Stable Nonlinear Adaptive Control With Growing Radial Basis Function Networks

Copyright © IFAC Adaptive Systems in Control and Signal Processing. Budapest. Hungary. 1995

STABLE NONLINEAR ADAPTIVE CONTROL WITH GROWING RADIAL BASIS FUNCTION NETWORKS Visakan Kadirkamanathan· and Simon Fabri· ·Dep4rtm.et~t of AUtOm.4tic CO'll.trol €J Sy,tem., E'II.gi'll.eeri'll.g, U'II.i1ler,ity of Skeffield, PO Bo: 600, M4ppi'll. Street, Skeffield S1 "DU, UK

Abstract. A:n. adaptive control technique, wing dynamic atructure Gawaian radial basia function neural networks, that grow in time based on the location of the ayatem'a atate in apace, ia preaented for the affine clasa of nonlinear ayatem. having unknown or partially known dynamica. The method reaulta in a network that ia 'economic' in tenna of network aize, for casea where the atate apans only a amall aubaet of atate·apace, by utilizing leaa basia functions than would have been the case if basia functions were centred on diacrete locations covering the whole, relevant region of atate.apace. Additionally, the ayatem ia augmented with aliding control ao as to ensure global atability if and when the atate movea outside the region of at ate-apace apanned by the basia functions, and to ensure robwtneaa to diaturbancea that ariae due to the network inherent approximation erron. Key Words. Adaptive control, Feedback lineariaation, Neural networlta, Radial Basia FUnction network, Nonlinear control, Lyapunov atability

ically exhibits the property that during operation, state x(t) will only scan a subset of the space in 'X.M. Most of the basis functions that lie outside this subset are not used and their parameters are barely adapted. This has previously lead to the development of growing RBF networks in time-series analysis and system identification (Platt, 1991; Kadirkamanathan, 1991; Kadirkamanathan and Niranjan, 1993) and in classification (Kadirkamanathan and Niranjan, 1992). These networks begin with no basis functions (or nodes) and 'grows' by activating the relevant nodes on-line. A growing Gaussian RBF network adaptive controller, similar to the above ideas, was suggested in (Sanner and Slotine, 1992). The idea here is to place basis functions only on mesh points that lie in the vicinity of the regions where the state has been present whilst the system is evolving, thereby utilising only the most relevant basis functions - referred to as selective node activation.

1. INTRODUCTION The use of Gaussian radial basis function (RBF) neural networks (NN) (Broomhead and Lowe, 1988) for stable adaptive tracking control of the affine class of nonlinear systems (Slotine and Li, 1991; Sastry and Bodson, 1989; Isidori, 1989) has been recently developed (Polycarpou and loannou, 1991; Sanner and Slotine, 1992; TzirkelHancock and Fallside, 1992; Tzirkel-Hancock, 1992). In these schemes, NNs are used to approximate the unknown, or partially known, system dynamics which are nonlinear functions of the Mdimensional system state x(t). The control signals are generated based on feedback linearisation techniques. The NN parameters are adjusted online via laws derived from considerations similar to Lyapunov-based model reference adaptive control techniques (Narendra and Annaswamy, 1989; Sastry and Bodson, 1989), that ensure asymptotic boundedness of the tracking error.

The presence of network approximation errors introduces disturbances that could lead to parameter drift (Narendra and Annaswamy, 1989). This problem is overcome by augmenting the control law with low-gain sliding control whilst x E 'X.M (Tzirkel-Hancock and Fallside, 1992). Furthermore, as in (Sanner and Slotine, 1992), when x rt. 'X.M, high-gain sliding control is used to force the state quickly back into 'X.M whenever it drifts out, for example, during transient periods.

