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From Differential Equations Through LyapunovFunctions to Sliding Mode Control (v1.0)

Ogbeide Imahe

Abstract—This paper gives a holistic introduction to slidingmode control, and presents it as a suitable ’advanced’ controlmethod to teach in an introductory controls course. Sliding modecontrol is a robust control method with very good rejection ofdisturbance signals that act or appear to act through the controlinput channel. The main idea and design task is to encapsulate astable dynamic system that obeys the system model dynamics intoa variable known as the switching function. This new variable istreated as an objective function to be minimised using Lyapunovstability theory. Restatements of that theory, known as reachingconditions/laws, were derived and a general form given. Theselaws were used to derive control laws that explicitly contained thesignum function, and were applied to some nonlinear systems.The stability behaviour of a simplified version of the so-calledsupertwisting sliding mode control algorithm provided insightinto the stability behaviour of Proportional-Integral (PI) control,since they were shown to be analogous. Their similarity leadsto a generalised view of Proportional-Integral-Derivative (PID)control that suggests that the sliding mode control structure maybe used to replace or modify PID controllers in some existingand future industrial controllers.

Index Terms—nonlinear, Lyapunov stability theory, slidingmode, introduction, tutorial, supertwisting, power-rate, reachinglaws, reachability condition.

CONTENTS

I Introduction 1

II Key Concepts 2

III Lyapunov Statement Of Stability 2III-A Stability And Lyapunov Functions . . . 2

III-A1 Norms . . . . . . . . . . . . 3III-B Asymptotic Stability Illustration . . . . 3

IV Evolution of Sliding Mode Control 3IV-A From Differential Equations To Lya-

punov Functions . . . . . . . . . . . . . 3IV-B From Differential Equations to Sliding

Surfaces . . . . . . . . . . . . . . . . . 3IV-C From Lyapunov Functions to Sliding

Mode Control . . . . . . . . . . . . . . 4IV-C1 Dealing with Input Disturbance 4IV-C2 Reaching Laws . . . . . . . 5IV-C3 A Simplified Supertwisting

Algorithm . . . . . . . . . . 5

V An Illustrative Example 6V-A Cruise Control . . . . . . . . . . . . . . 6

VI Further Examples 8VI-A Single Link Manipulator . . . . . . . . 8VI-B Ball and Beam System . . . . . . . . . 9

VII Improvements and Enhanced Implementations 10VII-A Sliding Dynamics . . . . . . . . . . . . 10VII-B Implementations And Dealing With Un-

matched Disturbances . . . . . . . . . . 10

VIII Conclusion and Recommendations 11

References 12

I. INTRODUCTION

A large group of systems have disturbance signals ormodel uncertainty that appear primarily, or entirely, to affectthe system through the same integrator or channel as theinput signal. We call the disturbance/uncertainty matched inthis case, otherwise, unmatched. Sliding mode control maytherefore be suitable to control these systems given its notedinvariance/insensitivity, in the ideal case, to matched distur-bance when the so-called sliding mode is in effect. Mechanicaland electro-mechanical systems are typical examples. [1], [2],[3], [4]

Within the above group, there is a large class of systemsthat are SISO (single input with single output) and havemodels that are representable in the controllable canonicalform (1)—or that are linearisable to yield the same form. Thismodel lends itself to easy creation of the switching functionfor sliding mode control, and so is suited to an introductorynote like this one.

The controllable canonical form for nonlinear systems, withmatched disturbance d, can be given as

xi = xi+1

xn = f(xi,n) + g(xi,n)(u+ d)(1)

where n is the dimension of the system; xi, xn are the systemstate variables (i = 1, . . . , n − 1); u, the control input, andeither or both f(x) and g(x) are linear or nonlinear.

That there are many SISO systems, with matched uncer-tainty, that are representable in the controllable canonical form(as described above) makes sliding mode control a useful con-trol method to learn in an introductory controls course. Hence,in addition to being an introductory note, the presentation and

c© 2015 Ogbeide Imahe

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organisation (as shown in the table of contents) is such as todemonstrate a simple and easily understandable flow of ideasthat may be regarded as pedagogically useful. Additionally,links with classic PID control and other control methods arenoted, so as to show that sliding mode control could be usedto motivate these topics.

The paper shows how stable linear systems could be usedto derive Lyapunov functions, highlights that sliding modesare stable trajectories, and then combines these with Lya-punov stability theory to derive sliding mode controllers. Inthe process we promote the view of controllers as dynamicoptimisation engines. A view that is clearly demonstrated bythe idea known as model predictive control (MPC): whereoptimisation is explicitly used throughout the system responsetrajectory to derive a control signal that achieves an objective.We may thus see that the switching function could be used tospecify the objective for MPC.

Reaching laws are laws or conditions that may be derivedform Lyapunov stability theory that ensure that sliding modeis achieved and maintain. They were presented together asa construct in [4], but without derivation. Their derivationis simple and is shown in this paper in addition to givingreasons why the parameter value, α, in the so-called power-rate reaching law is commonly given a value of 0.5 [5].

Stability is fundamental to control, since control may beunderstood to be a quest to avoid unstable behaviour in thelong run; typically while also attaining another objective.It is the subject of the next section. After that, we wouldevolve sliding mode control from basic ideas and discuss howthe classic Proportional-Integral (PI) controller could operatein a region of instability that doesn’t end up harmful—forthe right gain parameters and disturbance magnitude limit.Subsequently, several design and simulation examples arepresented to give a sense of the resulting control signals andassociated performance, and to highlight other considerations.In these examples, all system states were assumed to beavailable (measured, or determinable) for use in control.

