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Page 1: [IEE IEE Colloquium on Symbolic Computation for Control - Birmingham, UK (17 June 1999)] IEE Colloquium on Symbolic Computation for Control - Symbolic algebra toolbox for the design

SYMBOLIC ALGEBRA TOOLBOX FOR THE DESIGN OF DYNAMICAL ADAPTIVE BACKSTEPPING CONTROLLERS Alan S. I. Zinober' and Miguel Rios-Bolivar2

Keywords: Adaptive control, backstepping, uncertain systems, nonlinear control, symbolic al- gebra computation.

Abstract A symbolic algebra toolbox is described for the design of dynamical adaptive nonlinear control- lers for regulation and tracking tasks of a class of observable minimum phase uncertain nonhear systems. This toolbox also allows the design of non-adaptive controllers for systems without un- certainty, and adaptive sliding mode controllers (SMC) to provide robustness in the presence of disturbances. The design procedure employs the basic ideas of the adaptive backstepping algorithm with tuning functions via inpuboutput linearization, and is applicable to both triangular and non- triangular systems.

1 Introduction

The computer technology advances of recent years have allowed the development of a number of computer software systems intended for numerical and symbolic computation, such as MACSYMA, MAPLE and MATHEMATICA. The availability of these packages has allowed the development of useful toolboxes for the systematic analysis and design of feedback control systems. For instance, some of the toolboxes developed thus far include analysis and control design for affine and non- affine systems [2, 31, modelling and nonlinear control design [l], and analysis and design based on flatness [12]. These toolboxes simplify the use of systematic and recursive control design methods SO that the design of stabilizing controllers may be carried out more efficiently.

The various backstepping control design algorithms [4, 5, 61 recently compiled in [7], provide a systematic framework for the design of tracking and regulation strategies suitable for large classes of nonlinear systems. The adaptive backstepping algorithm has enlarged the class of nonlinear systems controlled via a Lyapunov-based control law to uncertain systems transformable into the parametric strict feedback (PSF) form and the parametric pure feedback (PPF) form. In general, local stability is achieved for systems in the PPF form, whilst global stability is guaranteed for systems in the PSF form [5]. These two forms can be seen as special structural triangular forms of nonlinear systems which are adaptively input-output linearizable with the linearizing output -51. A more general algorithm has been developed by Rios-Bolivar et a1 [9], which allows one

to design dynamical adaptive controllers following an input-output linearization procedure based upon the backstepping approach with tuning functions [6], and is applicable to both triangular and nontriangular uncertain nonlinear systems.

This paper is organized as follows: Section 2 outlines the generalized backstepping algorithm. Some features of the symbolic toolbox are given in Section 3. Section 4 presents examples of application of the dynamical adaptive backstepping algorithm using the implemented symbolic toolbox, and conclusions are presented in Section 5.

'Department of Applied Mathematics, The University of Sheffield, Sheffield S10 2TN 'Departamento de Sistemas de Control, Facdtad de Ingenieria, Universidad de Los Andes M&da 5101, Venezuela

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2 Dynamical Adaptive Control Design

The Dynamical Adaptive Backstepping (DAB) algorithm proposed in [9] is based upon a combina- tion of dynamical input-output linearization and the adaptive backstepping algorithm with tuning functions [SI. Since it has been developed in a general context, without the use of canonical forms, its applicability to both triangular (PSF and PPF forms) and nontriangular systems is guaranteed, but it requires that the controlled plant be abservable and minimum phase. The observability condition is required to guarantee the existence of a local nonlinear mapping which transforms the plant into a convenient form of the error system, as shown below. The need for the minimum phase property is to allow the applicability of the systematic algorithm presented here, and to guarantee stability of the closed-loop system. This general algorithm includes as a particular case the adaptive backstepping algorithm with tuning functions [6] developed for systems in PSF and PPF forms.

