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February  2022, 11(1): 169-175. doi: 10.3934/eect.2020106

Some results on the behaviour of transfer functions at the right half plane

Tahir Aliyev Azeroğlu 1, , Bülent Nafi Örnek 2,, and  Timur Düzenli 2,3,

1. 

Department of Mathematics, Gebze Technical University, Gebze, Kocaeli, Turkey
2. 

Amasya University, Technology Faculty, Department of Computer Engineering
3. 

Amasya University, Technology Faculty, Department of Electrical and Electronics Engineering, Amasya, Turkey

* Corresponding author: Bülent Nafi Örnek

Received  February 2020 Revised  September 2020 Published  February 2022 Early access  December 2020

In this paper, an inequality for a transfer function is obtained assuming that its residues at the poles located on the imaginary axis in the right half plane. In addition, the extremal function of the proposed inequality is obtained by performing sharpness analysis. To interpret the results of analyses in terms of control theory, root-locus curves are plotted. According to the results, marginally and asymptotically stable transfer functions can be determined using the obtained extremal function in the proposed theorem.

Keywords: Positive real function, control system, extremal function, sharpness analysis, marginal stability, transfer function.
Mathematics Subject Classification: Primary: 32A10, 32A05.
Citation: Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations and Control Theory, 2022, 11 (1) : 169-175. doi: 10.3934/eect.2020106
References:
[1]

M. Corless, E. Zeheb and R. Shorten, On the SPRification of linear descriptor systems via output feedback, IEEE Transactions on Automatic Control, 64 (2019), 1535-1549.  doi: 10.1109/TAC.2018.2849613.

[2]

G. Fernández-Anaya, J.-J. Flores-Godoy and J. Álvarez-Ramírez, Preservation of properties in discrete-time systems under substitutions, Asian Journal of Control, 11 (2009), 367-375.  doi: 10.1002/asjc.114.

[3]

J.-S. Hu and M.-C. Tsai, Robustness analysis of a practical impedance control system, IFAC Proceedings Volumes, 37 (2004), 725-730.  doi: 10.1016/S1474-6670(17)31695-6.

[4]

S. S. Khilari, Transfer Function and Impulse Response Synthesis Using Classical Techniques, Master Thesis, University of Massachusetts Amherst, 2007.

[5]

E. Landau and G. Valiron, A deduction from Schwarz's lemma, Journal of the London Mathematical Society 4, (1929), 162–163. doi: 10.1112/jlms/s1-4.3.162.

[6]

M. Liu and et al., On positive realness, negative imaginariness, and h-$\inf$ control of state-space symmetric systems, Automatica J. IFAC, 101 (2019), 190-196.  doi: 10.1016/j.automatica.2018.11.031.

[7]

F. Mukhtar, Y. Kuznetsov and P. Russer, Network modelling with Brune's synthesis, Advances in Radio Science, 9 (2011), 91-94.  doi: 10.5194/ars-9-91-2011.

[8]

R. H. Nevanlinna, Eindeutige Analytische Funktionen, Springer-Verlag, Berlin, 1953.

[9]

A. Ochoa, Driving point impedance and signal flow graph basics: a systematic approach to circuit analysis, in Feedback in Analog Circuits, Springer, Cham, 2016, 13–34.

[10]

Y. Pan, M. J. Er, R. Chen and H. Yu, Output feedback adaptive neural control without seeking spr condition, Asian Journal of Control, 17 (2015), 1620-1630.  doi: 10.1002/asjc.966.

[11]

F. Reza, A bound for the derivative of positive real functions, SIAM Review, 4 (1962), 40-42.  doi: 10.1137/1004005.

[12]

M. Şengül, Foster impedance data modeling via singly terminated LC ladder networks, Turkish Journal of Electrical Engineering & Computer Sciences, 21 (2013), 785-792. 

[13]

A. Sharma and T. Soni, A review on passive network synthesis using cauer form, World J. Wireless Devices Eng., 1 (2017), 39-46. 

[14]

W. Sun, P. P. Khargonekar and and D. Shim, Solution to the positive real control problem for linear time-invariant systems, IEEE Transactions on Automatic Control, 39 (1994), 2034-2046.  doi: 10.1109/9.328822.

[15]

M. S. Tavazoei, Passively realisable impedance functions by using two fractional elements and some resistors, IET Circuits, Devices & Systems, 12 (2017), 280-285.  doi: 10.1049/iet-cds.2017.0342.

