Direct adaptive neural control for affine nonlinear systems

Indrani Kar *, Laxmidhar Behera

Department of Electrical Engineering, Indian Institute of Technology, Kanpur, 208016 UP, India

Applied Soft Computing 9 (2009) 756–764

A R T I C L E I N F O

Article history:

Received 15 March 2007

Received in revised form 8 October 2008

Accepted 12 October 2008

Available online 25 October 2008

Keywords:

Feedback linearization

Adaptive control

Neural network

Lyapunov stability

A B S T R A C T

This paper presents a direct adaptive neural control scheme for a class of affine nonlinear systems which

are exactly input–output linearizable by nonlinear state feedback. For single-input–single-output (SISO)

systems of the form x ¼ f ðxÞ þ gðxÞu, the control problem is comprehensively solved when both f ðxÞ and

gðxÞ are unknown. In this case, the control input comprises two terms. One is an adaptive feedback

linearization term and the other one is a sliding mode term. The weight update laws for two neural

networks, which approximate f ðxÞ and gðxÞ, have been derived to make the closed loop system Lyapunov

stable. It is also shown that a similar control approach can be applied for a class of multi-input–multi-

output (MIMO) systems whose structure is formulated in this paper. Simulation results for both SISO- and

MIMO-type nonlinear systems have been presented to validate the theoretical formulations.

� 2008 Elsevier B.V. All rights reserved.

Contents lists available at ScienceDirect

Applied Soft Computing

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1. Introduction

The input–output behavior of a nonlinear system can be madelinear under certain assumptions by means of nonlinear statefeedback. The controller can be proposed in such a way that theclosed loop error dynamics becomes linear as well as stable [1]. Themain problem with this control scheme is that the cancellation ofthe nonlinear dynamics depends upon the exact knowledge ofsystem nonlinearities. When the system nonlinearities are notknown completely but some bounds on them are known, thenonlinearities can be approximated either by neural networks orby fuzzy systems. The controller then uses these estimates tolinearize the system. This concept was used in several papers [2–6]dealing with the study of feedback linearizable systems. Sannerand Slotine [7] have studied the applications of Gaussian networksfor direct adaptive control. Spooner and Passino [8] proposed bothdirect and indirect adaptive control schemes which use Takagi–Sugeno fuzzy systems to provide asymptotic tracking when theplant dynamics are poorly understood. Choi and Farrell [9]proposed a non-parametric stable adaptive control approach usinga piecewise linear approximator. Lewis et al. [10] designed a directadaptive control for robot manipulators using radial basis andmulti-layer networks. Zang et al. [11] proposed a modifiedLyapunov function to show the closed loop system stability. Wangand Hill [12] presented a new learning mechanism by which anadaptive neural controller is capable of learning the unknown

* Corresponding author. Tel.: +91 512 2597854.

E-mail addresses: indranik@iitk.ac.in (I. Kar), lbehera@iitk.ac.in (L. Behera).

1568-4946/$ – see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.asoc.2008.10.001

nonlinear systems dynamics while controlling the system. In arecent paper [13], we have proposed variable gain controllerswhere the nonlinear systems have been approximated in theframework of T–S fuzzy model.

In this paper, we propose a direct adaptive control scheme toachieve output tracking of affine nonlinear systems. The mainadvantage of a direct adaptive control scheme over an indirectadaptive control scheme is that in a direct adaptive control schemethere is no need for explicit system identification. In indirectadaptive control scheme, the system is generally identified off-linefrom its input–output data and the controller is designed based onthe identified system model. The identified model should beaccurate enough for better performance of the controller. More-over stability is a critical issue in indirect adaptive control. But indirect adaptive control, the controller is designed in such a waythat the closed loop stability is maintained while the tracking errorconverges to 0 with time. The affine systems have a form x ¼f ðxÞ þ gðxÞu where f ðxÞ and gðxÞ are two nonlinear functions. Forgeneral applications, there may happen two distinct cases. First iswhen the system nonlinearity f ðxÞ is unknown but the inputnonlinearity gðxÞ is known. Second is when both f ðxÞ and gðxÞ areunknown. The systems are assumed to have a relative degree of n,i.e. after n differentiation, the input will appear in the output. Oneshould note that for a system to be feedback linearizable jgðxÞjshould be greater than 0, i.e. gðxÞ can be either positive or negative.The nonlinearities can be approximated using neural networks orfuzzy systems. Since the adaptive controller has a formu ¼ ð1=gðxÞÞ½� f ðxÞ þ Ke�, the control problem becomes difficultwhen the nonlinearity gðxÞ is unknown because of the fact that theapproximation of gðxÞ can be 0 at times which makes the controller

Fig. 1. An n input and 1 output RBF network.

