Open access peer-reviewed chapter - ONLINE FIRST

An In-Depth Analysis of Sliding Mode Control and Its Application to Robotics

By İhsan Ömür Bucak

Submitted: March 10th 2020Reviewed: May 26th 2020Published: July 16th 2020

DOI: 10.5772/intechopen.93027

Downloaded: 21

Abstract

In this study, a sliding mode control scheme with a bounded region and its convergence analysis are explained to the finest detail and are applied to robotic manipulators which represent the best examples for strongly coupled, highly nonlinear, time-varying dynamical systems. Simulation studies have been applied separately to two different control systems in order to demonstrate the feasibility, performance, and effectiveness of the proposed control methodology through the design of the sliding mode controller: firstly, the position control of an armature-controlled dc servo motor subject to a varying external disturbance, and secondly, a two-link robot manipulator that were also analyzed in terms of its robustness by adding extra mass to one of the joints to be able to maintain the trajectory in the sliding surface. Simulations show that a fast convergence rate, and therefore quick response, the ability to reject the varying external disturbances, and the robustness against the model uncertainty assumed to be unbounded and fast-varying have all achieved its purpose entirely. This study also examines the advantages of SMC and PID comparably. The results given here do not contradict the view that one can use it instead of the other without losing too much performance, and confirm the success of the presented approach.

Keywords

  • robotic manipulators
  • sliding mode control
  • convergence analysis
  • trajectory tracking
  • robustness
  • two-link planar robot manipulator
  • PID control

1. Introduction

Unknown plant parameters or, more generally, plant uncertainty and the preferences in which the system dynamics are purposely represented by simplifications, such as the use of linearized friction model, lead to model imprecision [1]. Control engineering classifies the model inaccuracies, which were used here as synonym of imprecision, into two main categories as structured and unstructured uncertainties. The first one implies inaccuracies within the model and the second one corresponds to inaccuracies on the system order (i.e., underestimated system order). Modeling inaccuracies can have adverse effect on nonlinear control systems [1].

Robotic manipulators represent the best examples for strongly coupled, highly nonlinear, time-varying dynamical systems [2]. These qualities alongside structured uncertainties caused by model imprecision of link parameters and payload variation, and unstructured uncertainties produced by unmodeled dynamics such as nonlinear friction, compliance in gearing, sensor noise, external disturbances, and the high-frequency part of the dynamics turn the motion control of rigid-link manipulators into a complex problem [2]. Particularly, robotic manipulators suffer so much from these structured and unstructured uncertainties. The consequence of having to deal with various uncertainties in their dynamics and the necessity to manage the various tools and, hence, the variation of dynamic parameters during operation make it difficult for robots to introduce a mathematical model suitable for employing model-based control strategies.

The theory of conventional sliding mode control (SMC) as a simple robust nonlinear control scheme has been applied to robotic manipulators successfully since the 1980s. In those studies, the advantages of the SMC properties such as its robustness against disturbances and variation of parameters, and its fast dynamic response have been utilized greatly. Two important approaches, such as robust control and adaptive control, can deal with modeling uncertainty [1]. Particularly, adaptive control is effective in solving the structured and unstructured uncertainties and is capable of maintaining a uniformly good performance over a limited range.

SMC as a special class of the variable structure systems (VSS) has been preferred in practical applications for over 50 years due to its simplicity and robustness against parameter variations and disturbances [3]. VSS concept was first evolved from the pioneering work of Emel’yanov and Barbashin in the early 1960s in Russia [4]. Especially, VSS and SMC have received a great attention by the control research community worldwide since the published 1977 article [4]. SMC methodology is used to design a control law that imposes all system trajectories to converge on a surface in the state space, the so-called sliding surface St. The designer chooses the dynamics of this surface so that all trajectories will asymptotically converge to the set point. When the trajectory lies inside the sliding surface, the system operates in so-called sliding mode and is sensitive to parametric variations and external disturbances [5]. Control action in sliding mode is discontinuous by nature and can stimulate high-frequency dynamics [6]. Discontinuous nature of the control action serves to maintain a resulting superior system performance of VSS and SMC by switching between two incomparably different system structures such that sliding mode with this feature is also referred to as a new type of system motion in a manifold, or in another substantial terms, in the vicinity of a prescribed switching manifold, the velocity vector of the controlled state trajectories is always directed toward the switching manifold by such motion induced by imposing discontinuous control actions [3, 6]. This system performance is expected to exhibit insensitivity to parameter variations as well as demonstrate complete disturbance rejection [6].

Despite its advantages such as simplicity and robustness, SMC suffers from a rather widespread and well-known chattering problem, which is generally perceived as motion oscillating about the predefined switching manifold(s) [3, 6]. There are two reasons behind the chattering phenomenon: first, under the absence of switching nonidealities such as delays, that is, in a situation where the switching device ideally switches at an infinite frequency, the presence of parasitic dynamics in series with the plant causes a small amplitude high-frequency oscillation to occur around the sliding manifold. If the closed loop pole locations are well defined or the closed loop poles are well assigned with the aid of the pole placement design technique, these parasitic dynamics which represent the fast actuator and sensor dynamics are often neglected in the open loop model used for control design in control applications. In general, the motion of the real system is closer to an ideal system where the parasitic dynamics are neglected, and the difference between the ideal and the real motion, which is at negligible time constants, shows a rapid decline. However, the parasitic dynamics interacted with variable structure control (VSC) in particular produce a nondecreasing oscillatory component with a finite amplitude and a frequency, referred previously to as chattering phenomenon already. Second, the switching nonidealities alone can cause such high-frequency oscillations around the sliding surface St[3, 6]. These may include small time delays due to sampling (e.g., zero-order hold), execution time required to calculate the control, and transmission delays in networked control systems [3]. As time delays cause the resulting chattering phenomenon, and the delay type of switching nonidealities is the most relevant to any electronic implementation of the switching device whether it includes analogue or digital circuits, delay type nonidealities are considered in general for the design approaches, and discrete-time control design techniques are the most commonly used design approaches by control engineers to mitigate the chattering caused by the switching delays [6]. Many design methodologies have been applied so far to mitigate or reduce the chattering. To eliminate the chattering, a boundary layer around the sliding surface Sthas been introduced [1]. Inside the boundary layer, the switching function is replaced by a high-gain proportional control [5]. In order to reduce the chattering level, several switching functions such as saturation functions, relay functions, hyperbolic functions, and hysteresis saturation functions have been used in literature. The use of these functions can minimize or, if desired, completely eliminate the chattering, but it turns perfect tracking into a tracking with a guaranteed precision problem, meaning that a steady-state error is always maintained.

SMC design procedure is split into two major steps corresponding to the two main phases [3]: reaching phase is defined to derive the system state from initial state to reach the switching manifolds in finite time; and sliding-mode phase is defined to induce the system into the sliding motion on the switching manifolds like an attractor.

No matter how active the research on SMC has been during the last 50 years, the key technical challenges such as chattering, the elimination of the effects caused by unmodeled dynamics, disturbances and uncertainties, adaptive learning, and improved robustness can still remain to be addressed to reach a perfect solution [3]. An ideal sliding mode can only be achieved when the dynamic equation governing the sliding mode is satisfied by the system state for all time. This implies an infinite switching to assure the sliding motion [3]. Although the switching rate of the switching control device of the SMC system (design) is infinite ideally, it is much lower than that in practice due to the physical limitations of switching [6]. Physical limitations of switching have been tried to be explained in the previous paragraphs.

Usually, intelligent control approaches can mitigate the effects of structured parametric uncertainty and unstructured disturbance with their effective learning ability without requiring a detailed knowledge of the controlled plant within the design process. SMC research has recently been integrated with intelligent control approaches such as neural networks, fuzzy logic, genetic algorithms, and probabilistic reasoning, just a few of them, to make it more intelligent [3, 7, 8, 9, 10, 11]. Another goal behind the combination of the intelligent control with the attractive features of this traditional control is to create more powerful control algorithms. Nevertheless, it appears that many intelligent control algorithms do not take into account actuator dynamics in robot control systems, which play a critical role in overall robot dynamics and their negligence can cause adverse effects, especially in the case of high-speed torque, respectable load variations, friction, and actuator saturations [2]. Electrical actuators are very much controllable than others and are more suitable for driving robot manipulators [2, 12].

