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The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system
with a state process x {\displaystyle x} , an output process y {\displaystyle y} and a control u {\displaystyle u} , where w {\displaystyle w} is a vector-valued Wiener process, x {\displaystyle x} is a zero-mean Gaussian random vector independent of w {\displaystyle w} , y = 0 {\displaystyle y=0} , and A {\displaystyle A} , B 1 {\displaystyle B_{1}} , B 2 {\displaystyle B_{2}} , C {\displaystyle C} , D {\displaystyle D} are matrix-valued functions which generally are taken to be continuous of bounded variation. Moreover, D D ′ {\displaystyle DD'} is nonsingular on some interval {\displaystyle }. The problem is to design an output feedback law π : y ↦ u {\displaystyle \pi :\,y\mapsto u} which maps the observed process y {\displaystyle y} to the control input u {\displaystyle u} in a nonanticipatory manner so as to minimize the functional
where E {\displaystyle \mathbb {E} } denotes expected value, prime denotes transpose. and Q {\displaystyle Q} and R {\displaystyle R} are continuous matrix functions of bounded variation, Q {\displaystyle Q} is positive semi-definite and R {\displaystyle R} is positive definite for all t {\displaystyle t}. Under suitable conditions, which need to be properly stated, the optimal policy π {\displaystyle \pi } can be chosen in the form
where x ^ {\displaystyle {\hat {x}}} is the linear least-squares estimate of the state vector x {\displaystyle x} obtained from the Kalman filter