identification and control of nonlinear systems...

IDENTIFICATION AND CONTROL OF NONLINEAR SYSTEMS USING NONLINEARLY PARAMETERIZED ARNN

F. M. Raimondi*, T. Raimondi* and P. Barretta*

*Automation and System Engineering Department University of Palermo

Viale delle scienze 90128, Palermo Italy Tel.++3991481119, ++3991427940

e-mail: [email protected], [email protected] Keywords: Recurrent neural networks, Adaptive control, Gaussian radial basis functions, Lyapunov stability.

Abstract

In this paper, a new algorithm is described for on-line identification and adaptive control of MIMO affine nonlinear systems having unknown dynamics a priori, by using a nonlinearly parameterized additive recurrent neural network (ARNN). The ARNN uses the radial basis functions (RBF) as activation functions. However, some adjustable parameters (centers and variances) in RBF appear nonlinearly and the determination of the adaptive law for such parameters is a nontrivial task. Then, we propose a new method in order to determine the training laws of all the RBF and ARNN parameters which allow to reduce the identification and control error for tracking tasks of time trajectories. Additionally, the system is augmented with sliding control to offset the higher-order terms in the Taylor series of RBF output. Such a development is necessary for the linearization of the RBF with respect to the parameters and, therefore, to obtain the training laws of the ARNN. The study of the total system stability is based on the Lyapunov theory. The theoretical results are verified through simulations executed on a simple nonlinear system.

1 Introduction

The recent years have seen a development in the use of neural networks for the identification and control of nonlinear systems. The neural network possesses powerful approximation capabilities and can therefore be used for identification of unknown, or rather partially unknown, nonlinear dynamic systems. In the last decade, the neural architectures used for identification and control are the MLP (MultiLayer Perceptron) networks and the RBF (Radial Basis Function) networks. Their application range goes from the indirect adaptive control [4], with heuristic approach, to the direct adaptive control [6], [9], and [10].

The RBF network is more suitable for on-line adaptation, being insensitive to the order of presentation of the signals used for adaptation. They also require less computation time for learning and have a more compact topology [3]. With the use of Gaussian activation functions, the RBF network forms a local representation (hyper-ellipsoids), as opposed to the

sigmoidal MLP (hyper-planes), where each basis function responds only to inputs in the neighbourhood of a unit center and the spread is determined by the unit variance.

The first to have introduced the use of neural networks in dynamical systems identification and control were Narendra and Parthasarathy in [4]. They employed static MLP networks connected either in series or in parallel with linear dynamical systems, where the synaptic weights were updated through a gradient learning algorithm. However, the stability of the total system was verified only through results of simulations. Sanner and Slotine [10] incorporate Gaussian radial basis function neural networks with sliding mode control and linear feedback, to formulate a direct adaptive tracking control architecture. Besides, they developed a systematic procedure for determination off-line the variances and centers of the basis functions that censure the network approximation accuracy to be uniformly bounded everywhere within a relevant and finite region of state space. However, in the practice, the RBF networks require a very high number of hidden neurons to approximate functions in wide intervals. Polycarpou e Ioannou [5] employed Lyapunov stability theory to develop stable adaptive laws for identification and control of SISO dynamical systems with unknown nonlinearities, using various neural network architectures. Lewis and Fierro [1] applied the MLP networks with a robustifying control signal, to guarantee tracking performance in robotic systems. They have also shown that the backpropagation rule alone is insufficient to assure stability of the whole system. Rovithakis and Christodoulou [8] presented indirect and direct adaptive control schemes based on a recurrent neural network model of the unknown system. As activation functions were used the logistic functions. Also in this case the Lyapunov technique was used to provide answers to the problems of stability, convergence, and robustness.

Generally, the most of above works impose restrictions on the forms of allowable nonlinearities and, furthermore, the control laws need the a priori knowledge of the upper bound on the modelling error and on the norm of the optimal parameter values of the used neural network. However, in many practical cases such bounds may not be known.

