Automatic Control

International Review of

Automatic Control (IREACO) Theory and Applications

Contents Using Uncertainty Bounds Technique in the Design of a Real -Time Type-2 Fuzzy Robust Regulator Around Operating Point of the Nonlinear System by Hadi Chahkandi Nejad, Rouzbeh Jahani, Sina Zarrabian, Assef Zare, Heidar Ali Shayanfar

1

Synchronization of Chaotic Systems with Uncertain Parameters Using Backstepping Algorithm by A. Ikhlef, N. Mansouri

6

Fuzzy Based Design Specification for Set Membership System Identification by Majda Ltaief, Ridha Ben Abdennour, Mohammad M'Saad

13

Designing Controller for Large Scale Non-Affine Nonlinear Systems: Decentralized Intelligent Adaptive Approach by Reza Ghasemi

21

Output Feedback Higher Order Sliding Mode Controller by Abderraouf Gaaloul, Faouzi M'Sahli

32

Application Possibilities of Non Parametric Identification Techniques in On-Line Process Monitoring by Tomi Roinila, Mikko Huovinen, Matti Vilkko

40

Output Tracking Control Design for Non-Minimum Phase Systems: Application to the Ball and Beam Model by Monia Charfeddine, Khalil Jouili, Houssem Jerbi, Naceur Ben Hadj Braiek

47

Droop Based Control of Parallel Connected Three Phase Inverters Using Modified Droop Method by M. Ramezani, M. Joorabian, S. Golestan

56

PD-Like and PI-Like Fuzzy Control Implementation Using FPGA Technology by A. Sakly, A. Azzouna, A. Trimeche, A. Mtibaa

65

Design of a Recurrent Neural Controller for PMSM Drive Based on Sliding Mode Torque Control by A. M. El-Sawy, A. A. Hassan, Y. S. Mohamed, E. G. Shehata

78

Modeling and Fuzzy Logic Control of a Submerged Arc Furnace by G. Shabib

86

(continued)

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Power System Stabilization Using a Combination of Static Var Compensator and Multi-Band Power System Stabilizer by Y. A. Mobarak

94

GA-Based Output Feedback Controller Design for PSS and TCSC to Damp the Power System Oscillations by Abolfazl Jalilvand, Reza Noroozian, Mohammad Reza Safari Tirtashi

102

Active Vibration Control of Self-Excited Vibration with Simplified Disk Brake Model by T. Jearsiripongkul

109

Using Fuzzy Logic Path Tracking for an Autonomous Robot by Ahmed Hechri, Fayçal Hamdaoui, Anis Ladgham, Abdellatif Mtibaa

115

Nonlinear Observers for the Estimation of Kinetic Parameters in an Alcoholic Fermentation Bioprocess by Dan Selişteanu, Constantin Marin, Emil Petre, Dorin Şendrescu

124

Technology Advances Open Up New Possibilities in Industrial Process Management by Mikko Huovinen

133


International Review of Automatic Control (I.RE.A.CO.), Vol. 4, N. 1 January 2011

Using Uncertainty Bounds Technique in the Design of a Real -Time Type-2 Fuzzy Robust Regulator Around Operating Point of the Nonlinear System Hadi Chahkandi Nejad1, Rouzbeh Jahani2, Sina Zarrabian2, Assef Zare3, Heidar Ali Shayanfar4 Abstract – This paper presents a type-2 fuzzy robust regulator for the nonlinear and uncertain systems. In this paper, first, state equations of the nonlinear system is linearized around operating point and then a fuzzy system is designed based on state vector of the linearized system and knowledge about the control system. The designed fuzzy system uses control signals to regulate outputs through measurements obtained from system state vector and basic fuzzy rules of the human knowledge. In this paper, to immediate use of the regulator, type-2 fuzzy system is designed based on uncertainty bounds technique. Finally, simulations are implemented for an individual inverted pendulum model in two cases: certain and uncertain equations of the system. In both cases, the simulation results show that proposed regulator has a better performance in convergence speed and robustness compare to type-1 fuzzy regulator. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Interval Type-2 Fuzzy System, Jacobian Matrix, Robust Regulator, Uncertainty Bounds, T1: Type-1, T2: Type-2, FS: Fuzzy Set

I.

Recently, many researches show that type-1 fuzzy systems have some problems and limits in handling and minimizing of uncertainty effects. Mendel showed that it is wrong to display logic statements by type-1 fuzzy sets [1] because statements are uncertain but a type-1 fuzzy set is completely certain and definite. Systems based on logical rules use membership functions (MF). These membership functions are often formulated by a series of pure mathematical functions. In fact, they are not linguistic. The word fuzzy has the connotation of uncertainty but type-1 fuzzy membership function is completely definite when its parameters are certain and it means a paradox [2]. At least four linguistic uncertainties can occur in type-1 fuzzy systems [3] because the fuzzy rule-based statements in these systems are pure and certain functions. Basically, there are two types of high level uncertainty: linguistic uncertainty and random uncertainty. Probability theory is used to handle random uncertainty and fuzzy sets are used to handle linguistic uncertainty, and sometimes fuzzy sets can also be used to handle both kinds of uncertainty, because a fuzzy system may use noisy measurements or operate under random disturbances. If we use type-1 fuzzy sets to handle random uncertainty, just first order moments of probability density function (pdf) will be used which would not be very useful because random uncertainty requires an understanding of dispersion about the mean, and this information is provided by variance. If fuzzy sets appear in random applications, then both types of

Introduction

Nonlinear systems control has been important issue from many years. Conventionally, control system design is obtained from mathematical models based on physical laws. But, in fact, most of the parameters and structures in a system are unknown due to ambient changes, modeling errors and dynamical parameters which cannot be modeled. An overview about such a systems is that they have unknown and uncertain equations. Actually, in addition to unknown mathematical model of the system, they are uncertain too. Many efforts have been made to control these systems i.e. a combination of neural networks and fuzzy logic methods (like if-then statements) or fuzzy logic and evolutionary algorithms. It is obvious that our world is based on probability, possibility, and uncertainty. Also, behavior of all systems in the world will change after a period of time. Therefore, researchers are finding new control methods to improve design aspects of control systems. It must be noted that in introduced systems with uncertainty characteristics, definition of fuzzy logic ifthen rules includes uncertainty. Today, fuzzy systems are the best choice to transform logic statements (like if-then) into automatic control strategies in order to design controllers for nonlinear systems. In fact, this capability of fuzzy logic systems leads to much better controller design in nonlinear systems. Manuscript received and revised December 2010, accepted January 2011


1

Hadi Chahkandi Nejad, Rouzbeh Jahani, Sina Zarrabian, Assef Zare, Heidar Ali Shayanfar

It must be noted that each point in state space can reach to an equilibrium point through change of variables. To design a fuzzy regulator, it is assumed that there is enough knowledge about control systems. Also, it is assumed that there is an accessible set of fuzzy if-then rules which can describe the behavior of control system. According to given explanation about type-2 fuzzy sets, in this paper, fuzzy rules are considered as type-2 to make a robust regulator. Therefore, fuzzy if-then rules are specified as follows:

uncertainty must be considered. A type-2 fuzzy set has a capability to provide proper estimation of dispersion in uncertain conditions [1]. Hence, it is found that a type-2 fuzzy set is capable to handle and minimize the effect of both linguistic and random uncertainties, simultaneously. A wide range of applications in type-2 fuzzy sets show that they provide much better solutions particularly in uncertain conditions [1]. A set of applications and papers about type-2 fuzzy sets have been presented in [10], [11]. According to above information and due to existing uncertainty in fuzzy rules, it is found that type-2 fuzzy sets provide more robust response compare to type-1 fuzzy sets. In the next section, a type-2 fuzzy regulator will be designed to provide an appropriate control signal for output regulation of nonlinear and uncertain systems. The regulator has been designed to regulate a nonlinear system in an individual operating point.

II.

if x1 is A ni then U is C i , i = 1,...,m

These rules extract control strategies from state vector to regulate system’s outputs moment by moment. In fact, system state is reported to tuned-fuzzy inference engine (FIE) and FIE sends the control range of U in order to regulate the output or state. With considering product t-norm for combination of antecedent sets and after applying singleton fuzzy for input sets, according to primary rules of fuzzy rules base which are related to designed fuzzy system, firing level is defined as follows:

Design of Type-2 Fuzzy Robust Regulator Based on Uncertainty Bounds Technique

Dynamic of most industrial processes and real systems is nonlinear. So, analysis and design of control systems in such cases is very difficult and application of nonlinear controllers in most practical cases is not necessary. Actually, experiments show that the linear control systems cover a wide control range of real systems and complicated industrial processes. So, it is very important and unavoidable to obtain accurate linear model from nonlinear systems in engineering sciences. A practical method in linearization of nonlinear equations is to use Taylor series expansion. According to this method, equations of n-order nonlinear system (i.e., equation (1)) can be linearized around its equilibrium point as equations (2): X = f ( X ,U ,t ) Y = g ( X ,U ,t ) X = AX ( t ) + BU ( t ) Y = CX ( t ) + DU ( t )

(3)

Ai =

n

∏ µ A ( x j ) , i = 1,...,m j =1

(4)

i j

Because of applying type-2 fuzzy membership functions in fuzzy rules base, after applying singleton i

fuzzy input, A will be an interval type-1 fuzzy set which is defined as a range. Then, equation (4) can be updated as follows: Ai ( x ) = ⎡⎣ Ai ( x ) , Ai ( x ) ⎤⎦

(5)

where, Ai ( x) and Ai ( x ) are defined as:

(1)

Ai ( x ) =

n

∏ µA ( x j )

(2) Ai ( x ) =

It is assumed that g and f are nonlinear and uncertain. Also, U and Y are process input and output,

n

∏ µA ( x j )

T

Also,

vector. A, B are Jacobian matrix of f, with respect to X,U, respectively. Also, C, D are Jacobian matrix of g, with respect to X, U, respectively. To calculate these matrixes, refer to [12] and [13]. This method shows how to apply linear control systems to practical problems and ensures that stable linear control design can provide a local stability around operating point in the main system [12]. By applying these robust control methods on linearized model of the system, it can be guaranteed that system is stable around operating point in spite of process changes.

µ Ai ( x j ) j

and

(7)

i j

j =1

respectively. X = ( x1 ,...,xn ) ∈ R n is a measurable state

(6)

i j

j =1

µ Ai ( x j ) j

are membership

( )

functions as a lower and upper bound for µ Ai x j . The j

next step is the calculation of centroid of consequent fuzzy set which is proportional to each rule. Centroid of consequent set for each rule is defined as: C i = ⎡⎣Cli ,C ri ⎤⎦ which is an interval type-1 fuzzy set [8]. Within regulator design,uncertainty bounds theory is used to obtain mapping of fuzzy inference engine [9]. In


International Review of Automatic Control, Vol. 4, N. 1

2


fact, system based on uncertainty bounds theory consists of two parallel processes obtained from type-1 fuzzy calculations which can be operated as a real time. Since type-reduction methods [4] are not used in this strategy, this inference speed is almost as equal as type-1 fuzzy inference engine. Therefore, this system is used more than typereduction methods specially in online and immediate applications [8] because type-reduction methods require using KM (Karnik-Mendel) iteration algorithm [6]. The calculations of uncertainty bounds-based inference engine have been shown in Fig. 1 where, Uˆ ( X ) is an

Fig. 3. Inverted pendulum system

The control purpose is to regulate the angular position of pendulum on its unstable equilibrium point independent of cart’s position. θ , θ are the inputs of fuzzy system and restoring force is considered as output. Nonlinear and linearized dynamic equations of the system around point (0, 0) have been presented in [13], section 1-2-2. Parameters of the model are considered as follows:

approximation of control signal. To calculating Uˆ ( X ) , four bounds containing U l , U l , U r , U r must be determined. For more information about this method, refer to [9]. The designed regulator has been shown in Fig. 2, briefly. It must be noted that system state and appropriate control signal are input and output of type-2 fuzzy system, respectively.

mc = 2kg , m = 0.1kg , l = 0.5,g = 10 m

s2

(The parameters are length and mass of the rod). Initial conditions are: X ( 0 ) = ⎡⎣θ = 0.1, θ = 0.2 , x = 0 , x = 0 ⎤⎦ Rule-based type-2 fuzzy membership functions which are considered in Table I, have been shown in Figures 4, 5 and 6.

Fig. 1. Calculation of an Interval T2 Fuzzy set using uncertainty bounds technique instead of type-reduction methods

Fig. 4. T2 fuzzy membership functions defined for θ ( ε = 0.03 )

Fig. 2. Fuzzy regulator

III. Simulation Results In this section, simulation results will be investigated and Comparison between type-1 and type-2 fuzzy sets to regulate inverted pendulum on unstable equilibrium point, will be implemented. To achieve this purpose, we apply fuzzy regulator to an inverted pendulum shown in Fig. 3.

Fig. 5. T2 fuzzy membership functions defined for θ ( ε = 0.09 )



3


Fig. 7. Comparison between T1 and T2 fuzzy regulator (certain process model)

Fig. 6. T2 fuzzy membership functions defined for control vector U ( ε = 0.09 )

Comparison index in this table is considered as sumsquared error between proper pendulum path and regulated pendulum path which takes values in the interval [0, 10] s. The membership functions as shown in Figures 4-6, are the same as presented in [7], section 3, part 4-1. The only difference is a tolerance (which is applied to center of these functions in order to make them indefinite (uncertain). Also, Table II shows the simulation results while the parameters of the model are indefinite. It must be noted that uncertainty is applied to the length and mass of the rod, separately and a random noise with normal distribution around zero point, is applied to the length and mass of the rod in scale of 0.05 and 0.01, respectively. SSE index has been separately inserted in Table II for the length and mass of the rod under uncertain condition.

Fig. 8. Performance of T1 fuzzy regulator (uncertainty in length)

TABLE I COMPARISON BETWEEN T1 AND T2 FS UNDER CERTAIN CONDITION Type-1 Fuzzy Control system modeling Type-2 Fuzzy Set Set (SSE)

0.1947

0.1399

Fig. 9. Performance of proposed T2 fuzzy regulator (Uncertainty in length)

TABLE II COMPARISON BETWEEN T1 AND T2 FS UNDER UNCERTAIN CONDITION Type-1 Fuzzy Control system modeling Type-2 Fuzzy Set Set Uncertainty in length 0.3617 0.2210 (SSE) Uncertainty in mass 0.3382 0.2032 (SSE)

IV.

Conclusion

It can be concluded from simulation results that proposed regulator (type-2 fuzzy regulator) has a better convergence speed and robustness compare to type-1 fuzzy regulator, in both certain and uncertain model of the system. The reason is that type-2 fuzzy system has a better performance in uncertain conditions rather than type-1 fuzzy system. Also, by applying uncertain type-2 membership functions, uncertainty of the model can be handled much better than before. It must be noted that more uncertain membership function does not lead to less uncertainty condition. But, it is very important to choose an optimal value according to each problem. In addition, an optimal selection for is an optimization problem, solely. Finally, it can be concluded that by applying a type-2 fuzzy regulator, the system can achieve

Finally, to examine the regulator performance, a comparison between type-1 and type-2 fuzzy system has been implemented in Figures 7 – 9. The performance of type-1 and type-2 fuzzy regulator has been shown in Fig. 7, under certain condition. The performance of type-1 and type-2 fuzzy regulator has been shown in Figures 8 and 9, respectively under uncertain condition in the length of the rod.



4


to more robust performance around operating point. Also, following new ideas are proposed to accomplish researches about the fuzzy systems and regulator design: 1) Applying type-1 and type-2 fuzzification in order to reduce noises in measurements. 2) Determination of optimal value in control problems of inverted pendulum. 3) Increase of convergence zone around operating point by applying optimal type-2 fuzzy membership functions.

Author's information 1

Electrical Engineering Department, Islamic Azad University, Birjand Branch, Birjand, Iran. 2 Electrical Engineering Department, Islamic Azad University, South Tehran Branch, Tehran, Iran. 3 Electrical Engineering Department, Islamic Azad University, Gonabad Branch, Gonabad, Iran. 4 Center of Excellence for Power System Automation and Operation, Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran.

Hadi Chahkandi Nejad received the B.Sc. degree from Islamic azad university of birjand , birjand, Iran, and M.Sc. degree from, Islamic azad university of gonabad ,gonabad, Iran in 2005 and 2009, respectively. In 2009, he was employed in Islamic Azad University, Branch of Birjand, Iran, as a lecturer in the department of computer and Electronics engineering. His research interests are in fuzzy systems and neural networks in system control.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15]

[16]

[17]

J.M. Mendel, “Advances in type-2 fuzzy sets and systems,” Information Sciences, Vol. 177, pp. 84-110, 2007. Klir, G. J. and T. A. Folger (1988). Fuzzy Sets Uncertainty, and Information, Prentice-Hall, Englewood Cliffs, NJ. 2004. J.M. Mendel, R.I. John, “Type-2 fuzzy sets made simple,” IEEE Trans. Fuzzy Syst. 10 (2002) 117-127. April 2002. N. N. Karnik and J. M. Mendel, “Type-2 fuzzy logic systems: Type-Reduction,” in IEEE Syst., Man, Cybern. Conf., San Diego, CA, Oct. 1998. Jerry M. Mendel, Robert I. Bob John, and Feilong Liu, “Interval Type-2 Fuzzy Logic Systems Made Simple,” IEEE Transaction on Fuzzy Systems, Vol. 14, No. 6, December 2006, pp. 808-821. N. N. Karnik and J. M. Mendel, “Centroid of a type-2 fuzzy set, “Inform. Sci., Vol. 132, pp. 195-220, 2001. M. Mokhtari and M. Marie, Engineering Applications of MATLAB 5.3 and SIMULINK 3.0 Berlin, Germany: SpringerVerlag, 2000. J.M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper-Saddle River, NJ, 2001. H.Wu and JM. Mendel, “Uncertainty Bounds and Their Use is the Design of Interval Type-2 Fuzzy Logic Systems,” IEEE Transactions on fuzzy systems, Vol. 10, pp.622-639, October 2002 www.type2fuzzylogic.org Castillo, O. and P. Melin, “Type-2 Fuzzy Logic Theory and applications,” Springer-Verlag, Berlin, 2008. J. E. Slotine, W. P. Li, “Applied Nonlinear Control,” PrenticeHall, 1991. Khaki Seddigh, A. “Modern Control Theory,” University of Tehran Press, 2005. S. J. P. S. Mariano, J. A. N. Pombo, M. R. A. Calado, L. A. F. M. Ferreira, Pole-Shifting Procedure to Specify the Weighting Matrices for an Optimal Voltage Regulator, International Review of Automatic Control (IREACO), Vol. 2. n. 6, pp. 685-692, November 2009. Hêmin Golpîra, Hassan Bevrani, Ali Hessami Naghshbandy, A Survey on Coordinated Design of Automatic Voltage Regulator and Power System Stabilizer, International Review of of Automatic Control (IREACO), Vol. 3. n. 2, pp. 172-182, March 2010. H. Chahkandi nejad, R. Jahani, M. Mohammad Abadi, A. Zare, J. Olamaei, H. A. Shayanfar, Synchronous Generator’s Dynamic Parameters Estimation by a Fuzzy Index and Neural-based Observer, International Review of of Automatic Control (IREACO), Vol. 3. n. 5, pp. 492-498, September 2010. Rouzbeh Jahani, Heidar Ali Shayanfar, Omid Khayat, GAPSOBased Power System Stabilizer for Minimizing the Maximum Overshoot and Setting Time, International Review of of Automatic Control (IREACO), Vol. 3. n. 3, pp. 270-278, May 2010.

Rouzbeh Jahani received his B.Sc. Degree from Amirkabir University of Technology (AUT), Tehran, Iran, 2009 and his M.Sc. Degree of electrical power engineering from Islamic Azad University (IAU), South Tehran Branch, Iran, 2010. His research interests include the Application of Robust Control, application of Artificial Intelligence in Power System Control and Design, Operation and Planning and Power System Restructuring, Optimization Problems in Electrical Power Systems. Corresponding Author. E-mail: [email protected] Tel: 009821-88904677 - Fax: 009821-88904193 Sina Zarrabian was born in Tehran, Iran, in 1985. He received the B.S. degree in Electrical Engineering from Islamic Azad University – Tehran South Branch, Tehran, in 2008, where he is currently working toward the M.S. degree in Electrical Engineering in Islamic Azad University – Tehran South Branch. His research interests are Power System Dynamics, FACTS controllers in power systems, and HVDC systems. Assef Zare received the B.S. degree in electronic engineering from sharif university Technology (SUT), iran in 1990 and obtain M.S. degree in control& automation system from engineering department of Tehran university in 1993 and Ph.D. in control& automation system from science& research Tehran in 2001.curently he is assistant professor in electrical department of Islamic azad university, Gonabad branch.His main research interest include robust control , adaptive control, optimal control and intelligent algorithms. Heidar Ali Shayanfar received the B.Sc. and M.Sc. degrees in Electrical Engineering in 1973 and 1979, respectively. He received his Ph.D. degree in Electrical Engineering from Michigan State University, U.S.A., in 1981. Currently, he is a Full Professor in Electrical Engineering Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran. His research interests are in the application of artificial intelligence to power system control design, dynamic load modeling, power system observability studies and voltage collapse. He is a member of Iranian Association of Electrical and Electronic Engineers and IEEE.



5


Synchronization of Chaotic Systems with Uncertain Parameters Using Backstepping Algorithm A. Ikhlef, N. Mansouri Abstract – In this paper a backsteppnig algorithm is proposed to synchronize tow identical chaotic systems with uncertain parameters. Backstepping control law is derived to make the state variables of the master and the slave systems asymptotically synchronized. The method is easy to implement and has excellent performances. To illustrate the effectiveness of the proposed approach, two illustrative examples are presented. Copyright © 2011 Praise Worthy Prize S.r.l. All rights reserved. Keywords: Chaos, Synchronization, Uncertain Parameters, Backstepping

I.

The paper is organized as follows: In section 2, the Backstepping algorithm is described, and Lorenz and Chen systems are tacked as examples. In order to show the effectiveness of the proposed approach, we give in section 3, the numerical simulation results obtained on Lorenz and Chen systems. The paper is concluded with some remarks in section 4.

Introduction

Several studies have showed that chaos can be useful or has great potential in many disciplines and most of the developed methods concern the chaotic synchronization [1]-[4]. Pecora and Carroll suggest that the phenomenon of chaos synchronism may serve as the basis for new ways to achieve secure communication [5]. The synchronization is always done between a system designed as master and another as slave. The principle of the synchronization is to apply on the slave a control function, such as the error between the two systems tends to zero. The problem of synchronization can then be expressed as a problem of control that consists in minimizing the error between the master and the slave by applying a control law. A wide variety of approaches such as: delay feedback method [6], adaptive control [7], sliding mode control [8] and so on, have been successfully applied to the chaos synchronization. The two most recently proposed methods are the active control [9], [10] and Backstepping control [11], [12]. Backstepping design represents a powerful technique based on the definition of Lyapunov functions and does not require complicated calculation. On the other hand, in real physical systems, chaotic systems may have some uncertain parameters. For that, studies of chaotic systems with uncertain parameters represent a great interest, so adaptive synchronization of uncertain chaotic systems has been studied mostly through linear or nonlinear feedback control [13]-[17]. In this work we are interested to the synchronization of chaotic systems with uncertain parameters using a Backstepping based algorithm where the number of controllers used is less than the dimension of the systems being synchronized. At each step we try to calculate the adaptation law of the parameters that belongs to the subsystem under study by guaranteeing the stability according to Lyapunov’s theory.

II. II.1.

Backstepping Algorithm

Synchronization of Uncertain Lorenz System Using Backstepping Algorithm

Consider the Lorenz system defined by: ⎧ x = a ( y − x ) ⎪ ⎨ y = − xz + rx − y ⎪ z = xy − bz ⎩

(1)

where a,r,b are the system parameters. For a = 10 ,b = 8 / 3,r = 28 , the system exhibits chaotic behaviors as shown in Fig. 1. 50 45 40 35

z

30 25 20 15 10 5 0 -20

-15

-10

-5

0

5

10

15

20

x

Fig. 1. Lorenz attractor for r = 28

Manuscript received and revised December 2010, accepted January 2011

6


A. Ikhlef, N. Mansouri

Assume that we have two systems and that the master system with the subscript m must drive the slave system with subscript s. The master system is given by:

we get: V1 = − a β12 < 0

Thus, the subsystem ( β1 ) is asymptotically stable.

⎧ xm = a ( ym − xm ) ⎪ ⎨ y m = − xm zm + rxm − ym ⎪ z = x y − bz m m m ⎩ m

We denote by β 2 the error between the state e y and the

(2)

estimation function α1 ( β1 ) :

β 2 = e y − α1

and the slave system is:

(3)

⎪⎧ β1 = a ( β 2 − β1 ) + ea ( ys − xs ) ⎨ ⎪⎩ β 2 = ( r − zm ) β1 − β 2 − xs ez + er xs + u2

where au ,ru ,bu are uncertain parameters to be estimated, and u1 ,u2 ,u3 are the control laws which are to be designed to achieve the synchronization. Let: ea = au − a, er = ru − r, eb = bu − b

In order to stabilize the subsystem ( β1 , β 2 ) , we consider the Lyapunov function V2 :

(4) V2 = V1 +

ex = xs − xm ,e y = ys − ym ,ez = zs − zm

β 22 2

+

er2 2

(13)

where its derivative is:

(5)

⎡( r − zm ) β1 − β 2 + ⎤ V2 = −a β12 + β 2 ⎢ ⎥ + er er ⎣ − xsα 2 + er xs + u2 ⎦

and its variation is:

)

⎧ex = a e y − ex + ea ( ys − xs ) + u1 ⎪⎪ ⎨e y = ( r − zm ) ex − e y − xs ez + er xs + u2 ⎪ ⎪⎩ez = ym ex + xs e y − bez − eb zs + u3

For α 2 = 0 ,u2 = − ( r − zm ) β1 and er = − β 2 xs , we

(6)

obtain: V2 = − a β12 − β 22 < 0

We choose β1 = ex . So:

(14)

So, under these conditions the subsystem ( β1 , β 2 ) is

)

β1 = a e y − β1 + ea ( ys − xs ) + u1

asymptotically stable: Let β3 = ez − α 2 the error between the state ez and the

(7)

estimation function α 2 ( β1 , β 2 ) .

where e y = α1 ( β1 ) is considered as a virtual controller.

We get the subsystem represented:

To study the stability of the subsystem ( β1 ) , we consider the following Lyapunov function: V1 =

(12)

where ez = α 2 ( β1 , β 2 ) is a virtual controller.

The synchronization error is given by:

(

(11)

We obtain the following new subsystem ( β1 , β 2 ) :

⎧ xs = au ( ys − xs ) + u1 ⎪ ⎨ y s = − xs zs + ru xs − ys + u2 ⎪ z = x y − b z + u s s u s 3 ⎩ s

(

(10)

β12 2

+

ea2 2

V1 = β1 ⎡⎣ −ea ( xs − ys ) − a β1 + aα1 + u1 ⎤⎦ + ea ea

⎧ β1 = a ( β 2 − β1 ) + ea ( ys − xs ) ⎪ ⎨ β 2 = − β 2 − xs β3 + er xs ⎪ β = y β + x β − bβ − e z + u m 1 s 2 3 b s 3 ⎩ 3

(8)

(15)

We choose the Lyapunov function V3 :

(9)

For:

V3 = V2 +

α1 = 0 ,u1 = 0 and ea = β1 ( xs − ys )

β32 2

+

eb2 2

(16)

Its variation is given by: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


7


⎧ xm = a ( ym − xm ) ⎪ ⎨ y m = ( c − a − zm ) xm + cym ⎪ z = x y − bz m m m ⎩ m

(19)

⎧ xs = au ( ys − xs ) + u1 ⎪ ⎨ y s = ( cu − au − zs ) xs + cu ys + u2 ⎪ z = x y − b z + u 3 s s u s ⎩ s

(20)

⎡ xs β 2 − b β 3 + ⎤ V3 = − a β12 − β 22 + β3 ⎢ ⎥ + eb eb + − + β y e z u 3⎦ ⎣ m 1 b s

For u3 = − ( xs β 2 + ym β1 ) and eb = β3 zs : and:

V3 = − a β12 − β 22 − bβ32 < 0

(17)

For that, the subsystem ( β1 , β 2 , β 3 ) is asymptotically stable. Thus the control laws that ensure the synchronization are:

u 2 = − ( r − z m ) ex

(

u3 = − xs e y + ym ex

where au ,bu ,cu are uncertain parameters and u1 ,u2 ,u3 are the control laws. The dynamics of the synchronization error is given by:

)

where the uncertain parameters adaptation laws are:

⎧ ex ⎪ ⎪⎪e y ⎨ ⎪ ⎪ ⎪⎩ ez

ea = ex ( xs − ys ) er = −e y xs eb = ez zs

II.2.

)

= ( c − a − z s ) ex + ce y − xm ez + + ec ( xs + ys ) − ea xs + u2 = ym ex + xs e y − bez − eb zs + u3

with:

Synchronization of Uncertain Chen System Using Backstepping Algorithm

ea = au − a,eb = bu − b,ec = cu − c

The Chen system is defined by: ⎧ x = a ( y − x ) ⎪ ⎨ y = ( c − a − z ) x + cy ⎪ z = xy − bz ⎩

(

= a e y − ex + ea ( ys − xs ) + u1

(21)

Let β 1 = e x . Then:

(18)

β1 = a ( e y − β1 ) + ea ( ys − xs ) + u1

a,b,c are system parameters. For with a = 35,b = 3,c = 28 the Chen system has a chaotic behaviour as shown in Fig. 2.

(22)

e y = α1 ( β1 ) is considered as a virtual controller.

