Sliding Mode Control Part 15 potx

479Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor... The least square methodleads to the following nominal mode

Fig 1 The structure of a multimodel discrete second order sliding mode control

(MM-2-DSMC)

where md is the number of the partial models The control applied to the system is given by

the following relation:

u(k) = v1(k)u1eq(k) + v2(k)u2eq(k) + +v md(k)umdeq(k) +u dis(k); (35)with

• v i(k): the validity of the i thlocal state model,

• u ieq(k) : the partial equivalent term of the 2-DSMC calculated using the i th local statemodel,

• u dis(k): the discontinuous term of the control

i=1v i(k)ueqi(k) +u dis(k) (36)

A stability analysis of this last control law is proposed in the following paragraph

479Multimodel Discrete Second Order Sliding Mode Control:

Stability Analysis and Real Time Application on a Chemical Reactor

3.3 Stability analysis of the MM-2-DSMC

Let’s consider the following non stationary system:

x(k+1) =A d x(k) + B d u(k) +Γ(k)

Γ(k)represents eventual non linearities and external disturbances

Considering the following notations: A m=md∑

Which gives:

S(k+1) +βS(k) = (S(k) + βS(k −1) +C T(Δ(k) −Δ(k−1)) −M ∗ sign(S(k) +βS(k −1))

(46)The relation (46) can be written:

σ(k+1) =σ(k) + C T(Δ(k) −Δ(k−1)) −M ∗ sign(σ(k)) (47)

In discrete time sliding mode control, instead of the sliding mode, a quasi sliding-mode isconsidered in the vicinity of the sliding surface, such that| σ(k)| < ε, where σ(k)is the slidingfunction andε is a positive constant called the quasi-sliding-mode band width.

Bartoszewicz, in (Bartoszewicz (1998)), gave the following sufficient and necessary conditionfor a system to satisfy a convergent quasi sliding mode:

⎧

⎨

⎩

σ(k) > ε ⇒ − ε ≤ σ(k+1) <σ(k) σ(k) < −ε ⇒ σ(k) < σ(k+1) ≤ε

∀ k, C T(Δ(k) −Δ(k−1))is supposed to be bounded such that:

T(Δ(k) −Δ(k−1)) <Δ0 (49)withΔ0being a positive constant

Théorème 0.1. Let’s consider the system (37) to which the MM-2-DSMC given by (36) is applied If the discontinuous term amplitude M is chosen such that:

Stability Analysis and Real Time Application on a Chemical Reactor

can be written as follows:

σ(k) + C T(Δ(k) −Δ(k−1)) −M ∗ sign(σ(k)) <σ(k) (55)Knowing thatσ(k) >0, the inequality (55) becomes:

−Δ0− C T(Δ(k) −Δ(k−1)) <σ(k) (60)This last inequality is true, knowing thatσ(k) > M ∗+Δ0>0

σ(k) + C T(Δ(k) −Δ(k−1)) <Δ0 (63)which gives:

σ(k) <Δ0− C T(Δ(k) −Δ(k−1)) (64)This last inequality is true becauseΔ0− C T(Δ(k) −Δ(k−1)) >0 andσ(k) <0

* Besides, it is evident thatσ(k) < σ(k) + C T(Δ(k) −Δ(k−1)) +M ∗, knowing that:

M ∗ >Δ0> C T(Δ(k) −Δ(k−1)) (65)

3 Let’s consider condition (53):

| σ(k)| < M ∗+Δ0⇒ | σ(k+1)| <M ∗+Δ0

* Ifσ(k) >0, then, the inequality

be used in these conditions

4 Experimentation on a chemical reactor

Stability Analysis and Real Time Application on a Chemical Reactor

presented in this section (a photo of the considered reactor is given by figure 2).

