adaptive fuzzy output tracking control of mimo nonlinear uncertain systems

8/7/2019 Adaptive Fuzzy Output Tracking Control of Mimo Nonlinear Uncertain Systems

http://slidepdf.com/reader/full/adaptive-fuzzy-output-tracking-control-of-mimo-nonlinear-uncertain-systems 1/8

ADAPTIVE FUZZY OUTPUT TRACKING CONTROL OF MIMO NONLINEAR

UNCERTAIN SYSTEMS

Document BySANTOSH BHARADWAJ REDDY

Email: [email protected]

Engineeringpapers.blogspot.comMore Papers and Presentations available on

above site

Abstract - A fuzzy logic controller equipped

with adaptive algorithm is proposed in this work

to achieve tracking performance for a class of

uncertain nonlinear single-input single-output

(SISO) systems. The extension of this method

for a class of uncertain multiple-input-multiple-

output (MIMO) nonlinear systems are suggested

as the next stage. Simulation examples are given

finally to illustrate the performance of the

proposed method (for the case of SISO systems

results are provided).

I. Introduction

Fuzzy theory has recently found extensive

application for a wide variety of industrial systems

and consumer products and has attracted the

attention of many control researchers due to its

model free approach. The fuzzy control systems

attempt to make use of the knowledge from the

human experts. The expert information is generally

represented using fuzzy terms, e.g., small, large, not

very large, etc., for convenience or lack of more

precise knowledge, ease of communication, and so

on [1-3].

* Subba Rao Puvvadi is with Department of Electrical

Engineering, NIT Calicut, India (e-mail: [email protected])

**Dr. Abraham T Mathew is with the Department of Electrical

Engineering, NIT Calicut, India (e-mail: [email protected]) This work

was supported in part by the Department of Electrical

Engineering, National Institute of Technology Calicut, Kerala,

India.

Fuzzy control methodology gives a promising way

to deal with the control problems of nonlinear

systems containing highly uncertain nonlinear

functions. It has been shown that fuzzy logic

systems can be used to approximate anynonlinear function over a compact region [4],

based on this observation, many systematic fuzzy

controller design methods have been developed to

solve output tracking control problems for SISO

systems with unknown non linearties as well as to

MIMO systems.

Adaptive fuzzy control system has been

developed to incorporate the expert information

systematically and the stability is guaranteed by

theoretical analyses [4], and [5]. An adaptive fuzzy

system is a fuzzy logic system equipped with a

training algorithm, in which the fuzzy logic system

is constructed from a collection of fuzzy IF-THEN

rules, and the training algorithm adjusts the

parameters of the fuzzy logic system according to

input-output data. Conceptually, adaptive fuzzy

systems combine linguistic information from

experts with numerical information from sensors.

Linguistic information can be directly incorporated,

since fuzzy logic systems are constructed from

fuzzy IF-THEN rules.

1

mailto:[email protected]




Stable adaptive fuzzy control schemes have

been introduced for SISO nonlinear systems. After

this the corresponding adaptive fuzzy control

scheme is extended to MIMO nonlinear systems.

The basic idea of this approach is to use the fuzzy

logic systems to approximate the unknown nonlinear

functions in systems and the adaptive controller is

designed by using Lyapunov stability theory [6] and

[7].

To avoid the problem of matching error

condition, the backstepping design technique is

used to design the adaptive fuzzy system. With the

backstepping design technique, fuzzy systems were

mostly applied to approximate the unmatched andunknown nonlinearities, and then implement

adaptive control using the conventional control

technology [8]. This backstepping technique is used

in designing the control algorithm for the case of

MIMO system.

For MIMO nonlinear systems, the control

problem is very complicated due to the couplings

among various inputs and outputs [9]. It becomes, in

general very difficult to deal with the control design

due to the presence of uncertain parameters and

unknown nonlinear functions in the input-output

coupling matrix. It has been noticed that in

comparison with the vast amount of results on

controller design for SISO nonlinear systems in the

control literature, there are relatively fewer results

available for the class of MIMO nonlinear systems.

Based on feedback linearization, several adaptive

control schemes have been proposed for certain

classes of MIMO nonlinear systems. One method

has been adopted and the investigation is in

progress.