To study the stability of the subsystem

( β1 ) we

choose the Lyapunov function V1 : 50 45

V1 =

40

β12 2

+

ea2 2

(23)

35

Its derivative is:

z

30 25

V1 = β1 ⎡⎣ −ea ( xs − ys ) − a β1 + aα1 + u1 ⎤⎦ + ea ea (24)

20 15 10 5 -30

If we choose: -20

-10

0

10

20

30

x

α1 = 0 ,u1 = 0

(25)

ea = β1 ( xs − ys )

(26)

Fig. 2. Chen attractor for c = 28

and: As below, we consider the following two Chen systems, the first is the master system and the second is the slave:



8


The subsystem ( β1 ) is asymptotically stable because: V1 = −a β12 < 0

⎧ β1 = a ( β 2 − β1 ) + ea ( ys − xs ) ⎪ ⎨ β 2 = − β 2 − xm ez + ec ( xs + ys ) ⎪ β = y β + x β − bβ − e z + u 3 3 m 1 s 2 b s ⎩ 3

(27)

The error between the state e y and the estimation

Choose Lyapunov function as follows:

function is described by:

β 2 = e2 − α1

(35)

V3 = V2 +

(28)

Thus, the new subsystem is given by:

β32 2

+

eb2 2

(36)

⎡ ym β1 + xs β 2 + ⎤ V3 = − a β12 − β 22 + β3 ⎢ ⎥ + eb eb ⎣ −bβ3 − eb zs + u3 ⎦

⎧ β1 = a ( β 2 − β1 ) + ea ( ys − xs ) ⎪ ⎨ β 2 = ( c − a − zs ) β1 + c β 2 − xm ez + ⎪ + ec ( xs + ys ) − ea xs + u2 ⎩

If u3 = − ( xs β 2 + ym β1 ) and eb = β3 zs , the subsystem

( β1 , β 2 , β3 ) is

asymptotically

stable

because

the

and ez = α 2 ( β1 , β 2 ) is the virtual controller. In order to

derivative of the Lyapunov function is negative definite.

find the stability conditions that stabilize the system ( β1 , β 2 ) , we consider the Lyapunov function V2 :

V3 = − a β12 − β 22 − bβ 32 < 0

e2 + c V2 = V1 + 2 2

β 22

Thus the control laws that ensure the synchronization are:

(29)

u2 = − ( c − a − zs ) ex − e y ( c + 1) + ea xs

(

V2 =

u3 = − xs e y + ym ex

⎡( c − a − z s ) β1 + c β 2 − xmα 2 + ⎤ = − a β12 + β 2 ⎢ ⎥ + (30) ⎢⎣ +ec ( xs + ys ) − ea xs + u2 ⎥⎦ +ec ec

)

With the following parameters adaptation laws: ea = ex ( xs − ys ) ec = −e y ( xs + ys )

For:

eb = ez zs

α 2 = 0,u2 = − ( c − a − zs ) β1 − β 2 ( c + 1) + ea xs (31)

III. Numerical Simulations

and: ec = − β 2 ( xs + ys )

(32)

V2 = − a β12 − β 22 < 0

(33)

In this section, we propose a series of numerical simulations to demonstrate the effectiveness of the proposed adaptive synchronization. The differential equations are integrated using fourth order Runge-Kutta method.

We get:

So, the subsystem

( β1 , β 2 ) is

III.1. Synchronization of Uncertain Lorenz System

asymptotically stable

The parameters values of the drive system are a = 10,b = 8 / 3,r = 28 . For the slave the initial values of uncertain parameters are au = 6,bu = 8,ru = 13 . The time step of integration equals to h = 0.002 and the initial conditions of the master and the slave system are:

because the derivative of the Lyapunov function is negative definite. Let β3 be the error between the state ez and the

estimated function α 2 ( β1 , β 2 ) :

β 3 = ez − α 2 The

(37)

obtained

new

subsystem

xm ( 0 ) = −5, ym ( 0 ) = −5,zm ( 0 ) = 10,

(34)

xs ( 0 ) = −5.01, ys ( 0 ) = −5.01,zs ( 0 ) = 10.01

( β1 , β 2 , β3 ) is

determined by: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


9


At the beginning the master and slave systems have different trajectories, It is due to the sensitivity to the initial conditions. The control laws calculated in section (II.1) are applied on the slave at t = 3s. On Figs. 3 to 6, we can notice that after only some seconds the trajectories of the two systems become practically identical and the error stabilizes to zero. Figs. 7 to 9 show the estimation of the uncertain parameters of the slave system. We remark, that the uncertain parameters stabilize precisely on the same values that the main system.

40 ex ey ez

30 20

ex,ey,ez

10 0 -10 -20 -30 -40 -50

0

5

10

15

20

25

30

35

40

time (s) 20 xm xs

15

Fig. 6. Synchronization errors 20

10

xm,xs

5

15

0

10 -5

5

au

-10 -15 -20

0 0

5

10

15

20

25

30

35

40

-5

time (s)

-10

Fig. 3. Synchronization of the variables xm and xs

0

5

10

15

20

25

30

35

40

time (s)

40

Fig. 7. Estimation of the parameter au

ym ys 30 35

20

ym,ys

30 25

10

20

0 15

ru

-10

10 5

-20

0

-30

0

5

10

15

20

25

30

35

-5

40

time (s)

-10 -15

Fig. 4. Synchronization of the variables ym and y s

0

5

10

15

20

25

30

35

40

time (s)

70

Fig. 8. Estimation of the parameter ru

zm zs 60 40

50

20

30

10 bu

zm,zs

30

40

20

0

10

-10

0

0

5

10

15

20

25

30

35

-20

40

time (s)

-30

0

5

10

15

20 time

25

30

35

40

Fig. 5. Synchronization of the variables z m and z s Fig. 9. Estimation of the parameter bu



10


III.2. Synchronization of Uncertain Chen System

70 zm zs

The parameters values of the drive system are a = 35,b = 3,c = 28 . For the slave the initial values of uncertain parameters are au = 30 ,bu = 1,cu = 18 . The time step of integration is equal to h = 0.002 and the initial conditions of the master and the slave systems are:

60

zm,zs

50

40

30

20

10

xm ( 0 ) = −5, ym ( 0 ) = −5,zm ( 0 ) = 10 ,

0

0

5

10

15

20

25

30

time (s)

xs ( 0 ) = −5.01, ys ( 0 ) = −5.01,zs ( 0 ) = 10.01

Fig. 12. Synchronization of the variables z m and z s

Before applying the control, the two trajectories are different because they evolve from two completely different initial conditions. When the control is applied, the trajectories of the master and slave systems begin to come closer and become identical after a reduced time. Figs. 10 to 13 represent the evolution of the synchronization for the three state variables. The estimation of the uncertain parameters is represented in Figs. 14 to 16.

30 ex ey ez

20

ex,ey,ez

10

0

-10

-20

-30

30

0

5

10

15

20

25

30

25

30

time (s)

xm xs

Fig. 13. Synchronization errors

20 45

10

xm,xs

40

0

35

-10

au

30

25

-20

20

-30

0

5

10

15

20

25

30 15

time (s)

Fig. 10. Synchronization of the variables xm and xs

10

0

5

10

15

20

time (s) 30

Fig. 14. Estimation of the parameter au

ym ys 20 35

10

ym,ys

30

0 25

cu

-10

20

-20 15

-30

0

5

10

15

20

25

30

time (s)

10

Fig. 11. Synchronization of the variables ym and y s

0

5

10

15

20

25

30

time (s)

Fig. 15. Estimation of the parameter cu



11


[12] Y. Yu and S. Zhang, Adaptive Backstepping Synchronization of uncertain chaotic system, Chaos, Solitons and Fractals, Vol. 21: 643-649, 2004. [13] J. Park, Adaptive Synchronization of Unified Chaotic System with Uncertain Parameter, Int J Nonlinear Science and Numerical Simulation, Vol. 6: 201-206, 2005. [14] H. Zhang, X. Ma and W. Liu, Synchronization of Chaotic Systems with Parametric Uncertainty using Active Sliding Mode Control, Chaos, Solitons and Fractals, Vol. 21: 1249-1257, 2004. [15] G. Cai, and W. Tu, Adaptive Backstepping of the Uncertain Unified Chaotic System, International Journal of Nonlinear Science, Vol. 4: 17-24. 2007. [16] C. M. Liu, Y. F. Peng and M. H. Lin, CMAC- Based Adaptive Backstepping Synchronization of Uncertain Chaotic Systems, Chaos, Solitons and Fractals, vol. 21: 981-988, 2009. [17] B. Wang and G. Wen, On the Synchronization of Uncertain Master-Slave Chaotic Systems with Disturbance, Chaos, Solitons and Fractals, Vol. 41: 145-151, 2009.

30

20

bu

10

0

-10

-20

-30

0

5

10

15

20

25

30

time (s)

Fig. 16. Estimation of the parameter bu

IV.

Conclusion

Authors’ information

In this work the synchronization of chaotic systems with uncertain parameters has been proposed. The synchronization is achieved using a Backstepping algorithm. The control laws and the estimation functions of the uncertain parameters are calculated from the Lyapunov stability theory where the system stability is guaranteed. The interest of this method is that the control laws are without derivative terms, which makes the implementation of the controller easier. Two numerical examples are provided to show the effectiveness of the proposed method.

Laboratory of Automatics and Robotic, Department of Electronics, Faculty of Engineer Sciences, University Mentouri, Constantine, Algeria. E-mails: [email protected] [email protected] Ikhlef Ameur was born in Constantine (Algeria) on February 1979. He received his bachelor’s degree in Electronic and Automation engineering in 2003 and the Magister degree in automation engineering, option control systems in 2007, from the University Mentouri Constantine, Algeria. Now he is preparing his PHD degree in the automatic control and robotics laboratory of the University Mentouri Constantine (Algeria). His research interests are control, synchronization and anti control of chaotic and non linear systems.

References [1]

Q. Zhang and J. Lu, Chaos Synchronization of New Chaotic System via Nonlinear Control, Chaos, Solitons and Fractals, Vol 37: 175-179, 2008. [2] Y. Zhao, and W. Wang, Chaos Synchronization in a Josephson Junction System via Active Sliding Mode Control, Chaos, Solitons and Fractals, Vol. 41: 60-66, 2009. [3] S. Bowong, Adaptive synchronization between two different chaotic dynamical systems, Communications in Nonlinear Science and Numerical Simulation,Vol. 12: 976-985, 2007. [4] M. T. Yassen, Controlling, Synchronization and Tracking Chaotic Liu System using Active Backstepping Design, Physics Letters A, Vol. 360: 582-587, 2007. [5] L. M. Pecora, and T. L. Caroll, Synchronization in Chaotic Systems, Physical Review Letters, Vol. 64: 821, 1990. [6] L. X. Yang, Y. D. Chu, J. G. Zhang, X. F. Li and Y. X. Chang, Chaos Synchronization in Autonomous Chaotic System via Hybrid Feedback Control, Chaos, Solitons and Fractals, Vol. 41: 214-223, 2009. [7] T. Gao, Z. Chen, Z. Yuan and D. Yu, Adaptive Synchronization of New Hyperchaotic System with Uncertain Parameters, Chaos, Solitons and Fractals, Vol. 33: 922-928, 2007. [8] M. Haeri, M. Tavazoei and M. Naseh, Synchronization of Uncertain Chaotic Systems using Active Sliding Mode Control, Chaos, Solitons and Fractals, Vol. 33: 1230-1238, 2007. [9] M. Ho and Y. Haung, Synchronization of two Different Systems by Using Generalized Active Control, Physics Letters A, Vol. 301: 424-431, 2002. [10] S. Du, B. J. V. Wyk, G. Qi and C. Tu, Chaotic Synchronization with an Unknown Master Model using Hybrid HOD Active Control Approach, Chaos, Solitons and Fractals, Vol. 42: 19001913, 2009. [11] X. Tan, Z. Jiye and Y. Yang, Synchronization Chaotic Systems using Backstepping Design, Chaos, Solitons and Fractals, Vol. 16: 37-45, 2003.

Mansouri Noura was born in Algeria, on August 1956. She received the DEA and the Docteur Ingenieur degrees in control and signal processing from the University of Technology of Compiègne, France, in 1981 and 1984 respectively, and the PHD degree in electrical engineering from the University Mentouri Constantine, Algeria, in 1999. Currently, she is full professor in the Department of Electronics at the University Mentouri Constantine, Algeria, and director of automatic control and robotics laboratory, in the same university. Her main research activities and publications include parameters identification in non linear problems, systems with uncertain parameters, chaotic systems.



12


Fuzzy Based Design Specification for Set Membership System Identification Majda Ltaief1, Ridha Ben Abdennour1, Mohammad M'Saad2 Abstract – This paper is mainly concerned with the parameter design specification for system identification in the presence of bounded disturbances. More specifically, a fuzzy based procedure is proposed to properly determine an adequate estimate of the disturbances bound providing thereby the key design parameter for set membership system identification. A robust parameter adaptation algorithm involving a suitable data weighting has been particularly considered for its fundamental as well as implementation simplicity. A case study involving an academic system identification problem is carried out to emphasize the engineering effectiveness of the proposed fuzzy based procedure. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Set Membership Identification, Robust Parameter Adaptation Algorithm, Parameter Design Specification, Fuzzy Based Disturbances Upper Bound Estimation

u(k) y(k) v(k) A(q-1),B(q-1) d

J ym(k) S

Control Signal System output External disturbances Polynomials System delay Regression vector Parameter vector Noise upper bound Coast function Model output Smallest parameter set

λ

Weighting factor

ε (k )

A priori estimation error

θˆ

Estimated parameter vector

P

Covariance matrix

In1 ( k ) , In2 ( k )

Supervision inputs

In1n ( k ) , In2n ( k )

Normalized supervision inputs

∆γ (k)

Noise bound variation

ε N (k ) Nce1i, Nce2i

Mean of the a priori estimation error Inputs center numbers

Ncsj

Output center numbers

Dj

Heights of the trapezes gotten by leveling of the output triangular membership functions Inputs fuzzy sets number

ϕ θ γ

ne

Output fuzzy sets number

ns

Nomenclature

I.

Introduction

A remarkable research activity has been devoted to set-membership based parameter adaptation for system identification, adaptive control and controller reduction [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The appealing feature consists in the possibility to deal with the identification in the presence of bounded disturbances without resorting to any modelling assumption unlike the usual predictive identification techniques [11], [12]. The underlying set-membership system identification problem consists in determining the smallest parameter set, consistent with the available measurements as well as the disturbances upper bound, containing the indistinguishable models of the system [10]. Two approaches have been pursued to address such a problem. • A geometrical approach which consists in computing a hypercubic set providing accurate bounds on the model parameters thanks to an important computational burden [7]. A recursive algorithm that minimizes the size of an ellipsoid enclosing the parameter set has been proposed in [5] to overcome the computing burden. Further improvements of this algorithm have been performed in [1]. Though suitable insights on the subtleties between hypercubic and ellipsoid bounding sets have been given in [1], the potential of the ellipsoid bounding algorithms has been emphasized in [8]. • A stability oriented approach that heavily borrows from the optimal bounding ellipsoid formulation as well as the equation error based parameter adaptation


13


Majda Ltaief, Ridha Ben Abdennour, Mohammad M'Saad

( )

[3] [4], [6]. The latter contribution has shown to be particularly attractive from an engineering point of view. It consists in simple least squares based parameter adaptation algorithm involving a suitable data weighting together with a stability robustness feature [4]. An extension to the multivariable systems has been made in [2] and an adequate reformulation for the output error based parameter adaptation algorithm has been investigated in [9]. Two remarks are worth to be pointed out for this approach. Firstly, the parameter estimates are projected on the boundary of an ellipsoid region compatible with the disturbances upper bound. Secondly, the parameter adaptation is automatically frozen whenever the incoming information is not likely to improve the parameter estimation process. In spite of the appealing features of the stability oriented approach and more particularly the genuine contribution [4], they require to specify an upper bound on the disturbances. This specification constitutes a challenging design issue. Indeed, the underlying parameter adaptation performances strongly depend on a proper specification of this design parameter as it was pointed out by the simulation results reported in [4] and [9]. A fuzzy based estimation procedure of a proper upper bound of the disturbances is proposed in this paper from the available fuzzy supervision results (See [13]). This addresses the applicability issue of set membership parameter estimation providing thereby the remaining underlying engineering feature. An academic system identification problem involving a realistic disturbance has been carried out to provide suitable insights about the engineering feature of the proposed fuzzy based estimation procedure. The paper is organized as follows. Section 2 presents the considered system identification algorithm with a particular emphasis on the practical features throughout a set of simulation results. Section 3 is devoted to the proposed fuzzy based estimation procedure of the upper bound on the disturbances. Simulation results are reported in section 4 to demonstrate the performance improvements that could be attained using the proposed fuzzy based estimation procedure. Some concluding remarks end the paper.

II.

B q −1 = b1q −1 + ... + bnb q − nb

where k denotes the sampling time instant, {u(k)}, {y(k)} and {v(k)} are respectively the control variable, the measured output and the external disturbances and modeling errors, d is the system delay in sampling periods, and q-1 is the shift operator. Such a model may be given the following appropriate form from system identification point of view: y ( k ) = ϕ ( k − 1) θ + v ( k )

(4)

⎡ − y ( k − 1) ⎤ ⎡ a1 ⎤ ⎢ ⎥ ⎢ ⎥ # ⎢ ⎥ ⎢ # ⎥ ⎢ − y ( k − na ) ⎥ ⎢ ana ⎥ ⎥ θ = ⎢ ⎥ and ϕ ( k − 1) = ⎢ ⎢ ⎥ b u k d 1 − − ( ) 1 ⎢ ⎥ ⎢ ⎥ ⎢ # ⎥ ⎢ ⎥ # ⎢ ⎥ ⎢ ⎥ b ⎣⎢ nb ⎦⎥ ⎣u ( k − d − nb ) ⎦

(5)

T

with:

The underlying system identification problem is well posed provided that the input, output and disturbance sequences are bounded and that a bound on the disturbances is known, i.e. There exists a known positive scalar γ such that:

v (k ) ≤ γ

() {

( )

( )

(7)

with: ym ( k ) = ϕ T ( k − 1)θ

(8)

()

where S θ represents a class of models for which one has for k>0: y ( k ) − ym ( k ) ≤ γ ( k )

(1)

(9)

In the following, one will briefly presents the considered identification algorithm with simulation results showing the importance of a proper specification of the disturbance bound γ.

with: A q −1 = 1 + a1q −1 + ... + ana q − na

}

2 S θ = θ : ( y ( k ) − ym ( k ) ) ≤ γ 2 ( k ) ; ∀k > 0

We are particularly concerned with the identification of sampled data systems which input-output behavior can be described by the following equation:

( )

(6)

Indeed, under these assumption, it is possible de determine the smallest parameter set, consistent with the available measurements as well as the disturbances upper bound, containing the indistinguishable models of the system, i.e.:

The Problem Statement

A q −1 y ( k ) = B q − 1 u ( k − d ) + v ( k )

(3)

(2)



14


II.1.

The Considered Identification Algorithm

II.2.

To this end, one will particularly use a simple least squares based parameter adaptation algorithm involving a suitable data weighting together with stability robustness feature that has been proposed in [4]. The underlying cost function is given by:

) ∑ λ ( i,k ) ( y (i ) − ϕ (i − 1)

(

k

J θ ( k ) =

T

θ ( k )

i =0

)

2

Consider the sampled data system:

(1 + 0.8q = ( 0.9q

the input-output sequence { ϕ ( k ) } is persistently excited in the usual sense [4]. The underlying parameter adaptation algorithm is given by:

θˆ ( k ) = ϕ

( k − 1) P ( k − 1) ϕ ( k − 1)

1

η (k )

(10b)

η ( k ) = ( ε ( k ) − γ ( k ) ) sign ( ε ( k ) ) T ε ( k ) = y ( k ) − ϕ ( k − 1) θˆ ( k − 1)

0.1

0.8

0

0.6

-0.2

0.2 0

(11)

k 0

100

300

400

-0.3

1

0 -0.2

k 0

100

200

100

300

400

200

300

400

300

400

b2(k)

-0.6

0.2 0

k 0

-0.4

b1(k)

0.4

-0.8 -1

k 0

100

200

Fig. 1. Evolutions of the estimated parameters

⎧0 if ϕ T ( k − 1) P ( k − 1) ϕ ( k − 1) = 0 ⎪⎪ σ (k ) = ⎨ (13) or ε ( k ) < γ ( k ) ⎪ ⎪⎩1 otherwise

1

0.2

0.8

0.1

0.6

0

a1(k)

0.4

-0.1

0.2 0

γ (k ) ε (k )

200

0.8 0.6

⎡ P ( k − 1) − σ ( k ) ⋅ ⎤ ⎥ (12) 1⎢ T = ⎢ P ( k − 1) ϕ ( k − 1) ϕ ( k − 1) P ( k − 1) ⎥ λ ⎢⋅ µ ( k )⎥ T ϕ ( k − 1) P ( k − 1) ϕ ( k − 1) ⎣ ⎦

a2(k)

-0.1

a1(k)

0.4

P (k ) =

µ (k ) = 1−

− 0.7 q −2

The system identification has been carried out using the above parameter identification with three specifications of the design parameter γ, namely γ = 0.295, γ = 0.2 and γ = 0.4. Figures 1, 2 and 3 show the respective evolutions of the parameter estimates. One can appreciate the identification accuracy resulting from a proper choice of the design parameter, i.e. γ = 0.295, the oscillatory behavior of the parameter estimates in the case of an underestimation of the disturbances bound, i.e. γ = 0.2, and the parameter bias occurring in the case of an overestimation of the disturbances bound, i.e. γ = 0.4.

( )

T

) )u (k ) + v (k )

+ 0.1q −2 y ( k ) =

with v ( k ) ≤ 0.295 .

(10a)

It is worth noticing that the time varying sequence {λ(i,k)} is mainly motivated by stability purposes and allows an exponential convergence of the parameter vector sequence { θˆ ( k ) } to the set S θˆ provided that

= θˆ ( k − 1) +

−1

−1

with λ ( i,k ) = λ k −i λ ( i ) where λ∈[0,1] and λ(k,k)=λ(k).

σ ( k ) P ( k − 1) ϕ ( k − 1)

Simulation Results

(14)

k 0

100

300

-0.3

1

0 -0.2

k 0

100

200

100

200

300

400

300

400

b2(k)

-0.6

0.2 0

k 0

-0.4

b1(k)

0.4

information is not likely to improve the parameter

400

0.8 0.6

It is worth noticing that the parameter adaptation is frozen when the estimation error is within a dead zone, namely ε ( k ) ≤ γ ( k ) , as well as when the incoming

200

a2(k)

-0.2

300

400

-0.8 -1

k 0

100

200

Fig. 2. Evolutions of the estimated parameters

adaptation process, e.g. ϕ ( k − 1) P ( k − 1) ϕ ( k − 1) = 0 . T

(underestimation of γ)

Such a feature is of fundamental importance from parameter estimation point of view, i.e. provide a model witch is consistent with the disturbances constraint. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


15


0.8

0.1

0.6

0

0.4

k 0

100

200

300

400

-0.3

1

0 -0.2

0.6

100

0.4

300

0

100

200

300

400

-1

(15)

measured in the same sliding window using the following relation:

b2(k)

-0.8

k

i = k − N +1

∑

400

-0.6

0.2

2

⎣⎡ε ( i ) − ε N ( k ) ⎦⎤ =

with ε N ( k ) is the mean of the a priori estimation error

-0.4

b1(k)

200

k

∑

⎞ 2 1⎛ = ⎜⎜ ε 2 ( i ) ⎟⎟ − ⎡⎣ε N ( k ) ⎤⎦ N ⎝ i = k − N +1 ⎠

k 0

1 N k

-0.2

0.8

0

a2(k)

-0.1

a1(k)

0.2 0

In1 ( k ) =

k 0

100

200

300

ε N (k ) =

400

Fig. 3. Evolutions of the estimated parameters (overestimation of γ)

⎞ 1⎛ k ε ( i ) ⎟⎟ ⎜ ⎜ N ⎝ i = k − N +1 ⎠

∑

(16)

III.1.2. The A Priori Estimation Error Boundary Constraint

In the following, one will exploit the fuzzy logic culture to design a supervisor to determine an on line appropriate estimate of disturbance upper bound. This estimate will be used to improve the accuracy of the system identification.

The second supervision input is formulated by the a priori estimation error boundary constraint which is resulting from the noise boundary constraint:

III. The Fuzzy Based Disturbances Upper Bound Estimation

In2 ( k ) = ε ( k ) − γ ( k )

The proposed fuzzy based procedure for the noise bound estimation consists to combine a fuzzy logic based supervisor with a set-membership based identification algorithm to ameliorate its estimation quality. A fuzzy logic based supervisor is inserted in the identification loop for the online adjustment of the noise bound γ(k) appealing to two online criterions, which are based on the a priori estimation error (first supervision input: In1(k)) and the noise constraint (second supervision input: In2(k)). The considered fuzzy supervisor, having as output the noise bound variation (∆γ (k)), has the scheme given by Figure 4.

This online performance criterion is used to control the existence of the a priori estimation error in the "dead zone" despite the online adjustment of the noise bound.

In1(k) In2(k)

Fuzzy supervisor

(17)

III.2. The Fuzzy Supervisor The fuzzy supervisor is based on three stages which are the fuzzfication, the fuzzy inference and the defuzzification [13], [14], [15], [16], [17], [18] , [19], [20].

III.2.1. Fuzzification The fuzzification consists to convert the normalized supervision real inputs (In1n(k), In2n(k)) into fuzzy inputs. Indeed, to each normalized supervision real input, we associate ne fuzzy sets. Each fuzzy set is described by an adequate linguistic term and can be represented by triangular membership functions centered on the numbers (Nce1i, Nce2i, i=1, ..., ne). The a priori estimation error variance (In1(k)) is normalized in the interval [0, 1]:

∆γ (k)

Fig. 4. The fuzzy supervisor scheme

III.1. Supervision Inputs III.1.1. The Variance of the a Priori Estimation Error

In1n ( k ) =

The variance of the estimation error is considered as the first supervision input. This performance criterion is used to evaluate the convergence of the considered identification algorithm (estimation accuracy). The variance of the a priori estimation error is measured in a sliding window (N iterations from max (0, k-N+1) to k) as follows:

In1 ( k ) − In1min In1max − In1min

Nce1i =

i −1 ne − 1

(18)

(19)

where i=1, ..., ne. The a priori estimation error boundary constraint (In2(k)) is normalized in the interval [-1, 1] and we have:



16


In2 n ( k ) =

In2 max − In2 min 2 In2 max − In2 min 2

In2 ( k ) −

Nce2i = −1 +

Medium

(20)

If In1n(k) is L and In2n(k) is P then ∆γ(k) is LP

i −1 ne − 1 2

(21a)

µ(In1n(k))

Small

the supervisor inputs, which are directly related to the a priori estimation error and the noise constraint. These rules are of the following type:

The inference table associates ns fuzzy sets (ns=7) to the supervisor output variable, which are descried, also, by triangular membership functions (see Figure 6). These functions are defined in the interval [-1, 1] and centered on the numbers Ncsj:

µ(In2n(k)) Negative Zero

Large

Nce11=0 Nce12=0.5 Nce13=1

In1n(k)

Ncs j = −1 +

Positive

Nce21=-1 Nce22=0 Nce23=1 In2n(k)

⎧⎪ ⎡ µi ( In1n ( k ) ) ,⎤ ⎫⎪ ⎥⎬ D j ( k ) = MAX ⎨ MIN ⎢ Rule j ⎢⎣ µl ( In2 n ( k ) ) ⎥⎦ ⎭⎪ ⎪⎩

In the present case, three fuzzy sets have been reserved for each supervision input (ne=3) represented by triangular membership functions as illustrated by the Figure 5.

The synthesis of the fuzzy supervisor requires a phase of experimentation permitting the development of a logical and methodological rules' basis. The stage of inference consists to apply the linguistic rules provided by the rules' basis to the fuzzy inputs descended of the fuzzification procedure to evaluate the supervisor output (∆γ(k)). The fuzzy inference rules that manage the online supervision of the noise bound is presented in the following decision table:

µ(∆γ(k))

LN

MN

SN

Z

SP

MP

LP

Ncs1=-1 Ncs2 =-2/3 Ncs3=-1/3 Ncs4=0 Ncs5=1/3 Ncs6=2/3

TABLE I FUZZY INFERENCE TABLE

In2n(k)

(22)

with: i, l = 1, …, ne ; j = 1, …, ns. Actually, the terms Dj correspond to the heights of the trapezes gotten by leveling of the output triangular membership functions.

III.2.2. Fuzzy Inference

∆γ(k)

Fig. 6. Membership functions related to the supervision output

In1n(k) N

(21b)

The MIN-MAX inference method is used for the evaluation of the fuzzy rules contribution. Therefore the supervisor's output is determined by calculation of the coefficients Dj given by the following formula:

Fig. 5. Membership functions related to the supervision inputs

∆γ(k)

j −1 ns − 1 2

S

M

L

Z

MN

LN

III.2.3. Defuzzification The defuzzification operation consists to calculate the numerical value of the supervision output from its fuzzy value and constitutes the last stage of supervision. This value (∆γ(k)) can be calculated using the formula of gravity center given by:

Z

Z

SN

MN

P

SP

MP

LP

The linguistic variables are given: • For In1n(k): S: Small ; M: Medium ; L: Large. • For In2n(k): N: Negative ; Z: Zero ; P: Positive. • For ∆γ(k): LN: Large Negative; MN: Medium Negative; SN: Small Negative; Z: Zero; SP: Small Positive; MP: Medium Positive; LP: Large Positive. The inference rules that are based on the know-how of human experts in identification, express in a linguistic form, the variation of the noise bound as a function of

ns

∆γ ( k ) =

∑ D j ( k ) Ncs j j =1

ns

(23)

∑ Dj (k ) j =1



17


At the end of every supervision cycle, the activation of the associated rules' set yields to the necessary noise bound variation, and the new noise bound is given by:

γ ( k ) = γ ( k − 1) + g ∆γ ( k )

Another interesting observation is the fact that, using this fuzzy supervision level, the considered identification algorithm can be exploited without the assumption about the availability of the noise bound. The adequate noise bound is supplied online using the supervision level (see Figure 8 presenting the evolutions of the fuzzy noise bound γ(k) and the fuzzy supervisor output ∆γ(k)). We consider in this section the case when the noise sequence varies online (see Figure 9). The observations picked out on the previous second order system are supposed to be corrupted by this variable noise sequence and we apply the considered identification algorithm together with the fuzzy based procedure to estimate the system parameters. The estimation results are presented in the following figures.

