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

Abstract

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 [5–8].

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., [9–16]). 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.

2. Problem Statement

Let us consider a second order sliding mode system in the form 𝑥̇₁=𝑥₂,𝑥̇₂=𝑓𝑡,𝑥₁,𝑥₂−𝑘sign𝑥₁(𝑡), (1) where 𝑓(𝑡,𝑥₁,𝑥₂) is smooth and 𝑘>0. The first issue to consider is getting sufficient conditions such that the system has a stable equilibrium point in 𝑥₁=0, 𝑥₂=0 and reached in finite time, considering a domain of attraction 𝐷𝐱=𝑥₁𝑥₂^𝑇∈𝐷. (2) Now, if conditions for stability are met for (1), when a delay is present in the control input as follows: 𝑥̇₁=𝑥₂,𝑥̇₂=𝑓𝑡,𝑥₁,𝑥₂−𝑘sign𝑥₁(𝑡−ℎ), (3) with 𝐱∈𝐷, ∀𝑡∈[−ℎ,0), 𝑥₁ 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 𝑥₁=0, 𝑥₂=0 in system (1) considering a domain of attraction.

Proposition 1. Consider system (1) with 𝑓(𝑡,0,0)=0. If 𝑘−|𝑓(𝑡,𝑥₁,𝑥₂)|≥𝑐₀>0 and, moreover, after 𝑥₁ first zero crossing, the following is met thereafter: 𝑐_𝑛𝑐_𝑛+1>𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂, (4) where 𝑡_𝑛 is the 𝑛th zero crossing time, 𝑐_𝑛 and 𝑐_𝑛+1 are the values for 𝑥₂(𝑡_𝑛) and 𝑥₂(𝑡_𝑛+1) in the 𝑛th and 𝑛+1th zero crossing time for 𝑥₁. Also, |𝑓_𝑛max(𝑡,𝑥₁,𝑥₂)| is the biggest value of |𝑓(𝑡,𝑥₁,𝑥₂)| in the interval 𝑡_𝑛≤𝑡<𝑡_𝑛+1 and |𝑓_𝑛+1max(𝑡,𝑥₁,𝑥₂)| is the biggest value of |𝑓(𝑡,𝑥₁,𝑥₂)| in the interval 𝑡_𝑛+1≤𝑡<𝑡_𝑛+2. Then, system (1) has a stable equilibrium point in 𝑥₁=0, 𝑥₂=0 which will be reached in finite time.

