Stabilization of Uncertain Nonlinear Systems via Immersion and Invariance based Sliding Mode Control Ankit Sachan, Shyam Kamal, Devender Singh Department of Electrical Engineering, Indian Institute of Technology (BHU) Varanasi, India Abstract This paper introduces a new methodology for sta- bilization of uncertain nonlinear system using immer- sion and invariance (I &I ) based sliding mode con- trol. The attractivity of the manifold ensures the re- flection of the desired system behavior by I &I princi- ple for the nominal system. Higher-order sliding mode of an arbitrary r th order is invoked to replace the sta- bilizing control law of an output regulator theory to derive the off-the-manifold with completely controlling the disturbance/uncertainties in finite-time. This makes the system more systematic to achieve the equilibrium point asymptotically. Finally, the validation of proposed strategy is tested on a magnetic-levitation system. 1. Introduction The stabilization of perturbed system has always been a challenging task and several methodologies were proposed in the control literature [1]. For regulation of nonlinear problem of output feedback stabilization, a new approach named Immersion and Invariance (I &I ) is introduced for the nonlinear system in [2] which cap- tures the dominating behavior by forming a reduced or- der dynamical system with the help of a suitable con- troller. Moreover, the general nonlinear system with a suitable control law guarantees the immersion of con- trolled system into a reduced system asymptotically. This relies on the solution of Francis-Byrnes-Isidori (FBI ) equation to compute the solution of regulator problem. More precisely, I &I principle relies on a non- linear regulator theory for generating an invariant set which shows its attractivity for closed-loop dynamics by feedback control and system immerses into reduced order dynamics. If the system lies outside the manifold region, the feedback law is enforced to move the sys- tem states again to the manifold. This process is done by defining an extended dynamical system by copy- ing the nonlinear plant and these new dynamics shows the characteristics of the off-the-manifold coordinates. Then, control feedback law for original system matches with extended dynamics and system states guarantees the convergence to zero asymptotically. With the motivation of I &I to avoid the solving of partial differential equations (PDE ) was explored for stabilization of various practical applications. In [3] the classical cart and pendulum system was reported to show the interlacing the step procedures of I &I to obvi- ate the PDE solutions. Additionally, constructive way of I &I was adopted for a class of unactuated mechani- cal system in the framework of Port-Hamiltonian in [4]. But these approaches were only restricted to the nomi- nal nature of the plant. As the uncertainty is added to the nonlinear plant, it shows the shifting of equilibrium point and its states converges at the value of uncertain dynamics asymptotically with the stabilizing control. Several elegant controllers have been elaborated for controlling the performance of the plant under uncer- tainty and robustness. Sliding mode control (SMC) [5] is one of the most promising techniques for controlling systems with uncertainties. The central features of this control-algorithm is that it renders finite-time conver- gence to off-the-manifold and model uncertainties be- comes insensitive to the closed-loop response. Specif- ically, the first-order sliding mode applied to the sys- tem has relative degree 1 for control input with respect to output. Moreover, the design of sliding mode for any arbitrary relative degree r, a new paradigm called “higher-order sliding mode” is proposed in [6–9]. This algorithm relies on the input-output relation of relative degree r which provides finite-time stabilization for off- the-manifold coordinates and its r − 1 derivatives. As the limitation of higher-order SMC is that it is specifi- cally structured in the form of chain-of-integrators for output and its derivatives to perform a control action. But it is not always straightforward to get the input- output relation in form of chain-of-integrators and one possible solution is to identify off-the-manifold coordi- nates of lower dimension which helps to find the solu- tion in the form of the chain-of-integrators. From the above specification of I &I principle and higher-order SMC, the combination of control proce- 2018 15th International Workshop on Variable Structure Systems (VSS) Graz University of Technology, Graz, Austria, July 9-11, 2018 978-1-5386-6439-1/18/$31.00 ©2018 IEEE 79 Authorized licensed use limited to: Indian Institute Of Technology (Banaras Hindu University) Varanasi. Downloaded on September 23,2020 at 11:08:06 UTC from IEEE Xplore. Restrictions apply.