(24)

with g is a positive coefficient. Generally, the choice of g must favor the convergence rapidity of the supervision level to the optimal noise bound without oscillation.

IV.

Simulation Results

To evaluate the contribution of the proposed fuzzy based procedure in term of estimation accuracy, we investigate the identification problem of the same second order system given in section II.2. The estimation results, using the considered identification algorithm, are illustrated by Figure 7. 1

0.2

0.8

0.1

0.6

0.3 0.25 0.2

k 0

100

200

a2(k)

-0.1

0.2 300

400

-0.3

1

0 -0.2

0.6

100

k 0

100

200

200

300

400

300

-1

k 0

100

200

300

400

Fig. 7. Evolutions of the estimated parameters (Fuzzy supervision of γ) 0.4 0.3

γ(k)

0.2 0.1 0

k 0

50

100

150

200

500

1000

k 1500

The Figure 10, presenting the estimated parameters evolutions demonstrates that the estimation accuracy is not influenced by the online variation of the noise sequence. Indeed, the fuzzy supervision level was able to detect this variation and it provides the adequate noise bound (see Figure 11) which leads to a, relatively, good estimation. The Figure 12 presenting the evolutions of the estimated parameters in the case of a constant noise bound, demonstrates the degradation in the estimation quality (especially in the interval where the noise sequence varies) by comparison with the case when the fuzzy noise bound is used. This is due to the online variation of the noise bound which may occur randomly (without any a priori knowledge) in practice case.

b 2(k)

-0.8

400

0

Fig. 9. Evolution of the noise sequence

-0.6

0.2

Noise sequence

k 0

-0.4

b1(k)

0.4

0.15

-0.2

0.8

0

0.35

0

a1(k)

0.4

0

0.4

250

300

350

400

250

300

350

400

1 0.5 0

∆γ(k)

-0.5 -1

k 0

50

100

150

200

Fig. 8. Evolutions of the fuzzy noise bound γ(k) and the fuzzy supervisor output ∆γ(k)

Referring to this figure we can notice the convergence rapidity and the estimation accuracy registered when appealing to the proposed fuzzy based procedure to determine the adequate estimate of the disturbances bound.

Fig. 10. Evolutions of the estimated parameters (variable noise sequence)



18


models but are not always possible to establish. This conclusion is particularly true as shown by the second simulation studies devoted to the process estimation performances when the proposed fuzzy supervisor is used. More specifically, it has been shown that the proposed estimation procedure is able to determine a proper upper bound on the disturbances providing thereby the key design parameter to get the best model that could be attained within the considered system identification framework. Finally we can note that the fuzzy supervisor performance is closely related to the judicious choice of its inputs and output discourse universe. Fig. 11. Evolutions of the fuzzy noise bound γ(k) and the fuzzy supervisor output ∆γ(k) (variable noise sequence)

References [1]

[2]

[3]

[4] [5]

[6] Fig. 12. Evolutions of the estimated parameters (variable noise sequence and constant noise bound)

V.

[7]

Conclusion [8]

An adequate fuzzy based procedure has been proposed to handle the crucial parameter design specification of a genuine parameter adaptation algorithm developed in a suitable set membership identification framework [4]. Two set of simulation studies have been carried out. The first one is mainly motivated by the parameter design specification issue of set membership system identification, namely how the performances are closely related to a proper choice of the upper bound of the disturbances? The following features have been emphasized. • The estimation process is insensitive to the system parameter variations when the disturbances upper bound is overestimated. • The estimation process exhibits an oscillatory behavior when the disturbances upper bound is underestimated. These features naturally yield to conclude that time varying bounds. Indeed, the fuzzy supervision level was able to detect this variation and it provides the adequate noise bound (see Figure 11) which leads to a, relatively, good estimation, would provide an admissible set of

[9]

[10] [11] [12] [13]

[14]

[15]

[16]


G. Belforte, B. Bona, and V. Cerone. Parameter estimation algorithms for a set-membership description of uncertainty. Automatica, Vol. 26(5):887-898, 1990. M. Boutayeb, M. Darouach, and H. Rafaralahy. Generalized statespace observers for chaotic synchronization and secure communication. IEEE Trans. on Circuits and Systems, TCS-49, 345-349, 2002. S. Dasgupta and Y. Huang. Asymptotically convergent modified least squares with data dependent updating and forgetting for systems with bounded noise. IEEE Transactions on Information Theory, Vol. 33(3): 383-391, 1987. C. C. de Wit and J. Carillo. A modified ew-rls algorithm for systems with bounded disturbances. Automatica, Vol. 26: 599606, 1990. E. Fogel and Y. Huang. On the value of information in system identification - bounded noise case. Automatica, Vol. 26(4):651677, 1982. R. Lozano and R. Ortega. Reformulation of the parameter identification problem for systems with bounded disturbances. Automatica, Vol. 23:247-250, 1987. M. Milanese and G. Belforte. Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models and estimators. IEEE Trans. Auto. Control, AC-27:408-414, 1982. J. Norton. Identification and application of bounded-parameter models. Automatica, Vol., 23(4):479-507, 1987. M. Pouliquen and M. M'Saad. Further stability and convergence analysis of a set membership identification. In Proceedings of the 2nd International Symposium on Control, Communications, and Signal Processing, Marrakech,, Morocco, 13-15 March 2006. E. Walter and L. Pronzato. Identification of Para-metric Models (Springer-Verlag, 1997). L. Ljung. System Identification: Theory for the User (Prentice Hall, 1997). T. Soderstrom and P. Stoica. System Identification (Prentice Hall, 1989). M. Ltaief, K. Abderrahim, R. Ben Abdennour, and M. Ksouri. A fuzzy fusion strategy for the multi-model approach application. WSEAS Transaction on Circuits and Systems, Vol. 2(2):686-691, 2003. R. Ben Abdennour, F. Bouani, M. Ksouri, and G. Favier. Etude comparative des commandes floue et prédictive généralisée : Application un procédé pilote. Journal Européen des Systèmes Automatisés, Vol. 32(2):1-19 1998. R. Ben Abdennour, G. Favier, and M. Ksouri. Fuzzy trace identification algorithms for non-stationary systems. Intelligent and fuzzy systems, Vol. (6):403-417, 1998. R. Ben Abdennour, M. Ksouri, and G. Favier. Appli-cation of fuzzy logic to the on-line adjustment of the parameters of generalized predictive controller. Intelligent Automation and Soft Computing, Vol. 4(3):197-214,1998.


19


[17] R. Ben Abdennour, M. Ltaief, and M. Ksouri. Un coefficient d'apprentissage flou pour les réseaux de neurones artificiels. APIIJESA, Vol. 35(6):1089-1103 , 2001. [18] P. Fedor, D. Perduková. Fuzzy Model-Based Black Box System Control. International Review of Automatic Control (I.RE.A.CO.). Vol. 3( 5):438-442, 2010. [19] Z. Souar, K. Chegroune, F. Olivie. Comparative Analysis of PI and Fuzzy Logic Controllers for a New Intelligent Control of Speed. International Review of Automatic Control (I.RE.A.CO.). Vol. 3(1):53-59, 2010. [20] Tomoaki Ishihara, Jun Yoneyama. H∞ Sampled-Data Control for Fuzzy Systems with Discrete and Distributed Delays. International Review of Automatic Control (I.RE.A.CO.). Vol. 2(6):654-660, 2009.

Ridha Ben Abdennour received the Doctorat de spécialité degree from the Ecole Normale Supérieure de l’Enseignement Technique in 1987, and the Doctorat d’Etat degree from the Ecole Nationale d’Ingénieurs de Tunis in 1996. He is Professor in Automatic Control at the National School of Engineering of GabesTunisia. He was chairman of the Electrical Engineering Department and the Director of the High Institute of Technological Studies of Gabes. He is the Head of the Research Unit of Numerical Control of Industrial Processes and is the President of the Tunisian Association of Automatic and Numerisation. His research is on Identification, Multimodel & Multicontrol approaches, Numerical Control and Supervision of Industrial Processes. He is the co-author of a book on Identification and Numerical Control of Industrial Processes and he is the author of more than 250 publications.

Authors’ information

Mohammed M'Saad received his PhD in 1978 in Rabat. He received his Doctorat d’Etat-es-Sciences Physiques from the Institut National Polytechnique de Grenoble in April 1987 and held a research position at the Centre National de Recherche Scientifique in March 1988. In September 1996, he is a Professor at the Ecole Nationale Supérieure d’ingénieurs de Caen where he is the Head of the Control Group at the GREYC. His main research areas are adaptive control theory, system identification and observation, advanced control methodologies and applications, computer aided control engineering.

1

University of Gabes, National School of Engineers of Gabes, 6029 Gabes, Tunisia. Research Unit: Numerical Control of Industrial Processes. 2 Equipe d’Automatique du GREYC (UMR 6072), ENSICAEN 6 Boulevard Maréchal juin-14050.

Majda Ltaief received the Engineering Diploma in Electric-Automatic from National School of Engineers of Gabes-Tunisia in 1996 and the DEA for the same speciality from National School of Engineers of Tunis-Tunisia in 1999. In 2005, she obtained the Ph. D. degree in Electric-Automatic Engineering from National School of Engineers of Tunis-Tunisia. She has been Assistant Professor (2004/2005) in the Electric Engineering Department on High Institute of Applied Sciences and Technology of Gabes-Tunisia. She is actually an Associate Professor in the Electric Engineering Department in National School of Engineers of Gabes-Tunisia. Her research interests include Multimodel and Multicontrol approaches, Fuzzy supervision and Numerical Control of complex systems.



20


Designing Controller for Large Scale Non-Affine Nonlinear Systems: Decentralized Intelligent Adaptive Approach Reza Ghasemi Abstract – This paper introduces a new decentralized fuzzy adaptive controller for a class of large scale non-affine nonlinear systems in which the subsystems dynamics and interconnections are represented by unknown nonlinear functions. The proposed controllers are mainly based on fuzzy concepts. Through Lyapunov stability analysis, stability of closed-loop system along with convergence of the tracking error to zero are guaranteed. Robust adaptive control has been used to avoid chattering in adaptation laws. To show the effectiveness of the proposed controller, a nonlinear system is chosen as a case study. Simulation results are very promising. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Stability, Robust Adaptive Control, Non-Affine Nonlinear Systems, Large Scale System, Fuzzy System

I.

To design stable FAC and linear observer for class of affine nonlinear systems are presented in [29], [30], [14], [13]. Fuzzy adaptive sliding mode controller is presented for class of affine nonlinear time delay systems in [2], [21], [6]. The output feedback FAC for class of affine nonlinear MIMO systems is suggested in [4]. The main drawbacks of these papers are those restricted conditions imposed on the system dynamics. For example, it is assumed the control gain is bounded to some known functions or constant values. In [27], a fuzzy adaptive controller has been developed for a class of non-affine nonlinear systems in which the parameter update laws are established based on optimal control theory. Accordingly, the boundedness of the controller parameters and the convergence of the tracking error to the residual set are guaranteed based on Lyapunov theorem. In [28], feedback linearization has been applied to design fuzzy adaptive controller for a class of canonical non-affine nonlinear systems. The controller includes three terms, the stabilizer of the linear term, the adaptive controller and the robust controller. The main limitation of these methods is that convergence of tracking errors to zero is not guaranteed. [38], [39] proposed a decentralized fuzzy model reference controller for a class of canonical nonlinear large scale system. The main limitation of these references is considering the interaction as a bounded disturbance. Stable adaptive controller for class of linear large scale system is proposed in the [3], [5], [8], [9]. [12] deals with designing FAC based on sliding mode for class of large scale affine nonlinear systems. [7] presents decentralized sliding mode fuzzy adaptive tracking for a class of affine nonlinear systems in large scale systems. [10] designed FAC for a class of affine nonlinear time

Introduction

Nowadays, fuzzy adaptive controller (FAC) has attracted many researchers to developed appropriate controllers for nonlinear systems especially for large scale systems (LSS) [1]-[39] because of the following reasons. Due to its tunable structure, the performance or the FAC is superior that of the fuzzy controller. Instead of using adaptive controller, FAC can use knowledge of the experts in the controller. In the recent year, FAC has been fully studied as follow: 1. The Takagi-Sugeno (TS) fuzzy systems have been used to model nonlinear systems and then TS based controllers have been designed with guaranteed stability [15], [16]. To model affine nonlinear system and to design stable TS based controllers have been employed in [17]. Designing of the sliding mode fuzzy adaptive controller for a class of multivariable TS fuzzy systems are presents in [20]. In [18], [19], the non-affine nonlinear function are first approximated by the TS fuzzy systems, and then stable TS fuzzy controller and observer are designed for the obtained model. In these papers, modeling and controller has been designed simply, but the systems must be linearizable around some operating points. 2. The linguistic fuzzy systems have been used to design controllers for nonlinear systems. [1], [25], [22], [36], [37] have considered linguistic fuzzy systems to design stable adaptive controller for affine systems based on feedback linearization. Stable FAC based on sliding mode is designed for affine systems in [26]. Designing of the FAC for affine chaotic systems are presented in [23], [24].


21


R. Ghasemi

dfu ( X i , ui )

delayed systems. In none of these papers, fuzzy adaptive controllers have been developed only for nonlinear nonaffine systems. In this paper, we propose a new method to design an asymptotically stable decentralized robust adaptive controller based on fuzzy systems for a class of large scale non-affine nonlinear systems. Comparison to previous studies which mainly focus on non-affine SISO systems and affine large scale systems, the proposed method is on non-affine nonlinear large scale systems. The controller is robust against uncertainties, external disturbances and approximation errors. This paper proposes a new method to encounter the unknown functions of system and interconnection between subsystems. Finally, the stability of the closed loop system is guaranteed. The rest of the paper is organized as follows. Section 2 gives problem statement. General concept of the fuzzy systems is formulated in section 3. To design Fuzzy adaptive controller is proposed in section 4. Section 5 shows simulation results of the proposed controller and Section 5 concludes the paper.

II.

dt

large

scale

Assumption 2: The desired trajectory yd ( t ) and its j time derivatives yd( ) ( t ) , j = 1, 2,...,ni , are all smooth and

bounded. The interactions can be considered as external inputs bounded by some constant time varying signal, which is in general a function of state tracking errors. To make it more suitable for the proposed controller derivation, the following assumption is used. Assumption 3: the interconnection term satisfies the following: mi ( X1 , X 2 ,..., X N ) ≤ ψ i 0 +

∑ j =1ψ ij ( bTj Pj e j ) N

(3)

where ψ i 0 and ψ ij ( .) is an unknown parameters and

ψ ij ( 0 ) = 0 . non-affine

⎧ xi,l = xi,l +1 l = 1, 2,...,ni − 1 ⎪ ⎨ xi,ni = fi ( X i ,ui ) + mi ( X 1 , X 2 ,..., X N ) + di ( t ) ⎪ ⎩ yi = xi,1

Assumption 4: the disturbance in the above equation is bounded by: di ( t ) ≤ d max (4) Define the tracking error vector as:

(1)

T

ei = ⎡⎣ ei ,1 , ei ,2 , . . . , ei,ni ⎤⎦ ∈ \ ni

where i=1,…N and xij is jth state of ith subsystem, T

(2)

f dm ∈ \ is known and constant.

Problem Statement

Consider the following nonlinear system:

≥ f dm

(5)

where: ei,1 = yd

X i = ⎡⎣ xi,1 ,... , xi ,ni ⎤⎦ ∈ \ is the state vector of the ith subsystem which is assumed available for measurement, ui ∈ R is the control input, yi ∈ R is the system output, ni

− yi

(6)

Taking the ni th derivative of both sides of the equation (6) we have:

( X i , ui ) is an unknown smooth nonlinear function, is an unknown nonlinear mi ( X 1 , X 2 ,..., X N ) interconnection term, and di ( t ) is bounded disturbance. fi

n n n ei,( 1i ) = yd( i ) − yi ( i ) = n = yd( i ) − fi ( X i ,ui ) − mi ( X1 , X 2 ,..., X N ) − di ( t )

The control objective is to design an adaptive fuzzy controller for system (1) such that the system output yi ( t ) follows a desired trajectory yd ( t ) while all

(7)

Use equation (5) to rewrite the above equation as: ⎧⎪ y ( ni ) − f ( X ,u ) + ⎫⎪ i i i ei = Ai 0 ei + bi ⎨ d ⎬ ⎪⎩ − mi ( X 1 , X 2 ,..., X N ) − di ( t ) ⎭⎪

signals in the closed-loop system remain bounded. In this paper, we will make the following assumptions concerning the system (1) and the desired trajectory yd ( t ) .

(8)

where Ai 0 and bi are defined below:

Assumption 1: without loss of generality, it is assumed that the nonzero function fu ( X i , ui ) = ∂f ( X i , ui ) ∂ui satisfies the following

⎡0 ⎢0 ⎢ Ai 0 = ⎢ # ⎢ ⎢0 ⎢⎣0

condition: fu ( X i , ui ) ≥ f min > 0 for all ( X i , ui ) ∈ \ ni ⋅ \


1 0 " 0⎤ ⎡0 ⎤ 0 1 " 0⎥⎥ ⎢0 ⎥ # # % # ⎥ ∈ \ ni ⋅ni and bi = ⎢ ⎥ ∈ \ ni (9) ⎢# ⎥ ⎥ 0 0 " 1⎥ ⎢ ⎥ ⎣1⎦ 0 0 " 0⎥⎦ International Review of Automatic Control, Vol. 4, N. 1

22

R. Ghasemi

Consider the vector ki = ⎡⎣ ki,1 , ki,2 , . . . , ki,ni ⎤⎦

T

constant λ in the range of 0 < λ < 1 , such that the

be

nonlinear function fi

coefficients of L ( s ) = s ni + ki ,ni s ni -1 + ... + ki,1 and chosen

u*i as:

so that the roots of this polynomial are located in the open left-half plane. This makes the matrix Ai = Ai 0 − bi kiT be Hurwitz. Thus, for any given positive definite symmetric matrix Qi , there exists a unique positive definite symmetric solution Pi for the following Lyapunov equation: AiT Pi + Pi Ai = −Qi

(

= fi X i ,

where

(

)

)

)+e

(15)

ui f iuλ

fiuλ = ∂f ( X i , ui ) / ∂ui |ui =uiλ

and

⎧eui fiuλ + mi ( X 1 , X 2 ,..., X N ) + ⎫ ⎪ ⎪ ei = Ai ei − bi ⎨ ⎬ (16) T ′ d t tanh b Pe v + + β ε + ( ) ⎪⎩ i ⎪ i i i i⎭

(

(11)

)

However, the implicit function theory only guarantees the existence of the ideal controller u*i ( X i ,vi ) for system

large positive constant, and ε is a small positive constant. By adding and subtracting the term T T ki ei + β tanh bi Pe i i ε + vi′ from the right-hand side of

(14), and does not recommend a technique for constructing solution even if the dynamics of the system are well known. In the following, a fuzzy system and classic controller will be used to obtain the unknown ideal controller.

)

equation (8), we obtain: ei = ⎧ f X ,u − v + ⎫ ⎪ i( i i) i ⎪ ⎪ ⎪ = Ai ei − bi ⎨+ mi ( X1 , X 2 ,..., X N ) + ⎬ ⎪ ⎪ T ⎪⎩ + di ( t ) + β tanh bi Pe i i ε + vi′ ⎪ ⎭

) (

Substituting equation (15) into the error equation (12) and using (14), we get:

where tanh ( .) is the hyperbolic tangent function, β is a

(

u*i

uiλ = λ ui + (1 - λ ) u*i .

Let vi be defined as:

(n )

(

can be expressed around

fi ( X i , ui ) = fi X i , u*i + ui − u*i fiuλ =

(10)

vi = yd i + kiT ei + β tanh biT Pe i i ε + vi′

( X i , ui )

III. Fuzzy Systems

Using assumption (1), equation (11) and the signal vi which is not explicitly dependent on the control input

Figure 1 shows the basic configuration of the fuzzy systems considered in this paper. Here, we consider a multi-input, single-output fuzzy systems: x ∈ U ⊂ R n → y ∈ V ⊂ R . Consider that a multi-output system can be separated into a group of single-output systems. The fuzzifier performs a mapping from a crisp input

ui , the following inequality is satisfied:

vector x = [ x1 ,x2 ,...,xn ] to a fuzzy set, where the label of

∂ ( fi ( X i ,ui ) − vi )

the fuzzy set are such as "small", "medium", "large", etc. The fuzzy rule base is consisted of a collection of fuzzy IF-THEN rules. Assume that there are M rules, and the lth rule is:

(

∂ui

=

(12)

)

∂fi ( X i ,ui ) ∂ui

T

>0

(13)

Invoking the implicit function theorem, it is obvious that the nonlinear algebraic equation fi ( X i ,ui ) − vi = 0

R l ( u ) : if x1 is A1l ... xn is A1l then y is B l

is

l = 1, 2,...,M

locally

soluble

for

the

arbitrary ( X i , vi ) .

input

ui for

(

an

( X i , vi ) ∈ \ n

i

fi

(

X i ,u*i

)

(17)

T

output of the fuzzy system, respectively. Alj and B l are fuzzy membership function in U j and V, respectively.

\:

⋅

(

where x = [ x1 ,x2 ,...,xn ] and y are the crisp input and

Thus, there exists some ideal controller * ui ( X i , vi ) satisfying the following equality for a given

)

)−v

i

=0

The fuzzy inference performs a mapping from fuzzy sets in U to fuzzy sets in V, based on the fuzzy IF-THEN rules in the fuzzy rule base. The defuzzifier maps fuzzy sets in V to a crisp value in V. The configuration of Figure 1 represents a general

(14)

As a result of the mean value theorem, there exists a



23

R. Ghasemi

framework of fuzzy systems, because many different choices are allowed for each block in Figure 1, and various combinations of these choices will construct different fuzzy systems [34]. Here, we use the sumproduct inference and the center-average defuzzifier. Therefore, the fuzzy system output can be expressed as: M

n

l =1 M

i =1 n

∑ yl ∏ µ Ai ( xi )

so that

IV.

(18)

∑ ∏ µ Ai ( xi ) l

l =1

n

l =1

i =1

l

Fuzzy Adaptive Controller Design

In Section 2, it has been shown that there exists an ideal control for achieving control objectives. In this section, we show how to develop a fuzzy system to adaptively approximate the unknown ideal controller. The ideal controller can be represented as:

l

y ( x) =

M

∑ ∏ µ Ai ( xi ) ≠ 0 for all x ∈ U .

i =1

u*i = fi ( z ) + u pid + ε iu

where µ Al ( xi ) is the membership degree of the input xi

(22)

i

to fuzzy set Alj and y l is the point at which the membership function of fuzzy set B maximum value.

l

where

achieves its

fi ( z ) = θi*1wi1 ( z ) , and θi*1 and wi1 ( z )

are

consequent parameters and a set of fuzzy basis functions, respectively. ε iu is an approximation error that satisfies

ε iu ≤ ε max and ε max > 0 . The u pid is the primary controller that developed properly to initially control the underlying system and parameters θi*1 are determined through the following optimization:

Fig. 1. Configuration of fuzzy system

θi*1 = arg min ⎡⎣sup θiT1 wi1 ( z ) − fi ( z ) ⎤⎦

The fuzzy systems in the form of (18) are proven in [35] to be a universal approximator if their parameters are properly chosen. Theorem 1 [34]: Suppose f ( x ) is a continuous

θi1

(23)

function on a compact set U. Then, for any ε > 0 , there exists a fuzzy system like (18) satisfying:

Denote the estimate of θi*1 as θi1 and uirob as a robust controller to compensate approximation error, uncertainties, disturbance and interconnection term to rewrite the controller given in (22) as:

sup f ( x ) − y ( x ) ≤ ε

ui = θiT1 wi1 ( z ) + u pid + uirob

(19)

x∈U

in which uirob is defined below:

The output given by (18) can be rewritten in the following compact form: y ( x) = w( x) θ T

(24)

biT Pe i i

uirob =

(20)

f min biT Pe i i

⋅

N T ⎛ˆ ⎞ i i uic + ⎜ψ i 0 + 2 bi Pe ⎟ ⎜ ⎟ 1 T ⎟ b Pe ⋅ ⎜ + ∑ Nj =1θˆ Tji 2 wi 2 biT Pe + i i i i i ⎜ 2 ⎟ ⎜ ⎟ ⎜⎜ + f min uicom + f min uir + vˆ i′ ⎟⎟ ⎝ ⎠

where θ = ⎡⎣ y1 y 2 ... y M ⎤⎦ is a vector grouping all consequent parameters, and

(

T

w ( x ) = ⎣⎡ w1 ( x ) w2 ( x ) ... wM ( x ) ⎤⎦ is a set of fuzzy basis functions defined as:

)

(25)

n

∏ µ Ai ( xi ) l

wi ( x ) =

i =1

M

∑ l =1

n

∏µ i =1

A

l

i

In the above,

(21)

( xi )

controller, ψˆ i 0 +

1 2

θiT1 wi1 ( z ) approximates the ideal

∑ j =1θˆ Tji 2 wi 2 ( biT Pei i ) b N

T i Pe i i

tries to

estimate the interconnection term and θ ji 2 ,wi 2 ( .) define

The fuzzy system (18) is assumed to be well defined

later, uicom ,uic compensate for approximation errors and



24

R. Ghasemi

( λmin ( Qi ) f min + λmin ( Pi ) f d m ) ≥ 0

uncertainties, uir is designed to compensate for bounded external disturbances, and vˆ i′ is estimation of vi′ . Define error vector θ = θ − θ * and use (24) and (25) to i1

i1

This in turn leads to the following inequality:

i1

rewrite the error equation (16) as:

(

1 ( λmin ( Qi ) f min + λmin ( Pi ) f d m ) ei fu2i

)

⎧ ( z ) + uirob − ε iu fiuλ + ⎪⎫ ⎪ ⎪ ⎪ ei = Ai ei − bi ⎨+ mi ( X 1 , X 2 ,..., X N ) + di ( t ) + ⎬ (26) ⎪ ⎪ T ⎪⎩+ β tanh bi Pe ⎪⎭ i i ε + vi′

θiT1 wi1

(

+

uicom = γ uicom ϑ −1 ( t ) biT Pe i i

vî′ =

2 f min

γ vˆ i′ f min

θi1 = Γ1ϑ ψˆ i 0 = θij 2 = where

ϑ −1 ( t ) biT Pe i i

γψ i 0 f min

(t )

biT Pe i i wi1

2 f min

(27)

,γ ,γ ˆ ′ > 0 icom uic v i

following equation,

2

(

)

constant

parameters.

and

svd max ( Ai ) ≤ −

In

(33)

fu 1 T β T ei Qi ei + 2i eiT Pe bi Pe i i + i i ≥0 fui fui fui

Qi ≤ AiT Pi + Pi Ai = 2 Pi Ai

(34)

(35)

Using the above equation, we get: Qi ≤ 2 Pi Ai = 2λmax ( Pi ) svd max ( Ai )

(28)

(36)

Use (30) and (36) to have the following which completes the proof:

Proof: From assumption (1) and β > 0 , we can have the following inequality has been satisfied: biT Pe i i ≥0

f dm λmin ( Pi ) 2 f min λmax ( Pi )

Proof: using equation (10) and after some algebraic manipulations, the following inequality is obtained:

eigenvalue and maximum singular value decomposition, respectively. Lemma 1: The following inequality holds if λmax ( Qi ) ≥ − ⎛⎜ f d m f ⎞⎟ λmax ( Pi ) : min ⎠ ⎝

fui

(32)

Lemma 2: based on lemma (1) and equation (10), the following inequality holds:

λmax( .) and svd max ( .) are maximum

β

2 1 ⎛⎜ λmin ( Qi ) f min ei + ⎞⎟ 2 ⎟ fu2i ⎜ + λ ( P ) f ⎝ min i d m ei ⎠

Q.E.D.

T ϑ −1 ( t ) biT Pe i i wi 2 bi Pe i i

are

fu2i

eiT Pe i i ≥

fu 1 T β ei Qi ei + 2i eiT Pe biT Pe i i + i i ≥0 fu fui i fui

( zi )

Γ1 = Γ1T > 0, Γ j 2 = ΓTj 2 > 0, γ uir > 0

γu

fui

Use (29) and (32) to have the following which completes the proof:

ϑ −1 ( t ) biT Pe i i

Γ j2

(31)

2

ϑ −1 ( t ) biT Pe i i −1

≥0

1 T ei Qi ei + fui

uir = γ uir ϑ −1 ( t ) biT Pe i i

γ uic N

2

After some algebraic manipulations, the following inequality is obtained:

)

Consider the following update laws:

uic =

(30)

svd max ( Ai ) ≤ −

f dm λmin ( Pi ) 2 f min λmax ( Pi )

(37)

Q.E.D.