Sliding mode control strategy is the simple approach to robust control. By intuition, controlling first-order systems is much easier than controlling general nth-order systems, even if they are nonlinear or uncertain. Therefore, an introduction of a notational simplification allows nth-order problems to be replaced by equivalent first-order problems. Thus, it is then easy to demonstrate in principle the achievement of perfect performance under the presence of arbitrary parameter inaccuracies for the transformed problems. However, such performance is only achieved at the expense of extremely high control activity. This is typically disputed by the other source of modeling uncertainty, such as the presence of neglected dynamics, which the high control activity can stimulate. This leads to a change in control laws aimed at achieving an effective trade-off between tracking performance and parametric uncertainty, given the acceptable control activity [1].

The concepts will first be presented for systems that have a single control input that allows us to develop an intuition about the fundamental aspects of nonlinear controller design.

2. Sliding surfaces

Consider the single-input dynamic system given below:

xn=fx+bxu,E1

where the scalar xis the output we are interested in, the scalar uis called the control input, and x=xx·xn1Trepresents the state vector. While the function fxin (1) is not exactly known, an upper bound on fxis set by a known continuous function of xto limit the size of the imprecision. Similarly, the control gain bxis not exactly known either, only its sign is known; therefore, it is also bounded by the known continuous functions of x. The control problem is to ensure that the state xtracks an explicit time-varying state xd=xdx·dxdn1Tin the presence of model imprecision on fxand bx. In order for the tracking task to be achieved using a finite control u, desired initial state xd0must be such that it ultimately validates the following relationship:

xd0=x0E2

For example, in a second-order system, the position or speed cannot bounce; as a result of this fact, any desired trajectory that can be workable from time t=0necessarily starts at the same position and speed as those of the plant.

2.1 A notational simplification

Let us define x=xxdas the tracking error in the variable x, and x=xxd=xx˙xn1Tas the tracking error vector. Let us also define a surface Stthat is time-varying in the state-space Rnwith the scalar equation sxt=0expressed as:

sxt=ddt+λn1x,E3

where λis a strictly positive constant. For example, if n=2, Eq. (3) takes the following form:

s=x˙+λx,

that is, it simply consists of a weighted sum of the position and the velocity errors; thus, we can express xfrom Eq. (3) as follows:

x=sxtddt+λn1E4

More specifically, a first-order stabilization problem in sis basically nothing other than a replacement for the problem of tracking the n-dimensional vector xd(i.e., the original nth-order tracking problem in x). Indeed, sneeds to be differentiated only once for the appearance of the input udue to the presence of the term xn1in the expression sin (3).

Additionally, the bounds on scan be evaluated directly as the bounds on the tracking error vector x, and therefore, the scalar sis considered a true measure of tracking performance. Specifically, under the assumption that x0=0(in the meantime, we make a note that the effect of non-zero initial conditions in xcan be added separately), we write:

t0,stΦt0,xit2λiε      i=0,,n1E5

where ε=Φ/λn1.Indeed, Eq. (3), or more precisely, Eq. (4) derived from (3) clearly indicates that the tracking error xcan be obtained from sthrough a sequence of first-order low-pass filters (see Figure 1, where p=d/dtis commonly known as the Laplace operator).

Figure 1.

Computing bounds on x∼.

In general, a first-order low-pass filter’s input-output relationship is given as follows:

OutputlpfInputlpf=Kf11+,

where Kfis filter gain and τis filter time constant. Let y1be the output of the first filter. We can express the output of the first filter in terms of a convolution integral in time domain by taking into account the fact that the input is already defined as s:

y1t=0teλtTsTdT.

Using sΦ, we get the following:

y1tΦ0teλtTdT=ΦλeλtT]0t=Φλ1eλtΦ/λ.

We can apply similar reasoning to the second filter, et cetera, until we reach yn1=x. Ultimately, we get

xΦ/λn1=ε.

Similarly, xican be considered to be acquired through the sequence of Figure 2.

Figure 2.

Computing bounds on x∼i.

One can easily make another similar relationship, zIΦ/λn1iby referring to the previous result. Here, zIis the output of the ni1thfilter. It is worth noting, however, that

pp+λ=p+λλp+λ=1λp+λ.

One sees that the remaining iblocks right after ni1blocks in the sequence of Figure 2 include numerators of pas can typically be seen in practical filter applications, and that the sequence of Figure 2 means that

xiΦλn1i1λp+λ1λp+λi blocks,

where the first multiplier to the right of inequality sign includes the first ni1blocks which do not have pas the numerator. When iblocks multipliers are arranged, the expression takes the form:

xiΦλn1i1.1i times+λ..λλi,

where term comes from the result derived for the sequential blocks each of which is represented by 1/p+λ,and 1/p+λi1/λi.Besides, the positive sign must be considered to obtain the prospective upper bound. In this case, the next step is:

xiΦλn1i1i+λiλi.

Finally, we can write,

xiΦλn1i1+λλi

since 1i+λiλi1+λλi. Thus, as a result, we reach the following statement:

xiΦλn1λi1+1i=Φλn12λi=2λiε,

i.e., the bounds of (5) are proven. Finally, in the case where x00, the bounds of (5) are obtained asymptotically, that is, within a short time constant n1/λ. Please note that since the single filter block has a time constant equal to 1/λas seen from the above analysis starting from (3), the sequence of n1filter blocks will have the time constants equal to n1/λ.

Hence, we, indeed, have replaced an nth-order tracking problem by a first-order stabilization problem, and have quantified with inequality (5) the transformations in which the performance measures correspond to.

Keeping the scalar sat zero, which is a simplified first-order problem, can now be achieved by choosing the control law uof (1) such that apart from St

12ddts2ηs,E6

where ηis a strictly positive design constant and ensures inequality (6) which is called η- reachability condition. A fundamental requirement is that the sliding mode dynamics must be attractive to the system state and there are many reachability conditions defined in the literature [13, 14, 15, 16, 17]. Basically, inequality (6) indicates that the squared “distance” to the surface, as measured by s2, decreases throughout entire system trajectories. Therefore, it restricts trajectories to head toward the surface St, as depicted in Figure 3. Particularly, when they are on the surface, the system trajectories remain on the surface. In other words, the fact that the surface is an invariant set indicates that condition (6) (a.k.a., sliding condition) is satisfied. Moreover, as we shall notice, inequality (6) also suggests tolerating some disturbances or dynamic uncertainties while still holding the surface stationary (i.e., an invariant set). In Figure 3, this graphically means that the trajectories away from the surface can “move” while still denoting the surface. Stverifying (6) is referred to as a sliding surface, and the system behavior that occurs on the surface is called sliding regime or sliding mode.

Figure 3.

The sliding condition.

Another appealing feature of the invariant set Stis that when the system trajectories are on the set, it is defined by the equation of the set itself, i.e.,

ddt+λn1x=0.

That is to say, the surface Strepresents both a place and a dynamics. This fact is clearly the geometric interpretation of our previous statement that definition (3), in fact, allows us to substitute a first-order problem for an nth-order one.

Finally, if condition (2) is not fully validated, i.e., if xt=0is indeed away from xdt=0, then, nevertheless, satisfying (6) gives a guarantee for reaching the surface Stat a finite time smaller than st=0/η. Indeed, assume, for instance, that st=0>0and define treachas the time needed to hit the surface s=0. Integrating (6) between the points t=0and t=treach(i.e., in the interval [0, treach]) gives rise to:

0st=0=st=treachst=0ηtreach0,

which means that

treachst=0/η.E7

This result can simply be proven to be true by starting to integrate both sides of (6) between 0and treachas follows:

t=0t=treach12ddts2dtt=0t=treachηsdt.

Making the necessary simplifications within the integrals, we get the following:

t=0t=treachdst=0t=treachηdt.

Now, the integrals are taken and evaluated for the lower and upper limits as shown below:

st=treachst=0ηtreach0.

Finally from here,

st=0ηtreach

is written, and the same result as (7) is hereby obtained. Even if st=0<0, a similar result would be obtained, and thus, writing the above inequality as follows would be a correct representation:

treachs(t=0/η.

Furthermore, definition (3) implies that once on the surface, the tracking error tends exponentially to zero, with a time constant n1/λ.

The typical system behavior implied by satisfying sliding condition (6) is shown in Figure 4 for n=2. A line with slope λand containing the time-varying point xd=xdx¨dTrepresents the sliding surface in the phase plane. The state trajectory reaches the time-varying surface in a finite time smaller than st=0/ηfrom any initial condition, and then slides across the surface towards xdexponentially, with a time constant equal to 1/λ.

Figure 4.

Graphically represented Eqs. (3) and (6) for n=2.

