Hafez Sarkawi (D1) Control System Theory Lab 2016.11.28 Optimal state-feedback and Proportional-Integral Controller Performance Comparison for Dc-dc Zeta Converter Operating in Continuous Conduction Mode Hafez Sarkawi (D1) Control System Theory Lab 2016.11.28 1/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work 2/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

Motivation and Scope of Research Motivation – To investigate and compare the performance of two different controllers design namely Optimal State-feedback Control and PI Control when subject to load perturbation. Scope of Research – This research only design the Optimal state-feedback control. The PI control parameters design is extracted from paper E. Vuthchhay and C. Bunlaksananusorn, “Modeling and Control of a Zeta Converter”, in Proc. IEEE Int. Power Electronics Conf., 2010, pp. 612-619. 3/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

Introduction A dc-dc zeta converter is a fourth order dc-dc converter that can step-up or step-down the dc input voltage. In order to stabilize the output voltage, a control subsystem needs to be deployed. Proportional-integral (PI) control: Commonly used due to its simple solutions but no definite way for tuning the parameters, and this requires ad-hoc approach. Optimal (LQR) control: Optimization of a cost function/performance index. The choice of cost function parameters is advantageous because it can minimize the ripple present in the feedback signal. 4/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

The Dc-dc Zeta Converter Modeling in CCM Fig. 1 A dc-dc zeta converter circuit. To ensure CCM operation: 5/17 Fig. 2 Equivalent circuit when S=1 (on). Fig. 3 Equivalent circuit when S=0 (off).

The Dc-dc Zeta Converter Modeling in CCM (Cont.) For a system that has a two switch topologies, the state equations can be describe as: Switch on: Switch off: For switch on for the time dT and off for (1-d)T, the weighted average of the equations are: (1) 6/17

The Dc-dc Zeta Converter Modeling in CCM (Cont.) By assuming that the variables are changed around steady-state operating point (linear signal), the variables can now be written as: (2) 7/17

The Dc-dc Zeta Converter Modeling in CCM (Cont.) During steady-state, the derivatives and the small signal values are zeros: By subs. (2) and (3) into (1) and neglecting nonlinear terms, the small-signal model can be written as: (3) where 8/17

The Dc-dc Zeta Converter Modeling in CCM (Cont.) The state and input vectors for the dc-dc zeta converter: In order to obtain a zero steady-state error between the reference voltage Vref and the output voltage vo, the model is augmented with an additional state variable: The new (augmented) state vectors can be written as: 9/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

The Dc-dc Zeta Converter Control The control law used for state-feedback is: where K is the feedback gain matrix. To optimally control the control effort within performance specifications, a compensator is sought to provide a control effort for input that minimizes a cost function: where Qw is a symmetric, positive semidefinite matrix and Rw is a symmetric, positive definite matrix. 10/17

The Dc-dc Zeta Converter Control (Cont.) To solve the optimization problem over an infinite time interval, the algebraic Ricatti equation is the most commonly used: where P is a symmetric, positive definite matrix. The feedback gain is solved by: (4) (5) 11/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

Table 1 The dc-dc zeta converter parameters. Simulation Results Table 1 The dc-dc zeta converter parameters. Fig. 4 The dc-dc zeta converter with state-feedback controller implementation. 12/17

Simulation Results (Cont.) The chosen weight matrices of the performance index: By solving equation (4) and (5) using the LQR Matlab command, the resulting controller gain vector is: * Chosen criteria: Integral action is enforced Control duty-cycle ripple is lower than 20% Integral constant match the one that use in PI control 13/17

Simulation Results (Cont.) Time (ms) Fig. 5 Output current transient response comparison between LQR and PI controller for the dc-dc zeta converter subject to load current perturbation of ± 4 A 14/17

Simulation Results (Cont.) Time (ms) Fig. 6 Output voltage transient response comparison between LQR and PI controller for the dc-dc zeta converter subject to load current perturbation of ± 4 A 15/17

Simulation Results (Cont.) Time (ms) Fig. 7 Control duty ratio transient response comparison between LQR and PI controller for the dc-dc zeta converter subject to load current perturbation of ± 4 A 16/17

Presentation Outline Motivation and Scope of Research Introduction The Dc-dc Zeta Converter Modeling in CCM The Dc-dc Zeta Converter Control Simulation Results Conclusion and Future Work

Conclusions and Future Work Although PI controller exhibits good output voltage response, its control duty-cycle response is highly undesirable of which make it impractical experimentally. Optimal state-feedback controller shows a balance between output voltage response and control duty-cycle. Future work will deal with robust LQR control by taking into account parameters uncertainty for dc-dc zeta converter. 17/17

References [1] R. W. Erickson, Fundamentals of Power Electronics. Norwell, MA: Kluwer, 1999. [2] H. Sarkawi, M. H. Jali, T. A. Izzuddin, M. Dahari, “Dynamic Model of Zeta Converter with Full-state Feedback Controller Implementation”, Int. Journal of Research in Eng. and Tech., Vol. 02, No. 08, pp. 34-43, Aug. 2013. [3] C. Olala, R. Leyva, A. E. Aroudi, I. Queinnec, “Robust LQR Control for PWM Converters: An LMI Approach”, IEEE Trans. Ind. Electron., Vol. 56, No. 7, pp. 2548-2558, July 2009. [4] European Cooperation for Space Standardization, Electrical and Electronic Standard: ECCS-E20A, 2004 [5] E. Vuthchhay and C. Bunlaksananusorn, “Modeling and Control of a Zeta Converter”, in Proc. IEEE Int. Power Electronics Conf., 2010, pp. 612-619. [6] C. Olala, I. Queinnec, R. Leyva, A. E. Aroudi, “Optimal State-feedback Control of Bilinear Dc-dc Converters with Guaranteed Regions of Stability”, IEEE Trans. Ind. Electron., Vol. 59, No. 10, pp. 3868-3880, Nov. 2012 [7] K. Ogata, Modern Control Engineering, 4th Edition, Prentice Hall, 2002.

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Appendix Fig. 8 The dc-dc zeta converter with PI controller implementation