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  1. 23 International Journal of Applied Mathematics and Computer Science
  2. 20 Bulletin of the Polish Academy of Sciences. Technical Sciences
  3. 9 Przegląd Elektrotechniczny
  4. 9 Systems Science
  5. 7 Archives of Control Sciences
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  1. 34 Kaczorek T.
  2. 19 Busłowicz M.
  3. 12 Kaczorek Tadeusz
  4. 2 Czornik A.
  5. 2 Kościelny J. M.
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  1. 2 2025
  2. 2 2024
  3. 5 2023
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1
Content available Discrete-time linear systems with desired poles and zeros of the transfer matrices
Kaczorek Tadeusz, Sajewski Łukasz
International Journal of Applied Mathematics and Computer Science
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2025
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Vol. 35, no. 1
59--68
EN
Methods for the design of discrete-time linear systems with desired poles and zeros of their transfer matrices are proposed. Conditions for the existence of the solution to the problem and the procedures for computation of the desired matrices are given. Reduction of the systems with controllable and observable pairs to those with nilpotent matrices is analysed. The procedures are illustrated by simple numerical examples of linear discrete-time systems.
2
Content available Observers for unobservable linear systems
Kaczorek Tadeusz
Archives of Control Sciences
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2025
|
Vol. 35, No. 1
145--154
EN
Observers for unobservable linear systems ẋ = 𝐴𝑥 + 𝐵𝑢, 𝑦 = 𝐶𝑥, 𝑥 = 𝑥(𝑡) ∈ ℜ𝑛, 𝑢 = 𝑢(𝑡) ∈ ℜm, 𝑦 = 𝑦(𝑡) ∈ ℜp are proposed. It is shown that there exist full-order and reduced-order observers for systems satisfying the condition rank [𝐴 𝐶] = 𝑛. Procedures for computation of the matrices of the observers are given and illustrated by numerical examples.
3
Content available Dirac delta as a useful technical Tool in modelling signals but hard to think about It as a physical signal itself
Borys A.
TransNav : International Journal on Marine Navigation and Safety of Sea Transportation
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2024
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Vol. 18 No. 2
387--392
EN
In this paper, we show that the Dirac delta is a useful technical tool in modelling signals but hard to think about it as a physical signal itself. This thesis is supported here by an example coming from the field of measuring physical quantities and measurement theory.
4
Content available Transformations of linear standard systems to positive asymptotically stable linear ones
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2024
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Vol. 34, no. 3
341--348
EN
New approaches to transformations of linear continuous-time systems to their positive asymptotically stable canonical controllable (observable) forms are proposed. It is shown that, if the system matrix is nonsingular, then the desired transformation matrix can be chosen in block diagonal form. Procedures for the computation of the transformation matrices are proposed and illustrated with simple numerical examples.
5
Content available Fractional time-invariant compartmental linear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
|
2023
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Vol. 33, no. 1
97--102
EN
Fractional time-invariant compartmental linear systems are introduced. Controllability and observability of these systems are analyzed. The eigenvalue assignment problem of compartmental linear systems is considered and illustrated with a numerical example.
6
Content available Transformations of the matrices of linear systems to their canonical form with desired eigenvalues
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2023
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Vol. 71, nr 6
art. no. e147342
EN
A new approach to the transformations of the matrices of linear continuous-time systems to their canonical forms with desired eigenvalues is proposed. Conditions for the existence of solutions to the problems were given and illustrated by simple numerical examples.
7
Content available Eigenvalues assignment in descriptor linear systems by state and its derivative feedbacks
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2023
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Vol. 71, nr 5
art. no. e145765
EN
The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples.
8
Content available Stability of convex linear combinations of continuous-time and discrete-time linear systems
Kaczorek Tadeusz
Archives of Control Sciences
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2023
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Vol. 33, No. 4
789--799
EN
The asymptotic stability of the convex linear combination of continuous-time and discrete-time linear systems is considered. Using the Gershgorin theorem it is shown that the convex linear combination of the linear asymptotically stable continuous-time and discretetime linear systems is also asymptotically stable. It is shown that the above thesis is also valid (even simpler) for positive linear systems.
9
Content available Standard and fractional discrete-time linear systems with zero transfer matrices
Kaczorek Tadeusz, Ruszewski Andrzej
Acta Mechanica et Automatica
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2023
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Vol. 17, no. 2
188--191
EN
The transfer matrix of the standard and fractional linear discrete-time linear systems is investigated. Necessary and sufficient conditions for zeroing of the transfer matrix of the linear discrete-time systems are established. The considerations are illustrated by examples of the standard and fractional linear discrete-time systems.