Gaussian RBFs, due to its localisation property, are typically centred on regular points of a square sampling mesh covering a relevant region of statespace, where the states of the dynamical system x(t) is known to be contained, represented by a compact set 'X.M C !R M - the network approximation region. The number of points, and hence the number of basis functions in 'X. M , increase exponentially with the increase in order of the system or M - known as the curse of dimensionality. As noted in (Sanner and Slotine, 1992; TzirkelHancock, 1992), in such cases, the system typ-

In this paper a growing network technique based on the proposal of (Sanner and Slotine, 1992) I

245

with the use of low-gain sliding control proposed in (Tzirkel-Hancock, 1992) ensuring robustness to disturbances is developed. This offers an advantage over the robust adaptive technique of (Sanner and Slotine, 1992), namely the use of simple deadzone adaptation, since the minimum number of basis functions that must be activated at any one time is independent of the tracking accuracy required, and hence the network growth-rate is more restricted. In practice, the only limitation on the minimum number of basis functions requiring activation arises from the need to restrain the maximum control bandwidth (Slotine and Li, 1991; Tzirkel-Hancock, 1992), so that excitation of high frequency unmodelled dynamics is avoided. Therefore, a compromise between the tracking accuracy and network growth-rate against the maximum control bandwidth must be settled for.

Since the Lie derivatlves JtX), gtX) are assumea unknown, adaptive control techniques are used to keep Zl bounded where NNs estimate /(x), §(x) which are used in the control law:

(2) where, U ..I (t )

=

- j(x) + tI(t) §(x)

(3)

is inspired from feedback linearisation control laws (Isidori, 1989; Sastry and Bodson, 1989; Slotine and Li, 1991). tI(t) = y~,.) - o,.e(,.-l) - ... - 0le is an auxiliary input, e = (y - Yd) is the output tracking error and coefficients 0i are chosen such that r(,,) = (,," + 0,.",.-1 + ... + 01) is a Hurwitz polynomial in terms ofthe Laplace variable ". The second component of the control law,

2. CONTROL OF AFFINE SYSTEMS

(4)

The controller design goal is to ensure that the system output y(t) tracks a desired output Yd(t), where y(t) = [y y(l) ... y(,.-l)jT and Yd(t) = [Yd y~l) ... y~"-l)]T. The affine class of nonlinear, single-input single-output (SISO) systems are described in an affine form:

y(") = f(x)

+ g(x)u(t)

represents an approximate sliding-mode component based on smoothed out boundary layer techniques (Slotine and Li, 1991) where, "at(r) is a saturation nonlinearity ("at( r) = r if Irl < I, "at(r) = 1 if r ~ 1 and -1 otherwise). e1(t) is a filtration of the tracking error e(t) such that e1(t) = /3,.e(,.-l) + ... + /31e or in the s-domain e1(") = ~(,,)e(,,) where ~(,,) is a Hurwitz polynomial and such that ~(,,)r-1(,,) is strictly positive real (Narendra and Annaswamy, 1989). k.. is a state-dependent sliding mode gain and tP is the boundary layer associated with the sliding mode term introduced to avoid chattering and limit the control signal bandwidth (Slotine and Li, 1991).

(1)

where f(x), g(x), are nonlinear functions (Lie derivativesof the system dynamics) of the state vector x(t): lR+ 1-+ lRM and r iss the relative degree of the system. y(i), y~i), are the i-th derivatives w.r.t time t of system output and desired output, assumed bounded 'rI i, 0 $ i $ r. u(t) iss the input to the system ie., the control signal. When the relative degree r is defined and r < n, a state diffeomorphism T converting states x to socalled normal states z can be proven to exist near any given point x = X o (!sidori, 1989; Sastry and Bodson, 1989; Slotine and Li, 1991), such that in the neighbourhood of x o , system (1) can be expressed in normal form:

Z11

=

Zl~,

Zl~

Zl~_l

=

Zl~,

Z2 Y

f(x)

3. GROWING RBF NETWORK

3.1. The neural networks Gaussian radial basis function (RBF) neural networks (NNs) are used to generate the approximations j(x), §(x), j(x; W/) g(x, wg)

+ g(x)u

I(Zl,Z2)