The perspective taken in this paper is holistic, placing slid-ing mode control, Lyapunov functions and stable (stabilized)systems in their relative contexts. It thereby highlights anintuitiveness to the associated mathematics. The presentationattempts to build intuition and facilitate heuristic design, so itprimarily requires only a basic understanding of calculus andordinary differential equations, and the concept of control tofollow.

II. KEY CONCEPTS

Two ideas make sliding mode control when combinedtogether. One is encapsulation, while the other is Lyapunovstability theory. Encapsulation involves abstracting a desired ordesigned dynamics based on the system states into a new statevariable. This new state variable is then used in the controllerdesign. The classic example of this is the error signal: thedifference between a desired reference signal and the actualoutput.

For sliding mode control, we call this new state variable theswitching function (which hints at the idea that the eventual

control signal may demonstrate switching or discontinuousbehaviour). It defines a space such that when it is eitherpositive or negative, its gradient can be designed, via a controllaw, to drive its value towards zero. And since it representsdynamics that the systems is made to obey by the controllaw, it is both an objective function and a constraint onthe behaviour of the states of the system to achieve totaldisturbance rejection where a zero value is maintained.

Lyapunov stability theory plays the role of ensuring thatthe value of the switching function is driven to zero when notequal to zero, and kept at that value subsequently. The controllaw derived from it thus serves as an optimisation engine toachieve and maintain the desired objective.

When the switching function takes the zero value andcontinually retains it, we say that the system is in slidingmode. The system dynamics is then characterised by thedesigned dynamic response when the switching function iszero. Necessarily, therefore, the switching function must bedesigned such that it ensures that the system is stable in thesliding mode, and that it results in acceptable performance.

For appropriately chosen controller parameters, the effectsof matched disturbance signals are absorbed by the dynamicsof the switching function. And if the system arrives and stayson the sliding mode, the system becomes invariant or immuneto those disturbances.

However, unmatched disturbance signals, where present,would still filter through to the output through the states andaffect the overall performance. In this case, sliding modecontrol may be combined with other control methods orparadigms suitable for this problem [6].

Two advantages to be obtained from sliding mode controlare thus robustness to matched uncertainty/disturbance andability to choose the system dynamic response. As noted in theintroduction, there is a sufficiently large group of controlledsystems where it is sufficient to deal only (or mainly) withmatched uncertainty/disturbance. We will focus on developingsliding mode control using the concepts highlighted in thissection, in the next two sections.

III. LYAPUNOV STATEMENT OF STABILITY

Stability refers to the quality of resisting change or deviationfrom a balance point or region. It also refers to a tendency toconverge to a finite point or region of behaviour, and remainingthere.

Newtons first law of motion says that a body continues in itsstate of rest or uniform motion in a straight line unless actedupon by an external force. It thus gives two examples of stablebehaviour while noting that regions of stable behaviour may bechanged by an external force. It is the objective of controllersto ensure the stability of a system and/or give the system theability to switch stable states (or equilibrium points).

We next describe asymptotic stability from a Lyapunov per-spective and show how discontinuous control signals emergeto satisfy it.

A. Stability And Lyapunov Functions

A Lyapunov function is a scalar valued differentiable arbi-trary function of the state variables of a systems that has the

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TABLE ISIGN CONDITIONS FOR ASYMPTOTIC STABILITY OF A SINGLE STATE

SYSTEM x

x x xx

-ve +ve -ve+ve -ve -ve

following characteristics:1) It has a zero value when all the state variables have zero

values.2) It always takes a positive value except when all the state

variables have zero values.3) It’s value would tend to infinity as the absolute values

of the states tend to infinity.We can infer from the items in the above list that the Lyapunovfunction has zero as it’s minimum possible value.

In order to prove a system stable, its first time derivativemust have a value that is either negative or zero. If its gradientis negative, the Lyapunov function itself would tend to zero;which in turn means that the state variables will tend to zero,making the system asymptotically stable. If this gradient iszero, the system would just be described as stable, with thestates maintaining their values. A positive gradient impliesthat the state variables are increasing, thus implying unstablebehaviour.

Hence, a controller derived based on the gradient of theLyapunov function that is negative will make the systemasymptotically stable; it will drive the variables that makeup the Lyapunov function towards zero. Therefore all thesystem states would ideally be used to construct the Lyapunovfunction.

1) Norms: A norm is a function that maps a set of numbersor variables to a positive number greater than or equal to zero;It would only yield zero for a domain or input set of all zeros.If the set is scaled, then the resulting norm is similarly scaledby the absolute value of the scalar. Norms obey the triangleinequality: the norm of a sum of input sets is less than or equalto the sum of its norms. We can thus see that every Lyapunovfunction is a norm, and norms could be made into Lyapunovfunctions.

B. Asymptotic Stability Illustration

Table I refers to Figure 1. For asymptotic stability, the ball,distance x from the fulcrum, should tend towards a positionabove the fulcrum specified to be the equilibrium point. Ifthe ball is to the left of the fulcrum (negative side), it shouldbe moved to the right (with positive velocity). Also, if theball is to the right of the fulcrum (positive side), it should bemoved to the left (with negative velocity). It becomes clearthat the product of the position and velocity must be negativeif asymptotic stability is to be achieved.

In sliding mode control, the sliding mode is defined to bethe equilibrium point. And the sliding mode controller is whatdrives the system states towards the sliding mode.

Fig. 1. Ball on a see-saw

IV. EVOLUTION OF SLIDING MODE CONTROL

This section highlights how sliding mode control, modelsand evolves from Lyapunov stability theory.

A. From Differential Equations To Lyapunov Functions

We show an example of how a Lyapunov function mayencapsulate a stable dynamic system by working from thesolution of a simple stable linear system to a Lyapunovfunction (equations 2 to 5). Define

x(t) = x(0)e−kt (2)

so that,x(t) = −kx(t) (3)

Equation (2) is an asymptotically stable path for x(t) withk > 0 and initial value x(0). It is the solution to the differentialequation (3), which must therefore be a stable system.