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764 757

unbounded. This problem has been addressed in different mannersby various researchers [14,9,11]. In [14], a second controlcomponent is added and a new design parameter is introducedto maintain the boundedness of the input. The region of operationis divided into two regions and the stability of the closed loopsystems is analyzed separately for the two regions. Choi and Farrel[9] have modified the update law using a projection algorithm. Apersistently exciting condition is imposed to show the stability ofthe closed loop system. In [11], a single approximator approx-imates f(x)/g(x) which is used in the control law. A modifiedLyapunov function is used to prove the closed loop stability.However, it is also assumed that the |f(x)| is bounded by a knownfunction and this information is used in a time varying designparameter. These modifications and assumptions make the designmore complex and restricted. In this paper we have solved thisproblem by keeping gðxÞ away from 0 using a projection algorithm.The effect of the projection algorithm on the closed loop stabilityhas been taken care of by introducing a sliding mode term in thecontroller. In many of the earlier works it is assumed that gðxÞ>0.In this paper, the assumption has been relaxed by consideringjgðxÞj>0. However it is assumed that gðxÞ is lower bounded by aknown constant gl. Two radial basis function (RBF) networks areused to approximate the nonlinearities f ðxÞ and gðxÞ. The weightupdate laws of the RBF networks are derived in such a way that theclosed loop system is Lyapunov stable and the output trackingerror converges to 0 with time. When the approximation error ofthe RBF networks are taken into account, the weight update lawsare modified in [10] for closed loop stability. But in this paper wehave shown that the same update law can maintain the closed loopstability with a bounded tracking error. Though the proposedadaptive control scheme is implementable only for a class ofnonlinear systems, however there are many practical systems likesingle link flexible joint manipulator, jet engine compressionsystems [1], MEMS devices [15] for which the proposed controlscheme can be applied successfully. Even if the system dynamicsare readily not available in affine strict feedback form, one canconvert them into the specified form using some transformation.We have further extended the application of the proposedcontroller to multi-input–multi-output (MIMO) systems whenf ðxÞ is unknown and gðxÞ is known. Many of the mechatronics kits,

manipulator systems are examples of affine MIMO systems forwhich the controller derived in this manuscript can be applied.

Rest of the paper is organized as follows. A brief introductionabout feedback linearization techniques for SISO systems followedby directive adaptive control scheme for SISO systems is presentedin Section 2. Section 3 presents the directive adaptive controltechnique for MIMO systems when f ðxÞ is unknown but gðxÞ isknown. Simulation results for three nonlinear systems arepresented in Section 4 with concluding remarks in Section 5.

2. Single-input–single-output affine systems

A large class of single-input–single-output nonlinear systemscan be represented by the following affine system:

x1 ¼ x2

x2 ¼ x3

..

.

xn ¼ f ðxÞ þ gðxÞuy ¼ x1

(1)

where x ¼ ½x1 x2 . . . xn�T 2Rn, y2R and u2R.The control problem: Find u such that xðtÞ follows a desired

trajectory xdðtÞ.

If we choose control input u ¼ ð1=gðxÞÞ½� f ðxÞ þ kvr þ l1eðn�1Þ þ� � � þ ln�1eð1Þ þ xnd� where e ¼ yd � y is the output tracking errorand r ¼ eðn�1Þ þ l1eðn�2Þ þ � � � þ ln�1e (power denotes respectivederivatives), the closed loop error dynamics becomes r ¼ �kvr

which is linear as well as stable. kv and l’s are positive designparameters. Such design techniques are known as feedbacklinearization techniques.

2.1. Direct adaptive control of SISO systems when f ðxÞ is unknown

but gðxÞ is known

The problem arises with feedback linearization controltechniques when the nonlinear functions f ðxÞ or gðxÞ or bothare unknown. For such cases different function approximators likeneural networks or fuzzy systems can be used to estimate thesenonlinear functions. In this section we have assumed that thefunction f ðxÞ is unknown while the function gðxÞ is known. f ðxÞ isapproximated by a radial basis function network (RBFN). Thepictorial diagram of the RBFN when approximating f ðxÞ is shownin Fig. 1. As can be seen from the figure, the RBFN has a singleoutput thus the network weight constitutes a vector W in this case.

Theorem 1. Suppose that the nonlinear function f ðxÞ of system (1) is

unknown while the function gðxÞ is known. Let f ðxÞ be approximated

as fðxÞ ¼ WTfðxÞ using a radial basis function network. Then the

control law u ¼ ð1=gðxÞÞ½� fðxÞ þ kvr þ l1en�1 þ � � � þ ln�1eð1Þ þxnd� will stabilize the system (1) in the sense of Lyapunov provided

W is updated using the update law ˙W ¼ �Ff r.

