I know we can forecast univariate time series using different models of exponential smoothing , but am searching for whether same can be extended to multivariate time series and if yes what are those equations?
If you have 1 endogenous series and 1 or more exogenous series the approach is to build and identify a Transfer Function Model as described here http://www.autobox.com/pdfs/WHY-WE-FILTER.ppt and here https://autobox.com/pdfs/PREFERRED.pdf and here http://www.autobox.com/pdfs/A.pdf leading to http://www.autobox.com/pdfs/SARMAX.pdf . Analysis will suggest "the form of the appropriate equation" .
EDITED TO BE MORE SPECIFIC AS TO HOW A SIMPLE EXPONENTIAL SMOOTHING MODEL CAN BE GENERALIZED TO INCLUDE PREDICTORS BOTH STOCHASTIC AND DETERMINISTIC
The simple exponential smoothing model is a particular case of an ARIMA model (0,1,1) with coefficient 1-A from https://people.duke.edu/~rnau/411arim.htm
Following is the simple exponential smoothing model by incorporating X's as in the SARMAX model we are accomplishing the task. Essentially the lag structure in Y is handled by the ARIMA (noise) component in the SARMAX model.
If you have more than 1 endogenous series then either a VARIMA model or VAR model would be appropriate.
AND restrict the arima structure for the NOISE component to a first difference ma(1) model .