Let us simply represent x(t) by x. If we multiply equation(3) by x, it gives (4), which has a strictly negative right-hand-side for x 6= 0.

xx = −kx2 (4)

Also, define the gradient of a Lyapunov function V , as V =xx. Integrating it with respect to t gives

V =1

2x2 (5)

It is a Lyapunov function since V (x) > 0 for x 6= 0, andV (0) = 0. Circularly, (3) is shown to be a stable system sincethe time derivative V is less than zero.

B. From Differential Equations to Sliding Surfaces

Generally, any system in the controllable canonical form (1)could have a linear switching function designed in the form(6), or equivalently, (7). The superscripts of x1 are derivativedegrees, n is the order of the system, and ai is constant, (i =1, . . . , n− 1).

S = x(n−1)1 (t) + an−1x

(n−2)1 (t) · · ·+ a1x1(t) (6)

S = xn(t) + an−1xn−1(t) + an−2xn−2(t)· · ·+ a1x1(t)

(7)

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The equation of the sliding mode, S = 0, may then be writtenas (8). For stability, bi (i = 1, . . . , n) is a chosen to be positiveand Real.

0 = (xn+ bn)(xn−1+ bn−1)(xn−2+ bn−2) . . . (x1+ b1) (8)

Equation (8) is an equation specifying the designed dynamicsfor the system in the sliding mode. This echoes [7], thatany stable differential equation in the states of a system thatrespects the relationships of the states in the system modelcould be the sliding mode.

C. From Lyapunov Functions to Sliding Mode Control

The function of the controller reduces to driving the systemstates to the point where they obey the sliding mode, andkeep them there. We could thus use a suitable Lyapunovfunction—considering the sliding mode as the equilibriumpoint of the system—to derive a family of controllers thatcould satisfy this objective.

Let us define a Lyapunov function, V = |S|; its firstderivative is V = S d|S|dS , thus V = S · sgn(S). For stability,set S · sgn(S) = −η; rewritable as (9).

S = −η · sgn(S), η > 0 (9)

Equation (9) would ensure that S → 0 in finite time. It is calledthe η-reachability condition [8] or the contant-rate reachinglaw [4]. It is the basis of classic sliding mode control.

Similarly, let V = 12S

2; its first derivative would beV = SS. For stability, we can set SS = −η|S|, which thenleads again to (9).

If we rephrase (9) as

dS

dx· x = −η · sgn(S), x = f(x, u) (10)

where x defines the system of differential equations that modelthe dynamical system, and u its control signal, we see that Smust be designed a function of all the states of the system.And since u is the manipulated variable, it must be a functionof sgn(S).

From vector calculus, dSdx is the vector perpendicular to S.

Therefore its inner product with any vector field (e.g. x) showswhether the vector field faces the same direction with it, or isopposite it in relation to S = 0. Therefore, for S = 0 to beattractive to the state trajectories that make it true, the controlsignal must ensure that if S > 0, then the left-hand-side of(10) must be negative, and vice versa. (The left-hand-side of(10) can also be described as the Lie derivative of S in thedirection of x).

We can also write

S = h(x) + j(x)u (11)

where h(·) and j(·) are functions obtained by taking thederivative of S, for instance of (7), given (1). Equation (11)shows that the system should be linear in the control so thatit could be easily made the subject of the equation. (It seemsfeasible to also have a system linear in an exclusive functionof the control, and still be able to put the control signalexclusively on one side of the control equation.)

Combining (9) with (11) , we get the controller

u =1

j(x)(−η · sgn(S)− h(x)) (12)

We see from (11) and (12) that j(x) 6= 0. Put differently, j(x)must be invertible.

The next subsection discusses how the size of η may be de-termined: we consider the bounds of all the non-signum termsof the controller (12), and any potential input disturbances.

1) Dealing with Input Disturbance: Matched disturbancesmodify S (13), which then affects S. But if regardless of this,S is driven to zero, we may then intuitively see why invarianceto input/matched disturbance exists as a characteristic ofsliding mode control.

Where the disturbance signal is unmatched, it means thatit affects a state directly, and outside of where a matchingcontrolled signal could be used to compensate for its effect.It may thus be seen as implicit in the variable processed bythe system; so that when S = 0, their effect on the subjectvariables remain.

Considering input disturbance d, we can rewrite equation(11) as

S = h(x) + j(x)u+ d (13)

Combine (12) and (13) to yield

S = −η · sgn(S) + d (14)

S = −sgn(S) (η − sgn(S)d) (15)

with η − sgn(S)d > 0, for asymptotic stability. Thus chooseη > |d| to this meet condition.

Disturbance inputs within the specified bound would there-fore appear as if they do not exist when the system is in slidingmode. This is the ideal behaviour.

If we choose to regard h(x) as a disturbance input also, thecontroller (12) becomes

u =1

j(x)(−η · sgn(S)) (16)

with the value of η adjusted as appropriate. The variable j(x)may be replaced by a constant, min(J(x)), or its mean value,with the value of η similarly adjusted to ensure asymptoticstability. This would transform (16) to the simpler structure

u = −η · sgn(S) (17)

Using (17) as derived in this section, effectively turns anynonlinear system in the form (1) to a linear system with addeddisturbance inputs.

Since computer simulations can be used to heuristicallydetermine η, designing a suitable S may be regarded as themain design activity for sliding mode control. It was shownin section IV that this is relatively easy for systems in thecontrollable canonical form.

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2) Reaching Laws: A reaching law is an equation like (9)that makes S = 0 achievable and maintainable; attractive to thestate trajectories. We elaborate here on the power-rate reachinglaw, and the general form of reaching laws stated in [4].

