Optimal feedback control as a theory of motor coordination: Supplementary Notes

All simulations described in the main text are instances of the following general model: 1 , 1 Dynamics Feedback Cost 0 k t t t i t i t i t t t T T t t t t t A B C H Q R ε + = = + + = + ≤ + ∑ x x u u y x ω x x u u (1) where the , i t ε terms are independent standard normal random variables, and i C are constant matrices. The sensory noise terms t ω are independent multivariate normal random variables with mean 0 and covariance matrix Ω ω. The initial state 1 x has multivariate normal distribution with mean 1 ˆ x and covariance 1 Σ. The optimal control problem is the following: given 1 1 1 t t t t − − = u π x u u y y which minimizes the expected cumulative cost () , 1 T T T t t t t t t E Q R = + ∑ ε ω x x u u over the time interval [1; T]. Time is expressed in units of 10msec, which is the discrete time step we use. When the system noise in Eq 1 is additive rather than multiplicative, the LQG problem has a well-known solution 1 , which involves recursive linear state estimation (Kalman filtering) and linear mapping from estimated statesˆt x to optimal control signals t u. In the case of multiplicative noise, we have derived 2 the following iterative algorithm for solving this problem. The state estimate is updated using a modified Kalman filter which takes into account the multiplicative noise. For a given control law t