Proof. The output tracking error is defined as e ¼ yd � y ¼ x1d � x1

where yd ¼ x1d is the desired output of the system. Let us define anew variable r as:

r ¼ eðn�1Þ þ l1eðn�2Þ þ � � � þ ln�1e (2)

where eðn�1Þ is the n� 1th derivative of e and so on. l1; . . . ;ln�1 arechosen such that the above system is stable. The aim is to achieve asatisfactory tracking result as well as to maintain boundedness ofthe error and neural network weight vector. The control law u is asfollows:

u ¼ 1

gðxÞ ½� fðxÞ þ kvr þ l1eðn�1Þ þ � � � þ ln�1eð1Þ þ xnd� (3)

Let us assume that there exists an ideal weight vector W such thatthe original function f ðxÞ can be represented as f ðxÞ ¼WTfðxÞ.According to universal approximation property of neural network,for any smooth function f ð�Þ, there exist weights such that f ðxÞ ¼WTfðxÞ þ e where e is the estimation error. Here we have assumedthat the estimation error e is 0. This assumption will hold good forcomparatively less complex nonlinear functions if one takes asufficiently large number of adjustable weights. Putting the control

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764758

law u (3) in the system (1) we get,

xn ¼ f ðxÞ þ gðxÞ � 1

gðxÞ½� fðxÞþkvr þ l1eðn�1Þ þ � � � þ ln�1eð1Þ þ xnd�

¼WTf� WTfþ kvr þ l1eðn�1Þ þ � � � þ ln�1eð1Þ þ xnd

(4)Defining W

T ¼WT � WT

we can write

xn ¼ WTfþ kvr þ l1eðn�1Þ þ � � � þ ln�1eð1Þ þ xnd

or; xnd � xn ¼ eðnÞ ¼ �WTf� kvr�

l1eðn�1Þ � � � � � ln�1eð1Þ(5)

Differentiating (2) we get,

r ¼ en þ l1eðn�1Þ þ � � � þ ln�1eð1Þ (6)Substituting en from (6) into (5),

r� l1eðn�1Þ � � � � � ln�1eð1Þ ¼ �WTf� kvr�

l1eðn�1Þ � � � � � ln�1eð1Þ

or;r ¼ �kvr � WTf

(7)

Consider a Lyapunov function candidate

V ¼ 1

2r2 þ 1

2W

TF�1W (8)

where F is a positive definite matrix. Differentiating (8),

V ¼ rrþ WTF�1 ˙W (9)

Substituting r from (7) into (9),

V ¼ rð�kvr � WTfÞ þ W

TF�1 ˙W (10)

Since W is constant, we can write ˙W ¼ W� ˙W ¼ � ˙W. Thus,

V ¼ �kvr2 � WTfr � W

TF�1 ˙W

¼ �kvr2 � WTðfr þ F�1 ˙WÞ

(11)

Equating the second term of (11) to 0, we get

fr þ F�1 ˙W ¼ 0

or; ˙W ¼ �Ffr(12)

Using update law (12), equation (11) becomes

V ¼ �kvr2 (13)

Since V >0 and V � 0, this shows stability in the sense of Lyapunovso that r and W (hence W) are bounded. Hence the proof. MoreoverZ 1

0�V dt<1

Again V ¼ �2rkvr and all signals on the right hand side of (7) verifythe boundedness of r and hence of V. Therefore V is uniformlycontinuous. Thus according to Barbalat’s Lemma, V!0; as t!1and hence r vanishes. Since (2) represents a stable dynamics, theoutput tracking error eðtÞ will also vanish with t. &

2.1.1. Inclusion of non-zero RBFN approximation error

Let us now assume that we have a non-zero RBFN approxima-tion error e which is bounded by a small positive number eN , i.e. theoriginal function f ðxÞ can be written as:

f ðxÞ ¼WTfðxÞ þ e

where kek< eN . With this modification the time derivative of theLyapunov function becomes:

V ¼ rð�kvr � WTf� eÞ � W

TF�1 ˙W (14)

Considering the same update law W ¼ �F f r, we get:

V ¼ �kvr2 � e r

V can be further expressed as:

V � �kvr2 þ ke rkor; V � �kvr2 þ eN krk

¼ �krkðkvkrk � eNÞ

The above expression will be negative if kvkrk � eN is positive. Inother words,

kvkrk � eN >0

or; krk> eN

kv

This implies that V is negative definite as long as krk> ðeN=kvÞ. Bychoosing a large kv, the filtered tracking error r can be madearbitrarily small which serves our purpose.

Remark. Direct adaptive control of SISO systems is easily achiev-able when f ðxÞ is unknown and gðxÞ is known.