The general form of a reaching law was given as

S = −Qsgn(S)−Kf(S) (18)

where f(0) = 0, and K,Q > 0. Multiplying (18) by S gives

SS = −Q|S| −KSf(S) (19)

Therefore, for the stability of (18), we design Sf(S) ≥ 0.This means that both S and f(S) are the same sign, so thatwe can rewrite (18) as

S = − (Q+K|f(S)|) sgn(S) (20)

Equation (20) is the η-reachability condition (9) with ηset as (Q+K|f(S)|), therefore (18) is a restatement of thiscondition. The import of this is that we can derive a familyof controllers with different behaviours by making η in (9)variable: specifically, by making it a suitable function of S,and more specifically, a suitable function of the state variables.

For instance, if we set Q = 0, K = 1, and |f(S)| = k|S|α,0 < α < 1, (20) yields the power-rate reaching law

S = −k|S|αsgn(S) (21)

And if α = 1, we get what is akin to classic proportional con-trol. (The sliding mode for the classic proportional controlleroccurs when the error signal equals zero.) Therefore, we caninfer that the power-rate reaching law is a generalised viewof proportional control. By this, we also see that proportionalcontrol, in the classic PID controller formulation, deals withdisturbances acting through the input channel.

We can now rewrite the η-reachability condition as

S = −η(S) · sgn(S) (22)

where η(S) ≥ 0, with η(S) = 0 iff S = 0, to moreclearly reflect a general view. Hence, we can from it definenew reaching laws that satisfy the conditions for asymptoticstability. For instance:

S = −k(|S|

|S|+m

)sgn(S) (23)

S = −k(1− e−a|S|

m)sgn(S) (24)

S = −k (a|S|+ |S|α) sgn(S) (25)

Equation (13) shows that the disturbance d filters throughto S by affecting S. If the controller is a function of |S|, thecontrol signal gets adjusted to compensate for changes in themagnitude of d, up to the limit it can handle, given the valuesof its parameters, the size of S, and actuator limits. This is why(21), (23)-(25) could still be effective for disturbance rejectioneven if they appear to contradict the idea of specifying η as afixed value to deal with disturbances, as was done for (17).

3) A Simplified Supertwisting Algorithm: The supertwistingsliding mode controller has been said to be one of the mostimplemented of sliding mode control algorithms [5]. It can bewritten in simple form as

u = −k|S|αsgn(S)−∫w · sgn(S), w, k > 0 (26)

From (26), we see that in its essence, it combines the power-rate reaching law with an integral term in the sign of theswitching function; thus mimicking the structure of the PIcontroller. It would be exactly like the classic PI controllerwith a varying proportional gain (k|S|α−1) if w = ki|S|(ki > 0 the integral gain). The ability of the integral termto accumulate its input may thus lend it a greater robustnessto disturbance signals.

Given the preceding discussion on reaching laws, its corre-sponding reaching law may chosen to be

S = −k|S|αsgn(S)−∫w · sgn(S) (27)

Multiplying (27) by S gives

SS = −k|S|α+1 − S∫w · sgn(S) (28)

This means that S∫w · sgn(S) ≥ 0, for (28) to represent

a Lyapunov stable system at S = 0 and an asymptoticallystable system for S 6= 0. Alternatively, for stability, k|S|α+1 ≥|S∫w · sgn(S)|. The corresponding Lyapunov function is

known to be V = 12S

2.Where S, S = 0, the system would be stable. It is also stable

where S and∫w · sgn(S) have the same sign; like the case

where S approaches zero from it’s initial value.As S crosses zero the first time, it changes sign to that

opposite of∫w · sgn(S), potentially making the system

unstable, while at the same time reducing the magnitude if theintegral term. (For very small S, after crossing, the system maystill be stable.) If the system leaves the stability region, therelative values of the parameters k and w, given the selectedα, play a role as to how quickly the system kicks back intostability as the size of integral term reduces, while the termwith |S|α+1 increases with S to the point where the alternativecondition for stability stated above is achieved.

If, however, S and∫w · sgn(S) have different signs, and

the sign of S remains the same, and long enough for theaccumulated value by the integral term to exceed k|S|α, thesystem tips into instability and remains there. (A Lyapunovfunction to directly assess the stability of the supertwistingsliding mode control algorithm was presented in [9]. It mayalso thus be useful for analysing PI control.)

Let S = q(x, t) + v(x, t)ust, 0 < m ≤ v ≤ M , and q ≤ pwith p > 0. Then the full supertwisting controller [5] is givenas

ust = u1 + u2

u1 = {−k|S0|αsgn(S) if |S|>S0

−k|S|αsgn(S) if |S|≤S0

u2 = {−ust if |ust|>usat−w·sgn(S) if |ust|≤usat

(29)

with parameters determined via k2 ≥ 4pM(w + p)

m3(w − p), w >

p

m,

and 0 < α ≤ 0.5. Observe that a limit usat is placed on the

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magnitude of the control signal such that the integral term u2is only in effect when the control signal is not saturated atusat. This would stop the control signal from growing due tothe integral term when it is saturated, alleviating the effectknown as ’integral windup,’ which may slow the responseof the controller when the sign of S changes. (Also knownto be potentially a serious issue with classic PI control.)Furthermore, notice that the magnitude of S is also givena limit S0 which is useful for managing the size of theproportional term u1 when S > S0 and is thus another waydesign for actuator limits.

Analogously, replacing S (or sgn(S)) with the classic errorsignal e turns the above controller into an ’advanced’ PIcontroller with anti-windup protection.

