Robust Indirect Adaptive Fuzzy Decentralized Control

2011 International Conference on Computer Control and Automation (ICCCA 2011)

Robust Indirect Adaptive Fuzzy Decentralized Control for a Class of Nonlinear Interconnected Systems Ibrahim Fahad Jasim Electrical and Electronic Engineering Department College of Engineering, University of Kabala Karbala, Iraq e-mail: [email protected] consideration. An indirect adaptive fuzzy decentralized control scheme is presented that guarantee global system stability performance. The rest of the paper is organized as follows. Section 2 gives the problem statement. In section 3, fuzzy logic approximation is briefly described. The suggested control approach is explained in section 4, and section 5 gives the concluding remarks.

Abstract— The problem of controlling a large scale uncertain nonlinear system consisting of interconnected subsystems is addressed. The interconnections between different subsystems are assumed to be bounded. Fuzzy logic systems are used to approximate the unknown dynamics of different subsystems. Then, robust indirect adaptive fuzzy decentralized controllers are derived. Moreover, the uncertainty bounds resulted from approximation are tuned adaptively so that global system stability is guaranteed and all closed loop signals are assured to be bounded. Simulation results are performed to validate the approach.

II.

Suppose that we have a large- scale nonlinear system is composed of N subsystems, and each subsystem described by:

Keywords-- Adaptive Control, Decentralized Fuzzy Systems, Nonlinear Systems

I.

PROBLEM STATEMENT

INTRODUCTION

The justification behind the introduction of decentralized control was to attack the problems encountered when controlling interconnected systems. Those includes the complexity in designing centralized control methodologies, physical limitations in information exchange, and other reasons that may cause centralized control to be more complex. So, decentralized control would be more suitable for interconnected systems. However, the concept of adaptive control would be inevitable when parameters uncertainty is involved which motivated researchers to use adaptive control techniques in the decentralized control approach. Decentralized model reference adaptive control was successfully used and promising results were obtained [1- 3]. Even though those approaches are limited to linear subsystem with possible nonlinear interconnections, they presented a good basis for later researches. Decentralized adaptive control techniques were successful used to handle the control of robotic manipulators [4, 5], and this was the spur for later researches on decentralized adaptive control of nonlinear subsystems (see [6, 7] and the references therein). All the schemes above require the exact knowledge of the system dynamics, and this was the justification for using the fuzzy logic as a universal approximator for the case of unknown dynamics [8]. Decentralized adaptive fuzzy control approaches were suggested and promising results were obtained even if the dynamics of the subsystems is unknown (see [9- 11] and references therein). However, till now the uncertainty resulted in functional approximation, in the decentralized adaptive fuzzy control, is not yet addressed. In this paper, we present a new control scheme that takes the uncertainty resulted from the unknown functional approximation in decentralized adaptive fuzzy control into

978-1-4244-9767-6/11/$26.00 ©2011 IEEE

…

:

∆

,

,…,

(2.1)

Where , ,…, is the -dimensional and state vector, is the control input, are unknown but continuous functions, and ∆ are the interconnections among the , ,…, subsystems 1,2, … , . The objective of this paper is to design robust indirect adaptive control laws u such that global stability is guaranteed and all closed loop signals are assured to be bounded. However, the following assumptions are needed to be satisfied: are assumed to be bounded and A1. All references available for measurement. The references have up to 1 bounded and measurable derivatives. Let us define the output error of the ith subsystem to be: (2.2) Then we can easily find the error vector of the ith subsystem to be: ,

,…,

(2.3)

Now, consider the following assumption concerning the interconnection.

222


approximating any continuous function, i.e. for any real functions : in a compact set and any positive real constants there exist fuzzy logic systems such that [8]:

A2. Let the scalars quantify the strength of the interconnection between the ith and jth subsystem. Then it is assumed that: ,

∆

,…,

∑

(2.4)

Where . is the Euclidean vector norm and are positive constants. So, according to (2.4), the interconnections are assumed to be bounded and their bounds are known, say . III.

(3.6)

From (3.6), we can easily deduce that: (3.7)

FUZZY LOGIC APPROXIMATION

Knowing that satisfying:

and are assumed to be Since the functions unknown, then we use the concept of fuzzy logic approximation to estimate those functions [8]. For each functions of and , consider an -input and a single- output fuzzy logic system with the product- inference rule, single tone fuzzifier, center average defuzzifier, and Gaussian membership function given by fuzzy IF- THEN rules:

are the approximation errors of (3.8)

are positive constants and parameter vectors: arg min sup

…

:

|

|

(3.9)

Similarly,

(3.1)

(3.10)

Where denotes the rth rule of the ith plant function to , ,…, and are the be approximated, input and output of the ith fuzzy logic respectively, is the fuzzy single tone for the output of the rth rule- ith plant function, and

are the approximation errors of

are positive constants and parameter vectors:

(3.2)

arg min sup

where and are the center and width of membership function . The output of the fuzzy logic system can be described as: …

satisfying: (3.11)

are fuzzy sets characterized by Gaussian … membership functions:

…

are the optimal

(3.3)

are the optimal (3.12)

and are unknown, then we Since can start with some known fuzzy approximators and such that:

with

(3.13) ∏ ∑

∏

(3.14)

(3.4) Knowing that:

Or in matrix form:

(3.15) (3.5) (3.16)

is the parameter vector, and : where the fuzzy basis functions. The fuzzy rules described by (3.1) is said to be complete, if for any , there is ∏ at least one fuzzy rule to be fired, i.e., ∑ 0. It is well known that the fuzzy logic systems described by (3.5) are universal approximator and capable of

and are positive constants. The nominal where fuzzy logic system can be chosen as:

223


⁄

.