10
Content available Some analysis problems of the linear systems
Kaczorek Tadeusz
Journal of Automation, Electronics and Electrical Engineering
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2022
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Vol. 4, No. 2
7--12
EN
New approaches to the transformations of the uncontrollable and unobservable matrices of linear systems to their canonical forms are proposed. It is shown that the uncontrollable pair (A,B) and unobservable pair (A,C) of linear systems can be transform to their controllable (𝐴̅ , 𝐵̅ ) and observable (𝐴̅ , 𝐶̅ ) canonical forms by suitable choice of nonsingular matrix M satisfying the condition 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] and [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ], respectively. It is also shown that by suitable choice of the gain matrix K of the feedbacks of the derivative of the state vector it is possible to reduce the descriptor system to the standard one.
PL
Zaproponowano nowe podejścia do transformacji niesterowalnych i nieobserwowalnych macierzy układów liniowych do ich postaci kanonicznych. Wykazano, że niesterowalna para (A,B) i nieobserwowalna para (A,C) układów liniowych może być przekształcona do ich postaci kanonicznych sterowalnych i obserwowalnych prze odpowiedni dobór nieosobliwej macierzy M spełniającej warunki 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] i [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ]. Pokazano,żeprzezodpowiednidobórmacierzyKsprzężeniazwrotnego od pochodnej wektora stanu jest możliwa redukcja układu deskryptorowegodoukładustandardowego.
11
Content available A Kalman filter with intermittent observations and reconstruction of data losses
Rhouma Taouba, Keller Jean-Yves, Abdelkrim Mohamed Naceur
International Journal of Applied Mathematics and Computer Science
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2022
|
Vol. 32, no. 2
241--253
EN
This paper deals with the problem of joint state and unknown input estimation for stochastic discrete-time linear systems subject to intermittent unknown inputs on measurements. A Kalman filter approach is proposed for state prediction and intermittent unknown input reconstruction. The filter design is based on the minimization of the trace of the state estimation error covariance matrix under the constraint that the state prediction error is decoupled from active unknown inputs corrupting measurements at the current time. When the system is not strongly detectable, a sufficient stochastic stability condition on the mathematical expectation of the random state prediction errors covariance matrix is established in the case where the arrival binary sequences of unknown inputs follow independent random Bernoulli processes. When the intermittent unknown inputs on measurements represent intermittent observations, an illustrative example shows that the proposed filter corresponds to a Kalman filter with intermittent observations having the ability to generate a minimum variance unbiased prediction of measurement losses.
12
Content available Eigenvalues assignment in uncontrollable linear systems
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2022
|
Vol. 70, nr 6
art. no. e141987
EN
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.
13
Content available Divisibility of the second-order minors of the nominators by minimal denominators of transfer matrices of cyclic fractional linear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
|
2021
|
Vol. 31, no. 4
627--633
EN
The divisibility of the second-order minors of the numerators of transfer matrices by their minimal denominators for cyclic fractional linear systems is analyzed. It is shown that all nonzero second-order minors of the numerators of the transfer matrices are divisible by their minimal denominators if and only if the system matrices of fractional standard and descriptor linear systems are cyclic. The theorems are illustrated by examples of fractional standard and descriptor linear systems.
14
Content available Partial observability of finite dimensional linear systems
Danine Mohamed Elkhail, Bernoussi Abdes Samed, Bel Fekih Abdelaziz
Control and Cybernetics
|
2021
|
Vol. 50, No. 2
269--300
EN
In this work, we consider the partial observability problem for finite dimensional dynamical linear systems that are not necessarily observable. For that purpose we introduce the so called ”observable subspaces” and ”partial observability” to find a way to reconstruct the observable part of the system state. Some characterizations of ”observable subspaces” have been provided. The reconstruction of the orthogonal projection of the state on the observable subspace is obtained. We give some examples to illustrate our theoretical approach.
15
Content available Convex linear combination of the controllability pairs for linear systems
Kaczorek Tadeusz, Klamka Jerzy
Control and Cybernetics
|
2021
|
Vol. 50, No. 4
541--551
EN
The convex linear combination of the controllability pairs of linear continuous-time linear systems is defined and its properties are discussed. The main result is obtained using pure algebraic methods. In the illustrative examples different cases of linear convex combinations are analyzed.
