming bacteria as a time-varying uncertain parameter. Only worst-case bounds are assumed to be available for the range of this parameter while no information is provided about its actual variation rate.     

The transformation method for the simulation and analysis of systems with uncertain parameters

In this paper, the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters. Based on the concept of α-cuts, the method represents a special implementation of fuzzy arithmetic that avoids the well-known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation. Systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results. As a by-product of the implementation scheme, the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output. By this, a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic. Finally, to show the efficiency of the transformation method, the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection.     

Interval techniques for enclosures of regions of reachability and controllability and for guaranteed state and parameter estimation of dynamical systems

Interval techniques are a powerful means for calculation of enclosures of the regions of reachability and controllability of dynamical systems with uncertainties during analysis and design of controllers. In this contribution, both discrete-time and continuous-time dynamical systems are considered. Using suitable algorithms, guaranteed state enclosures can be determined for systems with uncertain parameters, uncertain initial conditions, nonlinearities, and time-varying characteristics. Although both uncertain system parameters and bounded control variables are assumed to be represented by interval boxes in the following, they have to be distinguished in reachability and controllability analysis. Typically, robustness specifications for controllers of dynamical systems are given in terms of bounds on the system's time response which must not be violated for any possible operating condition. Hence, reachability as well as controllability of states have to be proven for all possible parameter values but for at least one admissible control sequence. Robust control strategies for nonlinear systems usually rely on knowledge of all current states. However, the complete state vector is not always directly accessible for measurement. In this case, observers are applicable to reconstruct non-measurable state variables. Furthermore, they can reduce the uncertainties of the measured quantities by model-based recursive computation of estimates and fusion of information gathered by different measurement devices. If guaranteed bounds of all uncertain parameters of a dynamical system (including the sensor characteristics) and conservative bounds of all disturbances can be specified, the presented interval observer provides guaranteed enclosures of all reachable states. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Guaranteed Nonlinear State Estimator for Cooperative Systems

This paper is about state estimation for continuous-time nonlinear models, in a context where all uncertain variables can be bounded. More precisely, cooperative models are considered, i.e., models that satisfy some constraints on the signs of the entries of the Jacobian of their dynamic equation. In this context, interval observers and a guaranteed recursive state estimation algorithm are combined to enclose the state at any given instant of time in a subpaving. The approach is illustrated on the state estimation of a waste-water treatment process.     

Symbolic Interval Inference Approach for Subdivision Direction Selection in Interval Partitioning Algorithms

In bound constrained global optimization problems, partitioning methods utilizing Interval Arithmetic are powerful techniques that produce reliable results. Subdivision direction selection is a major component of partitioning algorithms and it plays an important role in convergence speed. Here, we propose a new subdivision direction selection scheme that uses symbolic computing in interpreting interval arithmetic operations. We call this approach symbolic interval inference approach (SIIA). SIIA targets the reduction of interval bounds of pending boxes directly by identifying the major impact variables and re-partitioning them in the next iteration. This approach speeds up the interval partitioning algorithm (IPA) because it targets the pending status of sibling boxes produced. The proposed SIIA enables multi-section of two major impact variables at a time. The efficiency of SIIA is illustrated on well-known bound constrained test functions and compared with established subdivision direction selection methods from the literature.     

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

区间运算和静力区间有限元

用均值和离差两参数表征区间变量的不确定性,根据区间运算规则,论证了区间变量的运算特性.将区间分析和有限元方法相结合,提出了非概率不确定结构的一种区间有限元分析方法.将区间有限元静力控制方程中n自由度不确定位移场特征参数的求解归结为求解一2n阶线性方程组.实例分析表明文中方法是有效和可行的.     

在不确定观测下离散状态时滞系统的最优滤波

研究了在不确定观测下离散状态时滞系统的最优滤波问题,观测值的不确定性则通过一个满足Bernoulli分布且统计特性已知的随机变量来描述. 一般采用状态增广方法将时滞系统转换为无时滞随机系统, 再利用Kalman滤波器的设计方法解决最优状态估计问题, 但是当系统时滞较大时,转换后的系统状态维数很高, 这样增加了计算负担. 为此,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了一个新的滤波器设计方法, 而且保证了滤波器的维数与原系统相同.最后, 给出一个仿真例子说明所提方法的有效性.     

Robust Stability Criteria for Uncertain Neutral Systems with Interval Time-Varying Delay

In this paper, we consider the problem of robust stability of a class of linear uncertain neutral systems with interval time-varying delay under (i) nonlinear perturbations in state, and (ii) time-varying parametric uncertainties using Lyapunov-Krasovskii approach. By constructing a candidate Lyapunov-Krasovskii (LK) functional, that takes into account the delay-range information appropriately, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMI) to compute the maximum allowable bound for the delay-range within which the uncertain neutral system under consideration remains asymptotically stable. The reduction in conservatism of the proposed stability criterion over recently reported results is attributed to the fact that time-derivative of the LK functional is bounded tightly without neglecting any useful terms using a minimal number of slack matrix variables. The analysis, subsequently, yields a stability condition in convex LMI framework, that can be solved non-conservatively at boundary conditions using standard LMI solvers. The effectiveness of the proposed stability criterion is demonstrated through standard numerical examples.     