(29)

Theorem 2: consider the error dynamical system given in (26) for the large scale system (1) satisfying assumption (1), interconnection term satisfying assumption (3), the external disturbances satisfying assumption (4), and a desired trajectory satisfying assumption (2), then the controller structure given in

From assumption (1) and the assumption mentioned in lemma (1) liked the equation fd m P , it is obvious that: λmax ( Qi ) ≥ − λ f min max ( i )



25

R. Ghasemi

In equation (38), to choose Lyapunov function ρ1 ≤ ϑ ( t ) ≤ ρ 2 must be satisfied. The time derivative of the Lyapunov function becomes (39):

(24), (25) with adaptation laws (27) makes the tracking error and error of parameters estimation converge asymptotically to a neighborhood of origin. Proof: consider the following Lyapunov function (eq. (38):

⎞ fui T 1⎛ 1 T 1 T ⎜ ei Pe e Pe ei Pe + + i i i i i i i ⎟+ 2 ⎜ ⎟ fui fui i =1 2 ⎝ f ui ⎠ ⎛ T −1 ⎞ ψ ψˆ N θTji 2 Γ −j 12θ ji 2 + ⎟ ⎜ θi1Γ1 θi1 + i 0 i 0 + 1 j = γ uir ⎜ ⎟ +ϑ ( t ) ⎜ ⎟+ ⎜ + uir uir + uicom uicom + uic uic + vi′vˆ i′ ⎟ ⎜ γu ⎟ γ γ γ uicom uic vˆ i′ ir ⎝ ⎠ 2 ⎛ T −1 ⎞ ψ N θTji 2 Γ −j 12θ ji 2 + ⎟ ⎜ θi1Γ1 θi1 + i 0 + j 1 = γ uir ⎟ ϑ ( t ) ⎜ + ⎜ 2 ⎟ 2 2 2 2 ⎜ u ir u ic u icom vi′ ⎟ + + + + ⎜ γu ⎟ γ γ γ uic uicom vˆ i′ ir ⎝ ⎠

V =

V= ⎛ ⎛ T −1 ⎞⎞ ψ 2 ⎜ ⎜ θi1Γ1 θi1 + i 0 + ⎟⎟ γ uir ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟⎟ 2 N u ir ⎟ ⎟ N 1⎜ 1 T ⎜ T −1 ei Pe θ Γ θ + = + i i + ϑ (t ) ⎜ + ⎜ j =1 ji 2 j 2 ji 2 γ uir ⎟ ⎟ i =1 2 ⎜ f ui ⎜ ⎟⎟ 2 2 ⎜ ⎜ u 2 ⎟⎟ ′ u v ⎜ ⎜ + icom + ic + i ⎟⎟ ⎜ γu ⎟⎟ ⎜ γ uic γ vˆ i′ icom ⎝ ⎠⎠ ⎝

∑

θi1 = θi1 − θi*1 ,

uir = uir − d max f min ,

∑

∑

∑

where

N

∑

θ ji 2 = θ ji 2 − θ *ji 2 ,

uicom = uicom − ε max − δ max f min ,

and vi′ = vˆ i′ − vi′ .

Use (26), to rewrite above equation as (40):

(

)

⎛ 1 T T ⎞ ⎜ f ei Ai Pi + Pi Ai ei + ⎟ ⎛ N ⎛ θiT1 wi1 ( z ) + uirob − ε iu fu + ⎞ ⎞ ⎟ 1 ⎜ T 1 ⎜ ui i ⎜ ⎟⎟ + ′ V =∑ ⎜ b Pe v + + i i i i ⎟ f ⎜ f ⎟ 2 ⎜ ⎟ β u u i =1 i ⎜ + i eT Pe + ⎟ ⎝ + mi ( X 1 , X 2 ,..., X N ) + di ( t ) ⎠ ⎠ ⎝ bT Pe tanh biT Pe i i ε ⎜ fu2 i i i fu i i i ⎟ i i ⎝ ⎠ ⎛ T −1 ⎞ ⎛ T −1 ⎞ ψ i20 N ψ ψˆ N T −1 + ∑ j =1θTji 2 Γ −j 12θ ji 2 + ⎟ ⎜ θi1Γ1 θi1 + ⎜ θi1Γ1 θi1 + i 0 i 0 + ∑ j =1θ ji 2 Γ j 2θ ji 2 + ⎟ γ uir γ uir ⎟ ⎜ ⎟ ϑ (t ) ⎜ + +ϑ ( t ) ⎜ ⎜ ⎟ ⎟ 2 2 2 2 2 ⎜ u ir u ic u icom vi′ ⎟ ⎜ + uir uir + uicom uicom + uic uic + vi′vˆ i′ ⎟ + + + + ⎜ ⎟ ⎜ γu ⎟ γ γ γ γ γ γ γ ˆ ′ ′ ˆ uicom uic vi uir uic uicom vi ir ⎝ ⎠ ⎝ ⎠

(

(

Use

(

(b

T i Pe i i

tanh biT Pe i i ε

)) = b

T i i i Pe

)

)

(40)

and (10), to

rewrite (40) as follow (41): ⎞ fui T 1 T ⎛ ⎛ ⎞⎞ ⎛ θiT1 wi1 ( z ) + ⎞ e Q e e Pe + ⎟ − i i i 2 i i i N ⎜ ′ ⎜ ⎟⎟ v f + + fui ⎜ ⎟ f ⎟ 1 i ⎜ ui T ⎟ ui V = ⎟⎟ + i i⎜ ⎟ + f ⎜ bi Pe ⎝ + uirob − ε iu ⎠ β T ui ⎜ ⎜⎜ ⎟⎟ ⎟ i =1 ⎟ ⎜ ⎟ bi Pe + + m X , X ,..., X d t ( ) ( ) i i i N i 1 2 ⎝ ⎠⎠ ⎝ ⎟ fui ⎠ N ⎛ T −1 ⎞ ⎛ T −1 ⎞ ψ 2 N T −1 ψ ψˆ −1 T ⎜ θi1Γ1 θi1 + i 0 + θ ji 2 Γ j 2θ ji 2 + ⎟ ⎜ θi1Γ1 θi1 + i 0 i 0 + θ ji 2 Γ j 2θ ji 2 + ⎟ γ uir j =1 γ uir ⎟ ⎜ ⎟ ϑ ( t ) ⎜ j =1 + ϑ (t ) ⎜ + ⎜ ⎟ ⎟ 2 2 2 2 2 ⎜ + u ir + u ic + u icom + vi′ ⎟ ⎜ + uir uir + uicom uicom + uic uic + vi′vˆ i′ ⎟ ⎜ ⎟ ⎜ ⎟ γ γ γ γ γ γ γ γ uir uicom uic vˆ i′ uic uicom vˆ i′ ⎝ ⎠ ⎝ uir ⎠

∑

⎛ ⎜− 1⎜ 2 ⎜⎜ − ⎜ ⎝

∑

∑

Furthermore, expansion biT Pe i i φij

use

assumption

(3)

and

( ) ( b Pe ) . Use theorem1, φ ( b Pe ) to

T i i i

rewrite

(41)

(

Taylor

ψ ij

biT Pe i i

as

ij

T i i i

can be

)

(

)

T ˆT φ 2ji biT Pe i i = θ ji 2 wi 2 bi Pe i i + δi

where

δi

(42)

is approximation error and satisfying

δ i ≤ δ i max .

approximate by fuzzy system as follow:



26

R. Ghasemi

Using assumption (1) yields 1 fui ≤ 1 f min and by assumptions (3), (4) and equations (42), to rewrite (41) as follow: fu 1⎛ 1 T β T ⎜− ei Qi ei − 2i eiT Pe bi Pe i i − i i 2 ⎜ fui fui fui ⎝

N

∑

V ≤

i =1

biT Pe i i

−

f min

−

biT Pe i i

1 N ⎛ + ⎜ψˆ i 0 + biT Pe i i + 2 2 ⎝ uir −

biT Pe i i

⎛ψ i 0 biT Pe i i + ⋅⎜ f min ⎜ + ⎝

f min

⎞ biT Pe i i ⎟+ vi′ − biT Pe i i uicom + ⎟ f min ⎠

∑ j =1 θˆ Tji 2 wi 2 ( biT Pei i ) biT Pei i ⎟⎠ + ⎞

N

T T vˆ i′ + biT Pe i i ε max − bi Pe i iθ i1 wi1 ( z ) +

∑ j =1 bTj Pj e j φij ( biT Pei i ) N

(43)

⎞ biT Pe i i ⎟+ d max + ⎟ f min ⎠

2 2 2 ⎛ T −1 ⎛ T −1 ⎞ ψ ψˆ N Γ θ + ψ i 0 + u ir + u ic + T Γ −1θ + ⎟ θ θ ⎜ ⎜ θi1Γ1 θi1 + i 0 i 0 + i i 1 1 1 2 2 2 ji j ji j =1 γ uir γ uir γ uic γ uir ⎜ ⎟ ϑ ( t ) ⎜ +ϑ ( t ) ⎜ + ⎜ ⎟ 2 ⎜ u 2 v′2 N ⎜ + uir uir + uicom uicom + uic uic + vi′vˆ i′ ⎟ + j =1θTji 2 Γ −j 12θ ji 2 + icom + i ⎜ ⎜ ⎟ γ uir γ uicom γ uic γ vˆ i′ γ uicom γ vˆ i′ ⎝ ⎠ ⎝

∑

∑

(

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

)

To use sigmoid properties and αβ ≤ 0.5 α 2 + β 2 , the equation (43) can be rewritten as below (44): V ≤

N

∑ i =1

(

fu β T 1⎛ 1 T ⎜− ei Qi ei − 2i eiT Pe bi Pe i i − i i 2 ⎜ fui fui fui ⎝

)

⎞ biT Pe i i ⎟− vˆ ′i − v′ + i ⎟ f min

⎠ v′ i

−

−

biT Pe i i 2 f min

biT Pe i i

⋅

T θijT2 − θij*T2 ) wi 2 ( biT Pe i i ) bi Pe i i ∑ j =1 (

N

−

θijT2

uicom − ε max ) − (

uicom

N biT Pe i i

biT Pe i i

(

2

2 f min

uic − 1 − δ i max ) − biT Pe ( i i

uicom

)

ψˆ i 0 −ψ i 0 + f min

ψi 0

(

)

d T uir − max − biT Pe i iθ i1 wi1 ( z ) + f min

uir

(44)

2 ⎛ T −1 ⎞ ⎛ T −1 ψ ψˆ N Γ θ + ψ i 0 + N θˆ T Γ −1θ + ⎟ ˆ T Γ −1θ + ⎞⎟ θ θ ⎜ ⎜ θi1Γ1 θi1 + i 0 i 0 + i i ji 2 j 2 ji 2 1 1 1 2 2 2 ji j ji = j 1 = 1 j γ γ uir uir ⎟ ⎜ ⎟ ϑ (t ) ⎜ + ϑ (t ) ⎜ ⎟ ⎟+ 2 ⎜ 2 2 2 2 ⎜ + u ir + u ic + u icom + vi′ ⎟ ⎜ + uir uir + uicom uicom + uic uic + vi′vˆ i′ ⎟ ⎜ γu ⎟ ⎜ γu ⎟ γ γ γ γ γ γ ′ ˆ ˆ ′ v u u u u v i ir icom ic ⎝ ⎠ ir ic icom i ⎝ ⎠

∑

∑

Using (27), the above inequality rewrites as: ⎛ 1 T fu ⎜ ei Qi ei + 2i eiT Pe i i N fui 1 ⎜ fui V≤ − ⎜ β T i =1 2 ⎜ bi Pe + i i ⎜ f ⎝ ui

∑

⎛ T −1 ⎞ ⎞ ψ i20 N T Γ −1θ + ⎟ θ + ⎜ θi1Γ1 θi1 + +⎟ ji j ji 2 2 2 j =1 γ uir ⎟ ⎟ ϑ ( t ) ⎜ + ⎜ ⎟ ⎟ 2 2 2 2 2 ⎜ u ir u ic u icom vi′ ⎟ ⎟ + + + + ⎜ γ ⎟ ⎟ γ γ γ ˆ ′ uic uicom vi ⎠ ⎝ uir ⎠

∑


(45)


27

R. Ghasemi

Based the lemma 1 and ϑ ( t ) ≤ 0 , V ≤ 0 are satisfied. Using Barbalat's lemma, it is guaranteed the tracking error asymptotically to the origin. It completes the proof. Q.E.D.

(

Remark 1. The term β tanh biT Pe i i ε

) in the signal

vi compensates for the error modeling and disturbance and it is a smooth approximation of the discontinuous

(

term β sign biT Pe i i

)

usually used in robust controllers.

The ε is chosen so that the sign(.) function can be approximated by tanh(.). The sign(.) function is not used in the paper due to avoiding chattering in the response. Remark 2. It is very important to select properly the controller parameters to gain a satisfactory performance. Here at this stage, number of the rules and the input membership functions are obtained by trial and error.

θij 2 =

2

2 f min

(

+

α j1

sat ( u1 ) +

kr 2 sin ( x21 ) + d ( t ) j1

= x11

(47)

⎛ m gr kr 2 = ⎜⎜ 2 − 4 j2 ⎝ j2 +

α j2

⎞ kr (l − b ) + ⎟⎟ sin ( x21 ) + 2 j2 ⎠

sat ( u2 ) +

kr 2 sin ( x12 ) + d ( t ) j2

= x21

m

of the = 2.5kg masses, inertia,

is spring constant, r = 0.5m is the height of g = 9.81 m

shows the gravitational s2 acceleration, l = 0.5m is the natural length of spring, α1 ,α 2 = 25 are the control input gains and b = 0.4m presents distance between the pendulum hinges. Furthermore it is assumed d ( t ) = sin ( 200π t ) .

ˆ i0 ϑ −1 ( t ) biT Pe i i − σγψ i 0ψ

Γ j2

= x22

⎛ m gr kr 2 ⎞ kr = ⎜⎜ 1 − (l − b) + ⎟⎟ sin ( x11 ) + j 4 j 2 j1 1 ⎠ ⎝ 1

the pendulum,

( )

f min

⎧ x21 ⎪ ⎪ x ⎪⎪ 22 ⎨ ⎪ ⎪ ⎪ ⎩⎪ y2

k = 100 N

θi1 = Γ1ϑ −1 ( t ) biT Pi ei wi1 zi − σΓiθi1 γψ i0

= x12

where y1 , y2 are the angular displacements pendulums from vertical position. m1 = 2kg , m2 are the pendulum end j1 = 0.5kg , j2 = 0.62kg are the moment of

Remark 3. To guarantee the boundedness of the parameters in the presence of the approximation error, which is unavoidable, the proposed adaptive laws (27) is modified it by introducing a σ − modification term as follows:

ψˆ i 0 =

⎧ x11 ⎪ ⎪ x ⎪⎪ 12 ⎨ ⎪ ⎪ ⎪ ⎪⎩ y1

)

T ϑ −1 ( t ) biT Pe i i wi 2 bi Pe i i − σΓ 2θ i 2

uir = γ uir ϑ −1 ( t ) biT Pe i i − σγ uir uir

(46)

uicom = γ uicom ϑ −1 ( t ) biT Pe i i − σγ uicom uicom uic =

γ uic N 2 f min

2

ϑ −1 ( t ) biT Pe i i − σγ uic uic

γ vˆ ′ ˆ i′ vî′ = i ϑ −1 ( t ) biT Pe i i − σγ vˆ i′ v f min

V.

Fig. 2. Two inverted pendulum connected by a spring

Simulation Results

We consider the desired value of the outputs be zero ( yid = 0 for i=1, 2). As discussed in section (4) the following primary PI controller are obtained after some trials and errors:

In this section, we apply the proposed decentralized fuzzy model reference adaptive controller to a twoinverted pendulum problem [36] in which the pendulums are connected by a spring as shown in Figure 2. Each pendulum may be positioned by a torque input ui applied by a servomotor and its base. It is assumed that the angular position of pendulum and its angular rate are available and can be used as the controller inputs. The pendulums dynamics are described by the following nonlinear equations:

t uPI = 40 ⎛⎜ ei + 1 ∫ ei dτ ⎞⎟ 4 0 ⎝ ⎠

(48)

Figures 3 and 4 present the outputs of the system where only the controller defined in equation (48) is applied to the system.



28

R. Ghasemi

Fig. 5. Performance of the proposed controller in first subsystem

Fig. 3. Performance of the PI controller in first subsystem

Fig. 6. Performance of the proposed controller in second subsystem

Fig. 4. Performance of the PI controller in second subsystem

Obviously the primary controller by itself is not admissible. Now we applied the proposed controller defined in (24), (25). Initially the PI controller keeps the states of system xi1 ,xi 2 in the range of [ −1,1] ,[ −5,5] . Let T

T

X i = ⎡⎣ xi1 ,xi 2 ⎤⎦ , zi = ⎡⎣ xi1 ,xi 2 ,vi ⎤⎦ and vi are defined over [ −45, 45] . For each fuzzy system input, we define 6

membership functions over the defined sets. Consider that all of the membership functions are defined by the ⎛ ( χ − c )2 ⎞ Gaussian function µ j ( χ ) = exp ⎜ , where c 2⎟ ⎜ 2δ ⎟⎠ ⎝ is center of the membership function and δ is its variance. We assume that the initial value of θi1 ( 0 ) , θi 2 ( 0 ) , uir ( 0 ) , uicom ( 0 ) , and vˆ i′ ( 0 ) be zero.

Fig. 7. Control input u1

Furthermore, it has been assumed that f min = 1 , Γ1 = 10 , Γ 2 = 10 , γ ucom = 5 , γ ur = 5 , γ vˆ i′ = 5 . In equation (46) and remark (1), we assume that σ = 0.01 , ε = 0.01 . The f dm , f min and the vector parameters T

ki = ⎡⎣ ki,1 , ki,2 , . . . , ki,ni ⎤⎦ has been chosen so that the lemma 2 holds. As shown in Figures 3 and 4 and Figures 5, 6, it is obvious that the performance of the proposed controller is promising. Figures 7 and 8 show the total input of each subsystem. To verify the boundedness of the controller parameters, the trajectories of some of them are depicted in Figures 9 –14.

Fig. 8. Control input u2

Fig. 9. Time trajectory of the robust term in first subsystem( u r 1 )



29

R. Ghasemi

Fig. 10. Time trajectory of the robust term in second subsystem ( u r 2 ) Fig. 15. Output of the first subsystem

Fig. 11. Time trajectory of the compensation term in first subsystem( u1com ) Fig. 16. Output of the second subsystem

From Figures 5, 6, 15 and 16, it is clear that our method is faster than the method given in [36]. Robustness against external disturbances, the fast response, relaxing the conditions imposed on the subsystems functions and having a universal form of interconnections are advantages of the method proposed in the paper.

Fig. 12. Time trajectory of the compensation term in second subsystem( u 2 com )

VI.

Conclusion

This paper developed a new method to design a decentralized adaptive controller using fuzzy systems for a class of large-scale nonlinear non-affine systems with unknown nonlinear interconnections. The proposed adaptive controller guarantees the closed-loop stability and convergence of the tracking errors asymptotically to zero. The stability analysis is performed using the Lyapunov’s theory. Robustness against external disturbances and approximation errors, relaxing the conditions on the nonlinear function of the system are the merits of the proposed controller.

Fig. 13. Time trajectories of the fuzzy parameters in first subsystem( θ11 )

Acknowledgements This work was supported by Islamic, Azad University, Damavand Branch, Department of Electrical Engineering, Damavand, Tehran, Iran. Fig. 14. Time trajectories of the fuzzy parameters in second subsystem( θ 21 )

References

To compare the performance of our controller with those in [36], the proposed method in [36] has been applied to system (47) and the outputs of the subsystems are shown in Figures 15 and 16, respectively.

[1] [2]


G. Feng, S.G. Cao, N.W. Rees, “Stable Adaptive Control Of Fuzzy Dynamic Systems”, Elsevier Science, Fuzzy Sets And Systems 131, Pp. 217 – 224, 2002. G. Feng, “An Approach To Adaptive Control Of Fuzzy Dynamic Systems”, IEEE Transactions On Fuzzy Systems, Vol. 10, No. 2, Pp. 268-275, April 2002.


30

R. Ghasemi

[3] [4] [5] [6]

[7]

[8]

[9] [10] [11] [12] [13] [14] [15] [16]

[17] [18] [19]

[20]

[21]

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Y.C. Hsu, G. Chen, S. Tong, H.X. Li, “Integrated Fuzzy Modeling And Adaptive Control For Nonlinear Systems”, Elsevier Science, Information Sciences 153, Pp.217-236, 2003. C.C. Cheng, S.H. Chien, ”Adaptive Sliding Mode Controller Design Based On T–S Fuzzy System Models”, Elsevier Science, Automatica 42, Pp.1005-1010, 2006. N. Golea, A. Golea, K. Benmahammed, “Stable Indirect Fuzzy Adaptive Control”, Elsevier Science, Fuzzy Sets And Systems 137, Pp. 353-366, 2003. C.W. Park, M. Park,”Adaptive Parameter Estimator Based On T– S Fuzzy Models And Its Applications To Indirect Adaptive Fuzzy Control Design”, Elsevier Science, Information Sciences 159, Pp. 125-139, 2004. P. Ying-Guo And Z. Hua-Guang, “Design Of Fuzzy Direct Adaptive Controller And Stability Analysis For A Class Of Nonlinear System”, Proceedings Of The American Control Conference, Philadelphia, Pennsylvania, 24-26June 1998: 22742275. S. Jagannathan, “Adaptive Fuzzy Logic Control Of Feedback Linearization Discrete Time Dynamical Systems Under Persistence Of Excitation”, Automatica, Vol. 34, No. 11, Pp. 12951310, 1998. S. C. Tong, Q. Li, T. Chai, “Fuzzy Adaptive Control For A Class Of Nonlinear Systems”, Elsevier Science, Fuzzy Sets And Systems 101, Pp. 31-39, 1999. Tong, S. And Tang, J. And Wang, T, Fuzzy Adaptive Control Of Multivariable Nonlinear Systems, Elsevier Science, Fuzzy Sets And Systems 111 , Pp.153-167., 2000. Zhang, H. And Bien, Z., Adaptive Fuzzy Control Of MIMO Nonlinear Systems, Fuzzy Sets And Systems 115, Www.Elsevier.Com/Locate/Fss, Pp. 191-204, 2000 S. Labiod, M. S. Boucherit, T. M. Guerra, “Adaptive Fuzzy Control Of A Class Of MIMO Nonlinear Systems”, Elsevier Science, Fuzzy Sets And Systems 151, Pp.59–77, 2005. Y. Tang, N. Zhang, Y. Li, “Stable Fuzzy Adaptive Control For A Class Of Nonlinear Systems”, Elsevier Science, Fuzzy Sets And Systems 104,Pp. 279-288,1999. L. Chen, G. Chen, Y.W. Lee, “Fuzzy Modeling And Adaptive Control Of Uncertain Chaotic Systems”, Elsevier Science, Information Sciences 121, Pp. 27-37, 1999. H.F. Ho, Y.K. Wong, A.B. Rad, W.L. Lo, “State Observer Based Indirect Adaptive Fuzzy Tracking Control”, Simulation Modeling Practice And Theory 13, Pp. 646–663, 2005. L. Zhang, “Stable Fuzzy Adaptive Control Based On Optimal Fuzzy Reasoning”, IEEE, Proceedings Of The Sixth International Conference On Intelligent Systems Design And Applications (ISDA'06), 2006. S.Tong, H.X. Li, W. Wang, “Observer-Based Adaptive Fuzzy Control For SISO Nonlinear Systems”, Elsevier Science, Fuzzy Sets And Systems 148, Pp. 355–376, 2004. T. Shaocheng, C. Bin, W.Yongfu, “Fuzzy Adaptive Output Feedback Control For MIMO Nonlinear Systems”, Elsevier Science, Fuzzy Sets And Systems 156, Pp. 285–299, 2005. W.S Yu., “Model Reference Fuzzy Adaptive Control For Uncertain Dynamical Systems With Time Delays”, IEEE International Conference On Systems, Man And Cybernetics, Taiwan, 2004 : 5246-5251, Www.Elsevier.Com/Locate/Chaos C.C. Chiang, “Adaptive Fuzzy Sliding Mode Control For TimeDelay Uncertain Large-Scale Systems", Proceedings Of The 44th IEEE Conference On Decision And Control, And The European Control Conference, Pp. 4077-4082,Seville, Spain, December 1215, 2005. X. Jiang,W. Xu, Q.L. Han, “Observer-Based Fuzzy Control Design With Adaptation To Delay Parameter For Time-Delay Systems”, Elsevier Science, Fuzzy Sets And Systems 152, Pp. 637– 649, 2005. T. Yiqian, W. Jianhui, G. Shusheng, Q. Fengying, “Fuzzy Adaptive Output Feedback Control For Nonlinear MIMO Systems Based On Observer”, Proceedings Of The 5th World Congress On Intelligent Control And Automation Hangzhou, P.R. China, Pp. 506-510June 15-19, 2004. S. Labiod, T. M. Guerra, “Adaptive Fuzzy Control Of A Class Of SISO Non-affine Nonlinear Systems”, Elsevier Science, Fuzzy

Sets And Systems 158, Pp. 1126 –1137, 2007. [24] J.H. Park, G.T. Park, S.H. Kima, C.J. Moon, “Direct Adaptive Self-Structuring Fuzzy Controller For Non-Affine Nonlinear System”, Elsevier Science, Fuzzy Sets And Systems 153, Pp. 429– 445, 2005. [25] R. Ghasemi, M.B. Menhaj and A. Afshar, “A New Decentralized Fuzzy Model Reference Adaptive Controller for a Class of Largescale Non-affine Nonlinear Systems” European Journal of Control (5), p.p: 1–11, 2009. [26] R. Ghasemi, M.B. Menhaj and A. Afshar, “A decentralized stable fuzzy adaptive controller for large scale nonlinear systems”, Journal of applied sciences (9) 5, p.p: 892-900, 2009. [27] P.R. Pagilla, R.V. Dwivedula, N.B. Siraskar, “A Decentralized Model Reference Adaptive Controller For Large-Scale Systems”, IEEE/ASME Transactions On Mechatronics, Vol. 12, No. 12, No. 2, Pp. 154-163, April 2007. [28] P. Ioannou, M.D. Ponte Jr., “Adaptive Control Techniques For A Class Of Large-Scale Systems”, IEEE Conference, 1988. [29] L. Shi, S.K. Singh," Decentralized Adaptive Controller Design For Large Scale Systems With Higher Order Interconnections”, IEEE Transactions On Automatic Control, Vol. 37,No. 8,Pp. 1106-1113, AUGUST, 1992. [30] H. Yousef, M.A. Simaan, “Model Reference Adaptive Control For Large Scale Systems With Application To Power Systems”, IEE PROCEEDINGS-D, Vol. 138, No. 4, Pp. 321-327, JULY 1991. [31] C.C. Chiang, W.H. Lu, “Decentralized Adaptive Fuzzy Controller Design For Uncertain Large-Scale Systems With Unknown DeadZone”, IEEE Conf., Pp.1-6, 2007. [32] Y. Zhang, B. Chen, S. Liu, S. Zhang, ”Fuzzy Indirect Adaptive Sliding Mode Tracking Costrol For A Class Of Nonlinear Similar Composite Large-Scale Systems”, Proceedings Of The American Control Conference Anchorage. AK, Pp. 2961-2967, May 8-1 0. 2002. [33] H. Wu, “Decentralized Adaptive Robust Control For A Class Of Large-Scale Systems Including Delayed State Perturbations In The Interconnections”, IEEE Transactions On Automatic Control, Vol. 47, No. 10, Pp.1745-1751, October 2002. [34] L.X. Wang,” A Course In Fuzzy Systems And Control”, Prentice Hall PTR,1997. [35] L.X. Wang, J.M. Mendel, "Fuzzy Basis Function, Universal Approximation, And Orthogonal Least Square Learning", IEEE Trans. Neural Network 3 -5, Pp.: 807-814, 1993. [36] B. Karimi, M.B. Menhaj, I. Saboori ”Decentralized Adaptive Control Of Large Scale Non-affine Nonlinear Systems Using Radial Basis Function Neural Network”, IEICE TRANS. FUNDEMENTALS, Vol. E90-A, No.10, Pp.2239-2247, OCTOBER 2007. [37] H. Khalil, Nonlinear Systems, Prentice Hall, Second Edition, 1996. [38] H. J. Zimmermann, Fuzzy Set Theory- And Its Application, Kluwer Academic Publishers, 1996. [39] P. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996.

Authors’ information Reza Ghasemi (M'09) was born in Tehran, Iran in 1979. He received his B.Sc degrees in Electrical engineering from Semnan University in 2000 and M.Sc. degrees and Ph.D. from in control engineering from Amirkabir University of Technology, Tehran, Iran, in 2004, 2009. His research interests include large-Scale Systems, Adaptive Control, Robust Control, Nonlinear Control, and Intelligent Systems. Dr. Reza Ghasemi joined Islamic Azad University, Damavand Branch, the Department of Electrical Engineering, Damavand, Tehran, Iran, where he is currently a Assistant Professor of electrical engineering.



31


Output Feedback Higher Order Sliding Mode Controller Abderraouf Gaaloul1, Faouzi M'Sahli2 Abstract – Standard sliding mode controller leads, generally, to the appearing of an undesirable chattering phenomenon which can be overcome using higher-order sliding mode controllers. In this field, a novel approach was proposed, recently. However, it admits a serious drawback. Indeed, the control law depends explicitly on the whole system states assumed to be known. Nevertheless, such assumption is very hard to fulfill in practice. In this paper, we propose a convenient solution to deal with such problem through the design of an output feedback higher order sliding mode controller. This is accomplished using a finite time high gain observer to estimate the missing states. Then, the observer is incorporated into the original controller. Numerical simulations are developed to show the effectiveness of the proposed approach. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: High Gain Observer, Higher Order, Sliding Mode Controller, Stability

A number of approaches were proposed to face the chattering phenomenon from a theoretical viewpoint [10], [7]. The most interesting way to get rid of this phenomenon consists of enforcing a higher-order sliding mode (HOSM). In 1-SMC, the discontinuous control signal acts on σ to enforce a sliding motion on σ = 0 . The concept of SMC of order r has been developed as the generalization of 1-SMC. The main objective of SMC of order r (r-SMC) is to obtain a finite time convergence onto the manifold

Nomenclature m g u x xc t R

σ

Mass of the load Gravitational constant Torque Angular coordinate Reference trajectory Time variable Radius of the pendulum Sliding variable

I.

kg ms -2 Nm rad rad s

{

Sr = σ = σ =… = σ (

r −1)

}

= 0 . So, the control acts on

σ and its higher derivatives to force the sliding variable and its r − 1 first time derivatives to zero in finite time. Consequently, HOSM preserve the main properties of 1SMC and provides a natural solution to avoid the chattering phenomenon. The main results according to HOSM concern the second order SMC (2-SMC) [11], [12]. In [13], sliding mode controllers were introduced for an arbitrary order r > 2 . Such controllers provide actually for full realtime output control of uncertain SISO dynamic systems with known relative degree. In [14], an arbitrary-order real-time exact differentiator is combined together with the arbitrary order sliding mode controllers. An output feedback quasi- continuous HOSMC was proposed in [8] and provides for the finite-time stable sliding motion on the zero-dynamics manifold of high relative degree. In [15], authors presented an integral arbitrary order sliding mode controller. The proposed controller provides for the finite and well-known in advance convergence time. In [9], a novel finite-time convergent HOSMC was established. Nevertheless, such controller suffers from a serious deficiency. Indeed, it depends explicitly on the whole system states assumed to be

Introduction

During the past few decades, sliding mode principle has been extensively developed in the literature and it has been used in many fields such as state estimation [1], [2], fault detection [3], process control [4]-[6], etc. The main objective of Sliding Mode Control (SMC) is to reach a prescribed surface, called sliding surface and to force the system state to slide on. The most outstanding feature of SMC is its ability to achieve a robust control with respect to uncertainties and external disturbances. The robustness property is achieved by using a highfrequency switching to steer the states of a system into the sliding surface [7]. Standard SMC called also first order SMC (1-SMC) is bounded by some restrictions. Indeed, 1-SMC is applicable if the relative degree is 1, i.e. the first time derivative of the sliding variable σ depends explicitly on the control variable [8]. Besides, the high-frequency switching leads to an undesirable chattering of the control input. Such phenomenon may excite unmodeled high frequency modes, degrade the performances of the system and may lead to instability [9].


