Automatic Control

International Review of

Automatic Control (IREACO) Theory and Applications

Contents Fuzzy Feedback Linearization Adaptive Control for Nonlinear Systems Via a NN-Based Approach by M. Bahita, K. Belarbi

144

Design of Model Reference Adaptive Intelligent Controller Using Neural Network for Nonlinear Systems by R. Prakash, R. Anita

153

Modified GPC Controller for Control of Processes with Long Dead-Time and Integral Action by Danijel Jolevski, Ozren Bego, Ranko Goić

162

Adaptive Sliding Mode Observer for Interconnected Fractional Nonlinear Systems by Elleuch Dorsaf, Damak Tarak

170

Multivariable System’s Parameters Interaction and Robust Control Design by K. M. Yanev, G. O. Anderson, S. Masupe

180

Robust Exponential Stabilization of Uncertain Perturbed Systems with Multiple Time Varying Delays in State and Control Input by I. Amri, D. Soudani, M. Benrejeb

191

New Stability Conditions for Neutral and Retarded Time Delay Systems by S. Elmadssia, K. Saadaoui, M. Benrejeb

204

Current Regulation of Three-Phase Grid Connected Voltage Source Inverter Using Robust Digital Repetitive Control by M. Jamil, S. M. Sharkh, M. A. Abusara

211

A New Analysis of Sliding Mode Control Based on Setting Time Criteria for Third Order Induction Motor Drive by Ghazanfar Shahgholian, Seyed Jafar Salehi

220

Torque Ripple Minimization in Switched Reluctance Motor Using a Novel Passivity-Based Robust Adaptive Control by M. M. Namazi, S. M. Saghaiannejad, A. Rashidi

229

(continued)

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

Direct Pump Control Effects on the Energy Efficiency in an Electro-Hydraulic Lifting System by Tatiana A. Minav, Lasse I. E. Laurila, Juha J. Pyrhönen, Victor B. Vtorov

235

Adjusting PID Controllers Coefficients to Control Fuel Cell Using Genetic Algorithm by H. Nasir Aghdam, A. Ataei, N. Ghadimi, P. Farhadi

243

Computed Torque Control of a Puma 600 Robot by Using Fuzzy Logic by Ouamri Bachir, Ahmed-Foitih Zoubir

248

A Novel Method for State Estimation in Large Power Systems Using Phasor Measurement Units by I. Naziri, M. Karrari

253

Power Systems Voltage Stability Assessment Using STATCOM by Youssef A. Mobarak

259

An Estimation Rate of Change of Frequency Using Wavelet Transform by S. Avdakovic, A. Nuhanovic, M. Kusljugic

267

SHMP Problem Solution for Surface Treatment Line Using Genetic Algorithm by I. Mhedhbi, H. Camus, E. Craye, M. Benrejeb

273

A Hybrid Method Based on Genetic Algorithms and Tabu Search for the Single-Machine Scheduling Problem in Agro-Food Industry by A. Karray, M. Benrejeb, P. Borne

280


International Review of Automatic Control (I.RE.A.CO.), Vol. 4, N. 2 March 2011

Fuzzy Feedback Linearization Adaptive Control for Nonlinear Systems Via a NN-Based Approach M. Bahita, K. Belarbi

Abstract – In this study, we present an adaptive control based approach for a class of nonlinear systems. The method combines fuzzy logic control performance with neural network (NN) systems approximators. The NN using radial Basis function networks (RBF) is adopted to deal with the on line training and approximation of the control gain which is a nonlinear and unknown function in the nonlinear system. Fuzzy-logic control is designed by means of Tagagi–Sugeno (TS) type system to construct an adaptive law. The parameters of the adaptive controller are adapted and changed according to a law derived using Lyapunov stability theory. Asymptotic stability is established with the tracking errors converging to a neighborhood of the origin. Finally, the combined fuzzy neural (TS-RBF) adaptive system is then applied to control the inverted pendulum system. The simulation results show that the proposed method is able to control and stabilize a nonlinear system. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Adaptive Control, Control Gain, Nonlinear Systems, Radial Basis Function Network, Takagi-Sugeno

NC, more precisely NN control, and fuzzy control are two of the most popular intelligent control techniques [1]. They are similar in many ways. For example, both of them are basically model-free control techniques, both are able to store knowledge and use it to make control decisions, and both are able to provide robustness of control to certain extent with respect to system variations and external disturbances. However, the two techniques are different in their ways to obtain knowledge. NC acquires knowledge mainly through data training (or learning), but sometimes it could be a disadvantage if the training data set does not fully represent the domain of interest. Fuzzy control, in particular conventional fuzzy control, on the other hand mainly obtains qualitative and imprecise knowledge via an operator or expert’s perspective. As the two control techniques complement to each other, that is, NC providing learning capabilities and high computation efficiency in parallel implementation, and fuzzy control providing a powerful framework for expert knowledge representation, the combination or integration of the two techniques have attracted lots of attention from control community.

A typical combination of these two techniques is the so-called neuro-fuzzy control, which is basically a fuzzy control augmented by neural networks to enhance its characteristics like flexibility, data processing capability, and adaptability [2]. It has been proven that artificial neural networks can approximate any nonlinear functions to any desired degree [3]-[6]. Following the similar idea in neural networks [7] for their universal function approximation capability, it is shown in [8] that a fuzzy system is capable of approximating any smooth nonlinear functions over a convex compact region. Also, adaptive fuzzy logic systems and NN systems are widely used for this purpose, where most of the adaptive controllers involve certain types of function approximators in their learning mechanism. These function approximators can be constructed via a Fuzzy system or a NN system [9][13]. The last few years have witnessed a great deal of progress in the design of feedback control for nonlinear systems with fuzzy and NN control. The used fuzzy inference system or the NN system is introduced for approximating part or all the components of the control law. One can find very interesting works in [9], [14]-[18] In this paper, we introduce an alternative indirect adaptive controller for a class of nonlinear systems. The architecture is based on tow on line adapted systems: a fuzzy logic system of Takagi Sugeno (TS) type to approximate the ideal control law which can not be computed and a NN system of RBF type to estimate the virtual control gain which is an unknown nonlinear function in the nonlinear controlled system. On the other

Manuscript received and revised February 2011, accepted March 2011


Nomenclature NN NC RBF TS TS-RBF

Neural-Network Neuro-Control Radial Basis Function Takagi-Sugeno Takagi-Sugeno- Radial Basis Function

I.

