We consider general nonlinear systems with time varying input delay for which we design a predictor-based feedback controller. Under the assumptions of forward completeness of the open-loop system and the existence of a (possibly time-varying) stabilizing controller, in the absence of the input delay, our controller achieves global asymptotic stability in the presence of a time-varying input delay. We prove asymptotic stability based on a backstepping transformation which enables us to construct a Lyapunov functional. Our design is illustrated by a numerical example of a second order system.

DOI: 10.1109/MED.2011.5983098 .

or considering general nonlinear systems with time-varying input and state delays for which we design predictor based feedback controllers. Based on a time-varying infinite-dimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization.

DOI: 10.1115/1.4005278 .

Smith predictor may be a good solution pratically. According to my understanding, the idea of Smith predictor is very simple, which is to construct a parallel artificial virtual model, without the delay, as a counterpart of the delayed plant and use the signals generated by the model to synthesize the proper input.

View the delay as a time profile, rather than a single delay.

Instead of f(x(t-td)), one use \sum_{i=1}^m a_i y(t-i/m*Td). This way, the problem simplifies to identifying a_i's. Use adaptive control afterwards.

Delay time may (or not) be of first order. The magnitude of the error that results from identifying the observed/measured variable (e.g. concentration; affected by first order delay) with the corresponding actual (real) concentration, can be estimated accordingly eq. 2.58 of the following reference (p. 25).

Application example ― While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the recursive least squares algorithm (RLS) with forgetting factor (RLS-FF) to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...