Research on Disturbance Observer

Last updated on October 18, 2023

This article can be considered as an English version of http://post.cdsl.kr/archives/467.

The works done by CDSL have been summarized in the book:

Disturbance Observers
H. Shim
in the Encyclopedia of Systems and Control, Springer, 2020
https://doi.org/10.1007/978-1-4471-5102-9_100068-1
preprint available at http://arxiv.org/abs/2101.02859

and in the tutorial paper:

Research Timeline

1. The initial result. Stability and the performance limitation of DOB for linear plants are studied in the state-space using the singular perturbation theory and the role of the zero-dynamics is revealed.

“State space analysis of disturbance observer and a robust stability condition” 
H. Shim and Y.J. Joo
IEEE Conf. on Dec. and Control, pp. 2193-2198, Dec., 2007.
http://dx.doi.org/10.1109/CDC.2007.4434130

2. The above result can be more simply proved in the frequency domain, and the design procedure for the Q-filter is proposed that always guarantees the closed-loop robust stability against arbitrarily large but bounded parametric uncertainties.

“An Almost Necessary and Sufficient Condition for Robust Stability of Closed-loop Systems with Disturbance Observer”
H. Shim and N.H. Jo
Automatica, vol. 45, no. 1, pp. 296-299, 2009.
http://dx.doi.org/10.1016/j.automatica.2008.10.009

3. Although the way how to guarantee the robust stability of the closed-loop with the DOB has been revealed in the above, the question whether the DOB also guarantees the robust transient response still remains unanswered. This question turns out negative, but a simple modification of the classical DOB (using the saturation and deadzone function) is suggested which guarantees the robust transient response. This strategy is also applied to nonlinear systems. Finally, it is shown that the DOB controller intrinsically contains the high-gain observer that has been actively studied by Khalil and co-workers.

“Adding Robustness to Nominal Output Feedback Controllers for Uncertain Nonlinear Systems: A Nonlinear Version of  Disturbance Observer”
J. Back and H. Shim
Automatica, vol. 44, no. 10, pp. 2528-2537, 2008.
http://dx.doi.org/10.1016/j.automatica.2008.02.024

4. The above result is refined and extended to the MIMO nonlinear systems with simpler proof in

“An Inner-loop Controller Guaranteeing Robust Transient Performance for Uncertain MIMO Nonlinear Systems”
J. Back and H. Shim
IEEE Trans. on Automatic Control, vol. 54, no. 7, pp. 1601-1607, 2009.
http://dx.doi.org/10.1109/TAC.2009.2017962

5. The paper (no.1) also pointed out that a blind application of DOB controller may degrade the performance especially when the performance output variable is different from the measurement output, and it is one of the states in the zero-dynamics. Instead, the DOB may be applicable only to a portion of the plant dynamics. An example is

“Robust Tracking and Vibration Suppression for a Two-Inertia System by Combining Backstepping Approach with Disturbance Observer”
J.S. Bang, H. Shim, S.K. Park, and J.H. Seo
IEEE Trans. on Industrial Electronics, vol. 57, no. 9, pp. 3197-3206, Sept. 2010
http://dx.doi.org/10.1109/TIE.2009.2038398

6. All the above results are limited to the minimum phase systems. A trial has been made to apply the DOB approach to non-minimum phase systems.

“A New Disturbance Observer for Non-minimum Phase Linear Systems” 
H. Shim, N.H. Jo, and Y.I. Son
American Control Conference, pp. 3385-3389, Seattle, June, 2008.
http://dx.doi.org/10.1109/ACC.2008.4587015

or, its refinement:

“Disturbance Observer for Non-minimum Phase Linear Systems”
N.H. Jo, H. Shim, and Y.I. Son
Int. J. of Control, Automation, and Systems, vol. 8, no. 5, pp. 994-1002, Oct. 2010
http://dx.doi.org/10.1007/s12555-010-0508-x

7. Our stability condition has not been experimentally tested, but the following paper uses our proposed theorem to perform a real experiment:

“Disturbance-Observer-Based Hysteresis Compensation for Piezoelectric Actuators”
J. Yi, S. Chang, and Y. Shen
IEEE/ASME Trans. on Mechatronics, vol. 14, no. 4, 2009
http://dx.doi.org/10.1109/TMECH.2009.2023986

8. Since the same Q-filters are found in the loop, they could be merged into one in order to reduce the dimension of the DOB controller:

“Robust Tracking by Reduced-order Disturbance Observer: Linear Case”
J. Back and H. Shim
IEEE Conf. on Dec. and Control and European Control Conf. (CDC-ECC), Orlando, pp. 3514-3519, 2011
http://dx.doi.org/10.1109/CDC.2011.6161399

“Reduced-order Implementation of Disturbance Observers for Robust Tracking of Non-linear Systems”
J. Back and H. Shim
IET Control Theory & Applications, vol. 8, no. 17, pp. 1940-1948, 2014
http://dx.doi.org/10.1049/iet-cta.2013.1036