Proof. As a first step, let us demonstrate that there exists a finite time for 𝑥₁(𝑡) zero crossing. Without loss of generality, let us consider 𝑥₁(0)>0, therefore, in a neighborhood 𝑡⁺ of 𝑡=0 we will have 𝑥̇₁=𝑥₂,𝑥̇₂=𝑓𝑡,𝑥₁,𝑥₂−𝑘. (5) From hypothesis, 𝑘−|𝑓(0,𝑥₁(0),𝑥₂(0))|≥𝑐₀>0 and hence 𝑥̇₂≤−𝑐₀. (6) Therefore, 𝑥₂(𝑡)≤−𝑐₀𝑡+𝑥₂(0). (7) And so 𝑥₁(𝑡)𝑐≤−⁰𝑡²2+𝑥₂(0)𝑡+𝑥₁(0). (8) Hence, in a finite time, 𝑡₁, 𝑥₁(𝑡₁)=0, where 𝑡₁≤𝑥₂(0)+𝑥²₂(0)+2𝑐₀𝑥₁(0)𝑐₀. (9) 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 𝑛th 𝑥₁(𝑡) zero crossing goes from positive to negative, and therefore 𝑥₂(𝑡_𝑛)=𝑐_𝑛 with 𝑐_𝑛<0. Hence, for 𝑡_𝑛≤𝑡≤𝑡_𝑛+1, where 𝑡=𝑡_𝑛+1 is the next time for zero crossing, we will have 𝑥̇₁=𝑥₂,𝑥̇₂=𝑓𝑡,𝑥₁,𝑥₂+𝑘. (10) And therefore 𝑥₂(𝑡)≥𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂𝑡−𝑡_𝑛+𝑐_𝑛. (11) So that 𝑥₁(𝑡)≥𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂𝑡−𝑡_𝑛²2+𝑐_𝑛𝑡−𝑡_𝑛. (12) From the above, for the following zero crossing, 𝑥₁(𝑡_𝑛+1)=0, 𝑡_𝑛+1−𝑡_𝑛≤−2𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂. (13) To assure finite time for equilibrium point reaching, it only has to be guaranteed that (𝑡_𝑛+1−𝑡_𝑛)>(𝑡_𝑛+2−𝑡_𝑛+1) since if (𝑡_𝑛+1−𝑡_𝑛)=𝑇 and hence (𝑡_𝑛+2−𝑡_𝑛+1)≤𝑎_𝑛𝑇, where 0<𝑎_𝑛≤𝑎<1, total time will be ^∞_𝑛=1𝑡_𝑛+1−𝑡_𝑛≤𝑇+𝑎𝑇+𝑎²𝑇+⋯=𝑇^∞_𝑛=1𝑎^𝑛−11=𝑇.1−𝑎 (14) From (13), and a similar development for negative to positive zero crossing, the following can be deduced: 𝑡_𝑛+2−𝑡_𝑛+1+𝑡_𝑛+1−𝑡_𝑛≤2𝑐_𝑛+1𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂+2𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂. (15) Now, considering that to comply with the supposed sufficient condition for reaching the equilibrium point in finite time (𝑡_𝑛+1−𝑡_𝑛)=𝑇 and (𝑡_𝑛+2−𝑡_𝑛+1)≤𝑎_𝑛𝑇, the following must be complied as well: 1+𝑎_𝑛𝑡_𝑛+1−𝑡_𝑛≤2𝑐_𝑛+1𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂+2𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂. (16) So that, making 1𝑡_𝑛+1−𝑡_𝑛×2𝑐_𝑛+1𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂+2𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂<2, (17) it is guaranteed that 1+𝑎_𝑛<2.
Now, using again (13) together with (17) 𝑐_𝑛+1𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂+𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂<2𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂. (18) Therefore, it is sufficient 𝑐_𝑛+1𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂<𝑐_𝑛𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂. (19) In other words 𝑐_𝑛𝑐_𝑛+1>𝑘−𝑓_𝑛max𝑡,𝑥₁,𝑥₂𝑘−𝑓_𝑛+1max𝑡,𝑥₁,𝑥₂. (20)

Remark 2. If |𝑓_𝑛max(𝑡,𝑥₁,𝑥₂)| and |𝑓_𝑛+1max(𝑡,𝑥₁,𝑥₂)| are not known, instead of (20) we may use as a more conservative condition 𝑘−|𝑓(𝑡,𝑥₁,𝑥₂)|≥𝑐₀, together with 𝑐_𝑛𝑐_𝑛+1>1. (21) 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 𝑀ℎ≤min^′+𝑥1𝑀^′+_𝑥2,𝑀^′−_𝑥1𝑀^′−_𝑥2, (22) where 𝑀^′+_𝑥2=sup_{𝑡>0,𝐱∈𝐷+,0<𝛿<𝑘}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂𝑀+𝑘=2𝛿_′−𝑥2=inf_{𝑡>0,𝐱∈𝐷−,0<𝛿<𝑘}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂−𝑘=−2𝛿0<𝑀^′+_𝑥1≤𝑀⁺_𝑥1=sup𝑥₁>0∣𝑥₁𝑥₂=𝑀^′+_𝑥2∈𝐷0>𝑀^′−_𝑥1≥𝑀⁻_𝑥1=inf𝑥₁<0∣𝑥₁𝑥₂=𝑀^′−_𝑥2∈𝐷, (23) with the domain of attraction being 𝐱=𝑥₁𝑥₂^𝑇𝐷∈𝐷,₊=𝐷∣𝑥₁>0,𝑥₂𝐷≥0₋=𝐷∣𝑥₁<0,𝑥₂≤0. (24)