Introduction

144

M. Bahita, K. Belarbi

e = y − ym should be as small as possible. Define the error vector as:

hand, the main advantage of the RBF network is that their output depends linearly on the connection weights and thus the training becomes a linear optimisation problem [19]. However this is not possible for the centres. In this work, we propose to online adjust both the centres of the basis functions and the connection weights. The centres of the RBF network are adapted on line using the k-means algorithm [20]-[21] In most of these woks discussed above, the stability convergence is established and the control laws are derived using Lyapunov stability theory and based on some basic assumptions. In our present work, the Asymptotic stability of the resulting closed loop system is established and the control laws of the parameters (the connection weights of the RBF network and the parameters of the consequent part of the TS controller system) are derived using also Lyapunov stability theory, but in some aspects based on some additional mathematical developments in a different manner compared to the one used in works cited above (for example [9]) as we will see in section III. This paper is organised as follows: in section II, the problem formulation is introduced, in section III, the stability analysis is developed and the adaptive laws are derived, in section IV, the indirect adaptive TS-RBF controller is used in simulation for controlling the inverted pendulum system. Section V concludes the paper.

II.

(

( n −1) e = e,e,...,e

)

u• =

)

T

(3)

n x( ) = v

(4)

Step 2: We choose the artificial input v (an equivalent input) as a simple linear pole-placement controller v = ym( ) − K T ⋅ e that provides guarantee about the stability of the overall system, with the: n

K = ( k0 ,k1 ,...,kn −1 ) ∈ R n T

(5)

chosen so that the polynomial: (1) s n + kn −1 ⋅ s n −1 + ...... + k0 = 0

(6)

has all its roots strictly in the left-half complex plane. Then the optimal control law is:

nonlinear functions. We assume that the state vector

(

1 ( v − f ( x )) b ( x)

Substituting (3) into (1), we can cancel the nonlinearities and obtain the simple input-state relation:

the system respectively, f ( x ) and b ( x ) are unknown T

(2)

assumption) then from (1), the optimal control law is:

where u ∈ R and y ∈ R are the input and output of

( n −1) x = ( x1 ,x2 ,...,xn ) = x,x,...x

∈ Rn

assuming b ( x ) to be non zero (this is an usual

Consider a nonlinear system:

(

T

Step 1: We choose u to cancel the nonlinearities in a nonlinear system so that the closed-loop dynamics is in a linear form, and guarantees tracking convergence, this is called feedback linearization [22]. If the functions f ( x ) and b ( x ) are known and

Problem Formulation

n ( n −1) x( ) = f x,x,...x + b ( x ) u, y = x

)

∈ R n is available for

u• =

measurement. The control objective is to force y ( t ) to follow a given bounded reference signal ym ( t ) , under

(

1 n ym( ) − K T ⋅ e − f ( x ) b ( x)

)

(7)

Based on e = y − ym then:

the constraints that all signals involved must be bounded. More specifically, determine a feedback control u ( x,θ )

n n n e( ) = y ( ) − ym( )

approximating the term u in (1) and an estimation bˆ ( x ) = b ( x,θ b ) of the unknown nonlinear function or

(8)

Substituting (7) into (1), using (8) and based on y = x (see(1)) we have:

the control gain b ( x ) , and this is based on two systems: a fuzzy inference system of TS type and a NN system of RBF network type. Based on Lyapunov theory, determine also an adaptive law for adjusting the parameters vectors θ and θ b such that the closed-loop system must be globally stable in the sense that all variables must be bounded and .the tracking error,

n n −1 e( ) + kn −1 ⋅ e( ) + ...... + k0 ⋅ e = 0

This

implies

that

lim e ( t ) → 0

as

(9) t→∞

(exponentially stable dynamics), which is the main objective of control. Since f ( x ) and b ( x ) are unknown,


International Review of Automatic Control, Vol. 4, N. 2

145


the optimal control u • of (7) can not be implemented. Our purpose is to design a fuzzy inference system of TS type with output u ( x,θ ) to approximate this optimal

thus:

control law u • and based on this approximation we design an RBF network to approximate the unknown function b ( x ) (the control gain). The following sections

Based on y = x in (1), and using (2) and (8), equation (14) leads to the error system:

describe the TS fuzzy inference system and the RBF network and how to construct the combined TS-RBF adaptive controller.

(15)

(

e = Ac ⋅ e + bc u ( x,θ ) − u •

⎡ 0 ⎢ 0 ⎢ Ac = ⎢ ⎢ ⎢ 0 ⎢⎣ −k0 ⎡0 ⎤ ⎢ ⎥ ⎢0 ⎥ ⎥ bc = ⎢. ⎢ ⎥ ⎢0 ⎥ ⎢b x ⎥ ⎣ ( )⎦

The adaptive controller proposed is composed of two systems: a fuzzy controller based on a TS inference system and a NN estimator of RBF type III.1. The TS Fuzzy Adaptive Controller A TS fuzzy inference system with linear consequences is composed of rules of the form: R i : if z1 is A1i and ...zn is Ani then ui = a0i + a1i z1 + ... + ani zn

1 0

0 1

0 − k1

0 −k2

... ...

0 ⎤ 0 ⎥⎥ ⎥, ⎥ 1 ⎥ − kn −1 ⎥⎦

0 0

0 ... ... − kn − 2

(

optimal approximator control signal u x,θ •

(16)

)

of the

optimal control signal u* of (7). It has been proven that (10) (or the output of a TS fuzzy system) can approximate over a compact set Ω Z , any smooth function up to a given degree of accuracy [8]. It can thus be used to approximate the ideal control law u* as given in (7).

(10)

θ T = ⎡⎣θ1T θ 2T ...θ nT ⎤⎦ contains all adjustable parameters and ξ 1 ( x ) is a vector of fuzzy basis

where

(

)

It follows that: u* ≈ u x,θ • . Thus, the error equation

functions. Supposing that the control u in (1) is the TS control u ( x,θ ) , i.e., u = u ( x,θ ) , then (1) becomes:

= f ( x ) + b ( x ) u ( x,θ )

)

of the TS controller. Define the parameter vector θ • as the optimal parameter vector which corresponds to the

Aij are fuzzy sets. If we take θiT = ⎡⎣ a0i a1i ...ani ⎤⎦ as the vector of adjustable parameters of the consequence of rule R i . The output of a TS fuzzy system can be put in the following form:

x

(14)

Let’s now study the stability of the system in order to develop an adaptive law to adjust the parameter vector θ

where z1 … zn are functions of state variables.

