Fig. 5. Comparison of the open-loop gain transfer function with or without half-gain tuning.
5.2 Impact on Closed-Loop Characteristics
With the control loop signals denoted as in Figure 1, the “gang of six” transfer functions are defined as
providing frequency-domain insights for a two-degrees-of-freedom control loop as is the case with ADRC, cf. Åström and Murray (2008). For the four cases in our example, they are presented and discussed in Figure 6 and Figure 7.
While not shown here for brevity, a discrete-time implementation of “K/2” and “L/2” design was successfully tested as well, exhibiting the desired noise reduction in the control signal $u$.
6. CONCLUSION
A new “half-gain tuning” rule for linear active disturbance rejection control (ADRC) based on the so-called $\alpha$-controller design was introduced. Compared to the common “bandwidth parameterization” approach, similar closed-loop dynamics can be achieved with lower (halved) feedback gains, therefore reducing the noise sensitivity of ADRC.
In view of the examples presented in Section 5, a recommendation emerges to start with half-gain tuning for the observer. This has the least impact on the closed-loop dynamics compared to bandwidth parameterization, while already providing a significant reduction of control signal sensitivity to measurement noise.
While being the analytical solution of an algebraic Riccati equation, the proposed feedback gains can simply be obtained from a bandwidth parameterization design by
halving the gains, as proved in this paper, establishing a link between pole placement and optimal control.
ACKNOWLEDGEMENTS
Gernot Herbst would like to thank Michael Buhl for drawing his attention to the $\alpha$-controller approach.
REFERENCES
Åström, K.J. and Murray, R.M. (2008). Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press.
Buhl, M. and Lohmann, B. (2009). Control with exponentially decaying Lyapunov functions and its use for systems with input saturation. In 2009 European Control Conference (ECC), 3148–3153. doi: 10.23919/ECC.2009.7074889.
Fliess, M. and Join, C. (2013). Model-free control. International Journal of Control, 86(12), 2228–2252. doi: 10.1080/00207179.2013.810345.
Gao, Z. (2003). Scaling and bandwidth-parameterization based controller tuning. In Proceedings of the 2003 American Control Conference, 4989–4996. doi: 10.1109/ACC.2003.1242516.
Gao, Z. (2006). Active disturbance rejection control: A paradigm shift in feedback control system design. In Proceedings of the 2006 American Control Conference, 2399–2405. doi:10.1109/ACC.2006.1656579.
Gao, Z., Huang, Y., and Han, J. (2001). An alternative paradigm for control system design. In Proceedings of the 40th IEEE Conference on Decision and Control. doi: 10.1109/CDC.2001.980926.
Han, J. (2009). From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics, 56(3), 900–906. doi:10.1109/TIE.2008.2011621.
Herbst, G. (2013). A simulative study on active disturbance rejection control (ADRC) as a control tool for practitioners. Electronics, 2(3), 246–279. doi: 10.3390/electronics2030246.
Herbst, G. (2016). Practical active disturbance rejection control: Bumpless transfer, rate limitation, and incremental algorithm. IEEE Transactions on Industrial Electronics, 63(3), 1754–1762. doi: 10.1109/TIE.2015.2499168.
Madoński, R., Shao, S., Zhang, H., Gao, Z., Yang, J., and Li, S. (2019). General error-based active disturbance rejection control for swift industrial implementations. Control Engineering Practice, 84, 218–229. doi: 10.1016/j.conengprac.2018.11.021.
Sira-Ramírez, H., Luviano-Juárez, A., Ramírez-Neria, M., and Zurita-Bustamante, E.W. (2017). Active Disturbance Rejection Control of Dynamic Systems: A Flatness Based Approach. Butterworth-Heinemann.
Zheng, Q. and Gao, Z. (2018). Active disturbance rejection control: Some recent experimental and industrial case studies. Control Theory and Technology, 16(4), 301–313. doi:10.1007/s11768-018-8142-x.
Zhou, B., Duan, G., and Lin, Z. (2008). A parametric Lyapunov equation approach to the design of low gain feedback. IEEE Transactions on Automatic Control, 53(6), 1548–1554. doi:10.1109/TAC.2008.921036.