Least squares parameter estimation of continuous-time ARX models from discrete-time data

When modeling a system from discrete-time data, a continuous-time parameterization is desirable in some situations, In a direct estimation approach, the derivatives are approximated by appropriate differences. For an ARX model this lead to a linear regression. The well-known least squares method would then be very desirable since it can have good numerical properties and low computational burden, in particular for fast or nonuniform sampling. It is examined under what conditions a least squares fit for this linear regression will give adequate results for an ARX model. The choice of derivative approximation is crucial for this approach to be useful. Standard approximations like Euler backward or Euler forward cannot be used directly. The precise conditions on the derivative approximation are derived and analyzed. It is shown that if the highest order derivative is selected with care, a least squares estimate will be accurate. The theoretical analysis is complemented by some numerical examples which provide further insight into the choice of derivative approximation.