V. AN ILLUSTRATIVE EXAMPLE

In this section, we simulate the phenomenon of chattering[10] when the classic sliding mode controller is used. Andwe show graphically that for controllers using reaching lawslike (21) e.g., (23)-(25), chattering is minimal or non-existent.Also, we present PI control from the perspective of slidingmode control, highlight its relationship to the super-twistinglaw, and thus suggest a modification to the classic PI controllerfor improved disturbance rejection.

Generally, we attempt to show how the control signal affectsthe trajectory of the switching surface, and vice versa, byviewing the locus of the error signal for a simple first-ordersystem.

A. Cruise Control

We use a simple first order cruise control system for a carin fourth gear [11] for the illustrations. The reference speedis r = 25m/s, u is the controller, and the state variable thatrepresents its velocity is y. Choosing e = y − r, the systemmodel becomes

e = −0.0142e+ 1.38u (30)

If we consider the slope of the road, we can obtain the systemequation

e = −0.0142e+ 1.38u− g sin θ, g = 9.81m/s2 (31)

where θ is the slope of the road.The control objective is to drive e to zero, to make r = y.

To ensure that this first order system is stable (that e → 0),when e is positive, we would want u to be negative and viceversa. This results in a controller given by

u = −k · sgn(e) (32)

with k chosen heuristically to compensate for the disturbance,to and enhance system performance. Figure 2 shows theperformance of (32), the classic sliding mode control equation.

Observe that the error signal e behaves as a switchingfunction so that its corresponding sliding mode is a pointand not a dynamic (sliding) trajectory. (The switching functiontypically defines a dynamic system, but as done here, it couldbe defined to not exclude systems not described by differential

Fig. 2. Classic sliding mode cruise control: k = 0.95

equations. This view somewhat makes all controllers thatattempt to minimise the error signal sliding mode controllers.It is therefore unhelpful in delineating this control methodfrom others. The explicit use of the signum term in thecontroller and a recognition that the variable it maps to asign encapsulates a stable constrained ’version’ of the systemshould sufficiently distinguish sliding mode control from othermethods.)

Figures 3-6 show plots of the system response using aPI controller (with e = r − y), and several sliding modecontrollers. The classic PI and classic sliding mode controllersprovide a background to assess the performance of the othercontrollers designed using a power-rate law, a sigmoid basedlaw, and a super-twisting control law.

The controllers where manually tuned in the simulationsto ensure the control signals fell within the range 0 ≤ u ≤ 1without saturation. Performance may be improved if saturation(staying at u = 1) is allowed.

Figure 2 shows a chattering (rapidly varying) control signaland output. We can imagine that for most mechanical actua-tors and controlled systems, this input and response may beunrealisable given mechanical inertia and energy requirements.However, this controller would give the best disturbanceresponse compared to the others in this section.

The causes of chattering are noted to be finite switchingfrequency and the interaction of the switching control withparasitic dynamics (e.g. of unmodelled, fast, actuator andsensor dynamics) [1]. Therefore, sliding mode controllers thatuse continuous signals naturally tend to eliminate actuatorchattering.

Controllers derived via (21) (23)-(25) (27), do not force S tobe discontinuous across S = 0. This makes the sliding modecontroller continuous, and therefore allows it to be used withsystems that are sensitive to chattering control signals. Alterna-tive ways to eliminate chattering include to put the controllerin a model reference framework that isolates the chatteringfrom the actuator [1], and averaging the discontinuous signalvia integration or filters [8] [10].

Furthermore, the so-called higher-order sliding mode con-trollers involve switching on derivatives of S. Their integrals,

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Fig. 3. Classic PI cruise control: P = 0.8, I = 0.25.

Fig. 4. Power-rate reaching law cruise control: α = 0.5, k = 1.2.

which are continuous, thus appear in the control signals. Anexample is the super-twisting algorithm (section IV-C) becausethe integral term results from switching on the first derivativeof S|S| . It is specifically classed as a second-order sliding mode

controller since the order assigned is the derivative degree plusone [5], [13].

Additionally, there are cases where chattering control sig-nals do not lead to chattering actuators. This is because someactuators have a good filtering effect on the discontinuoussignals. In these cases, they yield a continuous input to thesystem under control still with the benefit of disturbancerejection. This is particularly the case with electric motors[3]. Thus, we can assert that chattering need not be an issuein sliding mode control implementations.

Regarding the power-rate control law, u = −η|e|αsgn(e)(used in figure 4), if α = 0, it yields the classic sliding modecontroller and a response like figure 2. Generally, the outputwould chatter for α very near zero. If α = 1, it yields theclassic Proportional controller. We note also that as α tendsfrom zero to one, the offset between the reference signal andthe steady state output increases while the rise time decreases.Thus α = 0.5, the mid-point, represents a sort of optimal valuefor a trade-off between rise time and output offset, which is

Fig. 5. Sigmoid law cruise control: ε = 0.25, k = 1.3.

Fig. 6. Super-twisting law cruise control: W = 0.05, k = 0.87.

one likely reason why it is regarded as a reasonably goodchoice in [5]. Also, if the power-rate law were exclusivelyapplied to a simple second-order system with the error signalas the switching function, α = 0.5 yields the smallest steadystate error without oscillation. Any α < 0.5 would produceoscillations. And as one goes from 0.5 to 1, the steady stateerror increases.

For α > 1, the shape of |S|αsgn(S) changes such that,in the limit, it approximates what is known as a deadzone.However, it is possibly to use an |S|ρ term in an addedpower-rate controller, with ρ ≥ 1, in a way that shortens thetransient period, particularly when |S| > 1. Generally, there issome flexibility in choosing α, and combining reaching laws.Our choices would then depend on performance requirementsand actuator capability, as would be illustrated in the ’furtherexamples’ section.