Consider a single-input single-output nonlinear system with linearly parameterized uncertainty

x = fo(z) + rn(z)O + (go(.) + *(z)B) U (1)

E/ = w where 5 E Rn is the state; U, y E W the input and output respectively; and 8 = [el,. . is a vector of unknown parameters. fo, go and the columns of the matrices *, q E RnxP are smooth vector fields in a neighbourhood Eo of the origin x = 0 with fo(0) = 0, go(0) # 0; and h is a smooth scalar function also defined in &.

The steps leading to the the design of the dynamical adaptive compensator follow an input- output linearization procedure in which, at each step, a control dependent nonlinear mapping and a tuning function are constructed. The parameter update law and the dynamical adaptive control law which stabilize the controlled plant, are designed at the final step. In order to characterize the class of nonlinear systems for which this procedure is applicable, we set up a nonlinear mapping by considering the output y ( t ) and its first n - 1 time derivatives as follows

Due to the presence of the unknown parameter vector 6 we rewrite (2)

where 8 is an estimate of 8, and the vector ~1 is defined as

W1 = - @(z) + U*($)) ax ah ( In other words, (3) may be rewritten as

0 = L k ( Z , e, U) = .&,e, U) + W l ( 8 - e) with

ah

The second time derivative of the output is

(4)

5/2

Page 3: [IEE IEE Colloquium on Symbolic Computation for Control - Birmingham, UK (17 June 1999)] IEE Colloquium on Symbolic Computation for Control - Symbolic algebra toolbox for the design

which can be rewritten as

,Q=L~(x,o,U,u) 2 * =~(x ,S ,U ,u )++z(o -e )

with

ax and

By proceeding successively this manner, we obtain the j-th time derivative of the output

y(’) = L{(z, 8, U, 6, . . . , U(’-’)) = E{ (x, e, U, U, . . . , U(’-’)) + wj (8 - 8)

with

and

The expression (11) is d i d if the relative degree is one. The general expression for systems with well-defined relative degree, i.e. 1 5 p 5 n, has the form

y(j) = L{ (Z , 8, U, i ~ , . . . , &-p)) = 2i(x, 8 , U, 6 ) . . . , d j - p ) ) + w j ( e - 6) (14)

with

In other words, the time derivatives of the output are obtained by the application of the following recursively defmed operator

LCg = h(x) (16)

which also characterizes the control dependent nonlinear mapping

Assumption 2.1 System (1) is locally observable, i.e. the mapping (17) satisfies the rank condi- tion a q . )

ax rank- = n

in a subspace RI c Ro c Rn.

513

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Assumption 2.2 System (1) is minimum phase in RI c I& c P.

For observable minimum phase nonlinear systems of the form (l), the general problem of ad- aptively trxking a bounded desired reference signal yF(t) with smooth and bounded derivatives can be solved through the DAB algorithm summarized as follows:

D A B Algori thm Coordinate transformation

21 := y - 97 ( t ) = do) (2) - Yr ( t ) (19) z k := - yik-"(t) f a k - l ( ' ) , 2 5 k 5 n

with

k

Parameter. upda te law e=rn=rwTz=r[uFW;.. .

Dynamical adaptive compensator

61 = 112 62 = ~3

with

Page 5: [IEE IEE Colloquium on Symbolic Computation for Control - Birmingham, UK (17 June 1999)] IEE Colloquium on Symbolic Computation for Control - Symbolic algebra toolbox for the design

where the ci's are constant design parameters and r = rT > 0 is the adaptation gain matrix. The control U is obtained implicitly as the solution of the nonlinear timevarying differential equation (25). The overall closed-loop error system has the form

A , + A T = - 2

i = A,z+W(B-8)

8 = rwTZ where the matrix A, has the following skew-symmetric form

0 cz ... 0 c1 0 ... 0

. . . . . 0 0 ...

(29) . . . .

A, =

with

- c1 1 0 ... 0 -1 -CZ 1+ez,3 ... ez+

e3,n 0 -1- &,3 -c3 ...

0 -e2,*-1 -e3+-1 ... l+en-l ,n 0 -ez,* -e3,,, . . . -c,

with the quadratic Lyapunov function 1, v=-z z 2

It has been proved in [9,10] that the stability of the overall system is guaranteed and also asymptotic tracking is achieved. These facts are summarized in the following theorem (see [lo]).