In this paper, a new algorithm is provided for on-line identification and adaptive control of MIMO affine nonlinear systems having unknown dynamics a priori, by using a

Control 2004, University of Bath, UK, September 2004 ID-211

mailto:[email protected]

mailto:[email protected]

nonlinearly parameterized additive recurrent neural network (ARNN). The ARNN uses the radial basis functions (RBF) as activation functions. However, some adjustable parameters (centers and variances) in RBF appear nonlinearly and the determination of the adaptive law for such parameters is a nontrivial task. Then, we propose new method for the determination of the training laws of all the RBF and ARNN parameters which allow to reduce the identification and control error for tracking tasks of time trajectories. Additionally, the system is augmented with sliding control to offset the higher-order terms in the Taylor series of RBF output. Such a development is necessary for the linearization of the RBF with respect to the parameters and, therefore, to obtain the training laws of the ARNN. The study of the stability of the total system is based on the Lyapunov theory. The theoretical results are verified through simulations executed on a simple nonlinear system.

2 Additive recurrent neural network

We consider the dynamic model of a neuron as shown in Fig.1. The terms represent the synaptic weights, the respective inputs x

1 2ˆ ˆ ˆ, , ,i i iw w wK n

1(t),x2(t),…,xn(t) represent the potentials and represent the input to the i-th neuron; n is the number of the state variables. Therefore, the filter input of the Fig.1 is:

( )iu t

1

ˆˆ ( ) ( )n

ij j i ij

w x t h u t=

+∑ , (1)

where ih is the input gain of the i-th neuron. Let vi(t) be the input potential to the nonlinear element g(·). Then, it is possible to express the dynamic of the order first filter through the following differential equation:

( ) ( )ii i

dv t k v tdt

+ , (2)

where 1 represents the time constant associated to the i-th neuron. Given the activation potential v

iki(t), we determined the

output of the i-th neuron using the nonlinear relation: ( ) ( ( ))i ix t g v t= , (3)

where g(·) is a continuous nonlinear function and then differentiable, so-called activation function. We now consider a recurrent neural network (that is, with feedback) constituted by interconnected n neurons, each of which described by the equations (1), (2) and (3). It is possible to define the dynamic of the network through the following system of coupled differential equations of the first order:

1

( ) ˆˆ( ) ( ) ( )n

ii i ij j i i

j

dv t k v t w x t h u tdt =

= − + +∑ , (4)

with . The model described by the equation (4) is called additive model; this terminology is used to distinguish it from the multiplied models where depends from x

1, 2,...,i = n

i

ˆ ijw j. In definitive, the (4) can be rewrite in the following form

which represents the dynamic model of the additive recurrent neural network (ARNN):

ˆˆ ( )Ti i i i iv k v h u= − + +& w g v , (5)

where v is the state vector (nx1) of model, is the i-th

vector of synaptic weights associated to the i-th neuron, and g(·) is the vector (nx1) of activation functions.

ˆ iw

)(⋅g

)(1 tx

)(2 tx

)(txn

( )iu t

1ˆ iw

2ˆ iw

ˆ inw

( )ix t( )iv t

∑

ih∑ filtro

M

Figure 1. Dynamic model of a neuron.

2.1 Identification of a nonlinear system

In this section, we consider the problem of identifying a continuous time nonlinear dynamic system of the form:

( ) ( )i i i ix f b u= +& x x , (6) where ix is the i-th element of the state vector (nx1) x, which is assumed to be available for measurement, u is the i-th element of the input vector (nx1) u, and , are the i-th elements, respectively, of the smooth vector fields f and b defined on real set to n dimensions.

i

if ib

The problem of identification consists of choosing an opportune model and modifying the parameter values of the model itself according to some training laws such that his response of the identification model approximates the response of the real system, both forced to the same input signal u.