16
Content available Fractional order tube model reference adaptive control for a class of fractional order linear systems
Balaska Hanane, Ladaci Samir, Djouambi Abdelbaki, Schulte Horst, Bourouba Bachir
International Journal of Applied Mathematics and Computer Science
|
2020
|
Vol. 30, no. 3
501--515
EN
We introduce a novel fractional order adaptive control design based on the tube model reference adaptive control (TMRAC) scheme for a class of fractional order linear systems. By considering an adaptive state feedback control configuration, the main idea is to replace the classical reference model with a single predetermined trajectory by a fractional order performance tube guidance model allowing a set of admissible trajectories. Besides, an optimization problem is formulated to compute an on-line correction control signal within specified bounds in order to update the system performance while minimizing a control cost criterion. The asymptotic stability of the closed loop fractional order control system is demonstrated using an extension of the Lyapunov direct method. The dynamical performance of the fractional order tube model reference adaptive control (FOTMRAC) is compared with the standard fractional order model reference adaptive control (FOMRAC) strategy, and the simulation results show the effectiveness of the proposed control method.
17
Content available Extremal properties of linear dynamic systems controlled by Dirac’s impulse
Białas Stanisław, Górecki Henryk, Zaczyk Mieczysław
International Journal of Applied Mathematics and Computer Science
|
2020
|
Vol. 30, no. 1
75--81
EN
The paper concerns the properties of linear dynamical systems described by linear differential equations, excited by the Dirac delta function. A differential equation of the form an x(n) (t) + ∙∙∙ a1 x’(t) + a0 x(t) = bm u (t) + ∙∙∙ + b1 u’(t) + b0 u(t) is considered with ai, bj >0. In the paper we assume that the polynomials Mn(s) = ansn + ∙∙∙ + a1s + a0 and Lm(s) = bmsm + ∙∙∙ + b1s + b0 partly interlace. The solution of the above equation is denoted by x(t, Lm, Mn). It is proved that the function x(t, Lm, Mn) is nonnegative for t ∊ (0, ∞) , and does not have more than one local extremum in the interval (0, ∞) (Theorems 1, 3 and 4). Besides, certain relationships are proved which occur between local extrema of the function x(t, Lm, Mn), depending on the degree of the polynomial Mn(s) or Lm(s) (Theorems 5 and 6).
18
Content available Numerical checking method for positive invariance of polyhedral sets for linear dynamical systems
Yang H., Hu Y.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2020
|
Vol. 68, nr 3
593--599
EN
Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in Lyapunov sense of equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily implemented numerically. The effectiveness of the proposed checking method is illustrated by examples. Compared with the now existing algebraic methods, numerical checking method is attractive and, importantly, easy to be implemented.
19
Content available An output sensitivity problem for a class of linear distributed systems with uncertain initial state
Larrache Abdelilah, Lhous Mustapha, Rhila Soukina Ben, Rachik Mostafa, Tridane Abdessamad
Archives of Control Sciences
|
2020
|
Vol. 30, No. 1
139--155
EN
In this paper,we consider an infinite dimensional linear systems. It is assumed that the initial state of system is not known throughout all the domain Ω C Rn, the initial state x0 ϵ L2(Ω) is supposed known on one part of the domain Ω and uncertain on the rest. That means Ω = ω1 U ω2 U... U ωt with ωi ∩ ωj = ∅, ∀i ≠ j ϵ {1,...,t}, i ≠ j where ωi ≠ ∅ and x0(θ) = αi for θ ϵ ωi, ∀i, i.e., x0(θ) = [wzór] (θ) where the values α1,...,αr are supposed known and αr+1,...,αt unknown and 1ωi is the indicator function. The uncertain part (α1,...,(α)rof the initial state x0 is said to be (ɛ1,...,ɛr )-admissible if the sensitivity of corresponding output signal (yi)i≥0 relatively to uncertainties (αk)1≤k≤r is less to the treshold ɛk, i.e., ∥∂yi)/(∂αk∥ ≤ ɛk, ∀i≥ 0, ∀k ϵ {1,...,r]. The main goal of this paper is to determine the set of all possible gain operators that makes the system insensitive to all uncertainties. The characterization of this set is investigated and an algorithmic determination of each gain operators is presented. Some examples are given.
20
Content available Fractional vector-order h-realisation of the impulse response function
Pawłuszewicz Ewa
Acta Mechanica et Automatica
|
2020
|
Vol. 14, no. 2
108--113
EN
The problem of realisation of linear control systems with the h−difference of Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type fractional vector-order operators is studied. The problem of existing minimal realisation is discussed.
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