We see from the figures 4 and 5 that there is an offsetbetween the achieved steady state output and the referencesignal—as would be obtained with classic proportional control.Hence we can infer that the sigmoid and power-rate lawsdemonstrate proportional action and are also proportionalcontrollers. Adding Integral-control would thus prove usefulto eliminate this situation since the control signal is not

8

necessarily zero when the switching function is zero.Figure 6 demonstrates a PI controller where the input is

chosen as sgn(e) rather than e; this is a super-twisting con-troller. (We ignore adding a derivative term with sgn(e) sinceit is constant except at the ideally instantaneous transitions.)

The superior performance of the supertwisting controller(with S = e)to the classic PI controller may be due to thestronger proportional action when 0 < S < 1, of the power-rate law (lower tracking offset). It therefore needs less integralaction to perfect the tracking. A lower offset means that theaccumulated integral value needs be less than would otherwisebe needed. Smaller integral term growth makes overshootminimal, as can be observed when we compare the top plotsin figures 3 and 6.

Also, the size of its integral term changes according tothe sign of the error signal rather than the size and sign. Ittherefore has a fixed change per time, so that its effect in morepredictably modifiable using its gain W . Hence, we may wantto replace, for general use, the classic Proportional controllerthat uses α = 1 in the power-rate law with one using, say,α = 0.5. A combination of the two might also be a goodoption. And where there’s an integral term, to use sign(e) inplace of e.

Given the above discussion, we can suggest that improve-ments to the controller structure of sliding mode control mightbe translatable to improved PID control, and vice versa.

VI. FURTHER EXAMPLES

The first example is the tracking control of a single linkmanipulator. It is representative of a simple robot arm, arudder, a vane or similar. The control objective is for the angleto a reference point to follow a planned path as best as possibledespite the presence of input or matched disturbances, like,say, undulating gusts of wind.

Because the model is in the controllable canonical form, wecan design a switching function in the states of the system byspecifying a stable linear system. Arbitrarily choosing polesthat are negative would suffice to result in a stable system inthe sliding mode.

A case where this isn’t directly feasible for stability inthe sliding mode is explored in the second example wherewe simulate set-point tracking control in a nonlinear ball andbeam system model. This example specifies the required stablepoles without the superfluous step of using the Routh stabilitycriterion as in [14] and [15], to achieve this.

We use a plug-play-tune approach for parameter selectionin a simulation environment, akin to manual PID tuning. Thisdemonstrates a comparable simplicity in use to PID control;the extra step, unlike with PID, being to design a dynamicsystem for the switching function. Once this is done, and thecontroller structure selected, parameters may then be modifiedto suit actuator limits and disturbance rejection ability withsimilar reasoning to manual PID tuning.

A. Single Link Manipulator

This example shows that sliding mode control may be usedto replace PID control in plants where facility exists, or may

Fig. 7. An Implementation Scheme for PI Control

be added, to replace the error signal e with |S|αsgn(S). Italso demonstrates a usefulness of this control method formotion control, and particularly where there is significant inputdisturbance present.

The model (33) from [16] is used, with the addition of adisturbance input signal:

x1 = x2x2 = −9.8sin(x1)− 3x2 + 0.5(u+ d)

(33)

Following from section IV-A, let us define a switching functionS = x2 + 30e, which, when S = 0, yields the first ordersystem x1 = −30(x1 − r) that has a steady state value ofr, the reference signal. We avoided using S = e + 30e asmight have been expected because it includes a derivative ofthe reference signal. This reduces the size of the spike in S,that would transfer to the control signal given the selectedcontroller design, if r suddenly changes value. (A better waymay be to shape/filter the input so that there are no suddenor steep transitions in states thereby reducing the need toconsider their impact on performance and controller design.)Additionally, the control signal was saturated at ±35, selectedbased on the example design in [16], because real controllershave limits.

We used a simpler control design in this example; one thatfollows from the derivation of (17). The controller designedin [16] had the structure of (12).

Also, we perturbed the system with a sinusoidal disturbancesignal of amplitude 10 and frequency 0.5π. (The magnitude ofthis disturbance signal was such that classic PI control and theclassic super-twisting sliding mode control algorithm wouldlikely not produce good tracking for any parameter selection,given the control input constraints and the desire to minimisechattering.)

An implementation structure for PI control was shown in[11] which effects u = k(e + 1

Ti

∫e). We show it in figure

7, and substitute |S|αsgn(S) for e as the input for the actualcontroller design (34). We effect it using

u = −k(u1 + 1Ti

∫u1)

u1 = |S|αsgn(S) (34)

which is the same structure as the super-twisting algorithm.Figure 8 shows its performance: good tracking of the referencesignal despite substantial disturbance—the main selling pointfor the use of sliding mode control.

The significance of figure 7 is also that it describes acontroller as an engine to dynamically optimise the value of

9

Fig. 8. Top: Tracking performance; Middle: Input signals; Bottom: Trackingerror. (k = 35, α = 0.75, T i = 0.2)

an input objective function, the optimisation of which achievesthe control objective. (Minimisation is the usual case.)

We had previously assessed the stability of the PI controlanalogue that is the supertwisting algorithm formulation, andso can get a sense of the contribution of the intergral term in(34) to its overall disturbance rejection performance.

B. Ball and Beam System

This article demonstrates sliding mode control for an un-deractuated mechanical system, and shows the effect of un-matched disturbances.