Theorem 2.1 The closed-loop adaptive system consisting of the plant (l), the dynamical controller defined by (25) and the update law (Z4), has a locally stable equilibrium at ( z , 8 - 8) = (0,O) and limt+m z(t) = 0, which means that asymptotic tracking is achieved, i.e.

Moreover, i f limt,, y!i'(t) = 0, i = 1,. . . , n and [@(O) + uQ(O)] = 0, then limt+, z ( t ) = 0.

Theorem 2.1 guarantees local asymptotic tracking in general. Nevertheless, global asymptotic tracking can be achieved if the relative degree is defined globally, and also Assumptions 2.1 and 2.2 are satisfied globally.

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2.1

A particularly important aspect in regulation and tracking tasks for uncertain systems is robust- ness in the face of disturbances and unmodelled dynamics. In [U] a solution to this problem has been proposed, which is based on the combination of the adaptive input-output linearization al- gorithm above and sliding mode control (SMC). It allows one to design dynamical adaptive sliding mode tracking controllers. The resulting control law achieves robust asymptotic stability with considerably reduced chattering.

To provide robustness, the DAB algorithm can be modified for the design of dynamical adaptive output tracking controllers (see [ll] for details). The modification is carried out at the final step of the algorithm by incorporating the following sliding surface defined in terms of the error coordinates

U = k1z1+ k2zz + . . . + kn-lz"-l + zn = 0

Dynamical Adaptive Sliding Mode Control

(33)

where the scalar coefficients k; > 0, i = 1,. . . , n - 1, are chosen in such a manner that the polynomial

in the complex variable s is Hurwitz. Additionally, the Lyapunov function is modified as follows

p(s) = kl + kzs + . . . + kn-1Sn-2 + sn-' (34)

1 1 2 2

n-1 v = - 2: + -2 + -(% - s)Tr-l(% - e)

2 i=l

By differentiating (35) we can obtain the update law

and the dynamical adaptive sliding mode output tracking controller

n-1

(35)

with K, > 0, > 0 and a, defined by

This dynamical adaptive sliding mode control yields

n-1 2 v = - c;z: + zn-1z, - KU - KWIUI

i = l

which guarantees asymptotic stability for suitably chosen design parameters [Ill.

(39)

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3 Symbolic Algebra Toolbox Since the DAB algorithm has been developed in a general context, its implementation via a sym- bolic algebra toolbox allows one to deal with various classes of systems. The most general class corresponds to observable minimum phase systems, which may be in triangular or nontriangular form. Triangular systems in PSF or PPF forms are particular subclasses of linearizable observable systems. On the other hand, the DAB algorithm may be used for the design of non-adaptive con- trollers for nonlinear systems without uncertainty by specifying null matrices @ and q. Also, the combined DAB-SMC algorithm of Section 2.1 may be implemented via a symbolic algebra toolbox.

We have developed the symbolic toolbox BACKDSMC which implements both the DAB and DAB-SMC algorithms, for the synthesis of tracking and regulating adaptive (and non-adaptive) controllers; requiring a minimum of effort by the user. It has the following features [lo]:

automatises the backstepping control design procedure

does not use transformations into canonical forms

allows the design of a number of adaptive and deterministic controllers

does not require the user to have expert knowledge of the backstepping design technique

automatically generates MATLAB code programs for computer simulation of the closed-loop systems.

The types of controllers designed by BACKDSMC for regulation and tracking problems include:

static and dynamical non-adaptive linearizing controllers for deterministic systems

static and dynamical adaptive backstepping controllers for uncertain systems

robust static and dynamical combined backstepping-Sliding Mode Control (SMC) for uncer-

The outputs generated by BACKDSMC consist of the feedback control law, the coordinate transformation placing the system into the error coordinates, the parameter update law for uncertain systems, the sliding surface for the combined backstepping-SMC design, and the MATLAB code programs for simulation purposes.