32


A. Gaaloul, F. M’Sahli

known. However, most industrial processes admit, generally, one or more states for which is impossible or prohibitively expensive to measure. This leads to a hard difficulty when implementing the controller in real time. In [16], we proposed a high gain observer based higher order sliding mode controller. Such controller, while diminished the chattering phenomenon, it suffers from stability results. In this paper, we propose an efficient solution to overcome the drawback devoted in [9] through the design of an output feedback higher order sliding mode controller. This is accomplished in two steps. First, a finite time high gain observer [17] is used to estimate the missing system states. Then, the observer is incorporated into the state feedback HOSMC in the spirit of the separation principle [18], [19]. The resulting algorithm is applied to control the angular position of a variable length inverted pendulum. Numerical simulations showed that, besides the capability of tracking a variable signal reference, the robustness of the closed loop system against uncertainties and step like disturbances is ensured. This paper is organized as follows. The problem formulation is presented in the next section. Section 3 is devoted to the arbitrary order sliding mode controller. Section 4 consists of the synthesis of an output feedback HOSMC where the main result of this contribution is given. The main features of the proposed approach are illustrated in section 5 through a simulation example. Conclusions are reported in the last section of the paper.

II.

Consider the following sliding variable: ⎛ σ1 ( t,x ) ⎞ ⎛ y1 ( t ) − y1,d ( t ) ⎞ ⎜ ⎟ ⎜ ⎟ σ ( t,x ) ⎟ ⎜ y2 ( t ) − y2 ,d ( t ) ⎟ = σ ( t,x ) = ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎜⎜ ⎟⎟ ⎜⎜ ⎟ ⎟ ⎝ σ m ( t ,x ) ⎠ ⎝ ym ( t ) − ym,d ( t ) ⎠

The relative degree vector r = [ r1 ,r2 ,… ,rm ]

T

of

system (1) with respect to σ is assumed to be constant and known. This means that the control input explicitly appears first time in the rth total time derivative of σ , i.e.: ⎡σ ( r1 ) ( x ) ,… ,σ ( rm ) ( x ) ⎤ = A ( x ) + B ( x ) u m ⎢⎣ 1 ⎥⎦

(3)

with: A ( x ) = ⎡ Lφr1 σ 1 ( x ) ,… ,Lφrm σ m ( x ) ⎤ ⎣ ⎦ 1 − r ⎡ Lψ Lφ1 σ1 ( x ) … Lψ Lφr1 −1σ 1 ( x ) ⎤ m ⎢ 1 ⎥ B ( x) = ⎢ ⎥ ⎢ ⎥ rm −1 rm −1 ⎢⎣ Lψ 1 Lφ σ m ( x ) … Lψ m Lφ σ m ( x ) ⎥⎦

The vector A ( x ) and the matrix B ( x ) are partitioned into nominal parts ( A ( x ) and B ( x ) ) and uncertain parts ( ∆ A ( x ) and ∆ B ( x ) ) as follows: ⎧⎪ A ( x ) = A ( x ) + ∆ A ( x ) ⎨ ⎪⎩ B ( x ) = B ( x ) + ∆ B ( x )

Problem Formulation

One seeks a smooth and admissible feedback control allowing resolving a tracking problem for nonlinear systems whose dynamical behavior can be described by the following representation: ⎧⎪ x = φ ( x ) + ψ ( x ) u ⎨ ⎪⎩ y = h ( x )

(2)

(4)

Uncertain parts are assumed bounded and satisfying the following properties: ⎧ ∆ A ( x ) − ∆ B ( x ) B −1 ( x ) A ( x ) ≤ κ ( x ) ⎪ ⎨ −1 ⎪ ∆B ( x) B ( x) ≤ 1− α ⎩

(1)

(5)

where κ ( x ) is a positive function, α is a positive

where x ∈ ℜn represents the system states, u ∈ ℜm is the system input, φ ( x ) and ψ ( x ) are sufficiently

constant ( 0 < α ≤ 1 ) and B ( x ) is assumed invertible. The r-SMC of (1) with respect to the sliding variable

differentiable uncertain smooth functions, y ∈ ℜm is the

σ is equivalent to the finite time stabilization of the

output vector, and h ( x ) is a smooth vector function.

following multivariable uncertain system (6):

The control problem considered therein consists in the convergence of the output y ( t ) of the system to a

⎧⎧ z1,i = z2 ,i ⎪⎪ ∀i = {1,… ,m} ⎪⎪⎨ ⎨⎪ zr −1,i = zr ,i i ⎪⎩ i T ⎪⎡ −1 −1 ⎪⎩ ⎣ zr1 ,1 ,… ,zrm ,m ⎤⎦ = I m + ∆ B B ω + ∆ A − ∆ B B A

desired trajectory yd ( t ) ∈ ℜm of the output despite the existence of uncertainties and disturbances affecting the process dynamics. So, one refers to the HOSM scheme.

(


)


33


with 1 ≤ j ≤ ri , z j ,i = σ i(

j −1)

zi = ⎡⎣ z1,i ,… ,zri ,i ⎤⎦

,

T

T

notation Sign ( s ) denotes ⎡⎣ sign ( s1 ) ,… ,sign ( sm ) ⎤⎦ .

and

The auxiliary function zaux ∈ ℜm used in the design of the sliding variable s is obtained by:

T

z = ⎡⎣ z1T ,… ,zmT ⎤⎦ .

zaux = −ωnom ( z )

III. Higher Order Sliding Mode Control HOSMC is a conveniently solution proposed to overcome the deficiencies of the standard SMC, especially in the purpose of eliminating or at least diminishing the chattering phenomenon. and our control goal is to fulfill the Being in mind ,,that constraint σ = 0 in finite time, one refers to the r-SMC approach proposed recently by Defoort et al. [9]. Such controller is given by: u = B −1 ( x ) (ωnom ( z ) + ωdisc ( z,zaux ) − A ( x ) )

Theorem 1 [9]. The higher order sliding mode controllers given by (7)-(13) for the case of uncertain systems (1) ensures the establishment of a higher order sliding mode with respect to σ in finite time. Remark 1: The successive derivatives of the sliding variable are estimated using a homogenous robust exact finite time convergent differentiator [14]. One can see that the implementation of the controller (7) requires the knowledge of the whole system state. However, most of industrial processes admit one or more unknown states. Such deficiency becomes a serious problem when implementing the controller in real time. So, we propose to overcome such problem in the sequel.

(7)

where A ( x ) and B ( x ) are obtained according to (4),

ωnom ( z ) , ωdisc ( z,zaux ) respectively, by:

and

ωnom,i ( zi ) = −k1,i sign ( z1,i ) z1,i

( )

... − kri ,i sign zri ,i

zri ,i

are

zaux

v1,i

given,

+…

vri ,i

IV.

(8)

such that the polynomial p ri + kri ,i p ri −1 + … + k2 ,i p + k1,i is Hurwitz, sign is the usual signum function, and v1,i ,… ,vri ,i satisfy: v j ,i v j +1,i 2v j +1,i − v j ,i

, j = 2 ,… ,ri

(9)

with vri +1,i = 1 and vri ,i = vi , vi ∈ (1 − ε i ,1) , ε i ∈ ( 0 , 1) :

ωdisc ( z,zaux ) = −G ( z ) Sign ( s )

∑ ∑

⎧ξ = ξ + m g (ξ ) u j 2 1 j =1 1, j ⎪ 1 ⎪ m g (ξ1 ,ξ 2 ) u j ⎪ξ 2 = ξ3 + j =1 2 , j ⎨ ⎪ ⎪ m ⎪ξ n = ϕ (ξ ) + g (ξ ) u j j =1 n, j ⎩

(10)

where the sliding variable s ∈ ℜm associated to the discontinuous part ωdisc of the controller is given by: T

s = ⎡⎣ zr1 ,1 ,… ,zrm ,m ⎤⎦ + zaux

(1 − α ) ωnom ( z ) + κ + η α

(14)

∑

where the state ξ ∈ ℜn , u ∈ ℜm and y = ξ1 designate the input and the output of the system, respectively, ϕ and gi, j , 1 ≤ i, j ≤ n , are nonlinear functions.

(11)

and the gain G ( z ) satisfies the following property: G (z) ≥

Output Feedback Control

Being in mind the non availability of all the system states, the output feedback control we are concerned by is obtained by simply combining a sate feedback higher order sliding mode controller with a high gain observer (HGO). We consider the particular case where system (1) is single output, i.e. y ∈ ℜ . The synthesis of the observer needs the following assumptions. (A1) There exists a diffeomorphism Φ : ℜn → ℜ n , x ξ = Φ ( x ) that puts system (1) under the form:

where k1,i ,… ,kri ,i , i = 1,… ,m , are positive reels chosen

v j −1,i =

(13)

(A2) The functions ϕ and gi, j , 1 ≤ i, j ≤ n are globally Lipschitz. (A3) The input u is bounded by u0 > 0 . For the nonlinear systems class (14), Ménard et al. [17] proposed a HGO given by the following form:

(12)

with η > 0 , κ and α are given according to (9), the



34


⎧ ξˆ ⎪ 1 ⎪ ⎪ ⎪ ⎪ξˆ ⎪ 2 ⎨ ⎪ ⎪ ⎪ξˆ ⎪ n ⎪ ⎪ ⎩

( )

m = ξˆ2 + ∑ j =1 g1, j ξˆ1 u j +

(

+ k1 sign ( e1 ) ⋅ e1

α1

+ ρ e1

(

where the components ωnom ( z ) , ωdisc ( z,zaux ) and zaux which depend on the measured states are given by (8), (10) and (13), respectively, A ( ˆx ) and B ( ˆx ) are

)

)

obtained in a similar way as A ( x ) and B ( x ) for

m = ξˆ3 + ∑ j =1 g 2 , j ξˆ1 ,ξˆ2 u j +

(

+ k2 sign ( e1 ) ⋅ e1

α2

+ ρ e1

( ) ∑ j =1 gn, j ( ) m

= ϕ ξˆ +

( )

ˆx = Φ −1 ξˆ .

(15)

)

Theorem 3. Consider the nonlinear systems class (1). The control law corresponding to the output feedback controller (18) leads to the establishment of a higher order sliding mode with respect to σ in finite time provided that assumptions (A1)-(A3) hold. Proof. The proof is established in the spirit of [20]. One shall begin by showing the finite time convergence of the observer. This part of proof was established by Ménard et al. [17]. Indeed, according to Theorem 2, authors proved that the estimated states converge in finite time to the true states. So, after a finite time T0 , one obtains x = ˆx and the proposed control law (18) becomes identical to the controller (7) and the control objective is fulfilled (see [9] for more details). Now, let us show that during the transient time of the convergence of the observer, the variables of the closed loop system remain bounded. The whole dynamics of the combined observer-controller system are given by (19):

ξˆ u j +

(

+ kn sign ( e1 ) ⋅ e1

αn

+ ρ e1

)

where ξˆ ∈ ℜ n is the system estimated states, sign is the usual signum function, e = ξ − ξˆ , the powers α and 1

1

1

i

the gains ki , i = 1,… ,n , are defined, respectively, by: ⎤

1 ⎡

α i = iτ + ( i − 1) , τ ∈ ⎥1 − ,1⎢ ⎦ n ⎣

[ k1 ,… ,kn ]

T

=

S∞−1

(θ ) C

(16)

T

where S∞ (θ ) is the unique solution of the following Riccati equation:

θ S∞ (θ ) + AT S∞ (θ ) + S∞ (θ ) A − C T C = 0

⎧⎧ z1,i = z2 ,i ⎪⎪ ∀i = {1,… ,m} ⎪⎨ ⎪⎪ z = zri ,i ⎪⎩ ri −1,i ⎨ T −1 ⎪ ⎣⎡ zr1 ,1 ,… ,zrm ,m ⎦⎤ = I m + ∆ B ( ˆx ) B ( ˆx ) ω ( z,zaux ) + ⎪ ⎪+∆ A ( ˆx ) − ∆ B ( ˆx ) B −1 ( ˆx ) A ( ˆx ) ⎪ ⎩

(17)

with θ > 0 is the design parameter of the observer, and matrices A and C are done, respectively, by: ⎡0 ⎢0 ⎢ A=⎢ ⎢ ⎢0 ⎢⎣ 0

1

0

0

1

0

0

0

0

(

… 0⎤ ⎥ ⎥ 0 ⎥ , C = (1, 0 ,… , 0 ) ⎥ 1⎥ … 0 ⎥⎦

Here, ω ( z,zaux ) = ωnom ( z ) + ωdisc ( z, zaux )

depends

on the measurable states (outputs of the system). The dynamics (19) can be rewritten in a condensed form as follows:

The parameter ρ is given by:

ρ=

)

z = Μ (z) + N

n 2θ 2 / 3 S1 + 1 2

(20)

with:

where S1 = max1≤i , j ≤ n S∞ (1)i , j ⋅ S∞−1 (1) j ,1 .

z2 ,i ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ M (z) = ⎢ ⎥ zri ,i ⎢ ⎥ ⎢ I m + ∆ B ( ˆx ) B −1 ( ˆx ) ω ( z ) ⎥ ⎣ ⎦

Theorem 2 [17]. There exists θ * > 0 and ε > 0 such that for all θ > θ * and τ ∈ ]1 − ε ,1[ , system (15) is a

(

global finite time observer for system (14). Now, combining the state feedback HOSMC (7) with the HGO (15) results in the establishment of the output feedback higher order sliding mode controller given by:

)

0 ⎡ ⎤ ⎢ ⎥ ⎥ N=⎢ ⎢ ⎥ 0 ⎢ ⎥ −1 ⎢⎣ ∆ A ( ˆx ) − ∆ B ( ˆx ) B ( ˆx ) A ( ˆx ) ⎥⎦

u = B −1 ( ˆx ) (ωnom ( z ) + ωdisc ( z,zaux ) − A ( ˆx ) ) (18)



35


On the other hand, the dynamics of the observer error e = ξ − ξˆ can be written as follows:

(

e = Ae − F ( K ,e ) − ρ S∞−1 (θ ) C T Ce + D ξ ,ξˆ ,u

)

⎡ zr ,1 ,… ,zr ,m ⎤ m ⎣ 1 ⎦

T

≤ (1 + γ ) ω ( z ) + β

This leads to:

(21)

z =Ψ z +β

with:

(

)

where Ψ is a positive constant. On the other hand, one has:

)

e = ϑ e + D ξ ,ξˆ ,u − F ( K ,e )

⎛ k sign ( e ) ⋅ e α1 ⎞ 1 1 ⎜ 1 ⎟ ⎜ ⎟ F ( K ,e ) = ⎜ ⎟ ⎜ k sign e ⋅ e α n ⎟ ( 1 ) 1 ⎟⎠ ⎜ n ⎝ 0 ⎛ ⎞ ⎜ ⎟ m ⎜ ⎟ ˆ ˆ D ξ ,ξ ,u = ⎜ ⎟ + ∑ g j (ξ ) − g j ξ u j ( t ) 0 ⎜ ⎟ j =1 ⎜ ϕ (ξ ) − ϕ ξˆ ⎟ ⎝ ⎠

(

(

(

)

(26)

(

)

(

with ϑ > 0 . Moreover, D ξ ,ξˆ ,u

( ))

)

(27)

and F ( K ,e ) are

bounded, respectively, as follows (see [17] for more details):

( )

(

)

D ξ ,ξˆ ,u ≤ χ e

(28)

Moreover, (20) may be rewritten as follows: z = M ( z,e ) + N ( e )

where χ = nlu0C1 C with S = max1≤i, j ≤ n S∞ (1)i, j and

(22)

l,C1 are two positive constants:

with M ( z,e ) and N ( e ) are obtained when substituting

F ( K ,e ) ≤ λ e

ˆx with Φ −1 (ξ − e ) in (20). Thus, the closed loop

dynamics are given by:

(29)

where λ is a positive constant which depend on θ . One obtains:

⎧⎪ z = M ( z,e ) + N ( e ) ⎨ ⎪⎩e = Ae − F ( K ,e ) − Σe + D (ξ ,ξ − e,u )

(23)

e =µ e

with µ = ϑ + χ − λ . Using (24) and combining (26) and (30) one obtains:

where Σ = ρ S∞−1 (θ ) C T C . Denoting X = [ z,e ] , one T

obtains: X = Γ( X )

(30)

X ≤ ς X +δ

(24)

(31)

where ς and δ are positive constants. Finally, applying the Gronwall Lemma [21], one has:

with: ⎡ M ( z,e ) + N ( e ) ⎤ Γ( X ) = ⎢ ⎥ ⎢⎣ Ae − F ( K ,e ) − Σe + D (ξ ,ξ − e,u ) ⎥⎦

X ( t ) ≤ X ( 0 ) exp (ς t ) +

Now, let us verify that z and e are bounded. As

δ ( exp (ς t ) − 1) ς

(32)

Thus, inequality (32) implies that the variables of the whole observer-controller system are bounded in finite time. This ends the proof. Remark 2: The controller (18) allows recovering the separation principle for the nonlinear systems class (1).

the observer is global finite time convergent, the observation error is bounded in finite time [17]. Consequently, properties (5) still verified for ˆx . One gets ⎧ ∆ A ( ˆx ) − ∆ B ( ˆx ) B −1 ( ˆx ) A ( ˆx ) ≤ β ⎪ (25) ⎨ −1 ⎪ ∆ B ( ˆx ) B ( ˆx ) ≤ γ ⎩

V.

Simulation Example

To show the effectiveness of the proposed approach, numerical simulations are carried out to deal with the control problem of a variable-length pendulum with motions restricted to some vertical plane (Fig. 1). A load

where β and γ are two positive constants. Using properties (25), one has: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


36


moves without friction along the pendulum rod. An engine transmits a torque u considered as the manipulated variable [12]. The model of the system is given by:

obtained for k1 = 1 , k2 = 1.5 , v = 3 / 4 , G = 0.5 , and z1 is the real-time estimation of σ obtained using the following differentiator:

R g 1 x = −2 x − sin ( x ) + u R R mR 2

⎧⎪ z = −3.35 z − σ 1 / 2 sign ( z − σ ) + z 0 0 0 1 ⎨ z = − 5 . 5 sign z − z ( 1 0) ⎪⎩ 1

(33)

where m = 1kg and g = 9.81ms-2 . The task is to enforce the angular coordinate x of the rod to track some function xc given in real time. In the simulations, the functions R and xc are given, by:

The estimated states ˆx1 and ˆx2 are obtained from the following finite time high gain observer:

(

⎧ ˆx = ˆx + 2θ sign ( e ) ⋅ e α1 + ρ e 2 1 1 1 ⎪ 1 ⎪ R g 1 ⎪ u+ ⎨ ˆx2 = −2 ˆx2 − sin ˆx1 + 2 R R mR ⎪ α2 ⎪ 2 ⎪⎩ +θ sign ( e1 ) ⋅ e1 + ρ e1

R ( t ) = 0.8 + 0.1 sin 8t + 0.3 cos 4t xc ( t ) = 0.5 sin 0.5t + 0.5 cos t

(

σ = x − xc is assumed to be As the relative degree of the

σ is r = 2 , a 2-SMC can be

the state variable x1 = x and

(a)

(b) 10 torque (Nm)

(34)

0 -0.5

xc

0

5 Time (s) (c)

angular velocity (rad/s)

0 -0.5

σ Dσ

-1

(35)

0 -5

0

5 Time (s)

-10

10

0.5

dynamics of the pendulum. One can easily verify that the model (34) belongs to the nonlinear system class (14). Now, combining the proposed HOSMC (18) and the finite time differentiator [14] achieve:

5

x

-1

where ∆ ( x ) is an additive disturbance affecting the

⎛ ⎞ R g u = mR 2 ⎜ τ + 2 ˆx2 + sin ˆx1 ⎟ R R ⎝ ⎠

(37)

)

0.5 angle (rad)

⎧ x1 = x2 + ∆ ( x ) ⎪ ⎨ R g 1 u ⎪ x2 = −2 x2 − sin x1 + R R mR 2 ⎩

)

where e1 = x − ˆx . Fig. 2(a) shows the full agreement between the desired and the actual trajectories after a short transient time. The corresponding control inputs depicted in Fig. 2(b) are smooth enough and quite feasible. In Fig. 2(c), one can show that the sliding variable and its first time derivative Dσ = dσ / dt converge to zero rapidly. Estimation results obtained with α1 = 0.9 and θ = 200 are depicted in Fig. 2(d). One can see the convergence of the estimated missing state ˆx2 to the simulated one.

Fig. 1. Variable length pendulum [12]

The sliding variable available in real time. system with respect to designed. By choosing x2 = x , one obtains:

(36)

10

0

5 Time (s) (d)

10

0.5

0 x2

-0.5

x 2e 0

5 Time (s)

10

Figs. 2. (a) Reference and actual trajectories; (b) Control input; (c) σ and its derivative; (d) x2 and its estimate

with:

In order to evaluate the robustness of the closed loop system, we assume that a bounded uncertainty affects the dynamics of the angle of the pendulum according to the following form:

⎧τ = τ nom + τ disc ⎪ 3/ 5 3/ 4 − 1.5sign ( z1 ) ⋅ z1 ⎪τ nom = − sign (σ ) ⋅ σ ⎨ ⎪τ disc = −0.5sign ( z1 + zaux ) ⎪ ⎩ zaux = −τ nom

∆ ( x ) = 0.2 sin ( 0.05 x1 )


(38)


37


finite time high gain observer into the original controller. The resulting output feedback controller has been used to deal with the control problem of a variable length pendulum. The robustness of the closed loop system with respect to uncertainties and step like disturbances is ensured and good agreement between the output and the reference trajectory was attained, as well as a successful behavior of the whole observer/controller structure.

Using the same design parameters as previously, results depicted in Figs. 3 demonstrate the robustness of the controller against uncertainties. (a)

(b) 10 torque (Nm)

angle (rad)

0.5 0 -0.5

xc x

-1 0

5 Time (s) (c)


0 -0.5

σ Dσ

-1

0

5 Time (s)

0 -5 -10

10

0.5

5

0

5 Time (s) (d)

References

10

[1]

0.5

[2]

0 x2

-0.5

10

x 2e 0

5 Time (s)

[3]

10

[4]

Figs. 3. (a) Reference and actual trajectories; (b) Control input; (c) σ and its derivative; (d) x2 and its estimate, in presence of uncertainties

[5]

Furthermore, we add to uncertainties (38) a step like disturbance of magnitude 0.1rad to the system output x1 . Simulations plotted in Figs. 4 show the robustness of the proposed controller against such perturbations. Indeed, the perturbations are well rejected and the observer estimate properly the states of the system. (a)

[7] [8]

(b)

[9]

10 torque (Nm)

0.5 0 -0.5

xc x

-1 0

5 Time (s) (c)

2 0

σ

-2

Dσ 0

5 Time (s)

10

5

[10]

0 -5 -10

10


angle (rad)

[6]

0

5 Time (s) (d)

[11]

10

[12]

1 0.5

[13]

0 x2

-0.5

x 2e

-1 0

5 Time (s)

[14] 10

[15] Figs. 4. (a) Reference and actual trajectories; (b) Control input; (c) σ and its derivative; (d) x2 and its estimate, in presence of step like disturbance

VI.

[16] [17]

Conclusion

In a recently proposed higher order sliding mode controller, the control law depends explicitly on the full states of the controlled system. As most industrial processes admit one or more unknown states, the implementation of the controller becomes hard to fulfill. In the present work, we have proposed a conveniently solution of such problem. Indeed, we have incorporated a

[18] [19] [20]


R. Aguilar-Lopez, R. Maya-Yescas, State estimation for nonlinear systems under model uncertainties: a class of sliding mode observers, Journal of Process Control, Vol. 15: 363-370, 2005. M. Mihoub, A.S. Nouri, R. Ben Abdennour, A Finite Time Convergent Chattering Free Second Order Sliding Mode Observer for non Stationary Systems, International Review of Automatic Control, Vol. 3 (No. 6), 2010. C. Edwards, S.K. Spurgeon, R.J. Patton, Sliding mode observers for fault detection and isolation, Automatica, Vol. 36: 541-553, 2000. C. Edwards, S.K. Spurgeon, Sliding mode control: Theory and applications (London: Taylor & Francis, 1998). R. Ben Khaled, C. Mnasri, M. Gasmi, A Combined Adaptive Backstepping-Sliding Mode Control of a Class of Uncertain Nonlinear Systems, International Review of Automatic Control, Vol. 3 (No. 5): 443-451, 2010. M. Mihoub, A.S. Nouri, R. Ben Abdennour, Real-time application of discrete second order sliding mode control to a chemical reactor, Control Engineering Practice, Vol. 17: 1089-1095, 2009. V. Utkin, Sliding modes in control and optimization (Berlin: Springer, 1992). A. Levant, Quasi-continuous high-order sliding-mode controllers, IEEE Trans. Automatic Control, Vol. 50 (No. 11) 1812-1816, 2005. M. Defoort, T. Floquet, A. Kokosy, W. Perruquetti, A novel higher order sliding mode control scheme, Systems & Control Letters, Vol. 58: 102-108, 2009. J.J. Slotine, S.S. Sastry, Tracking control of nonlinear systems using sliding surfaces with application to robot manipulator, International Journal of Control, Vol. 38 (No. 2): 465-492, 1983. G. Bartolini, A. Ferrara, E. Usai, V. Utkin, On multi-input chattering-free second-order sliding mode control, IEEE Trans. Automatic Control, Vol. 45 (No. 9): 1711-1717, 2000. A. Levant, Principles of 2-sliding mode design, Automatica, Vol. 43: 576-586, 2007. A. Levant, Universal SISO sliding-mode controllers with finitetime convergence, IEEE Trans. Automatic Control, Vol. 49 (No. 9): 1447-1451, 2001. A. Levant, Higher-order sliding modes, differentiation and outputfeedback control, International Journal of Control, Vol. 76 (No. 9/10): 924-941, 2003. S. Laghrouche, F. Plestan, A. Glumineau, Higher order sliding mode control based on integral sliding mode, Automatica, Vol. 43 (No. 3): 531-537, 2007. A. Gaaloul, F. M'Sahli, High gain observer-based higher order sliding mode controllers, 7th Asian Control Conference, HongKong, China, 2009. T. Ménard, E., Moulay, W. Perruquetti, Global finite-time observers for non linear systems, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China, 2009. A.N. Atassi, H.K. Khalil, A separation principle for the stabilization of a class of nonlinear systems, IEEE Trans. on Automatic Control, Vol. 44 (No. 9): 1672-1687, 1999. Y. Hong, G., Yang, L., Bushnell, H.O. Wang, Global finite-time stabilization: from state feedback to output feedback, 39th IEEE Conference on Decision and Control, Sidney, Australia 2000. F. Nollet, T, Floquet, W. Perruquetti, Observer-based second order sliding mode control laws for stepper motors, Control International Review of Automatic Control, Vol. 4, N. 1

38


Engineering Practice, Vol. 16: 429-443, 2008. [21] H.K. Khalil, Nonlinear systems (New York: MacMillan Publishing Company, 1992).

Faouzi M'Sahli was born in Beja, Tunisia in 1963. He received his Mastery of Sciences and the DEA from the ENSET, Tunisia, in 1987 and 1989, respectively. He obtained the Ph.D. and the HdR Degrees in Electrical Engineering from the National Engineering School of Tunis, Tunisia, in 1995 and 2001, respectively. He has published several technical papers in various conferences and journals. He is a co-author of the book "Identification et Commande Numérique des Procédés Industriels" Technip editions, Paris, France. His research interests include Modeling, Identification, Predictive and Adaptive Control of Linear and Nonlinear Systems. Prof. M’Sahli is currently Professor of automatic control at the National Engineering School of Monastir, Tunisia. He is the co-responsible of the research unit of Numerical Control of Industrial Processes and the vice director of the ENIM, Tunsia.

Authors’ information 1

Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, cité taffala, 4003 Sousse, Université de Sousse, Tunisia. E-mail: [email protected]

2

Ecole Nationale d'Ingénieurs de Monastir, rue Ibn Eljazzar, 5019 Monastir, Université de Monastir, Tunisia. E-mail: [email protected] Abderraouf Gaaloul was born in Mahdia, Tunisia, in 1980. He received the License degree in electrical engineering from the ESSTT and the Master’s degree in automatic control from the ENIM, Tunisia, in 2003 and 2005, respectively. He is now a Ph.D student at the national engineering school of Monastir, Tunisia. He is a member of the research unit of Numerical Control of Industrial Processes, ENI Gabès, Tunsia. His research interests include sliding mode control and nonlinear observers. Mr Gaaloul is currently an assistant professor at the High Institute of Applied Sciences and Technologies of Sousse, Tunisia.