The nonlinear ball and beam system model (35) is takenfrom [14], which took the model structure from [17] andparameters from [15]. This model mimics the ball on a see-sawdescribed earlier, giving the view that sliding mode control isa natural fit for it. Sliding mode control should be a naturalcontrol option for systems where balancing the output variableat a point (regulation) is a key feature—as with the invertedpendulum and flight controls in general.

x1 = x2x2 = 1

k4x1x

24 −

gk4sin(x3)

x3 = x4x4 = 1

mx21+k1

[k2u− (2mx1x2 + k3)x4

−(mgx1 + L2Mg)cos(x3)] + d

(35)

With the system states defined thus: x1, the ball position onthe beam; x2, the ball velocity; x3, the angle of the beamto the horizontal, with −π6 < x3 <

π6 ; and x4, the angular

velocity of the beam. The parameters are k1 = 0.16956kgm2,k2 = 1.00083333Nm/V , k3 = 16.55455208V/(rad/s),k4 = 1.4, L = 0.43m (beam length), g = 9.81m/s2

(gravitational acceleration), m = 0.07kg (ball mass), andM = 0.15kg (beam mass). The control input is the voltageu to the electric motor that drives the beam. The desiredequilibrium state is x2 = x3 = x4 = 0, and x1 = r; that is,that we want to be able to make the ball stationary virtuallyanywhere r on the beam.

Even though the system is not in the form (1), we can stillchose the switching function to be S = x4+px3+qx2+mx1and assess its suitability for the purpose. If S is driven to zero,we get

x4 = −px3 − qx2 −mx1so that the resulting dynamics is the nonlinear third ordersystem

x1 = x2x2 = 1

k4x1x

24 −

gk4sin(x3)

x3 = −px3 − qx1 −mx1Its performance and stability around an equilibrium point canbe designed using an appropriate selection of p, q, and m,despite that x3 has a nonlinear relationship with x2, andtherefore, also with x1.

It is known that if a linearised system, that is, a linearapproximation of a nonlinear system at an equilibrium point,is strictly stable, then the associated nonlinear system has thatequilibrium point asymptotically stable. Also, x3 ≈ sin(x3)where −π6 < x3 <

π6 . We may therefore linearise the above

system around the stationary point x4 = x3 = x2 = x1 = 0to give

x1 = x2x2 = − g

k4x3

x3 = −px3 − qx1 −mx1(36)

From this we get x3 = −k4g x2. The sliding mode equationcan then be rewritten as

...x1 + px1 −

g

k4qx1 −

g

k4mx1 = 0 (37)

with which we can do pole placement.Placing the poles at [−1,−5,−5] gives the characteristic

equation P 3+11P 2+35P +25 = 0, in a variable P . Hence,we have the sliding mode system as

...x1 + 11x1 + 35x1 + 25x1 = 0 (38)

Equating (37) and (38) provides the parameters for the switch-ing function, yielding it to be

S = x4 + 11x3 − 5x2 − 3.6x1

For tracking control, we replace x1 with e = x1 − r.Figures 9, 10 and 11 show the performance of the system

under different conditions. The control signal used is (34)with parameters as specified in their respective captions.Figures 9 and 10, respectively, show the performance withoutdisturbance, and with matched input disturbance equivalent toa constant 2V input. Such a disturbance signal might perhapsbe due to the beam being actually lopsided, with a centre ofgravity not at its middle.

The tracking performance shown in figure 10 (with distur-bance) is better than in figure 9 (without disturbance) becausethe controller parameters were selected with the assumptionthat the system would normally operate with disturbanceinputs.

Figure 11 illustrates the response to an unmatched distur-bance signal (arbitrarily applied at x3 and chosen to be asinusoidal signal of amplitude 0.5 rad/s and period 2π). Un-matched disturbance may be caused by some real extraneous

10

Fig. 9. A PI-Sliding mode control of a ball and beam system. (k = 7,Ti = 0.2, α = 0.75)

Fig. 10. A PI-Sliding mode control of a ball and beam system with asimulated disturbance input of 2 Volts. (k = 7, Ti = 0.2, α = 0.75)

Fig. 11. A PI-Sliding mode control of a ball and beam system withunmatched disturbance signal. (k = 7, Ti = 0.2, α = 0.75)

signals, model approximation, and parameter variation; all notentering the system through the same integrator as the input.

If this is significantly present for the system to be controlled,then sliding mode control by itself would be unsuitable. Thenext section briefly surveys improvements to sliding modecontrol as presented in this paper, including attempts to dealwith unmatched disturbance/uncertainty.

VII. IMPROVEMENTS AND ENHANCED IMPLEMENTATIONS

We consider that a key area for enhanced implementation isin dealing with systems that have unmatched uncertainty. Herewe briefly mention some attempts and proposed solutions.

While we do not consider reaching phase elimination sig-nificant for the increased and broader use of sliding modecontrol, it nevertheless represents an advancement in slidingsurface design.

We also briefly mention some implementations, and a wayto potentially design improved sliding dynamics comparedwith pole placement.

A. Sliding Dynamics

Since S determines the sliding mode dynamics, theparadigm used to design it may lead to better transient perfor-mance of the states while in the sliding mode. Bandyopadhyayet al, in [18], discuss nonlinear sliding surfaces for highperformance tracking and robustness. A nonlinear functionwas used in the switching function design such that, while inthe sliding mode, the system has a low initial damping ratio(for fast rising); this ratio increases over time to a relativelyhigh value as it approaches steady state. This approach wassaid to shorten the transient period while also preventingovershoot.

Removing (or reducing) the reaching mode implies thatthe disturbance rejection property is effective throughout (orfor most of) the system operation. It therefore represents animprovement in disturbance rejection performance to eliminatethe reaching phase. An approach to doing this uses a dynam-ically changing switching function. This involves a nominalor final sliding mode system added to a function to form theoverall sliding mode system. The value of the function is madeto disappear over time in such a way as to maintain a slidingmode until the nominal sliding mode is achieved.

A way to design nonlinear sliding surfaces that eliminatethe reaching phase by adding a nonlinear term to the nominalsiding surface was proposed in [18]. The integral sliding modeapproach [3] adds an integral term to the nominal slidingmode system. In practice, its performance would be adverselyaffected by noise in measurements [3] but good design choiceswould alleviate this issue [6].