The paper is organized as follows the

formulation of problem is given in section II. Fuzzy

logic systems are briefly discussed in section III

and adaptive fuzzy tracking control for the case of

SISO system given in the section IV. Then

simulation example is provided in section V for

illustration.

II. Problem Formulation

Consider the following nth order SISO system

( ) ( ) ( ).,.........,.,........., 11

x y

u x x x g x x x f x nnn

=

+=−−

(1)

where f and g are unknown but bounded

continuous functions and u ∈ R and y ∈ R are the

scalar input and scalar output of the system

respectively. Let nT n R x x x x ∈=

−),....,(

)1( be

the state vector of the system, which is assumed

to be completely available for measurement using

appropriate transducers. For the system of Eq. (1)

to be controllable, we require that 0)( ≠ x g for

an x in certain controllable region in the state

space.The control objective is to force y to follow

a given bounded reference signal ym, and θ is the

parameter of the system. Let us denote the output

tracking error e and the parameter tracking error

θ ~

*~

θ θ θ −=

−= y yem

(2)

for some parameter estimate θ of a system and

optimal parameter estimate ∗θ of fuzzy logic

system. Then our design objective is to impose an

adaptive fuzzy control algorithm so that following

asymptotically stable tracking

2



0..............)1(

1

)( =+++ − ek ek en

nn(3)

is achieved. Indirect adaptive fuzzy control algorithm

employed to control the nonlinear SISO system in

Eq (1) and the algorithm is given in the chapter IV.

III. Description of fuzzy logic system

As it is mentioned the fuzzy logic systems are

universal approximations [4] from the viewpoint of

human experts and can approximate nonlinear

continuous functions. The fuzzy logic systems in Fig.

1 qualified as building blocks of adaptive controllers

for nonlinear systems. Furthermore the fuzzy logic

systems are constructed from the fuzzy IF-THEN

rules using some specific inference, fuzzification,

and defuzzification strategies. Therefore, linguistic

information from human experts can be directly

incorporated into controllers.

Fig. 1. The basic configuration of fuzzy logic system

The basic fuzzy logic system is shown in

Fig. 1. The fuzzy logic system performs a mapping

from U ∈R to V ∈R . The fuzzy rule base consists of a

collection of fuzzy IF-THEN rules [3]

R (l) : IF x1 is F1 j, and...and, xn is Fn

j, THEN y is G j (4)

where U x x x xT n∈=

−).,.........,(

)1( and y∈V

are the input and output of the fuzzy logic system

respectively, and V is the crisp set. Fi j

and G j are

labels of fuzzy sets Ui and y, respectively, and i=1,

….M, V is the crisp set of output variable y. The

fuzzy inference engine performs a mapping from

fuzzy sets in U to fuzzy sets in R, based upon the

fuzzy IF-THEN rules in the fuzzy rule base and the

compositional rule of inference. The fuzzufier

maps a crisp point T n x x x x ),.....,,( )1( −= into

a fuzzy set in U. The defuzzifier maps a fuzzy set

in U to a crisp point in V.

The fuzzy logic systems with center

average defuzzifier, product inference and

singleton fuzzufier in the following form

( )

( )∑ ∏

∑ ∏

==

==

=M

l

n

i i F

M

l

n

i i F

l

x

x y

x y

l i

l i

11

11

)(

)(

)(

µ

µ

(5)

where yl is the point at which l G

µ achieves its

maximum value, and we assume that

.1)( =l

Gyl µ )( i F

x ji

µ is the membership of the

variable x i.

Eq. (5) can be written as

)()( x x yT ξ θ = (6)

where ( )T M y y ,........,1=θ is a parameter

vector, andT M

x x x ))(),.......,(()(1 ξ ξ ξ = is

a regressive vector with the regressor )( xl

ξ defined as

( )∑ ∏

∏

==

==

M

l

n

i i F

n

i i F l

x

x

x

l

i

l

i

11

1

)(

)()(

µ

µ ξ

(7)

Two main reasons arise for using the fuzzy logic

system by Eq. (7) as basic building block of

adaptive fuzzy controllers. First, the fuzzy logic

systems in the form of Eq. (7) are proven to be

universal approximaters, i.e., for any given real

continuous function f on the compact set U, there

exists a fuzzy logic system in the form of Eq. (7)

3



such that it can uniformly approximate f over U to

arbitrary accuracy. Second, the fuzzy logic systems

of Eq. (7) form are constructed from the fuzzy IF-

THEN rules of Eq. (4) using some specific fuzzy

inference, fuzzification, and defuzzification

strategies. Therefore, linguistic information from a

human expert can be directly incorporated into the

controllers.