This process is used to esterify olive oil The produced ester is widely used for the

Fig 2.The esterification reactor used for the experiments.

manufacture of cosmetic products A specific temperature profile sequence must be followed

in order to guarantee an optimal exploitation of the involved reagents’ quantities The oliveoil contains, essentially, a mono-unsaturated fatty acid that react with alcohol to give waterand ester as shown by the following reaction equation:

Acid + Alcohol 1

The final solution contains all the reagents and products in certain proportions To drivethe reaction equilibrium in the way 1 and, consequently, increase the ester’s proportion, weshould take away water from the solution This is done by vaporization The fatty acid (oleicacid) and the ester ebullition temperatures are approximately 300◦ C The chosen alcohol

(1-butanol) is characterized by an ebullition temperature of 118◦ C Consequently, heating

the reactor to a temperature slightly over 100◦ C will result in the vaporization of water only

(which is evacuated through the condenser)

The reactor is heated by circulating a coolant fluid through the reactor jacket This fluid

is, in turn, heated by three resistors located in the heat exchanger (Figure 3) The reactortemperature control loop monitors temperature inside the reactor and manipulates the powerdelivered to the resistors It is, also, possible to cool the coolant fluid by circulating cold waterthrough a coil in the heat exchanger Cooling is, normally, done when the reaction is over, inorder to accelerate the reach of ambient temperature

The process can be considered as a single input - single output system The input is the heating

power P(W) The output is the reactor temperature TR( ◦ C) The interface between the

process and the calculator is ensured by a data acquisition card of the type RTI 810 The dataacquisition card ensures the conversion of the analog measures of the temperature to digitalvalues and the conversion of the digital control value to an analog electric signal proportionalthe power applied to the heating resistors

Col d water

Col d water

Heating resistorsHeat exchanger

Agitator

Reactor

P omp

ReagentsTF

TR

Fig 3.Synoptic scheme of the real process.

The control law must carry out the following three stages :

• Bring the reactor’s temperature TR to 105 ◦ C.

• Keep the reactor’s temperature to this value until the reaction is over (no more waterdripping out of the condenser)

• Lower the reactor’s temperature

We chose, therefore, the set point given by figure 4

We represent, in figure 5, the static characteristic of the system The different coordinates aretaken relatively to the three stages of the process We notice that the system can be considered

as a linear one, though, with some approximations According to the step responses of the

system, the retained sampling step is equal to 180s A Pseudo-Random, Binary input Signal

(PRBS) is applied to the real system An identification of the system structure, based on theinstrumental determinants ratio method (Ben Abdennour et al (2001)), led to a discrete secondorder linear model

Due to the nature of the control law to be applied to the reactor, the needed model is a state

model The considered state variables are the reactor’s temperature TR( ◦ C) (noted x1(k)), which is at the same time the system’s output, and the coolant fluid temperature TF( ◦ C) (noted x2(k))

The state variables sequences x1(k) and x2(k) relative to the PRBS excitation input aremeasured and used for the parametric identification of the system The least square methodleads to the following nominal model:

485Multimodel Discrete Second Order Sliding Mode Control:

Stability Analysis and Real Time Application on a Chemical Reactor

x(k) +

00.0033



u(k) y(k) =1 0

The application of the multimodel approach and by using the least square method applied onthe input/states sequence relative to each reaction stage leads to three partial models of theform:

−0.1296 1.0651



; B1=

00.0036



• the reaction stage:

A2=

0.4114 0.5482

−0.0358 0.9787



; B2

00.0034

−0.0386 0.0.9912



; B3=

00.0032



The control performance and robustness of the previously mentioned control laws, withrespect to the model-system mismatch and external disturbance, are illustrated and comparedthrough the experimental results given in the following paragraph

4.2 Experimental results

In this paragraph, the performance of the MM-2-DSMC is shown by an experimentation onthe chemical reactor Firstly, the chattering reduction, obtained by exploiting the second ordersliding mode control, is illustrated by a comparison between the results obtained by the firstorder discrete sliding mode control with those realized by the 2-DSMC (Mihoub et al (2009b)).The nominal model (73) is used for both the DSMC and the 2-DSMC

k 2−DSMC proposée

(a) Heating power (input).

