Input delay estimation using quadratic programming

Several techniques have been developed to take into account the presence of time delays in the design of a controller. However, such techniques usually require a point or interval estimate of the delay, which motivates the development of suitable estimators. Within the scope of discrete-time systems, the estimation can be carried out by exhaustive evaluation of the possible delay values with respect to a suitable criterion. If the system model is linear and a quadratic cost function is adopted, such an approach can be cast into a binary quadratic programming (BQP) framework. The present work investigates potential advantages of relaxing the binary constraint by allowing the decision variables to be real-valued, which leads to a standard quadratic programming (QP) problem. The resulting estimate is then calculated as a weighted average of the possible delay values. Simulation results indicate that the proposed QP approach is less sensitive to measurement noise, especially in terms of the estimator variance, as compared to the “winner-take-all” BQP method. Such a finding may be ascribed to the averaging process involved in the calculation of the QP estimate, which tends to smooth the noise effects.