=

= lC(x)w (t)~ / (x) + fo(x) = lI:(x)w g (t)~g(x) + go(x)

f

(5)

and W/ , W9 are K -dimensional vectors containing the linear coefficients with K being the number of basis functions increasing with network growth. ~ / (x) (and ~ 9 (x)) is K -dimensional Gaussian RBF vectors whose k-th element is given by,

Z11

where z = [Zl z2]T, Zl(t) : lR+ 1-+ lR" (= y(t)) represents the output and its (r - 1) derivatives, Z2(t) : lR+ 1-+ lRn -,. represents the internal dynamics and I(Zl,Z2) is a vector function of Zl, Z2. If the zero-dynamics 1(0, Z2) are globally exponentially stable and I(Zl' Z2) have continuous and bounded partial derivatives in Zl, Z2, good tracking performance and stability is achieved for the affine class of systems when controlled by a law that keeps Zl bounded (Sastry and Bodson, 1989).

(6) such that m/" (and milt) is M-dimensional representing the coordinates of the mesh point on which the k-th basis function is centred. o} (and 246

0';) is the variance representing the 'spread' of

other are nearly linear dependent. Since we use NNs to approximate some underlying function, the set of basis functions used must span the space in which the underlying function lies. Thus, we must choose RBFs that are near the state tr~ jectory. This observation has been used to develop a growth criterion in which the angle between any two basis functions must be greater than a specified threshold, which can be specified in terms of the distance between the node centres of these RBFs (Kadirkamanathan, 1991; Kadirkamanathan and Niranjan, 1993). Hence, one could consider an activation criterion where, a node le (in XM) is activated at time t provided that it was not previously activated and that it satisfies the condition ~k(X(t)) ~ Drnin where Drnin represents an activation threshold (6rnin is constant and satisfies 0 < Drnin :s 1). From the definition of Gaussian RBFs, it follows that for an RBF centred at IDJ" or ID", to be activated,

RBFs. lo(x) and go(x) are known prior estimates of I(x) and g(x respectively. K(X) is a statedependent function included, as in (Sanner and Slotine, 1992), to utilise the NN outputs whilst x E XM' where XM is a slightly smaller subset of the network approximation region XM, and to consider only the prior estimates lo(x), go(x) when x f!. XM since the networks are incapable of proper approximation when x lies in this region. Thus K(X) is defined as, K(X) = 0 if x f!. XM, K(X) = 1 if x E XM' Given any uniform bounds (J and (" and the prior estimates lo(x), go(x), optimal parameter vectors wj, and an optimal number of basis functions K* exist, such that 'V x E XM, the network approximation errors denoted by 6J(x) and 6,(x), satisfy the conditions:

w;,

16/1 = Ir(x) 16,1 = Ig*(x) -

l(x)1 :s (/ g(x)1 :s (,

(7)

(8)

with K* defining the number of basis functions (nodes) required in XM to satisfy (7), and r(x) = io(x;wj,K*) and g*(x) = 9o(X;w;,K*). Hence K* and XM directly determine the distance between the points of the sampling mesh, ~, required to satisfy the approximation error bounds (7). The optimal parameters wj, are however unknown and, in the growing network scheme, the number of basis functions K (t) utilised at anyone time is less than K* - only the relevant nodes centred on particular mesh points in the vicinity of x(t) are used for function approximation via (5), the rest being neglected so as to limit the network size. The actual parameter vectors Vi J , Vi, need are adapted on-line while the system stability in the Lyapunov sense is ensured.

which is the inside of M -dimensional hyper-sphere centred on x(t) and with radius J-20'2ln (6rn in). Hence, all nodes centered on mesh points within this sphere must be activated if not activated previously. Note that the larger the activation threshold Drnin, the smaller is the size of the sphere, and the lesser the activated nodes are at anyone time, restricting the network growth-rate further. It turns out that, although marginally more nodes would be activated, it is more convenient to activate nodes that are located on mesh points within a hypercube centred on the nearest mesh point to x(t) with radius equal to l~, where I is the closest integer to J-20'2In(Drnin)/~,rather than the sphere. In this scheme the hypercube edges always coincide with the mesh points. The mesh points represent locations where a RBF could be centred if the node activation criterion is satisfied, rather than the position of RBFs actually located at all of these points, as in the fixed size network schemes of (Sanner and Slotine, 1992; Tzirkel-Hancock, 1992).

