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## Citation

Yi S, Ulsoy AG, Nelson PW. Design of observer-based feedback control for time-delay systems with application to automotive powertrain control. J Franklin Institute. DOI

## 10 Word Summary

DDEs have infinite eigenvalues, Lambert-W function places finite number of poles.

## Abstract

A new approach for observer-based feedback control of time-delay systems is developed. Time-delays in systems lead to characteristic equations of infinite dimension, making the systems difficult to control with classical control methods. In this paper, a recently developed approach, based on the Lambert W function, is used to address this difficulty by designing an observer-based state feedback controller via assignment of eigenvalues. The designed observer provides estimation of the state, which converges asymptotically to the actual state, and is then used for state feedback control. The feedback controller and the observer take simple linear forms and, thus, are easy to implement when compared to nonlinear methods. This new approach is applied, for illustration, to the control of a diesel engine to achieve improvement in fuel efficiency and reduction in emissions. The simulation results show excellent closed-loop performance.

## Notes

• Use an observer when not all states are directly accessible for measurement
• Delays lead to infinite dimensionality in the characteristic equation
• Pade approximations have limited accuracy and create a non-minimum phase system
• Smith-predictors (among others) turn delay systems into higher order approximations, but these require highly accurate models and are subject to modeling errors
• This approach does not require the integration state during finite intervals
• Separation principle is shown to hold for linear DDEs, so control and observer can be designed separately.
• The lambert W function is the key: $z = \mathbf{W}(z)e^{\mathbf{W}(z)}$
• The system: $\dot{x}(t) = (A-BK)x(t) + (A_d-BK_d)x(t-h)$
• Placed poles are determined from: $S_k = \frac{1}{h}\mathbf{W}_k(A_d h Q_k) + A$