This paper deals with a class of second order sliding mode systems. Based on the derivative of the sliding surface, sufficient conditions are given for stability. However, the discontinuous control signal depend neither on the derivative of sliding surface nor on its estimate. Time delay in control input is also an important issue in sliding mode control for engineering applications. Therefore, also sufficient conditions are given for the time delay size on the discontinuous input signal, so that this class of second order sliding mode systems might have amplitude bounded oscillations. Moreover, amplitude of such oscillations may be estimated. Some numerical examples are given to validate the results. At the end, some conclusions are given on the possibilities of the results as well as their limitations.

Sliding mode control has been effectively used in engineering for more than three decades. One of the major concerns for some applications is chattering in output or control signal. Along with other techniques, higher order sliding modes have been proposed to alleviate chattering. Most used algorithms for high sliding motion are the twisting controller [

When delays arise, sliding mode controller performance deteriorates. Such delays may occur in states and/or inputs and have been treated using several approaches (see e.g., [

Motivated by the aforementioned analysis, this paper presents sufficient stability conditions for a class of sliding mode system that does not require a measure of the derivative of the sliding surface in order to provide the control input. Also, sufficient conditions are given for stable oscillations to exist if a delay occurs in the input, as well as an estimate for their amplitude. The class of systems treated here is a more general class of systems than the one reported in [

Let us consider a second order sliding mode system in the form

In the following proposition, sufficient conditions are given for finite time reaching and, stability of the equilibrium point

Consider system (

As a first step, let us demonstrate that there exists a finite time for

Now, using again (

If

Considering system (

First, let us get an estimate for a bound on

Therefore, with

Therefore, there exists

Similarly, when

Considering

Hence, if

Behavior of

For a second order sliding mode system in the form (

If the second order sliding mode system is of the form

If the system is a first order sliding mode system with the form

For a first order sliding mode system with amplitude bounded oscillations, if an upper bound for the sliding surface derivative magnitude is known (see Remark

In this section, three examples are given to validate the results presented above.

Let us consider the following system:

Therefore,

So that, for this example, we choose

Up to now, first conditions for the Proposition

Let’s see if the system meets such a condition. Meanwhile,

Behavior of system (

Knowing that sliding modes are gotten for system (

Estimate for biggest value of

Hence, the oscillation amplitude for

Behavior of dynamics for system (

Considering a case where the equilibrium point of the involved system is globally stable such as

Behavior of globally stable system (

Stability analysis for a class of second order sliding mode systems, which does not need measuring or estimation of the derivative of the sliding variable for the control signal, was presented. Also, when delay in input is present, bounded oscillations are studied. The system can be linear or nonlinear. The class of systems treated is a more general class of the systems studied in [