w;

3.2. The node activation criterion

In Gaussian RBF, each basis function has localised field of influence and those nodes that are centred 'far away' in state space from the present location of x(t) barely contribute to the network output. Indeed, if the state never visits certain regions of space in Xn, any nodes located in these regions need not have existed at all. Hence, as proposed in (Sanner and Slotine, 1992), one could consider activating nodes that lie in the vicinity of the trajectory ie., nodes whose basis function ~k(X) has been significantly large at some time during the operation interval.

4. NEURAL ADAPTIVE CONTROLLER 4.1. The adaptation law

r

By definition of (x), g* (x) and (5) and assuming conditions (7) are satisfied, we get,

I(x) g(x)

The localisation property of the Gaussian RBF implies that if for two basis functions the node centres are far apart, these two basis functions will be nearly orthogonal (Kadirkamanathan, 1991; Kadirkamanathan and Niranjan, 1993). However, basis functions with node centres close to each

= K(X)W?~J - 6/ + 10

=

K(X)w;T~, - 6,

+ go

(9)

where ~ J' ~ 9 represent the K* -dimensional RBF vectors representing all nodes in XM, ie., both, nodes that have been activated and those representing basis functions that have not yet been activated, referred to as passive nodes. 247

Consider indexing wj, w;, the optimal full-size NN parameters, such that the first series of elements are from the active nodes used in the growing network scheme, denoted by K-dimensional wj .. , w;.. and the next series of elements are from the full-network passive nodes, denoted by wjp' w;p' Similarly, ~j [~r.. ~rpJT and ~g = [~r.. ~rp]T where sub-vectors ~j .. (x), ~g .. (x) contain the active nodes and ~jp(x), ~gp(x) contain the passive nodes. Re-expressing (9) in terms of these sub-vectors and subtracting from (5),

• Parameter resetting mechanism which maintains §.. (x) bounded below by gl(X) - f·(X):If wr.. (t)~g .. (x) + go ~ g,(x) - f·(X),

wg.. (t+) = wg.. (t) +(g/ - g..... )II~g .. (x)"-2~g .. (x)

=

gl(X) is a known lower bound on g(x) g/(x) ~ go't/ x fI. XM' f· > 0 is small such that g,(x)f·(X) > 0 't/ x E XM and t+ denotes the time just after parameter resetting. Then, the Lyapunov function candidate

i(x) - I(x) = lC(x)(wr.. ~j .. - wj;~Jp)

eft. 1 -T1 -TV =-2-+-2WjWj+-2WgW g

+l:1j (10)

=

7]j

=

where Wj .. Wj .. -wj.. , wg .. wg .. -w;.. denote the parameter errors of active nodes.

implyingboundedness ofelt.(t), w/", w,.. , where, Wj = [(Wj .. - wj .. )T ,-wj;JT, wg = [(w g.. w;.. f, _w;;]T are the parameter errors. Boundeness of elt.(t) implies boundedness of el(t) and e(t), so that x(t) is bounded if conditions required for good tracking performance are satisfied. In turn, boundedness ofwj.. , wg .. implies that j, 9 are bounded and since g(x) is bounded away from zero by the parameter resetting mechanism, u..r(t) is bounded. Thus all terms on the right of (12) are bounded so that dd(t) and k.. are bounded, ensuring boundedness of u,/(t). el can be shown to be bounded, and by Barbalat's Lemma (Narendra and Annaswamy, 1989; Slotine and Li, 1991), elt.(t) -+ 0 as t -+ 00 implying that el(t) asymptotically converges to ± 0 then as shown in (Slotine and Li, 1991), in the steady state !e(i)1 ~ 2i~i-r+l
(11)

where dd(t) represents a disturbance term, d d (t) =

ICW .T~ jp ~jp

+ ICW gp .T~

~gpu

../ (12)