In this paper, we consider series-parallel identification scheme based on the ARNN that uses as activation potential the Gaussian radial basis function (RBF). The identification model is described from (5), or (4), with:

( 2 2ˆ ˆ ˆ( , , ) expi i i i i )ˆg a a= − −v c v c , (7)

where ĉi e âi are, respectively, the vector (nx1) representing the centers and the value representing the variance associated to the i-th element of the vector of activation functions . Then, the parameters characterizing the vector of activation functions are ĉ=[ĉ

g

1 ĉ2 …ĉn ]T and â=[â1 â2… ân]T, that is ˆ ˆ( ) ( , , )=g v g v c a . It has been proven [5] that the RBF network

satisfies the conditions of the Stone-Weierstrass theorem and is capable to uniformly approximate any real continuous nonlinear function on the compact set . This involves that RBF networks are universal approximator. How it is possible to observe, the vector of the RBF is nonlinear with respect to the parameters ĉ

n⊆ ℜX

i and âi, we thus call it a nonlinearly parameterized approximator.

The real nonlinear system (6) can be expressed as: ( , , )T

i i i i i i ix k x h u L= − + + + ∆& w g x c a , (8) where ix is the i-th element of the state vector (nx1) x of the not well known system, and , c, a, are the optimal estimation values concerning the i-th element of the state. At last,

iw ih

iL∆ is the modelling error concerning the i-th element of the state. We note that they are defined as the values of the parameters that minimize the modelling error. Assumption 1. The optimal estimation values and the


modelling error are bounded in norm. Defined the identification error and

using the equations (5) and (8), the dynamic of the identification error can be written:

( ) ( ) ( )i i ie t x t v t= −

ˆ ˆ( , , ) ( , , )T Ti i i i i i ie k e h u L= − + − + + ∆%& w g x c a w g v c a

ˆ%i

i

, (9) where . In order to obtain an adaptive law for the parameters it’s convenient to consider the first order approximation of the vector as activation functions. Using the Taylor series expansion of

i ih h h= −

( , , )g x c a around the point we obtain:

ˆ ˆ( , )c a

ˆ ˆ( , , ) ( , , ) ( ) ( ) ( , , )g g= + + +% % %g x c a g x c a J c c J a a Ο x c a% , (10) where , are the estimates of the optimal centers and variances respectively, and represents the higher order terms of the expansion. The Jacobiane matrices present in (10) are given from:

ˆ= −%c c c ˆ= −%a a a( , , )% %Ο x c a

1 21 1 2 2

1 2

ˆ ˆ ˆ ˆ ˆˆ ˆ( ) ( , , ) ( , , ), ( , , ), , ( , , )ng n nT T T T

n

gg gdiag a a a

∂∂ ∂∂= = ∂ ∂ ∂ ∂

KgJ c x c a x c x c x cc c c c

ˆ ,

(11) 1 2

1 1 2 21 2

ˆ ˆ ˆ ˆ ˆˆ ˆ( ) ( , , ) ( , , ), ( , , ), , ( , , ) ,ng n nT

n

gg gdiag a a aa a a

∂∂ ∂∂= =

∂ ∂ ∂∂ K

gJ a x c a x c x c x ca

ˆ

(12) with:

2

2

3

( )( , , ) 2 ( , , ),

( , , ) 2 ( , , ).

i ii i

i i

iii i i i

i i

ga g

a

ga g

a a

∂ −=

∂

−∂=

∂

x cx c x cc

x cx c x c

ia

a (13)

Now, by replacing (10) in (9) we obtain: ˆ ˆ ˆ ˆ( , , ) ( ) ( )

ˆ ( ) ( ) ,

Ti i i i g g

Ti g g i i i

e k e

h u E

= − + − − + + + + + ∆

%&

%% %

w g v c a J c c J a a

w J c c J a a (14)

where represents the i-th vector of the synaptic weight estimation error, and represents the disturbance terms expressed as:

ˆi i= −%w w wi

iE∆

[ ]ˆ ˆ ˆ ˆ( , , ) ( , , ) ( , , )

( ) ( ) .