39


Application Possibilities of Non Parametric Identification Techniques in On-Line Process Monitoring Tomi Roinila, Mikko Huovinen, Matti Vilkko Abstract – The systems in process industries are typically very large and complex. There can be hundreds or thousands of I/O variables, and the sub processes are typically linked with strong interactions. Therefore, continuous monitoring of such processes, and ensuring their desired operation may become challenging. This paper proposes the use of non parametric identification methods for continuous monitoring of industrial processes. In order to get information-rich identification data and to deal with inherent process nonlinearities the Inverse-Repeat Binary Sequence (IRS) is applied as a stimulus signal, and the system-characterizing frequency responses are estimated through cross-correlation technique. The proposed methods are verified by experimental data from a physical process emulating the traditional headbox of a paper machine. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Industry Automation, Monitoring, Frequency Response, Excitation Signal Design

process models, have been developed, e.g. methods utilizing process data and model identification. Even though specific system identification tests are often needed to evoke the underlying process characteristics, they have several undesired properties which make conducting identification tests an unattractive option. System-identification experiments are often expensive and time-consuming due to typically slow process dynamics. Furthermore, during typical plant tests, production is disrupted which may cause offspecification production. Therefore, developing new identification methods that do not disturb normal operation is of great interest for process industry. Yet an astonishing fact is that most of the developed identification techniques are not used by industrial control engineers, although there is an urgent need for efficient and effective identification methods. It has been claimed that one reason for this failure of technology transfer is that too many people concentrate on parameter estimation and convergence analysis, while test design and model validation are neglected even though it is this part that is close to model applications [1]. Today, the typical identification practice is still very much based on single-variable type of thinking although industrial processes are typically very complex and highly disturbed. The basic identification experiments are called step tests, in which each manipulating variable (MV) is stepped separately and some clear step responses are expected for modeling each transfer function. This approach may provide sufficient information on the dynamics of a system but this is not always the case. Very often, external noise masks the step response data. In order to avoid this problem, the size of the step may have to be increased to an

Nomenclature y u g v Ruu Ruv

α δ ˆ G Ts Rˆ

uy

ak P n hi K

Output signal Input signal Impulse response Disturbance signal Auto correlation of input signal Cross correlation between input and disturbance Variance of u Kronecker delta function Estimated frequency-response function Sampling interval Estimated cross correlation of input and output PRBS Period length of PRBS Length of shift register Kernel Length of doubled MLBS

I.

Introduction

Process monitoring usually requires a process model which is generally hard to come by due to poorly-known and complicated processes, poor instrumentation level, and/or resource limitations in the process creation. Consequently, 1st-principle analytical models based on fundamental understanding of the process and its underlying characteristics are often out of the question. Thus, several monitoring methods, not requiring detailed Manuscript received and revised December 2010, accepted January 2011

40


T. Roinila, M. Huovinen, M. Vilkko

unacceptably large level, thus evoking nonlinear process characteristics. In addition, inherently such tests produce information only on very limited frequency bandwidth. Thus, the models derived from this approach may not be accurate for complex and highly disturbed processes [2]. An alternative to the traditional time-domain analysis described above is to analyze the processes in the frequency domain. In practical measurements, the system-characterizing models can be found either by parametric or non parametric system identification methods [3]. Parametric methods require a selection of an input stimulus [4], a priori selection of a parameterized model structure including system order and number of zeros, construction of a suitable prediction error equation and loss function, and methods to minimize the loss function. These methods return the parameters of the system model such as the coefficients of the system difference or differential equations, transfer function, or state-space model and are useful e.g. for complex controller design. Non parametric methods on the other hand do not assume a system model and require only a selection of a stimulus, and therefore provide a cost effective alternative for model creation. Non parametric methods return frequency-response-function (FRF) data directly and are useful e.g. for quality assessment procedures. Cross-correlation techniques [5] with carefully selected stimulus sequences provide efficient means to measure the system-characterizing frequency responses. Very often, a stimulus signal with a broadband spectrum is applied. The stimulus signal with a broadband spectrum has energy at several frequencies thus allowing to measure frequency responses at those frequencies simultaneously. The signals having a broadband spectrum are generally divided into binary, near-binary, and non binary sequences, each having many attractive properties. For continuous monitoring of the industrial processes it is important that the system is not disturbed too much in order to guarantee the normal operation. Thus, the stimulus signal has to be selected such that its amplitude in the time domain is kept small but the energy, i.e. the amplitude in the frequency domain, is maximized. One method to analyze this quality of a stimulus is to measure the signal peak factor, also known as crest factor [6]. The smaller the peak factor is, the greater the energy is in relation to the signal amplitude. It can be shown that the binary signals have the lowest possible peak factor [7]. Therefore, the binary sequences can be considered as one of the most effective stimulations for continuous FRF-measurements of industrial processes. Another advantage of the binary sequences over the near- or non binary signals is that they can be generated with low-cost applications whose outputs can only cope with a small number of signal levels. One of the most popular binary signals used in FRF measurements is the periodic pseudorandom binary

sequence (PRBS) [8]. One special class of these signals is the maximum-length binary sequence (MLBS) [9]. The sequence is very popular in the applications of system identification due to its straight-forward implementation, low peak factor, attractive frequency content, and other useful properties. However these sequences may evoke the inherent nonlinear properties of the system under inspection. To combat this, the InverseRepeat Binary Sequence (IRS) has been developed. The IRS is generated by doubling the MLBS and toggling every other digit of the doubled sequence [10]. The IRS cancels the even-order kernels in the system Volterra series [11] leaving the linear term and higher order infinitesimals thus providing more accurate estimate about the underlying linear dynamics. Several technological advances have opened up new possibilities for process monitoring in the process industries [12] but applications utilizing the frequency domain have not been documented or at least are hard to find. Thus, the purpose of this paper is to study the use of an online process-monitoring algorithm based on non parametric identification and utilization of the IRS stimulus signal. The IRS signal is continuously or periodically added to the normal control signal to evoke the underlying process characteristics. The process characteristics are monitored by comparing the obtained frequency response to the nominal response obtained under normal operating conditions. It is also intended to study how to design the stimulus signal to be powerful enough to provide information-rich data but at the same time delicate enough to allow normal process operation while identification is executed. The ultimate goal is to obtain a cheap and effective online monitoring practice. The rest of the paper is organized as follows. The characteristics of a typical industrial process are discussed in Section 2. Section 3 reviews the basic theory applied in this paper; cross-correlation technique in system identification, and the IRS excitation and its design procedure are briefly introduced. Experimental evidence based on a physical process emulating the traditional headbox of a paper machine is presented in Section 4 supporting the theoretical findings. Finally, the conclusions are drawn in Section 5.

II.

Characteristics of Process Industry

Processes are usually composed of several unit processes, for example, tanks, mixers, heat exchangers, furnaces, reactors, and pumps and pipes. Thus, a process is a collection of interconnected unit processes usually with strong interactions, and the performance of a process depend on a complex way on the performance of its unit processes. The unit processes are typically instrumented by number of measurement devices and actuators, and controlled by well-tuned controllers in order to obtain a stable operation of the entire process. If the process is subject to remarkable disturbances, the intermediate



41


tanks or storages are used to filter and adequately sized actuators are used to compensate the disturbances. The operation of unit processes are coordinated by supervisory control that gives the optimal operation points to unit processes. However, the large intermediate reservoirs and over-sized actuators make the processes both expensive to build, and make the operation of the processes inconvenient to respond increasingly repeating changes in operation conditions. An alternative way to handle disturbances is to apply sophisticated modelbased control methods. Models are used to optimally generate control signals that minimize the effects of disturbances, even before the consequences of the disturbances can be noticed on the measured output signals. The development of computing and communication technologies has allowed the development of very largescale integrated and interconnected production systems, for example, energy-transfer and distribution systems, very large process plants, and telecommunication systems. The operation of these large-scale processes has to fulfil concurrent production, security, energy management, and environmental requirements. These sometimes conflicting requirements can be taken into account only if the operational state and the capabilities of the processes are well known and adequately communicated to all the responsible parties. This is best accomplished by well-established mathematical models of the dynamics of the processes together with trustworthy measurement information. However, because such models are rarely available, alternative approaches have to be considered. A recent trend in process industry is the reduction of manpower and a trend towards a more lean organization. The staffs responsible for operation have to monitor ever larger number of unit processes and are responsible for larger integrated systems. Therefore, the operators need more elaborate information of the state of operations and condition of the devices. This, also, sets new incentives to develop reliable and accurate descriptions of the dynamic behaviour of the processes. The large integrated systems of processes are supervised by increasing number of separate companies. The companies might not even own the processes but they provide process operation as a service offering and specialize to the competencies required in different parts of value chain and phases of production. Besides better and more cost-effective process management tools, the new kind of business models require more stringent service level agreements from each participants.

III.1. Cross-Correlation Technique In steady state, for small-signal disturbances, a typical industrial process can be considered as a linear timeinvariant system. The sampled system can be described as: m

y ( m) = ∑ g ( k ) u ( m − k ) + v ( m)

(1)

k =1

where y ( m ) is the sampled output signal, u ( m ) the input signal (excitation), g ( m ) the system impulse response and v ( m ) represents disturbances, such as measurement and quantization noises. The crosscorrelation between the input and output signal can be given by: Ruy ( m ) = =

m

∑ u ( k ) y (k + m) = k =1

(2)

m

∑ g ( k ) Ruu ( m − k ) + Ruv ( m ) k =1

where Ruu ( m ) is the auto correlation of the input signal and Ruv ( m ) the cross correlation between the input and disturbance signals. In the case of white noise as an input signal the following characteristics hold: ⎧⎪ Ruu ( m ) = αδ ( m ) ⎨ ⎪⎩ Ruv ( m ) = 0

(3)

where α denotes the variance of u ( m ) , and δ ( m ) the Kronecker delta function. Thus, the auto correlation of the input signal is a delta function and the crosscorrelation of the input and disturbance signal is zero. Under the assumption of (3), the cross correlation in (2) can be represented as: Ruy ( m ) = α g ( m )

(4)

Hence, the cross correlation between the measured input and output signals yields the system impulse response. Using finite-length signals an estimate is obtained for Ruy ( m ) yielding an estimated impulse response. The response can be converted to the frequency domain and presented as FRF by applying discrete Fourier transform (DFT) [13]:

III. Theory This chapter presents the main theory applied in the paper. Frequency-response-measurement technique based on cross-correlation technique is shortly reviewed, and the theory behind the IRS excitation sequence is discussed.

(

)

ˆ e jωTs = 1 G


α

M −1

∑ Rˆ uy ( m ) e− jkωT

s

(5)

k =0


42


where M denotes the total length of collected data, ˆ e jωTs the estimated FRF, T sampling interval, and G s

(

For a nonlinear system, the output response can be represented as a Volterra series expansion [16], [17]:

)

Rˆ uy ( m ) the estimated cross correlation between the

y ( n) =

measured input and output signals. The requirement in (5) is that the system is perturbed with a signal resembling white noise. This requirement can be approximately met by applying the PRBS signal [10].

k =0

+ +

n

mod ( 2 )

M

∑ ∑ h2 ( k1 ,k2 ) u ( n − k1 )u ( n − k2 ) + " + M

(7)

M

∑ ∑ hi ( k1 ,..,ki ) u ( n − k1 )"u ( n − ki ) "

k1 = 0

One special class of PRBS signals is maximum length binary sequence (MLBS). PRBS ak is a maximum length sequence if and only if it satisfies a linear recurrence:

∑ ci ak −i

M

k1 = 0 k2 = 0

III.2. Inverse-Repeat Binary Sequence

ak =

M

∑ h1 ( k ) u ( n − k ) +

k2 = 0

where u ( k ) represents the system input and M is the length of total data sequence of interest. Each of the discrete convolutions contains a kernel, either linear ( h1 ) or nonlinear ( h2 ,...,hi ), which represents the behavior of the system. In the absence of nonlinearities, the system can be fully described by means of the linear kernel. The attractive property of the IRS excitation can be found from its antisymmetry, expressed as:

(6)

i =1

where ci has a value of 1 or 0 and ak has a period of P = 2n − 1 [14]. The length of the period depends on the values of ci and with appropriate choice the sequence

u ( n ) = −u ( n + K / 2 )

has a maximum length. The MLBS can be generated efficiently by an n-bit shift register with exclusive or (XOR) feedback. In practice, the values 0 and 1 generated by the shift register are mapped to +1 and -1, respectively, to produce a symmetrical maximum length sequence with mean close to zero. The MLBS has similar spectral properties as true random white noise. Due to its deterministic nature, it can be repeated precisely. It is therefore possible to increase the signal-to-noise ratio by synchronous averaging of the response periods. An assumption made while designing the MLBS is that the process under consideration is linear. However, industrial processes often suffer from a various nonlinear phenomena. Thus, the use of the MLBS may produce false information about the process linear dynamics. A system with nonlinearities can be modeled basically in two ways. One method is to identify the system including all of its nonlinearities [15]. The other way is to identify only the linear portion of the model, which requires that the nonlinearities are suppressed. The latter technique is useful for systems where the nonlinearities are typically difficult to detect and model. One method to minimize the effect of the nonlinearities is to use a carefully selected excitation signal. One such signal is the IRS which is generated by doubling the MLBS and toggling every other digit of the doubled sequence [10]. The IRS suppresses effectively the system secondorder nonlinearities thus providing more accurately the estimate of the underlying linear dynamics compared to the MLBS.

(8)

where K denotes the length of the doubled MLBS. As a consequence, by injecting the IRS into a system, all the even-order kernels cancel out from (7) leaving only the linear term and terms involving odd-order kernels, shown more in detail for instance in [18]. Thus, the estimation of the system linear part can be obtained more accurately. The contributions of the higher-order kernels on the output are usually of negligible amplitude compared to the contributions of the lower-order kernels. Hence, the nonlinear effect caused by the secondorder kernel may be assumed to be dominating.

IV.

Experimental Verification

The applied part of the research was done in a physical process emulating the traditional headbox of a paper machine, shown in Fig. 1. The headbox is a paper machine subprocess designed to ensure an even flow of pulp from the headbox to the wire by dampening disturbances. An essential part in this is ensuring a constant feed pressure by using pressurized and controlled volume of air above the volume of pulp. The pulp escapes the headbox from the bottom through the outflow nozzles called the slice opening, and the constant overall pressure in the headbox provides a constant flow. The second tank in Fig. 1 is only needed in the laboratory equipment for recycling the liquid in the process, i.e. it is not present or needed in an actual headbox process.



43


The amplitude of the excitation was experimentally adjusted such that the standard deviations of the process outputs do not increase over a specified limit, i.e. the process is not disturbed too much. The excitation was added into the pressure control signal which was measured together with the surface level. IV.1. Experiment 1 The first experiment was performed under normal operation conditions in order to obtain reference values for the frequency responses. These results are essential for the other experiments where the aim is to detect the possible fault conditions in the process. Fig. 2(a) shows the frequency response from the pressure control signal to surface level when the abovedescribed excitation sequence was injected into the system and (5) was applied. Fig. 2(b) shows a sample of perturbed and non perturbed surface-level signal. The standard deviation of the perturbed signal stays below an appropriate value such that the normal operation of the process is guaranteed.

Fig. 1. The process

The headbox has two main actuators, an air compressor providing air flow and a pump providing pulp flow to the headbox. The compressor is used to control the surface level and the pump is used to control the pressure level inside the headbox. The process variables are strongly interconnected (e.g. if one wants to change the pressure it surely affects the surface level). Thus, a MIMO controller is used to minimize the effects of these interactions. Besides the measurements needed for the controllers, there is also the measurement of pulp flow from the headbox which is also the output of main interest. As the slice opening is kept at constant level, the outflow is a function of only the overall pressure inside the headbox. However the function is nonlinear. The system is controlled by a basic PC using Matlab/simulink environment. The data is collected directly in the Matlab environment. In the experiments an additional excitation signal is added to the output of the pressure controller. The signal characteristics are determined from the process characteristics such as process gains, dominant time constants and measurement characteristics. The process characteristics are inherently nonlinear and dependent on the operating point (i.e. pressure and surface levels). Therefore the used signal should be adjusted for different operating points or designed so that it is applicable in all the used operating points. It should also be remembered that any results obtained may be sufficiently accurate only locally. All the experiments were performed with a 1022-bitlength IRS (i.e. applying 9-bit-length shift register) with a generation frequency of 0.1 Hz. At this point the stimulus signal was designed to be executed as a single-input-variable experiment, i.e. only one stimulus signal is applied in a test.

(a)

(b)

Figs. 2. (a) Frequency response from pressure control to surface level, and (b) sample of perturbed and non perturbed surface-level signal

IV.2. Experiment 2 The second experiment shows an example how the presented method can be used in on-line monitoring purposes to detect possible fault conditions. To illustrate this, the original set up was modified by manipulating the pump control signal with a backlash operator which can be used to represent equipment wear. Due to the equipment wear (i.e. the presence of backlash) the devices ability to follow the control signal will decrease. The excitation signal was injected into the pressure control signal as in the first experiment. The frequency response was measured using a sliding window every 30 seconds. Each frequency-response computation was performed by using a data vector of a length of one excitation period. At some point the fault condition was switched on. Fig. 3 illustrates how the data was sampled during the experiment.



44


V.

Continuous monitoring and evaluation of processes during normal system operation is of great interest in various fields of industry. Typically the process experiments in the process industry are expensive and time consuming. In addition, usually the experiments cannot be performed on-line due to interruptions in the production. This paper presented methods to measure and evaluate an industrial process through non parametric identification methods. Inverse-repeat binary sequence (IRS) was applied as an excitation signal, and a frequency response was computed through crosscorrelation technique. The IRS is generated by doubling the traditional maximum-length-based PRBS and toggling every other digit of the doubled sequence. The IRS suppresses effectively the even-order Volterra kernels yielding more accurate approximation of the linear part. The key point of the paper was to present a technique which can be applied on-line with minimal disturbances to the process. This approach requires that the system under test is not disturbed too much by the external excitation signal, thus ensuring the normal system operation. The proposed method was verified by experimental measurement from a physical process emulating the traditional headbox of a paper machine. The results confirmed that frequency-domain analysis is a valid method for monitoring purposes. It is emphasized that the presented experiments were all performed as a singleinput-variable experiments. It may be obvious, however, that for a more sufficient monitoring of a typical industrial process, multi-input-variable experiments are required. This, in turn, requires methods to identify multiple-input multiple-output (MIMO) systems, and will be one of the future works of the authors. The presented methods can be used in various frequencyresponse-based applications in process industry. Possible applications could include system validation, controller design, and system monitoring.

Fig. 3. The frequency response is computed every 30 seconds using corresponding data segments with length of one excitation period

Fig. 4 shows some of the measured frequency responses. The figure clearly shows the difference between fully functioning and worn equipment, especially at low frequencies. Furthermore, the results clearly indicates that the fault condition can be detected relatively fast once occurred. The legend numbers indicates the measurement time. For instance, 1500 s means that the frequency response was measured 1500 s after the fault condition was switched on. The computation time for a single frequency response was approximately 2 s. Hence, the measurements can be easily performed online. The differences in the frequency responses can be explained by studying further how the signal is modified by the backlash operator. Applying simple Fourier analysis it can be seen that the backlash operator reduces the response amplitude, especially at low frequencies, thus modifying correspondingly the measured frequency response. The change would be relatively difficult to observe from the time-domain data, and even if detected, it would be difficult to tell if the change originates from other process units or if the pump characteristics have changed. -5

0s 30 s 1500 s 10220 s

-10

Magnitude(dB)

-15

References [1]

-20

[2]

-25

[3]

-30 -35

[4]

-40 -45

[5] -4

10

-3

10 Frequency(Hz)

Conclusion

-2

10

[6]

Fig. 4. Some of the measured frequency responses in experiment 2

[7]


Y. Zhu, Multivariable Process Identification for MPC: the Asymptotic Method and its Applications, Journal of Process Control, vol. 8 n. 2, 1998, pp. 101 – 115. B. Liu, J. Zhao, J. Qian, Design and analysis of test signals for system identification, International Conference on Computational Science, 2006, pp.593 – 600. R. Pintelon, J. Schoukens, System Identification – A Frequency Domain Approach (Institute of Electrical and Electronics Engineers, Inc., 2001). K. Godfrey, A. Tan, H. Barker, B. Chong, A Survey of Readily Accessible Perturbation Signals for System Identification, Control Engineering Practice, vol. 13, 2005, pp. 1391 – 1402. L. Ljung, System Identification – Theory for the User (Prentice Hall PTR, 1999). D. Wulich, Comments on the Peak Factor of Sampled and Continuous Signals, IEEE Communications Letters, vol. 4 n. 7, 2000, pp. 213 – 214. K. Godfrey, Design and Application of Multifrequency Signals,


45


[8] [9] [10] [11] [12] [13] [14] [15] [16]

[17] [18]

Computing and Control Engineering Journal, vol. 2, 1991, pp. 187 – 195. W. Davies, System Identification for Self-Adaptive Control (Wiley-Interscience, a division of John Wiley and Sons Ltd., 1970). K. Godfrey, H. Barker, A. Tucker, IEEE Trans. on Control Theory and Applications, vol. 146 n. 6, 1999, pp. 535 – 548. K. Godfrey, Perturbation Signals for System Identification (Prentice Hall, UK, 1993). R. Tymerski, Volterra Series Modeling of Power Conversion Systems, IEEE Trans. on Power Electronics, vol. 6 n. 4, 1991, pp. 712 – 718. M. Huovinen, Technology Advances Open Up New Possibilities in Industrial Process Management, The International Review of Automatic Control, January 2011,In press. G. James, Advanced Modern Engineering Mathematics, third edition (Pearson Education Limited, 2004). S. W. Golomb, Shift Register Sequence (Holden-Day, San Francisco, 1967). O. Nelles, Nonlinear System Identification (Springer-Verlag, 2001). C. Evans, D. Rees, L. Jones, M. Weiss, Periodic Signals for Measuring Nonlinear Volterra Kernels, IEEE Trans. on Instrumentation and Measurement, vol. 45 n. 2, 1996, pp. 362 – 371. F.J. Doyle III, R.K. Pearson, B.A. Ogunnaike, Identification and Control Using Volterra Models (Springer-Verlag, 2002). T. Roinila, M. Vilkko, T. Suntio, Frequency Response Measurement of Switched-Mode Power Supplies in the Presence of Nonlinear Distortions, IEEE Trans. on Power Electronics, vol. 25 , 2010, pp. 2179 – 2187.

Mikko Huovinen received the M. Sc. (Tech) and the D. Sc. (Tech) degrees in automation engineering from Tampere University of Technology, Tampere, Finland, in 2005 and 2010, respectively. He has worked at the Department of Automation Science and Engineering, Tampere University of Technology, in the field of process automation since 2005. His current research interests include process monitoring, process control, modeling, simulation and analysis. Matti Vilkko received the M.Sc. degree in electrical engineering in 1989, Lic.Tech degree in 1993 in electrical engineering and Dr.Tech degree in 1999 in automation engineering from Tampere University of Technology (TUT), Tampere, Finland. From 1989 to 1999 he was a researcher in Institute of Automation and Control, Tampere university of Technology. His research focused on scheduling and optimization of hydro-thermal power production. From 2000 to 2003 he had a research and development management positions in Patria Ailon Inc and Ailocom Inc. In 2003 he became senior researcher and a Full Professor in process control in 2010 in Department of Automation Science and Engineering at TUT. His current research interest is in the areas of process control, modeling, simulation and system identification.

Authors’ information Department of Automation Science and Engineering, Tampere University of Technology. Tomi Roinila was born in Raisio, Finland, in 1980. He received the M. Sc. (Tech) and the D. Sc. (Tech) degrees in automation and control engineering from the Tampere University of Technology (TUT), Tampere, Finland, in 2006 and 2010, respectively. Since 2006 he has worked at TUT, Department of Automation Science and Engineering as a researcher. His current research interests include system modeling, simulation and analysis.



46


Output Tracking Control Design for Non-Minimum Phase Systems: Application to the Ball and Beam Model Monia Charfeddine, Khalil Jouili, Houssem Jerbi, Naceur Ben Hadj Braiek Abstract – In this paper, we suggest a new a control technique based on the formalism of the exact input-output linearization. The fulfilled developments exploit the disturbance vanishing theory. The framework of the study that we present utilizes a nonlinear model with non-minimum phase of the "ball and beam" system and aims at ensuring an output trajectory tracking. In spite of the variation of amplitude, there is a perfect tracking of the output trajectory. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Trajectory Tracking, Non-Minimum Phase Systems, Exact Input-Output Feedback Linearization, Disturbance Vanishing, Lyapunov Theory

I.

Afterwards, a simulation study aiming at evaluating the obtained performances will be carried out in section V. We, lastly, end up this paper by a conclusion emphasizing the developed study.

Introduction

The formalism of the exact input-output feedback linearizing control arises as a natural frame for the synthesis of control laws, which ensure the tracking of an output trajectory. This technique, as a matter of fact, was successfully established in various practical applications such as the "ball and beam" system [1]-[3]. The control of this system is a delicate task owing to the fact that it is a nonlinear model with non-minimum phase and that it is also characterized by a instable behavior the of the zero dynamics of [4]-[7]. For these reasons, the nonlinear control based on an approximation method is a well adapted method for the stability of this system [8]-[9]. These last years, a growing interest has been devoted to the problem of stabilization of the "ball and beam" system. Most of the works dealing with this problem study the stability of the controlled system in a closed loop [1]-[2][8]-[9]. Nevertheless, the performances concerning the track of the variable-amplitude-output trajectory are scarcely evoked. These performances are in fact, strongly considered in this paper and will be the focal point of our study. Actually, the fundamental idea of our work is to develop an approximate linearizing control technique exact input-output based on the disturbance vanishing theory [10], [11]. This paper is divided into six sections: section II, firstly, is devoted for the description of the "ball and beam" system as well as the formulation of the control problem. In section III, we, secondly, show the principle of the feedback linearization as well as the disturbance vanishing theory. The establishment of a linearizing approximate exact input-output control based on the disturbance vanishing theory is, then considered in section IV.

II.

Description of the "Ball and Beam" System

The "ball and beam" system is an unstable nonlinear one which is well-known in automation; therefore, it is regarded as a perfect bench test for the design of the control-laws for the non-minimum phase nonlinear systems [1]-[2]-[3]. The "ball and beam" system is composed of a rigid bar carrying a ball. The latter is characterized by an horizontal axis and the moment of inertia J . The rotation angle θ compared to the horizontal one is controlled by a motor with direct current applying a couple τ . A ball is placed on the beam where it is able to move with a certain freedom under the effect of gravity, as it is illustrated in Fig. 1.

rb

θ

Fig. 1. Synoptic diagram of the "ball and beam" system


47


M. Charfeddine, K. Jouili, H. Jerbi, N. Ben Hadj Braiek

neighborhood Ω of x0 such as f ( x ) = 0 , and given by

The objective of the studied system control is to ensure that the ball always keeps contact with the beam and that its movement is carried out without slip what imposes a mechanical constraint on the acceleration of the beam. There was always some discussion for this typical system about carrying out a trajectory among a class of acceptable trajectories while satisfying the above-mentioned constraints. The dynamic model governing the behavior of the "ball and beam" system in open loop can be expressed by the following equations [1]:

the following equation system: ⎧⎪ x = f ( x ) + g ( x ) u ⎨ ⎪⎩ y = h ( x )

where x∈ℜ n is the state vector, u∈ℜ is the control input, y∈ℜ is the system output. f ( x ) , g ( x ) and h ( x ) are smooth vector functions.

The problem of the exact input-output linearization is based on the search for a nonlinear transformation which transforms model (3) into a linear one using a control by return state. The property of linearity, in this case, should be established between a new control v and the output y . For the case of single-input single-output systems, the approach consists in defining a linear input-output relation by successively deriving the output until the input appears which can be expressed by:

⎧⎛ J b ⎞ 2 ⎪⎜ ⎪⎝ R 2 + M ⎟⎠ rb + MG sin (θ ) − Mrbθ = 0 (1) ⎨ ⎪ Mr 2 + J + J θ + 2 Mr r θ + MGr cos (θ ) = τ b b b b b ⎩⎪

(

)

If we consider that the system output is such as u = θ the state representation will be given by [1]: ⎧⎡ x ⎤ ⎡ x2 ⎪⎢ 1 ⎥ ⎢ 2 ⎪ x2 ⎢ Bb x1 x4 − G sin x3 ⎪ ⎢⎢ ⎥⎥ = ⎢ ⎨ x3 x4 ⎢ ⎪⎢ x ⎥ ⎢ ⎢ ⎥ 0 ⎪⎣ 4 ⎦ ⎣ ⎪y = x ⎩ 1

(

)

⎤ ⎡0⎤ ⎥ ⎢ ⎥ ⎥ ⎢0⎥ ⎥ + ⎢0⎥ u ⎥ ⎢ ⎥ ⎥ ⎣1 ⎦ ⎦

⎧ y = Lf h ( x) ⎪ ⎪⎪ y = L2f h ( x ) ⎨ ⎪ ⎪ (r ) r r −1 ⎪⎩ y = L f h ( x ) + Lg L f h ( x ) u

(2)

with:

(

Definition of the relative degree [11]

)

In the case of nonlinear systems (3), one may say that a system has a relative degree r in x0 , if it verifies the two following conditions: i. Lg Lkf −1h ( x ) = 0 for any x the neighbor of x0 and

The numerical parameters of the "ball and beam" system are recapitulated in the following Table I [1]- [3].

0 < k < ( r − 1)

ii. Lg Lrf−1h ( x0 ) ≠ 0

TABLE I PARAMETERS AND NUMERICAL VALUES OF THE "BALL AND BEAM SYSTEM" Notation Description Numerical values 0. 05 kg M Ball mass

Jb

(4)

where r is the system relative degree.

