Comparative study of the saturated sliding modeand anti-windup controllers

Chaker ZaafouriResearch Unit c3s ESSTT,

University of Tunis, TUNISIAchaker.zaafouri@enib.rnu.tn

Anis SellamiResearch Unit c3s ESSTT,

University of Tunis, TUNISIAanis.sellami@esstt.rnu.tn

Germain GarciaLAAS-CNRS

University of Toulouse, FRANCEGarcia@laas.fr

Abstract—The saturation problem is the one of the mostcommon handicaps for applying linear control to real applica-tions, especially the actuator saturation. This paper focus on acomparative study between the classical anti-windup regulatorand robust saturated sliding mode control. In the first step,we present a design methodology of SMC of a class of linearsaturated systems. We introduce the structure of the saturation,them we perform the design of the sliding surface as a problem ofroot clustering, which leads to the development of a smooth andnon-linear control law that ensures to reach the sliding surface.The second step is devoted to present briefly the anti-windupcontroller technique. Finally, we use an example of a quarter ofvehicle system to give simulation results.

Keywords: Variable Structure Control; Sliding Mode; Anti-windup; Satureted Systems; LMI.

I. INTRODUCTION

Most industrial processes operate in the areas caracterisedby many physical and technological constraints (saturation,limit switches...). The implementation of the control lawdesigned without considering these limitations can have direconsequences for the system.The problem of the control of saturated systems is a subjectof great interest for applications. Used in early days, manyrigorous design methods are available to provide guaranteeproperties on systems stability. Let us quote of these methods,the anti-windup design ([1], [2]...), and many other methodswhich introduce conditions on systems containing saturationfunctions ([1], [3], [4], [5], [6]...). In this paper, we areinterested the problem of sliding mode control of linear systemwith saturation on the entries. The sliding mode representsa very significant transitory mode for the Variable StructureControl (VSC) in terms of robustness. Early work in this areawas mainly done by Soviet control scientists ([7], [8]...). Inrecent years, more and more research in this area has beendone ([6], [9], [10...).This paper is organized as follows: in the beginning, we give ashort introduction on the structure of the saturation constraintreported on the control vector and its implementation in thesystem. We will then present a design procedure of robustsaturated sliding mode control. To validate the theoreticalconcepts of this work, we treated an application of a quarter ofvehicle system where we will highlight a comparison betweenthe sliding mode control and the anti-windup controller.

II. SYSTEM DESCRIPTION AND PRELIMINARIES

Let us consider that the structure of the saturation constraintis described by figure 1:

Sat(U)

U

Usat

-Usat

-UsatUsat

Fig. 1. The Structure of the saturation constraint

Assumption: The control vector is subjected to constantlimitations in amplitude. It’s defined by:

𝑢 ∈ ℜ𝑚 = {𝑢 ∈ ℜ𝑚/− 𝑈𝑠𝑎𝑡 ≤ 𝑢𝑖 ≤ 𝑈𝑠𝑎𝑡; 𝑈𝑠𝑎𝑡 > 0} (1)

The term of saturation has the following form

𝑠𝑎𝑡(𝑢) =

⎧⎨⎩

𝑈𝑠𝑎𝑡 if 𝑢𝑖 > 𝑈𝑠𝑎𝑡

𝑢𝑖 if − 𝑈𝑠𝑎𝑡 < 𝑢𝑖 < 𝑈𝑠𝑎𝑡

−𝑈𝑠𝑎𝑡 if 𝑢𝑖 < −𝑈𝑠𝑎𝑡

, ∀𝑖 = 1, ...,𝑚

(2)we can write

𝑠𝑎𝑡(𝑢)=𝛽𝑢 (3)

the elements of 𝛽𝑖 are expressed as follows

𝛽𝑖 =

⎧⎨⎩

𝑈𝑠𝑎𝑡

𝑢𝑖if 𝑢𝑖 > 𝑈𝑠𝑎𝑡

1 if − 𝑈𝑠𝑎𝑡 < 𝑢𝑖 < 𝑈𝑠𝑎𝑡−𝑈𝑠𝑎𝑡

𝑢𝑖if 𝑢𝑖 < −𝑈𝑠𝑎𝑡

, ∀𝑖 = 1, ...,𝑚

(4)The saturated system can be written as:

�� (𝑡) = 𝐴𝑥 (𝑡) +𝐵𝛽𝑢 (𝑡) (5)

with 𝐴 ∈ ℜ𝑛×𝑛, 𝐵 ∈ ℜ𝑛×𝑚, 𝑥 ∈ ℜ𝑛 and𝑢 ∈ ℜ𝑚

Assumption: The pair (𝐴,𝐵) is controllable,B has full rankm, and 𝑛 > 𝑚.

