Delay Compensating Control

Delay compensating feedback stabilization entails finding formulas for feedback controls whose values can be calculated from time-lagged (instead of current) measurements from dynamical systems. Feedback controls are forcing functions for dynamical systems, and so are useful for modeling forces that can be applied to engineering systems. Normally, feedback controls are chosen to ensure prescribed stabilization properties, like the global asymptotic stability property that is commonly taught in basic differential equations courses, or to maintain forward invariance of regions of interest. Three commonly used delay compensating feedback stabilization approaches are emulation, reduction model and exact predictor controls, and chain predictors. Emulation entails finding formulas for feedback controls that would ensure that prescribed stabilization objectives are realized if current measurements from the systems had been available, and then determining how large a time lag the feedback control can tolerate in its measurements, while still ensuring the prescribed stabilization property. However, emulation can sometimes produce bounds on the magnitudes of the allowable time lags that are too small for applications. Reduction model and exact predictors can compensate for arbitrarily long time lags, but can lead to computational challenges that result from their being only implicitly defined by integral equations. Chain predictors can sometimes address these challenges, by compensating for arbitrarily long delays without producing the distributed terms that arise in other delay compensation methods.

Delay compensating controls are widely used for biological, electronic, and mechanical systems where time consuming information gathering or delayed responses of dynamical systems can be modeled by feedback controls that are computed from time-lagged system measurements. For instance, in the study of bioreactors, delay compensating feedback control can model the delays between the times nutrients are provided to organisms and the time that growth of the organisms occurs. Delay compensating feedback control has also been applied to neuromuscular electrical stimulation to compensate for delayed muscle response to electrical stimuli. In the study of grid-tied inverters that are used for grid integration of renewable energy sources and in other power electronics applications, the limited processing capability of digital signal processors (or DSPs) and control scheme complexity can result in a delay between the time sample measurements are received and the time control actions are made by the DSP. When comparing different types of mouse acceleration for computers, the delays can represent delayed human reactions to visual stimuli. In underwater marine robotics, the delays can model the time lags between the time a force is applied to a vehicle and the time a vehicle responds through movement. Below are references on delay compensating feedback control. Click on the titles to see the presentation or paper.

References:

1. Bhogaraju, I., M. Farasat, M. Malisoff, and M. Krstic, "Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties," Systems and Control Letters, Volume 152, June 2021, Paper 104933.
2. Karafyllis, I., M. Malisoff, M. de Queiroz, M. Krstic, and R. Yang, "Predictor-based tracking for neuromuscular electrical stimulation," International Journal of Robust and Nonlinear Control, Volume 25, Issue 14, 2015, pp. 2391-2419.
3. Mazenc, F., M. Malisoff, and I. Bhogaraju, "Sequential predictors for delay compensation for discrete time systems with time-varying delays," Automatica, Volume 122, December 2020, Paper 109188.
4. Mazenc, F., M. Malisoff, and S.-I. Niculescu, "Reduction model approach for linear time-varying systems with delays," IEEE Transactions on Automatic Control, Volume 59, Issue 8, August 2014, pp. 2068-2082.
5. Mazenc, F., M. Malisoff, and S.-I. Niculescu, "Stability and control design for time-varying systems with time-varying delays using a trajectory based approach," SIAM Journal on Control and Optimization, Volume 55, Issue 1, 2017, pp. 533-556.
6. Mazenc, F., G. Robledo, and M. Malisoff, "Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties," Discrete and Continuous Dynamical Systems Series B, Volume 23, Issue 4, June 2018, pp. 1851-1872.
7. Varnell, P., M. Malisoff, and F. Zhang, "Stability and robustness analysis for human pointing motions with acceleration under feedback delays," International Journal of Robust and Nonlinear Control, Volume 27, Issue 5, March 2017, pp. 703-721.