10 30 40 50 70 80 90 100 110 120

1−DSMC y(k)

k

y c (k) 2−DSMC proposée

(b) Reactor temperature (output).

90 92 94 96 98 100 102 104 106 108 110

y c (k) y(k)

k

1−DSMC

2−DSMC proposée

(c) Zoom on the reactor temperature evolution.

Fig 6 Comparison between DSMC and 2-DSMC

We observe that the chattering of the control (u(k)) is remarkably reduced (figure 6.a) A better

set point tracking is, consequently, obtained as shown by figures 6.b and 6.c, which represent,respectively, the evolution of the reactor temperature and a zooming of this last one in theneighborhood of 105◦C As mentioned above, the reaction takes place essentially during thisphase If the temperature reactor overshoots 105◦ C, a large amount of alcohol is evaporated

487Multimodel Discrete Second Order Sliding Mode Control:

Stability Analysis and Real Time Application on a Chemical Reactor

and wasted and if it does not reach 105◦ C, the reaction kinetics are slowed down So, the2-DSMC results in a better efficiency relatively to the first order DSMC.

20 30 40 50 60 70 80 90 100 110

MM−2−DSMC y(k)

k

y

c (k) 2−DSMC

(b) Reactor temperature (output).

92 94 96 98 100 102 104 106 108

MM−2−DSMC y(k)

k

y

c (k)

2−DSMC

(c) Zoom of the reactor temperature evolution.

Fig 7 Comparison between 2-DSMC and MM-DSMC

Secondly, the multimodel approach is combined with the 2-DSMC in order to enhance thereaching phase The MM-2-DSMC and the 2-DSMC are represented together in figure 7(Mihoub et al (2009a)) It can be observed that the sliding function overshoots due to a badreaching phase in the case of the 2-DSMC are reduced thanks to the multimodel approach (seefigure 7.a) A better set point tracking is then obtained, as shown by figures 7.b and 7.c Anamelioration of the efficiency of the chemical reactor is, consequently, obtained

5 Conclusion

In this work, the problems of the discrete sliding mode control are discussed A solution

to the chattering problem can be given by the second order sliding mode To enhance thereaching phase, the multimodel approach is exploited A combination of the 2-DSMC andthe multimodel approach is, then, used A stability analysis of the multimodel second orderdiscrete sliding mode control is proposed in this work An experimentation on a chemicalreactor is considered On the one hand, a comparison between the results obtained by thefirst order DSMC and those obtained by the 2-DSMC showed the chattering reduction offered

by the second order approach On the other hand, a comparison between the results of the2-DSMC and those of the MM-2-DSMC, illustrated both an enhancement of the reaching phaseand a notable reduction of the chattering phenomenon A better efficiency of the reactor is,therefore, obtained

6 References

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Ben Abdennour, R., Borne, P., Ksouri, M & M’sahli, F (2001) "Identification et comm&e

numérique des procédés industriels" Editions Technip, Paris.

Chiu., S L (1994) "Fuzzy model identification based on cluster estimation" Journal of

Intelligent & Fuzzy Systems, 2:267–278.

Decarlo, R A., Zak, H S & Mattews, G.P (1988) "Variable structure control of nonlinear

multivariable systems: A tutorial" Proceeding IEEE, 73:212–232.

Emelyanov, S V., Korovin, S K & Levantovsky, L.V (1986) "Higher order sliding modes in

the binary control systems" Soviet Physics, Doklady, 31:291–293.

Filippov, A (1960) "Equations différentielles à second membre discontinu" Journal de

Mathématiques, 51(1):99–128.

Gao, W., Wang, Y & Homaifan, H (1995) "Discrete-time variable structure control systems"

IEEE Trans Ind Electronics, 42(2):117–122.

Jimenez, T S (2004) "Contribution à la comm&e d’un robot sous-marin autonome de type torpille".

Thèse de doctorat, Université Montpellier II

Ksouri Lahmari, M (1999) "Contribution à la comm&e multimodèle des processus complexes" PhD

thesis, USTL, Lille

Levant, A (1993) "Sliding order & sliding accurcy in sliding mode control" International

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Heterogeneous Systems Materials of the seminar, 32–43, Moscow.