U.S. Government work not protected by U.S. copyright

The sliding mode occurs when the state reaches and remainsin the surface given by :

𝑆 =𝑚∩𝑗=1

𝑆𝑗 = {𝑥 ∈ ℜ𝑛/𝐶𝑥 = 0} (6)

The sliding mode occurs when the state reaches and remains inthe surface intersection S of the m hyperplanes, geometricallythe subspace S is the null space of C.

Differentiating with respect the time,

�� = 𝐶𝐴𝑥+ 𝐶𝐵𝛽𝑢 = 0 (7)

if (𝐶𝐵𝛽)−1 exists, then

𝑢𝑒𝑞 = − (𝐶𝐵𝛽)−1

𝐶𝐴𝑥 = −𝐾𝑥 (8)

with 𝐾 = (𝐶𝐵𝛽)−1𝐶𝐴

�� = (𝐼𝑛 − 𝛽𝐵 (𝐶𝐵𝛽)−1

𝐶)𝐴𝑥 = 𝐴𝑒𝑞𝑥 (9)

the dynamics �� (equation 9) describes the motion on thesliding surface and depends only on the choice of C.

III. DESIGN OF THE SLIDING SURFACE

The canonical form used in Reference [11] for VSC designto select the gain matrix C that gives a good and stable motionduring the sliding mode.

By assumption, the matrix B has full rank m; as a result,there exists an (n x n) orthogonal transformation matrix T

such that: 𝑇𝐵 =

[0𝐵2

], where 𝐵2 is (m x m) and non-

singular. Note that the choice of an orthogonal matrix T avoidsinverting T when transforming back to the original system. Asthe transformed state variable vector is defined as:

y = 𝑇𝑥 (10)

the state equation becomes

y(t) = T𝑥(t) = T𝐴𝑥(𝑡) + 𝑇𝐵𝛽𝑢(𝑡) (11)

If the transformed state is partitioned as 𝑦𝑇 =[𝑦𝑇1 𝑦𝑇2

];

𝑦1 ∈ ℜ𝑛−𝑚; 𝑦2 ∈ ℜ𝑚

then{��1(𝑡) = 𝐴11𝑦1(𝑡) +𝐴12𝑦2(𝑡)

��2(𝑡) = 𝐴21𝑦1(𝑡) +𝐴22𝑦2(𝑡) + 𝛽𝐵2𝑢(𝑡)(12)

Since the sliding condition is

𝐶𝑥 = 𝐶𝑇𝑇 𝑦 = 0 (13)

witch

TAT𝑇=

[𝐴11𝐴12

𝐴21𝐴22

],CT𝑇=

[𝐶1 𝐶2

](14)

We can the new defining sliding condition

𝐶1𝑦1 + 𝐶2𝑦2 = 0 (15)

Assumption: CB is non-singular then 𝐶2 must be non-singular.

The sliding mode condition becomes

𝑦2 = −𝐶−12 𝐶1𝑦1 = −𝐹𝑦1 (16)

wich 𝐹 = 𝐶−12 𝐶1 being an [m x (n-m)] matrix. The sliding

mode is then governed by the equations{

��1 = 𝐴11𝑦1 +𝐴12𝑦2𝑦2 = −𝐹𝑦1

(17)

representing an (𝑛−𝑚)𝑡ℎ order system with 𝑦2 playing the roleof a state feedback control.The closed-loop system will thenhave the dynamics ��1 = (𝐴11 − 𝐴12𝐹 )𝑦1 This indicates thatthe design of a stable sliding mode requires the selection of amatrix F such that ��1 = (𝐴11−𝐴12𝐹 )𝑦1 has (𝑛−𝑚) left half-plane eigenvalues. Performances are taken into account viaroot clustering of the closed-loop dynamic matrix in a regionof the complex plane. The area Ω(𝛼,−𝑞, 𝑟, 𝜃) consideredhere is defined in Figure 2,which ensures a minimum decayrate 𝛼 < 0, a minimum damping ratio 𝜉 = cos 𝜃, and forrelative stability and speed limitation can be made to placethe eigenvalues in a circle in the left half complex plane.

