The phrase "assigned itself" is correct and usable in written English.
It can be used in contexts where something has taken on a role or responsibility without external assignment.
Example: "The project was so engaging that it assigned itself to the top of my priority list."
Alternatives: "took on" or "designated itself".
The Turkish military assigned itself a similar supervisory role after its 1980 coup.
Indeed, the most complex and potentially perilous tasks that the United States has assigned itself still lie ahead.
In the end, though, "The Lone Ranger" can't quite pull off the daredevil feats it has assigned itself.
To him, the steep price Google has assigned itself is proof that the stock market has not shaken off the excesses of the bubble.
"Stopping these last cases of polio in these last countries is among the most difficult tasks the world has ever assigned itself.
She reminded Mr. Putin that the Organization for Security and Cooperation in Europe, of which the United States and Russia are members, had recently assigned itself a role in resolving conflicts within countries.
In other words, each vehicle should assign itself to a target located within the Voronoi region associated with the vehicle.
Unaware of what is going on deep down inside, the conscious mind assigns itself the starring role.
It is also that when men and women make decisions together they generally make different and better decisions than when one sex assigns itself all executive responsibilities.
Washington might like to assign itself full responsibility for remaking Iraq, but the political and economic costs of doing so are sure to be severe.
The defense secretary made clear that the long months of debate within the North Atlantic Treaty Organization at the cold war's twilight over whether to assign itself missions outside the boundaries of its member nations had been overtaken.