Lopez, P & Nouri, A.S (2006) "Théorie élémentaire et pratique de la comm&e par les régimes

glissants" Mathématiques et applications 55, SMAI, Springer - Verlag.

Ltaief, M.,Ben Abdennour, R., Abderrahim, K & Ksouri, M (2003) "A new systematic

determination approach of a models base for the representation of uncertain

systems" In SSD’03, Sousse, Tunisie.

Ltaief, M., Abderrahim, K., Ben Abdennour, R & Ksouri, M (2003) "A fuzzy fusion strategy

for the multimodel approach application" WSEAS Trans on circuits & systems,

2(4):686–691

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of a models base for the multimodel approach: Experimental validation" WSEAS Trans on Electronics, 1(2): 331–336.

Mihoub, M., Nouri, A S & Ben Abdennour, R (2008) "The multimodel approach for a

numerical second order sliding mode control of highly non stationary systems" In

ACC’08, American Control Conference, Seattle, W.A USA.

Mihoub, M., Nouri, A.S & Ben Abdennour, R (2009) "A real time application of discrete

second order sliding mode control to a semi-batch reactor: a multimodel approach"

Int Journal of Modelling, Identification & Control, 6(2):156–163.

Mihoub, M , Nouri, A.S & Ben Abdennour, R (2009) "Real time application of discrete

second order sliding mode control to a chemical reactor" Control Engineering Practice,

17:1089–1095

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undergoing sliding motion" Int J Systems Sci., 21(4):665–674.

489Multimodel Discrete Second Order Sliding Mode Control:

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Talmoudi, S., Ben Abdennour, R., Abderrahim, K & Ksouri, M (2002) "Multimodèle et

multi-comm&e neuronaux pour la conduite numérique des systèmes non linéaires

et non stationnaires" In CIFA’02, France.

Talmoudi, S., Ben Abdennour, R., Abderrahim, K & Borne, P (2002) "A systematic

determination approach of a models’base for uncertain systems: Experimental

validation" In SMC’02, Conference on Systems, Man & Cybernetics, Tunisie.

Talmoudi, S., Abderrahim, K., Ben Abdennour, R & Ksouri, M (2003) "A new technique of

validities’computation for multimodel approach" In WSEAS03 (ICOSMO03), Greece Utkin, V I (1992) "Sliding Mode in Control Optimisation" Springer-Verlag, Berlin.

Utkin, V.I., David Young, K & Ozguner, U (1999) "A control engineer’s guide to sliding mode

control" IEEE Transactions on Control Systems Technology, 7:328–342.

