A Hybrid Gradient Algorithm for Linear Regression with Hybrid Signals

Given a linear input/output relationship involving unknown parameters, we propose a hybrid gradient descent algorithm to estimate the unknown parameters when the inputs and the outputs are hybrid signals. These signals are allowed to change continuously during ordinary time - or flow - and to change discretely - or jump - at isolated time instances. To estimate the unknown parameters, we develop a gradient descent algorithm that updates the estimates continuously during flows and instantaneously at jumps. The proposed hybrid gradient algorithm generalizes the existing gradient descent algorithms in the continuous-time and the discrete-time settings. Under a relaxed (hybrid) version of the well-known persistence of excitation condition, the proposed hybrid gradient descent algorithm estimates the parameters exponentially fast. An illustrative example is presented, showing the capabilities of our approach while classical algorithms fails to ensure the convergence.