Fig. 2. LMI Region:intersection of three elementary regions.

Where F must be selected, that those (n-m) eigenvaluesof the system are in the region Ω(𝛼,−𝑞, 𝑟, 𝜃), it can bedetermined by using root clustering with LMI concept. C isgiven by them :

𝐶 = [𝐹 I𝑚]𝑇 (18)

IV. SATURATED CONTROL LAW DESIGN

Once the existence problem has been solved that is thematrix C has been determined, attention must be turned tosolving the reachability problem. This involves the selectionof a feedback control function 𝑢(𝑥) which ensures that tra-jectories are directed towards the switching surface from anypoint in the state space. The control strategy used here will bederived from that of Reference [12] which originated from thework of Gutman [13], and it consists of the sum of a linear

control law 𝑢𝐿 and a nonlinear part 𝑢𝑁 . The general form is:

𝑢(𝑥) = 𝑢𝐿(𝑥) + 𝑢𝑁 (𝑥) = 𝐿𝑥+ 𝜌𝑁𝑥

∥𝑀𝑥∥+ 𝛿(19)

where L is an (n-m) matrix, the null spaces of the matrices N;M; and C are coincident, and 𝛿 is a small positive constant toreplace the discontinuous component by a smooth nonlinearfunction, yielding chattering-free system response.

Starting from the transformed state y, we form a secondtransformation 𝑇2 : ℜ𝑛 → ℜ𝑛 such that

𝑍 = 𝑇2𝑦 = 𝑇2𝑇𝑥; 𝑍𝑇 =[𝑧𝑇1 𝑧𝑇2

](20)

And 𝑧1 ∈ ℜ𝑛−𝑚; 𝑧2 ∈ ℜ𝑚

where

𝑇2 =

[𝐼𝑛−𝑚

𝐹0𝐼𝑚

](21)

𝑇2 is non-singular as its inverse is given by

𝑇−12 =

[𝐼𝑛−𝑚

−𝐹0𝐼𝑚

](22)

The new state variables are then{𝑧1 = 𝑦1𝑧2 = 𝐹𝑦1 + 𝑦2

(23)

and the transformed system equation becomes{��1 =

∑1 𝑧1 +

∑2 𝑧2

��2 =∑

3 𝑧1 +∑

4 𝑧2 + 𝛽𝐵2𝑢(24)

with ⎧⎨⎩

∑1 = 𝐴11 −𝐴12𝐹∑2 = 𝐴12∑3 = 𝐹

∑1 −𝐴22𝐹 +𝐴21∑

4 = 𝐴22 +𝐴12𝐹

(25)

In order to attain the ideal sliding mode, it is necessary toforce 𝑧2 and ��2 to become identically zero. To this end, thelinear control law part 𝑢𝐿 is formulated as

𝑢𝐿(𝑧) = − (𝛽𝐵2)−1

[Σ3𝑧1 (Σ4 − Σ∗4)𝑧2] (26)

where∑∗

4 ∈ ℜ𝑚∗𝑚 is any design matrix with stable eigen-values. In particular, we may set

∑∗4 = 𝑑𝑖𝑎𝑔 (𝜇𝑖) such that

Re (𝜇𝑖) < 0 fot i = 1 to m. Transforming back into the originalx-space yields

𝑢𝐿(𝑥) = 𝐿𝑥 = − (𝛽𝐵2)−1

[Σ3 (Σ4 − Σ∗4)]𝑇2𝑇𝑥 (27)

𝐿 = − (𝛽𝐵2)−1

[Σ3 (Σ4 − Σ∗4)]𝑇2𝑇 (28)

Before presenting the nonlinear control law part 𝑢𝑁 letting thematrix 𝑃2 denote the positive definite unique solution of theLyapunov equation