In conclusion, the idea behind (3) and (6) is to obtain an appropriate function of the tracking error, s, in accordance with (3), and then choose the feedback control law uin (1) such that s2continues to be used as a Lyapunov-like function of the closed-loop system, in spite of the presence of model imprecision and of disturbances. Then, the controller design is a two-step procedure. First, the selection of a feedback control law uis performed to verify sliding condition (6). However, it is required that the control law be discontinuous throughout Stto take into account the presence of modeling imprecision and of disturbances. Since the execution of the associated control switchings is not necessarily perfect (for instance, in practice, switching is not instantaneous, and the value of sis not known with infinite precision), this causes chattering (Figure 5). Now, with a few important exceptions, chattering is practically undesirable, because it contains high control activity and can trigger neglected high-frequency dynamics during modeling (such as unmodeled structural modes, neglected time-delays, etc.). Thus, in a second step, the discontinuous control law uis smoothed accordingly to reach an optimal compromise between control bandwidth and tracking precision: while the first step explains parametric uncertainty, the second step ensures robustness to high-frequency unmodeled dynamics.

Figure 5.

Chattering caused by the switching delays.

As mentioned previously, the discontinuous control law causes chattering of the trajectories to take place around the surface s=0. This problem can be eliminated by smoothing out the discontinuities in the vicinity of the sliding surface through the introduction of a boundary layer thickness. An adaptation of saturation nonlinearity instead of signum nonlinearity in a position control system in which it is represented by Eqs. (83) and (84) in order to decrease the chattering phenomenon caused by sliding mode control law is the result of the same effort of smoothing out the discontinuities with the introduction of the boundary layer thickness as illustrated in Figure 6.

Figure 6.

Boundary layer with thickness ∅.

To maintain the system work in the sliding surface, a switching action term, usw, is added to the control law, and is defined by

usw=Ksats,

and overall control law can be expressed as:

u=û+usw=f̂+x¨dλx˙Low frequency controlKsatsHigh frequency control,

which will be explained in more detail in Section 5.1. Here, the nonlinear saturation function sats, which is the replacement for nonlinear signum function sgns, is defined by

sats=1,s<00,s<1,1,s<1

where is the boundary layer thickness.

3. Sliding mode control design

A continuous-time dynamical equation of an n-link robot manipulator is defined as follows:

Mqq¨+Cqq˙q˙+Fq˙+Gq=τ,E8

where qRnx1denotes the joint configuation variables (translational or rotational) representing the generalized position coordinates (alias the joint positions) of the robot links. Similarly, q˙,q¨Rnx1represent the joint velocity and acceleration of the robot links, respectively. MqRnxnis the symmetric, bounded, positive definite inertia mass matrix. Cqq˙Rnxnis the matrix of Coriolis and centripetal forces and Fq˙Rnx1is the vector of viscous friction. Furthermore, the vector of GqRnx1represents the gravity terms, and finally, τRnx1is called the control torque vector, or the vector of applied joint torques.

Sliding surface defined below is considered in the design of SMC controller:

s=e˙+λe,E9

where e=q=qqdrepresents the error vector and λis assumed to be a symmetric positive definite matrix such that s=0would evolve into a stable surface. The reference velocity vector q˙ris the same as the definition in [1]:

q˙r=q˙dλe.E10

Hence, one can define the sliding surface as follows:

s=q˙q˙r.E11

Now, the following lemma refers to the sliding mode controller design.

Lemma 1. Let us consider the system consisting of (8) through (10). If the following control rule is selected:

τ=τ̂KsgsgnsE12

such that

τ̂=Mq¨r+Ĉq¨r+GE13

and

KsgiCq˙r+Γi,E14

then the following sliding condition [1],

12ddtsTMs<ηsTs1/2,η>0E15

is satisfied. In (14), Ksgistands for the element of sliding gain vector Ksgand Γis a design parameter vector that must be chosen to ensure the inequality ΓiFup+ηi.

Proof. Let Lyapunov function candidate be given as follows:

V=12sTMs.E16

Since Mis positive definite and sis different from zero (s0), Vis always greater than zero (V>0) and by taking time derivative of (16) and taking into account the symmetric property of M, it takes the following form:

V˙=12s˙TMs+12sTM˙s+Ms˙=12s˙TMs+12sTM˙s+12sTM˙s=sTMs˙+12sTM˙sE17

Using (11), we get:

V˙=sTMq¨q¨r+12sTM˙s=sTMq¨Mq¨r+12sTM˙sE18

Taking q¨from (8) and replacing it in (18), we have:

V˙=sTMM1τCq˙GFMq¨r+12sTM˙s.

Then, taking q˙from (11) and replacing it above yields:

V˙=sTτCq˙rGFMq¨r+sTM˙2Cs2.

In the above equation, the second term is zero due to the asymmetry property; therefore, it disappears. In this new case,

V˙=sTτCq˙rGFMq¨r.E19

Next, applying (12) and (13) successively for τand τ̂in Eq. (19), and proceeding step by step, the following result is reached:

V˙=sTĈCq˙rKsgsgnsF=sTΔCq˙rKsgsgnsF.E20

In robot modeling, the terms Mqand Gqcan be well and accurately determined, but in most cases it is not easy to have the parameters Cqq˙and Fq˙precisely. Therefore, in this work, the matrix Cis considered

C=Ĉ+ΔC,E21

where Ci,jstands for the elements of the matrix C. Also, the vector Fis assumed an external disturbance with the upper bound defined as,

FFup,E22

where the operator .denotes Euclidian norm [2]. Now, by rearranging (20) as shown step by step below, we get the following:

V˙=sTΔCq˙r+FsTKsgsgns=sTΔCq˙r+Fi=1nKsgisi,E23

where si=siTsgnsi.

At this point, we can briefly verify that the terms on the right side of (14) are positive. First of all, it is easiest to say that the first term on the right, ΔCq˙r, is positive in any case, because the Euclidian norm is used. The other term Γiis also positive. Because, as we have already stated in (14) that ΓiFup+ηi, where Fupis an unknown upper bound defined as FFupin (22), which also indicates that it is absolutely positive, and ηis a strictly positive design constant and ensures inequality (6). Hence, Γi0. Now, if the inequality Ksgigiven by (14) is substituted in Eq. (23) under the acceptance of its equality sign for a moment, we can rewrite Eq. (23) by extending it as follows:

V˙=sTΔCq˙r+Fi=1nΔCq˙rsii=1nFupsii=1nηisi.

The first, second and third terms on the right side of the equation above are negative in varying amounts and contribute to the final term, which is i=1nηisi, more negatively. Therefore, we can easily conclude that

V˙i=1nηisi.E24

This shows that Vis a Lyapunov function and the satisfaction of sliding condition in (15) is proven.

4. Achievement of the control law for robot manipulators for the adapted reaching mode

VSC systems include a group of different, generally fairly simple, feedback control laws and a decision rule. Depending on the system condition, a decision rule, usually called the switching function, determines which control law is “on-line” at any time. The transient dynamics of VSC systems consists of two modes: a “reaching mode” (or “non-sliding mode”), and a subsequent “sliding mode”. Hence, VSC design involves two stages: the first one involves the design of the appropriate ndimensional switching function sxfor a desired sliding mode dynamics. The second one involves a control design for the reaching mode where a reaching condition is met. The desired sliding mode dynamics usually includes a fast and stable error-free response without overshoot. In sliding mode, an asymptotic convergence to the final state will be accomplished. The desired response in the reaching mode, in general, is to reach the switching manifold defined as

sx=ψTx=0,E25

in a finite time with a small amount of overshoot with regard to the switching manifold [18].

The reaching law is a differential equation that determines the dynamics of a switching function sx. If sxis an asymptotically stable differential equation, then, it is solely a reaching condition. Further, the parameter selection in the differential equation controls the dynamic quality of the VSC system in the reaching mode. The reaching law can be expressed practically in general form as follows [18]:

s˙=QsgnsKhs,E26

where Q=diagq1qn, qi>0; sgns=sgns1sgnsnT; K=diagk1kn, ki>0; hs=h1s1hnsnT; and sihisi>0, hi0=0.

