Stability Analysis of a Class of Second Order Sliding Mode Control Including Delay in Input

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 slidingmode systemsmight have amplitude bounded oscillations.Moreover, amplitude of such oscillationsmay 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.


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 [5][6][7][8].
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., [9][10][11][12][13][14][15][16]).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.

Problem Statement
Let us consider a second order sliding mode system in the form where (,  1 ,  2 ) is smooth and  > 0. The first issue to consider is getting sufficient conditions such that the system has a stable equilibrium point in  1 = 0,  2 = 0 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 x ∈ , ∀ ∈ [−ℎ, 0),  1 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.

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  1 = 0,  2 = 0 in system (1) considering a domain of attraction.
Therefore, there exists  =  2 , where  2 ( 2 ) = 0, and accordingly when  =  2 ≤  1 +  2 ( 1 )/ 0 ,  1 arrives to a maximum and in the interval  2 <  <  3 , 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 x ∈  + .inside the domain of attraction may be estimated as Now, with a similar reasoning as above, it can be shown that once  1 () < 0, a time will come such that sign( 1 ( − ℎ)) = −1, and if the dynamics are in the domain of attraction,  1 will reach a minimum and begin moving toward the opposite direction remaining bounded.Therefore, an estimate for the minimum value of  2 is Now, since in the interval  0 ≤  ≤  1 (see Figure 1), we have , and considering that we have,  1 ( 1 ) ≤  +  1 ≤  +  2 ℎ max .Therefore, it is clear that there exists ℎ  ≤ ℎ max such that  +  1 =  +  2 ℎ  .And then for ℎ ≤ ℎ  , we have ℎ ≤  +  1 / +  2 .So that for this delay, the system dynamics remains in the domain of attraction for positive values of  1 .
Similarly, when  1 < 0, ẋ 1 =  2 ≥  −  2 , and for ℎ max =  4 −  3 , where  4 is the time when , so that, for this delay, the system dynamics remains in the domain of attraction for negative values of  1 .
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, ℎ, and noticing that the maximum  1 amplitude is obtained when sign( 1 ()) = sign( 1 ( − ℎ)) and  2 = 0, an estimate for the oscillation amplitude,   1 , can be obtained with where    2 is the lowest magnitude value of  2 for the interval when sign( 1 ()) = − sign( 1 ( − ℎ)).Now, noticing that ẋ 2 = 0 is an extreme for  2 in some cases, such lowest value for those cases may be obtained from such that ṡ =  (x) ..
so that x = 0 is stable for ℎ = 0, with the domain of attraction being x ∈  and an estimate of the maximum delay for steady oscillations in the sliding surface is where , where it can be seen as ℎ <  1 −  0 and maximum value for Remark 6.If the system is a first order sliding mode system with the form so that x = 0 is stable for ℎ = 0, with the domain of attraction being x ∈  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, ℎ, an upper bound for the oscillation amplitude,   , of the sliding surface, , can be obtained with

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

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 5 sign( 1 ) 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  1 > 0. Since  = 5, ẋ 2 > − = −5 must be complied and also ẋ 2 ≤ − 0 , such that  2 > −5 +  2 (0), and From this expression, it can be observed that the time when the biggest value for ẋ 2 (less negative) is  = 0, since for |(,  1 ,  2 )| < 5, it is required that | 2 | < 5, then from  = 0, ẋ 2 goes more negative while  1 > 0. Similarly, it can be shown that if the initial condition is  1 < 0, the lowest value for ẋ 2 occurs at  = 0, such that  − |(,  1 ,  2 )| ≥  0 .Hence, a sure way to choose the initial conditions would be | 1 (0) − 5 2 (0)| < 5.
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   ≤  ≤  +1 , the maximum value for |(,  1 ,  2 )| is at   , since this is the initial time in this interval.Then (  , 0,   ) = −5  and at  =  +1 , ( +1 , 0,  +1 ) = −5 +1 .From Proposition 1, it is needed This is an asymptotically stable first order difference equation for   dynamics, since  +1 >   , giving the eigenvalue magnitude less than one and guaranteeing that | +1 | < |  |, complying with the condition for reaching the surface in finite time.Eventually, when  +1 =   , the equilibrium point is reached,   = 0. Therefore, system dynamics are stable and reach the surface in finite time without need of using a measure or estimate of  2 for the control signal.In Figure 2, the behavior of  1 and  2 is shown using  1 (0) = 4,  2 (0) = 0.
It is seen that effectively,  1 approaches smoothly and in finite time to zero and chattering is confined to  2 , which also approaches in finite time to zero.

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: (50) The domain of attraction is shown in Figure 3, together with all possible values for    2 .It is noticed that at  1 = 4.5,    2 = 1.9 intersects the upper limit of the domain of attraction.Therefore In this example, ℎ = 2.36 was used along with initial conditions set to  1 = 2 and  2 = 0, in order to ensure the initial crossing by zero since for  ∈ [0, ℎ] no control is available.To estimate the oscillation amplitude, Remark 4 of Hence, the oscillation amplitude for  1 is bounded with a value between 2.36 and 5.In Figure 4, it can be verified that ℎ = 2.36 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.

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   1 with    2 = 5/5 = 1 is   1 > ℎ   2 = ℎ.If ℎ = 5, then  > 5.In Figure 5, the system response for  1 is seen.The oscillation amplitude is a little bigger than 5, verifying the estimated value.Also, since  2 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  1 using Remark 4 cannot be obtained, since  1 is not explicit in (, , 0) −  = −2.

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.

Figure 2 :Figure 3 :
Figure 2: Behavior of system (43) dynamics showing that  1 = 0 and  2 = 0 are reached in finite time.As expected for second order sliding mode, chattering is not present in  1 .
1 ,  2 )| ≥  0 > 0 and, moreover, after  1 first zero Remark 2. If |  max (,  1 ,  2 )| and | +1 max (,  1 ,  2 )| are not known, instead of (20) we may use as a more conservative 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.