Distribution Control of Systems with Random Parametric
5.2 Future Works
5.2.2 Control of Probability Density Function
The feedback control signal in Chapter 4 is calculated using the augmented state variables and the uncertainties under the assumption that these values are available. However, in real applications, this may not be the case. Moreover, generally, convergence in distribution cannot imply the convergence of probability density functions. Therefore, it is non-trivial to consider controller design with direct feedback of the actual state variables. In this case, the actual probability density function of the state variables have to be estimated in real time and be expanded in gPC expansion. The convergence of the gPC coefficients of the estimated probability density function to those of the desired probability density function will result in the convergence of the actual probability density function to the desired one.
Estimation algorithms of probability density functions can be found in [145].
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