The design principle of the SMC law for the plants of arbitrary order is to force a variable’s error and its derivative to zero. Tracking of a desired motion qdtis the main task of the robot arm. Here, let us start first by defining a 2n-dimensional error vector [18]:

e=e1e2=qdx1q˙dx2=qdqq˙dq˙,E27

and then, an n-dimensional vector of switching function:

se=Ψe=ΛIe1e2=Λe1+e˙1,E28

where e˙represents the tracking speed error and:

Λ=diagλ1λn,λi>0,

that determines the system bandwidth. Next, the time derivative of (28) is taken as follows [18]:

s˙e=Λe˙1+e˙2=Λe˙1+q¨dq¨.E29

Now, constant plus proportional rate reaching law as represented by

s˙=QsgnsKsE30

is adapted. Substituting (30) into (29) and setting q¨apart yields:

q¨=Qsgns+Ks+Λe˙1+q¨d.E31

Finally, substituting (31) into the non-linear plant of continuous-time dynamic model of robot systems in (8) results in:

MqQsgns+Ks+Λe˙1+q¨d+Cqq˙q˙+Fq˙+Gq=τ.E32

This is also known as the final control law.

5. Proofs of the boundedness and convergence properties of smooth sliding mode controllers

In this section, the proofs of the boundedness and convergence properties of the smooth sliding mode controllers are introduced. In particular, the convergence analysis of smooth sliding mode controllers will be explained and discussed to the finest detail. Lyapunov’s direct method is used to handle the finite-time convergence of the tracking error vector to the boundary layer. Also, once in the boundary layer, the tracking error vector is said to have exponentially converged to a bounded region, as proven analytically.

5.1 Problem statement

Consider the following non-linear system class of n-th order:

xn=fx+bxu,E33

where uis the control input, xn, is the n-th order derivative of the interested scalar output variable xwith respect to time t0. Here, also, x=xx˙xn1Trepresents the system state vector, and both f(x) and b(x), such that f,b:RnR, denote nonlinear functions.

The following assumptions will be made in terms of the dynamic system presented in (33).

Assumption 1. fis an unknown function such that it is bounded by a known function x, i.e., f̂xfxFx, where f̂is an estimate of f.

Assumption 2. Input gain bxis an unknown function such that it is positive and bounded, i.e., 0<bminbxbmax.

In the proposed state space control problem, the xstate vector must be able to follow a desired trajectory xd=xdx¨dxnn1, even under the presence of parametric uncertainties and unmodulated dynamics.

The following assumptions should also be made during the development of the control law.

Assumption 3. The state vector xhas availability.

Assumption 4. The desired trajectory xdis differentiated once in time. Moreover, each element of the vector xdas well as xdn, is available and has known bounds.

Now, let x=xxdbe defined as the tracking error for the variable x, and x=xxd=xx˙xn1as the tracking error vector.

Let us define a sliding surface Sin the state space by the equation sx=0in which sis the function mapping from n-dimensional real space Rnto one-dimensional real space R, i.e., s:RnR, and satisfying the following equation:

sx=ddt+λn1x,

which can be plainly rewritten as

sx=cTx,E34

where c=cn1λn1++c1λc0with cirepresenting binomial coefficients as follows:

ci=n1i=n1!ni1!i!,i=0,1,,n1E35

which makes cn1λn1++c1λ,c0a Hurwitz polynomial.

It can be easily verified from (35) that c0=1, for n1. Therefore, the time derivative of swill be expressed in the following form:

s˙=cTx˙=cn1λn1c1λc0cTx˙x¨xnx˙=0cn1λn1c2λ2c1λc¯Txx˙xn1x˙+xn

i.e.,

s˙=cTx˙=xn+c¯TxE36

where, here, as used for the first time above, there is a definition in the form of c¯=0cn1λn1c1λ. At this point, let us evaluate Eqs. (34) and (36) for n=3, i.e.,

sx=ddt+λ2x=x¨+2λx˙+λ2x,

from which cappears to be as c=c2λ2c1λc0. Then, sx=cTxor to create a polynomial, cxT=c2λ2c1λc0xx·x¨=c2λ2x+c1λx˙+c0x¨with c0=1as always, and c2λ2+c1λ+c0is a Hurwitz polynomial. That is, it is defined as the polynomial with its coefficients (i.e., ci) that are positive real numbers, and its zeros are located in the left half-plane –i.e., the real part of every zero is negative– of the complex plane.

Now, Let the problem of controlling the uncertain nonlinear system expressed by (33) be handled for review through the classical sliding mode approach that defines a control rule consisted of an equivalent control û=b̂1f̂+xdnc¯Txand a discontinuous term Ksgnsas follows:

u=b̂1f̂+xdnc¯TxKsgns.E37

where b̂=bmaxbminrepresents the estimated value of b, and Krepresents a positive gain. Furthermore, the sign or signum function represented by sgnsabove

sgns=1,ifs<00,ifs=0.1,ifs>0

Based on Assumptions 1 and 2 given above and taking into account the fact that β1b̂/bβ, where β=bmax/bmin, the gain Kmust be determined in such a way as to ensure the following inequality:

Kβb̂1η+F+β1b̂1f̂+xdnc¯Tx,E38

where ηis a strictly positive constant of the reaching time. Now, in this step, let us reaffirm the validity of the lower and upper bounds of busing the b̂and βdefinitions given: first of all, Let the definitions of b̂and βbe placed in the expression β1b̂/bβgiven above. In this case,

1bmax/bminbmaxbminbbmax/bmin.

If each side is multiplied by 1/bmaxbmin, the following inequality is obtained:

1bmax1b1bmin.

In this inequality, if inversion is applied to all terms, inequalities will be completely displaced, that is to say, it will become bmaxbbmin. This is a necessary initial acceptance. Therefore, when we turn b’s upper and lower bounds into inequality, we once again confirm the correctness of the definitions for b̂and β. Since the control rule will be designed to be robust against the inequality β1b̂/bβ, that is, a bounded multiplicative uncertainty, taking advantage of the similarity to the terminology used in linear control, we can call βthe gain margin of the design.

In order to ensure xxdsystem tracking, we define a sliding surface s=0according to sxt=ddt+λn1x, that is,

s=ddt+λx=x˙+λx.

When we derive the expression s, we obtain the following:

s˙=x¨+λx˙=x¨x¨d+λx˙=f+ux¨d+λx˙.

For s˙=f+ux¨d+λx˙=0to be realized, other terms outside of umust be determined equal to the opposite sign of û, which is the best approximation of a continuous control rule uthat can implement s˙=0, that is,

û=f̂+x¨dλx˙.

In fact, to see this result, the first thing to do is to draw ufrom the equation s˙=f+ux¨d+λx˙=0. Hence, u=f+x¨dλx˙is obtained. Then, from here, in order to obtain the approximate value of u, searching for the approximation of the function on the right side of the equation, and representing this approximated function by f̂symbolically are sufficient to lead us to the correct result, as seen above.

The control rule u=b̂1ûksgnswith predefined sand û, and kdefined by the inequality kβF+η+β1û—as will be explained little below—meets the sliding condition. Indeed, when we substitute this control rule in the expression s˙=f+bux¨d+λx˙by choosing the use of x¨=f+bu, which is more specific to this type of structure, instead of x¨=f+uused only just two above, we obtain the following,

s˙=f+bb̂1ûksgnsx¨d+λx˙.

Once the previously determined ûis replaced above, the following equation is reached:

s˙=f+bb̂1f̂+x¨dλx˙ksgnsx¨d+λx˙.

The organized form of this statement will be as follows:

s˙=fbb̂1f̂+1bb̂1x¨d+λx˙bb̂1ksgns.

Such that kshould meet the following condition,

kb̂b1ff̂+b̂b11x¨d+λx˙+ηb̂b1E39

We can really achieve this condition by following the steps below:

s˙=0=fbb̂1f̂+1bb̂1x¨d+λx˙bb̂1ksgns
bb̂1ksgns=fbb̂1f̂+1bb̂1x¨d+λx˙
ksgns=b̂b1ff̂+b̂b11x¨d+λx˙
ksgns=b̂b1f̂+ff̂f̂+b̂b11x¨d+λx˙.

Here, it was previously described that ff̂F. But, when determining k, it will be necessary to take into account the reaching time η. Therefore, we will allow F+ηto be written instead of F. Removing the term sgns, kwill be determined as follows as a result of our compensation through expressing, with an absolute value, the effect of its reciprocations being of which the negative svalues relative to the positive svalues and of which only the sign changed:

kb̂b1F+ηb̂b1+b̂b11f̂x¨d+λx˙.