𝑃2Σ∗4 +Σ∗

4𝑃2 + I𝑚 = 0 (29)

then 𝑃2𝑧2 = 0 if and only if 𝑧2 = 0, and we may take

𝑢𝑁 = −𝜌(𝛽𝐵2)

−1𝑃2𝑧2

∥𝑃2𝑧2∥+ 𝛿(30)

Transforming back into the original x-space, we obtain

𝑢𝑁 = −𝜌(𝛽𝐵2)

−1[0 𝑃2]𝑇𝑇2𝑥

∥[0 𝑃2] 𝑇𝑇2𝑥∥+ 𝛿(31)

since the existence of the nonlinear component is checkedthem we can deduce the matrices N and M

𝑁 = − (𝛽𝐵2)−1

[0 𝑃2]𝑇𝑇2 (32)

𝑀 = [0 𝑃2] 𝑇𝑇2 (33)

V. ANTIWINDUP CONTROL

The windup is a phenomenon in which a response becomesunstable or oscillated when the control input is saturated by alimiter in a control system which has integral terms. The maincause of this phenomenon is surplus integration of the integralterms in the controller when the input saturation occurs. As aresult, errors appear in the state variable of the controller andit causes a windup phenomenon.The integral term can be separated from PI/PID controllerbecause these structures are comparatively easy. Thereforea common practice in the PI/PID controller is to stop theintegration part while the input saturation is occurring.

Anti-windup is a traditional approach to dealing with ac-tuator saturation. The idea is to augment the closed-loopsystem that was designed without taking actuator saturationinto consideration so that the negative effect of actuatorsaturation is weakened. Earlier works on anti-windup designtry to minimize the effect of saturation in a direct way byreducing the difference between the input and output of theactuators (see, for example, [14], [15]).There have been many suggested ways of implementing Anti-windup control schemes, one of the best known is the so-calledback information anti-windup scheme whose block diagram isdepicted in Figure 3.

Fig. 3. Anti-windup control scheme

As displayed in Figure 3, the anti-windup control scheme in-troduces a new feedback signal "𝑒𝑡" (saturation error) definedas:

𝑒𝑡 = 𝑢− 𝑣 (34)

where v is the magnitude of the control action requested bythe control system, whereas u is the magnitude of the samecontrol signal coming out from the saturation element.

In Figure 3, 𝑇𝑡 stands for a saturation time constant whoseaim is to speed up the incoming of the anti-windup scheme.As a rule of thumb, normally 𝑇𝑡 < 𝑇𝑖

Under saturated conditions the integral part I of the controlsystem will be given by :

𝐼 =𝐾𝑝

𝑇𝑖

∫ 𝑡

0

𝑒(𝑡) 𝑑𝑡+1

𝑇𝑡

∫ 𝑡

0

𝑒𝑡(𝑡) 𝑑𝑡 (35)

and𝑉 (𝑡) = 𝐾𝑝𝑒(𝑡) +𝐾𝑝𝑇𝑑

𝑑𝑑𝑡𝑒(𝑡)+

𝐾𝑝

𝑇𝑖

∫ 𝑡

0𝑒(𝑡) 𝑑𝑡+ 1

𝑇𝑡

∫ 𝑡

0𝑒𝑡(𝑡) 𝑑𝑡

(36)

Therefore, from the above equation, we see that the aim ofthe anti-windup control scheme is to modify the value of theintegral control action in a way that leads to a faster recoveryfrom saturation conditions.It is clear that when no saturation problems occur v = u.therefore 𝑒𝑡 = 0 and the action of the anti-windup controlscheme will be canceled recovering the conventional feedbackcontrol structure.

VI. NUMERICAL APPLICATION

We consider a two degree of freedom vibrating system withone actuator describes by figure 4 The state equation of the

Fig. 4. Two degrees of freedom vibrating system with one actuator

system is given by,[16]:

��(𝑡) = 𝐴𝑥(𝑡) +𝐵𝑢(𝑡) (37)

with

𝐴 =

⎛⎜⎜⎝

0 0 1 00 0 0 1

−𝑘1+𝑘2

𝑚1

𝑘2

𝑚1−𝐶1+𝐶2

𝑚1

𝐶2

𝑚1𝑘2

𝑚2− 𝑘2

𝑚2

𝐶2

𝑚2− 𝐶2

𝑚2

⎞⎟⎟⎠

𝐵 =

⎛⎜⎜⎝

001

𝑚1

0

⎞⎟⎟⎠

Simulation is achieved under the following condition:

