Stochastic Robust Control Design Using Density Ratio Estimation

p(x): probability density distribution r(x): probability density ratio psuccess(x): density distribution for satisfactory samples pall(x): density distribution for all samples (x|μ,Σ): normal distribution with mean μ and covariant matrix Σ k : index of components for mixture model π k: mixing coefficient for k-th mixture component μk : mean for k-th mixture component ∑k: covariant matrix for k-th mixture component x: states of aircraft dynamics u: control input y: output w: velocity in z-direction (body axis) q: pitch angular velocity α: angle of attack αc: angle of attack command k: adjustable parameter vector Kα: feedback gain for AoA (adjustable parameter) Kq: proportional feedback gain for pitch angular velocity (adjustable parameter) KIα: integral feedback gain for AoA (adjustable parameter) Kδe: additional elevator loop gain (uncertain parameter) TD: delay time (uncertain parameter) σ: standard deviation ( )  : estimated value