Note here that F0and η>0(absolute positive). For this reason, there is no need to take absolute values of these terms. Here again, using the definition b̂b1β, for k, we get the expression,

kβF+η+β1û.E40

Remark 1. To avoid any confusion, if we wanted to verify (40) by proceeding from (39), since ηhas already been taken into account in (39), we would not need to take F+ηinstead of F. We would proceed directly with ff̂F.

Remark 2. Considering the fact that the Fvalue can be faced with moments where the estimation problem will be relatively large by nature and similarly with the moments when ηreaching time will be relatively larger, it is possible to state precisely that it is the right choice or necessity to take the direction of inequality kgreater than or equal to.

Thus, it can be easily verified that the control rule u=b̂1f̂+xdnc¯TxKsgnsis sufficient to impose the shift condition,

12ddts2ηs,

which indeed guarantees the convergence of the tracking error vector to the sliding surface Sand consequently its exponential stability in a finite-time. In response to the uncertainty of fon dynamics, we add a discontinuous term to ûacross the surface s=0to meet the slip condition given above, that is:

u=ûksgns.

Here we can now guarantee that the sliding condition will be verified by choosing k=kxx˙sufficiently large. Indeed,

12ddts2=s˙s=f+ûksgnsx¨d+λx˙s=ff̂ksgnss.

This last operation is important; because we have reached this point by using the equations u=f+x¨dλx˙, û=f̂+x¨dλx˙, and u=ûksgnsas follows:

s˙=f+ux¨d+λx˙=f+ûksgnsx¨d+λx˙=f+f̂+x¨dλx˙ksgnsx¨d+λx˙=ff̂ksgns.

If we continue where we were,

12ddts2=ff̂ksgnss=ff̂sksgnss.

Therefore, since sgnss=s, the following expression is reached,

12ddts2=s˙s=ff̂ksgnss=ff̂sks.

So that, when k=F+ηis selected, the above statement follows,

12ddts2=s˙s=ff̂sFsηs.

However, although the definition of f̂fFis given at the beginning of the section, recalling that we prefer the form ff̂Fto be used in the case study below, Let us give the statement its final form:

12ddts2=s˙s=ff̂sff̂sηs.E41

In fact, note that here the expression ff̂,which we substitute for Frepresents the smallest value that Fcan take. We generally know that Fis greater than this value.

We will now carry out the following case studies for the Eq. (41):

Case 1. If ff̂and sare both negative or positive, as such, ff̂sff̂s=0. However, it is known to be Fff̂, hence, ff̂sFs0, i.e., it will always be negative.

Case 2. However, if ff̂and sopposite signs; ff̂swill always be negative. Fswill also be negative. Hence, ff̂sFswill always be more negative as compared to Case 1.

As a result,

12ddts2ηs,

is always true.

However, the presence of a discontinuous term (i.e., Ksgns) in the control rule leads to the well-known chattering effect. To prevent these unwanted high-frequency oscillations of the controlled variable, Slotine had proposed the idea of adopting a thin boundary layer Saround the switching surface [1]:

S=xRnsx.E42

Here is an absolute positive constant, which represents the boundary layer thickness.

The boundary layer is accomplished by replacing the sign function with a continuous interpolation in S. It should be emphasized that this smooth approximation referring to the flatness or smoothness of the interpolating curve and its derivatives, which will be called φs, here, will definitely act as a sign function outside the boundary layer.

Various options are available to smooth out the ideal switch. But the closest choices are the saturation function expressed by

sats=sgns,ifs1s,ifs<1,E43

and the hyperbolic tangent function expressed by tanhs. Thus, the smooth sliding mode control rule can be expressed as follows:

u=b̂1f̂+xdnc¯Txs.E44

5.2 Convergence analysis

The attractiveness and invariance properties of the boundary layer are introduced in the following theorem:

Theorem 1. Consider four previously made assumptions with the uncertain nonlinear system given in (33). Therefore, the smooth sliding mode controller defined by (38) and (44) provides the finite-time convergence of the tracking error vector to the boundary layer Sdefined by (42).

Proof. Let a positive-definite Lyapunov function candidate Vbe defined as,

Vt=12s2.E45

Here, as a measure of the distance of the current error to the boundary layer, scan be computed as follows:

s=ssats.E46

Noting that s=0in the boundary layer, it is shown that V˙t=0inside S. It is also possible to easily verify that s˙=s˙outside the boundary layer through (43) and (46), and in this case, V˙can be written as follows:

V˙t=ss˙=ss˙=scTx˙=sxn+c¯Tx=xnxdn+c¯Txs=f+buxdn+c¯Txs.

Next, considering that the control rule given by (44) is written as

u=b̂1f̂+xdnc¯TxKsgns,

outside the boundary layer and noting that f=f̂f̂f, we get the following result:

V˙t=f̂fbb̂1f̂+xdnc¯Txf̂+xdnc¯Tx+bKsgnss.

Thus, by taking the Assumptions 1 and 2 into consideration, and defining Kaccording to (38), V˙can be written as follows,

V˙tηs.

Because the Lyapunov function candidate, which we initially defined with (45) as positive definite, essentially inspired by the inequality in the form of 12ddts2ηswhich we have always correctly demonstrated above, may well be represented by a similar structure to the form, 12ddts2ηs.It will be seen from here that V˙t=ss˙ηs,as well. Hence, the inequality V˙tηswill imply that VtV0and therefore sis bounded. Moreover, from the definitions of sand sexpressed in (35) and (46), respectively, it can be verified that xis bounded. Therefore, Assumption 4 and (36) imply that s˙is also bounded.

Finite-time convergence of the tracking error vector to the boundary layer can be shown remembering the expression,

V˙t=12ddts2=ss˙ηs.

Then, dividing both sides into sabove and integrating them between 0 and twill refer to the following result:

0tsss˙0tηdτ.

Remark 3. Here, considering the ratio s/sas the ratio of two numbers of the same size and therefore assuming it disappeared, that is, since it has no effect in size, substantially it is a correct approach to consider the integral as an equivalent to 0ts˙. This produces the result st0t.Consequently, knowing the fact that in the situation before taking this approach, the product ss˙which appears in the numerator of the integral to the left of inequality is essentially equal to the derivative of the positive-definite VLyapunov candidate function and is therefore positive again, it is essential to show the terms on the left side of the inequality with absolute value. That is to say, it is important to see that ss˙s>0.Then, the next step to ensure this will turn into the form sts0ηt.In this way, considering treachas the time required to reach sand noting that streach=0, we have the expression,

treachs0η

guaranteeing the convergence of the tracking error vector to the boundary layer in a time interval less than s0/η.

Remark 4. If both sides of sts0ηtare multiplied by −1, s0stηtis obtained, that is, briefly, the inequality is displaced. If tis left alone in the next step, ts0stηis obtained. Hence, it is guaranteed to be treachs0/η. That is, the right-hand side will act as the largest value achievable for treach. In other words, it will appear as a guaranteed upper value. Then, the value of s0stηis expected to be less than this guaranteed value of s0/η.

Therefore, to keep the reaching time, treach,as short as possible, the value of the positive constant ηcan be chosen appropriately. We clearly see from Figure 7 that the time evolution of sis bounded by the linear equation st=s0ηt.

Figure 7.

Time evolution of the distance of the current tracking error to the boundary layer s∅.

Lastly, the proof of the boundedness of the tracking error vector is based on Theorem 2.

Theorem 2. Let the boundary layer Sbe defined according to (42). Then, once inside S,the tracking error vector will exponentially converge to an n-dimensional box defined according to xiζiλin+1ϕ,i=0,1,,n1,with ζisatisfying

ζi=1,fori=01+j=0i1ijζj,fori=1,2,,n1.E47

Proof. Considering the fact that sxcan be rewritten as sxwith the definition of sgiven in (34), the expression below

sx=cTx=cn1λn1++c1λc0xx˙xn1=c0xn1+c1λxn2++cn1λn1x.

Thus,

sx=c0xn1+c1λxn2++cn1λn1x,E48

or the following,

ddt+λn1xE49

can be written. If (49) is multiplied by eλt, the following statement is reached:

eλtddt+λn1xeλteλt.