IV. Adaptive Fuzzy Controller For A SISO System

Adaptive control is a useful approach in maintaining

the consistent performance of a system in the

presence of the uncertainties [10]. An adaptive fuzzy

system is a fuzzy logic system equipped with an

adaptive algorithm to maintain a consistentperformance under plant uncertainties. The most

important advantage of adaptive fuzzy control over

conventional adaptive control is that adaptive fuzzy

controllers are capable of incorporating linguistic

fuzzy information from human operators, whereas

conventional adaptive controllers are not [10]. This is

especially important for those systems with a high

degree of uncertainty. For examples, in chemical

processes and in aircraft industry, they are difficult to

control from a control theory point of view and are

often successfully controlled by human operators.

However, human operators successfully control

such complex systems without a mathematical

model in their mind but only according to a few

control rules in fuzzy terms and some linguistic

descriptions regarding the behavior of the system

under various conditions, which are, of course, in

fuzzy terms. They provide very important information

about how to control the system and how the system

behaves. Adaptive fuzzy control provides a tool for

making use of the fuzzy information in a systematic

and efficient manner for certain control systems.

Conventional adaptive controllers are of

two types i.e. direct and indirect adaptive

controllers in the similar way the adaptive fuzzy

control is also divided into two categories as direct

and indirect adaptive fuzzy control. An adaptive

fuzzy controller, which uses fuzzy logic systems as

controllers, is a direct adaptive fuzzy controller. A

direct adaptive fuzzy controller can incorporate

fuzzy control rule directly into itself. An adaptive

fuzzy controller, which uses fuzzy logic systems as

a model of the plant, is an indirect adaptive fuzzy

controller. An indirect adaptive fuzzy controller can

directly incorporate fuzzy descriptions about the

plant (in terms of fuzzy IF-THEN rules) into itself [5]. Indirect adaptive fuzzy control algorithms are

tried in this work to facilitate the control of

uncertain nonlinear system of Eq. (1) to achieve

the desired tracking performance. The design

procedure as given below.

If )( x g is known in Eq. (1), the direct

adaptive fuzzy control has attempted to directly

approximate the following control law

])([)(

1 )(ek y x f

x g u

T n

m++−= (8)

with this control law we can achieve the following

error dynamics of the system similarly as Eq. (3)

0..............)1(

1

)(=+++

− ek ek e n

nn

the results in Eq. (8) and Eq. (3) are possible only

while )( x f and )( x g in nonlinear system Eq.

(1) are well known. However )( x f and )( x g

are unknown in our problem. Obtaining a control

algorithm similar to Eq. (8) is impossible. In this

situation approximation by fuzzy logic systems in

section III is employed to treat this tracking control

4



design problem. We replace )( x f and )( x g in Eq.

(8) by the fuzzy logic systems )(ˆ f x f θ and

)(ˆ g x g θ as Eq. (6) i.e.,

f T T

f f x x x f θ ξ ξ θ θ )()()(ˆ == (9)

f

T T

f f x x x f θ ξ ξ θ θ )()()(ˆ == (10)

where )( xξ is a vector of fuzzy bases, f θ and

g θ are corresponding parameters of fuzzy

systems. Consequently the following controller is

obtained

[ ]ek y x f

x g

uT n

m f

g

++−= )()(ˆ

)(ˆ

1θ

θ (11)

applying Eq. (11) to Eq. (1), after some

manipulations, we obtain the tracking error dynamic

equation as

g x g x f x f ek e g f

T n ()/(ˆ[)]()/(ˆ[)( −+−+−= θ θ

(12)

or equivalently

()/(ˆ())()/(ˆ[( g x g x f x f bee g f cc −+−+Λ= θ θ

(13)where

−−−

=Λ

− 11 ..........

10....0000

................

00....0100

00....0010

k k k nn

c

=

1

..

..

..

