# Extensions of Kharitonov Theorem to Positive Fractional Linear Systems

Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)

## Abstract

The asymptotic stability of interval positive continuous-time linear systems of integer and fractional orders is investigated. The classical Kharitonov theorem is extended to the interval positive continuous-time linear systems of integer and fractional orders. It is shown that:
1. (1)

The interval positive linear system is asymptotically stable if and only if the matrices bounding the state matrix are Hurwitz Metzler.

2. (2)

The interval positive fractional system is asymptotically stable if and only if bounding the state matrix are Hurwitz Metzler.

3. (3)

The interval positive of integer and fractional orders continuous-time linear systems with interval characteristic polynomials are asymptotically stable if and only if their lower bounds of the coefficients are positive.

It is shown that the interval positive fractional discrete-time linear systems are asymptotically stable if and only if the lower and upper bounds of the state matrices are asymptotically stable. The classical Kharitonov theorem is extended to the discrete-time interval positive fractional linear systems.

## Keywords

Interval Positive Fractional Linear Continuous-time Discrete-time System Stability

## Notes

### Acknowledgement

This work was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.

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