2.2. Direct adaptive control when both f ðxÞ and gðxÞ are unknown

It is difficult to achieve closed loop stability when both f ðxÞ andgðxÞ are unknown. It is assumed that jgðxÞj>0; 8 x. An additionalsliding mode term is added with the control input to maintain theclosed loop stability.

Theorem 2. Given that both nonlinear functions f ðxÞ and gðxÞ of

system (1) are unknown, let f ðxÞ be approximated as fðxÞ ¼ WTfðxÞ

and gðxÞ be approximated as gðxÞ ¼ PTcðxÞ using two radial basis

function networks. Let u1 ¼ ð1=gðxÞÞ½� fðxÞ þ kvr þ l1eðn�1Þ þ � � � þln�1eð1Þ þ xnd� and u2 ¼ ðjgj=glÞju1jsgnðrÞ. Then the control law u ¼u1 þ u2 will stabilize the system (1) in the sense of Lyapunov provided

W and P are updated using the update laws ˙W ¼ �Ff r and

˙P ¼ 0 when g� gl <0 and c u1 r>0;¼ �Gcu1r otherwise

Proof. Assume that there exist two ideal weight vectors W

and P such that the original functions f ðxÞ and gðxÞ can be repre-sented as

f ðxÞ ¼WTfðxÞ; gðxÞ ¼ PTcðxÞ (15)The control law is:

u ¼ u1 þ u2 (16)

where u1 is defined as:

u1 ¼1

gðxÞ ½� fðxÞ þ kvr þ l1eðn�1Þ þ � � � þ ln�1eð1Þ þ xnd� (17)

xnd can be written in terms of u1 as

xnd ¼ gðxÞu1 þ fðxÞ � kvr � l1eðn�1Þ � � � � � ln�1eð1Þ (18)

Using (15) and (16), system (1) can be rewritten as

xn ¼WTfþ PTcu

¼WTfþ PTcu1 þ PTcu2

(19)

Since xn ¼ xnd � eðnÞ, substituting xnd and eðnÞ from (18) and (6) into(19), we get

gðxÞu1 þ fðxÞ � kv � l1eðn�1Þ � � � � � ln�1eð1Þ � r

þl1eðn�1Þ þ � � � þ ln�1eð1Þ ¼WTfþ PTcu1 þ PTcu2

Fig. 2. An 2n input and n output RBF network.

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764 759

Rewriting the above equation,

r ¼ �kvr �WTfþ WTf� PTcu1

�PTcu1 � PTcu2

or; r ¼ �kvr � WTf� P

Tcu1 � PTcu2

(20)

where W ¼W � W and P ¼ P � P.

To prove the closed loop system stability, let us consider aLyapunov function

V ¼ 1

2r2 þ 1

2W

TF�1Wþ 1

2P

TG�1P (21)

where F and G are two positive definite matrices. Differentiating(21),

V ¼ rrþ WTF�1 ˙Wþ P

TG�1 ˙P (22)

Substituting rðtÞ from (20) into (22),

V ¼ rð�kvr � WTf� P

Tcu1 � PTcu2Þ

þWTF�1 ˙Wþ P

TG�1 ˙P

Since W and P are constants, we can write ˙W ¼ W� ˙W ¼ � ˙W and˙P ¼ P� ˙P ¼ � ˙P. Thus,

V ¼ �kvr2 � WTfr � W

TF�1 ˙W� P

Tcu1r

�PTcu2r � PTG�1 ˙P

¼ �kvr2 � WTðfr þ F�1 ˙WÞ

�PTðcu1r þ G�1 ˙PÞ � PTcu2r

(23)

Equating the second and third terms of (23) to 0, we get

fr þ F�1 ˙W ¼ 0

or; ˙W ¼ �Ffr(24)

and

cu1r þ G�1 ˙P ¼ 0

or; ˙P ¼ �Gcu1r(25)

From (17) it is clear that u1 becomes unbounded when gðxÞ!0. Toavoid this singularity problem, different approaches have beentaken so far. Yesilderik and Lewis [14] used an extra controlcomponent which made the overall control action bounded evenwhen gðxÞ approaches 0. Another way is to keep gðxÞ away from 0using some projection algorithms [8,9]. Using this concept ofprojecting g inside a set where g 6¼0, we modify the update law forP in such a way that when the estimate g is less than the lowerbound gl and at the same ˙P is negative, then we do not update P.Thus the modified update law becomes:

˙P ¼ 0 when g� gl <0 and c u1 r>0;¼ �Gcu1r otherwise

(26)

Using update laws (24) and (25), (23) becomes

V ¼ �kvr2 � PTcu2r¼ �kvr2 � gu2r

(27)

