Xinjia Chen

A probabilistic approach to robust control

Considers a probabilistic approach to robust control. We construct explicitly the probability density function for the associated distribution of a random sample set of the robust performance index function which itself is a function of a vector of random variables. Using this probability density function, we derive several bounds on the minimum sample size required to estimate a robust performance index function with prescribed probability and confidence. These bounds show that the probabilistic approach to the robust control analysis and synthesis based on this framework has low complexity. Furthermore, a way of constructing the distribution function of the performance index function is proposed which gives the tradeoff between the performance and the risk.

Order statistics and probabilistic robust control

Order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution. Also, the concept of distribution-free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution. To overcome the limitations imposed by the continuity assumption in the existing order statistics theory, we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied. By applying this inequality, we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound (or lower bound) and the range of a quantity. We also give conditions, which are much weaker than the absolute continuity assumption, for the existence of such minimum sample size. Furthermore, the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established. This guideline can be applied in general without continuity assumption.

Constrained optimal synthesis and robustness analysis by randomized algorithms

We extend the existing order statistics distribution theory to the general case in which the order statistics is associated with certain constraints and the distribution of population is not assumed to be absolutely continuous. In particular, we derive an inequality on distribution for related order statistics. Moreover, we also propose two different approaches in searching reliable solutions to the robust analysis and optimal synthesis problems under constraints. Furthermore, minimum computational effort is investigated and bounds for sample size are derived.

Fast Construction of Robustness Degradation Function

We develop a fast algorithm to construct the robustness degradation function, which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set. This function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system.

On the probabilistic characterization of model uncertainty and robustness

This paper considers the probabilistic characterization of model uncertainties and stability robustness on the Nyquist diagram. For any given /spl epsi/>0, any frequency /spl omega/, we describe the boundary of a domain D/sub /spl epsi/,/spl omega// such that the frequency response of an uncertain plant at that frequency /spl omega/ is guaranteed in the domain with a probability of at least 1-/spl epsiv/ where /spl epsiv//spl les//spl epsi/ is a function of /spl epsi/ and frequency /spl omega/ and a upper bound of /spl epsiv/ can be estimated easily. Then, by analyzing the evolution of the boundary of D/sub /spl epsi/,/spl omega// as a function of the frequency /spl omega/, a combined curve C is constructed such that it includes the boundary of a domain D/sub /spl epsi// covering the Nyquist plot with a probability of at least 1-/spl epsiv/ for a range of frequencies.

Probabilistic Robustness Analysis—Risks, Complexity, and Algorithms

It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function, and we show that such an evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity that is linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of todays computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase.

Constrained robustness analysis by randomized algorithms

This paper shows that many robust control problems can be formulated as constrained optimization problems and can be tackled by using randomized algorithms. Two different approaches in searching reliable solutions to robustness analysis problems under constraints are proposed, and the minimum computational efforts for achieving certain reliability and accuracy are investigated and bounds for sample size are derived. Moreover, the existing order statistics distribution theory is extended to the general case in which the distribution of population is not assumed to be continuous and the order statistics is associated with certain constraints.

A new family of unitary space-time codes with a fast parallel sphere decoder algorithm

In this paper, we propose a new design criterion and a new class of unitary signal constellations for differential space-time modulation for multiple-antenna systems over Rayleigh flat-fading channels with unknown fading coefficients. Extensive simulations show that the new codes have significantly better performance than existing codes. We have compared the performance of our codes with differential detection schemes using orthogonal design, Cayley differential codes, fixed-point-free group codes, and product of groups and for the same bit-error rate, our codes allow smaller signal-to-noise ratio (SNR) by as much as 10 dB. The design of the new codes is accomplished in a systematic way through the optimization of a performance index that closely describes the bit-error rate as a function of the SNR. The new performance index is computationally simple and we have derived analytical expressions for its gradient with respect to constellation parameters. Decoding of the proposed constellations is reduced to a set of one-dimensional closest point problems that we solve using parallel sphere decoder algorithms. This decoding strategy can also improve efficiency of existing codes.

Fast universal algorithms for robustness analysis

In this paper, we develop efficient randomized algorithms for estimating probabilistic robustness margin and constructing robustness degradation curve for uncertain dynamic systems. One remarkable feature of these algorithms is their universal applicability to robustness analysis problems with arbitrary robustness requirements and uncertainty bounding set. In contrast to existing probabilistic methods, our approach does not depend on the feasibility of computing deterministic robustness margin. We have developed efficient methods such as probabilistic comparison, probabilistic bisection and backward iteration to facilitate the computation. In particular, confidence interval for binomial random variables has been frequently used in the estimation of probabilistic robustness margin and in the accuracy evaluation of estimating robustness degradation function. Motivated by the importance of fast computation of binomial confidence interval in the context of probabilistic robustness analysis, we have derived an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is extremely simple and very tight in comparison with classic Clopper-Pearsons approach.

Explicit Formula for Constructing Binomial Confidence Interval with Guaranteed Coverage Probability

In this article, we derive an explicit formula for constructing the confidence interval of binomial parameters with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduces unknown errors in applications. Moreover, the formula is very tight in comparison with classic Clopper–Pearsons approach from the perspective of interval width. Based on the rigorous formula, we also obtain approximate formulas with an excellent performance of coverage probability.