( n)

)

with:

III. The Adaptive Controller

u ( x,θ ) = θ T ⋅ ξ 1 ( x )

(

n n x( ) − ym( ) + K T ⋅ e = b ( x ) u ( x,θ ) − u •

(15) can be rewritten as:

(

(

e = Ac ⋅ e + bc u ( x,θ ) − u x,θ •

(11)

))

(17)

Based on (10) we have:

Now adding and subtracting b ( x ) u* to (11) we will have: n x( ) = f ( x ) + b ( x ) u ( x,θ ) + b ( x ) u • − b ( x ) u • (12)

u ( x,θ ) = θ T ⋅ ξ1 ( x )

(18)

(

(19)

)

u x,θ • = θ *T ⋅ ξ1 ( x )

Substituting (7) into (12), we obtain: x

(n)

= f ( x ) + b ( x ) u ( x,θ ) − b ( x ) u +

Let ϕ = θ − θ * and using (18) and (19), thus (17) becomes: e = Ac ⋅ e + bc ⋅ ϕ T ⋅ ξ1 ( x ) (20)

•

+ ym( ) − K T ⋅ e − f ( x ) n

(13)



146


from (28), we see that θ = −γ ⋅ b ( x ) eT ⋅ Pn ⋅ ξ1 ( x ) is a

Define the Lyapunov function candidate: V=

1 T 1 e ⋅ P ⋅ e + ϕTϕ 2 2γ

function of b ( x ) , and b ( x ) is not known,

then it

(21)

remains to develop an estimation of b ( x ) using an RBF

where γ is a positive constant and P is a solution of the Lyapunov equation:

network in order to can adapt the parameters vector θ of the TS fuzzy controller. Based on a first initialized value (different from zero) of b ( x ) , we can use the adaptive

AcT ⋅ P + PAc = −Q with Q > 0

law (28) to adjust the parameters vector θ of the controller as a first step, and then the ideal control signal

(22)

u • (or in the general case the control input u ) is given a first value using the TS fuzzy system with

Differentiate V with respect to time:

output u ( x,θ ) = θ T ⋅ ξ1 ( x ) ,

1 1 1 1 V = eT ⋅ P ⋅ e + eT ⋅ P ⋅ e + ϕ T ϕ + ϕ T ϕ (23) 2 2 2γ 2γ

parameter vector θ b output b ( x,θ b )

value

eT ⋅ P ⋅ bc = eT ⋅ Pn ⋅ b ( x )

The RBF network can be considered as a two-layer network with only one hidden layer. The output depends linearly on the weights, then the training is simply a linear optimization problem [19]. More explicitly, the RBF network performs the transformation:

)

1 1 V = − eT Qe + ϕ T b ( x ) γ ⋅ eT ⋅ Pn ⋅ ξ1 ( x ) + ϕ (26) γ 2

In order to force or to make V ≤ 0 (according to Lyapunov stability theory to guarantee the stability convergence), setting the second term of V in (26) equals to zero, i.e.:

)

f r : Rn → R

with: b ( x,θ b ) =

(27)

*

∑ ξiθbi = θ Tb ⋅ ξ 2 ( x ) , i =1

2

)

(30)

x is the input vector, ψ is a non linear function,

•

parameter vector θ is constant and obviously its

called radial basis function, θbi are connection weights (parameters) between the hidden layer and the output layer, ci are centres of basis functions, nr is the number of basis functions. The most used basis function is the Gaussian function:

*

derivative is zero, i.e., θ = 0 , thus ϕ = θ and based on this, from (27) we obtain the adaptation law:

(28)

⎛ −r 2 ⎞ 2 ⎟ ⎟ ⎝ 2σ ⎠

ψ ( r ) = exp ⎜⎜

Then from (26), it follows that: 1 V = − eT ⋅ Q ⋅ e ≤ 0 2

nr

ξi = ψ ( x − ci

Recalling that ϕ = θ − θ = θ , because the optimal

θ = −γ ⋅ b ( x ) eT ⋅ Pn ⋅ ξ1 ( x )

⋅ ξ 2 ( x ) using an adaptive law as we

III.2. The RBF Adaptive Estimator

Substituting (25) into (24), we have:

ϕ T b ( x ) γ ⋅ eT Pn ⋅ ξ1 ( x ) + ϕ = 0

of the RBF network with

we give a brief description of the RBF network.

(25)

(

= θ Tb

will see later. All this is in order to approximate the virtual control gain b ( x ) in the adaptive law (28). First,

Let Pn be the last column of P, and using (16) we obtain:

(

first

a first value of b ( x,θ b ) and in general to adjust the

1 1 V = − eT ⋅ Q ⋅ e + eT P ⋅ bc ⋅ ϕ T ⋅ ξ1 ( x ) + ϕ T ϕ (24) γ 2

γ

the

of u ( x,θ ) exists. Then, we will use this fact to compute

Using (20) and (22), we have:

1

i.e.,

(29)

with r = x − ci

2

(31)

, ci is the centre of ψ ( r ) , σ is an

associated constant to the function ψ ( r ) and represents

From (29), we can see that V ≤ 0 , which is the main objective according to Lyapunov stability theory. Now,

the width of the Gaussian function. In control applications, online training is only concerned with the



147


connection weights between the hidden and output layer, whereas the centres are fixed off line in most works. In this work, we propose to online adjust both the centres of the basis functions and the connection weights. The kmeans algorithm is used for the centres adjustment. The connection weights are the concerned parameters (of the RBF network estimator) to be adapted and are represented by the vector θ b as mentioned previously. The k_means algorithm [20]-[21] is an unsupervised training method for data clustering. It consists in dividing the input space into nr classes as follows: 1. Choose a number of classes (nr basis functions in our case). 2. Initialise the centres of the basis functions. 3. Compute the Euclidean distances between the centres of each basis function and the input vector x , i.e.: dist ( i ) = x − ci

2

i=1 to nr

with u ( x,θ ) ≠ 0 . The case when u ( x,θ ) = 0 will be discussed later in remark 1. We chose an artificial input vb as in step 2 of section II (Problem formulation), i.e.: n vb = ym( ) − KbT ⋅ e

with the vector K b chosen so that the polynomial s n + kbn −1 ⋅ s n −1 + ...... + kb 0 = 0 has all its roots strictly in the left-half complex plane. Then equation (36) can be written as: b• ( x ) =

)

(33) This

where j is the index of the basis function which corresponds to the minimum Euclidean distance dist ( j ) which tends to zero as t → ∞ . One adaptation law for this parameter as given in [23] is the following:

δ ( t − 1)

( nr )

(39)

n n −1 e( ) + kbn −1 ⋅ e( ) + ...... + kb 0 ⋅ e = 0

(40)

implies

as

t→∞

b ( x ) . In the equation (35), we can then replace b ( x ) by its estimation bˆ ( x ) , thus (35) becomes: n x( ) = f ( x ) + bˆ ( x ) u ( x,θ )

