The size of the membership-set in a probabilistic framework

In this paper, we study the size of the membership-set for system identification in a probabilistic framework. Assuming that the regressors are persistently exciting and the measurement noise is a sequence of independent, identically distributed bounded random variables, lower and upper non-asymptotic probability bounds on the membership-set diameter are obtained. These bounds are used in the computation of the confidence intervals for interpolatory estimators.     

Tight bounds for the trace of a matrix product

We propose a family of new upper and lower bounds for the trace of the matrix product AB when A, or B is symmetric. Those bounds depend on a scalar parameter, and both converge monotonically to tr(AB) when this parameter vanishes, thus providing arbitrary close approximations. Even large values of the parameter yield very good bounds     

Scenario-based verification of uncertain parametric MDPs

We consider parametric Markov decision processes (pMDPs) that are augmented with unknown probability distributions over parameter values. The problem is to compute the probability to satisfy a temporal logic specification with any concrete MDP that corresponds to a sample from these distributions. As solving this problem precisely is infeasible, we resort to sampling techniques that exploit the so-called scenario approach. Based on a finite number of samples of the parameters, the proposed method yields high-confidence bounds on the probability of satisfying the specification. The number of samples required to obtain a high confidence on these bounds is independent of the number of states and the number of random parameters. Experiments on a large set of benchmarks show that several thousand samples suffice to obtain tight and high-confidence lower and upper bounds on the satisfaction probability.

     

Robustness of globally exponential stability of delayed neural networks in the presence of random disturbances

This paper analyzes the robustness of globally exponential stability of time-varying delayed neural networks (NNs) subjected to random disturbances. Given a globally exponentially stable neural network, and in the presence of noise, we quantify how much noise intensity that the delayed neural network can remain to be globally exponentially stable. We characterize the upper bounds of the noise intensity for the delayed NNs to sustain globally exponential stability. The upper bounds of parameter uncertainty intensity are characterized by using transcendental equation. A numerical example is provided to illustrate the theoretical result.     

Computational length in pattern recognizers

The computational time of an absolute comparison recognizer is a random variable related to a measure of computational length. These variables depend on the choice of an algorithm which carries out the recognition. The expectation and the variance of the computational length are examined in detail with reference to the probability distribution of the classes to be discriminated. In particular, upper and lower bounds of the smallest mean computational length are found in easily evaluable forms.     

Optimal control of a linear system subject to partially specified input noise

One of the most basic problems in control theory is that of controlling a discrete‐time linear system subject to uncertain noise with the objective of minimizing the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent, and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence and show that, in this case, computing the optimal control inputs becomes infeasible. In the next step, we assume only the knowledge of lower and upper bounds on the conditional expectation of the noise and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise.     

Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems

We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree.     

Covariance bounds for discrete-time linear systems with time-varying parameter uncertainty

This paper deals with the computation of upper bounds for the state covariance matrix of discrete-time linear systems subject to stochastic excitation and additive time-varying uncertainty in the system dynamic matrix. Such upper bounds are obtained as the stabilizing solutions of suitable H ∞ -type Riccati equations. A necessary and sufficient condition for the existence of such solutions is given in terms of the H ∞ -norm of a suitable transfer function. As for the computation of the optimal bound, it is demonstrated that the bounds are a convex function of a scalar parameter, so that efficient numerical schemes can be worked out.     

Robust fusion time‐varying Kalman estimators for multisensor networked systems with mixed uncertainties