Define the sliding mode term as u2 ¼ ðjgjÞ=ðglÞju1j sgnðrÞ. Eq. (27)then becomes

V ¼ �kvr2 � gjgjgl

ju1jr sgnðrÞ (28)

Since we have assumed that jgj>0 and the lower bound gl isknown to us, the term ðgjgjÞ=ðglÞju1j is always positive. Moreoverr sgnðrÞ is always positive. Thus V is negative definite. Since V >0and V � 0, this shows stability in the sense of Lyapunov so that r, W

(hence W) and P (hence P) are bounded. Hence the proof. Again if V

is uniformly continuous then according to Barbalat’s Lemma,V!0; as t!1. From (28) it is clear that V ¼ 0 only when r ¼ 0,thus r vanishes as t!1.

One should note that to make the control input bounded, theupdate law for P is modified according to Eq. (26). It follows fromEq. (26) that in absence of the sliding mode term u2, V in Eq. (23)equals �kvr2 � P

Tcu1r when ˙P ¼ 0. Since �P

Tcu1r can take both

positive and negative values, negative definiteness of V cannot beensured in this case. To cope with this fact the sliding mode term u2

is introduced in the control law (16). The modification in theupdate law for P suggests that the initial value of P should bechosen such that g> gl. &

3. Direct adaptive control of MIMO systems

In this section we have extended the application of theproposed controllers to MIMO systems. A general MIMO systemcan be written as

x ¼ f ðxÞ þ gðxÞuy ¼ Cx

(29)

where x2Rn, f ðxÞ 2Rn, u2Rm, gðxÞ 2Rm�n, y2R p, C 2R p�n. SomeMIMO systems can also be written in the following form:

x1 ¼ x2

x2 ¼ f 1ðxÞ þ g11ðxÞu1 þ � � � þ g1mðxÞum

x3 ¼ x4

x4 ¼ f 2ðxÞ þ g21ðxÞu1 þ � � � þ g2mðxÞum

..

.

x2n�1 ¼ x2n

x2n ¼ f nðxÞ þ gn1ðxÞu1 þ � � � þ gnmðxÞum

y ¼ ½x1 x3 � � � x2n�1�

For such cases, the system equations can be re-written as:

z1 ¼ z2

z2 ¼ f ðzÞ þ gðzÞuy ¼ z1

(30)

where z1 ¼ ½ x1 x3 � � � x2n�1 �, z2 ¼ ½ x2 x4 � � � x2n �,f ðzÞ ¼ ½ f 1 � � � f n �

T 2Rn,gðzÞ ¼ ½ g11 � � � g1m � � � gn1 � � � gnm � 2Rn�m, z ¼ ½ z1 z2 �T 2R2n.The output error can be defined as:

e ¼ yd � y ¼ z1d � z1

where yd ¼ z1d is the desired output vector. Let us define a variabler as r ¼ eþLe, where L is a diagonal matrix with positive diagonalelements.

� In

case of MIMO systems, the function f ðxÞ or f ðzÞ is a vectorvalued function. � T hus the RBF network has multiple outputs as shown in Fig. 2. � T he network weights constitute a matrix (W) in this case.

t Computing 9 (2009) 756–764

Theorem 3. Suppose that the nonlinear function f ðzÞ of system (30)is unknown while the function gðzÞ is known. Suppose also that f ðzÞcan be approximated as fðzÞ ¼ W

TfðzÞ using a radial basis function

network. Then the control law u ¼ gTðzÞðggTÞ�1½� fðzÞ þ Kvr þL1eþz2d�will stabilize the system (30) in the sense of Lyapunov provided W

is updated using the update law ˙W ¼ �Ff rT.

I. Kar, L. Behera / Applied Sof760

Proof. The control law u is defined as follows:

u ¼ gTðzÞðggTÞ�1½� fðzÞ þ Kvr þL1eþ z2d� (31)

Kv is a positive definite diagonal matrix. Here W is the weightmatrix of appropriate dimension. Let us assume that there exists anideal weight matrix W such that the original vector f ðzÞ can berepresented as f ðzÞ ¼WTfðzÞ. The RBFN approximation error isassumed to be 0. Putting the control law u (31) in the systemEq. (30) and after simplification we get,

z2 ¼WTf� WTfþ Kvr þL1eþ z2d (32)

Defining WT ¼WT � W

Twe can write

z2 ¼ WTfþ Kvr þL1eþ z2d (33)

Again

r ¼ eþL1e ¼ z2d � z2 þL1e (34)

Combining Eqs. (33) and (34),

r ¼ �Kvr � WTf (35)