Now, we continue the development of the adaptive law of the parameter vector θ b of the RBF network with

(41)

Now adding and subtracting b• ( x ) ⋅ u ( x,θ ) to (41),

output b ( x,θ b ) = θ Tb ⋅ ξ 2 ( x ) as mentioned previously.

then (41) becomes:

So, from (11) we have:

n x( ) = f ( x ) + bˆ ( x ) u ( x,θ ) +

(35)

+b• ( x ) u ( x,θ ) − b• ( x ) u ( x,θ )

Supposing that f ( x ) is known and as the first value of u ( x,θ ) exists (as mentioned previously), then from

(42)

or equivalently:

(35), the optimal expression of b ( x ) is: 1 b ( x) = ⎡ vb − f ( x ) ⎤⎦ u ( x,θ ) ⎣

lim e ( t ) → 0

implemented. Our second purpose is to construct an RBF system to approximate the optimal expression b• ( x ) of

(34)

where t is the time, nr is the number of basis functions, and int is the integer part of ( t nr ) .

•

that

the TS fuzzy approximation of the optimal control, but f ( x ) is not known, then b• ( x ) in (38) can not be

1 + int t

n x( ) = f ( x ) + b ( x ) u ( x,θ )

n e( ) + K bT ⋅ e = 0

(exponentially stable dynamics), which is always the main objective of control and estimation. From (38), u ( x,θ ) is known (the first value is computed), i.e., it is

and δ ( t ) is a gain belonging to the interval [0 1], and

δ (k ) =

(38)

Or equivalently:

law [20]:

(

1 ⎡ (n) ym − KbT ⋅ e − f ( x ) ⎤ ⎦ u ( x,θ ) ⎣

Substituting (38) into (35), and using y = x (see (1)) and (8), we will have:

(32)

Then adjust the vector of centres ci which corresponds to the minimum distance dist ( j ) = min x − ci 2 using the following adaptation c j ( t ) = c j ( t − 1) + δ ( t ) x ( t ) − c j ( t − 1)

(37)

(

)

n x( ) = f ( x ) + bˆ ( x ) − b• ( x ) u ( x,θ ) +

+b• ( x ) u ( x,θ )

(36)


(43)


148


Let’s now develop an adaptive law to adjust the θb of the RBF network. parameter T bˆ ( x,θ b ) = θ b ⋅ ξ 2 ( x ) .

Substituting (38) in (43), we can obtain:

(

)

n x( ) = f ( x ) + bˆ ( x ) − b• ( x ) u ( x,θ ) +

+

1 ⎡ ( n) ym − K bT ⋅ e − f ( x ) ⎤ u ( x,θ ) ⎦ u ( x,θ ) ⎣

(44)

We chose a candidate Lyapunov function: Vb =

or:

(

)

n n x( ) = ym( ) − K bT ⋅ e + bˆ ( x ) − b• ( x ) u ( x,θ )

(45)

Define now the parameter vector as the optimal parameter vector which guarantees the optimal estimation of b• ( x ) , We recall that it has been proven

AbT ⋅ Pb + Pb Ab = −Qb with Qb > 0

that an artificial NN can approximate any nonlinear function to any desired degree [3]-[7]. So (30) (or the output of the RBF system) can thus be used to approximate the optimal term b• ( x ) of (38), then we

1 1 1 T 1 Vb = eT ⋅ Pb ⋅ e + eT ⋅ Pb ⋅ e + φ φ + φ T φ (54) 2 2 2α 2α

Using (50) and (53), then (54) becomes:

)

(46)

(

)

Let Pbn be the last column of Pb , using part 2 of (51) we have:

bˆ ( x,θ b ) = θ bT ⋅ ξ 2 ( x ) and bˆ x,θ b• = θ •bT ⋅ ξ 2 ( x ) (47)

eT ⋅ Pb ⋅ u b = eT ⋅ Pbn ⋅ u ( x,θ )

If we define:

φ = θ b −θ b

•

(48)

n e( ) + K bT ⋅ e = φ T ⋅ ξ 2 ( x ) u ( x,θ )

1 Vb = − eT Qb e + 2 1 T + φ ⋅ α ⋅ eT ⋅ Pbn ⋅ u ( x,θ ) ⋅ ξ 2 ( x ) + φ

(49)

e = Ab ⋅ e + φ ⋅ ξ 2 ( x ) u b

Vb equals to zero, i.e.:

(50)

where:

1 ...

0

0

1

...

0

0

0

...

0

− kb1

− kb 2

... − kbn − 2

)

(57)

In order to make Vb ≤ 0 , setting the second term of

T

0

(

α

or equivalently:

1

(56)

Substituting (56) into (55), we obtain:

Based on y = x in (1), and using (8), (46), (47) and (48), equation (45) leads to the error system:

⎡ 0 ⎢ 0 ⎢ Ab = ⎢ ⎢ ⎢ 0 ⎢⎣ − kb 0

)

1 1 Vb = − eT Qb e + φ T α ⋅ eT ⋅ Pb ⋅ u b ⋅ ξ 2 ( x ) + φ (55) 2 α

Based on (30), we have:

(

(53)

Differentiate now Vb with respect to time, then:

can write:

(

(52)

where α is a positive constant and Pb is a solution of the Lyapunov equation:

θ •b

b• ( x ) ≈ bˆ x,θ •b

1 T 1 T e ⋅ Pb ⋅ e + φ φ 2 2.α

⎤ 0 ⎥⎥ ⎥, ⎥ 1 ⎥ − kbn −1 ⎥⎦

α

0

⎡0 ⎤ ⎢ ⎥ ⎢0 ⎥ ⎥ u b = ⎢. ⎢ ⎥ ⎢0 ⎥ ⎢u x,θ ⎥ )⎦ ⎣ (

(

)

(58)

θb = −α ⋅ eT ⋅ Pbn ⋅ u ( x,θ ) ξ 2 ( x )

(59)

φ T α ⋅ eT ⋅ Pbn ⋅ u ( x,θ ) ξ 2 ( x ) + φ = 0

Using (48), we can obtain:

Remark 1: As we can see from (59) that the adaptation of θ b is a function of u ( x,θ ) , then when u ( x,θ ) =0,

(51)

(the case mentioned after equation (36)), the parameters vector θ b will not be adapted using equation (59) and will only keep its previous value (its last value) before the term u ( x,θ ) becomes zero. Besides, the case



149


u ( x,θ ) = 0 in (36 or 38) will not cause any problem,

⎡1130.8 4.2 ⎤ P=⎢ 30.8⎥⎦ ⎣ 4.2

because the optimal value b• will not be considered for computation using equation (36) or (38), but its computation will be replaced or achieved using the RBF network

with

output

b ( x,θ b )

= θ Tb

⋅ ξ 2 ( x ) as

The

fuzzy controller has 2 ⎡ z = [ z1 z2 ] = ⎢θ m K T e − θ m( ) ⎤⎥ , ⎣ ⎦ ⎡ ⎤ e = [ e e] = ⎣θ − θ m θ − θ m ⎦ .

an

alternative as mentioned previously.

