RACT: Randomized Algorithms Control Toolbox for MATLAB

A probabilistic solution of robustH∞control problem with scaled matrices

“…degree in automation, M.S. degree and Ph.D. degree in electrical engineering from Northwestern Polytechnical University, Xi'an, Shaanxi, China in 2004, 2007, and 2010 respectively. From 2010 to 2011, he was an Engineer in Shaanxi Aero Electric Co., Ltd.,China.…”

“…The scaled H ∞ control problem becomes convex for the fixed sample of the scaling matrix D, and a optimal solution corresponding to the fixed sample is obtained. Then, a probabilistic optimal solution is obtained by using the randomised algorithm based on a finite number N optimal solutions, which are obtained for the N samples of the scaling matrix D. In addition, both YALMIP (Juradoa et al, 2013) and RACT (Tremba et al, 2010) tools are helpful and powerful to implement the proposed algorithm. The computational cost analysis shows that its worst case complexity is a polynomial.…”

“…Definition 1 (RA for probabilistic optimal performance (Tremba et al, 2010)): Let p * ∈ (0, 1), δ ∈ (0, 1) be assigned probability level. The RA should return with probability at least than 1 − δ a probabilistic optimal performance level γ N such that…”

“…The Equation (22) can be transformed into the following form: Suppose both of the matrices L and J obey the uniform distribution in the norm ball B D (1). They are expressed as L, J ∼ U B D (0, 1), where U B D (0, 1) denotes the random matrix distributed in ∞ induced norm ball of radius 1 (Tremba et al, 2010).…”

Rigorous error control methods for estimating means of bounded random variables

“…Clearly, the failure probability p is equal to the mean of X. That is, $p=\mathrm{double-struckE}\left[X\right]=\mathrm{double-struckE}\left[\mathrm{frakturX}\right(\mathrm{bold-italic\Delta }\left)\right]$ For estimating the failure probability p, randomized algorithms have been implemented in a widely used software package RACT [22], in which an absolute error criterion is used for estimating p. Specifically, for a priori ɛ, δ ∈ (0, 1), the objective is to obtain an estimator $\hat{\mathit{p}}$ such that $\mathbb{P}\left\{\left|\hat{\mathit{p}}-p\right|<\epsilon \right\}>1-\delta$ holds regardless of the value of the p ∈ (0, 1). The estimator is defined as $$\hat{\mathit{p}}=\frac{1}{N}true\sum _{i=1}^N\mathfrak{X}\left({\mathrm{bold-italic\Delta }}_i\right),$$where N is the sample size and Δ 1 , Δ 2 , ⋯, Δ N are i.i.d.…”