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Robust Controller of Buck Converter Feeding Constant Power Load

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Abstract

In this paper, we propose a robust PID controller based on Kharitonov theorem to overcome stability issues in DC power systems. These issues are caused by increasing number of constant power loads (CPLs), uncertainty in the input voltage and disturbances which affect the system performance. We designed this controller for achieving the desired tracking performance and closed-loop stability of DC buck converter that feeds constant impedance load and CPL. The drawback of small signal stability methods was handled as the system is linearized around all the possible operating points which is the main contribution of this paper. To assure stability over the operating range, Hermit Biehler theorem is used to find the stabilization sets of the PID controller. Stability analysis reveals that the proposed method is robust for uncertainties. The simulation results show promising performance of the proposed algorithm of the PID controller.

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Abbreviations

L :

Inductance

\( r_{L} \) :

Inductor resistance

\( i_{L} \) :

Inductor current

C :

Capacitance

R :

Resistance of the resistive load

\( v_{\text{o}} \) :

Output voltage

\( v_{\text{s}} \) :

Supply voltage

\( u \in \left( { 0 , 1} \right) \) :

Control signal

P :

Power of the CPL

\( V_{\text{ref}} \) :

Desired reference voltage

\( V_{\text{s}} ,U,V_{\text{o}} \,{\text{and}}\,I_{L} \) :

Average values of \( v_{\text{s}} ,u,v_{\text{o}} \). and \( i_{L} \), respectively

\( \tilde{v}_{\text{s}} , \tilde{u}, \tilde{v}_{\text{o}} \,{\text{and}}\, \tilde{i}_{L} \) :

Small signal disturbances of \( v_{\text{s}} ,u,v_{\text{o}} \). and \( i_{L} \), respectively

\( \delta \left( s \right), \delta \left( {j\omega } \right) \) :

Characteristic polynomial in frequency domain

s :

Normalized characteristic polynomial

\( \sigma_{i} \left( \delta \right) \) :

Imaginary signature of the characteristic polynomial

\( \delta_{\text{e}} \left( {s^{2} } \right),\delta_{\text{o}} \left( {s^{2} } \right) \) :

Even and odd components of \( \delta \left( s \right) \)

\( N\left( s \right), \, D\left( s \right) \) :

Nominator and denominator of plant transfer function

\( p\left( \omega \right),q\left( \omega \right) \) :

Real and imaginary components of \( \delta \left( {j\omega } \right) \)

\( p_{f} \left( \omega \right),q_{f} \left( \omega \right) \) :

Real and imaginary components of \( \delta_{f} \left( {j\omega } \right) \)

\( \theta (\omega ) \) :

Phase angle of \( \delta_{f} \left( {j\omega } \right) \)

\( \Delta_{0}^{\infty } \theta \) :

Net change in the phase angle \( \theta \left( \omega \right) \)

\( l(\delta ),r(\delta ) \) :

Number of roots of \( \delta_{f} \left( {j\omega } \right) \) in left and right half plane, respectively

I :

String contains all possible signs of \( p_{f} \left( \omega \right) \)

\( \gamma \left( I \right) \) :

Imaginary signature related to string I

\( K_{\text{p}} ,K_{\text{i}} ,K_{\text{d}} \) :

Proportional, integral and derivative gains of PID controller

\( n_{c} \left( s \right),d_{c} \left( s \right) \) :

Nominator and denominator of PID Transfer function

\( \underline{{b_{0} }} ,\overline{{b_{0} }} ,\underline{{a_{1} }} ,\overline{{a_{1} }} ,\underline{{a_{0} }} ,\overline{{a_{0} }} \) :

Lower and upper bounds of b and a, respectively

K(s):

Kharitonov polynomial

\( NS\left( s \right) \) :

Kharitonov segments of N(s)

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Authors and Affiliations

  1. Department of Control Engineering, Mechanical and Electrical Research Institute, National Water Research Center, El-Kanater El-Khayrya, Cairo, Egypt

    Mahmoud S. Ali & Said A. F. Hawash

  2. Electrical Engineering Department, Faculty of Engineering at Shoubra, Benha University, Cairo, Egypt

    M. Soliman & Ahmed M. Hussein

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  1. Mahmoud S. Ali
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  2. M. Soliman
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  4. Said A. F. Hawash
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Correspondence to Mahmoud S. Ali.

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Ali, M.S., Soliman, M., Hussein, A.M. et al. Robust Controller of Buck Converter Feeding Constant Power Load. J Control Autom Electr Syst 32, 153–164 (2021). https://doi.org/10.1007/s40313-020-00660-2

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