IV.

x2 = f ( x ) + b ( x ) u ( t ) ,

(60)

inputs with

2

(

2

(

2

µ P ( zi ) = exp − ( zi − cP ) / 2σ P

)

(63)

)

(64)

)

(65)

where zi stands for the input number i . The widths of the membership functions are σ N = σ Z = σ P = 0.25 for the first and the second inputs z1 , z2 . The centres are all set to cN = −0.05 , cZ = 0 and cP = 0.05 for z1 and z2 . This gives nine rules of the form: R i :if z1 is A1i and z2 is A2i

cos ( x1 ) M +m b ( x) = ⎛ 4 m cos 2 ( x1 ) ⎞ l⎜ − ⎟ ⎜3 M + m ⎟⎠ ⎝

then ui = a1i z1 + a2i z2

(61)

2) The parameters of the RBF estimator are α = 0.0039 and Kb = [ kb 0

2

T

T

Qb = diag (127 ,127 ) > 0 . Then by solving (53) we

can obtain: ⎡376.90 70.55 ⎤ Pb = ⎢ ⎥ ⎣ 70.55 26.81.⎦

and besides, we terminate by balancing the pole to the vertical position ( x1 ,x2 ) = ( 0 , 0 ) , i.e., AM = 0 . This last case (balancing the pole to the vertical position) is treated in [24, 25]. Clearly, the derivatives of the reference ym exist and are bounded. The step size is dt = 0.01 . 1) the parameters of the TS fuzzy inference system are T

kb1 ] = [ 0.9 5] in order to have all

roots of s 2 + kb1 ⋅ s + kb 0 = 0 in the open left-half plane. We chose Qb in (53) as

We use g = 9.8m/s 2 , M = 1 kg is the mass of the cart, m = 0.1kg is the mass of the pole and l = 0.5 m is the half length of the pole. The control objective is to make the pole of the pendulum track a sine wave trajectory θ m = ym = AM ⋅ sin ( t ) with different amplitudes AM ,

k1 ] = [36 5]

(66)

We have 18 parameters in the vector θ to tune. All parameters are initialised to zero.

x1 = θ is the angular position of the pendulum (see Fig. 1, x = θ is the angular velocity of the pendulum.

T

(

µ Z ( zi ) = exp − ( zi − cZ ) / 2σ Z

mlx22 cos ( x1 ) sin ( x1 ) g sin ( x1 ) − M +m f ( x) = ⎛ 4 m cos 2 ( x1 ) ⎞ l⎜ − ⎟ ⎜3 M + m ⎟⎠ ⎝

chosen as γ = 35 , and k = [ k0

)

µ N ( zi ) = exp − ( zi − cN ) / 2σ N

y = x1

and:

(

two

All two inputs are fuzzified with three fuzzy sets and similar membership functions given by:

Simulation Results

In this section, we test the performance of the proposed fuzzy Neural TS-RBF adaptive control approach on the inverted pendulum system depicted in Fig. 1. The Dynamic equations of the inverted pendulum system are: x1 = x2 ,

where:

TS

(62)

(67)

The RBF network estimator has two inputs with: the error e = θ − θ m = y − ym and the variation of error de = e = θ − θ and five radial basis functions. The m

parameters (5 connection weights) θ b are all initialised to 0.5. Other choices have been tried and these last values have given a satisfactory transient performance. The centres of the basis functions are uniformly distributed in the interval [-0.66 0.66]. The width σ of each basis function in the RBF network is set to 0.47.

in

order to have all roots of s + k1s + k0 = 0 in the open Q in (22) left-half plane. We chose 2

as Q = diag ( 300,300 ) > 0 . Then by solving (22) we can obtain: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


150


The following initial conditions for the inverted pendulum

( x1 ( 0 ) , x2 ( 0 ) )

T

= ( − 0.2 rad , 0 rad/s )

are

used in the simulation, where the results for different amplitudes of the reference signal are shown in Figs. 2 to 6. The system output y ( t ) (pole angle) is in continuous while the reference signal ym ( t ) is in dotted. Fig. 2 shows the response curve of the pole angle from the initial position (-0.2, 0) and the corresponding desired values with amplitudes AM = π / 30 during the time interval t ∈ [ 0 12.5] s , and AM = π / 15 during the time interval t ∈ [12.5 25] s , and finally with

Fig. 3. The velocity of the pole angle

amplitude AM = 0 which represents a regulation case as done [24], [25] during the remaining time interval t ∈ [ 25 30] s . We can see from this Fig. 2 that the system state x1 ( t ) = θ tracks the desired trajectory ym = θ m perfectly. Fig. 3 shows the velocity of the pole angle. Fig. 4 shows that the tracking error is converging very rapidly to a value close to zero. Fig. 5 represents the corresponding control input which peaks at t = 12.5 s (the first amplitude variation from AM = π / 30 to AM = π / 15 ), and at t = 25 s (the second amplitude variation from AM = π / 15 to AM = 0 ).

Fig. 4. The tracking error e = y − ym

Fig. 5. The control input

Fig. 1. The inverted pendulum system

Fig. 6. The estimation error when estimating b ( x ) by the RBF system

Fig. 2. The pendulum angles



151


Fig. 6 shows that the estimation error (between the real value of the virtual control gain b ( x ) in the

[8]

pendulum system and the one provided by the RBF network estimator b ( x,θ b ) ) is smooth, confirming the

[9]

smoothing property of RBF systems. It also remains bounded and converges very quite rapidly to a value close to zero. From these figures, we can see that the controlled system behaves well in all situations: tracking case and also including the regulation case treated in [24], [25]. As a comparison concerning this last case, we can see that our controller behaves in a best manner than the controller behaviour in [24], [25].

V.