T Ti i i

Ti g g i

E

L

∆ = + − +

+ + +

% %

%

w Ο x c a w g x c a g v c a

w J c c J a a ∆

n

(15)

Lemma 1. Given the disturbance terms , there exist an vector , with

iE∆,0 1 2 3[ , , , ]T

i i i i il l l l=l 1, 2,i = K , such that: ˆl+ a aT= lE l l∆ ≤ +0 1 2 3ˆ ˆ ˆ ˆ( , , )i i i i i i i i il+w c s w c , (16)

where ˆ ˆ ˆ ˆ ˆ ˆ( , , ) 1, , ,T

i i i= s w c a w c a . Proof. Using (10), since the RBF and partial derivatives are superiorly and lowerly bounded, the following inequality is verified:

0 1 2( , , ) d d d≤ + +% % % %Ο x c a c a , (17) where d0, d1 e d2 are positive constants. Given the disturbance terms (13), using (15), we obtain:

0 1 2( )ˆ ˆ ˆ ˆ( , , ) ( , , )

( ) ( ) .

i

g g i

E d d d

L

∆ ≤ + +

+ −

+ + +

% %

%

w c a

w g x c a g v c a

w J c c J a a ∆

n

(18)

Considering the assumption 1, opportunely picking up the constant terms present in (16), we obtain (14), with

. Q.E.D. 1, 2, ,i = K

We consider the Lyapunov function candidate:

1 3 4

1 12 2

nT

ii

V Vη η=

= + +∑ % % % %c c a aT , (19)

with: 2 2

1 2

1 1 12 2 2

Ti i i i

i iV e h

η η= + + %% %w w i

4

, (20)

where 1 2 3, , ,i iη η η η are positive constants. Differentiating (16) with respect to the time and using (12) we obtain:

[ ]{ }

[ ]

21

1

22

1 ˆ ˆ ˆ ˆ( , , ) ( ) ( )

1 ˆ ˆ ( ) ( ) .

Ti i i i i i g g

i

Ti i i i i i g g i i i

i

V k e e

h h u e e e E

ηη

ηη

= − − − − −

− − − + + ∆

&& %

&% % %

w w g v c a J c c J a a

w J c c J a a

ˆ i

(21) Using the following training laws of the parameters:

[ ]1

2

ˆ ˆ ˆ ˆ( , , ) ( ) ( ) ,

ˆ , 1, 2, , ,

i i g g

i i i i

e

h u e i

η

η

= − −

= =

&

&K

w g v c a J c c J a a i

n (22)

the derivative of (17) results: 2

1 1

33 1

44 1

1 ˆ ( )2

1 ˆ ˆ( )2

n n

i i i ii i

nT T T

g ii

nT T T

ˆ i

g i ii

V k e e E

e

e

ηη

ηη

= =

=

=

= − + ∆

− −

− −

∑ ∑

∑

∑

&

&%

&%

c c J c w

a a J a w

. (23)

Now using the following relations:

31

41

ˆ ˆ( ) ,

ˆ ˆ( ) ,

nT Tg i i

in

T Tg i i

i

e

e

η

η

=

=

=

=

∑

∑

&

&

c J c w

a J a w (24)

the (19) is modified to:

( 2

1

n

i i i ii

V k e e=

= − − ∆∑& )E . (25)

2.2 Control algorithm In this section, we determine a control algorithm for tracking tasks of time trajectories. The objective is to determine the control law u such that the state of the first order system (4) (or (5)) can track a desired trajectory . Now we consider the following input control:

( )t( )d tx

1 ˆ ˆ ˆ( , , )ˆT

i di i di ii

u x k xh = + − & w g v c a , (26)

where dix is the i-th element of the desired trajectory vector . Note that the derivation operations on the desired

trajectory don’t present any problem, after this last one is analytically well known. The error between the estimated state v t and

dx

( )i ( )dix t is given as: ( ) ( ) ( )di i die t v t x t= − , (27)

and is necessary obtaining the dynamic equation of the tracking error ( ) ( ) ( )ci i die t x t x t= − to evaluate the property of convergence of the real system state on the desired state. To


such a purpose, by differentiating (25) and substituting the (24) in (5) we obtain:

di i die k e= −& . (28) We note that, the tracking error can be written as

where is the identification error, with .

