1. 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
1/17

2. 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

3. 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

4. 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
3/17

5. 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

6. 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

7. 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

8. 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).

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. References
[1] R. W. Erickson, Fundamentals of Power Electronics. Norwell, MA: Kluwer, [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 , Aug [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 , July [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 [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 , Nov [7] K. Ogata, Modern Control Engineering, 4th Edition, Prentice Hall, 2002.

25. Thank you for listening!
Q&A session

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