The user needs to provide the nonlinear functions of the mathematical model of the system written in the general form (l), and the symbolic desired output. Depending on the nature of yr , the problem to be solved is either a regulation or tracking problem. Thus, when yT is constant the designed controller is for regulation, otherwise yr is a timedependent function and the controller is designed for tracking tasks.

tain systems (see [Is]).

4 Examples

The following three different examples illustrate the use of the symbolic toolbox BACKDSMC cor- responding to various classes of systems. The results are presented in the form of m-files (MATLAB code programs), which can be used directly for numerical integration.

4.1 Consider the third order system without uncertainties

Example 1: 1/0 Linearizable System

( 2 ) = ( ” ’ ~ ~ ~ ” ” ) + (8) 0

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Marino and Tomei [8] have shown that the system (40) is locally inpuboutput linearizable. The linearizing output is

y = z1 exp(-zz). (41) The symbolic expressions which characterize the mathematical model (40) are defined by the fol- lowing series of MATLAB commands:

f=sym(’[x2+~1*~3;~3;0]’) % f(x) g=sym(’[O;O;l]’) % g(x) phi=sym(’[O;O;O]’) % Phi(x) psi=sym(’[O;o;O]’) % Psi(x) h=’xl *exp(-x2) ’ % h(x) yd=’O’ % desired output yd

The linearizing static controller is obtained by the command line

[c,tau,z]=backdsmc(f,g,h,yd,phi,psi,’iolineam’,’iolinear’)

which also generates the m-files ‘i0liieam.m’ and ‘io1inear.m’ automatically. The m-file ‘io1ineam.m’ contains the system equations and the control law as shown

funct ion xdot=iolineam(t ,x) ; global c ; % 1 Control l a w % u ~ = ( - ~ ( ~ ) * c ( i ) / e x p ( x ( ~ ) ) * x ~ 2 ~ - 2 / e x p ~ x ~ 2 ~ ~ * x ~ 2 ~ + x ~ 3 ~ * c ~ l ~ / e x p ~ x ~ 2 ~ ~ * x ( 2 ~ - . . . x(3)*c(i)/exp(x(2)~-x~3~~2/exp(x(2))*x~2~+2*x(3~~2/exp(x(2))tx(3)*c(2)/. . . exp(x(2)) *x(2)-x(3) * c W /exp(x(2) )-c(3) *c(l)/exp(x(2) ) *x(2)+c(3) *x(3) / . . . exp(x(2))*x(~)-c(3)*x(3)/exp(x(2))-c~3~*c~2~/exp~x~2~~*x~2~-c(3~*c(2)*. . . c(i)*x(i) /exp (x(2) )-c(3) *x(l)/exp(x(2)) -c(l)*x(l)/exp(x(2)) )/ . . . (-exp (-x (2) ) *x (2) +exp (-x (2) ) ) ; x X System equations x xdot(l)=x(2)+x(l)*x(3) ; xdot(2)=x(3); xdot(3)=ul;

As shown by Marino and Tomei [8], the value 2 2 = 1 is a singular value for the above linearizing feedback control.

The m-file ‘io1inear.m’ is generated to run the numerical simulation of the closed-loop system in ‘io1ineam.m’ and allows one to specify the initial conditions, the parameter design values and the initial and final times for simulations, as follows:

% X This program runs io1ineam.m h global c ; % % Parameter values % c=c; ;I ;

Page 9: [IEE IEE Colloquium on Symbolic Computation for Control - Birmingham, UK (17 June 1999)] IEE Colloquium on Symbolic Computation for Control - Symbolic algebra toolbox for the design

t o = ; % I n i t i a l time tf=; % Final time A % I n i t i a l conditions 1 xO=[, ,I : [t ,XI =ode23 ( ’ iolineam’ , t O ,tf ,xO) ;

4.2

Consider the third order nonlinear system

Example 2: Uncertain PSF System

x1 = x 2 + 0 x : x 2 = x3

x3 = U

(42)

where 0 is an unknown scalar parameter. This system is already in the PSF form; therefore global adaptive regulation is achieved. The results obtained by BACKDSMC include the coordinate transformation z, the update law 0 for the unknown parameter, and the control law U.