9. One of the standing assumption is to know the relative degree of the uncertain plant. But, a series of study is performed when this is not the case. The so-called higher-order root locus technique, that is rarely used in other applications, is employed in

“Can a Fast Disturbance Observer Work Under Unmodeled Actuators?”
N.H. Jo, Y. Joo, and H. Shim
Int. Conf. on Control, Automation and Systems (ICCAS), pp. 561-566, Seoul, Korea, 2011
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6106423

“A Note on Disturbance Observer with Unknown Relative Degree of the Plant”
N.H. Jo, Y. Joo, H. Shim, and Y.I. Son
IEEE Conf. on Dec. Control, pp. 943-948, Maui, HI, 2012
https://doi.org/10.1109/CDC.2012.6426142

“A Study of Disturbance Observers with Unknown Relative Degree of the Plant”
N. H. Jo, Y. Joo, and H. Shim
Automatica, vol. 50, no. 6, pp. 1730-1734, 2014
http://doi.org/10.1016/j.automatica.2014.04.015

10. Disturbance rejection by DOB is an approximate one. But, by embedding some internal models, perfect rejection is possible for a class of disturbance signals.

The initial result is about polynomial-in-time disturbances:

“Rejection of Polynomial-in-time Disturbances via Disturbance Observer with Guaranteed Robust Stability”
G. Park, Y. Joo, H. Shim, and J. Back
IEEE Conf. on Dec. Control, pp. 949-954, Maui, HI, 2012
https://doi.org/10.1109/CDC.2012.6425973

In addition to such type of signals, sinusoidal disturbances also can be dealt with in

“Embedding Internal Model in Disturbance Observer with Robust Stability”
Y. Joo, G. Park, J. Back, and H. Shim
IEEE Trans. on Autom. Control,  vol. 61, no. 10, pp. 3128-3133, Oct 2016
https://doi.org/10.1109/TAC.2015.2503559

This property is combined with the nominal performance recovery so that, when the disturbance is a sum of unknown signals whose generating model is known and another signals that are completely unknown, the modeled disturbance is completely rejected while the unmodeled disturbance is approximately rejected with arbitrarily error precision.

“Asymptotic Rejection of Sinusoidal Disturbances with Recovered Nominal Transient Performance for Uncertain Linear Systems”
G. Park, Y. Joo, and H. Shim
IEEE Conf. on Dec. Control, pp. 4404-4409, Los Angeles, CA, 2014
https://doi.org/10.1109/CDC.2014.7040076

“Recovering nominal tracking performance in asymptotic sense for uncertain linear systems”
G. Park, H. Shim, and Y. Joo
SIAM J. Control Optim., vol. 56, no. 2, pp. 700-722, 2018
http://dx.doi.org/10.1137/17M1122657

11. The conventional DOB consists of two same Q-filters, yet they play different roles for nominal performance recovery; one works for disturbance rejection, while the other is to estimate the plant’s state (similar to the high-gain observer). This finding raises a question about stability and performance when the Q-filters have different forms. It is answered by

“Reduced Order Type-k Disturbance Observer Based on a Generalized Q-filter Design Scheme”
Y. Joo and G. Park
Int. Conf. Control, Automation, and Systems, Seoul, Korea, pp. 1211-1216, 2014.
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6987743&tag=1

12. Most of the previous works have focused only on the disturbance rejection. A study on attenuating both disturbance and noise via DOB is newly introduced:

“Robust Stabilization via Disturbance Observer with Noise Reduction”
N. H. Jo and H. Shim
European Control Conf., pp. 2861-2866, Zurich, Switzerland, 2013.
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6669565&tag=1

“Noise reduction disturbance observer for disturbance attenuation and noise suppression”
Nam Hoon Jo, Chanyoung Jeon, and Hyungbo Shim,
IEEE Trans. on Industrial Electronics, vol. 64, no. 2, pp. 1381-1391, 2017
http://dx.doi.org/10.1109/TIE.2016.2618858

13. While all the above results are discussed in the continuous-time domain, for implementing it into digital devices, DOB will be constructed in the discrete-time domain and be applied to a sampled-data system. At first glance, stability seems to remain guaranteed whenever a discretization of a well-designed continuous-time DOB is used with fast sampling. Interestingly, this is not the case and the sampling process may hamper stability of the DOB controlled system with the so-called sampling zeros. Some works are made to clarify this phenomenon in a theoretical sense, and to propose a new design guideline for the discrete-time DOB:

“On Robust Stability of Disturbance Observer for Sampled-data Systems Under Fast Sampling: An Almost Necessary and Sufficient Condition”
G. Park, Y. Joo, C. Lee, and H. Shim
IEEE Conf. on Dec. and Control, 2015.
http://dx.doi.org/10.1109/CDC.2014.7040076