Consider a Lyapunov function candidate

V ¼ 1

2rTr þ trace½1

2W

TF�1W� (36)

where F is a positive definite matrix. Since the Lyapunov functionshould be a scalar function but W is a matrix in this case, we havetaken trace of ð1=2ÞWT

F�1W. Differentiating (36),

V ¼ rTrþ trace½WTF�1 ˙W� (37)

Substituting r from (35) into (37),

V ¼ rTð�Kvr � WTfÞ þ trace½WT

F�1 ˙W� (38)

Since Wis a constant matrix, we can write ˙W ¼ W� ˙W ¼ � ˙W. Thus,

V ¼ �rTKvr � rTWTfþ trace½�W

TF�1 ˙W� (39)

Using the properties of trace, rTWTf ¼ trace½WT

frT�, we can furtherwrite

V ¼ �rTKvr þ trace½�WTfrT � W

TF�1 ˙W� (40)

Equating the second term of (40) to 0, we get

frT þ F�1 ˙W ¼ 0

or; ˙W ¼ �FfrT(41)

Using this update law, (40) becomes

V ¼ �rTKvr (42)

Since V >0 and V � 0, this shows stability in the sense of Lyapunovso that r and W (hence W) are bounded. Hence the proof.FurthermoreZ 1

0�V dt<1

V ¼ �2rTKvr and all signals on the right hand side of (35) verify theboundedness of r and hence of V. Therefore V is uniformly

continuous. Thus according to Barbalat’s Lemma, V!0; as t!1and hence r vanishes. &

3.1. Inclusion of non-zero RBFN approximation error

As in the SISO case, if we consider a non-zero approximationerror e, the derivative of Lyapunov function becomes:

V ¼ �rTKvr þ e r

To proceed further we will use following properties of matrices:For a positive definite matrix Kv,

lmin ðKvÞkrk2 � rTKvr � lmax ðKvÞkrk2

where lmax ð�Þ and lmin ð�Þ denote the maximum and minimumeigenvalues of a matrix. Thus we can write,

V � �lmin ðKvÞkrk2 þ ke rk� �lmin ðKvÞkrk2 þ eN krk

where kek< eN .The negative definiteness of V can be ensured as long as

krk> ðeNÞ=ðlmin ðKvÞÞ. To improve the tracking accuracy Kv shouldbe chosen such that lmin ðKvÞ is a large number.

4. Simulation results

The performance of the proposed controller is demonstratedthrough simulation results. Two nonlinear systems have beentaken for this purpose. The first example considered here is a SISOsystem while the second example is a MIMO one.

4.1. Example 1

The first example taken for the simulation is same as that of[10]. The dynamics of the nonlinear SISO system to be controlled isgiven by the following equations:

x1 ¼ x2

x2 ¼ f ðxÞ þ gðxÞuy ¼ x1

(43)

where x ¼ ½ x1 x2 �T

f ðxÞ ¼ 4sin ð4px1Þ

px1

� �sin ðpx2Þ

px2

� �2

gðxÞ ¼ 2þ sin ð3pðx1 � 0:5ÞÞ

We use a sinusoidal trajectory of unit amplitude and 1 Hzfrequency as the reference trajectory. The output tracking error isdefined as:

e ¼ yd � y ¼ x1d � x1

When f ðxÞ is unknown and gðxÞ is considered to be known, thecontrol law is:

u ¼ 1

g½� fþ kvr þ x2d þ l1eð1Þ�

where r ¼ eð1Þ þ l1e, eð1Þ ¼ e. The parameters kv and l1 are chosenas 20 and 15, respectively. The number of neurons for the RBFnetwork is taken as 30. The centers of the RBF network are chosenrandomly between 0 and 1 and weights are initialized to very smallvalues. The parameter matrix F is taken as the diagonal matrix ofappropriate dimension with the diagonal elements 1. The outputtracking result and the corresponding control input are shown inFig. 3.

Fig. 3. Left: output tracking result when f ðxÞ is unknown and gðxÞ is known and right: corresponding control input u.

Fig. 4. Left: output tracking result for when f ðxÞ and gðxÞ both are unknown and right: corresponding control input u.

Table 1Parameters used in Eqs. (44) and (45).

Parameters Symbol Value

Area (mm2) A 100

Permitivity (C2=N mm2) e 1

Initial gap (mm) g0 1

Mass (mg) m 1

Damping constant (mg/s) b 0.5

Spring constant (mg=s2) k 1

Resistance (V) R 0.001

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764 761

The control law when both f ðxÞ and gðxÞ are unknown:

u ¼ u1 þ u2; where;

u1 ¼ 1

g½� fþ kvr þ x2d þ l1eð1Þ� and

u2 ¼ jgjgl

ju1j sgnr

where r, kv and l1 are same as that of previous case. The lowerbound of g is set to gl ¼ 1. The number of neurons for both RBFnetworks is taken as 30. The centers of the RBF networks arechosen randomly between 0 and 1. The weights of the networks areinitialized such that the initial estimate of g is greater than thelower bound gl. The parameter matrices F and G are taken asdiagonal matrices with diagonal elements 5 and 1, respectively.Tracking result and the corresponding control input are shown inFig. 4.

