having been done 是过去分词的完成时态,属于非谓语动词.分词短语的逻辑主语是否省略,则要看分词短语的逻辑主语是否和主句的主语一致.如果一致,则分词短语的逻辑主语可以省略；如果不一致,则分词短语的逻辑主语要表示出来,这种分词短语也叫独立主格结构.请看例句：
Being sick, I stayed at home=Because I was sick,I stayed at home.( 现在分词短语的逻辑主语是 I,和主句的主语I一致,可省略） 我因病呆在家中.
The monitor being ill, we had better put off the meeting off.=Because the monitor was ill, we had better put off the meeting .(现在分词短语的逻辑主语是the monitor, 和主句的主语we不一致,所以要补充出来.）班长病了,我们最好还是延期开会吧.
Gone from home so long, they joyously embraced their mates of boyhood.=Since they had gone from home so long, they joyously embraced their mates of boyhood(过去分词短语gone from...的逻辑主语是they, 和主句的主语they一致 可省略） 他们久离家乡,高兴地拥抱他们儿时的伙伴.
This done, we went home.=After this had been done, we went home( 过去分词done的逻辑主语是this, 和主句的主语we不一致,所以要补充出来.）做完此事,我们就回家了.
问题补充回复：
在After this had been done, we went home这个句子中, we went home（我们回家）是过去发生的动作,用的是过去式,从句表示在回家以前这个事情已经被做完this had been done,从句用的是过去完成式的被动式,表示过去的过去（过去完成式）和动作被完成.过去分词done有两个意思,一个意思是表示动作已经完成,另一个意思是表示被动,所以 过去分词done 在上述的句子中就表示 had been done.在把从句After this had been done变成分词短语时,第一要去掉After,第二had been done 中要去掉 had been 就变成了过去分词done,变成分词短语就是This done.因此
This done, we went home.=After this had been done, we went home