Proof. First, let us get an estimate for a bound on 𝑥₂ in the domain of attraction. For 0≤𝑡₀<𝑡₁<𝑡₂, without loss of generality, let us consider 0<𝑥₁(𝑡₁)≤𝑀⁺_𝑥1, 𝑥₁(𝑡₂)>𝑥₁(𝑡₁)>0>−𝛽=𝑥₁(𝑡₀), 𝛽→0, and 𝑡₁−𝑡₀=ℎ_max<𝑡₂−𝑡₀. Where ℎ_max is the maximum delay such that the system remains in the domain of attraction, so that ℎ≤ℎ_max. Then, in the interval 𝑡₁<𝑡≤𝑡₂, we have sign(𝑥₁(𝑡))=sign(𝑥₁(𝑡−ℎ))=1 and in the interval 𝑡₀<𝑡<𝑡₁, sign(𝑥₁(𝑡−ℎ))=−sign(𝑥₁(𝑡))=−1.
Therefore, with ℎ=ℎ_max and while 𝑡₀<𝑡<𝑡₁, we will have 𝑥₁(𝑡)>0 and 𝑥₁(𝑡−ℎ)<0, so that 𝑥̇₂=𝑓(𝑡,𝑥₁,𝑥₂)+𝑘. Moreover, since 𝑘−|𝑓(𝑡,𝑥₁,𝑥₂)|≥𝑐₀>0, it can be seen that 𝑥₂(𝑡)≥𝑐₀(𝑡−𝑡₀)+𝑥₂(𝑡₀), and accordingly 𝑥₁ will grow while sign(𝑥₁(𝑡−ℎ))=−1, but when 𝑡>𝑡₁ and since 𝑥₁>0, we have sign(𝑥₁(𝑡−ℎ))=1, and therefore 𝑥̇₂=𝑓(𝑡,𝑥₁,𝑥₂)−𝑘. Hence, 𝑥₂(𝑡)≤−𝑐₀(𝑡−𝑡₁)+𝑥₂(𝑡₁).
Therefore, there exists 𝑡=𝑡₂, where 𝑥₂(𝑡₂)=0, and accordingly when 𝑡=𝑡₂≤𝑡₁+𝑥₂(𝑡₁)/𝑐₀, 𝑥₁ arrives to a maximum and in the interval 𝑡₂<𝑡<𝑡₃, it will diminish until being zero at 𝑡=𝑡₃≤𝑥₂𝑡₁+𝑥₂𝑡₁²+2𝑐₀𝑡₁𝑥₂𝑡₁+𝑥₁𝑡₁𝑐₀+𝑡₁. (25) This happens provided the system is in the domain of attraction, so the maximum value must occur for 𝐱∈𝐷⁺. An estimate for the maximum 𝑥₂ reached value can be gotten noticing that 0<𝑐₀≤𝑘, since 𝑘−|𝑓(𝑡,𝑥₁,𝑥₂)|≥𝑐₀>0. Therefore, the biggest possible value for |𝑓(𝑡,𝑥₁,𝑥₂)|, inside the domain of attraction, occurs when 𝑐₀→0, making feasible that |𝑓(𝑡,𝑥₁,𝑥₂)|→𝑘. On the other hand, if 𝑐₀→𝑘, |𝑓(𝑡,𝑥₁,𝑥₂)| might only take the smallest possible value, |𝑓(𝑡,𝑥₁,𝑥₂)|→0. Hence, from the aforementioned, 0<𝑓(𝑡,𝑥₁,𝑥₂)+𝑘<2𝑘, and the maximum value for 𝑥₂, 𝑀^′+_𝑥2 inside the domain of attraction may be estimated as 𝑀^′+_𝑥2=sup_{𝑡>0,𝐱∈𝐷+,0<𝛿<𝑘}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂+𝑘=2𝛿. (26) Now, with a similar reasoning as above, it can be shown that once 𝑥₁(𝑡)<0, a time will come such that sign(𝑥₁(𝑡−ℎ))=−1, and if the dynamics are in the domain of attraction, 𝑥₁ will reach a minimum and begin moving toward the opposite direction remaining bounded. Therefore, an estimate for the minimum value of 𝑥₂ is 𝑀^′−_𝑥2=inf_{𝑡>0,𝐱∈𝐷−,0<𝛿<𝑘}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂−𝑘=−2𝛿. (27) Now, since in the interval 𝑡₀≤𝑡≤𝑡₁ (see Figure 1), we have 𝑥̇₁=𝑥₂≤𝑀^′+_𝑥2, and considering that 0<𝑀^′+_𝑥1≤𝑀⁺_𝑥1=sup𝑥₁>0∣𝑥₁𝑥₂=𝑀^′+_𝑥2∈𝐷, (28) we have, 𝑥₁(𝑡₁)≤𝑀^′+_𝑥1≤𝑀^′+_𝑥2ℎ_max. Therefore, it is clear that there exists ℎ^′≤ℎ_max such that 𝑀^′+_𝑥1=𝑀^′+_𝑥2ℎ^′. And then for ℎ≤ℎ^′, we have ℎ≤𝑀^′+_𝑥1/𝑀^′+_𝑥2. So that for this delay, the system dynamics remains in the domain of attraction for positive values of 𝑥₁.
Similarly, when 𝑥₁<0, 𝑥̇₁=𝑥₂≥𝑀^′−_𝑥2, and for ℎ_max=𝑡₄−𝑡₃, where 𝑡₄ is the time when 𝑥₁(𝑡₄)≥𝑀^′−_𝑥1≥𝑀^′−_𝑥2ℎ_max.
Considering 0>𝑀^′−_𝑥1≥𝑀⁻_𝑥1=inf{𝑥₁<0∣𝑥₁𝑥₂=𝑀^′−_𝑥2∈𝐷} and since 𝑀^′−_𝑥2<0, there exists ℎ^′′≤ℎ_max  such that 𝑀^′−_𝑥1=𝑀^′−_𝑥2ℎ^′′. Therefore, for ℎ≤ℎ^′′, ℎ≤𝑀^′−_𝑥1/𝑀^′−_𝑥2, so that, for this delay, the system dynamics remains in the domain of attraction for negative values of 𝑥₁.
Hence, if 𝑀ℎ≤min^′+𝑥1𝑀^′+_𝑥2,𝑀^′−_𝑥1𝑀^′−_𝑥2, (29) 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.

