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Eigenvalues assignment in uncontrollable linear systems

Kaczorek Tadeusz

Bulletin of the Polish Academy of Sciences. Technical Sciences

2022

Vol. 70, nr 6

art. no. e141987

It is shown that in uncontrollable linear system x = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A − BK has the desired eigenvalues. The procedure is illustrated by simple example.

The problem of imputation of the missing data from the continuous counts of road traffic

Spławińska M.

Archives of Civil Engineering

2015

Vol. 61, nr 1

131--145

Missing traffic data is an important issue for road administration. Although numerous ways can be found to impute them in foreign literature (inter alia, the most effective method, that is Box-Jenkins models), in Poland, still only proven and simplified methods are applied. The article presents the analyses including an assessment of the completeness of the existing traffic data and works related to the construction of SARIMA model. The study was conducted on the basis of hourly traffic volumes, derived from the continuous traffic counts stations located in the national road network in Poland (Golden River stations) from the years 2005 – 2010. As a result, the proposed model was used to impute the missing data in the form of SARIMA (1.1,1)(0,1,1)168. The newly developed model can be used effectively to fill in the missing required days of measurement for estimating AADT by AASHTO method. In other cases, due to its accuracy and laboriousness of the process, it is not recommended.