Stabilization of Neural Control Systems Using the Conicity Criterion

Copyright © IFAC Control of Industrial Systems, Belfort, France, 1997

STABILIZAnON OF NEURAL CONTROL SYSTEMS USING THE CONICITY CRITERION

Fernandez de Caiiete, J.; Garcia-Cerezo, A.;Garcia-Gonzlilez, A.; Macias, C. ; Garcia-Moral, I.

Dpt. Ing:~nieria de Sistemas y Automatica. E. TS. de Ingenieros en Informatica Campus de Teatinos. 29071 MALAGA (SPAIN) e_mail: [email protected]

Abstract: Stable designs of neural based controllers have relied on the existence of a Lyapunov energy function or have been based upon the local controllability of the system under consideration. In this paper a stability analysis is realized using the conicity criterion, and based upon this, a RBF neural controller is adjusted, so that the closed loop dynamic is stable. A stabilization technique is formulated and application to an inherently unstable system, i.e. an inverted pendulum is also described, demonstrating through simulation the effectiveness of the proposed method. Keywords: Neural Control, Stabilization, Design Method, Input-Output Criterion, Gaussian.

In this paper the conicity input-output stability criterion is applied as a design method to stabilize a RBF neural controller for a unknown nonlinear dynamical system. Firstly, the neural controller is adapted using backpropagation of the output error through the plant provided a set of input-output training data uniformly distributed to cover the system output space. Secondly, using the conicity criterion, a new training data set is elicited so that a stable neural controller is obtained. Application to an inverted pendulum demonstrates through simulation the effectiveness of the proposed method.

1. INTRODUCTION Radial Basis Function networks (RBF) have been shown to be universal approximators of nonlinear functions (Hartrnan and Keeler 1990). Little work has been reported about using RBF networks applied to identification and control of dynamic systems. In such cases, invertibility of controlled plant (Hunt and Sbarbaro 1991) or feedback linearizable systems (Sanner and Slotine 1992) are assumptions usually made. Stable designs of neural based controller have relied on the existence ofa Lyapunov energy function (Yabuta and Yamada 1990; Carelli et al 1995) or have been based upon the local controllability of the system (Levin and Narendra 1993). On the other hand the conicity stability criterion (Safonov 1980) has been successfully applied to the design of stable fuzzy controllers (Garcia-Cerezo and Ollero 1994).

2. PROBLEM STATEMENT The conicity criterion defmes a sufficient condition for the stability of a closed loop system formed by a linear plant G (s) and a nonlinear static controller H (y) where r and v correspond to the reference and

451

A design technique is presented to stabilize a previous neural controller adapted using backpropagation of the output error through the plant. Identification techniques are necessary in this process.

x

3. PROBLEM SOLUTION Consider now an-dimensional nonlinear plant" given by

Fig. 1. Linear system with nonlinear controller

x= disturbance signals respectively, x being the system output (Fig. 1).

u

f(x) + b (x) . u

= NNe(e)

e

= r-x

(6) (7)

The system can be rewritten as Defmining a cone (C, r) where C and r are the centre and the radius respectively, the stability is given by the following three conditions (Safonov 1980):

x=

In case of reference r == 0 it is possible to realize a separation, since e = -x yielding to

(1 )

x=

1

- -G- (s) is outside the cone (C, r) which is

equivalent to

H(e) -1

F(s) = G (s) . (l + Co G (s»

is stable (2)

F (s) being

the linear feedback formed substituting H(y) for C.

r ~ rh (C)

= max y .. 0

- Co yll

IlYII

r
-1

= A·e+f(-e)

G(s)

system

+b(-e) ·NNe(e)

=

(sI-A)

-1

H(y) = A ·y+f(-y) +b(-y) 'NNe(y)

(9)

(10)

(11) (12)

and conicity criterion can be applied. A suitable change of coordinates x o = x - r is enough to study the stability at r:;t 0 .

The problem is to fmd in the space of all cones (C, r) a suitable one (Co' r o) for which the three conditions hold. The conditions (1) and (3) yield respectively

IIH (y)

A ·x+H(e)

Under these conditions G (s) and H (y) of fig. 1 are defmed by

(3)

maxwlIF(jw) 11 . r < 1

(8)

where A is the jacobian of f(x) at the origin.

- H(y) is inside the cone (C, r)

IIH (y) - Co yll < r . IlYll

A ·x-A ·x+f(x) +b(x) ·NNe(e)

Once obtained the conicity domains rh (C) and rg (C) applying the conicity criterion, if the closed loop system is not stable ( rh (C) n r (C) = ~ ) it should be convenient to alter the nonl~ear protocol H (y) for those values;' fulfilling

(4) (5)

IIHeY) - C* . ;'11

From Fig. 2 an interpretation of the conicity criterion is that for some stable feedback C (condition (2) limits a subset Se on (C, r) ), the robustness r g (C) of G (s) with feedback C must be greater than the conic deviation rh (C) of the neural control law given by z = H(y) from the linear law z = C· y .