(a)

(b)

(c)

Figure 1 

Behavior of 𝑥1 for different forms of 𝑓(𝑡,𝑥1,𝑥2), where it can be seen as ℎ<𝑡1−𝑡0 and maximum value for 𝑥2=𝑀+𝑥2, when 0<𝑥1≤𝑀+𝑥1.

Remark 4. For a second order sliding mode system in the form (1), with a specified delay, ℎ, and noticing that the maximum 𝑥₁ amplitude is obtained when sign(𝑥₁(𝑡))=sign(𝑥₁(𝑡−ℎ)) and 𝑥₂=0, an estimate for the oscillation amplitude, 𝐴_𝑥1, can be obtained with 𝑚^′_𝑥2ℎ≤𝐴_𝑥1≤maxsup_{𝑡>0,𝐱∈𝐷+,0<𝛿<𝑘}𝑥₁∣𝑓𝑡,𝑥₁,0−𝑘=−2𝛿,sup_{𝑡>0,𝐱∈𝐷−,0<𝛿<𝑘}𝑥₁∣𝑓𝑡,𝑥₁,,0+𝑘=2𝛿 (30) where 𝑚^′_𝑥2 is the lowest magnitude value of 𝑥₂ for the interval when sign(𝑥₁(𝑡))=−sign(𝑥₁(𝑡−ℎ)). Now, noticing that 𝑥̇₂=0 is an extreme for 𝑥₂ in some cases, such lowest value for those cases may be obtained from 𝑚^′_𝑥2=mininf_{𝑡>0,𝐱∈𝐷+∣𝑥1(𝑡−ℎ)<0,𝑥2>0}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂+𝑘=0,inf_{𝑡>0,𝐱∈𝐷−∣𝑥1()𝑡−ℎ>0,𝑥2<0}𝑥₂∣𝑓𝑡,𝑥₁,𝑥₂,−𝑘=0 (31) verifying that 𝑥^..₂>0 when sign(𝑥₁(𝑡))=1 and 𝑥^..₂<0 when sign(𝑥₁(𝑡))=−1.