The symbolic expressions which characterize the mathematical model (42) are defined by the following series of MATLAB commands

f=sym(’[x2;~3;0]’) % f(x)

phi=sym(’[xl-2;0;0]’) % Phi(x) psi=sym(’[0;0;0] ’) % Psi(x) h=’xl’ % h(x) yd=’O’ % desired output yd

g=sym(’[O;O;l]’) 7% g(x)

The regulating adaptive controller is obtained by the command line

[c,tau,z]=backdsmc(f,g,h,yd,phi,psi,’psfmod’)

which generates the m-file ’psfm0d.m’ for simulation purposes.

function xdot=psfmod(t ,x) : global c ad t h e t a : % % Auxiliary var iables 1 thl=x(4) : % % Update law % taul=x(l)~3*ad(l)+~x~2)+thl*x(l)~2+c(l~*x~L))*ad~l~*x(1~~2*~2*thl*x~1~+ . . . c( 1) )+(2*thl*x( 1) *x(2)+2*thlA2*x( 1)^3+c(l) *x(2)+c(l)*thl*x(l)^2+~(3)+. . . x(l)~2~(x(l)~3*ad(l~+~x(2)+thl*x(l)”2+c~l~*x(l~~*ad(l)*x~l~~2*~2*thl*. . . X( 1) +c( 1)) ) + c ( 2 ) * ( ~ ( 2 ) + t h l * x ~ l ~ ~ 2 + c ~ l ~ * x ~ l ~ )+x(i) 1 *ad(i) *x(1)-2*. . . (2~thl~x(2)+6~thl~2*x(l)~2+2*c(l)~thl*x~l~+5*x~l)~4*ad~l~+lO*x~l~~4* . . . a d ( l ) ~ t h l ~ x ( 2 ) + 4 ~ x ( l ) ^ 3 + a d ~ l ~ * c ( l ~ * x ~ 2 ~ + l 4 * x ( l ~ ~ 6 * a d ~ l ~ * t h l ~ 2 + l 8 * . . . x (1) “5*ad( 1) *c (1) cthl+5*x( 1)-4*ad( 1) *c(l) “2+2*c (2) *thl*x (1) + c ( 2 ) *c (1) +I) : %