Tandem Behaviour of a Telecommunication System with Repeated Calls: A Markovian Case with Buffers

The performance of a telecommunication system consisting of a set of transmitters with finite capacity buffers is modelled with a Markovian queueing network, and its tandem behaviour is approximated, in steady state. In this system, a fraction of the units which, at the instants of their arrival at each transmitter, find it busy may retry to be processed by merging with the incoming arrival units at the same transmitter after a fixed delay time. The performance of this system is approximated by a recursive algorithm, in steady state. Furthermore, the approximation outcomes are compared against those from a simulation study. In summary, our numerical results indicate that approximating the non-renewal superposition arrival, the non-renewal overflow and the non-renewal departure processes at each node of the network can be approximated with compatible Poisson processes.     

Robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay

This paper studies the problem of robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay. The state variables on the impulses are assumed dependent on the present state variables as well as delayed state variables. Based on the Razumikhin techniques and Lyapunov functions, some robust mean-square exponential stability criteria are derived in terms of linear matrix inequalities. The results show that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. Furthermore, the robust delayed-state-feedback controllers that mean-square exponentially stabilize the uncertain impulsive stochastic systems are proposed. Finally, several numerical examples are given to show the effectiveness of the results.     

MC方差减小技术在算术平均亚式外汇期权定价中的应用

建立了利率和汇率波动率均为随机情形下算术平均亚式外汇期权的定价模型.由于其定价问题求解十分困难,运用蒙特卡罗(Monte Carlo)方法并结合控制变量方差减小技术进行模拟,有效地减小了模拟方差,得到了期权定价问题的数值结果.     

Applying fuzzy GERT with approximate fuzzy arithmetic based on the weakest t-norm operations to evaluate repairable reliability

In general, the fuzzy Graphical Evaluation and Review Technique (GERT) usually evaluates/analyzes variables with interval arithmetic (α-cut arithmetic) operations, especially those with complicated fuzzy systems. Thus the interval arithmetic operations may occur accumulating phenomenon of fuzziness in complicated systems, and the accumulating phenomenon of fuzziness may make decision-maker that cannot effectively evaluate problems/systems under vague environment. In order to overcome the accumulating phenomenon of fuzziness or credibly reduce fuzzy spreads, this study adopts approximate fuzzy arithmetic operations under the weakest t-norm arithmetic operations (T_ω) to evaluate fuzzy reliability models based on fuzzy GERT simulation technology. The approximate fuzzy arithmetic operations employ principle of interval arithmetic under the weakest t-norm arithmetic operations. Therefore, the novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating, under vague environment. In numerical examples the approximate fuzzy arithmetic operations has evidenced that it can successfully calculate results of fuzzy operations as interval arithmetic, and can more effectively reduce fuzzy spreads. In the real fuzzy repairable reliability model the performance also shows that the approximate fuzzy arithmetic operations successfully analyze the reliability problem and obtain more confident fuzzy results.     

A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.     

An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels

This paper proposes a new design optimization framework for suspension systems considering the kinematic characteristics, such as the camber angle, caster angle, kingpin inclination angle, and toe angle in the presence of uncertainties. The coordinates of rear inner hardpoints of upper control arm and lower control arm of double wishbone suspension are considered as the design variables, as well as the uncertain parameters. In this way, the actual values of the design variables will vary surrounding their nominal values. The variations result in uncertainties that are described as interval variables with lower and upper bounds. The kinematic model of the suspension is developed in software ADAMS. A high-order response surface model using the zeros of Chebyshev polynomials as sampling points is established, termed as Chebyshev metamodel, to approximate the kinematic model. The Chebyshev meta-model is expected to provide higher approximation accuracy. Interval uncertain optimization problems usually involve a nested computationally expensive double-loop optimization process, in which the inner loop optimization is to calculate the bounds of the interval design functions, while the outer loop is to search the optimum for the deterministic optimization problem. To reduce the computational cost, the interval arithmetic is introduced in the inner loop to improve computational efficiency without compromising numerical accuracy. The numerical results show the effectiveness of the proposed design method.     

The use of approximate factorization in stiff ODE solvers

We consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration.     

不确定区间隶属度语言变量及其应用

基于不确定语言变量和区间模糊数,提出了不确定区间隶属度语言变量的概念,定义了不确定区间隶属度语言变量的运算规则、大小比较方法,给出了不确定区间隶属度语言变量的加权算术平均算子、加权几何平均算子及其相应性质,并将这些算子应用于属性权重确知且属性值以不确定区间隶属度语言变量形式给出的不确定多属性群决策问题中,通过示例验证了基于不确定区间隶属度语言变量信息的多属性群决策方法的有效性和可行性。