⎡ x1 ⎤ ⎡ rb ⎤ ⎢ ⎥ ⎢r ⎥ x2 b x = ⎢ ⎥ = ⎢ ⎥ , y = h ( x ) and Bb = M / J b / R 2 + M ⎢ x3 ⎥ ⎢θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ x4 ⎦⎥ ⎣⎢θ ⎦⎥

R

(3)

Ball radius Ball inertia

J

Beam inertia

G

Acceleration due to gravity

Bb

Constant

The relative degree, thus, of an output is in fact the order of minimal derivation of the necessary output to explicitly reveal the input. Consequently, the system is linearisable by feedback and one can define the transformation of a nonlinear state T ( x ) as:

0. 01m 2 × 10

−6

kg m

0. 02 kg m 9.81 m s

2

2

⎡h x ⎤ ⎡ T1 ( x ) ⎤ ⎢ ( ) ⎥ ⎢ ⎥ ⎢ Lf h ( x) ⎥ ⎢ T2 ( x ) ⎥ ⎢ ⎥ ⎥ =⎢ T ( x) = ⎢ ⎥ ⎢ ⎥ ⎢ r −1 ⎥ L h x ⎢Tr −1 ( x ) ⎥ ⎢ f ( ) ⎥ ⎢ ⎥ ⎢ r ⎣ Tr ( x ) ⎦ ⎢ L f h ( x ) ⎥⎥ ⎣ ⎦

2

0. 7143

III. Scope and Mathematical Preliminaries III.1. Basic Results on the Exact Input- output Feedback Linearization

(5)

By making use of the transformation (5) and by using the notation z as a variable of state, system (3) becomes:

Let us consider the single-input single-output nonlinear affine input system, defined in the Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


48


⎧ z1 = z2 ⎪z = z 3 ⎪ 2 ⎪⎪ ⎨z = z r ⎪ r −1 ⎪ z = Lr h ( x ) + L Lr −1 ( x ) u f g f ⎪ r ⎪⎩ = v (r )

r −1

(

∂V ( x ) ∂x

∂V ( x )

(6)

∂x

with c1 and c2

with v = yd + ∑ ki +1 yd − Ti +1 ( x ) , where yd is the i =0

Lg Lrf−1

( v − L h ( x )) ( x) r f

IV.

(7)

In this part, Lyapunov stability results will be applied to the analysis of stability of the disturbed autonomous systems. Let us consider the following dynamic system: (8)

where f ( x ) represents the nominal dynamics, with f ( 0 ) and ∆ ( x ) represents the disturbed dynamics.

f and ∆ are Lipschitz in x .

Since ∆ ( x ) is Lipschitz then: ∃δ > 0: ∆ ( x ) ≤ δ x

Synthesis of an Approximate InputOutput Linearizing Control Based on the Disturbance Vanishing Theory

in the reduction of the Lyapunov function, which, alternatively, can be used to adapt to the disturbances ∆ ( x ) .

(9)

The disturbance vanishing theory is based on the following assumption: ∆ ( 0) = 0

∆ ( x ) ≤ δ x with δ is a

In this section, we present the principle of the approximate input-output linearizing control based on the disturbance vanishing theory. Indeed, the idea of this approach is to neglect a part of the system dynamics in order to make approximate feedback input-output of the linearized system. The neglected part is then regarded as a disturbance. A linear regulator is designed for the non-linearities which did not compensate but simply ignored. The stability is analyzed by using the disturbance vanishing theory [10]. If we count on Lyapunov analysis [12], the theorem above sufficiently provides exponential conditions of stability for the disturbed system (8). If a nominal system x = f ( x ) is exponentially stable, then there is a margin

III.2. The disturbance Vanishing Theory

⎧⎪ x = f ( x ) + ∆ ( x ) ⎨ ⎪⎩ x ( 0 ) = x0

(12)

c1 , the origin ( x = 0 ) is a c2 balance point of the exponentially stable disturbed system (8).

are the solutions of Hurwitz polynomial [11]. The expression of the feedback will then be given by: 1

(11)

positive constant. If δ
0

(

)

Lnf h T −1 ( z ) ≤ δ1 z ∃δ 2 > 0

(

,

)

Lg Lnf−1h T −1 ( z ) ≥ δ 2

- The gains ki , ( i = 0 , … , n − 1) are the solutions of

(24)

Hurwitzs polynomial. - Let us consider the function ∆1 ( z ) defined by:

with:

(

)

Then we replace it in (20) whence we have ∆ ( 0 ) = 0 .

The resulting system (19)-(20) is composed of a linear one, and corresponds to an exact approximate linearisation input-state of system (3). Consequently, Brunovsky linear control is given by: r yd( )

(

in (15) then we Lnf −1h T −1 ( z ) = 0 .

⎡ ⎢ 0 ⎢ ⎢ ∆1 ( z ) = ⎢ Lg Lrf−1h T −1 ( z ) ⎢ ⎢ ⎢ n−2 −1 ⎢⎣ Lg L f h T ( z )

(25)

The primary advantage of the developed control method is that the resulting disturbance (25) disappears. (Which means that it tends to the equilibrium towards zero).

(

)

(

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(26)

and a matrix P which is the solution of Lyapunov T equation P ACL + ACL P = − I where I is the identity matrix and also the following inequality is being satisfied:



51


∆1 ( z ) 0 Φ 0

a) One observer for the unknown specific rates, T

(

Pˆ = ϕˆ 1 − DP + ω2 P − Pˆ

Ω = diag { − ωi } , Γ = diag γ j , ωi , γ j ∈ ℜ+ (14) j =1,2

⎡⎣ ρ1 ( t ) ρ 2 ( t ) ⎤⎦

ρ (t ) =

(

⎡0 1 ⎤ ⎡0⎤ where K R = ⎢ ,FR = ⎢ ⎥ . ⎥ ⎣1 0 ⎦ ⎣0⎦ The choice of design matrices Ω and Γ must be done such that the algorithm to be stable and convergent. The properties of stability and convergence for this estimator have been discussed at length in [1], [14]. A typical choice for the matrices Ω and Γ is of the following diagonal form:

i =1,2

for

Xˆ = ϕˆ 2 − DX + ω1 X − Xˆ

T

OBE T

ξ R = [ X P ] , then a reduced OBE can be designed. The equations of reduced estimator are derived from (12):

ξˆ R = K R H (ξ ) ⋅ ρˆ ( t ) − D ⋅ ξ R + FR − Ω ξ R − ξˆ R

an

= ⎣⎡ϕ1 ( t ) ϕ2 ( t ) ⎦⎤ can be obtained (with H (ξ ) = 1 ):

state vector. The error ρ − ρˆ is directly reflected by the estimation error ε . Ω is a gain matrix, and the injection matrix Γ is chosen such that the matrix ΩT Γ + ΓΩ is negative defined. The estimator (12) was designed by taking into consideration a full state vector. However, it is possible to design a reduced order OBE, by selecting a part of state-space equations, with the obvious condition that this subsystem contains the unknown kinetics which will be estimated. For example, if the next partition is chosen:

ρˆ = ⎡⎣ K R ⋅ H (ξ ) ⎤⎦ ⋅ Γ ⋅ ξ R − ξˆ R

(15)

b) If all reaction rate vector is considered as unknown,

)

T

) )

µˆ = X γ 1( X − Xˆ )

In (12) ρˆ is the on-line estimate of the unknown vector of parameters (the specific growth rates or the reaction rates). The first equation is a state observer, used for updating ρˆ , and not for state estimation. The update is generated by the estimation error ε = ξ − ξˆ , where ξˆ is the on-line estimation of the

(

( (

Xˆ = µˆ X − DX + ω1 X − Xˆ

information needed for updating the estimates of parameters. The OBE can be written as [1], [14]:

T

ρ ( t ) = ⎡⎣ ρ1 ( t ) ρ 2 ( t )⎤⎦ = ⎡⎣ν P ( t ) µ ( t ) ⎤⎦ .

with Ψ ( t ) the regressor matrix, and Φ ( t ) a gain matrix.

⎡X 0 ⎤ In this case the matrix H (ξ ) = ⎢ ⎥ and the ⎣0 X⎦ detailed equations of the observer-based estimator are as follows:

The forgetting factor λ ∈ ( 0 ,1) and β ∈ ℜ+ are design parameters. The stability and convergence properties of the regressive estimator are widely discussed and proven in [1], [2], [10].



127

D. Selişteanu, C. Marin, E. Petre, D. Şendrescu

ξa = K a ⋅ H (ξ a ,ξb ) ⋅ ρ ( t ) − D ⋅ ξ a + Fa

Again, it is possible to design a reduced order estimator, by taking into consideration a partition of the state vector, ξ R , previously defined. Then, the reduced order estimator is derived from (17): = − β Ψ + K H (ξ ) Ψ R Ψ 0 = − β Ψ 0 + ( β − D ) ξ R + FR T

By using this factorization, a high-gain observer can be implemented. The design of high-gain observers is done in [7], [8], with supplementary assumptions regarding global Lipschitz conditions, the boundedness of H (ξ ) diagonal elements’ away from zero, etc. The

T

(

ρˆ = ΦΨ ξ R − Ψ 0 − Ψ T ρˆ

(18)

)

equations of the observer for (5) are obtained as [12]:

= −ΦΨΨ T Φ + λΦ , Φ ( 0 ) = Φ > 0 Φ 0

(

) ( −1 ρˆ = −θ 2 ⋅ ⎡ K a ⋅ H (ξâ ,ξb ) ⎤ ⋅ (ξâ − ξ a ) ⎣ ⎦

ξâ = K a H ξâ ,ξb ρˆ − Dξâ + Fa − 2θ ξâ − ξ a

with the regressor matrix of the form: ⎡ 0 1 ⎤ ⎡ψ 1 0 ⎤ ΨT = ⎢ ⎥ ⎥⎢ ⎣1 0 ⎦ ⎣ 0 ψ 2 ⎦

(19)

we

have

⎡⎣ ρ1 ( t ) ρ 2 ( t ) ⎤⎦ 0⎤ ; X ⎥⎦

ρ (t ) =

T

T

⎡⎣ ρ1 ( t ) ρ 2 ( t ) ⎤⎦ = ⎡⎣ϕ1 ( t ) ϕ2 ( t ) ⎤⎦ , and H (ξ ) = 1 . Both regressive estimators require on-line information about the concentrations X and P. If X is not on-line measurable, then the estimates X est provided by the asymptotic observer (11) will be used in (18).

In

The third algorithm is based on a high-gain approach [7], [8]. Again, we will suppose that all states are measured or on-line estimated. In order to design highgain observers, the model (5) will be used. Since the matrix of yield coefficients K is full rank, a full rank arbitrary submatrix K a can be considered. Subsequently the following partitions are obtained:

case

⎡ Xˆ ⎤ ⎡ 0 1 ⎤ ⎡ Xˆ ⎢ ⎥=⎢ ⎥⎢ ⎢ ˆ ⎥ 1 0 ⎦ ⎢⎣ 0 ⎣P⎦ ⎣

ρ ( t ) = ⎣⎡ν P ( t ) µ ( t ) ⎦⎤

T

⎡ Xˆ ⎤ ⎡ Xˆ − X ⎤ 0 ⎤ ⎡νˆ P ⎤ ⎥ ⎢ ⎥ − D ⎢ ⎥ − 2θ ⎢ ⎥ ˆ ˆ Xˆ ⎦⎥ ⎣ µˆ ⎦ ⎣⎢ P ⎦⎥ ⎣⎢ P − P ⎦⎥

⎡ ⎡ 0 1 ⎤ ⎡ Xˆ ⎡νˆP ⎤ 2 ⎢ ⎥ = −θ ⋅ ⎢ ⎢ ⎥⎢ ⎢⎣ ⎣1 0 ⎦ ⎢⎣ 0 ⎢⎣ µˆ ⎥⎦

( K a ,Kb ) ,(ξ a ,ξb ) ,( Fa ,Fb )

and

−1

0 ⎤ ⎤ ⎡ Xˆ − X ⎤ ⎥⎥ ⋅ ⎢ ⎥ Xˆ ⎥⎦ ⎥⎦ ⎢⎣ Pˆ − P ⎥⎦

(23)

b) A second estimator for the reaction rates. In this case

If we choose:

ρ ( t ) = ⎡⎣ϕ1 ( t ) ϕ 2 ( t ) ⎤⎦

T

and

H (ξ ) = 1 .

The

equations of the high-gain observer are:

⎡0 1 ⎤ Ka = ⎢ ⎥ , Kb = [ −1 / YP / S ⎣1 0 ⎦

−1 / YX / S ]

⎡ Xˆ ⎤ ⎡ 0 1 ⎤ ⎡ ϕˆ ⎤ ⎡ Xˆ ⎤ ⎡ Xˆ − X ⎤ 1 ⎢ ⎥=⎢ − D ⎢ ⎥ − 2θ ⎢ ⎥ ⎢ ⎥ ⎥ ˆ⎥ ˆ −P⎥ ⎢ Pˆ ⎥ ⎣1 0 ⎦ ⎣ϕˆ 2 ⎦ P P ⎢ ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 1 − ⎡ ϕˆ1 ⎤ ⎡ Xˆ − X ⎤ 2 ⎡0 1 ⎤ ⎢ ⎥ = −θ ⎢ ⎢ ⎥ ⎥ ⎣1 0 ⎦ ⎢⎣ Pˆ − P ⎥⎦ ⎢⎣ϕˆ2 ⎥⎦

then the next partitions are obtained: P ] , ξb = S , Fa = [ 0 0] , Fb = DSin (20) T

this

⎡X 0 ⎤ H (ξ ) = ⎢ ⎥ . The equations of the high-gain ⎣0 X⎦ observer are:

III.4. A High-gain Observer for the Unknown Kinetics

ξa = [ X

(22)

Remark 2. Note that ξâ is an “estimate” of ξ a , provided by the algorithm in order to be compared with the real state ( ξ a is measured or provided by a state observer), and the resulting error to be used in (22). For the alcoholic fermentation process, two high-gain estimators can be derived from (22): a) An estimator for the specific rates.

⎡X T = ⎡⎣ν P ( t ) µ ( t ) ⎤⎦ , and H (ξ ) = ⎢ ⎣0 b) An observer for the unknown kinetic rates: ρ ( t ) = T

)

The observer (22) provides on-line estimates ρˆ for the unknown kinetics; this observer is in fact a copy of the process model, with a corrective term. The observer is simple and the tuning of the gain can be done by modifying only one design parameter: θ .

In fact, as in previous subsection, two estimators can be implemented by using (18): a) An observer for the unknown specific rates. In this case,

(21)

ξb = Kb ⋅ H (ξ a ,ξb ) ⋅ ρ ( t ) − D ⋅ ξb + Fb

T

(24)

The high-gain observers (23), and (24) respectively

Then (5) can be written as: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


128


100

need the measurements of X and P. If X is not on-line measurable, then the estimates X est provided by the asymptotic observer (11) will be used in (23) and (24).

(g/l)

90 80

S

70

IV.

Discussions and Simulation Results

P

60 50

The behavior and the performance of proposed observers were tested using numerical simulations. The fed-batch fermentation bioprocess has been simulated for the next process parameters [10], [12]:

40 30 20

X

10

µ max = 0.54 h -1 , ν max = 2.1 h -1 , K S 2 = 5 g/l K I 2 = 201 g/l , K S1 = 9 g/l , K I 1 = 297 g/l Pm = 70 g/l , YX / S = 1.5 , YP / S = 0.43 Sin = 160 g/l , Fin = 2 l/h

0

0

5

10

15

Time (h) 20

25

30

35

40

Fig. 1. Time evolution of the substrate, biomass and product concentrations 8

The time evolution of state variables (i.e. the concentrations) was obtained by numerical integration of the basic dynamical model equations (2), (3). Fig. 1 depicts the profiles of these concentrations. The proposed estimators were simulated by using some realistic scenarios. All the algorithms used the estimates of the biomass concentration provided by the asymptotic state observer (11). Also, in order to test the robustness of estimators with respect to noise, the measurements of S and P are vitiated with an additive Gaussian noise (zero mean and 3% of the nominal values). Fig. 2 presents the evolution of the estimate of biomass concentration versus its “real” profile. It can be observed that the asymptotic observer provide good estimates, and the influence of the noise is quite low. The kinetic estimators were implemented and the simulations were performed considering that the specific rates and the reaction rates respectively are unknown. The “true” values of the specific rates (3) are used only for the simulation of measured data from the process. (i) First, the observer-based estimators (15) and (16) were implemented. The tuning parameters were set to ω1 = ω2 = 5,γ 1 = γ 2 = 3 . Fig. 3 shows the results obtained with the first estimator (the specific rates versus their estimates). In Fig. 4 the results for the second OBE are depicted: the reaction rates and their estimates. In all figures, the estimates are represented with dashed line, and the time profiles of “real” rates with solid line. (ii) Second, the linear regressive parameter estimator (18) was used to obtain the estimates of specific rates and the reaction rates, respectively. The tuning parameters are β = 2 , λ = 0.8 . Fig. 5 and Fig. 6 present the obtained results, i.e. the profiles of the specific rates and of the reaction rates, in the same conditions as in the case of OBEs. (iii) Finally, the high-gain observers (23) and (24) were implemented, with the single tuning parameter set to θ = 5 . Fig. 7 and Fig. 8 show the simulation results for the estimation of unknown specific rates and of the reaction rates, respectively.

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The obtained results illustrate that the OBEs and highgain observers provide accurate estimates of the kinetic rates. It can be seen that the measurement noise induces some noisy estimates of the kinetics, but the noise effect is limited (this effect can be reduced for lower values of the tuning parameters). 1.6

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The linear regressive estimator is not so accurate, but the effect of noisy measurements seems to be lower than in the case of observer-based estimators and high-gain observers. Another drawback of the linear regressive estimator is that a double number of equations need to be implemented (in comparison with the OBEs and highgain observers).

The advantage of the high-gain observer is that only one tuning parameter is needed. The estimation error can be made as small as wished if we choose greater values of θ . For example, Fig. 9 shows the simulation results for the estimation of specific rates with a high-gain observer, with θ = 10 , for free-noise measurements.



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The problem for a large value of θ is that the observer becomes noise sensitive. The value of the tuning parameter is therefore a compromise between a good estimation and the noise rejection.

V.

[4]

[5]

Conclusion

[6]

Several on-line estimation strategies for the imprecisely known kinetics of an alcoholic fermentation bioprocess were implemented. The bioprocess takes place inside a Fed-batch Bioreactor. Three kinetic parameter estimation algorithms were designed: an observer-based estimator, a linear regressive estimator and a high-gain observer. These estimators needed the measurements of state variables; therefore, the biomass concentration, which is not on-line available, was estimated by using an asymptotic observer. Two kinds of kinetics were estimated: the so-called specific rates (specific growth rate and specific production rate), and the reaction rates of bioprocess, respectively. Various simulations were performed in order to study the behavior and the performance of proposed estimators. The advantages of the observer-based estimator are the simplicity of design, the good convergence and stability properties, and the accuracy of estimates. On the other hand, the number of tuning parameters and the behavior for noisy measurements can be considered drawbacks of this strategy. The linear regressive estimator has a good behavior with respect to noisy measurements, but it is not so accurate like the OBE; also, the design and implementation of estimator are more difficult. The high-gain observer needs a single tuning parameter. The estimation results can be improved if this parameter is chosen higher in value, but only for free-noise measurements. Nevertheless, the simulation results need to be verified through extensive real-life experiments. In practice, the hardware implementation of the nonlinear observers consists in the discretization of the proposed algorithms, followed by a proper implementation of the discrete algorithms inside a microcontroller or a process computer. The proposed observers can be used for the design of advanced control strategies for alcoholic fermentation bioprocesses.

[7] [8] [9]

[10] [11]

[12]

[13] [14]

Authors’ information Department of Automatic Control, University of Craiova, A.I. Cuza 13, Craiova 200585, Romania. E-mails: {dansel, cmarin, epetre, dorins} @automation.ucv.ro Dan Selişteanu received his electrical engineering degree (1989) and the Ph.D. in automatic control (1999) from the University of Craiova, Romania. In 2005 he was awarded an AUF (Agence Universitaire de la Francophonie) Research Fellowship which he spent at HEUDIASYC CNRS, Compiègne, France. Since 1991 he is with the University of Craiova, where he is currently Professor in the Department of Automatic Control. He has been and currently is involved in some national and EU-funded projects in the field of automatic control, robotics and digital signal processing, with various European partners, such as SUPELEC, Gifsur-Yvette, France, Ecole Supérieure d’Ingénieurs en Electronique et Électrotechnique (ESIEE), Noisy le Grand, Paris, France. His present research interests are on adaptive and sliding-mode control of bioprocesses. He has published more than 90 journal and conference papers, and he is author or co-author of 4 books. Prof. Selişteanu is member of IEEE, SRAIT (Romanian Society of Automation and Technical Informatics) and of ARR (Romanian Robotics Association).

Acknowledgements This work was supported by CNCSIS–UEFISCSU, project PNII – IDEI ID786/2007, Romania.

References [1] [2] [3]

S. Kamoun, M. Kamoun, A new parametric estimation algorithm for large-scale systems described by state-space mathematical models, International Review of Automatic Control, Vol. 2(No. 1): 17-26, 2009. K. Schugerl, Progress in monitoring, modelling and control of bioprocesses during the last 20 years, Journal of Biotechnology, Vol. 85(No. 2):149-173, 2001. H.K. Khalil, High-gain observers in nonlinear feedback control, Int. Conference on Control, Automation & Systems ICCAS 2008, pp. xlvii-lvii, Seoul, Korea, 2008. M. Farza, K. Busawon, and H. Hammouri, Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors, Automatica, Vol. 34(No. 3):301-318, 1998. J.P. Gauthier, H. Hammouri, and S. Othman, A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. in Automatic Control, Vol. 37(No. 6):875-880, 1992. D. Selişteanu, E. Petre, C. Marin, D. Şendrescu, High-gain observers for estimation of kinetics in a nonlinear bioprocess,” ICROS-SICE Joint Conf., pp. 5236-5241, Fukuoka, Japan, August 2009. E. Petre, A nonlinear adaptive controller for a fed-batch fermentation process, Control Engineering and Applied Informatics, Vol. 7(No. 4):31-40, 2005. D. Selişteanu, E. Petre, M. Roman, C. Ionete, D. Popescu, Estimation strategies for kinetic parameters of an alcoholic fermentation bioprocess, SICE Annual Conference 2010, pp. 3560-3565, Taipei, Taiwan, August 2010. D. Selişteanu, E. Petre, M. Roman, D. Şendrescu, E. Bobaşu, Online estimation of states and kinetic rates in an alcoholic fermentation fed-batch bioprocess, Int. Carpathian Control Conference, pp. 509-512, Eger, Hungary, May 2010. D. Dochain, P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes (IWA Publishing, 2001). R. Marino, P. Tomei, Nonlinear Control Design (Prentice-Hall, 1995).

G. Bastin, D. Dochain, On-line Estimation and Adaptive Control of Bioreactors (Elsevier, 1990). D. Dochain (Ed.), Automatic Control of Bioprocesses (ISTE Publ. and Wiley & Sons, 2008). R.B. Messaoud, N. Zanzouri, M. Ksouri, Robust state observers for nonlinear systems, International Review of Automatic Control, Vol. 3(No. 5): 452-456, 2010.



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Constantin Marin received the B.S. and M.Sc. degrees (1965), both in automatic control, and the Ph.D. degree in control systems (1977) from the Politehnica University of Bucharest, Romania. Since 1966 he is with the University of Craiova, where he is currently Professor in the Department of Automatic Control. He is involved in national and international research projects in the field of identification and control. His present research interests are on identification of nonlinear continuous systems. He has published more than 100 journal and conference papers, and he is author or co-author of 10 books. Prof. Marin is member of IEEE, SRAIT and of ARR.

Dorin Şendrescu received his B.S. and M.Sc. degrees (1998) in automatic control, and the Ph.D. degree in control systems (2007) from the University of Craiova, Romania. In 2005 he was a Marie Curie Fellow at Université de Technologie de Compiègne, France. Since 1998 he is with the University of Craiova, where he is currently Associate Professor in the Department of Automatic Control. He has been and currently is involved in national and international projects in the field of automatic control and robotics. His present research interests are on identification of nonlinear systems. He has published more than 60 journal and conference papers, and he is author or co-author of 3 books. Dr. Şendrescu is member of IEEE, SRAIT and of ARR.

Emil Petre received the B.S. and M.Sc. degrees (1977), both in automatic control, and the Ph.D. degree in control systems (1997) from the University of Craiova, Romania. Since 1981 he is with the University of Craiova, where he is currently Professor in the Department of Automatic Control. He is involved in national and international research projects in the field of modelling, identification and control of bioprocesses. His present research interests are on real-time systems and adaptive control of bioprocesses. He has published more than 100 journal and conference papers, and he is author or co-author of 5 books. Prof. Petre is member of IEEE, SRAIT and of ARR.



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Technology Advances Open Up New Possibilities in Industrial Process Management Mikko Huovinen Abstract – Modern industrial processes are increasingly complex and managed with limited personnel resources. In order to ease the task of process management new networked communication and monitoring technologies have been developed and their implementation in production plants is increasing in the near future. The objective of this paper is to discuss the current state of process automation in regard to process monitoring and to review some of the related methods. The favorable features of hybrid monitoring systems consisting of several methods complementing each other are highlighted using examples. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Automation, Distributed Fault Detection, Fault Detection, Monitoring, Process Automation

I.

For example emission of pollutants in a combustion process can increase drastically if an optimal operating point is crossed. Thus in such cases optimization requires that the process is operated strictly in the designated operating areas and proper functioning of process devices can be assured by early failure detection. In a production environment failure detection is often achieved by operators continuously monitoring the operation of their system and by observing sensor data generated by that system [3]. However this approach requires additional resources which are typically very limited to begin with. To make matters worse often the information needed to manage the processes is based on years of experience and the know-how is only available in the minds of operators and process engineers which is problematic in many ways. For example training and breaking-in of new operators can be difficult and the knowledge may be lost when the operator changes employment or retires. There is also the problem with large volumes of sensor and operational data resulting in a cognitive overload for the operator, increasing the likelihood of the operator reaching an incorrect decision. Furthermore direct observation of operations and available sensor data may not be sufficient to detect incipient failure in the first place. Advanced modeling tools are particularly beneficial in enabling detection and prediction of failures by discovering ways that can indicate a future failure [3]. The need for new management tools and practices is emphasized also by the changes in the basic set up of the whole delivery chain. Products are no longer manufactured into stock but just in time to fulfill orders. This is more efficient in terms of bound capital but it also makes the whole delivery chain more sensitive to disturbances as there is limited buffer for disturbance

Introduction

Tightening competition in the process industries is forcing the companies to constantly look for improvements in competitiveness. As a result the manufacturers have decreased the number of personnel operating and maintaining their processes while at the same time the requirements for higher production rates and quality have increased. Modern industrial processes are also increasingly challenging to manage, e.g. the interactions are typically strong and the amount of buffer capacity small due to reduced inventory and use of recycle streams and heat integration [1]. In other words increasingly large and complex production lines are managed by fewer personnel. Therefore providing the human operators and maintenance personnel assistance is an increasingly current topic and the people in the process industries view the automation of Abnormal Event Management as the next major milestone in control systems research and application [2]. Industrial statistics have shown that even though major catastrophes and disasters from chemical plant failures may be infrequent, minor accidents are very common, occurring on a day to day basis [2]. Early detection of such emerging problems is helpful for process management and may provide invaluable information with which accidents and serious process upsets can be avoided. This is increasingly important as tighter efficiency and quality demands often require that the processes are operated nearer to their optimal operating points which can be risky in a sense that the optimal operating point is often near to hazardous operating areas.


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attenuation. Another effect of the changing marketplace is the automation and device suppliers’ shift of focus from traditional delivery projects towards life cycle services which is an increasingly important part of modern business models. Life cycle management includes among other things design, start-ups, support services and maintenance but also continuous optimization and process development throughout the product life cycle. Whereas traditionally after delivery of a project the end-users have been responsible for operating, maintaining and optimizing the equipment mostly by themselves, nowadays various activities are outsourced and partners are increasingly involved in dayto-day operations of the plant [4],[5]. Outsourcing of factory services, the resulting increase in the number of involved organizations and the introduction of enterprise wide management systems set new requirements for the availability and quality of data to support decision making. The data should be available on-demand and refined to a level that suits the varying needs of the end users. This places new requirements to not only the algorithms producing the data, but also to the systems responsible for storing and disseminating the data to the end users. Various information systems, such as automation systems and MES’s (Manufacturing Execution Systems), need to be properly integrated to ensure the availability of the data also for remote users. Thus it is becoming obvious that the traditional methods and tools of process management are no longer adequate and there are big challenges to be met. On the other hand new technologies open up new possibilities to respond to the challenges. As a result of the new technologies process industries and process equipment suppliers are increasing the intelligence level of process equipment and management tools. Applications such as fault detection, fault diagnostic and performance monitoring systems are used to support process and life cycle management. These are useful for both the equipment suppliers and the end-users [4], [6], [7].

