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In navigation systems it is important to align a system to the vertical. This can be accomplished by measuring the vertical acceleration and controlling the platform so that the measured acceleration is zero. A simplified one-dimensional version of the problem can be modeled by de dt u = -ky, y=p+w, where is the alignment error, u the control signal, y the measured signal, and w the measurement noise, which is assumed to be white noise with zero mean and covariance function E(w(s)wT (t)) = rwo(t-s). The initial misalignment is assumed to be a random variable with zero mean and the covariance Po. Determine a time- varying gain k(t) such that the error goes to zero as fast as possible. Compare this with a constant gain. (Vertical alignment) = U