519

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X Control law x ul=-8*thl-2*~(2) *x( 1)̂ 2-2*~( L)*thl*x( 1) *~(2)-2*thl*~( 1)-2-2*~(2)-C(l)*. . . x( 1)-1O*x( 1)-4*ad( l)*thl*x(2)̂ 2-(2*x(i) *x(2)+4*thl*x(1)̂ 3+c(l) *x(1)-2+. . . x(l)~2*(x(l~^4*ad(l)*~2*thl*x(l~+c~l~~+2*~x~2~+thl*x~l~~2+c~l~*x~l~~*. . . ad( 1) *x( 1) -3)+c(2)*x(1)"2) *(x(l)̂ 3*ad(l)+(x(2)+thl*x(l)-2+c(l) *x(l) )* . . . ad(l)*x(i)~2*(2*thl*x(l)+c(l))+~2*thl*x(l)*x(2)+2*thl~2*x(l)~3+c~~~* . . . x(2)+c(l)*thl~x(i)"2+x(3)+x(l)~2*~x(l)^3*ad(l)+(x(2~+thl*x(l)"2+c~i~*. . . x(l))~ad(l)~x(l)"2~(2*thl*x(l)+c(l)))+c(2)*~x(2)+thl*x(l~~2+c(l~*x~l~~+. . . x(l))*ad(l~*x~l)~2*(2*thl*x~2)+6*thl~2*x(l~~2+2*c~l~*thl*x~l~+5*x~l~~4... ~ad(i)+l0~x(l)~4*ad(l)*thl*x(2~+4*x(l)~3*ad(l)*c(l~*x~2~+l4*x~l~~6*. . . ad( 1) *th1^2+18*x (1) "5*ad( 1) *c (1) *thl+5*x (1)̂ 4*ad(l) *c (1)-2+2*c (2) *thl* . . . X( l)+c(2) *c(l)+l))-2*x(3) *thl*x(l)-5*x(l) "4*ad(l)*x(2)-~(2)*c(l) *x(2)-. . . 2~c(i)*thl~2*x(l)~3-5*x(1~~6*ad(i)*thl-i4*x~1~^8*ad~l~*thl~3-2*c~2~*... thl^2*x (1)̂ 3- (x (2) +thi*x( 1)-2+c (1) *x (1) ) *x( i)-4*ad( 1) * (2*thl*x(2) +6*. . . th1^2*x (1) ̂2+2*c( 1) *thl*x (1) +5*x( 1) -4*ad( 1) +lO*x(l) 4̂*ad( 1) *thl*x(2) + . . . 4~x(l)~3~ad(i)*c(l)~x(2)+l4*x(l)~6*ad(i)*thl~2+l8*x~l~~5*ad(l~*c~l~* . . . thi+5*x (1)-4*ad( 1) *c (1)̂ 2+2*c (2) *thi*x( l)+c (2) *c (1) +1) -c (3) * (2*thl*. . . x (1) *x(2) +2*thiA2*x (1) ̂ 3+c (1) *x(2) +c (1) *thl*x(l) -2+x(3) +x( 1) -2*. . . (x(l)~3*ad~l~+~x(2)+thl*x(l)"2+c(l)*x(l)~*ad~l~*x~l~~2*(2*thl*x~l~+. . . c(l)))+c(2)*(x(2)+thl*x(l)^2+c~l~*~(l))+X~l))-2*thl*~~2~~2-6*thl~3* ... x (1)-4-x(3) *c (1) -x (3) *c(2) -24*x( 1)-6*ad( 1) *th1̂ 2*x(2) -4*x( 1)-3*ad( 1) * . . . c (1) *x(2)-2-22*x( 1)̂ 5*ad(l) *c(I)*x(2) *thl-L8*x(l)̂ 7*ad(l) *c(l) *th1-2-5*. . . x(i)~6*ad(l~*c(i)~2*thi-co*c~2)*c(l)*thl*x(l~~2-5*x~l~~4*ad~i~*c~l~~2*... x(2)-2~c(2)*thl*x(i)~x(2)-2*x(3)*x(l)~5*ad~l~*thl-x(3~*x(l~~4*ad~i~*c~l~ ; x Y, System equations x xdot ( 1) =x (2) +theta( 1) *x( 1) -2; xdot(2)=x(3) ; xdot(3)=ul; x Y, Parameter estimate equations % xdot(4)=taui;

Note that thl and ad(1) in the expressions obtained by BACKDSMC correspond to the parameter estimate 8 and the adaptation gain respectively.

4.3

Consider the third order uncertain nontriangular system

Example 3: Uncertain Nontriangular System

51 = -x + e(x2Z3 + 1) xz = x 3 + u

23 = -x - x1 + ex; (43)

where 0 is an unknown scalar parameter. This system is not transformable into the PSF or the PPF form. Nevertheless, it is globally stabilizable to the equilibrium point ( x , 8 ) = (e, 0, -@,e) by

5/10

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choosing y = $2 as the output. This output ensures that Assumptions 2.1 and 2.2 are satisfied globally. Therefore global stabilization of the equilibrium point is achieved. The dynamical adaptive controller is obtained by

[c,tau,z]=backdsmc(f,g,h,yd,phi,psi,’nontrian’)

which generates the m-file ’n0ntrian.m’ for simulation purposes.

