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Abstract

Hybrid systems (HS) are roughly described as a set of discrete state transitions and continuous dynamics modeled by differential equations. Parametric HS may be constructed by having parameters on the differential equations, initial conditions, jump conditions, or a combination of the previous ones. In real applications, the best solution is obtained by a set of metrics functional over the set of solutions generated from a finite set of parameters. This paper examines the choice of parameters on delta-reachability bounded hybrid systems.We present an efficient model based on the tool pHL-MT to benchmark the HS solutions (based on dReach), and a non-parametric frontier analysis approach, relying on multidirectional efficiency analysis (MEA). Three numerical examples of epidemic models with variable growth infectivity are presented, namely: when the variable of infected individuals oscillates around some endemic (non-autonomous) equilibrium; when there is an asymptotically stable non-trivial attractor; and in the presence of bump functions.
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Bibliography

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[10] S. Kong, S. Gao, W. Chen and E. Clarke: dReach: δ-reachability analysis for hybrid systems. In Proc. International Conference on Tools and Algorithms for the Construction and Analysis of Systems, 2015. Available: http://link.springer.com/10.1007/978-3-662-46681-0.
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Authors and Affiliations

Eugénio Miguel Alexandre Rocha
1
Kelly Patricia Murillo
1

  1. Center for Research and Development in Mathematics and Applications, and Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
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Abstract

This paper focuses on the invariance of the reachability and observability for fractional order positive linear electrical circuits with delays and their checking methods. By derivation and comparison, it shows that conditions and checking methods of reachability and observability for integer and fractional order positive linear electrical circuits with delays are invariant. An illustrative example is presented at the end of the paper.
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Bibliography

[1] Dorf C.R., Svoboda A.J., Introduction to electric circuits, John Wiley & Sons (2010).
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[4] Kaczorek T., Selected problems of fractional systems theory, Springer (2011).
[5] Xin Z., Wenru L. et al., Application of fractional calculus in iterative sliding mode synchronization control, Archives of Electrical Engineering, vol. 69, no. 3, pp. 499–519 (2020).
[6] Piotrowska E., Analysis of linear continuous-time systems by the use of the conformable fractional calculus and Caputo, Archives of Electrical Engineering, vol. 67, no. 3, pp. 629–639 (2018).
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Authors and Affiliations

Tong Yuan
1
ORCID: ORCID
Hongli Yang
1

  1. Shandong University of Science and Technology, China
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Abstract

Conditions for the positivity of linear electrical circuits composed of resistances, coils, capacitors and voltage (current) sources are established. It is shown that: 1) the electrical circuit composed of resistors, coils and voltage source is positive for any values of their resistances, inductances and source voltages if and only if the number of coils is less or equal to the number of its linearly independent meshes, 2) the electrical circuit is not positive for any values of its resistances, capacitances and source voltages if each its branch contains resistor, capacitor and voltage source, 3) the positive n-meshes electrical circuit with only one inductance in each linearly independent mesh is reachable if all resistances of branches belonging to two linearly independent meshes are zero.

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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

The concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.
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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

Necessary and sufficient conditions for the reachability and observability of the positive electrical circuits composed of resistors, coils, condensators and voltage sources are established. Definitions of the input-decoupling zeros, output-decoupling zeros and input-output decoupling zeros of the positive electrical circuits are proposed. Some properties of the decoupling zeros of positive electrical circuits are discussed.

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Authors and Affiliations

Tadeusz Kaczorek
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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

It is shown that 2(n + 1) is the upper bound for the reachability index of the n-order positive 2D general models.

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Authors and Affiliations

T. Kaczorek
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Abstract

New tests (criterions) for checking the reachability and the observability of positive linear-discrete-time systems are proposed. The tests do not need checking of rank conditions of the reachability and observability matrices of the systems. Simple sufficient conditions for the unreachability and unobservability of the systems are also established.

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Authors and Affiliations

T. Kaczorek

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