-l:1j - I:1 g u..,

that enters the error dynamics due to: • Inherent approximation errors I:1j(x), I:1 g (x) due to NN (always present), and • Selective node activation, which ignores the passive nodes, giving rise to non-zero terms .T~ w .T~ jp ~jp, w gp ~gp'

If the error e is filtered by the transfer function

'1f(8), where '1fr- l (8) = 1/(8 + kd) and kd > 0, then el(t) = '1f(8)e(t) defines a suitable slid-

ing surface (Sanner and Slotine, 1992; TzirkelHancock and Fallside, 1992). The sliding mode control gain of u,,(t) could then be defined such that el (t) remains bounded even if only crude upper bounds on dd(t) are known (Slotine and Li, 1991; Sanner and Slotine, 1992). Consider the use of the following:

4.2. The sliding gain

Determination of the sliding gain k.. , involves knowledge of the disturbance bound dd. From (12) and (7),

• Adaptation laws:-

Wt..

wg..

7]jlC(x)elt.(t)~j

=

.. (x)

7]glC(x)elt.(t)~g .. (x)u..,(t)

(16)

(17)

r-l(g(x)u,,(t) - ICwr.. (t)~j,,(x) -ICWr.. (t)~g .. (x)u../(t) + dd(t))

7]g

results in

Substituting control law (2) into system dynamics (1) and using definitions for v(t), e(t) and (10), e=

(15)

(13)

elt.(t) = 0 if lell <

0 are adaptation constants, • Sliding-mode gain:-

Iddl ~ IClwj;~jpl

+1C(fj

+ IClw;;~gpllu .. d + fglu.. ,1} + (1- 1C)(70 + golu.. d) (18)

where to' go are known bounds of the network ap£.roximation error when x fI. XM, ie., 1/0 - l(x)1 ~ lo(x) and Igo - g(x)J ~ go(x). All the terms in (18) are known except for IWj;~jp(x)1 and Iw;;~,p(x)l. However, it can be shown that IWj;~jp(x)1 ~ Wlb¥jb := 61 following arguments similar to Sanner and Slotine (1992), where,

(14)

248

WJ~ ~ max(\wjp.1) is a bound on the full network optimal parameters and "¥J~ is the bound on the sum of contributions from the ignored passive nodes. Similarly, IW;;~gp(x)1 $ Wg~"¥g~ := 6g. Bounds wJ~' Wg~ could be estimated by subjecting the system to a short period of a persistently exciting input (Narendra and Annaswamy, 1989; Sastry and Bodson, 1989). In practice, an overestimate based on this value is used, to ensure that max(lwjpl) is less than this estimate and use this value for wJ~' Wg~. Hence, from (18h dd := 1C(6J + 6g lu..,1) + IC( EJ + Eg lu.., I) + (1 - 1C)(f 0 + 90 lu.., I).

f(x) = cos(7(:cr + :c~)) exp( -(:cr g(x) = 2 + COS(7:Cl:C2)

The desired state is bounded well within the interval [-1, 1] x [-1, 1] (along :Cl, :C2)' During the transient period the actual state may overshoot these bounds, and hence the network approximation region is taken to be larger, namely XM = [-1.5, 1.5] x [-1.5, 1.5]. This ensures that the state is bounded within XM, thus avoiding the use of crude, high-gain sliding control. XM' being a subset of XM is set to XM = [-1, 1] x [-1, 1]. The known prior estimates to the functions be0, go 2 and as ing approximated are fo shown in (Tzirkel-Hancock and Fallside, 1992), full network inherent approximation error bounds EJ Eg 0.005 would be obtained with RBFs having UJ, ug = 0.03 located on a mesh of spacing jJ. = 0.05 within XM. Bounds /0' go are set to 2 and g,(x) g, 0.895. The activation threshold is set to 0.2. The control law is given by v(t) Yd-5(Y-Yd) and el(t) e(t). '7J, '7g = 25 and EO = 0.01. "¥J~ = "¥g~ = 0.062.

