Online Optimal Control

The problem of designing an Extended Regulator is defined as designing an algorithm that can adjust the optimal regulator gains for a time-variant linear system appropriately to match the variations in system parameters with time. Due to conditions of positive definiteness on the unknown variable in the Riccati equation for quadratic optimal control it is possible to show that repetitive iterations of the Riccati equation will always converges to a steady state value. Furthermore if small changes in system parameters result in small changes in the extremal solution then recalculating the optimal solution using traditional backward integration or ordered Schur decomposition for every such change would be computationally inefficient. A more efficient approach could be to use the already available optimum solution as an initial guess to arrive at a close approximation of the neighboring optimal solution. This report aims to investigate in detail computationally efficient methods to arrive at close approximation of the neighboring optimal solution for the Riccati equation found in linear optimal control theory. Nomenclature A,B,C = State, input and observations matrices Ad,Bd = Updated state transition and input matrices K = Kalman filter gain matrix G = LQR gain matrix L,J = Scalar cost functions N = Time horizon for LQR recursion P = Covariance matrix of the state error Q,R = LQR state and input weighing matrices S = Riccatti solution matrix FF,GG = Process and noise covariance matrices w(t),v(k) = Process and measurement noise functions f, g = Generic functions x f = Partial derivative w.r.t variable x k = Discrete time index t = Time u = Control vector x = State vector y = State measurement vector z = Discretized State measurement vector λ = Lagrange Multiplier t Δ = Discrete time step Suband Superscripts H = Transpose R = Inverse x ~ = Predicted variable x̂ = Corrected variable xa = Augumented variable Acronyms CARE = Continuous Algebraic Riccati Eq. DARE = Inverse MIMO = Predicted variable