The Optimization of a Relay Controlled System Subjected to Random Disturbances

adjoint variable approximating sum assumed assumption average value bounded control law control scheme correlation cost D'eF d§dt defined determined differentiated equation Dirac delta function double integral Du(T dy(T dynamical system equivalent system error evaluated expression FIGURE final state variance find the optimal follows Fx(t gaussian noise given by Eq Hamiltonian initial conditions input noise integral of Eq involving the autocorrelation J₁ known linear filter matrix Maximum Principle mean and variance minimization necessary noise is gaussian noise power noise process oly(T optimal control signal optimal value output p₁(t P₂(t parameters performance criterion performance index plant plu(T ply(T probability distribution problem random variable relay control respect to E[u(T right hand side scheme of Section Section 5.2 side of Eq standard deviation substituted into Eq T T SS term involving value of u(T variance given vector written x₂ x₂(T zero mean Ττ ན ན ན