The boundary layer ifJ, introduced in the sliding control term to limit the bandwidth of the control signal of the system, is to avoid excitation of any high frequency unmodelled dynamics (Slotine and Li, 1991). Using the same analysis as in (Tzirkel-Hancock, 1992), adapted to our growing RBF network case,

=

+ 6glu.. rlmu + EJ +E g lu.. rlmu)

=

..J..J.-(6 J 11" •••

(19)

= =

=

=

The system was initially subjected to a 40s persistently exciting input to obtain the order of magnitude for bounds of the optimal parameters, from which overestimates wJ~ Wg~ 1.6 were deduced. This allowed the following setting: 6J = 6g = 0.1. Using Ivl = ly~l)1 $ 5, !hm = 2, 91'. 4, 11, = 200011' rad/8, IIm ..", 400 rad/8, gives ifJmin ::::: 0.01 as the boundary layer. The persistently exciting input is used to obtain 6J , 6g only; once these values are deduced, network activation is started afresh, with no active nodes.

=

The larger the activation threshold 6min , the smaller is the network growth-rate. This however, requires larger sliding gain I .. to offset the effect of the larger approximation errors. This appears as larger 6J' 6g in (19) and so ifJmin increases for the same IImax . Since ifJ affects the value of the asymptotic tracking error, a compromise between the steady-state tracking accuracy and the size of the growing RBF networks against the maximum allowed control bandwidth must be struck. Note that this trade-off is brought about through bandwidth restrictions and not through stability considerations, because the system is ensured to remain robust to the disturbance term via the use of low-gain sliding control in XM. Hence, given high enough IImu , 6min could be set as large as possible to give the smallest network size.

=

=

=

The results of the simulation are shown in Figure 1. The system remains stable and the tracking error converges asymptotically to the expected value of 0.01, the size of the boundary layer. Moreover, as shown in Figure 1, the desired input forces the states to a subset of the space in Xn and thus the selective node activation technique places basis functions only in the neighbourhood of this subset (marked with a +). After 150 seconds of opera.tion, the growing networks contained 556 nodes each, which is 85% less than the 'full-size' networks of 3721 nodes used in (Tzirkel-Hancock and Fallside, 1992).

5. SIMULATION RESULTS The system used in this example simulation is (Tzirkel-Hancock and Fallside, 1992),

= f(x) + g(x)u = :Cl - :C2

=

= =

where 91'. is an upper bound on gh(X), such that g(x) $ gh(X) "Ix, g, is a lower bound on g,(x). lu.. dmu = (Am +v)/g, such that !hm is an upper bound on !h(X), where !h(X) ~ 1!(x)1 and v is an upper bound on v(t), v ::::: y~), the latter being an upper bound on Y~f'») and IImax is the maximum allowable control bandwidth. Hence ifJmin denotes the smallest possible value of ifJ necessary to ensure this bandwidth.

:Cl :C2

(21)

For this system, order n = 2 and degree r = 1. The dynamics are in global normal form and I(:Cl' :c2) = :Cl - :C2 has continuous and bounded partial derivatives. The zero dynamics, Z2 = -:C2 is globally exponentially stable. The reference input, Yd, is a zero-average, 0.9 amplitude, 0.4 H z square wave filtered by 1/ (1 + {ot

4.3. The boundary layer

ifJmin

+ :c~))

(20) 249

which the network size of the growing NN system is significantly smaller compared to a corresponding fixed size network scheme - typical of probl~ms where curse of dimensionality affects the reqUlred number of basis functions - while giving a similar control performance.

