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Kemin Zhou

Publications and source records attributed to Kemin Zhou.

16 recordsLinked to original sources
arXiv2024

Multi-Objective Complementary Control

This paper proposes a novel multi-objective control framework for linear time-invariant systems in which performance and robustness can be achieved in a complementary way instead of a trade-off. In particular, a state-space solution is first established for a new stabilizing control structure consisting of two independently designed controllers coordinated with a Youla-type operator ${\bm Q}$. It is then shown by performance analysis that these two independently designed controllers operate in a naturally complementary way for a tracking control system, due to the coordination function of ${\bm Q}$ driven by the residual signal of a Luenberger observer. Moreover, it is pointed out that ${\bm Q}$ could be further optimized with an additional gain factor to achieve improved performance, through a data-driven methodology for a measured cost function.

eess.SY↗
arXiv2008

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.

math.OC↗
arXiv2008

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.

math.OC↗
arXiv2008

Constrained Optimal Synthesis and Robustness Analysis by Randomized Algorithms

In this paper, we consider robust control using randomized algorithms. We extend the existing order statistics distribution theory 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. 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.

math.OC↗
arXiv2008

On the Binomial Confidence Interval and Probabilistic Robust Control

The Clopper-Pearson confidence interval has ever been documented as an exact approach in some statistics literature. More recently, such approach of interval estimation has been introduced to probabilistic control theory and has been referred as non-conservative in control community. In this note, we clarify the fact that the so-called exact approach is actually conservative. In particular, we derive analytic results demonstrating the extent of conservatism in the context of probabilistic robustness analysis. This investigation encourages seeking better methods of confidence interval construction for robust control purpose.

math.OC↗
arXiv2008

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 computing 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-Pearson's approach.

math.OC↗
arXiv2008

Sample Reuse Techniques of Randomized Algorithms for Control under Uncertainty

Sample reuse techniques have significantly reduced the numerical complexity of probabilistic robustness analysis. Existing results show that for a nested collection of hyper-spheres the complexity of the problem of performing $N$ equivalent i.i.d. (identical and independent) experiments for each sphere is absolutely bounded, independent of the number of spheres and depending only on the initial and final radii. In this chapter we elevate sample reuse to a new level of generality and establish that the numerical complexity of performing $N$ equivalent i.i.d. experiments for a chain of sets is absolutely bounded if the sets are nested. Each set does not even have to be connected, as long as the nested property holds. Thus, for example, the result permits the integration of deterministic and probabilistic analysis to eliminate regions from an uncertainty set and reduce even further the complexity of some problems. With a more general view, the result enables the analysis of complex decision problems mixing real-valued and discrete-valued random variables.

math.OC↗
arXiv2008

Fast Parallel Frequency Sweeping Algorithms for Robust ${\cal D}$-Stability Margin

This paper considers the robust ${\cal D}$-stability margin problem under polynomic structured real parametric uncertainty. Based on the work of De Gaston and Safonov (1988), we have developed techniques such as, a parallel frequency sweeping strategy, different domain splitting schemes, which significantly reduce the computational complexity and guarantee the convergence.

math.OC↗
arXiv2007

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 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 signal to noise ratio. 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.

cs.IT↗
arXiv2007

A Statistical Theory for the Analysis of Uncertain Systems

This paper addresses the issues of conservativeness and computational complexity of probabilistic robustness analysis. We solve both issues by defining a new sampling strategy and robustness measure. The new measure is shown to be much less conservative than the existing one. The new sampling strategy enables the definition of efficient hierarchical sample reuse algorithms that reduce significantly the computational complexity and make it independent of the dimension of the uncertainty space. Moreover, we show that there exists a one to one correspondence between the new and the existing robustness measures and provide a computationally simple algorithm to derive one from the other.

stat.AP↗
arXiv2007

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 evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity 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 today's 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.

stat.AP↗
arXiv2007

Explicit Formula for Constructing Binomial Confidence Interval with Guaranteed Coverage Probability

In this paper, we derive 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 very tight in comparison with classic Clopper-Pearson's approach from the perspective of interval width. Based on the rigorous formula, we also obtain approximate formulas with excellent performance of coverage probability.

math.ST↗
arXiv2007

Risk Analysis in Robust Control -- Making the Case for Probabilistic Robust Control

This paper offers a critical view of the "worst-case" approach that is the cornerstone of robust control design. It is our contention that a blind acceptance of worst-case scenarios may lead to designs that are actually more dangerous than designs based on probabilistic techniques with a built-in risk factor. The real issue is one of modeling. If one accepts that no mathematical model of uncertainties is perfect then a probabilistic approach can lead to more reliable control even if it cannot guarantee stability for all possible cases. Our presentation is based on case analysis. We first establish that worst-case is not necessarily "all-encompassing." In fact, we show that for some uncertain control problems to have a conventional robust control solution it is necessary to make assumptions that leave out some feasible cases. Once we establish that point, we argue that it is not uncommon for the risk of unaccounted cases in worst-case design to be greater than that of the accepted risk in a probabilistic approach. With an example, we quantify the risks and show that worst-case can be significantly more risky. Finally, we join our analysis with existing results on computational complexity and probabilistic robustness to argue that the deterministic worst-case analysis is not necessarily the better tool.

math.OC↗
arXiv2007

A statistical theory for the measurement and estimation of Rayleigh fading channel

In this paper, we propose a statistical theory on measurement and estimation of Rayleigh fading channels in wireless communications and provide complete solutions to the fundamental problems: What is the optimum estimator for the statistical parameters associated with the Rayleigh fading channel, and how many measurements are sufficient to estimate these parameters with the prescribed margin of error and confidence level? Our proposed statistical theory suggests that two testing signals of different strength be used. The maximum likelihood (ML) estimator is obtained for estimation of the statistical parameters of the Rayleigh fading channel that is both sufficient and complete statistic. Moreover, the ML estimator is the minimum variance (MV) estimator that in fact achieves the Cramer-Rao lower bound.

math.ST↗
arXiv2006

A Short Note on The Volume of Hypersphere

In this note, a new method for deriving the volume of hypersphere is proposed by using probability theory. The explicit expression of the multiple times convolution of the probability density functions we should use is very complicated. But in here, we don't need its whole explicit expression. We just need the only a part of information and this fact make it possible to derive the general expression of the voulume of hypersphere. We also comments about the paradox in the hypersphere which was introduced by R.W.Hamming.

cs.IT↗