II.

manufacturing which makes thing even more challenging. The monitoring solutions have to meet many requirements in this challenging environment, e.g. the number of missed and false alarms should be minimal. On one hand the application should in an optimal case always detect serious problem situations. On the other hand the alarms created by the application should always represent real problem situations as high number of false alarms decreases the efficiency of operating personnel and generally creates mistrust towards the application. The interface is also an important part as the key to the user acceptance, and therefore must be intuitive and easy to use. A successful interface provides a summary of problem areas that may exist in the plant, as well as a detailed presentation of the data collected and the analysis done. Thus the level of information should be controllable; in a steady situation abstract reports are often enough, unlike in an abnormal situation, where specific details may be needed. Along with intuitive operation user configurability is also a desired feature [8], [9]. Besides the application itself, the success of any industrial monitoring system is also dependent on the skill and commitment level of the personnel utilizing the application. Therefore it is important to clearly establish whose responsibility it is to learn and systematically use the monitoring system, to train the personnel, to establish the work practices and to make sure the personnel is committed to the tasks. These issues are often left with little attention and can be surprisingly difficult e.g. due to human resistance to change and sticking to old work routines. Unfortunately process monitoring systems are often seen as only auxiliary systems by the potential customers as they are not totally necessary for plant operation. This is accentuated by the typical overemphasis on steadystate situations, while it is the abnormal situations where support is mostly needed. Devices and control systems as well as their monitoring have been built for continuous steady usage and their operation has been optimized for a steady state. Not enough attention has been focused on operations in change situations in terms of monitoring. These situations have been traditionally treated as special cases handled with special attention. However, this is not feasible anymore, as systems have become larger and more complex, and there are not enough personnel available on-site to handle the change situations [9], [10]. Also, although the long term benefits would far outweigh the initial investment costs, the potential customers might not be convinced to the investment due to general lack of documented references. In general there seems to be little articulation in the literature about the benefits that can be accumulated through deployment of monitoring/diagnostic systems which hinders the development work and makes it harder to get the industry to commit to research and development projects. There are some general guidelines based on experience

Challenges and New Possibilities

The process industries are a very challenging environment for monitoring applications. One of the major problems is maintaining a feasible cost structure in a very complex and dynamic environment with large process entities. The individual processes are typically highly non-linear and interconnected with strong process interactions. To further complicate things each plant and process is typically individual in a sense that extensive customization work is usually needed when a monitoring application is deployed. The processes are also inherently dynamic in a sense that their behavior can change according to various factors such as ambient conditions (e.g. weather) and equipment wear. Significant control or production line modifications and/or retrofits are not that uncommon in modern production either. The production lines are also often capable of multiproduct



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on the economic impact due to abnormal situations, but there are no case studies that analyze specific benefits achievable through implementation of diagnostic systems. More research is needed on this issue in line with the work that has been carried out analyzing the benefits of advanced control systems [11]. Although there seems to be many factors against successful development of monitoring systems there are also factors promoting it. Recent technological advances are improving the possibilities in many respects. E.g. new sensors and monitoring methods are constantly being developed offering new kind measurement information. Fast development of microprocessor technology and parallel computing power has greatly expedited the computation speed and as a result many off-line algorithms can now be executed in an on-line mode [12]. Advances in electronics, communications and software techniques have had a favorable effect enabling e.g. relatively cheap storing and manipulation of large data files. Nowadays plants are often equipped with large process databases collecting and storing long periods of process data and the exploitation of process databases is a critical component in successful operation of industrial processes [13]. Cheaper and more powerful electronic components have enabled embedding of processors in individual field devices and thus resulted in so called intelligent field devices. This development has further promoted the use of decentralized periphery in which the intelligent devices can provide more detailed and new kind of information about the processes and individual devices. Thus it can be argued that intelligent field devices offer a new platform for embedded and more detailed diagnosis systems. The benefits of decentralized periphery include e.g. easier management of the automation entity (division into smaller parts) and more accurate control (measurements can be manipulated and control signals computed locally). Distributed functionality offers higher flexibility with the plant design and extensions of a plant by the use of independent and reusable modules. The independence of the modules enables component testing before commissioning and thereby accelerates the integration process [14]. In a broader sense decentralization offers several other potential benefits. For example some applications can be better solved using a distributed solution approach - especially tasks that are inherently distributed in space, time, or functionality. Also, if a system is solving subproblems in parallel, the overall task completion time is potentially reduced. In general, any system consisting of multiple, sometimes redundant entities, offers possibilities of increasing the robustness and reliability of the solution due to the ability of one entity to take over from a failing entity. Also, for many applications multiple, more specialized entities capable of sharing the workload offers the possibility of reducing the complexity of the individual entities, while in contrast, creating a

monolithic entity that can address all aspects of a problem can be very expensive and complex [15]. Agent technologies are one of the potential development directions enabled by increased decentralization. Agents are defined as decentralized, individual either physical or purely software entities with a powerful autonomous character, which enables them to make independent decisions and deliberated planning concerning their actions based on the nature of the environment they are responsible of. They are also characterized as intelligent due to their flexibility and fast adaptability. Their actual power, however, comes with their social behavior and their ability to organize and coordinate themselves in a hierarchical society. The most suitable application areas of agents in process industry consist of tasks requiring cooperative distributed problem solving. Especially the local decision making capability could be used for particular automation functions, such as self-diagnostics of intelligent devices, which could lead to more accurate analysis than with external diagnosis [16]. It has also been claimed that if data and resources are distributed, or if a number of legacy systems must be made to work together, agents are an appropriate technology for building such a system [17]. Although automated monitoring and diagnostics functionalities usually don’t directly intervene with control, and therefore are not generally restricted by the reliability limitations of control operations, the applications suffer from other limitations. The limitations of current Process Automation Systems lies in the inability to support complicated negotiations between distributed entities. Furthermore there is limited support for abnormal situation handling, e.g. device faults. Another problem is the lack of flexible connections between different automation systems and computational entities as they are mostly fixed, stable and defined at design-time [16]. With respect to supporting process automation functionalities, monitoring systems have been suffering from decisions that were made in the past, e.g. alarm handling and processing is based on a rather simple approach developed when the first digital control systems were introduced. The strong emphasis on reliability and performance requirements in automation systems has resulted in systems that, especially in monitoring operations, are not sufficient any more with larger and more complex processes. Fortunately with digital communication and almost fully software-based operation there are no big barriers any more to what functionalities may be developed and thus offered to users. This will result in a whole new set of functionalities and services for users working with process automation. New software technologies have been taken into use in the implementation of automation systems which are becoming increasingly distributed, networked and complex software systems [9], [10]. The new technologies in this respect include e.g. recently developed integration techniques, such as OPC



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UA, SOA and FDT, which have enabled unification of previously separate entities and process assets into more uniform and easier-to-manage entities. Integration of basic office products and process information systems is relatively easy and by using OPC process data can be presented e.g. in Excel sheet and stored in process history databases. With the new technologies the information from production sites is easier to access and can be utilized in many ways. For example life cycle management of process equipment will be easier as information from the processes and individual devices is readily available to the device suppliers and it can be utilized in many ways in tasks such as commissioning, life cycle costing, maintenance, optimization and product development [4]. Advances have also been made in the area of interoperable data formats and the use of additional metadata describing the actual data. This along with the new communication techniques opens up new possibilities in the data exchange between various information systems and by so enables more efficient exploitation of available data. In this respect XML seems to be the most prominent technique. The advances in networking, sensor and integration technologies have resulted to a range of Internet-based services provided by the process device manufacturers. Forward-looking manufacturers have started to develop network capabilities and integrate Internet-based services with their products, transforming them into platforms for service delivery. As a result of the new technologies, closer customer relationships will be formed throughout the product’s lifetime. “Service” usually means “repair” or “maintenance” today, but might mean something quite different with the new opportunities presented. The device suppliers need to consider how factories will be run in the future and the implications of this fundamental transformation to their business models. Machinery and equipment manufacturers need to change their existing business models and internal structure to deal with the new relationship with the customer. This growth in the importance and value of customer relationship is prompting a need to form partnerships with Information Technology or Internet services suppliers, application service providers and other industries. As a result, manufacturers need to change their concept of Customer Relationship Management from one that effectively ends with the sale to one that promotes an ongoing involvement with the customer and builds on continued service to develop valuable relationships. The manufacturers will also need to re-evaluate their corporate governance, decision-making control and supply chain strategies in this new environment given the increased high-technology content in their products. By taking advantage of the developing technologies new services can be created and product differentiation achieved [4]. Although the discussed technological advances offer several new possibilities, it should be also noted that they also pose new problems. E.g. the negative effects of

distributed intelligence can be said to include more difficult and complex management of the complete system (due to the lack of centralized control or of a centralized repository of global information), requirement for more communication to coordinate all the entities in the system, increased interference between entities and increased uncertainty about the state of the system as a whole [15]. The increased orientation on software development and communication also exposes the systems to problems such as software platform issues, IT-security in general and software versioning and updates.

III. Process Monitoring Overview There are several methods developed for process monitoring but the procedure is often the same. The first phases are fault detection and fault identification. The last parts are fault diagnosis and process recovery. Fault detection means determining if a fault has occurred. Identification means identifying the most relevant variables for diagnostic purposes. Diagnostic part determines which fault has occurred, i.e. the reason for abnormal observations. In process recovery the fault effects are removed. Not all these have to be implemented or automated. For example diagnostic part can be left for the operators [18]. Monitoring/diagnostic algorithms are often classified into three general categories which are quantitative model based methods, qualitative model based methods, and process history based methods. In model based methods (eg. [19], [20]) the model is usually developed based on some fundamental understanding of the physics of the process. In quantitative models this understanding is expressed in terms of mathematical functional relationships between the inputs and outputs of the system. In contrast, in qualitative model equations these relationships are expressed in terms of qualitative functions. Process history based methods (eg. [21], [22], [23]) on the other hand don’t assume a priori knowledge about the model, instead (theoretically) only the availability of large amount of historical process data is assumed. Various methods are then used to transform the data and present it as a priori knowledge to a diagnostic system [2], [18]. It should be noted that no single method is necessarily adequate to handle all the requirements for a monitoring/diagnostic system. Though all the methods are only as good as the quality of information provided to them, some methods might better suit the knowledge available than others. Some of the methods can complement one another resulting in better systems overall. Integrating complementary features is one way to develop hybrid methods that could overcome the limitations of individual solution strategies. Hence, hybrid approaches where different methods work in conjunction to solve parts of the problem are seen as attractive. For example, fault explanation through a



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causal chain might be done through the use of digraphs while fault isolation might be very difficult using digraphs due to the qualitative ambiguity. Analytical model-based methods might be superior in this respect and hence, combined hybrid methods might provide a powerful problem-solving platform [11]. Besides the division based on the used monitoring methods, monitoring applications can also be examined from the perspective of the complexity and size of the monitored process. The challenges of monitoring are clearly different when comparing monitoring of e.g. a single control loop and larger process entities. The division between small and large scale processes is of course vague but the differences in the challenge are significant. Monitoring of simple single-input-singleoutput (SISO) control loops is well established in the process industries but the SISO approach has shortcomings as the control loops are typically interconnected. The basic idea of process control is to divert unwanted variability from key process variables into places capable of accommodating the variability such as buffer tanks. Unfortunately this is not often accomplished and the disturbance appears somewhere else. This is increasingly the case in modern industrial processes which have strong process interactions due to characteristics such as low buffer capacity and increased use of recycle streams and heat integration [1]. A plant-wide approach means that the distribution of a disturbance can be directly mapped out and the location and nature of the cause of the disturbance determined. The alternative is a time consuming procedure of testing each control loop in turn until the root cause is found. Some key requirements are detection of the presence of one or more periodic oscillations, detection of nonperiodic disturbances and plant upsets and also determination of the locations of the various oscillations/disturbances in the plant and their most likely root causes [24], [25]. An interesting possibility to relieve the problems of plant-wide approach is to capture and make automated use of available process information (i.e. engineering data) resulting in a qualitative model. It is believed that qualitative models of processes will in the future become almost as readily available as the historical data. The technology for generating such models is already in place in Computer Aided Engineering tools such as ComosPT (Innotec) and Intools (Intergraph) [1]. XML has recently started to be used in such intelligent PI&D (Piping and Instrumentation Diagram) tools for export and exchange of process drawings and PI&Ds. This development offers new possibilities for qualitative modeling and large scale monitoring as the plant topology in a process diagram can now be exported into a vendor independent XML-based data format, giving a portable text file that describes all relevant items, their properties, the connections between them and the directions of those connections [26].

Besides the aforementioned methods, there are also more traditional methods using specific health assessment measurements, such as vibration and acoustic emission analysis. These are often associated with prognostics (ability to computationally predict future condition, i.e. the time between very early detection of incipient faults and progression to actual system or component failure states). One way to explore the relationship between predictive prognostics and diagnostics is to envision an initial fault to failure progression timeline with three points; incipient fault, failure and catastrophic failure. Component maintenance will often be delayed until the early incipient fault progresses to a more severe state but before an actual failure event occurs in order to maximize the benefits of continued operational life of a system. This area between very early detection of incipient faults and progression to actual system or component failure states is the realm of prognostics while diagnostic capabilities have traditionally been applied at or between the initial detection of a failure and complete system catastrophic failure [27]. The approaches to prognostics include e.g. experience-based prognostics (e.g. fitting statistical distributions to data from legacy systems and using this information for future prediction), evolutionary prognostics (gauging the proximity and rate of current condition to known degradation or faults), feature progression and artificial intelligence based prognostics (using artificial intelligence, e.g. neural networks, to predict fault or feature progression), state estimator prognostics (using tracking filters, e.g. Kalman filters, to extract states and using state transition equations to predict their future values) and physics based prognostics (a technically comprehensive modeling approach) [28]. Even though recent diagnostics technologies are enabling detections to be made at much earlier incipient fault stages than with more traditional diagnostic systems and some prognostic applications have been reported especially in the field of aviation, the related technology and theory seem to not be mature enough for large scale deployment in process industry where the challenges are quite different. Although component/parts prognostic techniques have been developed and investigated extensively, the techniques such as neural networks, time/stress measurement devices, vibration monitoring, etc. are by their very nature equipment-specific and very few applications have been implemented. Moreover, they are primarily point solutions and are often too expensive and risky for system application [27], [29]. Closely related area to process monitoring/diagnostics is the concept of fault tolerant control. In contrast to process disturbances and modeling uncertainties, process faults are more severe changes whose effect on the plant cannot usually be suppressed by a fixed controller. Fault tolerant control aims at changing the control law so as to cancel or minimize the effects of faults. Actions that try to keep the system in operation with failed equipment can be classified as fault accommodation and control



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reconfiguration. The previous deals with the autonomous adaptation of the controller parameters to faulty plant behavior while the latter includes the selection of a new control configuration and the online re-design of the controller. The first task of fault tolerant control is to detect and identify existing faults, fault locations and magnitudes by using the diagnosis functions. Reliable fault isolation and identification are essential, especially in safety-critical systems such as aeroplanes, where the goal is to incorporate “self-repairing” capabilities. This is in contrast to safety systems in process automation where the information about the existence of a fault is often sufficient [30], [31]. Nevertheless, if the controller adaptation is to be made autonomous, the importance of diagnosis reliability cannot be overlooked in any environment. Despite the attractive features FTC has not been deployed in large numbers in the process industries. It is probably the lack of affordable and reliable enough online diagnostics that is the limiting factor in deployment of FTC applications in the process industry. The situation is less complicated in application areas where there are a number of identical processes for whom it is reasonable to construct extensive fault tolerance and to duplicate the application. In comparison the uniqueness of processes in the process industry limits the use of FTC to critical processes where faults are relatively common and continued operation of faulty systems results in significant savings. On the other hand typically slow dynamics in process industry allow FTC more response time to activate when compared to e.g. aviation. Reasons for limited numbers of FTC implementations in the process industry can also be found in the research itself. The research efforts have been primarily focused on fault detection/diagnostic schemes and control reconfiguration mechanisms individually. In comparison relatively little work has been done to integrate the developed techniques from these areas to form an effective active fault-tolerant control system [31]. The increasing use of MPC (Model Predictive Control) systems may offer new possibilities as more detailed system models that can be used also for diagnosis purposes become available.

IV.

readily available engineering data. Similarly, process data can be acquired and used in a modeling tool to identify other process parameters. This way it will be relatively easy to develop generic frameworks for modeling of the most common process units. For example power plants have several common structures and features which enables the creation of generic model framework which simply describes what information and measurements are needed for the model creation. With the tools and metadata definitions the difficulties in creating detailed enough and extensive enough process models with reasonable costs can be relieved and process monitoring can take a major leap towards wider commercial exploitation. However, currently the availability of design data on a unified XML-platform is still scarce and thus other methods for assisting model creation are still needed. The Disturbance Propagation Path Identification (DPPI) method introduced in [32] is an interesting possibility. The basic idea is to use time series measurement data from industrial operations to deduce the causal directions and process delays between the process variables. A cause-and-effect matrix can be obtained by a simple pairwise hypothesis that any pair of measured variables could have a causal relationship and with some assumptions about the structure of the system, the matrix leads to a causal graph. The algorithm uses crosscorrelation to create the pair-wise hypotheses. The result for each pair includes (i) a direction (whether X precedes Y or Y precedes X), (ii) an estimate of the delay, and (iii) the strength of the delayed correlation. For detecting the cause-and-effect relations the cross correlation method was chosen over other methods (e.g. transfer entropy) due to some favorable features such as intuitive use [32]. In the first phase basic cross correlation routines are used to identify significant variable correlations and the delay between them. After the initial correlation analysis various routines test the significance of the calculated correlations to omit false results. As the cross correlation analysis produces minimum and maximum correlation results, the directionality must be confirmed by checking that one has a significantly larger magnitude than the other. An ambiguous result is obtained with similar magnitudes and no decision can be made. A directionality index which measures the difference between minimum and maximum correlation is calculated for this purpose. Secondly, the statistical significance of the results must be checked. These significance tests are based on the null hypothesis that the tested data sets are unrelated random sequences. The threshold for correlation significance is dependent on the amount of available data samples N and is determined from the probability distribution function of correlations between random sequences with sample size N. The threshold for directionality index is also dependent on the amount of available data samples. After determination of process interactions and delays a corresponding causality matrix showing the detected

Cases

IV.1. Model Creation utilizing Process Data A generic example of the benefits of modern information systems and data integration is the use of process engineering data along with time series process data to create more efficient process monitoring tools. Engineering data such as process connections and process parameters can now be directly imported from engineering tools to a modeling/simulation environment as long as a common data format is used. XMPlant offers such a platform and is currently used by several major process plant vendors. By using such techniques the basic structure for a process model and some process parameters can easily be obtained directly from the Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


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time delays is constructed. The entry in the second column of the first row, for example, is the estimated time delay from variable 1 to variable 2. The matrix is then arranged in a way that the number of entries above the main diagonal is at its maximum. The effect of the rearrangement is that the variable ending up in the first row is the candidate for the root cause as all other variables have time delays relative to that one. The rearrangement algorithm was presented in [33]. The causality matrix then has the following form: ⎡ − λ1,2 ⎢ − Λ=⎢ ⎢ ⎢ ⎢⎣

...

improves the model significantly. The user interface allows the user to select proper criteria and to determine the automation level of the identification process, i.e. how much user interaction is desired. The removal of a candidate variable is a similar process but naturally the criterion for removal is different from the addition. The basic stepwise linear regression algorithm was modified to include process delays between the variables and thus making better use of the underlying process dynamics. While the stepwise regression algorithm produces a linear regression model of the process by itself, the final process model can, however, be any other model type. The main task of the combination of the DPPI algorithm and stepwise linear regression is thus to perform the task of selecting appropriate input variables but also identifying some of the underlying process dynamics. It must be noted that the cross correlation and LR-methods are linear and therefore may lead to inappropriate results in a highly nonlinear processes if it is not taken into account. Piecewise analysis or nonlinear methods, such as the transfer entropy, may be considered in such cases. The developed tool is intended to be used for creating a number of soft sensors in a power plant environment to enable efficient monitoring of measurement accuracy through the Reconciliation and Rectification (RR) method [34]. The RR method is based on simple but very intuitive measurement balance calculations (e.g. in the long run the inflow of a process part must equal the outflow). The method relies on utilizing several interrelated measurements to create a number of balance equations and the method’s performance improves with the increase of related measurement positions, i.e. measurement redundancy. Thus, the case is ideal for using the presented tool that assists the creation of a number of soft sensors by utilizing process data and variable interactions. As the name suggests, the RR method not only checks if the balance equations hold, but it also calculates rectified estimates for the measurements which can then be used to replace the measurements most likely to be producing incorrect results. Furthermore, in [35] it was documented how this information was successfully used to determine which measurement instruments were in need of maintenance when there were contradictions between 14 measurements in a balance equation. The method is particularly useful for measurement performance monitoring as usually the problem is not how to detect malfunctioning measurements but to detect and rectify measurements which are affected by incipient and slowly developing problems. A good example of such situation is measurement drift due to instrument soiling and consequent measurement bias. Such situations can often go unnoticed for long periods of time which can cause non-optimal performance and even machine breaks. For example in one case a 30% measurement error was detected in a water treatment plant which had caused a 30% increase in the use of

λ1,p ⎤

⎥ ... ⎥ ... λ p −1,p ⎥ ⎥ − ⎥⎦ ...

where λm,n represents the time delay from variable xm to xn. A time delay matrix with positive entries remaining below the main diagonal indicates that the process has a recycle. The consistency of the matrix can be checked by acknowledging that the delay in row m, column q can be expressed as the sum of the already estimated time delays in row m, column n and row n, column q. The user can define a tolerance parameter for the consistency check as the delay estimates are not precise. A qualitative signed digraph (SDG) process model with the interaction signs and delays can then be constructed with relative ease using the causality matrix. Such models are useful e.g. for diagnostics, training and decision support purposes. The basic idea of the DPPI algorithm was utilized and further developed in a research project where the aim was to develop easy-to-deploy monitoring models to difficult process environments with plenty of variables. A tool was developed whose idea was to create simple qualitative process models utilizing process data and then utilize this refined information in creation of more detailed quantitative models. Essentially the tool identifies correlations and delays to create qualitative models which in turn are used to create soft sensors. Potential predictor variables of the model can be identified by studying a causality matrix column corresponding to a variable that needs to be modeled. Thus the DPPI algorithm can be used as an input candidate and corresponding delay selector for modeling purposes while stepwise linear regression is used for the final predictor variable selection. The basic idea of the stepwise linear regression algorithm is adding and removing candidate variables to and from the initial model on a trial-and-error basis. The choice of an additional variable candidate is based on inspecting the correlation of the model error with the part of the candidate variable that does not correlate with the variables already in the model. An F-distribution test is then conducted along with potentially other test indicators, such as R2 index or estimate error variance index, to decide if the addition of the candidate variable Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


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chemicals resulting in annual losses of hundreds of thousands of euros. In another case the efficiency of an oil refinery process was estimated to be too good due to a faulty flow measurement which resulted in an investment on a similar process based on a false investment payback time estimate [35]. Thus it is obvious that the importance of accurate measurements in critical areas is of utmost importance and should be verified systematically and with sufficient time periods. Currently the RR method is being used in manual offline-mode for periodically checking measurement accuracy in the pilot plant but further developments are being made to modify it to function as an online application. However a problem arises from the online use of the RR algorithm as the execution of the algorithm requires process data from relatively long time periods and the application is located in a separate PC in a separate network. Thus the execution momentarily increases the workload of the process database server and the related networks. This is fine for the offline use, but the online version might increase the workload to an unacceptable level. The solution was found in the integration of basic office products and the process database system. The process data required in the previous RR execution is stored in an MSAccess database (commonly found in PCs with the Office software package) which is used as a sort of cache memory. This way when the RR method is executed the next time, it is only necessary to request and store data from the period between the last execution and the current execution time, thus significantly reducing the workload of the server and the networks. Besides the improved fault detection, the online version will also utilize integration techniques to make the rectified process measurements available to the automation system and the operators, and therefore enables their utilization in process control and operation.

decentralized diagnostic system might in this case be agents representing the valve, the control loop, the particular subprocess and a supervisory agent. These can also be physically distributed; e.g. the valve agents can be executed locally in an intelligent positioner.

Fig. 1. Hierarchical monitoring system

Fig. 2. Control loop measurement and performance data

IV.2. Distributed Hierarchical Diagnostics System

In this case the control loop agent utilizing a CLPA tool is the first to detect the abnormal situation and therefore initiates the task of problem solving. Control loop level data is presented in Fig. 2 where the four performance indexes represent the amount of control activity (CTI), oscillation (OI) and control error (VI and IAE). The subprocess level agent informs that the situation has no or little effect on the overall performance of the subprocess as the controlled temperature remains inside the set limits and no other deviations are observed. However solving the matter is preferred considering the lifecycle wear of the valve. Also in some cases oscillation around the setpoint is worse than small static control error. Both the control loop and valve agents notice that the operating area of the valve has moved to an unfavorable area near closing the valve completely where the valve’s control performance is weak. This suggests that the valve is oversized for this particular operating point. However,

The second example features a distributed hierarchical diagnostic solution. The case is partly speculative although it is based on a real research project conducted at a paper mill’s stock preparation process. The research focused on the observability of various events in a largescale hierarchical monitoring system and the possibilities of utilizing intelligent field devices as a part of it. Besides the intelligent field devices the system also includes the control loop level, subprocess level and the plant level monitoring modules. A simplified structure of the hierarchical monitoring system is presented in Fig. 1. In the example case a decrease in control loop performance is detected utilizing a Control Loop Performance Assessment (CLPA) tool, Fig. 2. This particular loop is a temperature control loop manipulating the flow of a cooling fluid and its characteristics are affected by ambient conditions such as fluid temperature. The potential participants in a



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if the valve has operated satisfactorily in the past, also some relatively small change could cause the situation due to the increased disturbance sensitivity in this operating point. Thus the performance would not be affected in other operating points and the diagnostic system might not be able to detect it until the process is in the operating point in question. As the disturbance sensitivity of the control increases in the low end of the valve’s control area, the diagnostic parameters might be adaptively retuned to accommodate the situation. For example if the stiction of the valve has a slight increase, it might cause the performance degradation in this particular operating point even though it might not affect the performance at all in other operating points. Therefore once problems have been noticed in the low end of the valve’s control area, a new valve diagnosis check using stricter alarm limits might be requested by the control loop agent or the valve agent might initiate it autonomously. If the new diagnosis indicates a problem with increased stiction, a supervisory agent can authorize or initiate and supervise a parameter change to adapt to the new situation. For example a PI-controller might be changed to P-only controller with which the oscillation should disappear in a non-integrating process for a shortterm solution. Naturally a more permanent fix should be ordered and satisfactory plant-wide performance assured in the mean time. The subprocess or supervisory agent might also negotiate with other agents for alternative ways to manipulate the controlled variable. If such alternatives are known, the control scheme can be changed to accommodate the new situation. The agents can utilize any type of monitoring/diagnostic method depending on the case, but currently the positioner diagnostics are limited by the available power transmitted through the fieldbuses and other resource limitations. Therefore the embedded diagnostics are implemented using fairly simple statistical methods but using generic first principle valve models is one of the future possibilities. The CLPA tool in this case is a statistical method utilizing algorithms presented in [36]. The higher level diagnostics modules must in practice be based on more detailed quantitative process models or process knowledge to assure proper results for the diagnostics and control modifications. Qualitative topology models might also be utilized on the highest coordinating level to manage control modifications on the plant-wide level. For more deterministic solution the agents could be replaced with more traditional monitoring modules and an expert system containing the necessary information to deal with abnormal situations. In the presented case distributed problem solving will result in more precise diagnostics and more adaptable control than with traditional methods. The benefits of using adaptive diagnostic parameters are apparent but there are also other benefits. First of all the embedded valve/positioner diagnostic system has access to more

precise diagnostics data than it is possible to transmit through fieldbuses, whose capacity is mainly reserved for control purposes. Also the fieldbus traffic is more controllable as less detailed information is sufficient for storage and analysis in normal situations while more detailed information can be transmitted on request. The manipulations in the control system are naturally a more sensitive area and must be executed with extreme care. To meet hard control requirements alternative controller configurations can be designed off-line for the most important faults but due to the typically large number of possible faults, some authors see that online re-design is more reasonable from a practical point of view. Also the new techniques enabling automatic creation of plant topology models promote this point of view. However, as in practice no method can guarantee a 100 % fault tolerance anyways, it can also be argued that fault tolerant control should start with the selection of the most critical faults and continue with the investigation of fault tolerance against these faults [30], [31] which seems reasonable as this approach increases deterministic features of the system. In respect to the changes in the process control the plant wide control perspective is also an important aspect and requires a priori knowledge as changes in one subprocess often have effects on other subprocesses.

V.

Conclusion

The technological advances have opened up new possibilities for monitoring and managing industrial processes. The research in the described areas is active and the numbers of actual industrial applications are increasing. Whereas the problem used to be how to collect needed process data, nowadays the problem is more focused on what to do with the abundance of process data. In fact, it may often be the case that the data overflow may even further complicate matters. It is therefore necessary to further develop methods for refining the data into more useful form and to set up efficient practices to systematically improving process management. It is clear that there is plenty of untapped potential in this area. For example various efficiency and other monitoring measurements are often only used for periodic reporting and maintenance purposes and not for the control of processes. This is unfortunate as such measurements may offer valuable information that would be very useful in improving process control. For example the RR method produces rectified process measurement which may be used for more accurate process control but at the moment is only used for maintenance purposes. Naturally the matter of manipulating variables and using them for process control is a sensitive area but if the reliability of the system can be reasonably assured, the benefits of the system would far outweigh the initial investment.



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[4]

Similarly, the potential in hierarchically distributed process monitoring is not fully exploited, which may be contributed to the increased nondeterministic behavior of such systems. However it is easy to think up situations where such systems could improve the overall performance optimization and fault tolerant control in a fault situation. For example, if the monitoring system is considered reliable enough, more stringent controller performance criteria can be used instead of using a static controller that is expected to perform over different operating regions. After detection and identification of problems in the process behavior some retuning may first be performed on-line. If this does not correct the problem the loop could be taken off line and a complete redesign performed. However the monitoring and correction of control loop problems can become a very time consuming task as the number of loops under an engineer’s supervision increases. This is where a supervisory control system (which falls in scope between regulatory control and planning) could be used to assist in the decision process. The information available from the fault diagnosis system would be used by the supervisory control system to check and monitor the loops in the regulatory control system. If changes are detected in the control loops, the supervisory control system would then look for different control configurations or set points that would improve the process operations [11]. Regardless of the used methods, extensive research and pilot cases in industrial environments are needed to ensure that the used methods are mature enough for industrial deployment. In this respect the use of techniques based on functions that can be understood and validated is often demanded. In practice this requirement often eliminates the possibility to use black-box methods such as neural networks. The requirements for diagnostics reliability are further increased if the system directly affects the process control and the control modifications are to be made online.

[5]

[6] [7] [8] [9]

[10]

[11]

[12] [13] [14]

[15] [16]

[17] [18] [19]

Acknowledgements [20]

This work has been funded by the Academy of Finland and the Finnish Funding Agency for Technology and Innovation.

[21]

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Authors’ information Department of Automation Science and Engineering, Tampere University of Technology. Mikko Huovinen received the M.Sc. (Tech) and the D.Sc. (Tech) degrees in automation engineering from Tampere University of Technology, Tampere, Finland, in 2005 and 2010, respectively. He has worked at the Department of Automation Science and Engineering, Tampere University of Technology, in the field of process automation and especially process monitoring since 2005. His current research interests include process monitoring, process control, modeling, simulation and analysis.



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