0

cb (14)

since cΛ is a stable matrix, we know that there

exists a unique positive definite symmetric matrix

nn× matrix P which satisfies [12]

Q P P cT c −=Λ+Λ (15)

Our design objective involves specifying

the control u and adaptive laws f θ and g θ , so

that output tracking performance is achieved.

First, let us define the optimal parameter estimates

* f θ and *

g θ as follows:

])()/(ˆ[supminarg x f x f f x f x f f −= Ω∈Ω∈

∗θ θ θ

(16)

])()/(ˆ[supminarg x g x g g x g g x g −= Ω∈Ω∈

∗θ θ θ

(17)

where f Ω , g Ω ,and xΩ denote sets of

suitable bounds on f θ , g θ and x respectively.

We assume that the parameters never exceed

their bounds. Then the tracking error Eq. (13) will

become

(ˆ)/(ˆ())/(ˆ

)/(ˆ

[(*

x g x g x f x f bee g f f cc θ θ θ −+−+Λ=

(18)

From Eq. (9) and Eq. (10), Eq. (18) can be written

as

]~

)(~

)([ u x xbee g T

f T

cc θ ξ θ ξ ++Λ= (19)

where *~

f f f θ θ θ −= and *~

g g g θ θ θ −= . And the

above system to be stable the Lyapunov function

V to be positive definite and its derivative should

be a negative definite, let the Lyapunov function

be chosen as

g T g f

T f

T e P eV θ θ γ

θ θ γ

~~

2

1~~

2

1

2

1

21

++= (20)

5



For this Lyapunov function to be positive definite and

its derivate to be negative definite the adaptive laws

chosen as

e P b xT

c f )(1ξ γ θ −= (21)

ue P b xT

c g )(2ξ γ θ −= (22)

Summarizing all the above points to implement an

indirect adaptive fuzzy control for a given SISO

system the following design procedure has to be

followed

Design procedure:

Step 1: Select membership functions )( xi µ for

i =1,2,…..M and compute the fuzzy basis

functions )( xξ as in Eq. (7).

Step 2: Select matrix Q, γ and r .

Step 3: Specify the coefficients k i, i = 1,2,..,n.

Step 4: Solve the equation Eq. (15) to get matrix P.

Step 5:Compute the indirect adaptive fuzzy control

law as

[ ]ek y x f x g

u T nm f

g

++−=)()(ˆ

)(ˆ

1θ

θ

and the adaptive laws are implemented as in Eq.

(21)and Eq. (22).

V. Simulation Example

For illustrating the above-mentioned method the

control problem of inverted pendulum has been

taken as an example.

Fig. 2. Inverted pendulum system

Consider the inverted pendulum system

as in Fig. 2. The dynamic equations of the inverted

pendulum system are [5]

1

1

2

1

1

2

11

2

212

21

1

)cos

3

4(

cos

)cos

3

4(

sincossin

,

x y

d umm

xml

mm

x

mm

xml

mm

x xmlx x g x

x x

x

cc

cc

=

+

+−

++

+−

+−=

=

=

θ

(23)

where g=9.8m/s2 is the acceleration due to gravity,

mc is the mass of the cart, m is the mass of the

pole, l is the half-length of the pole, u is the

applied force. The reference signal assumed to be

a sinusoidal signal with amplitude as 1. Also we

assume that mc = 10kg, m = 1kg, l =3m.the

external disturbance d is assumed to be a square

wave of amplitude ± 0.05 and the period 2π .

The system is simulated without

considering a controller, and also by making the

system as linear taking the x 1 as very small value.

The result is given in Fig. 3. A fuzzy controller is

implemented to track the output of the system and

the result is shown in Fig.4. Then an adaptive

fuzzy controller designed for this inverted

6



pendulum following the below steps and the result is

shown in Fig. 5.

Step 1:Selected the membership functions )( xi µ

for i =1,2 and computed the fuzzy basis

functions )( xξ as in Eq. (7). Here

membership functions taken as the

Gaussian membership functions.

Step 2:Selected Q as 1210 × I ,γ 1=0.1, γ 2=0.01

and r =0.08.

Step 3: Coefficients selected as k1=2 and k2=1.

Step 4: Eq. (15) has solved for getting the matrix P.