(3.17)

⁄

.

(3.18)

Where and (3.4).

and

(4.4) (4.5) (4.6)

are the parameters vectors, and are the regissor vectors described by .

⁄

IV.

⁄

.

ROBUST ADAPTIVE CONTROL DESIGN

(4.7)

To achieve the stated control objective, let us define the ith filtered tracking error corresponding to the ith subsystem:

.

⁄

⁄

.

(4.1) with 0 and rewritten as:

(4.8)

. Equation (4.1) can be

Where: 1

Λ

.

(4.2)

where:

.

Λ

,

1

⁄

.

, … ,1

(4.9)

Taking the time derivative for equation (4.2), we obtain: Λ

and

(4.3)

1

(4.10)

where: Λ

0,

,

1

,…,

achieve global system stability and guarantee the boundedness of all closed loop signals involved. , , , and are positive constants, and Where | | .

1

Equation (4.3) can be rewritten as: Λ

∆

,

Proof: Consider the Lyapunov candidate:

,…, (4.4)

∑

Note: It has been shown that (4.2) has the following properties: (i) the equation 0 defines a time- varying hyperplane in , on which the tracking error vectors decay exponentially to zero, (ii) if 0 0 and | | Ω with constant , then 2

,

1,2, …

| 0 0 and | within a time- constant

for

Taking the time derivative of (4.11), we obtain: .

0, and (iii) if

.

.

.

.

.

.

will converge to Ω , then [12, 13].

(4.12)

Substituting (4.4) into (4.12), we obtain:

Theorem: For the interconnected nonlinear systems given in (2.1) satisfying assumptions A1 and A2, the controllers given in (4.4) and the update laws given in (4.5, 4.6, 4.7, and 4.8): 1 ⁄

(4.11)

. Λ ∆

⁄

.

224

,

,…, .

.

.

(4.13)


Using the facts given in equations (3.7), (3.10), (3.13), and (3.14), then we can easily deduce that: ⁄

.

∑

(4.15)

Substituting the control law given in (4.4) and equations (4.14), (4.15), (3.17), and (3.18) into (4.13), we obtain:

.

Λ

.

∆

.

.

.

,

.

. (4.18)

From (4.18) it is clear that and , , , . Since , , , are , , , are also bounded. From bounded, then equation (4.4) (4.14) and (4.15), we can easily conclude that . Since and , then invoking to the Barbalat’s lemma, we can deduce that 0 as ∞. This would make converging to Ω as ∞.

.

.

∆.

. ∑

⁄

. Λ

(4.17)

Using the parameters updates laws given in equations (4.5), (4.6), (4.7), and (4.8) in equation (4.17), we get:

(4.14)

⁄

2

,…,

V.

.

CONCLUSION

In this paper, we have presented a new control approach for controlling nonlinear interconnected systems with unknown dynamics. We suggested indirect adaptive fuzzy decentralized control for attacking the control problem in hand. Fuzzy function approximators were used to approximate the unknown dynamics of the subsystems. The uncertainty arising from the functional approximation was compensated and an efficient approach is resulted.

(4.16)

After several mathematical manipulations on (4.16) and using the facts that:

REFERENCES [1]

P. A. Ioannou, “Decentralized adaptive control of interconnected systems,” IEEE Trans. Autot. Contr., vol. AC-30, no. 4, pp. 291- 298, 1986. [2] D. T. Gavel and D. D. Silijak, “Decentralized adaptive control: structural conditions for stability,” IEEE Trans. Autom. Contr., vol. 34, no. 4, pp. 413- 426, 1989. [3] A. Datta, “Performance improvement in decentralized adaptive control: a model reference scheme,” IEEE Trans. Autom. Contr., vol. 38, no. 11, pp. 1717- 1722, 1993. [4] H. Seraji, “Decentralized adaptive control of manipulators: theory, simulation, and experiment,” IEEE Trans. Robot. Autom., vol. 5, no. 2, pp. 183- 201, 1989. [5] L. C. Fu, “Robust adaptive decentralized control of robot manipulators,” IEEE Trans. Autom. Contr., vol. 37, no. 1, pp. 106110, 1992. [6] S. Sheikholeslam and C. A. Desoer, “Indirect adaptive control of a class of interconnected nonlinear systems,” Int. J. Control, vol. 57, no. 3, pp. 743- 765, 1993. [7] Z. P. Jiang, “New results in decentralized adaptive nonlinear control with output- feedback,” Proc. Of IEEE Conf. on Decision and Control, vol. 5, pp. 4772- 4777, 1999. [8] L. X. Wang, Fuzzy Systems and Control: Design and Stability Analysis, Englewood Cliffs, NJ: Prentice- Hall, 1994. [9] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques, John Wiley & Sons, 2002. [10] S. Tong, H. X. Li, and G. Chen, “Adaptive fuzzy decentralized control for a class of large- scale nonlinear systems,” IEEE Trans. Man, Machine, Cybern.., vol. 34, no. 1, pp. 770- 775, 2004.

we can obtain: .

.

⁄ ∆

,

Λ

. ,…,

1

. ⁄

.

⁄

225


[13] J.-J. E. Slotine, and J. A. Coetsee, "Adaptive Sliding Control Synthesis for Nonlinear Systems", International Journal of Control, 43, pp. 1631- 1651, 1986.

[11] A. Errahmani, H. Ouakka, M. Benyakhlef, and I. Boumhidi, “Decentralized adaptive fuzzy control for a class of nonlinear systems,” WSEAS Trans. On Syst. And Contr., issue 8, vol. 2, pp411418, 2007. [12] J.-J. E. Slotine, "Sliding Controller Design for Nonlinear Systems", International Journal of Control, 40, pp. 435-448, 1984.

226