In fact, this expression is equal to

eλtdn1dtn1xeλteλt.E50

That is to say,

ddt+λn1xeλt=dn1dtn1xeλt.E51

We can confirm this form of (51) for small nvalues. Namely, if binomial expansion is applied for ddt+λn1,

ddt+λn1=k=0n1n1kdkdtkλn1k

is written. Hence,

sx=ddt+λn1x=k=0n1n1kdkxdtkλn1k.E52

At this point, we can make a confirmation by taking n=3:

For n=1, it becomes sx=00d0xdt0λ0=x. The binomial coefficient of this single term is 1, and this number is at the top of the Pascal triangle. For n=2,it becomes sx=10d0xdt0λ10+11dxdtλ11=λx+dxdt. Here, the coefficients of both terms are 1. It gives the numbers of one-down row from the top of the Pascal triangle. For n=3, it becomes

sx=20d0xdt0λ20+21dxdtλ21+22d2xdt2λ22=λ2x+2λdxdt+d2xdt2.E53

Here, the coefficients of the three terms from left to right are 1, 2, 1. This gives the elements of the two-down row from the top of the Pascal triangle. If the expression sxis multiplied by eλt,

eλtsxeλteλt

is obtained. If the result for n=3in Eq. (52), or its equivalent (53) is substituted above,

eλtλ2eλtx+2λeλtdxdt+eλtd2xdt2eλt,

or the following expression is obtained:

eλtd2dt2eλtxeλt.

This verifies the multiplication of sxwith eλtfor n=3. In other words, the equation ddt+λ2xeλt=d2dt2xeλtis satisfied. Once this statement is generalized for n, the validity of Eq. (51) is proven.

If the inequality (50) is integrated between 0 and t,

0teλτ0tdn1dtn1xeλt0teλt,

and one step later,

λeλτ0tdn2dtn2xeλτ0tλeλτ0t,

and finally the following expression is reached:

λeλt+λdn2dtn2xeλtdn2dtn2xeλtt=0λeλtλ.

When the term dn2dtn2xeλtt=0is added to each side of this expression, it takes the form below:

λeλt+λ+dn2dtn2xeλtt=0dn2dtn2xeλtλeλtλ+dn2dtn2xeλtt=0.E54

Since we can always write,

dn2dtn2xeλtt=0dn2dtn2xeλtt=0anddn2dtn2xeλtt=0dn2dtn2xeλtt=0

as a result of replacing the derivative terms in the inequality (54) with their equivalents expressed with an absolute value one above, the inequality conditions will be preserved exactly as the term with the absolute value will be smaller than the term that satisfies the “less than or equal to” condition on the left and greater than the term that provides “greater than or equal to” condition on the right in the equality (54). Furthermore, aside from the absence of a violation, the conditions of inequality have been further reinforced. Therefore, it is possible to write the following under these conditions,

λeλt+λdn2dtn2xeλtt=0dn2dtn2xeλtλeλtλ+dn2dtn2xeλtt=0.

Also, since both and λare initially defined as positive definite constants, we take /λinstead of the leftmost /λand /λinstead of the rightmost /λ, which can be more safely adapted to existing inequality conditions without loss of generality will be preferred at this stage. Hence, the inequality (54) will turn into a rewritten appropriate form given below:

λeλtdn2dtn2xeλtt=0+λdn2dtn2xeλtλeλt+dn2dtn2xeλtt=0+λ.

The same reasoning can be applied repeatedly until the n1thintegral is reached on the inequality (50). Once (50) is integrated, recall that (54) is obtained. If we apply a second integral on (54) or, alternatively, a first integral to the form of (54) given immediately above, the following expression is obtained:

0tλeλtdtParta0tdn2dtn2xeλtt=0+λdtPartb0tdn2dtn2xeλtdtPartc0tλeλtdtParta+0tdn2dtn2xeλtt=0+λdtPartb.

In determining the generalized cases below, we would like to state in advance that we do not focus on other terms that will appear in the shape of increasing powers of tin the form of tnn!especially in Parts (a) and (c), and we do not show them in the generalized statements. For that matter, as shown a little below, if Eq. (55) is divided by eλtand the limit is taken as tgoes to infinity (i.e., t), those terms will eventually disappear completely, since the denominator will go to infinity faster than the numerator. After this essential explanation,

For Part (a):

0tλeλtdt=λ2eλt0t=λ2eλtλ2,n=2forthe2ndintegral
λ3eλtλ3,n=3forthe3rdintegral
λn2eλtλn2,n=n2for then2th integral
λn1eλtλn1,n=n1forthen1th integral:Generalized form

For Part (b):

0tdn2dtn2xeλtt=0constant+λconstantdt=dn2dtn2xeλtt=0+λt,n=2forthe2ndintegral
dn2dtn2xeλtt=0+λt22,n=3forthe3rdintegral
dn2dtn2xeλtt=0+λt36,n=4forthe4thintegral
dn2dtn2xeλtt=0+λtn3n3!,n=n2forthen2thintegral
dn2dtn2xeλtt=0+λtn2n2!,n=n1forthen1thintegral:Generalized form

For Part (c):

Starting with (50), when the term in the middle of inequality, dn1dtn1xeλt, is integrated n1times in a row, it is obvious that only the result, xeλt,will be found. Therefore,

0tdn1dtn1xeλtdt=dn2dtn2xeλt0t=dn2dtn2xeλtdn2dtn2xeλtt=0,n=11stintegral
0tdn2dtn2xeλtdt=dn3dtn3xeλt0t=dn3dtn3xeλtdn3dtn3xeλtt=0,n=22ndintegral
0td2dt2xeλtdt=ddtxeλt0t=ddtxeλtddtxeλtt=0,n=n2n2thintegral
0tddtxeλtdt=xeλt0t=xeλtx0,n1n1thintegral

is written. However, due to the reason we have explained above, we would like to remind that we do not take into account other terms that will appear in the shape of increasing powers of tin the form of tnn!once again. Therefore, when the determinated generalized terms are put in place,

λn1eλt+λn1dn2dtn2xeλtt=0+λtn2n2!xeλtx0λn1eλtλn1+dn2dtn2xeλtt=0+λtn2n2!+

is obtained. Based on the previous similar practice, the term x0is added to each side, and once again reminding that and λare positive definite constants and that x0x0, x0x0, if the inequalities, λn1λn1, λn1λn1, are used,

λn1eλtdn2dtn2xeλtt=0+λtn2n2!x0+λn1xeλtλn1eλt+dn2dtn2xeλtt=0+λtn2n2!++x0+λn1E55

is written. Also, if (55) is divided into eλtand tis taken to infinity, the following result is obtained:

λn11eλtdn2dtn2xeλtt=0+λtn2n2!1eλtx0+λn1xtλn1+1eλtdn2dtn2xeλtt=0+λtn2n2!++1eλtx0+λn1.

From here, it can be easily verified that

λn1xtλn1.E56

Taking into account the n2thintegral of (50),

λn2eλtdn2dtn2xeλtt=0+λtn3n3!x˙0+λn2ddtxeλtλn2eλt+dn2dtn2xeλtt=0+λtn3n3!++x˙0+λn2,E57

and the derivative expression,

dxeλt/dt=x˙eλt+xλeλt,

by having (56)‘s bounds accepted to (57) and dividing it back into eλtfor t,

λn2tn3n3!eλt1eλtx˙t+xtλλn2+tn3n3!eλt++1eλt,

and finally from here,

λn2x˙t+λxtλn2E58

is obtained. However, in order to determine the bounds of (58) based on only x˙t, the bounds corresponding to the term λxtin addition to x˙tmust be determined exactly. For this, (56) is used and if each side in this inequality is multiplied by λ,

λn2λxtλn2E59

expression is obtained. Now then, if the effect of λxtin the inequality of (58) is substituted by the bound determination ascertained by (59) above, the inequality (59) turns into

2λn2x˙t2λn2.E60

Similarly, taking into account the n3thintegral of (50),

λn3eλtdn3dtn3xeλtt=0+λtn4n4!x¨0+λn3d2dt2xeλtλn3eλt+dn3dtn3xeλtt=0+λtn4n4!++x¨0+λn3,E61

and the derivative expression,

dx˙eλt+xλeλt/dt=x¨eλt+λeλtx˙+x˙λeλt+xλ2eλt=x¨eλt+2x˙λeλt+xλ2eλt,

by imposing the bounds of (56) and (60) on (61) and dividing this expression once again to eλtfor t, the following step is obtained first:

λn3eλtx¨+2x˙λ+xλ2λn3eλt.E62

Now, the bounds for xλ2and 2x˙λare respectively determined with,

λn3λ2xtλn3E63

by multiplying each side of the inequality of (56) by the term λ2, and with,

4λn32x˙λ4λn3E64

by multiplying each side of inequality of (60) by the term 2λ. Once these bounds determined by the inequalities (63) and (64) are imposed on (62), the expression,

λn3λn34λn3x¨λn3+λn3+4λn3,

and hence in brief, the result,

6λn3x¨6λn3E65

is concluded. As in obtaining (56), (60) and (65), the following general conclusion is reached if the similar procedure is applied sequentially until the bounds of xn1are achieved:

1+i=0n2n1iζixn11+i=0n2n2iζi.E66

Here, the coefficients ζii=01n2are related to the pre-acquired bounds of each xiand are summarized in Theorem 2.

