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1

Fractional time-invariant compartmental linear systems

Kaczorek Tadeusz

International Journal of Applied Mathematics and Computer Science

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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.

2

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.

3

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.

4

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.

5

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.

6

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.

7

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

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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.

8

Eigenvalues assignment in uncontrollable linear systems

Kaczorek Tadeusz

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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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.

9

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

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2021

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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.

10

Partial observability of finite dimensional linear systems

Danine Mohamed Elkhail, Bernoussi Abdes Samed, Bel Fekih Abdelaziz

Control and Cybernetics

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2021

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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.

11

Convex linear combination of the controllability pairs for linear systems

Kaczorek Tadeusz, Klamka Jerzy

Control and Cybernetics

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2021

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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.

12

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

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2020

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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.

13

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

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2020

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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 a_n x⁽ⁿ⁾ (t) + ∙∙∙ a₁ x’(t) + a₀ x(t) = b_m u (t) + ∙∙∙ + b₁ u’(t) + b₀ u(t) is considered with a_i, b_j >0. In the paper we assume that the polynomials M_n(s) = a_nsⁿ + ∙∙∙ + a₁s + a₀ and L_m(s) = b_ms^m + ∙∙∙ + b₁s + b₀ partly interlace. The solution of the above equation is denoted by x(t, L_m, M_n). It is proved that the function x(t, L_m, M_n) 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, L_m, M_n), depending on the degree of the polynomial M_n(s) or L_m(s) (Theorems 5 and 6).

14

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

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2020

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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.

15

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

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2020

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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.

16

Fractional vector-order h-realisation of the impulse response function

Pawłuszewicz Ewa

Acta Mechanica et Automatica

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2020

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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.

17

State estimators and observers for continuous and discrete linear systems. Part 2. Integral observers for exact state reconstruction

Byrski Jędrzej, Byrski Witold

Science, Technology and Innovation

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2019

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Vol. 5, no. 2

23--33

EN

In the paper, the exact state observers will be presented. The state estimators and observers can be used in technical processes for many purposes like the fault detection and diagnosis, the implementation of the state controllers, and soft reconstruction of inaccessible for measurements variables of the system. As the standard, for continuous systems the differential estimators of Kalman filter or Luenberger type observer are commonly used. However, if the initial conditions of the real state are unknown, both estimators guarantee only an asymptotic quality of the real state tracking. The paper presents another type of the state observers, which for continuous system have the structure given by two integral operators. Based on measurements of the system input and output signals on some predefined finite time interval T, they can reconstruct the initial state exactly. In on-line version, the exact state reconstruction is performed continuously for every t, based on special procedure executed within two moving windows of width T, on sliding time interval [t-T, t].

18

Event-based feedforward control of linear systems with input time-delay

Rodríguez Carlos, Aranda-Escolástico Ernesto, Guinaldo Maria, Guzmán José Luis, Dormido Sebastián

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 3

541--553

EN

This paper proposes a new method for the analysis of continuous and periodic event-based state-feedback plus static feedforward controllers that regulate linear time invariant systems with time delays. Measurable disturbances are used in both the control law and triggering condition to provide better disturbance attenuation. Asymptotic stability and L2-gain disturbance rejection problems are addressed by means of Lyapunov–Krasovskii functionals, leading to performance conditions that are expressed in terms of linear matrix inequalities. The proposed controller offers better disturbance rejection and a reduction in the number of transmissions with respect to other robust event-triggered controllers in the literature.

19

Nowa wkładka igus zgodna z normami FDA i UE dla systemów liniowych

Świat Obrabiarek i Narzędzi

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2018

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R. 13, nr 9-10

22

PL

Zintegrowane, bezobsługowe i odporne na zużycie wkładki stosowane w łożyskach liniowych umożliwiają przesuwanie prowadnic liniowych bez smarowania. Firma igus aktualnie opracowała wkładkę zgodną z wymogami UE i FDA, wykonaną z wysokowydajnego tworzywa sztucznego iglidur A160, które jest idealne do specjalnych wymagań higienicznych przemysłu spożywczego. Charakteryzuje się on przede wszystkim długą żywotnością w przypadku wałków ze stali nierdzewnej zgodnej z normami FDA i EU oraz w obszarach mokrych.

20

State estimators and observers for continuous and discrete linear systems. Part 1. Differential asymptotic state estimators

Byrski J., Byrski W.

Science, Technology and Innovation

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2018

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Vol. 3, no. 2

62--68

EN

In the paper an overview of state estimators and state observers used in linear systems, will be presented. The state estimators and observers can be used in many applications like the state reconstruction for the control purposes or for the diagnosis and fault detection in technical processes or for the virtual measurements of inaccessible variables of the system as well as for the best filtration of the differential equation solution. As the standard most commonly the Kalman filter and Luenberger type observers are used. Although the Kalman filter guarantees optimal filtering quality of the state, reconstructed from the noisy measurements, both Kalman filter and the Luenberger observer guarantee only asymptotic quality of the real state changes and tracking, basing on the current measurements of the system output and input signals. Unfortunately, the value of the estimation error at any moment of time cannot be calculated. The discussion on differences between continuous and two types of discrete Kalman Filter will be presented. This paper is planned to be the introduction to presentation of another type of the state observers which have the structure given by the integral operators. Based on measurements of the system output and input signals on some predefined finite time interval, they can reconstruct, after this interval, the observed state exactly.