This paper addresses the problem of designing robust fusion time‐varying Kalman estimators for a class of multisensor networked systems with mixed uncertainties including multiplicative noises, missing measurements, packet dropouts, and uncertain‐variance linearly correlated measurement and process white noises. By the augmented approach, the original system is converted into a stochastic parameter system with uncertain noise variances. Furthermore, applying the fictitious noise approach, the original system is converted into one with constant parameters and uncertain noise variances. According to the minimax robust estimation principle, based on the worst‐case system with the conservative upper bounds of the noise variances, the five robust fusion time‐varying Kalman estimators (predictor, filter, and smoother) are presented by using a unified design approach that the robust filter and smoother are designed based on the robust Kalman predictor, which include three robust weighted state fusion estimators with matrix weights, diagonal matrix weights, and scalar weights, a modified robust covariance intersection fusion estimator, and robust centralized fusion estimator. Their robustness is proved by using a combination method, which consists of Lyapunov equation approach, augmented noise approach, and decomposition approach of nonnegative definite matrix, such that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. The accuracy relations among the robust local and fused time‐varying Kalman estimators are proved. A simulation example is shown with application to the continuous stirred tank reactor system to show the effectiveness and correctness of the proposed results.     

On the Complexity of Searching for a Maximum of a Function on a Quantum Computer

We deal with the problem of finding a maximum of a function from the Hölder class on a quantum computer. We show matching lower and upper bounds on the complexity of this problem. We prove upper bounds by constructing an algorithm that uses a pre-existing quantum algorithm for finding maximum of a discrete sequence. To prove lower bounds we use results for finding the logical OR of sequence of bits. We show that quantum computation yields a quadratic speed-up over deterministic and randomized algorithms.     

Probabilistic Upper Bounds for the Matrix Two-Norm

We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a user-chosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be computed efficiently. Together with an approximation that is a guaranteed lower bound, this may result in a small probabilistic interval for the matrix norm of large matrices within a fraction of a second.     

Image noise induced errors in camera positioning

The problem of evaluating worst-case camera positioning error induced by unknown-but-bounded (UBB) image noise for a given object-camera configuration is considered. Specifically, it is shown that upper bounds to the rotation and translation worst-case error for a certain image noise intensity can be obtained through convex optimizations. These upper bounds, contrary to lower bounds provided by standard optimization tools, allow one to design robust visual servo systems.     

Kalman filtering over a packet-delaying network: A probabilistic approach

In this paper, we consider Kalman filtering over a packet-delaying network. Given the probability distribution of the delay, we can characterize the filter performance via a probabilistic approach. We assume that the estimator maintains a buffer of length D so that at each time k, the estimator is able to retrieve all available data packets up to time k−D+1. Both the cases of sensor with and without necessary computation capability for filter updates are considered. When the sensor has no computation capability, for a given D, we give lower and upper bounds on the probability for which the estimation error covariance is within a prescribed bound. When the sensor has computation capability, we show that the previously derived lower and upper bounds are equal to each other. An approach for determining the minimum buffer length for a required performance in probability is given and an evaluation on the number of expected filter updates is provided. Examples are provided to demonstrate the theory developed in the paper.     

Parameter optimization of stochastic automata operating in random environments

Two methods of determining the lower bounds of the rate of convergence of finite stochastic automata are presented. The rate of convergence, defined as the percentage decrease in the distance between the transient probability distribution and the equilibrium probability distribution in each step, is determined as a function of the probability transition matrix. Formulas for parameter optimization for a class of stochastic automata for fast convergence and maximum expediency are derived and illustrative examples of fourth-order systems are given.     

Finding the likely behaviors of static continuous nonlinear systems

This paper describes two methods for predicting the likely behaviors of static continuous nonlinear systems with varying input values. The methods use a parameterized equation model and upper or lower bounds on the joint input density to bound the likelihood of a behavior, such as a state variable being inside a numeric range. Using a bound on the density instead of the density itself is desirable because the density's parameters and shape are not exactly known. The first method is limited to using lower density bounds. It finds rough bounds at first, and then refines them as more iterations of the method are allowed. The second method is a hit-or-miss version of sample-mean Monte Carlo. Unlike the first method, the second method can also handle upper density bounds, which are more useful than lower density bounds, but the generated probability bounds are only approximate. However, standard deviations on the bounds are given and become small as the sample size increases. In contrast to other researchers' methods, the two methods described here (1) find all the possible system behaviors, and tell how likely they are, (2) do not just approximate the distribution of possible outcomes without some measure of the error magnitude, (3) do not use discretized variable values, which limit the events one can find probability bounds for, (4) can handle density bounds, and (5) can handle such criteria as two state variables both being inside a numeric range.     

