Relying on the possibility of generating a second-order sliding motion by using, as control, the first derivative of the control signal instead of the actual control, a new solution to the problem of chattering elimination in variable structure control (VSC) is presented. Such a solution, inspired by the classical bang-bang optimal control strategy, is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second-order uncertain system with unavailable velocity. Then, the applicability of the algorithm is extended, via suitable modifications, to the case of nonlinear systems with uncertainties of more general types. The proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold.

The effective application of sliding mode control to mechanical systems is not straightforward because of the sensitivity of these systems to chattering. Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable, so that the latter results are continuous. Generally, this approach requires the availability of a number of time derivatives of the sliding variable, and, in the presence of noise, this requirement could be a practical limitation. A class of second-order sliding mode controllers, guaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control, could be helpful both in reducing the number of differentiator stages in the controller and in dealing with unmodelled actuator dynamics. In this paper different second-order sliding mode controllers, previously presented in the literature, are shown to belong to the above cited class, and some challenging control problems involving mechanical systems are addressed and solved. Simulations and experimental results are provided throughout the paper.

A solution to the problem of eliminating the chattering effect, which is always associated with practical implementations of variable structure control, is presented with reference to a class of uncertain multi-input nonlinear systems. The solution procedure relies on the application of an original control approach capable of enforcing a second-order sliding mode (i.e., a sliding regime on a surface s[x(t)]=0 in the system state space, with s/spl dot/[x(t)] identically equal to zero, a regime enforced by a control signal depending on s[x(t)], but directly acting only on s/spl uml/[x(t)]). Such an approach, in its original formulation, only applies to single-input nonlinear systems with particular types of uncertainties. In the present paper, its validity is extended to multi-input nonlinear systems characterized by uncertainties of more general nature, covering a wide class of real processes.

A systematic approach to the chattering analysis in systems with second-order sliding modes is developed. The neglected actuator dynamics is considered to be the main cause of chattering in real systems. The magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second-order sliding-mode control generalized suboptimal (2-SMC G-SO) algorithms is evaluated. Sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding Poincare maps. For linear systems driven by 2-SMC G-SO algorithms, analysis tools based on the frequency-domain methods are developed. The first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions. The second technique represents a modified Tsypkin's method and provides for a relatively simple, theoretically exact, approach to evaluate the periodic motion parameters. Examples of analysis and simulation results are given throughout this paper.