Step 5:Computed the direct adaptive fuzzy control

law as in Eq. (11) and the adaptive control

laws as in Eq. (21) and Eq. (22).

Fig. 3. Output of the inverted pendulum system

without any controller

Fig. 4. Output of the inverted pendulum system with

fuzzy controller

Fig. 5. Output of the inverted pendulum system

with adaptive fuzzy controller

First the system is simulated with out any

controller the output (Fig. 3) of this system is not

following the input, and then the system is made

linear by approximating the x 1 as very small

quantity simulated the system by using the fuzzy

controller and the result (Fig. 4) shows that theoutput tracking the input. But in this case if we

apply input with combining two waves it is not

following the input. The system is simulated with

the adaptive fuzzy controller in the loop. The result

(Fig. 5) shows that the output following the input

almost without any disturbance and if we apply

even a disturbance in the system output is tracking

the input. Even if we give the input as combination

of the waves the output is seen tracking the given

input.

The adaptive fuzzy control algorithm is

applied to the MIMO system and the simulation of

the system will be done in the next stage of work.

Conclusion

Adaptive fuzzy tracking control method for

the case of SISO system has been presented and

also the method was applied to the inverted

pendulum by considering it as a nonlinear system.

The results of the inverted pendulum for two cases

linear and nonlinear model are shown. The

unknown nonlinear functions were approximated

using fuzzy logic and result shows that output is

7



tracking the input exactly even in the presence of

disturbance in the system. Application of this

adaptive fuzzy control method to MIMO system is

the future work pending.

References

[1] C.C.Lee, “Fuzzy logic in control systems: Fuzzy

logic controller, Part I,” IEEE Trans. on syst.,

Man, and Cybern., Vol 20, No.2, pp.404-

418,1990.

[2] C.C.Lee, “Fuzzy logic in control systems: Fuzzy

logic controller, Part II,” IEEE Trans. on syst.,

Man, and Cybern., Vol 20, No.2, pp.419-

435,1990.

[3] Drainkov, Hellendoorn, and Reinfrank, ”Anintroduction to fuzzy control,” Narosa publishing

house, 1993.

[4] L. X. Wang and J. M. Mendel, “Fuzzy basis

functions, universal approximation, and

orthogonal least squares learning,” IEEE Trans.

Neural Networks, vol. 3, no. 5, pp. 807–814,

Sep. 1992.

[5] B. S. Chen, C. H. Li, and Y. C. Chang, “H∞

tracking design of uncertain nonlinear SISO

systems: Adaptive fuzzy approach,” IEEE

Trans.Fuzzy Systems, vol. 4, no.1, pp. 32–43,

Feb. 1996.

[6] L. X.Wang, “Stable adaptive fuzzy control of

nonlinear systems,” IEEE Trans. Fuzzy

Systems, vol. 1,no. 2, pp. 146–155, May 1993.

[7] Kumpati S Narendra, Anuradhu K

Annaswamy,”stable adaptive systems”, Prentice

Hall , 1989.

[8] B.Cheng, X. Liu and S. C. Tong, “Adaptive fuzzy

output tracking control of MIMO nonlinear

uncertain systems,” IEEE Trans. Fuzzy systems,

vol. 15, no. 2, pp. 287–300, April 2007.

[9] X. P. Liu, A. Jutan, and S. Rohani, “Almost

disturbance decoupling of MIMO nonlinear

systems and application to chemical

processes,” Automatica, vol. 40, no. 3, pp.

465–471, 2004.

[10] C. Y. Sue and Y. Stepanenko, “Adaptive

control of a class of nonlinear systems with

fuzzy logic,” IEEE Trans. Fuzzy Syst ., vol. 2,

no. 1, pp.285–295, Feb. 1994.

[11] S. S. Ge and C. Wang, “Adaptive neural

control of uncertain MIMO nonlinear systems,”

IEEE Trans. Neural Networks, vol. 15, no. 3,

pp. 674–692, May 2004.

[12] Khalil, Hassan K “Nonlinear systems,”Prentice Hall, 1996.

Document By

SANTOSH BHARADWAJ REDDY

Email: [email protected]

Engineeringpapers.blogspot.com

More Papers and Presentations

available on above site

8



adaptive fuzzy output tracking control of mimo nonlinear uncertain systems

Documents