Higher Order Super-Twisting for Perturbed Chains of Integrators

Abstract

In this paper, we present a generalization of the supertwisting algorithm for perturbed chains of integrators of arbitrary order. This Higher Order Super-Twisting (HOST) controller is homogeneous with respect to a family of dilations and is continuous. It is built as a dynamic controller (with respect to the state variable of the chain of integrators) and the convergence analysis is performed by the use of a homogeneous strict Lyapunov function which is explicitly constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four, first pure then perturbed, where we compare the performances of two HOST controllers.

When is it reasonable to implement the discontinuous sliding‐mode controllers instead of the continuous ones? Frequency domain criteria

“…Consider, for instance, c = 1.1 , then the chattering parameters predicted by HB, amplitude (24), frequency (25) and AP (26), take the form…”

“…Higher‐Order Sliding‐Mode Controllers (HOSMC) drive the output of the system and its ( r − 1) successive time derivatives to zero in finite‐time by means of discontinuous control, for example, Quasi‐continuous and Nested algorithms (see also the recent works of Ding et al and Cruz‐Zavala and Moreno). More recently, Higher‐Order Super‐Twisting and High‐Order Continuous Twisting algorithms were announced as extensions of CSMC for systems of arbitrary relative degree (see also the work of Laghrouche et al). CSMC ensure finite‐time stabilization of the output and its r successive time derivatives by acting with a switching function on the ( r + 1) time derivative of the sliding variable, using the same information as discontinuous HOSMC.…”

Continuous sliding‐mode control for singular systems

“…where u 1 (t) is the CT control (21). Having defined the sliding output (2) (17) for (s) being a Hurwitz polynomial.…”

“…finite-time theoretically exact compensation of Lipschitz P/U; 2. sliding accuracy of order (r+1) with respect to the output ((r+1)th order of precision) in the face of time discretizations of the control input and actuator time constant; 3. robust convergence of the output and its first r time derivatives to the origin in finite time ((r + 1)-sliding motion) while assuming only the information of the output and its first (r − 1) time derivatives.Examples of CSM controllers for systems of relative degree two are: continuous twisting (CT) algorithm, 17 continuous terminal SM, 18 and discontinuous integral controller. 19 For an arbitrary relative degree, continuous algorithms have also been derived (eg, see other works [19][20][21][22] ).…”

“…Examples of CSM controllers for systems of relative degree two are: continuous twisting (CT) algorithm, 17 continuous terminal SM, 18 and discontinuous integral controller. 19 For an arbitrary relative degree, continuous algorithms have also been derived (eg, see other works [19][20][21][22] ).…”