26

Two Dimensional Sliding Mode Control

Hassan Adloo1, S.Vahid Naghavi2, Ahad Soltani Sarvestani2 and Erfan Shahriari1

Iran

1 Introduction

In nature, there are many processes, which their dynamics depend on more than one independent variable (e.g thermal processes and long transmission lines (Kaczorek, 1985)) These processes are called multi-dimensional systems Two Dimensional (2-D) systems are mostly investigated in the literature as a multi-dimensional system 2-D systems are often applied to theoretical aspects like filter design, image processing, and recently, Iterative Learning Control methods (see for example Roesser, 1975; Hinamoto, 1993; Whalley, 1990; Al-Towaim, 2004; Hladowski et al., 2008) Over the past two decades, the stability of multi-dimensional systems in various models has been a point of high interest among researchers (Anderson et al., 1986; Kar, 2008; Singh, 2008; Bose, 1994; Kar & Singh, 1997; Lu, 1994) Some new results on the stability of 2-D systems have been presented – specifically with regard to the Lyapunov stability condition which has been developed for RM (Lu, 1994) Then, robust stability problem (Wang & Liu, 2003) and optimal guaranteed cost control of the uncertain 2-D systems (Guan et al., 2001; Du & Xie, 2001; Du et al., 2000 ) came to be the area of interest In addition, an adaptive control method for SISO 2-D systems has been presented (Fan & Wen, 2003) However, in many physical systems, the goal of control design is not only to satisfy the stability conditions but also to have a system that takes its trajectory in the predetermined hyperplane An interesting approach to stabilize the systems and keep their states on the predetermined desired trajectory is the sliding mode control method Generally speaking, SMC is a robust control design, which yields substantial results in invariant control systems (Hung et al., 1993) The term invariant means that the system is robust against model uncertainties and exogenous disturbances The behaviour of the underlying SMC of systems is indeed divided into two parts In the first part, which is called reaching mode, system states are driven to a predetermined stable switching surface And in the second part, the system states move across or intersect the switching surface while always staying there The latter is called sliding mode At a glance in the literature, it is understood that there are many works in the field of SMC for 1-D continuous and discrete time systems (see Utkin, 1977; Asada & Slotine, 1986; Hung et al., 1993; DeCarlo et al., 1988; Wu and Gao, 2008; Furuta, 1990; Gao et al., 1995; Wu & Juang, 2008; Lai et al., 2006; Young et al., 1999; Furuta & Pan, 2000; Proca et al., 2003; Choa et al., 2007; Li & Wikander, 2004; Hsiao et al., 2008; Salarieh & Alasty, 2008) Furthermore SMC has been contributed to various control methods (see for example Hsiao et al., 2008; Salarieh & Alasty, 2008) and several experimental works (Proca et al., 2003) Recently, a SMC design for a 2-D system in RM

model has been presented (Wu & Gao, 2008) in which the idea of a 1-D quasi-sliding mode

(Gao et al., 1995) has been extended for the 2-D system Though the sliding surfaces design

problem and the conditions for the existence of an ideal quasi-sliding mode has been solved

in terms of LMI

In this Chapter, using a 2-D Lyapunov function, the conditions ensuring the rest of

horizontal and vertical system states on the switching surface and also the reaching

condition for designing the control law are investigated This function can also help us

design the proper switching surface Moreover, it is shown that the designed control law can

be applied to some classes of 2-D uncertain systems Simulation results show the efficiency

of the proposed SMC design The rest of the Chapter is organized as follows In Section two,

Two Dimensional (2-D) systems are described Section three discusses the design of

switching surface and the switching control law In Section four, the proposed control

design for two numerical examples in the form of 2-D uncertain systems is investigated

Conclusions and suggestions are finally presented in Section five

2 Two dimensional systems

As the name suggests, two-dimensional systems represent behaviour of some processes

which their variables depend on two independent varying parameters For example,

transmission lines are the 2-D systems where whose currents and voltages are changed as

the space and time are varying Also, dynamic equations governed to the motion of waves

and temperatures of the heat exchangers are other examples of 2-D systems It is interesting

to note that some theoretical issues such as image processing, digital filter design and

iterative processes control can be also used the 2-D systems properties

2.1 Representation of 2-D systems

Especially, a well-known 2-D discrete systems called Linear Shift-Invariant systems has been

presented which is described by the following input-output relation

Similar to the one-dimensional systems, the 2-D systems are commonly represented in the

state space model but what is makes different is being two independent variables in the 2-D

systems so that this resulted in several state space models

A well-known 2-D state space model was introduced by Roesser, 1975 which is called

Roesser Model (RM or GR) and described by the following equations

( , )

h v

Two Dimensional Sliding Mode Control 493

where x h (i, j) ∈ R n and x v (i, j) ∈ R m are the so called horizontal and vertical state variables

respectively Also u(i, j) ∈ R p is an input and y(i, j) ∈ R q is an output variable Moreover, i

and j represent two independent variables A1, A2, A3, A4, B1, B2, C 1 and C 2are constant

matrices with proper dimensions To familiar with other 2-D state space models (Kaczorek,

1985)

2.2 Stability of Rosser Model

One of the important topics in the 2-D systems is stability problem Similar to 1-D systems,

the stability of 2-D systems can be represent in two kinds, BIBO and Internally stability