[10] [11]

[12]

[13]

Conclusion

[14]

In this paper, we developed an adaptive control scheme for a class of unknown nonlinear systems. We used for this purpose two on-line systems: a fuzzy system of TS type to approximate the ideal control law and an RBF-NN system to estimate the virtual control gain in the controlled nonlinear system. We online adjusted both the centres of the basis functions and the connection weights in the RBF network. The k-means algorithm was used for the centres adjustment. The connection weights of the RBF network and the parameters of the consequent part of the TS controller system are adapted and changed according to a law derived using Lyapunov stability theory, and based on some additional mathematical proofs and developments. The proposed method could guarantee the stability of the resulting closed-loop system in the sense that all signals involved were bounded. Finally, we used the proposed approach TS-RBF to control the inverted pendulum system in both tracking (with different magnitudes) and regulation cases. Simulation results showed that our TSRBF control technique could confirm its good and smooth properties.

[15]

[16]

[17]

[18]

[19] [20]

[21]

[22] [23]

[24]

References [1]

[2]

[3]

[4] [5]

[6]

[7]

[25]

G. Feng, A Survey on Analysis and Design of Model-Based Fuzzy Control Systems, IEEE Transactions on Fuzzy Systems Vol 14 (Issue 5): 676-697, 2006. M. Boroushaki, M. B. Ghofrani, C. Lucas, M. J. Yazdanpanah, Identification and control of a nuclear reactor core (VVER) using recurrent neural networks and fuzzy systems, IEEE Trans. Nucl. Sci., vol 50 (Issue 1): 159–174, February, 2003. T. P. Chen, H. Chen, Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks, IEEE Trans. Neural Networks, Vol 6 (Issue 4): 904–910, 1995. T. Poggio, F. Girosi, Networks for approximation and learning, Proc. IEEE, Vol. 78, pp. 1481–1497, 1990. K. L. Funahashi, On the approximate realization of continuous mapping by neural Networks, Neural Networks Vol 2: 183-192, 1989. K. Hornik, Stinchombe M., H. White, Multilayer feedforward networks are universal approximators. Neural netwoks, Vol 2: 1083-1112, 1989. R. M. Sanner, J. E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Netw. Vol 3(Issue 6): 837–863, Jun,

1992. L. X. Wang, J. M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least squares learning, IEEE Trans. Neural Netw, Vol 3(Issue 5): 807–814, September, 1992. M. Hojati, S. Gazor, Hybrid Adaptive Fuzzy Identification and Control of Nonlinear Systems, IEEE Trans. Fuzzy Syst Vol 10 (Issue 2): 198-210, 2002. L. X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Syst Vol 1 (Issue 2): 146-155, 1993. S. C. Tong, T. Wang, J. T. Tang, Fuzzy Adaptive Output Tracking Control of Nonlinear Systems, Fuzzy Sets and Systems, 169-182, 2000. C. Y. Su, and Y. Stepanenko, Adaptive Control of a Class of Nonlinear Systems With Fuzzy Logic, IEEE Trans. Fuzzy Syst. Vol 2: 285-294, 1994. C. H. Wang, H. Liu, T. Lin, Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems, IEEE Trans. Fuzzy Syst Vol 10 (Issue 1): 39-49, 2002. Y. Diao, K. M. Passino, Adaptive neural/fuzzy control for interpolated nonlinear systems, IEEE Trans. Fuzzy Syst. Vol 10 (Issue 5): 583-595, 2002. Y. Lee, H. Zak, Uniformly ultimately bounded fuzzy adaptive tracking controllers for uncertain systems, IEEE Trans. Fuzzy Syst., Vol. 12(Issue 6): 797-811, 2004. D. Vélez-Díaz, Y. Tang, Adaptive robust fuzzy control of nonlinear systems, IEEE Trans. Systems man Cybernetics, SMCB, Vol 34 (Issue 3): 1596 -601, 2004. A. Wu, P. K. S. Tam, Stable fuzzy neural tracking control of a class of unknown nonlinear systems based on fuzzy hierarchy error approach, IEEE Trans. Fuzzy Syst., Vol 10 (Issue 6): 779789, 2002. H. N. Nounou, K. M. Passino, Stable auto-tuning of adaptive fuzzy/neural controllers for nonlinear discrete-time systems, IEEE Trans. Fuzzy Syst, Vol 12 (Issue 1): 70-83, 2004. S. Haykin, Neural Networks, A Comprehensive Foundation (Prentice-Hall, 1994). C. Darken, J. Moody, Fast Adaptive k-means Clustering: Some empirical Results, International Joint conf on Neural Networks Vol. 2, pp. 233-238, 1990. Zheru Chi, Hong. Yan, Image segmentation using fuzzy rules derived from k-means Clusters, Journal of Electronic imaging, Vol 4 (Issue 2): 199-206, 1995. J. J. E. Slotine, L. Weiping, Applied Nonlinear control, (PrenticeHall. 1991). S.Chen, S.A.Billings, P.M.Grant, Recursive hybrid algorithm for nonlinear systems identification using Radial Basis Function Networks, Internat. journal of Control Vol 55, 1051-1070, 1992. P. C. Chen, C. W. Chen, W. L. Chiang, GA-based modified adaptive fuzzy sliding mode controller for nonlinear systems, Expert Systems with Applications, Vol 36, 5872–5879, 2009. Xiaojun. Ban, X. Z. Gao, Xianlin. Huang, A. V. Vasilakos, Stability analysis of the simplest Takagi–Sugeno fuzzy control system using circle criterion, Information Sciences, Vol 177, 4387–4409, 2007.

Authors’ information Mohamed Bahita obtained his “Ingénieur“ and his Master Degree both in control Engineering from the University of Constantine, Algeria. He is working towards his PhD at the same University. He is currently a lecturer with the Faculty of Hydrocarbons and Chemistry (FHC), University of BOUMERDES, Algeria. His main interests are in artificial intelligence and adaptive control of nonlinear systems. Khaled Belarbi obtained his “Ingénieur “ degree from “Ecole polytechnique, Algiers, Algeria, and MSc and PhD in control Engineering both from Control System Center UMIST, Manchester, UK . He is currently a professor with the Faculty of Engineering, University of Constantine, Algeria. His current interests are in predictive control and fuzzy and neural control.