Figs. 3 and 4 show that the tracking performances aresatisfactory for both cases. The RMS error is found to be 0.001for the first case while for the second case it is 0.005.

4.2. Example 2

The next problem considered in this paper is simulation studiesof a MEMS device namely electrostatic actuator [15,16]. Thegoverning equations are

Q� 1

RVin �

Qg

eA

� �¼ 0 (44)

mgþ bgþ kðg � g0Þ þQ2

2eA¼ 0 (45)

where Q denotes the charge, g the gap between the plate and baseand g the rate of change of the gap when the plate moves. Vin is theinput voltage used to move the plate to a desired position. Variousparameters associated with the dynamics are given in Table 1.

We would now transfer the system dynamics into affine strictfeedback form so that the proposed controller can be applied.Defining the state variables x1 ¼ g and x2 ¼ g, we can write fromEq. (45),

x1 ¼ x2

x2 ¼ � 1

m

Q2

2eAþbx2 þ kx1 � kg0

!(46)

Fig. 5. Tracking results for electrostatic actuator: left, position g; right, charge Q; an output disturbance of magnitude 0.2 is applied at t ¼ 20 s.

Fig. 6. Electrostatic actuator: control input Vin.

Fig. 7. Position tracking result for electrostatic actuator: a sinusoidal output

disturbance of magnitude 0.1 is applied for 0.3 s.

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764762

If we denote the third state variable as x3 ¼ x2, then,

x3 ¼ � 1

m

Q2

2eAþbx2 þ kx1 � kg0

!

Thus; x3 ¼ � 1

m

d

dt

Q2

2eAþbx2 þ kx1 � kg0

! (47)

Since the objective is to control the gap between the plate andbase by varying the voltage Vin, the output variable is taken asy ¼ g ¼ x1 and the input variable is taken as u ¼ Vin. Thus,combining Eqs. (46) and (47), the system state equations become

x1 ¼ x2

x2 ¼ x3

x3 ¼ 2x1

meAðkg0 � kx1 � bx2 �mx3Þ �

k

mx2 �

b

mx3

�ffiffiffi2p

u

mRffiffiffiffiffiffieAp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkg0 � kx1 � bx2 �mx3Þ

qy ¼ x1

(48)

The above equations represent the system dynamics in a strictfeedback form so that controllers (3) and (16) can be applied. Thecontrol objective is to maintain the gap g at a desired value. Asgiven in [16], the desired value of g is taken as 0:5 mm. The desiredvalue of x3 is taken as 0 which yields Q ¼ 10 at steady state.Controller (3) is applied to the system for which the filteredtracking error comes out to be r ¼ eð2Þ þ l1eð1Þ þ l2e. Thecontroller parameters are heuristically chosen as kv ¼ 30,l1 ¼ 15, l2 ¼ 10. The number of neurons for the RBF network istaken as 30. The centers of the RBF network are chosen randomlybetween 0 and 1 and weights are initialized to very small values.The parameter matrix F is taken as the diagonal matrix ofappropriate dimension with the diagonal elements 0.5.

In [16], an optimal control algorithm is used to achieve thecontrol requirement. For comparison purpose, we have used linearquadratic regulator (LQR) to design a suitable linear controller forthe system. The optimal regulator is designed for the linearizederror dynamics where the desired or steady state value of thestates is chosen as ½Q g g � ¼ ½10 0:5 0 �. The details can befound in [16]. The comparative results are shown in Figs. 5 and 6.To establish the disturbance rejection property of the proposedcontroller, a one time output disturbance of magnitude 0.2 isapplied at t ¼ 20 s. It can be observed from Fig. 5(left) that for theproposed control scheme the output (position g) settles down soonafter the disturbance is applied whereas for LQR it takes more timefor the output to stabilize. This scenario is more prominent when asinusoidal output disturbance of magnitude 0.1 is applied for 0.3 s.

The position (g) tracking result for this case is shown in Fig. 7 wherethe better disturbance rejection capability of direct adaptivecontrol is clearly visible.

4.3. Example 3

The dynamics of a two-link manipulator has been taken as anexample of MIMO systems. For a two-link robotic manipulator (thesecond and third link of a PUMA 560 robot) [17], the dynamicalequations which relate the joint torques ½t1; t2� to the joint angles

Fig. 8. Tracking results for two-link manipulator: link angles u1 and u2.