In this way, by examining Eqs. (56), (60), (65), and (66) and, as much as other skipped boundaries, the integrals of (50), the tracking error will be kept within the bounds of xiζiλin+1, i=0,1,,n1, where ζiis defined by (47).

Remark 5. It should be noted that an n-dimensionally separated partition defined according to the boundaries mentioned earlier is not entirely within the boundary layer. Considering the attractiveness and invariant properties of Sproved in Theorem 1, the convergence region can be expressed as the intersection of an n-dimensional separated partition and boundary layer defined in Theorem 2. Thus, the tracking error vector will converge exponentially to a closed region Φ=xRnsxandxiζiλin+1i=01n1,where ζiis defined by (47).

Figure 8 describes the Φconvergence region defined according to Remark 5 for a second-order (n=2) system.

Figure 8.

Convergence region Φ in the case of a second-order system.

6. Numerical experimentation and simulation examples

6.1 Position control system by an armature-controlled dc servo motor

6.1.1 Positioning system description

The plant is an armature-controlled dc servo motor, the scheme of which is given in Figure 9 [19].

Figure 9.

DC motor schematic diagram.

In order to derive the state-space mathematical model from the physics of the motor, we first start by writing Kirchoff’s voltage equation around the armature current:

eat=iatRa+Kbdθmdt,E67

where eatis the armature input voltage to the dc amplifier that drives the motor, iatis the armature current, Rais the armature resistance, Kbis a constant of proportionality called the back emf constant, or briefly, the armature constant, and θmis the angular displacement of the armature.

The torque, Tmt, delivered by the motor is expressed separately in relation to both the armature current and the load seen by the armature as follows:

Tmt=Ktiat=Jm+d2θmdt2+Dmdθmdt,E68

where Jmand Dmboth represent the equivalent inertia and viscous damping seen by the armature, respectively. Solving Eq. (68) for iatand substituting the result into Eq. (67) yields

eat=RaJmKtd2θmdt2+DmRaKt+Kbdθmdt.E69

Defining the state variables x1and x2as

x1=θm,E70
x2=dθmdt,E71

and substituting into Eq. (69), we get

eat=RaJmKtdx2dt+DmRaKt+Kbx2.E72

Solving for dx2/dtyields

dx2dt=1JmDm+KtKbRax2+KtRaJmeat.E73

Using Eqs. (70), (71) and (73), the state equations are written as

dx1dt=x2E74
dx2dt=1JmDm+KtKbRax2+KtRaJmeatE75

Hence, in vector-matrix form,

x˙t=0101JmDm+KtKbRax+0KtRaJmeat.

Now, let us consider a position control system and assume a case of varying external disturbance torque to the dc motor. In other words, we assume that a varying external disturbance can enter into the system in the form of varying torque τton the shaft of the motor. Therefore, in this derivation, taking account of the external disturbance term which will appear as an extra term in the state equation, we finally describe the state-space representation (a.k.a. the state differential equation) as follows:

x˙t=x´1tx´2t=010αx1tx2t+0κeat0γτt,E76

where xt=x1tx2tThas as components the angular position x1tand the angular velocity x2tand where α1JmDm+KtKbRaand κKtRaJm, and γ=1/J, with Jthe moment of inertia of all the rotating parts. For the controller variable we choose the angular position:

x1t=10xt.E77

The numerical values of motor’s parameters have been taken from the case study in [20]:

α=4.6 s1,
κ=0.787 rad/Vs2,γ=0.1 kg1m2.

Finally, we try several varying external disturbance torque Nmoptions such as

  • sinat, γsinat, or msinat, where a=3,4,and mis any appropriate multiplier that is compatible and proportional to the size of the system variables and parameters.

  • sin2πt, γsin2πt, or msin2πt,where mis defined as one above.

  • sinx1, γsinx1, or msinx1.

as examples, and we decide that the torque of τt=sin3t Nmis the one we choose for the numerical experimentation and simulations as trade-off after we weigh the pros and cons upon several performance tests and considerations. In addition, the simulations given for the DC motor have been run in the form of script code using Matlab [21].

6.1.2 Design of the sliding mode controller for the position control system

Sliding mode surface is defined as:

st=cet+e˙t,E78

where et=rtx1t, e˙t=r˙tx2t, and rtis a desired input signal and defined as rt=Asin2πFtin the simulations where Aand Fare suitably determined, and cis a positive constant. Therefore, once we get a derivative of the expression above, we get the following:

s˙t=ce˙t+e¨t=ce˙t+r¨tx˙2t.

Taking the second row from Eq. (76) and replacing it with x˙2tabove yields:

s˙(t)=ce˙(t)+r¨(t)+αx2(t)κea(t)+γτ(t).E79

Next, we obtain the control law eatby equalizing s˙tto zero and then using γ=1/Jand τt=Ksgnstin Eq. (79) as follows:

ea(t)=1κ[ce˙(t)+r¨(t)+αx2(t)+Ksgn(s(t))J].E80

Once Eq. (80) is substituted into Eq. (79), we obtain the following:

s˙t=τtJKsgnstJ.E81

where Kis a positive constant. Letting K=maxτt+ηJ, where η>0, and multiplying (81) by styields:

sts˙t=τtJstmaxτt+ηJJst,

Here, the following can always be written:

sts˙t=τtJstmaxτtJstηstηst.E82

Remark 6. It is not difficult to briefly demonstrate the validity of the inequality (82) from a practical point of view: if stis greater than zero (i.e., st>0), then Zterm, which we define as τtJstmaxτtJstfrom (82), becomes less than zero (i.e., Z term<0); because maxτtJst>τtJst. Since the larger term has a negative sign, the difference between the terms will also be negative. As a result, some negative termηstηstis always valid and correct. If st<0, then τtJst<0; in addition, maxτtJst>0<0, therefore the addition of two negative terms will make the overall addition more negative; hence, more negative termηstηstwhich means that the inequality is still valid and in the right direction. Therefore, 12ddts2tηst.

In order to decrease the chattering phenomenon caused by sliding mode control law, saturation function is adapted in this work, and the controller becomes

eat=1κce˙t+r¨t+αx2t+KJsatstφinstead of sgnst,E83

where saturation function satstφis defined as,

satstφ=stφstφsgnstst>φ,E84

where φis a very small positive constant (Figures 1018).

Figure 10.

Position tracking under SMC of the position control system.

Figure 11.

Speed tracking under SMC of the position control system.

Figure 12.

Tracking error under SMC of the position control system.

Figure 13.

Torque vs. speed curve under SMC of the position control system.

Figure 14.

Sliding mode surface under SMC of the position control system.

Figure 15.

Control input under SMC of the position control system.

Figure 16.

Phase trajectory under SMC of the position control system.

Figure 17.

Position tracking under PD control of the position control system.

Figure 18.

Speed tracking under PD control of the position control system.

Let us also see how the results will change if PID control is used as an alternative to the SMC. Although, in the comparisons given in the literature, the pros and cons of both strategies are mentioned, it is generally observed that SMC performs better than PID [22, 23, 24, 25]. Nevertheless, PID control can still be used as an alternative to SMC. The results given here do not contradict the view that one can use it instead of the other without losing too much performance. In the case where only the PD control strategy is applied, let us state that we need to emphasize the following points for the tracking error performance indicated by Figure 19.

Figure 19.

Tracking error under PD control of the position control system.

We can prefer PD control strategy mostly to advance faster between intermediate points of the entire trajectory, i.e. from waypoint to waypoint at which course is changed for following a reference trajectory in which we have to move end-effector along a predefined path. Speaking of which, the end-effector is crucial for the entire trajectory tracking problem in catching up a desired position within shortest time. In other words, accuracy is highly desirable for the end-effector to be positioned accurately under unknown disturbances and payload variations. Basically, the desired position is a function of time and continuously changes with respect to time. Therefore, conventional PD control strategy does not always exhibit good accuracy and robustness properties for trajectory tracking problem. However, we can still choose the PD control strategy because of the advantages it offers [26]. We should emphasize that the errors between the actual points and the waypoints, each of which can also be viewed as intermediate setpoints, do not necessarily have to be eliminated completely. As a result, we have decided to use the PD control because of the advantages it offers and to move faster between the waypoints by tolerating or neglecting the steady-state error computations that would bring extra computational burden (Figures 2022).