Remark 5. If the second order sliding mode system is of the form 𝑥̇₁=𝑥₂⋮𝑥̇_𝑛=𝑓𝑡,𝑥₁,𝑥₂−𝑘sign𝑠𝑥₁(𝑡−ℎ),…,𝑥_𝑛−1(𝑡−ℎ), (32) such that 𝑠̇=𝜂_..(𝐱)𝑠=𝜑(𝑡,𝐱)−𝑚sign(𝑠(𝑡−ℎ)), (33) so that 𝐱=0 is stable for ℎ=0, with the domain of attraction being 𝐱∈𝐷 and 𝐷^𝑠+={̇},𝐷𝐷∣𝑠>0,𝑠≥0_𝑠−={𝐷∣𝑠<0,𝑠̇≤0}, (34) an estimate of the maximum delay for steady oscillations in the sliding surface is 𝑀ℎ≤min^′+𝑠𝑀^′+_̇𝑠,𝑀^′−_𝑠𝑀^′−_̇𝑠, (35) where 𝑀^′+_̇𝑠=sup_{𝑡>0,𝐱∈𝐷𝑠+,0<𝛿<𝑚}̇()𝑀𝑠∣𝜑𝑡,𝐱+𝑚=2𝛿_′−̇𝑠=inf_{𝑡>0,𝐱∈𝐷𝑠−,0<𝛿<𝑚}𝑠̇∣𝜑(𝑡,𝐱)−𝑚=−2𝛿0<𝑀^′+_𝑠≤𝑀⁺_𝑠=sup𝑠>0∣𝑠𝑠̇=𝑀^′+_̇𝑠∈𝐷0>𝑀^′−_𝑠≥𝑀⁻_𝑠=inf𝑠<0∣𝑠𝑠̇=𝑀^′−_̇𝑠∈𝐷. (36)

Remark 6. If the system is a first order sliding mode system with the form 𝑥̇₁=𝑥₂⋮𝑥̇_𝑛=𝑓𝑡,𝑥₁,𝑥₂−𝑘sign𝑠𝑥₁(𝑡−ℎ),…,𝑥_𝑛(𝑡−ℎ), (37) such that 𝑠̇=𝜓(𝑡,𝐱)−𝑝sign(𝑠(𝑡−ℎ)), (38) so that 𝐱=0 is stable for ℎ=0, with the domain of attraction being 𝐱∈𝐷 and 𝐷^𝑠+={}𝐷𝐷∣𝑠>0_𝑠−={𝐷∣𝑠<0}, (39) an estimate for the maximum delay where steady oscillations in the sliding surface are obtained may be gotten with 𝑀ℎ≤min^′+𝑠𝑀^′+_̇𝑠,𝑀^′−_𝑠𝑀^′−_̇𝑠, (40) where 𝑀^′+_̇𝑠=sup_{𝑡>0,𝐱∈𝐷𝑠+,0<𝛿<𝑝}̇()𝑀𝑠∣𝜓𝑡,𝐱+𝑝≤2𝛿_′−̇𝑠=inf_{𝑡>0,𝐱∈𝐷𝑠−,0<𝛿<𝑝}𝑠̇∣𝜓(𝑡,𝐱)−𝑝≥−2𝛿0<𝑀^′+_𝑠≤𝑀⁺_𝑠=sup𝑠>0∣𝑠̇=𝑀^′+_̇𝑠∈𝐷0>𝑀^′−_𝑠≥𝑀⁻_𝑠=inf𝑠<0∣𝑠̇=𝑀^′+_̇𝑠∈𝐷. (41)

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, 𝐴_𝑠, of the sliding surface, 𝑠, can be obtained with 𝐴_𝑠≤max𝑀^′+_̇𝑠ℎ,𝑀^′−_̇𝑠ℎ. (42)

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: 𝑥₁̇=𝑥₂,𝑥₂̇=𝑥₁−5𝑥₂−5sign𝑥₁. (43) It is easy to verify that system dynamics are unstable if 5sign(𝑥₁) 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 𝑥₁>0. Since 𝑘=5, 𝑥̇₂>−𝑘=−5 must be complied and also 𝑥̇₂≤−𝑐₀, such that 𝑥₂>−5𝑡+𝑥₂(0), and 𝑥₁≤−𝑐₀𝑡²/2+𝑥₂(0)𝑡+𝑥₁(0).