( ) ( ) ( )ci i die t e t e t= +1, 2, ,i n= K

( )ie t

Theorem 1. Consider the nonlinear system of the first order (6). We assume that the control law is given by:

ai i siu u u= + , (29) with:

1 ˆ sgn( )ˆT

si i ii

uh

= − l s ie . (30)

Furthermore, we assume that the training laws of the parameters are given by:

[ ]1

2

31

41

5

ˆ ˆ ˆ ˆ( , , ) ( ) ( ) ,

ˆ ,

ˆ ˆ( ) ,

ˆ ˆ( ) ,

ˆ sgn( ),

i i g g i

i i ai i

nT Tg i i

i

nT Tg i i

i

i i i i i

e

h u e

e

e

e e

η

η

η

η

η

=

=

= − −

=

=

=

=

∑

∑

w g v c a J c c J a a

c J c w

a J a w

l s

&

&

&

&

&

ˆ

i

(31)

where 1 2 3 4 5, , , ,i iη η η η η( )ci t

are positive constants. Then, the tracking error e , with 1, 2, ,i n= K , asymptotically converges to zero and the estimate error of the parameters is uniformly bounded. Proof. We consider the Lyapunov function candidate:

2

1 1 5

1 1 1 02 2

n nT

e di ii i i

V V eη= =

= + + ≥∑ ∑ % %l li

)

si

)

)

, (32)

where the scalar function V is defined by (17). Differentiating (32) with respect to time, using (28, 29, 30) and observing:

(2

1

n

i i i i sii

V k e e E u=

= − − ∆ − ∑& , (33)

we obtain:

( )2 2

1

sgn( )n

Te i i i di i i i i i i

i

V k e k e e e e E u=

= − + + − ∆ − ∑ %& l s .(34)

Substituting the robustifying term (31) in (34) we get:

(2 2

1

sgn( )n

Te i i i di i i i i i

i

V k e k e e E e=

= − + − ∆ −∑ %& l s . (35)

The (35) is bounded by:

(

(

2 2

1

2 2

1

2 2

1

sgn( )

sgn( )

0,

nT T

e i i i di i i i i i iin

Ti i i di i i i i i

in

i i i dii

V k e k e e e

k e k e e e e

k e k e

=

=

=

≤ − + − −

≤ − + − −

≤ − + ≤

∑

∑

∑

l s l s

l s

&

)

(36) Applying the results found in the section 4, we obtain the feedback linearising control based on the ARNN obtained in the phase of identification. To ensure the convergence of the tracking error to zero, the control law (24) is augmented by a sliding mode control term (29). The initial values of the variables present in (29) are shown in table I. To remove the members in high frequency (chattering effect) imposed by the robust term, in the present simulation we consider the logarithmic function to the place of the sign function.

since: sgn( ) 0i i ie e e− = . (37)

By (36) we obtain that all the error are bounded. Therefore by integrating (36) we have:

{ } [ ]2 2

01

1,

1( ) ( ) (0)min

n

i di ei i

i n

e e d V Vk

τ τ τ∞

∞=

=

+ ≤ − ∑∫K

< ∞ , (38)

Which implies that e e 2,i di L∈ . By (12) and (28), since the desired trajectory dix and the time derivative they assume bounded, we obtain that . Then, using the Barbalat’s Lemma, the identification error and the tracking error

,i die e L∞∈& &

( )t( )ie t

( ) ( )i die t ecie t = +( ) 0t ie t→∞

asymptotically converge to zero, that is lim = and lim ( )t cie t 0→∞ = , with

1, 2, ,i n= K . Q.E.D.

3 Simulation experiments

In this section, we presented the results obtained in the simulation of the control algorithm applied to the following nonlinear system:

1 1 2 1

2 2 1

5 3sgn( ) ,10 2sgn( ) ,2

x x x ux x x= − + += − + +

&

& u (39)

with 1(0) 2x = and 2 (0) 2x = − . Our algorithm can be separated into two phases: the first

takes into consideration only the identification of the system, and the second, in case of success of the first, determines the control laws that allow the system to tracking a desired trajectory.