[16]

A. D. Wunsch and S.-P. Hu, A closed-form expression for the driving-point impedance of the small inverted L antenna, IEEE Transactions on Antennas and Propagation, 44 (1996), 236-242.  doi: 10.1109/8.481653.

[17]

C. Xiao and D. J. Hill, Concepts of strict positive realness and the absolute stability problem of continuous-time systems, Automatica J. IFAC, 34 (1998), 1071-1082.  doi: 10.1016/S0005-1098(98)00049-1.

show all references

References:
[1]

M. Corless, E. Zeheb and R. Shorten, On the SPRification of linear descriptor systems via output feedback, IEEE Transactions on Automatic Control, 64 (2019), 1535-1549.  doi: 10.1109/TAC.2018.2849613.

[2]

G. Fernández-Anaya, J.-J. Flores-Godoy and J. Álvarez-Ramírez, Preservation of properties in discrete-time systems under substitutions, Asian Journal of Control, 11 (2009), 367-375.  doi: 10.1002/asjc.114.

[3]

J.-S. Hu and M.-C. Tsai, Robustness analysis of a practical impedance control system, IFAC Proceedings Volumes, 37 (2004), 725-730.  doi: 10.1016/S1474-6670(17)31695-6.

[4]

S. S. Khilari, Transfer Function and Impulse Response Synthesis Using Classical Techniques, Master Thesis, University of Massachusetts Amherst, 2007.

[5]

E. Landau and G. Valiron, A deduction from Schwarz's lemma, Journal of the London Mathematical Society 4, (1929), 162–163. doi: 10.1112/jlms/s1-4.3.162.

[6]

M. Liu and et al., On positive realness, negative imaginariness, and h-$\inf$ control of state-space symmetric systems, Automatica J. IFAC, 101 (2019), 190-196.  doi: 10.1016/j.automatica.2018.11.031.

[7]

F. Mukhtar, Y. Kuznetsov and P. Russer, Network modelling with Brune's synthesis, Advances in Radio Science, 9 (2011), 91-94.  doi: 10.5194/ars-9-91-2011.

[8]

R. H. Nevanlinna, Eindeutige Analytische Funktionen, Springer-Verlag, Berlin, 1953.

[9]

A. Ochoa, Driving point impedance and signal flow graph basics: a systematic approach to circuit analysis, in Feedback in Analog Circuits, Springer, Cham, 2016, 13–34.

[10]

Y. Pan, M. J. Er, R. Chen and H. Yu, Output feedback adaptive neural control without seeking spr condition, Asian Journal of Control, 17 (2015), 1620-1630.  doi: 10.1002/asjc.966.

[11]

F. Reza, A bound for the derivative of positive real functions, SIAM Review, 4 (1962), 40-42.  doi: 10.1137/1004005.

[12]

M. Şengül, Foster impedance data modeling via singly terminated LC ladder networks, Turkish Journal of Electrical Engineering & Computer Sciences, 21 (2013), 785-792. 

[13]

A. Sharma and T. Soni, A review on passive network synthesis using cauer form, World J. Wireless Devices Eng., 1 (2017), 39-46. 

[14]

W. Sun, P. P. Khargonekar and and D. Shim, Solution to the positive real control problem for linear time-invariant systems, IEEE Transactions on Automatic Control, 39 (1994), 2034-2046.  doi: 10.1109/9.328822.

[15]

M. S. Tavazoei, Passively realisable impedance functions by using two fractional elements and some resistors, IET Circuits, Devices & Systems, 12 (2017), 280-285.  doi: 10.1049/iet-cds.2017.0342.

[16]

A. D. Wunsch and S.-P. Hu, A closed-form expression for the driving-point impedance of the small inverted L antenna, IEEE Transactions on Antennas and Propagation, 44 (1996), 236-242.  doi: 10.1109/8.481653.

[17]

C. Xiao and D. J. Hill, Concepts of strict positive realness and the absolute stability problem of continuous-time systems, Automatica J. IFAC, 34 (1998), 1071-1082.  doi: 10.1016/S0005-1098(98)00049-1.

Figure 1.  Root-locus curves for the transfer function $ H(s) = \sum\limits_{i = 1}^{n}\frac{\alpha _{i}}{s-s_{i}}+i\beta $. It is assumed that $ \alpha_{i} $'s equal to 1 and $ \beta $ is zero. The figures are presented for different $ n $ values: (a) $ n = 1 $, (b) $ n = 2 $, (c) $ n = 3 $, (d) $ n = 4 $
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