𝑚1 = 𝑚2 = 1, 𝑘1 = 𝑘2 = 1, 𝐶1 = 𝐶2 = 0.01,

−1 ≤ 𝑢 (𝑡) ≤ 1

The initial condition is given by 𝑥0 =[0 0 0 1

]𝑇

The figure (5) represents the poles of the reduced order systemin an area defined by Ω(𝛼,−𝑞, 𝑟, 𝜃) = Ω (−0.7, 0, 2, 𝑝𝑖/8) inthe complex plan.

−3 −2 −1 0−3

−2

−1

0

1

2

3

Fig. 5. Poles of the reduced system

After 3 iterations, the algorithm gives the stabilizing gain F ofthe reduced system

F = [−3.0734 −1.5077 2.7477]

According to (18)

C = [3.0734 −1.5077 −1.0000 2.7477]

Figure (6) and figure (7) present, respectively, the evolutionof the control input and state variables (- -: Anti-windupcontroller, –: sliding mode control).

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

Fig. 6. Evolution of the control law (- -: Anti-windup, –: sliding mode)

Figure (6) presents the evolution of the saturated control input.It’s clear that the two controllers are saturated and alwaysinferior to its maximal value in the two cases, but we cancheck that the Anti-widup controller have a more transientmode and the convergence is more slowly than that of thefirst control , what proves that the robust stability of the SMCis checked.

0 5 10 15 20 25 30−0.5

0

0.5

1

t

x1

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

x2

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

x3

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

x4

Fig. 7. System response (x1 to x4), (- -: Anti-windup, –: sliding mode)

Figure (7) compares displacements of x1, x2, x3 and x4 withsaturation. It shows a typical stable sliding mode convergenceof the system using sliding mode control. As consequence,the state variables dynamics of the system with anti-windupcontroller have a more transient mode and the convergence ismore slowly than that of the first control.

To analyze the robustness of the control techniques to para-metric variations, we repeated the same simulation in theprevious case but we have introduced variations of dampingcoefficients.Figure (8) and figure (9) present, respectively, the evolution ofthe control input and state variables of the uncertain system(- -: Anti-windup controller, –: sliding mode control).

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Fig. 8. Evolution of the control law of uncertain system (- -: Anti-windup,–: sliding mode control)

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

x1

0 5 10 15 20 25 30−1

0

1

2

t

x2

0 5 10 15 20 25 30−1

0

1

2

t

x3

0 5 10 15 20 25 30−2

−1

0

1

t

x4

Fig. 9. Uncertain system response (x1 to x4), (- -: Anti-windup, –: slidingmode control)

Figures (8) and (9) show that the anti-windup controlis sensitive to parametric variations. There is a significantperformance degradation, especially the emergence of a strongoscillation reaction.We found that the performance of the anti-windup controlis limited only to systems without uncertainty . Indeed, theresults obtained from parametric variations are completelyunacceptable in an application context.contrarily two figures (8) and (9) show that the behavior ofthe system in both cases (ideal system or uncertain system)

are almost similar. This confirms the robustness of the slidingmode control.

VII. CONCLUSION

In this paper, we proposed a comparative study of the slidingmode and anti-windup controllers for linear time invariantsaturated systems. The structure of the saturation constraintis reported on the control input and being of constant lim-itations in amplitude. The design of the sliding surface isformulated as a pole assignment of a reduced system in an LMIregion.The non-linear saturated control scheme is introduced,will be ensure the elimination of the undesirable chatteringphenomenon and ensure a stable sliding mode motion. Afterthat, we briefly had the principle and results of anti-windupcontroller and we will use the same structure of saturationpresented. To verify the performance of the proposed SMC,we presented the simulation results for two controllers, appliedto the "Two degrees of freedom vibrating system with oneactuator". Indeed these simulation results show that the anti-windup controller is stable and acceptable,but the convergenceis slowly and it’s sensitive to parametric variations. The SMCcan remove the transient mode had been the main defectof the anti-windup control, and has the better performance.Consequently, we verified that the proposed SMC had thebetter robustness performance than the anti-windup control.

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