3.1 Phase 1 (identification)

In the identification phase, we consider the ARNN described by the equation (5) with 2n = , and the input is given by two sinusoidal functions of gain equal to 12. The parameters of the ARNN are trained according to the equations (18) and (20) with 1,2i = . The sampling time is fixed to the value of 0.001 s. The initial values of the variables (parameters and state) as well as the value of all design constants of the ARNN are random and shown in tables I, respectively. The obtained results are presented in fig. 2.

Constants of ARNN Value 1 2 3 4 5i i iη η η η η= = = = 10 ik 8

Table I. Constant values of ARNN.

3.2 Phase 2 (control)

The problem is to develop a control law such that the state of the system (30) can track a reference trajectory given by:

[ ]Td tt )4.0cos(),4.0sin(=x . (40)


0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3actual and estimate state x1

time (in seconds)

actual

estimate

a)

0 2 4 6 8 10 12 14 16 18 20-2

-1.5

-1

-0.5

0

0.5

1

1.5actual and estimate state x2

time (in seconds)

actual

estimate

b)

Figure 2. Time evolution: (a) identification of x1; (b) identification of x2.

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

2

time (in seconds)

actual and desired tracking trajectory x1

actual

desired

a)

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

time (in seconds)

actual and desired tracking trajectory x2

actual

desired

b)

Figure 3. Time evolution of the reference and real trajectories: (a) x1 and (b) x2.

The results obtained during the control phase are shows in fig. 3. The moving of the real trajectory of the system from the respective reference shows that the characteristics of tracking of the ARNN scheme are very good, as it is predicted by the theoretical analysis.

4 Conclusion

In this paper, a new algorithm for the identification and control of MIMO affine nonlinear systems characterized by unknown dynamics has been presented. It’ is used an additive recurrent neural network nonlinear with respect to the parameters. The considered activation functions for the ARNN are the Gaussian radial basis functions with the update parameters (centers and variances). In particular, the control algorithm does not need an off-line learning or training phase. It’ is showed that the ARNN uses a minimum hidden neuron (only two) number to approximate the function taken into consideration. A robustifying control term is also needed to overcome higher-order modelling error terms.

References [1] R. Fierro and F. L. Lewis, “Control of a nonholonomic mobile

robot using neural networks”, IEEE Trans. Neural Networks, vol. 9, no. 4, pp. 589-600, 1998.

[2] T. Fukao, H. Nakagawa and N. Adachi, “Adaptive tracking control of a nonholonomic mobile robot”, IEEE Trans. Robot. Autom., Kyoto, Japan, vol. 16, no. 5, pp. 609-615, 2000.

[3] R. I. W. Lang, “A future for dynamic neural networks”, Dept. Cybernetics, University of Reading, UK, pp. 1-26, 2000.

[4] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks”, IEEE Trans. Neural Networks, vol. 1, no. 1, pp. 4-27, 1990.

[5] T. Poggio and F. Girosi, “Networks for approximation and learning”, Proc. IEEE, vol. 78, pp. 1481-1497, 1990.

[6] M. M. Polycarpou and P. O. Ioannou, “Neural networks as on-line approximators of nonlinear systems”, IEEE Conf. Decision Contr., Tucson, AZ, pp. 7-12, 1992.

[7] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems”, IEEE Trans. Automat. Contr., vol. 41, no. 3, pp. 447-451, 1996.

[8] G. A. Rovithakis and M. A. Christodoulou, “Adaptive control of unknown plants using dynamical neural networks”, IEEE Trans. Syst. Man. Cybern., vol. 24, pp. 400-412, 1994.

[9] N. Sadegh, “A perceptron network for functional identification and control of nonlinear systems”, IEEE Trans. Neural Networks, vol. 4, pp. 982-988, 1993.

[10] R. M. Sanner and J. J. Slotine, “Gaussian networks for direct adaptive control”, IEEE Trans. Neural Networks, vol. 3, no. 6, pp. 837-864, 1992.


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