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764 763

½u1; u2� of the links are given as

t1 ¼ ½a1 þ a2 cos u2� u1 þ a3 þa2

2cos u2

h iu2 þ a4 cos u1

�ða2 sin u2Þ u1u2 þu2

2

2

!þ a5 cos ðu1 þ u2Þ

t2 ¼ a3 þa2

2cos u2

h iu1 þ a3 u2

þða2 sin u2Þu1

2

2þa5 cos ðu1 þ u2Þ

(49)

where a1 ¼ 3:82, a2 ¼ 2:12, a3 ¼ 0:71, a4 ¼ 81:82, and a5 ¼ 24:06.The above system can be re-written as,

u1

u2

� �¼ 1

Dm22 �m12

�m21 m11

� �t1 � v1

t2 � v2

� �(50)

where,

m11 ¼ a1 þ a2 cos u2

m12 ¼ a3 þa2

2cos u2

m21 ¼ m12; m22 ¼ a3

v1 ¼ a4 cos u1 � ða2 sin u2Þ u1u2 þu2

2

2

!þ a5 cos ðu1 þ u2Þ

v2 ¼ ða2 sin u2Þu1

2

2þa5 cos ðu1 þ u2Þ

D ¼ m11m22 �m12m21

Fig. 9. Two-link manipulator: control torques t1 and t2.

Considering the state variables as x1 ¼ u1, x2 ¼ u1, x3 ¼ u2, x4 ¼ u2,one can write

x1 ¼ x2

x2 ¼ 1

D½m22ðt1 � v1Þ �m12ðt2 � v2Þ�

x3 ¼ x4

x4 ¼ 1

D½�m21ðt1 � v1Þ þm11ðt2 � v2Þ�

Let us define z1 ¼ ½ x1 x3 � and z2 ¼ ½ x2 x4 �. The system equationscan be written in terms of these variables as:

z1 ¼ z2

z2 ¼ f ðzÞ þ gðzÞt

where

f ðzÞ ¼ f 1

f 2

� �¼ ð1=DÞ �m22v1 þm12v2

m21v1 �m11v2

� �

and

gðzÞ ¼ g11 g12

g21 g22

� �¼ ð1=DÞ m22 �m12

�m21 m11

� �

The output is y ¼ z1. The reference output trajectories are taken as:

z1d ¼x1d

x3d

� �¼

p6

sin ð2tÞp6

cos ð2tÞ

264

375

The control input is generated using (31). The controller and neuralnetwork parameter vectors are taken as

Kv ¼30 00 30

� �; L ¼ 20 0

0 20

� �

The parameter matrix F is taken as the diagonal matrix withdiagonal elements 20. The number of neurons for the network istaken as 30. The centers of the RBF network are chosenrandomly between 0 and 1 and weights are initialized to verysmall values. The tracking results for the link angles are shownin Fig. 8. Corresponding input torques are shown in Fig. 9. TheRMS error for the link u1 is found to be 0.0004 whereas for u2 itis 0.0006.

5. Conclusion

Feedback linearization techniques have been successfully usedover the past decades to design controllers for affine nonlinearsystems. The controller design becomes difficult when the systemis not known completely. In this paper, both the systemnonlinearity and the input nonlinearity have been assumed to

I. Kar, L. Behera / Applied Soft Computing 9 (2009) 756–764764

be unknown and two RBF networks have been used to approximatethose nonlinearities. A direct adaptive controller along with theweight update laws for the networks were derived to establishstability of the closed loop error dynamics. A sliding mode termwas added with the adaptive control law to maintain theboundedness of the controller. It is also shown that many MIMOsystems can be represented by a particular affine form, formulatedin this paper, for which the adaptive control law, derived for theSISO systems, has been directly extended.

Three nonlinear systems have been taken for simulation study.The first one is a SISO system which is same as that of [9]. Thesecond SISO system is a MEMS device namely electrostaticactuator. The dynamics of a two-link manipulator has been takenas the third example which represents a MIMO system of the classpresented in this paper. Simulation results show the satisfactoryperformance of the control schemes.

The proposed control scheme is applicable only for affinesystems in strict feedback form. Though there are manypractical systems like single link flexible joint manipulator,jet engine compression systems, MEMS devices for which theproposed control scheme can be applied successfully, the futurestep would be to increase the applicability of the proposedscheme for a more general class of nonlinear systems. For MIMOsystems, the proposed control law involves the computation ofmatrix inversion which can be replaced by a recursive formwhich is kept as a future scope of this work. When both thesystem and input nonlinearities are unknown, the derivation ofthe control law remains a open research problem for MIMOsystems.

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