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Abstract

The paper studies the fault identification problem for linear control systems under the unmatched disturbances. A novel approach to the construction of a sliding mode observer is proposed for systems that do not satisfy common conditions required for fault estimation, in particular matching condition, minimum phase condition, and detectability condition. The suggested approach is based on the reduced order model of the original system. This allows to reduce complexity of sliding mode observer and relax the limitations imposed on the original system.
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Bibliography

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[14] V. Filaretov, A. Zuev, A. Zhirabok, and A. Protcenko: Development of fault identification system for electric servo actuators of multilink manipulators using logic-dynamic approach. J. Control Science and Engineering, 2017 (2017), 1–8, DOI: 10.1155/2017/8168627.
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[18] R. Hmidi, A. Brahim, F. Hmida, and A. Sellami: Robust fault tolerant control design for nonlinear systems not satisfying matching and minimum phase conditions. Int. J. Control, Automation and Systems, 18 (2020), 1–14, DOI: 10.1007/s12555-019-0516-4.
[19] H. Rios, D. Efimov, J. Davila, T. Raissi, L. Fridman, and A. Zolghadri: Non-minimum phase switched systems: HOSM based fault detection and fault identification via Volterra integral equation. Int. J. Adaptive Control and Signal Processing, 28 (2014), 1372–1397, DOI: 10.1002/acs.2448.
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[27] A. Zhirabok: Nonlinear parity relation: A logic-dynamic approach. Automation and Remote Control, 69 (2008), 1051-1064, DOI: 10.1134/ S0005117908060155.
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[30] A. Zhirabok, A. Zuev, and A. Shumsky: Methods of diagnosis in linear systems based on sliding mode observers. J. Computer and Systems Sciences Int., 58 (2019), 898–914, DOI: 10.1134/S1064230719040166.
[31] A. Zhirabok, A. Zuev, andV. Filaretov: Fault identification in underwater vehicle thrusters via sliding mode observers. Int. J. Applied Mathematics and Computer Science, 30 (2020), 679–688, DOI: 10.34768/amcs-2020-0050.
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Authors and Affiliations

Alexey Zhirabok
1 2
Alexander Zuev
2
Vladimir Filaretov
3
Alexey Shumsky
1

  1. Far Eastern Federal University, Vladivostok 690091, Russia
  2. Institute of Marine Technology Problems, Vladivostok, 690091, Russia
  3. Institute of Automation and Processes of Control, Vladivostok, 690014, Russia
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Abstract

New equivalent conditions of the asymptotical stability and stabilization of positive linear dynamical systems are investigated in this paper. The asymptotical stability of the positive linear systems means that there is a solution for linear inequalities systems. New necessary and sufficient conditions for the existence of solutions of the linear inequalities systems as well as the asymptotical stability of the linear dynamical systems are obtained. New conditions for the stabilization of the resultant closed-loop systems to be asymptotically stable and positive are also presented. Both the stability and the stabilization conditions can be easily checked by the so-called I-rank of a matrix and by solving linear programming (LP). The proposed LP has compact form and is ready to be implemented, which can be considered as an improvement of existing LP methods. Numerical examples are provided in the end to show the effectiveness of the proposed method.

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Authors and Affiliations

H. Yang
Y. Hu
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Abstract

The invariant properties of the stability, reachability, observability and transfer matrices of positive linear electrical circuits with integer and fractional orders are investi- gated. It is shown that the stability, reachability, observability and transfer matrix of positive linear systems are invariant under their integer and fractional orders.

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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

The paper deals with the problems of designing observers and unknown input observers for discrete-time Lipschitz non-linear systems. In particular, with the use of the Lyapunov method, three different convergence criteria of the observer are developed. Based on the achieved results, three different design procedures are proposed. Then, it is shown how to extend the proposed approach to the systems with unknown inputs. The final part of the paper presents illustrative examples that confirm the effectiveness of the proposed techniques. The paper also presents a MATLAB® function that implements one of the design procedures.

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Authors and Affiliations

J. Korbicz
M. Witczak
V. Puig
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Abstract

Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.

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Authors and Affiliations

Mehmet Emir Koksal
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Abstract

The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.

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Authors and Affiliations

Mikhail I. Gomoyunov
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Abstract

A complete parametric approach is proposed for the design of the Luenberger type function Kx observers for descriptor linear systems. Based on a complete parametric solution to a class of generalized Sylvester matrix equations, parametric expressions for all the coefficient matrices of the observer are derived. The approach provides all the degrees of design freedom, which can be utilized to achieve some additional design requirements. An illustrative example shows the effect of the proposed approach.

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Authors and Affiliations

G.R. Duan
A.G. Wu
W.N. Hou
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Abstract

Spectral properties of nonnegative and Metzler matrices are considered. The conditions for existence of Metzler spectrum in dynamical systems have been established. An electric RL and GC ladder-network is presented as an example of dynamical Metzler system. The suitable conditions for parameters of these electrical networks are formulated. Numerical calculations were done in MATLAB.

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Authors and Affiliations

W. Mitkowski
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Abstract

This paper addresses weighted L2 gain performance switching controller design of discrete-time switched linear systems with average dwell time (ADT) scheme. Two kinds of methods, so called linearizing change-of-variables based method and controller variable elimination method, are considered for the output-feedback control with a supervisor enforcing a reset rule at each switching instant are considered respectively. Furthermore, some comparison between these two methods are also given.

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Authors and Affiliations

Wei He
Wei Xie
Weilin Wu
Langwen Zhagn
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Abstract

The paper addresses the problem of constrained pole placement in discrete-time linear systems. The design conditions are outlined in terms of linear matrix inequalities for the Dstable ellipse region in the complex Z plain. In addition, it is demonstrated that the D-stable circle region formulation is the special case of by this way formulated and solved pole placement problem. The proposed principle is enhanced for discrete-lime linear systems with polytopic uncertainties.

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Authors and Affiliations

Dušan Krokavec
Anna Filasová

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