B. Implementations And Dealing With Unmatched Distur-bances

In [12], the error between the plant and a reference modelwas made to exhibit a sliding mode thereby providing refer-ence model tracking by the system under control. The use ofsliding mode control in model reference control systems was

11

said to enable simpler reference model tracking while alsodealing with system uncertainties/disturbances [19].

Using a similar concept, sliding mode observers for stateestimation is discussed in [20], where the observer dynamicsare made to exhibit a sliding mode. A benefit of observerbased sliding mode control in disturbance rejection is basedon the fact that it can be used for both state and disturbanceestimation [1].

We had assumed that all the state values required for controlare available (measurable or determinable). See [8] for anapproach to use output feedback, where only the output isavailable for control determination.

Some systems have unmatched disturbance inputs. Sincesliding mode control by itself is only suited to matcheddisturbances, it may be combined with other suitably designedcontrollers to deal with the unmatched case. Castanos andFridman in [6] combined what is known as H∞ robust controlwith sliding mode control to deal with both matched andunmatched disturbance inputs.

An alternative approach was attempted in [21] for underac-tuated mechanical systems, where the switching function wasconstructed hierarchically in a similar general philosophy tothe so-called backstepping paradigm of deriving stabilisingcontrollers. However, as presented, their approach can notdeal with output disturbance. A backstepped sliding modecontroller with the possibility of dealing with unmatchedinputs at any part of the model was presented in [22]. Thisapproach is viable because backstepping creates virtual controlsignals that attempt to directly control the rate of change ofeach state.

For discrete-time systems: [23] presented an investigationof the implementation of sliding mode control for such sys-tems, touching on the basic theory and design techniques.And a discrete-time sliding mode control implementation thathandles unmatched disturbances was proposed in [24] for adiscrete-time linear time-invariant system.

VIII. CONCLUSION AND RECOMMENDATIONS

This paper has attempted to present sliding mode controlsimply and intuitively with the aim of providing an introduc-tion to the subject, generating increasing interest in the methodand its increased utility, and demonstrating that it is teachableand pedagogically useful in an introductory course in controlsystems engineering.

We’ve considered sliding mode control as involving a vari-able that is an encapsulation of a dynamical system in thestate variables of a model. Using this view, we highlightedan analogy with PID control, and determined the power-ratereaching law to be a generalised proportional-controller.

The power-rate reaching law has a parameter, α, that canbe chosen such that better performance may be obtained. Thiscan thus be used to replace classic Proportional control inindustrial controllers.

We’ve demonstrated that a simplified supertwisting slidingmode control structure with the sign of the switching func-tion replaced by the switching function itself could createan enhanced version of it with superior input disturbance

rejection performance. Conversely, for the classic PI controller,replacing the error signal with its sign produced a superiorversion of it. This version could plausibly be used to replaceclassic PI control, particularly where robustness to matcheddisturbance is a key requirement.

Generally, sliding mode control, as presented here, couldbe used as an alternative to, or a replacement for PID control.Industrial PID controllers could be designed to accommodatea generalised view of Proportional-Integral control, to providefacility for specifying an objective/switching function (theerror signal in the classic case) to be minimised, and forgains that vary with the switching function. This shouldthus facilitate the expanded use of the sliding mode controlperspective, particularly in industries where PID is commonlyused.

The analysis of the stability of the supertwisting slidingmode control algorithm provided intuition as to the conditionsfor stability of the classic PI controller. It demonstrated usingthis analogy, how a PI controlled system may be stable;highlighting that the relative sizes of the proportional andintegral gains in the PI controller matter to the overall stabilityof the controlled system. Discussions on the stability behaviourof PID controllers appear to be rare despite its simplicity, theubiquity of its treatment in controls courses, and popularity inindustry.

A PI control implementation structure was shown to modelthe idea of a controller as an engine to dynamically minimisean input objective function. That is, that a controller is afunction that dynamically progressively maps an objectivefunction variable towards the value zero. This is the viewexpressed by sliding mode control, reducing the job of findinga control signal for any system to determining appropriateobjective functions and an adequate mapping or optimisationfunction that sufficiently minimises the objective.

Because reaching laws/conditions are a rephrase of Lya-punov stability theory, and switching functions encapsulatestable dynamics, the principles of sliding mode control may beextended to other areas of knowledge where Lyapunov stabilitybehaviour have been applied or observed.

We assert that the chattering phenomenon is no longer anissue in the implementation of sliding mode controllers giventhat there are continuous controllers that can effect the slidingmode. And chattering has typically not being an issue withdiscrete-time controllers.

In regards to pedagogy in introductory controls courses:Most introductions to control systems engineering seem toleave Lyapunov stability theory for a second course or ad-vanced course. This paper is of the view that an introductorycourse in control engineering ought to include treatment ofLyapunov stability theory and some treatment of nonlinearsystems in order to present a more complete overview of thecontrols landscape. As we have seen—and probably alreadyknow—Lyapunov stability theory provides an intuitive anduseful view of the stability of linear and nonlinear systems,particularly in relation to controller synthesis.

PID control is a ubiquitous control method that is commonlyincluded in introductory control engineering courses. Giventhe intuitiveness and simplicity of sliding mode control, that

12

PID control may be extrapolated from it, that is has utility forboth linear and nonlinear systems, and that it can in almostsimilar ways be applied like PID control, there is a casefor its use (or increased use) in first/introductory courses incontrols. It may also used as a lead in for related advancedconcepts/coverage/topics in control like feedback linearisation,and Lie derivatives. The content of this article serves towardsthese objectives.

Finally, it is likely the case that there is already literaturerelated to an application area you might be interested in. Theliterature on the subject is large, as a cursory search on theinternet might show. This introduction is like the proverbialtip of the iceberg.

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