Ph. D. Project
Fast and robust estimation of dynamical systems: a hybrid approach
2020/10/01 - 2023/09/30
Other supervisor(s):
Prof. Nesic Dragan
Estimation is a central theme of control systems engineering. It consists in estimating variables, which
we do not measure with sensors by exploiting other available measurements and a mathematical
model of the system under consideration. When the mathematical model of the plant is given by a
linear finite-dimensional ordinary differential equation, solutions are available since the 60's [K60,
L66]. When the model is non-linear, solutions exist only for specific classes of systems, or exhibit major
issues. Indeed, generic solutions for nonlinear systems like high-gain observers [KP14] are very
sensitive to noise, which can be redhibitory in practice. In general, observers almost always exhibit a
trade-off between the speed of convergence, which is essential to quickly know the desired
unmeasured variables, and the accuracy in presence of measurement noises, which are inevitable in

The purpose of this PhD topic is to overcome this paradox by exploiting hybrid techniques, that is,
observers, which exhibit continuous-time dynamics and jumps. Recent results along those lines have
recently appeared in the literature [PTZ12, APTZ16, AZ18]. The approach we envisioned is different
and should be applicable to much broader classes of systems. Our idea is the following. Given a
system, linear or nonlinear, we assume that we know how to design an observer and we have several
choices for its parameters. Some values lead to fast convergence but high sensitivity to noises, and
others are more robust but generate a slow convergence. We propose to switch among different
values of these parameters to make the best out of them using hybrid techniques [GST09]. This
approach is sometimes used in an ad-hoc way in applications. The objective here is to provide
rigorous, robust analytical tools to do it.

Hybrid techniques have been proved to overcome fundamental limitations in the context of control
e.g. [ZNTH19], but, as far as we know, no such results exist for estimation. This PhD topic will give us
the opportunity to demonstrate the power of hybrid tools for estimation. We will rely for this purpose
on our expertise on (nonlinear) estimation, hybrid dynamical systems and, in particular, on our recent
results in [APN20] where we already demonstrated the benefits of hybrid techniques for estimation in
a different context and for a different objective. The challenge here will be to carefully mathematically
define performances and how to switch between the observers to make the best use out of them. We
will also give us the flexibility to pursue an alternative hybrid route for the same goal and with the
same philosophy, namely, the design of supervisory observers, which we initially proposed in [CNPK15,
CPKN17] for adaptive estimation problems, to improve performance.

[APN20] Astolfi, D., Postoyan, R., & Nesic, D. (2020). Uniting observers. IEEE Transactions on Automatic
[APTZ16] Andrieu, V., Prieur, C., Tarbouriech, S., & Zaccarian, L. (2016). A hybrid scheme for reducing
peaking in high-gain observers for a class of nonlinear systems. Automatica, 72, 138-146.
[AZ18] Alessandri, A., & Zaccarian, L. (2018). Stubborn state observers for linear time-invariant
systems. Automatica, 88, 1-9.
[BDZGF09] Boulkroune, B., Darouach, M., Zasadzinski, M., Gillé, S., & Fiorelli, D. (2009). A nonlinear
observer design for an activated sludge wastewater treatment process. Journal of Process Control,
19(9), 1558-1565.
[BPRBD19] Blondel, P., Postoyan, R., Raël, S., Benjamin, S., & Desprez, P. (2019). Nonlinear circle-
criterion observer design for an electrochemical battery model. IEEE Transactions on Control Systems
Technology, 27(2), 889-897.
[CNPK15] Chong, M. S., Nešić, D., Postoyan, R., & Kuhlmann, L. (2015). Parameter and state estimation
of nonlinear systems using a multi-observer under the supervisory framework. IEEE Transactions on
Automatic Control, 60(9), 2336-2349.
[CPKN17] Chong, M. S., Postoyan, R., Khong, S. Z., & Nešić, D. (2017, December). Supervisory observer
for parameter and state estimation of nonlinear systems using the DIRECT algorithm. In 2017 IEEE 56th
Annual Conference on Decision and Control (CDC) (pp. 2089-2094). IEEE.
[GST09] Goebel, R., Sanfelice, R. G., & Teel, A. R. (2009). Hybrid dynamical systems. IEEE Control
Systems Magazine, 29(2), 28-93.
[K60] Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of basic
Engineering, 82(1), 35-45.
[KP14] Khalil, H. K., & Praly, L. (2014). High‐gain observers in nonlinear feedback control. International
Journal of Robust and Nonlinear Control, 24(6), 993-1015.
[L66] Luenberger, D. (1966). Observers for multivariable systems. IEEE Transactions on Automatic
Control, 11(2), 190-197.
[PTZ12] Prieur, C., Tarbouriech, S., & Zaccarian, L. (2012, December). Hybrid high-gain observers
without peaking for planar nonlinear systems. In 2012 IEEE 51st IEEE conference on decision and
control (CDC) (pp. 6175-6180). IEEE.
[ZNTH19] Zhao, G., Nešić, D., Tan, Y., & Hua, C. (2019). Overcoming overshoot performance limitations
of linear systems with reset control. Automatica, 101, 27-35.
estimation, hybrid systems, Lyapunov stability, nonlinear systems, robustness
Duration: 3 years
We are looking for a strongly motivated candidate with a Master degree in control engineering or applied
Control Identification Diagnosis