“A Generalized Framework for Robust Stability Analysis of Discrete-time Disturbance Observer for Sampled-Data Systems: A Fast Sampling Approach”
G. Park and H. Shim
Int. Conf. of Control, Automation, and Systems, 2015
http://dx.doi.org/10.1109/ICCAS.2015.7364926

The above finding also can be interpreted in the state space by using the discrete-time singular perturbation theory:

“State-space Analysis of Discrete-time Disturbance Observer for Sampled-data Control Systems”
H. Yun, G. Park, H. Shim, H. J. Chang
American Control Conf., 2016
https://doi.org/10.1109/ACC.2016.7525587

14. In order to handle external disturbance in the discrete-time domain, the question should be answered whether the continuous-time disturbance can be expressed by its discrete-time counter part. Its answer is not straightforward even if it looks simple at first glance. For the details, refer to:

“Can continuous-time disturbance be represented by sampled input disturbance?”
Gyunghoon Park, H. Shim, and Kyungchul Kong
In Proc. of 16th International Conference on Control, Automation and Systems (ICCAS 2016)
http://dx.doi.org/10.1109/ICCAS.2016.7832474

15. What about robustness of the DOB-based controllers for non-parametric uncertainties? We already pointed out relatively weak robustness to the unmodelled dynamics, but if it goes to gain/phase margins, the DOB-based controller has very good robustness as pointed out by the following paper.

“Arbitrarily large gain/phase margin can be achieved by DOB-based controller”
H. Kim, G. Park, H. Shim, N.H. Jo
Int. Conf. of Control, Automation, and Systems, 2016
http://dx.doi.org/10.1109/ICCAS.2016.7832358

16. More applications of DOB:

“SoC regulator and DOB-based load frequency control of a microgrid by coordination of diesel generator and battery storage”
Takuto Hiranaka, H. Shim, Toru Namerikawa
In Proc. of IEEE Conference on Control Applications (CCA) in Multi-Conference on Systems and Control, Buenos Aires, Argentina, Sept. 2016.
http://dx.doi.org/10.1109/CCA.2016.7587823

“On Improving the Robustness of Reinforcement Learning-based Controllers using Disturbance Observer”
Jeong Woo Kim, Hyungbo Shim, and Insoon Yang
In Proc. of IEEE 58th Conf. Decision and Control, Nice, December, 2019

“Disturbance Observer Approach for Fuel-Efficient Heavy-Duty Vehicle Platooning”
Gyujin Na, Gyunghoon Park, Valerio Turri, Karl H. Johansson, Hyungbo Shim
and Yongsoon Eun
Vehicle System Dynamics, Special Issue on Connected and Automated Vehicles, 2020
https://doi.org/10.1080/00423114.2019.1704803

Guaranteeing almost fault-free tracking performance from transient to steady-state: A disturbance observer approach
Gyunghoon Park and Hyungbo Shim
SCIENCE CHINA Information Sciences, Special Issue on Advances in Disturbance/Uncertainty Estimation and Attenuation Techniques With Applications, vol. 61, July 2018
http://dx.doi.org/10.1007/s11432-017-9435-3

Stealthy Adversaries against Uncertain Cyber-Physical Systems: Threat of Robust Zero-Dynamics Attack
Gyunghoon Park, Chanhwa Lee, Hyungbo Shim, Yongsoon Eun, and Karl H. Johansson
IEEE Transactions on Automatic Control, 64 (12), pp. 4907-4919, 2019.
https://doi.org/10.1109/TAC.2019.2903429

Selected Experiments

1. Effect of embedding internal models into the DOB:
   
   This experiment is done by Prof. Juhoon Back and his student. See D. Kim, “Disturbance observer based robust motor speed control by embedding disturbance model,” Master Thesis, Kwangwoon University, 2015, for more details.
2. DOB used for drone control:
   
   This experiment is done in

Robust control of an equipment-added multirotor using disturbance observer
Suseong Kim, Seungwon Choi, Hyeonggeun Kim, Jongho Shin, H. Shim, and H. Jin Kim
IEEE Transactions on Control Systems Technology, 2017
http://dx.doi.org/10.1109/TCST.2017.2711602

Computer-aided design of DOB

• A preliminary version of DO-DAT (Disturbance Observer – Design & Analysis Toolbox) is available at MATLAB Central (https://mathworks.com/matlabcentral/fileexchange/66248-do-dat). While this is still on-going project, we have an initial outcome:

DO-DAT: A MATLAB toolbox for design & analysis of disturbance observer
Hamin Chang, Hyuntae Kim, Gyunghoon Park, and Hyungbo Shim
In Proc. of 9th IFAC Symp. on Robust Control Design (ROCOND), Brazil, 2018
https://doi.org/10.1016/j.ifacol.2018.11.130