152

International Review of Automatic Control (I.RE.A.CO.), Vol. 4, N. 2 March 2011

Design of Model Reference Adaptive Intelligent Controller Using Neural Network for Nonlinear Systems R. Prakash, R. Anita Abstract – In this paper a new approach to a neural network based intelligent model reference adaptive controller is proposed. In this scheme, the intelligent supervisory loop is incorporated into the conventional model reference adaptive controller framework by utilizing an online growing multilayer back propagation neural network structure in parallel with it. The idea is to control the plant by conventional model reference adaptive controller with a suitable single reference model, and at the same time respond to plant by online tuning of a multilayer Back propagation neural controller. In the conventional MRAC scheme, the controller is designed to realize plant output converges to reference model output based on the plant which is linear. This scheme is for controlling linear plant effectively with unknown parameters. However, using MRAC to control the nonlinear system at real time is a difficult. In this paper, it is proposed to incorporate a Neural Network (NN) in MRAC to overcome the problem. The control input is given by the sum of the output of conventional MRAC and the output of NN. The NN is used to compensate the nonlinearity of the plant that is not taken into consideration in the conventional MRAC. The proposed NN-based model reference adaptive controller can significantly improve the system behavior and force the system to follow the reference model and minimize the error between the model and plant output. The effectiveness of the proposal control scheme is demonstrated by simulations. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Model Reference Adaptive Controller (MRAC), Artificial Neural Network (ANN), Backlash and Dead Zone

In the adaptive literature, the question of control of nonlinear systems with present day sophistication and complexities has often been an important research area due to the difficulties in accurately modeling, estimating system nonlinearities and uncertainties. Model Reference Adaptive Control (MRAC) is one of the main schemes used in adaptive system. Recently MRAC has received considerable attention, and many new approaches have been applied to practical process [1], [2]. In the MRAC scheme, the controller is designed to realize plant output converges to reference model output based on the assumption that the plant can be liberalized.

Therefore this scheme is effective for controlling linear plants with unknown parameters. However, it may not assure for controlling nonlinear plants with unknown structure. In recent years, an Artificial Neural network (ANN) has become very popular in many control applications due to their higher computation rate and ability to handle nonlinear system. Some of the relevant research work including ANN as a part of control scheme is illustrated next. A robust Adaptive control of uncertain nonlinear system using neural network is discussed in [3].Various types of NN have been efficiently utilized in identification of nonlinear systems [4]-[5]. A variety of algorithms are utilized to adjust the weight of the NN. In a typical multilayered NN, the weights in the layers can be adjusted as to minimize the output error between the NN output and the observed output. The back propagation algorithm for efficiently updating the weight is useful in many applications such identification of non linear systems. Off-line iterative algorithm can be employed in such care of identification or modeling. However, in the aspect of control, the NN should work in on line manner. In the control system structure, the output of NN is the control input to the nonlinear controlled system. i.e., there is the unknown nonlinear system between the NN and the output error. In this case, in order to apply any learning rules, we need the

Manuscript received and revised February 2011, accepted March 2011


Nomenclature up yp r ym θ Γ U Umr

ω

Plant input Plant output Reference model input Reference model output Controller gain Positive gain Control input Adaptive controller output Regressor vector

I.

Introduction

153

R. Prakash, R. Anita

[26]. Naoki Uchiyama et al. [27] presented a Model Reference Control for Collision Avoidance of a HumanOperated Robotic Manipulator. A Multi-machine system, Vector control and model reference adaptive control is discussed in [28]. In this paper a proposal neural network-based model reference adaptive controller is designed from a multilayer back propagation neural network in parallel with a model reference adaptive controller. The control input is given by the sum of the output of adaptive controller and the output of neural network. The neural network is used to compensate the nonlinearity of the plant that is not taken into consideration in the conventional MRAC. The role of model reference adaptive controller is to perform the model matching for the uncertain linearized system to a given reference model. The network weights are adjusted by multilayer back propagation algorithm which carried out in online. Finally to confirm the effectiveness of proposed method, we compared the simulation results of the conventional MRAC with the proposed method. The paper is organized as follows section II proposes the problem statement and section III discusses the structure of an MRAC design. Section IV describes the proposed approach. Section V analysis the result and discussion of the proposed scheme and the conclusions are given in section VI. In this paper a proposal MRAC is designed from a multilayer back propagation neural network in parallel with a model reference adaptive controller.

derivatives of the system output with respect to the input [6]. Kawalo et al [7] presented a simple structure of NN based feed forward controller which is equivalently an inverse of the controlled system after the NN completes learning of the weights which are adjusted to minimize the feedback error. Narendra et al [8] has shown in general indirect approach to nonlinear discrete time neuro – control scheme which consists of identification and adaptive control by using the NN Chen [9], and Liu et al [10] that the NN – based adaptive control algorithm can cooperate well with identification of the nonlinear functions to realize a nonlinear adaptive control when the non linear adaptive control when the nonlinear control scheme is feedback linearizable. Kamalasudan [11] presented a fighter aircraft pitch controller evolved from a dynamic growing RBFNN in parallel with a model reference adaptive controller. The abilities of a neural network for nonlinear approximation and development for nonlinear approximation and the development of a nonlinear adaptive controller based on neural networks has been discussed in many works [12]-[13]. In particular, the adaptive tracking control architecture proposed in [14] evaluated a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of uncertainty is either unknown or impossible. The use of neural networks for identification and control of nonlinear system has been demonstrated in [15] discusses a direct adaptive neural network controller for a class of non linear system. An online radial basis-function NN (RBFNN) in parallel with a model reference adaptive controller (MRAC) is discussed in [16]. A neuro-sliding mode approach based on model reference adaptive control (MRAC) is proposed in [17]. An adaptive-neuro-fuzzybased sensorless control of a smart-material actuator is presented in [18]. An adaptive inverse model control system (AIMCS) is designed for the plant, and two radial basis function (RBF) neural networks are utilized in the AIMCS discussed in [19]. Xiang-Jie Liu et al. [20] discussed an adaptive inverse model control system (AIMCS) is designed for the plant, and two radial basis function (RBF) neural networks are utilized in the AIMCS. A model reference adaptive control (MRAC)-based current control scheme of a PM synchronous motor with an improved servo performance is presented in [21]. Fadali, et al. [22] presented a robust adaptive control approach using model reference adaptive control (MRAC) for autonomous robot systems with random friction. An adaptive output-feedback control scheme is developed for a class of nonlinear SISO dynamic systems with time delays [23]. An adaptive inverse model control system (AIMCS) is designed for the plant, and two radial basis function (RBF) neural networks are utilized in the AIMCS [24]. A. Karami et al. [25] discussed Variable Structure Model Reference Adaptive Control for Vehicle Steering System. A Model Reference Adaptive Predictive Control Scheme for General Nonlinear Systems is presented in

II.

Statement of the Problem

To consider a Single Input Single Output (SISO), Linear Time Invariant (LTI) plant with strictly proper transfer function: G (s) =

yP ( s )

up (s)

= KP

Z p (s) RP ( s )

(1)

where up is the plant input and yp is the plant output. Also, the reference model is given by: Gm ( s ) =

ym ( s ) r (s)

= Km

Zm ( s ) Rm ( s )

(2)

where r and ym are the model’s input and output. Define the output error as: e = y p − ym

(3)

Now the objective is to design the control input u such as that the output error e goes to zero asymptotically for arbitrary initial condition, where the reference signal r(t) is piecewise continuous and uniformly bounded.