Tractong .,rOl- phPO 01

o.• ~----~-=~="':::':""=~---------, o.~

0.3

0.25 0.2 0.1S

7. REFERENCES

0.1

Broomhead, D. S. and D. B. Lowe (1988). Multivariable functional interpolation and adaptive networks. Complex Systems, 2, 321-355. Isidori, A (1989). Nonlinear Control Systems: An Introduction. Springer-Verlag, Berlin. Kadirkamanathan, V (1991). Sequential Learning in Artificial Neural Networks, PhD Thesis, University of Cambridge, UK. Kadirkamanathan, V. and M. Niranjan (1992). Application of an architecturally dynamic network for speech pattern classifica.tion. Proc. Inst. Acoustics, 14(6), 343-350. Kadirkamanathan, V. and M. Niranjan (1993). A function estimation approach to sequential learning with neural networks. Neural Computation, 5, 954-975. Miller, W. T., R. S. Sutton and P. J. Werbos, eds. (1990). Neural Networks for Control. MIT Press, Cambridge, MA. Narendra, K, S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, Englewood Cliffs, NJ. Platt, J. C (1991). A resource allocating network for function interpolation. Neural Computation, 3(2), 213-225. Polycarpou, M. and P. Ioannou (1991). Identification and Control of Nonlinear Systems Using Neural Network Models:Design and Stability Analysis. Tech. Rep. 91-09-01, Dept. Electrical Engineering-Systems, University of Southern California, LA, USA. Sanner, R. M. and J. J. E. Slotine (1992). Gaussian Networks for Direct Adaptive Control, IEEE Trans. Neural Networks, 3(6). Sastry, S. and M. Bodson (1989). Adaptive Control:Stability, Convergence and Robustness. Prentice-Hall, Englewood Cliffs, NJ. Slotine, J. J. E. and W. Li (1991). Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ. Tzirkel-Hancock, E. and F. Fallside (1992). Stable Control of Nonlinear Systems using Neural Networks. Int. J. Robust and Nonlinear Control, 2, 63-86. Tzirkel-Hancock, E (1992). Stable Control of Nonlinear Systems using Neural Networks. PhD Thesis, University of Cambridge, UK.

-o·,0L-------:':~-----~IOO::-----~1",

:: :·:::::i:::::: 0.4

~

:

:

::: :::;±;:+::::++++++++++++++++++++++++ :::.+: +++++.. .. + ++ + ++ .. + ++++++.. +++.. ++

~:. ~ ~ ~ fIIIII:~ :~ ::::~ :~ ~ :ii ~ ±i : ~ : 0:: :;:::::: ::+++++++:;;;:+::+:::::::+::

0.2 :

:::

:

:: I:~~:::::::~~~~~~~~~~~~~!~~:!;~;~;·:!: ........................

-o.S : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

-0.8 -!1

: : : : : : : : : : : ::

~~~~~~~~~~~~~~~~~~~~~~~~ -0.8

-0.8

-0.4

-0.2

0 .1

_ +

.

0.2

S~I"ajoc1ay

.

AcWenodes

Pa~w nodes 0.4

0.8

0.8

Fig. 1. Growing RBF network adaptive control (a) Tracking error (b) Active nodes in state space.

6. CONCLUSIONS

An adaptive control scheme based on growing Gaussian radial basis function (RBF) neural networks (NN) for affine nonlinear systems has been presented, based on Lyapunov stability considerations. The growing NN is used to approximate the underlying functions and generate control signals based on a feedback linearisation control law . The work presented here is based on the system developed in (Tzirkel-Hancock, 1992; Tzirkel-Hancock and Fallside, 1992), which was extended and modified to utilise the growing NN developed in (Platt, 1991; Kadirkamanathan, 1991) and proposed in (Sanner and Slotine, 1992). The scope behind the use of growing networks is to keep the size of the neural networks limited, by ignoring basis functions that would be centred on mesh points located beyond the active regions of state-space. The system is augmented by sliding control, as in (Sanner and Slotine, 1992; TzirkelHancock, 1992) to ensure global stability and robustness to the presence of a disturbance term, arising from the inherent approximation errors in the growing network scheme. The effectiveness of the system was demonstrated by a simulation in 250