Therefore, 𝑥̇₂≤−𝑐₀𝑡²/2+(25+𝑥₂(0))𝑡+𝑥₁(0)−5𝑥₂(0)−5. From this expression, it can be observed that the time when the biggest value for 𝑥̇₂ (less negative) is 𝑡=0, since for |𝑓(𝑡,𝑥₁,𝑥₂)|<5, it is required that |𝑥₂|<5, then from 𝑡=0, 𝑥̇₂ goes more negative while 𝑥₁>0. Similarly, it can be shown that if the initial condition is 𝑥₁<0, the lowest value for 𝑥̇₂ occurs at 𝑡=0, such that 𝑘−|𝑓(𝑡,𝑥₁,𝑥₂)|≥𝑐₀. Hence, a sure way to choose the initial conditions would be |𝑥₁(0)−5𝑥₂(0)|<5.

So that, for this example, we choose 𝑥₁(0)=4, 𝑥₂(0)=0. Therefore, at 𝑡=0, |𝑓(𝑡,𝑥₁,𝑥₂)|<5, and just afterwards, 𝑥₂ is and remains negative while 𝑥₁>0. Therefore 𝑥₁ goes down until reaching the first zero crossing. Knowing that 𝑥₁(𝑡)=−.9642𝑒^0.19258𝑡−0.03328𝑒^{−5.1958𝑡}+5, (44) it is easy to see that 𝑥₁=0, when 𝑥₂=−.93.

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 𝑡_𝑛. In the interval 𝑡_𝑛≤𝑡≤𝑡_𝑛+1, the maximum value for |𝑓(𝑡,𝑥₁,𝑥₂)| is at 𝑡_𝑛, since this is the initial time in this interval. Then 𝑓(𝑡_𝑛,0,𝑐_𝑛)=−5𝑐_𝑛 and at 𝑡=𝑡_𝑛+1, 𝑓(𝑡_𝑛+1,0,𝑐_𝑛+1)=−5𝑐_𝑛+1. From Proposition 1, it is needed 𝑐_𝑛𝑐_𝑛+1>𝑘−𝑓𝑡_𝑛,0,𝑐_𝑛𝑘−𝑓𝑡_𝑛+1,0,𝑐_𝑛+1=5−5𝑐_𝑛5−5𝑐_𝑛+1. (45) Resulting in the condition |𝑐_𝑛|>|𝑐_𝑛+1|.