First, a BIBO stability condition for RM is stated

Theorem 1: A zero inputs 2-D system in RM (3) is BIBO stable if and only if one of the

following conditions is satisfied

circle Thus, from Theorem 1, it can be easily shown that a 2-D system in RM is unstable if

A 1 or A 2 is not stable

Similar to 1-D case, the Lyapunov stability for 2-D systems has been developed such that we

represented in the following theorem

Theorem 2: Zero inputs 2-D system (3) is asymptotically stable if there exist two positive

definite matrices P 1 ∈ R n and P 2 ∈ Rm such that

x i j and ( , ) x i j converge to zero when i+j→∞ v

Remark 2: The equality (3) is commonly called 2-D Lyapunov equation As stated in the

theorem 2, the condition for stability of 2-D systems in RM model is only sufficient not

necessary and the Lyapunov matrix, P, is a block diagonal while in the 1-D case, the stability

conditions is necessary and sufficient and the Lyapunov matrix is a full matrix

However, it is worthy to know that the Lyapunov equation (3) can be used to define the 2-D

Lyapunov function as shown below

1 00

2

0( , )

where (X=⎡⎣x i h +1, )j x i j v( , +1)⎤⎦ Now, we can state following fact T

Theorem 3: 2-D system (3) is asymptotically stable if there are the Lyapunov function, (6)

and the delayed function (7) such that

( , ) ( , ) ( , ) 0

V i j V i j V i j

As a result, the Theorem 3 can be used to design a 2-D control system

3 Sliding mode control of 2-D systems

In this section, we review some prominence of the 1-D sliding mode control and then

present the 2-D sliding mode control for RM

3.1 One dimensional (1-D) Sliding Mode Control

Generally speaking, Sliding Mode Control (SMC) method is a robust control policy in which

the control input is designed based on the reaching and remaining on the predetermined

state trajectory This state trajectory is commonly called switching surface (or manifold)

Usually, first the switching surface is determined as a function of the state and/or time, and

Fig 1 State trajectory for some different initial conditions

then the control action is designed to reach and remain the state trajectory on the switching

surface and move to the origin Therefore, it can be appreciated that the switching surface

should be contained the origin and designed such that the system is stabile when remaining

on it Three main advantages of the SMC method are low sensitivity to the uncertainty (high

robustness), dividing the system trajectory in two sections with low degree and also easily

in implementation and applicability to various systems

s(k) = 0

s(k) = 0

Two Dimensional Sliding Mode Control 495

To make easier understanding the 1-D SMC, let consider a simple example in which a

discrete time system is given as follows

where u x k e( , )= −x k1( ) 0.5 ( ) 0.5 ( )− x k2 − s k It is clear that the control input is u e (x, k) when

the system remain on the surface (in other word when s(k) = 0) Fig 1 illustrates the state

trajectories of the system for some different initial conditions such that they converge to the

surface and move to the origin in the vicinity of it As it is shown in Fig 1, the state

trajectories switch around the surface when they reach the vicinity of it The main reason of

this phenomenon comes from the fact that the system dynamic equation is not exactly

matched to the switching surface (Gao, 1995) In fact, the control policy in the SMC method

is to reduce the error of the state trajectory to the switching surface using the switching

surface feedback control It is worthy to note that in the SMC method, the system trajectory

is divided to two sections that are called reaching phase and sliding phase Thus, the control

input design is commonly performed in two steps, which named equivalent control law and

switching are control law design We want to use this strategy to present 2-D SMC design

3.2 Two dimensional (2-D) sliding mode control

Consider the 2-D system in RM model as stated in (3)

In this chapter it is assumed that the 2-D system (3) starts from the boundary conditions that

are satisfied following condition

where x h (0, k) and x v (k, 0) are horizontal and vertical boundary conditions Before

introducing 2-D SMC method, some definitions are represented

Definition 1: The horizontal and vertical linear switching surfaces denoted by s h (i, j) and

s v (i, j), are defined as the linear combination of the horizontal and vertical state of the 2-D

system respectively as shown below