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.

1. Introduction

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 [1], suboptimal controller [2], and the super twisting controller [3]. These algorithms need to measure or estimate the derivative of the sliding surface. The super twisting algorithm is used to get robust differentiators, so that it can be used together with the other algorithms to get such a derivative. Recently [4], homogeneity has been used to ease parameter design for second order sliding mode controllers. Some recent examples of engineering applications using second order sliding mode controllers are [58].

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., [916]). In [12], a good review is given for different control techniques up to 2003. When only a delayed measure is available for the control signal, two techniques have been used to compensate delay effects: predictors and integral sliding mode control. Some instances are [11, 12, 14], respectively. On the other hand, concerning analysis for existence and amplitude of steady oscillations due to delays, very few works have been reported, such as [17], where sufficient conditions are given for steady oscillations to exist in a first order sliding mode system; describing function and Poincaré map approach are used in [18] to analyze oscillations for a system with hysteresis. In [19], a different approach for perturbed first sliding mode systems is reported. For second order sliding mode systems with delay, [20] reported sufficient conditions for steady oscillations to occur in a class of systems; in [21], an analysis is given about oscillations for the suboptimal algorithm considering delay in the input.

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 [20] and using a different approach. In Section 2, the problem is formulated. Section 3 presents the main results by way of two propositions and their proofs, giving sufficient conditions for finite time sliding motion stability without delay and stability for oscillations when delay in input occurs. Next, in Section 4, some numerical examples are given and conclusions for the reported results are related.

2. Problem Statement

Let us consider a second order sliding mode system in the form where is smooth and . The first issue to consider is getting sufficient conditions such that the system has a stable equilibrium point in , and reached in finite time, considering a domain of attraction Now, if conditions for stability are met for (1), when a delay is present in the control input as follows: with , , might have steady oscillations if the time delay is sufficiently small, such that the system remains in the attraction domain. Otherwise, its dynamics will be unstable. Hence, it is important to get sufficient conditions for delay size, such that the system remains in the domain of attraction and an estimate for the amplitude of the oscillations. Next section shows sufficient conditions to reach the surface in finite time without delay and for existence of bounded amplitude oscillations in presence of delay.

3. Sufficient Conditions for Second Order Sliding Modes without Delay and Amplitude Bounded Oscillations with Delay

In the following proposition, sufficient conditions are given for finite time reaching and, stability of the equilibrium point , in system (1) considering a domain of attraction.

Proposition 1. Consider system (1) with . If and, moreover, after first zero crossing, the following is met thereafter: where is the th zero crossing time, and are the values for and in the th and th zero crossing time for . Also, is the biggest value of in the interval and is the biggest value of in the interval . Then, system (1) has a stable equilibrium point in , which will be reached in finite time.

Proof. As a first step, let us demonstrate that there exists a finite time for zero crossing. Without loss of generality, let us consider , therefore, in a neighborhood of we will have From hypothesis, and hence Therefore, And so Hence, in a finite time, , , where Now, as a second step, let us demonstrate by induction that in a finite time the equilibrium point is reached. Without loss of generality, assume that the th zero crossing goes from positive to negative, and therefore with . Hence, for , where is the next time for zero crossing, we will have And therefore So that From the above, for the following zero crossing, , To assure finite time for equilibrium point reaching, it only has to be guaranteed that since if and hence , where , total time will be From (13), and a similar development for negative to positive zero crossing, the following can be deduced: Now, considering that to comply with the supposed sufficient condition for reaching the equilibrium point in finite time and , the following must be complied as well: So that, making it is guaranteed that .
Now, using again (13) together with (17) Therefore, it is sufficient In other words

Remark 2. If and are not known, instead of (20) we may use as a more conservative condition , together with When a delay exists in the discontinuous control signal, the equilibrium point cannot be reached. If the dynamics are in the domain of attraction, amplitude bounded oscillations are obtained. Next proposition presents sufficient conditions for an upper bound on the delay in order to get amplitude bounded oscillations.

Proposition 3. Considering system (1), complying with conditions of Proposition 1, a sufficient condition such that system (3) has steady oscillations is where with the domain of attraction being

Proof. First, let us get an estimate for a bound on in the domain of attraction. For , without loss of generality, let us consider , , , and . Where is the maximum delay such that the system remains in the domain of attraction, so that . Then, in the interval , we have and in the interval , .
Therefore, with and while , we will have and , so that . Moreover, since , it can be seen that , and accordingly will grow while , but when and since , we have , and therefore . Hence, .
Therefore, there exists , where , and accordingly when , arrives to a maximum and in the interval , it will diminish until being zero at This happens provided the system is in the domain of attraction, so the maximum value must occur for . An estimate for the maximum reached value can be gotten noticing that , since . Therefore, the biggest possible value for , inside the domain of attraction, occurs when , making feasible that . On the other hand, if , might only take the smallest possible value, . Hence, from the aforementioned, , and the maximum value for , inside the domain of attraction may be estimated as Now, with a similar reasoning as above, it can be shown that once , a time will come such that , and if the dynamics are in the domain of attraction, will reach a minimum and begin moving toward the opposite direction remaining bounded. Therefore, an estimate for the minimum value of is Now, since in the interval (see Figure 1), we have , and considering that we have, . Therefore, it is clear that there exists such that . And then for , we have . So that for this delay, the system dynamics remains in the domain of attraction for positive values of .
Similarly, when , , and for , where is the time when .
Considering and since , there exists   such that . Therefore, for , , so that, for this delay, the system dynamics remains in the domain of attraction for negative values of .
Hence, if the system dynamics will stay inside the domain of attraction; so this is a sufficient condition in the delay size for amplitude bounded oscillations to exist.