154


III. Structure of an MRAC Design

where

III.1. Relative Degree n =1

( SI − F )

( n − 1)* ( n − 1) stable

kp km

matrix such that

T

(5)

is a vector of adjustable

parameters, and is considered as an estimate of a vector of unknown system parameters θ* . The dynamic of tracking error: e = Gm ( s ) p*θT ω

(7)

kp

and θ = θ ( t ) − θ * represents parameter km error. Now in this case, since the transfer function between the parameter error θ and the tracking error e is Strictly Positive Real (SPR) [1], the adaptation rule for the controller gain θ is given by:

Where P* =

( )

θ = −Γe1ω sgn p*

(8)

IV.

where Γ is a positive gain.

In the standard adaptive control scheme, the control u is structured as: T

(

where θ = [θ1 ,θ 2 ,θ3 ,C0 ]

T

)

(9)

is a vector of adjustable

parameters, and is considered as an estimate of a vector of unknown system parameters θ * . The dynamic of tracking error is: e = Gm ( s ) ( s + p0 ) p*θT φ

Proposed Approach

To make the system adaptable to more quickly and efficiently than conventional MRAC system, a new idea is proposed and implemented. The new idea which is proposed in this paper is the neural network-based model reference adaptive controller. In this scheme, the controller is designed by using parallel combination of conventional MRAC system and neural network controller. The block diagram of the proposed neural networkbased model reference adaptive controller is shown in Fig. 1. The theoretical basis for the proposed scheme is as follows. The state model of linear time invariant system is given by the following form:

III.2. Relative Degree n =2

u = θ T ω + θ Φ = θ T ω − θ T Γφ e1 sgn K p / K m

(11)

where e1= yp-ym and Γ is a positive gain. The adaptive laws and control schemes developed are based on a plant model that is free from disturbances, noise and unmodelled dynamics. These schemes are to be implemented on actual plants that most likely to deviate from the plant models on which their design is based. An actual plant may be infinite in dimensions, nonlinear and its measured input and output may be corrupted by noise and external disturbances. It is shown by using conventional MRAC that adaptive scheme is designed for a disturbance free plant model and may go unstable in the presence of small disturbances. When the disturbances and non linear component are added to the input of the plant of the conventional model reference adaptive controller such that the tracking error has not come to zero and the output of the plant output is not tracked with the reference model plant output. The large amplitude of oscillations will come entire period of the plant output and the tracking error has not come to zero .The disturbance is considered as a random noise signal. To improve the system performance, the neural network-based model reference adaptive controller proposed, the controller is designed by using parallel combination of conventional MRAC system and neural network controller. The control signal coming from the MRAC is added to the neural network controller and then given to the plant input.

In the standard adaptive control scheme, the control u is structured as: u = θTω (6) where θ = [θ1 ,θ 2 ,θ3 ,C0 ]

represent the

θ = Γφ e1 sgn ( K p / K m )

the zeros of the reference model and that (F, g) is a controllable pair. We define the “regressor” vector: T

θ = θ ( t ) − θ *

Strictly Positive Real (SPR). Now in this case, since the transfer function between the parameter error θ and the tracking error e is Strictly Positive Real (SPR), [1], the adaptation rule for the controller gain θ is given:

is a Hurwitz polynomial whose roots include

ω = ⎡⎣ω1T ,ω2T , y p ,r ⎤⎦

and

parameter error. Gm ( s ) ( s + p0 ) is strictly proper and

In Ref [1] the following input and output filters are used: ω 1 = F ω1 + gu p , (4) ω 2 = F ω2 + gy p where F is an

P* =

(10)



155


X ( t ) = AX ( t ) + BU ( t ) , Y ( t ) = CX ( t ) + DU ( t )

U d = c −1 ( x2 d − ax1 + bx2 d )

(12)

which is the same as:

(

U d = D y p ,x2 d ,x2 d

)

(19)

where D is functional relation between states, control and output. Thus it possible to have a system response equals to desired value if the controller Ud can effectively inverse the system dynamics. In other words the controller U should track the system such that e =0. However due to system dynamics, the error equation has to be written as:

Fig. 1. Block diagram of the proposed MRAC

This scheme is restricted to a case of Single Input Single Output (SISO) control, noting that the extension to Multiple Input Multiple Output (MIMO) is possible. To keep the plant output yp converges to the reference model output ym, it is synthesize the control input U by the following equation: U = U mr + U nn

(18)

e = ( xd − x ) = 0

Thus the controller U should be written as: U = c −1 ( x2 d − ax1 + bx2 d ) + U mr

(13)

(20)

The neural network control law now becomes:

where Umr is the output of the adaptive controller:

(

U d = D −1 y p ,x2 d ,x2 d

)

(21)

T

U mr = θ ω ,

θ = [θ1 ,θ 2 ,θ3 ,C0 ] , T

ω = ⎡⎣ω1 ,ω2 , y p ,r ⎤⎦

where yp is the plant output .from the above discussion it can be seen that the input to the neural network should be: (22) X = ⎡⎣ y p ,x2 d ,x2 d ⎤⎦

(14)

T

Stability of the system and adaptability are then achieved by an adaptive control low Umr tracking the system state x to a suitable reference model such that error e = yp – ym = 0 asymptotically. The controller design concept is illustrated using the following state equation of the second order system, which can be expanded to higher order system comfortably:

x1 = x2 , x2 = ax1 + bx2 + cU

The design procedure multilayer back propagation neural network controller and derivation are discussed next.

IV.1. Learning of NN The relations between inputs and output of NN is expressed as:

(15)

Z −inj = Voj +

and let the output: y p = x1

(16)

P

U = c −1 ( x2 − ax1 + bx2 )

(

(24)

)

(25)

Yk = F (Y−ink )

(26)

Z j = F Z _inj

(17)

(23)

i =1

Y−ink = W01 + ∑ j =1 z jW j1

Differentiating: y p = x1 = x2 = ax1 + bx2 + cU ,

n

∑ xiVij

where F (.) is the activation function. We chose sigmoid function for the activation function as follow:

Suppose a controller Ud can be established which should track a desired signal say x2d then the controller equation can be written as: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


156


F ( x) =

2a −a 1 + exp ( − µ x )

where:

(27)

∂E = − ym − y p ∂y p

(

where µ > 0 , a is a specified constant such that a ≤ 0 , and F(x) satisfies: –a

Automatic Control

Automatic Control

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