Switching Restrictions for Stability despite Switching Delay: Application to Switched Tracking Tasks in Parkinson’s Disease

Switched nonlinear systems with delay in the switching instant could be destabilized, despite stable dynamics in each mode, if the delay is long enough. We identify a restriction on the switching scheme to assure stability despite a finite delay in switching instant. The restriction partitions the state-space in a time-varying manner for a known switching delay, and converges to a steady-state partition that can be determined from the intersection of Lyapunov functions in each mode. We apply this technique to experimental data from a manual pursuit tracking task performed by 14 subjects with Parkinson’s disease, and 10 control subjects. Each subject manually tracks a moving target through a joystick-controlled cursor, with sudden changes in the tracking dynamics. The tracking task can be modeled as a 3-mode switched system. By calculating the maximal time delay for each mode pair and for each subject, we obtain a measure of relative stability that can be compared across groups and across tasks. Using the derived stability measure, subjects with Parkinson’s disease were shown to be relatively less stable than control subjects.