I~II

> r g (C·)

(13)

with C· E Se given ~y C· = mine (rh (C) -rg(C». These pairs (y, H (y») outside the critical cone ( C·, r (C·») are modified so that the new value - g H(y) for each y corresponds to the closer point if eY) located on the surface of this cone. In this way we force the nonlinear protocol H(y) to be inside the critical cone minimally altering its original value, so output response characteristics are maintained while stability is assured.

In this paper a reformulation is introduced for applying this criterion to nonlinear plants controlled by neural networks.

Changes on HeY) are reflected in the neural controller provided that the system's dynamic is not altered, so backpropagation of the error if eY) - HeY) through H (y) as defmed by (12) is used to produce the new stable neural controller. C (dim n)

Identification techniques using RBF networks have been used to estimate the true values of f(x) and b (x) . Starting from the off-line identification of the

Fig. 2. An interpretation ofthe conicity criterion

452

plant ;

=

'"

(14)

NN;(:x,u)

.g

i':E-,.. - -.·.1

estimates of b (x) and f(x) are given by aNN;

b(x) =

a;;-

~~5

(15)

~ ~

-1

0

50

100

150

Time

(16)

J(x) = ;-b(x) . u

1

nE---------

4. APPLICA nON

°_10-

A fourth order nonlinear inverted pendulum has been chosen to demonstrate the effectiveness of the proposed method. In this case C is a 4x4 matrix, therefore there are multiple combinations when the conicity criterion is applied. Visualization in 3D requires two elements Cij to be varied while the rest remain fixed.

o

50

100

150

Time

Fig. 5. State trajectories with modified neural controller initiated at x = (0.2,-2 ).

Fig. 3 shows the system trajectories during regulation around the origin for the neural RBF controller trained initially with backpropagation. Graphs of conicity domains rh (C) and r g (C) show that the feedback control system is unstable, provided that backpropagation by itself doesn't assure stability (Fig. 4).

29J

. ': -':J::::::: ~.: : : ::::J:::::::: :::::r:H::::::::': :::::::::,:

.'

./

200

...

:... :

,

.'; ..~ .. _.. -. -~- -. _. -::' .

<

1Sl 100 -200

so o -150

10' i·::E--------J .I .s:'e>

-1

50

0

100

Stabilization of this scheme has been realized modifying those training patterns which yield the nonlinear protocol to be outside the critical cone. In Fig. 5 it is shown how after this process the system stabilizes around the origin, due to the intersection between the conicity domains (Fig. 6).

150

rE-~1 o

50

~.

100

100

Fig. 6. Domains ofConicity

Time

O~o------

so

150

Time

5. CONCLUSIONS

Fig. 3. State trajectories with neural controller initiated at x = (0.2,-2).

A stability analysis has been realized based upon the conicity criterion extended to nonlinear plants. A design procedure has been described to stabilize the overall close loop system by modifying the previously trained neural controller.

/:<:r:::::::::::::::::j::::::::f::::::::;::::::: . .

. 300

•

--~"-""--i-----

.. -t..-

Future directions are going toward the application of the proposed technique to the design of stable neural controllers for experimental plants.

29J

'Sl 100

-200

50

REFERENCES

o -150

so

Carelli, R., E. Camacho and D. Patifio (1995). "A Neural Network Based Feedforward Adaptive Controller for Robots". IEEE Trans. on Systems,

100

Fig. 4. Domains of Conicity

453

Man and Cybernetics, Vol. 25, pp 1281-1288. Garcia-Cerezo, A. and A. Ollero (1994). "Design of Stable Fuzzy Control Systems from Experimental Data". Proceedings ofthe IV European Forum on Intelligent Techniques. pp 1175-1182. Hartman E. 1. and J.D. Keeler (1990). "Layered Neural Networks with Gaussian Hidden Units as Universal Approximators". Neural Computation Vol. 2, pp. 210-2 J5. Hunt K. 1. and D. Sbarbaro (1991) ."Neural Networks for Nonlinear Internal Model Control". Proceedings ofthe IEEE. Vol. 138, no. 5. pp. 431-438. Levin, A. and K Narendra (1993). "Control of Nonlinear Dynamical Systems Using Neural Networks: Controllability and Stabilization". IEEE Trans. on Neural Networks. Vol. 4, pp 192-206. Safonov, M. (1980). Stability and Robustness ofMultivariable Feedback Systems. M.LT. Press. Sanner, R.M. and J. E. Slotine (1992). "Gaussian Networks for Direct Adaptive Control". IEEE Trans. on Neural Networks. VoI. 6, pp 837-864. Yabuta, T. and T. Yamada (1990). "Possibility of Neural Networks Controller for Robot Manipulators" IEEE Int. Conference on Robotics and Automation, pp 1686-1691.

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