Figure 20.

Torque vs. speed curve under PD control of the position control system.

Figure 21.

Control input under PD control of the position control system.

Figure 22.

Phase trajectory under PD control of the position control system.

6.2 Modeling and simulation of A two-link (2-DOF) planar robot manipulator

A manipulator consists of an open kinematic chain of rigid links. Power is supplied to each degree of freedom of the manipulator by independent torques. The dynamical equations of motion of an n-link (i.e., n-degree-of-freedom) robot manipulator using the Lagrangian formulation has already been described in Section 3 by the Eq. (8). The robot model there is characterized by the following structural properties which are important for our sliding mode controller design of the tracking problem [27]:

Property 1. A vector αRmwith components that depend on manipulator parameters (masses, moments of inertia, etc.) exists, such that

Mqv˙+Cqq˙v+Gq=Φqq˙vv˙α,E85

where ΦRnxmis called the regressor, vRnis a vector of smooth functions. This property implies that the dynamic equation can be linearized according to a specially selected manipulator parameter set, hence constituting the basis for the linear parameterization approach.

Property 2. Both Mqand Cqq˙in Eq. (8), using a properly defined matrix Cqq˙, satisfy

xTM˙2Cx=0,xRnE86

with xTthe transposition of x.That is, M˙2Cis a skew-symmetric matrix. Property 2 simply states that the so-called fictitious forces, defined by Cqq˙q˙, do not work on the system.

The model given in Figure 23 is known as a two-Link (2-DOF) planar robot, as it corresponds to the two-dimensional special case, where n=2is taken in the n-link robot manipulator [27].

Figure 23.

A two-link robot manipulator model.

The dynamic model chosen for the simulations is given by

Mθθ¨r+Fθθ˙rθ˙r+Gθ=τ,

and the dynamic equation is given by

M11M12M12M22θ¨1θ¨2+F12θ˙2F12θ˙1+θ˙2F12θ˙10θ˙1θ˙2+G1gG2g=u1u2,

where

M11=m1+m2r12+m2r22+2m2r1r2cosθ2
M12=m2r22+m2r1r2cosθ2
M22=m2r22
F12=m2r1r2sinθ2
G1=m1+m2r1cosθ2+m2r2cosθ1+θ2
G2=m2r2cosθ1+θ2

6.2.1 No boundary layer

  • The desired joint trajectory for each joint (i) is given by [27, 28] as:

θdt=90o+52.51cos1.26t

  • Initial conditions:

θ10=45°andθ20=30°.

  • The parameter values used are selected as in [27, 28]:

m1=0.5kg,m2=0.5kg
r1=1.0m,r2=0.8m

  • Matlab-Simulink implementation options used in the simulations (Figures 2431) as in [28]:

Figure 24.

Tracking error of Joint 1 displacement under SMC of the robot manipulator.

Figure 25.

Tracking error of Joint 2 displacement under SMC of the robot manipulator.

Figure 26.

Tracking error of Joint 1 velocity under SMC of the robot manipulator.

Figure 27.

Tracking error of Joint 2 velocity under SMC of the robot manipulator.

Figure 28.

Torque at Joint 1 under SMC of the robot manipulator.

Figure 29.

Torque at Joint 2 under SMC of the robot manipulator.

Figure 30.

Phase portrait of Joint 1 under SMC of the robot manipulator.

Figure 31.

Phase portrait of Joint 2 under SMC of the robot manipulator.

Sampling timeTs=1kHz,fixedstep,ode5

  • Joint 1: K=14, λ=4, ϕ=0.

  • Joint 2: K=8, λ=4, ϕ=0.

6.2.2 Introducing boundary layer

The same parameters and initial conditions for the simulations have been chosen as in Section 6.2.1 except for the following ones which include the boundary layer thickness in particular:

  • Joint 1: K=14, λ=4, ϕ=0.02.

  • Joint 2: K=8, λ=4, ϕ=0.02.

Please note that due to the space constraint, we will be able to give only the figures whose effect is clearly observed, not eight figures as given in Section 6.2.1 (Figures 3234).

Figure 32.

Torque at of Joint 1 under SMC with a boundary layer.

Figure 33.

Torque at of Joint 2 under SMC with a boundary layer.

Figure 34.

Phase portrait of Joint 1 under SMC with a boundary layer.

For tracking a desired trajectory by two-link rigid planar robotic manipulator, PID control strategy will not work well under unknown disturbances and payload changes, and hence will not be represented here. In addition, the values of control input will get higher as in the case of DC motor position control and that would complicate the realization of such high gains through the proper actuators. On the other hand, SMC provides robustness against parameter uncertainties and unmodeled disturbances so long as the observed undesirable chattering effect is overcome through some modifications by simply replacing nonlinear signum function with nonlinear saturation function and introducing boundary layer thickness in there as explained in earlier sections. In order to realize this, the boundary layer has been introduced for the first time in Section 6.2.2 simulations, and consequently, no switching or chattering effect has been observed as can be verified by the phase portrait in Figure 34.

Later, the robustness of the SMC will be analyzed by adding an extra mass of 0.5kgto Joint 2, and we have not observed any performance degradation of the trajectory to be maintained in the sliding surface. Therefore, the controller is said to be robust enough. However, it is expected that switching will reappear to maintain the trajectory in the sliding surface.

As a rule of thumb, It is possible to do tracking with more load by reducing the boundary layer to allow more switching to occur. Now, we reduce the boundary layer thickness from 0.02 to 0.005 and add the extra mass to Joint 2 by 0.75 kg to a final of 1.25 kg and we can still observe that SMC will be able to do the tracking by observing the reemerged chattering effect as can be seen in the following simulations (Figure 35):

  • Joint 1: K=14, λ=4, ϕ=0.005.

  • Joint 2: K=8, λ=4, ϕ=0.005.

  • m1=0.5kg, m2=1.25kg(an extra mass of 0.75kg was added to Joint 2).

  • The rest of the parameters and the IC’s were kept the same as before.

Figure 35.

Phase portrait of Joint 1 under SMC with a robustness test including more load.

7. Conclusions

In this study, a sliding mode control scheme with a bounded region and its convergence analysis are explained to the finest detail. In particular, it can easily be said that the work done here is a field study that specifically gives the relevant subject with such meticulous detail. It is our claim that this study has a guiding identity for the researchers who are interested in this control method or want to present it with the intelligent and modern control methodologies with its understandability and clarity targeted here. In this regard, the design of SMC including its finite-time convergence is handled by using Lyapunov’s direct method. The tracking error vector converges exponentially to the bounded region once in the boundary layer as proven analytically. Two examples were used for simulation studies to demonstrate the feasibility and effectiveness of the proposed control problems, i.e., the position control of a dc motor subject to a varying external disturbance, and a two-link robot manipulator. Simulations show that a fast convergence rate, and hence quick response, the ability to reject the varying external disturbances, and the robustness against the model uncertainty assumed to be unbounded and fast-varying have all achieved its purpose entirely. Chattering is eliminated by using the boundary layer whose attractiveness and invariance properties of the boundary layer were also introduced. This study also examines the advantages of SMC and PID comparably. Although, in the comparisons given in the literature, the pros and cons of both strategies are mentioned, it is generally observed that SMC performs better than PID. Nevertheless, PID control can still be used as an alternative to SMC. When the PID control strategy does not work well under unknown disturbances and payload changes, SMC provides robustness against parameter uncertainties and unmodeled disturbances so long as the observed undesirable chattering effect is overcome through some modifications as described in the text. Robustness analysis has been performed and successfully applied to the two-link rigid planar robotic manipulator. We have not observed any performance degradation of the trajectory to be maintained in the sliding surface. The results given here do not contradict the view that one can use it instead of the other without losing too much performance. Finally, a two-step simulation has been carried out, testing all the features mentioned above, and the results have confirmed the success of the presented approach. However, it is meaningful and challenging to develop new SMC theories and methods for nonlinear systems due to its broad application potentials in today’s world.

Conflict of interest

The authors declare no conflict of interest.

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İhsan Ömür Bucak (July 16th 2020). An In-Depth Analysis of Sliding Mode Control and Its Application to Robotics [Online First], IntechOpen, DOI: 10.5772/intechopen.93027. Available from:

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