Remark 4. For a second order sliding mode system in the form (1), with a specified delay, , and noticing that the maximum amplitude is obtained when and , an estimate for the oscillation amplitude, , can be obtained with where is the lowest magnitude value of for the interval when . Now, noticing that is an extreme for in some cases, such lowest value for those cases may be obtained from verifying that when and when .

Remark 5. If the second order sliding mode system is of the form such that so that is stable for , with the domain of attraction being and an estimate of the maximum delay for steady oscillations in the sliding surface is where

Remark 6. If the system is a first order sliding mode system with the form such that so that is stable for , with the domain of attraction being and an estimate for the maximum delay where steady oscillations in the sliding surface are obtained may be gotten with where

Remark 7. 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 6), together with the delay, , an upper bound for the oscillation amplitude, , of the sliding surface, , can be obtained with

4. Examples

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

4.1. Reaching Equilibrium Point in Finite Time for Second Order Sliding Modes

Let us consider the following system: It is easy to verify that system dynamics are unstable if is not present. In order to verify if the dynamics of (43) reach in finite time the equilibrium point in zero, Proposition 1 is used. Let’s consider . Since , must be complied and also , such that , and .

Therefore, . From this expression, it can be observed that the time when the biggest value for (less negative) is , since for , it is required that , then from , goes more negative while . Similarly, it can be shown that if the initial condition is , the lowest value for occurs at , such that . Hence, a sure way to choose the initial conditions would be .

So that, for this example, we choose , . Therefore, at , , and just afterwards, is and remains negative while . Therefore goes down until reaching the first zero crossing. Knowing that it is easy to see that , when .

Up to now, first conditions for the Proposition 1 are met in order to get the first zero crossing. Now, to see if the condition for finite time surface reaching is met, let us consider an arbitrary time of zero crossing from positive to negative . In the interval , the maximum value for is at , since this is the initial time in this interval. Then and at , . From Proposition 1, it is needed Resulting in the condition .

Let’s see if the system meets such a condition. Meanwhile, in the interval , with Now, considering that in the following zero crossing, , and solving (48), using expressions for and in (46), This is an asymptotically stable first order difference equation for dynamics, since , giving the eigenvalue magnitude less than one and guaranteeing that , complying with the condition for reaching the surface in finite time. Eventually, when , the equilibrium point is reached, . Therefore, system dynamics are stable and reach the surface in finite time without need of using a measure or estimate of for the control signal. In Figure 2, the behavior of and is shown using , . It is seen that effectively, approaches smoothly and in finite time to zero and chattering is confined to , which also approaches in finite time to zero.

4.2. Maximum Permissible Delay in the Second Order Sliding Mode System

Knowing that sliding modes are gotten for system (43), now a delay in the control is introduced as shown below: Now, using (22), , which gives the maximum with The domain of attraction is shown in Figure 3, together with all possible values for . It is noticed that at , intersects the upper limit of the domain of attraction. Therefore In this example, was used along with initial conditions set to and , in order to ensure the initial crossing by zero since for no control is available. To estimate the oscillation amplitude, Remark 4 of Proposition 3 is used. To obtain , we have . And so . Therefore, . Now, the biggest value for is obtained from , so that .

Hence, the oscillation amplitude for is bounded with a value between 2.36 and 5. In Figure 4, it can be verified that is a sufficient condition for bounded oscillations to occur. Also, the amplitude is seen to be a little bigger than 3, which is inside the estimated limits. For this case the describing function approach cannot be used to estimate the amplitude, because of the instability of the associated linear system.

4.3. Amplitude Estimate for a Globally Stable System

Considering a case where the equilibrium point of the involved system is globally stable such as Using Remark 4 of Proposition 3, the inferior bound for amplitude with is . If , then . In Figure 5, the system response for is seen. The oscillation amplitude is a little bigger than 5, verifying the estimated value. Also, since is almost constant, the estimated amplitude is very close to the real one. If the describing function approach was used, the estimated amplitude would be around 4.44, which is lower than the minimum estimated with the proposed method. In this particular case, an estimate of the superior bound for using Remark 4 cannot be obtained, since is not explicit in .

5. Conclusions

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 [20], where sufficient conditions are given for stationary oscillations to exist using another approach and no means is given to estimate amplitude of oscillations. The analysis presented in this paper is based on two propositions. One of them gives sufficient conditions for sliding mode existence for a class of second order sliding mode systems. The other can be used as a method based on maximum derivative to get a delay estimate on the control signal for a class of second order sliding mode systems, giving sufficient conditions to maintain amplitude bounded oscillations on the sliding surface. Also, for a given delay in input, oscillation amplitude for the sliding surface dynamics can be estimated if the system state is in the attraction region. On the other hand, when oscillation amplitude is known, such as hysteresis happening in the discontinuous input, frequency of oscillation can be estimated. Numerical examples for second order sliding modes were presented to illustrate the proposed approach, first two examples have unstable dynamics when no input is applied. Third example has global stability for the considered input. It is clear that very conservative results using this method would arise for systems where the expected sliding surface derivative is much lower than the maximum derivative. For linear stable processes, the describing function method has been used in the literature to estimate oscillations amplitude. In the second example such approach cannot be used since the linear process is unstable. In the third example, the simulation shows the amplitude just a little bigger than the minimum estimated by the approach presented here. For this example, the describing function method gives a lower estimate than the minimum given by the proposed approach.