Closed-loop response of tremor by application of the proposed adaptive control methodology under Case I.

<p>The controller is applied at . By application of the proposed controller, tremor does not converge to zero, because a proper value of nominal stimulation amplitude does not exist for this case. (a) Plot of tremor signal , (b) plot of , which does not converge to due to saturation.</p

Comparison of estimation error and tremor power in Cases I and II.

<p>In Case I, the difference between and its estimation is higher. Correspondingly, the tremor power is also higher. In Case II, the estimation of nominal stimulation amplitude is correct; therefore, the tremor power is nearly zero.</p

Closed-loop response of tremor using the proposed adaptive feedback-control methodology in Case II.

<p>The controller is applied at . By application of the proposed controller, tremor converges to zero, because a proper value of nominal stimulation amplitude exists in this case. Hence, the proposed methodology by Algorithm 1 can be used for estimation of DBS parameters. (a) Plot of tremor signal , (b) plot of , which converges to .</p

Effect of on estimation error under constant .

<p>The estimation error is decreasing with increasing . However, more time for estimation is required for higher values of . Hence, selection of has critical effects on estimation time and estimation error.</p

Proposed methodology for feedback control of tremor.

<p>The filtered tremor signal (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062888#pone.0062888-Mera1" target="_blank">[16]</a>) can be used to calculate the control signal using an adaptive control methodology. The amplitude of the control signal is bounded using a saturation block. The saturated control signal is passed through a rate-limiter to filter high-frequency variations in stimulus amplitude. The resultant signal can be used as the stimulation amplitude to generate a stimulus signal.</p

Filtered tremor acceleration under different values of stimulation amplitude.

<p>By considering time-varying parameters of the linear oscillator-based tremor model, we can reproduce the experimental results. In the present case, the results in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062888#pone.0062888-Mera1" target="_blank">[16]</a> on the tremor of a patient under DBS with constant stimulation frequency and pulse-width are reproduced for different values of stimulation amplitude. The selected parameters of the model are as follows: (a), (b) , (c) , (d) , (e) , (f) , (g) .</p

Process of deep brain stimulation (DBS) and tremor measurement.

<p>Values of DBS parameters are provided to the stimulus generator. The stimulus generator produces electrical impulses, which are applied to the brain through an electrode. The tremor signal generated at the hand and fingers is measured using a position, velocity or acceleration sensor (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062888#pone.0062888-Mera1" target="_blank">[16]</a>). This tremor signal can be filtered to isolate noise.</p

Windup effect without AWC.

This paper proposes a static anti-windup compensator (AWC) design methodology for the locally Lipschitz nonlinear systems, containing time-varying interval delays in input and output of the system in the presence of actuator saturation. Static AWC design is proposed for the systems by considering a delay-range-dependent methodology to consider less conservative delay bounds. The approach has been developed by utilizing an improved Lyapunov-Krasovskii functional, locally Lipschitz nonlinearity property, delay-interval, delay derivative upper bound, local sector condition, L2 gain reduction from exogenous input to exogenous output, improved Wirtinger inequality, additive time-varying delays, and convex optimization algorithms to obtain convex conditions for AWC gain calculations. In contrast to the existing results, the present work considers both input and output delays for the AWC design (along with their combined additive effect) and deals with a more generic locally Lipschitz class of nonlinear systems. The effectiveness of the proposed methodology is demonstrated via simulations for a nonlinear DC servo motor system, possessing multiple time-delays, dynamic nonlinearity and actuator constraints.</div

Closed-loop response with AWC.

This paper proposes a static anti-windup compensator (AWC) design methodology for the locally Lipschitz nonlinear systems, containing time-varying interval delays in input and output of the system in the presence of actuator saturation. Static AWC design is proposed for the systems by considering a delay-range-dependent methodology to consider less conservative delay bounds. The approach has been developed by utilizing an improved Lyapunov-Krasovskii functional, locally Lipschitz nonlinearity property, delay-interval, delay derivative upper bound, local sector condition, L2 gain reduction from exogenous input to exogenous output, improved Wirtinger inequality, additive time-varying delays, and convex optimization algorithms to obtain convex conditions for AWC gain calculations. In contrast to the existing results, the present work considers both input and output delays for the AWC design (along with their combined additive effect) and deals with a more generic locally Lipschitz class of nonlinear systems. The effectiveness of the proposed methodology is demonstrated via simulations for a nonlinear DC servo motor system, possessing multiple time-delays, dynamic nonlinearity and actuator constraints.</div

Closed-loop response without AWC.

This paper proposes a static anti-windup compensator (AWC) design methodology for the locally Lipschitz nonlinear systems, containing time-varying interval delays in input and output of the system in the presence of actuator saturation. Static AWC design is proposed for the systems by considering a delay-range-dependent methodology to consider less conservative delay bounds. The approach has been developed by utilizing an improved Lyapunov-Krasovskii functional, locally Lipschitz nonlinearity property, delay-interval, delay derivative upper bound, local sector condition, L2 gain reduction from exogenous input to exogenous output, improved Wirtinger inequality, additive time-varying delays, and convex optimization algorithms to obtain convex conditions for AWC gain calculations. In contrast to the existing results, the present work considers both input and output delays for the AWC design (along with their combined additive effect) and deals with a more generic locally Lipschitz class of nonlinear systems. The effectiveness of the proposed methodology is demonstrated via simulations for a nonlinear DC servo motor system, possessing multiple time-delays, dynamic nonlinearity and actuator constraints.</div