Let’s see if the system meets such a condition. Meanwhile, 𝑥₁<0 in the interval 𝑡_𝑛≤𝑡≤𝑡_𝑛+1, 𝑥₁(𝑡)=𝐴𝑒^{0.19258(𝑡−𝑡𝑛)}+𝐵𝑒^{−5.1958(𝑡−𝑡𝑛)}𝑥−5₂(𝑡)=0.19258𝐴𝑒^{0.19258(𝑡−𝑡𝑛)}−5.1958𝐵𝑒^{−5.1958(𝑡−𝑡𝑛)}, (46) with 𝐴=5−0.1958𝑐_𝑛,1.037𝐵=5+5.1958𝑐_𝑛.27.963 (47) Now, considering that in the following zero crossing, 𝑡=𝑡_𝑛+1, 0=𝐴𝑒^{0.19258(𝑡𝑛+1−𝑡𝑛)}+𝐵𝑒^{−5.1958(𝑡𝑛+1−𝑡𝑛)}𝑐−5,_𝑛+1=0.19258𝐴𝑒^{0.19258(𝑡𝑛+1−𝑡𝑛)}−5.1958𝐵𝑒^{−5.1958(𝑡𝑛+1−𝑡𝑛)}, (48) and solving (48), using expressions for 𝐴 and 𝐵 in (46), 𝑐_𝑛+1=0.963−0.963𝑒^{−5.1958(𝑡𝑛+1−𝑡𝑛)}−𝑒^{−5.1958(𝑡𝑛+1−𝑡𝑛)}𝑐_𝑛. (49) This is an asymptotically stable first order difference equation for 𝑐_𝑛 dynamics, since 𝑡_𝑛+1>𝑡_𝑛, giving the eigenvalue magnitude less than one and guaranteeing that |𝑐_𝑛+1|<|𝑐_𝑛|, complying with the condition for reaching the surface in finite time. Eventually, when 𝑡_𝑛+1=𝑡_𝑛, the equilibrium point is reached, 𝑐_𝑛=0. Therefore, system dynamics are stable and reach the surface in finite time without need of using a measure or estimate of 𝑥₂ for the control signal. In Figure 2, the behavior of 𝑥₁ and 𝑥₂ is shown using 𝑥₁(0)=4, 𝑥₂(0)=0. It is seen that effectively, 𝑥₁ approaches smoothly and in finite time to zero and chattering is confined to 𝑥₂, which also approaches in finite time to zero.

Figure 2 

Behavior of system (43) dynamics showing that 𝑥1=0 and 𝑥2=0 are reached in finite time. As expected for second order sliding mode, chattering is not present in 𝑥1.

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: 𝑥̇₁=𝑥₂𝑥̇₂=𝑥₁−5𝑥₂−5sign𝑥₁(𝑡−ℎ). (50) Now, using (22), 𝑀^′_𝑥1−5𝑀^′_𝑥2+5=2𝛿, which gives the maximum with 𝛿=0𝑀^′_𝑥2≤𝑀^′_𝑥1+55. (51) The domain of attraction is shown in Figure 3, together with all possible values for 𝑀^′_𝑥2. It is noticed that at 𝑥₁=4.5, 𝑀^′_𝑥2=1.9 intersects the upper limit of the domain of attraction. Therefore 𝑀ℎ<^′𝑥1𝑀^′_𝑥2=4.51.9=2.368. (52) In this example, ℎ=2.36 was used along with initial conditions set to 𝑥₁=2 and 𝑥₂=0, in order to ensure the initial crossing by zero since for 𝑡∈[0,ℎ] no control is available. To estimate the oscillation amplitude, Remark 4 of Proposition 3 is used. To obtain 𝑚^′_𝑥2, we have 𝑥₁−5𝑥₂+5=0. And so 𝑚^′_𝑥2=min_𝑥1>0((5+𝑥₁)/5)=1. Therefore, 𝐴_𝑥1≥𝑚^′_𝑥2ℎ=2.36. Now, the biggest value for 𝑥₁ is obtained from 𝑥₁−5=−2𝛿, so that 𝐴_𝑥1≤max_0≤𝛿<5(5−2𝛿)=5.

Figure 3 

Estimate for biggest value of 𝑥2 in the system with delay (50) inside the domain of attraction.

Hence, the oscillation amplitude for 𝑥₁ 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.

Figure 4 

Behavior of dynamics for system (50) with delay ℎ=2.36.

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 𝑥̇₁=𝑥₂,𝑥̇₂=−5𝑥₂−5sign𝑥₁(𝑡−ℎ). (53) Using Remark 4 of Proposition 3, the inferior bound for amplitude 𝐴_𝑥1 with 𝑚^′_𝑥2=5/5=1 is 𝐴_𝑥1>ℎ𝑚^′_𝑥2=ℎ. If ℎ=5, then 𝐴>5. In Figure 5, the system response for 𝑥₁ is seen. The oscillation amplitude is a little bigger than 5, verifying the estimated value. Also, since 𝑥₂ 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 𝑥₁ using Remark 4 cannot be obtained, since 𝑥₁ is not explicit in 𝑓(𝑡,𝑥,0)−𝑘=−2𝛿.

Figure 5 

Behavior of globally stable system (53) with a delay ℎ=5.

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.