Super Twisting Based Lyapunov Redesign for Uncertain Linear Delay Systems

Closing the gap from nominal to robust asymptotic stabilization of time delay systems

“…Within the framework of Sliding Mode Control (SMC), preliminary results addressing the above problem have been recently presented, 16,17 where the departing assumption is that the time-delay system is nominally stabilizable and that there exists a Lyapunov-Krasovskii functional of a certain class associated with the stable nominal system, for example,…”

“…They can be found numerically; 15,22 the approach in Reference 22 being within the framework of delay Lyapunov matrices. In contrast with the departing assumptions in previous work, 16,17 Assumption 1 already enables constructing a complete type functional associated to system (2), that satisfies the upper and lower bounds in Lemma 1.…”

“…We prove that, under the theorem conditions, the time derivative of v along the solutions of ( 17) is negative. Let us rewrite (17) as…”

“…Proof The structure of control (14) is similar to the one proposed in Reference 17; eq. 11, enabling us to rely on the reported results therein.…”

“…Within the framework of Sliding Mode Control (SMC), preliminary results addressing the above problem have been recently presented, 16,17 where the departing assumption is that the time‐delay system is nominally stabilizable and that there exists a Lyapunov‐Krasovskii functional of a certain class associated with the stable nominal system, for example, $$$$ v\left({x}_t\right)=x{(t)}^T Px(t)+{\int}_{t-h}^t{x}^T\left(\theta \right) Qx\left(\theta \right) d\theta, P>0,Q>0. $$$$ The time derivative of $$$ v $$$ along the solutions of (1) makes evident that, similar to the delay‐free case, a natural choice for the sliding manifold is $$$ \left\{x:2{x}^T PB=0\right\} $$$.…”