Information fusion robust guaranteed cost Kalman estimators with uncertain noise variances and missing measurements

ABSTRACT

For multisensor systems with uncertain noise variances and missing measurements, it can be converted into one only with uncertain noise variances by introducing fictitious measurement white noises. According to the minimax robust estimation principle and parameterisation representation of perturbances of uncertain noise variances, based on the worst-case system with conservative upper bounds of uncertain noise variances, the two classes of guaranteed cost robust weighted fusion Kalman estimators with matrix weights, diagonal matrix weights, scalar weights, and covariance intersection fusion matrix weights are presented. One class is the construction of a maximal perturbance region of uncertain noise variances, in which for all admissible perturbances, the accuracy deviations are guaranteed to remain within the prescribed range. The other class is the finding of minimal upper bound and maximal lower bound of accuracy deviations over the given perturbance region of uncertain noise variances. Two problems can be converted into the optimisation problems with constraints. Their optimal analytical solutions can simply be found respectively by the Lagrange multiplier method and the linear programme method. The guaranteed cost robustness is proved by the Lyapunov equation approach. A simulation example applied to the mass-spring system is provided to demonstrate the correctness and effectiveness of the proposed results.     

带不确定参数和噪声方差的鲁棒观测融合Kalman滤波器

对带不确定参数和噪声方差的多传感器定常系统,引入虚拟白噪声补偿不确定参数,可将其转化为带已知参数和不确定噪声方差系统.应用极大极小鲁棒估值原理和加权最小二乘法,基于带噪声方差保守上界的最坏情形保守系统,提出了鲁棒加权观测融合Kalman滤波器,并证明了它与集中式融合鲁棒Kalman滤波器是等价的,且融合器的鲁棒精度高于每个局部滤波器鲁棒精度.一个Monte-Carlo仿真例子说明了如何寻求不确定参数的鲁棒域和如何搜索保守性较小的虚拟噪声方差上界.     

基于边状态枚举计算多状态网络可靠度动态界

为降低计算多状态网络可靠度的复杂性,综合考虑网络中具有多态性的边处于各中间状态的概率及从某中间状态转换到相邻状态对网络性能的影响,提出了一种基于边状态枚举计算多状态网络可靠度上下界的算法.该算法首先令网络中各边仅取完全工作和完全失效两种状态,将处于中间状态的概率分别叠加到完全工作和完全失效状态的概率上,得到可靠度上下界的初始值;而后按照对可靠度影响递减的顺序迭代枚举边的中间状态,通过集合间的比较,计算可靠度上下界的改变值,同时获得不断减小的可靠度上界和不断增加的可靠度下界,使其最终收敛于可靠度精确值.该算法不需提前求取网络d-最小割(路)集,且枚举较少的网络状态即可得到紧凑的可靠度上下界.相关引理的证明及算例分析验证了该算法的正确性和有效性.     

Misclassification Probability Bounds for Multivariate Gaussian Classes

Classification systems based on linear discriminant analysis are employed in a variety of communications applications, in which the classes are most commonly characterized by known Gaussian PDFs. The performance of these classifiers is analyzed in this paper in terms of the conditional probability of misclassification. Easily computed lower and upper bounds on this error probability are presented and shown to provide corresponding bounds on the number of Monte Carlo trials required to obtain a desired level of accuracy. The error probability bounds yield an exact and easily computed expression for the error probability in the case where there are only two classes and a single hyperplane. In the special case where misclassification into a nominated class is independent of all other misclassifications, successively tighter upper and lower bounds can be computed at the expense of successively higher-order products of the individual misclassification probabilities. Finally, bounds are provided on the number of Monte Carlo trials required to improve, with suitably high confidence level, on the confidence interval formed by the error probability bounds.     

Event-based input and state estimation for linear discrete time-varying systems

In this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.