function xdot=nontrian(t ,x) ; global c ad theta: % % Auxiliary variables I, ui=x(5) : u2=x(6) ; thl=x(4) : % % Update law 79 taul=(x(3)+ui+c(l)*x(2))*ad(l)*x(2)~2+~c~l~*x(3)+c(l~*ul-x~3~-X~~~+. . . t h l * ~ ( 2 ) ~ 2 + u 2 + c ~ 2 ~ * ~ x ~ 3 ~ + u i ~ c ~ l ~ * x ~ 2 ~ ~ + x ~ 2 ~ ~ * a d ~ l ~ * ~ - x ~ ~ ~ * x ~ 3 ~ - l + . . . x(2) 2̂*c(l)-x(2) -2+x(2)-2*c(2)) ; % % Control law % control=-2*x( 1) -thl*x(2) *x (3) +thl-2*thi*x (2) *ul-c (2) *c (1) *x(3) -c (2) * . . . c(l)~ul-3*~(3)-2*ul+~(l)*~(3)+~(l)*~(l)-~(l)*thi*~(2~~2+thl*~~2~~2+ ... x(2)-2+( C( 1) *x(3) +c( l)*~l-x(B)-~( 1) +thl*x (2)̂ 2+U2+c (2) * (X(3)+Ul+c ( 1) * . . . x(2))+x(2) )*ad(l)~(-x(2)*x(3)-l+x(2)^2*c~i)-x(2~~2+~(2~~~*~~~~ ))-c(3). . . *(c(i)*x(3)+c(l)*ul-~(3)-~(l)+thl*~(2)~2+U2+~(2)*(~(3)+Ul+~(l)*~~2~~+. . .

c(~)*x(~)+c(~)*x( i)-c(2) *thl*x(2) “2-x(2)-2* ((x(3)+ui+c(i)*x(2) ) *ad(l) * . . .

x(2))-c(i)*x(2) ; % % System equations I, xdot(l)=-~(1)+theta(l)*(x(2)*x(3)+1) : xdot (2)=x(3)+ul: xdot (3)=-~(3)-x(l)+theta(l)*x(2)̂ 2; b % Parameter estimate equations % xdot (4) =tau1 : 79 % Dynamic control equations % xdot(5)=x(6): xdot(b)=control;

Numerical simulations were carried out for this dynamically controlled system for a nominal “un- known” parameter value 6 = 1. Figure 1 shows the global asymptotic stabilization achieved with the design parameters c1 = 5, e2 = 2, c3 = 3 and y = ad(1) = 2.

5/11

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5 Conclusion

A symbolic algebra toolbox for the design of both static and dynamic adaptive backstepping con- trollers has been presented. The toolbox has been developed using the MATLAB Symbolic Toolbox and implements a general adaptive backstepping algorithm with tuning functions. It is applicable to a large class of observable minimum phase systems in triangular or nontriangular forms. Recent research has extended the design technique by incorporating the second order sliding approach [13].

References

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[2] B. de Jager, “Symbolic analysis and design for (nonaffine) nonlinear control systems”, PTO- ceedings 13th IFAC World Congress, San Francisco, F, pp. 289-294, (1996).

[3] A. Glumineau and L. Graciani, “Symbolic nonlinear analysis and control package”, Proceedings 13th IFAC World Congress, San Francisco, F, pp. 295-298, (1996).

[4] Z. P. Jiang and L. Praly, “Iterative designs of adaptive controllers for systems with nonlinear integrators”, Proceedings of the 30th IEEE Conference on Decision and Control, Brighton,

[5] I. Kanellakopoulos, P.V. KokotoviC and A. S. Morse, “Systematic Design of Adaptive Control- lers for Feedback Linearizable Systems”, IEEE Transactions on Automatic Control, 36, pp.

[6] M. KrstiC, I. Kanellakopoulos, and P.V. KokotoviC, “Adaptive Nonlinear Control without

[7] M. KrstiC, I. Kanellakopoulos, and P.V. KokotoviC, Nonlinear and Adaptive Control Design,

[SI R. Marino and P. Tomei, Nonlinear Control Design, London: Prentice-Hall, (1995).

[9] M. Rios-Bolivar, H. Sira-Ramirez and A. S. I. Zinober, “Output Tracking Control via Adaptive Input-Output Linearization: A Backstepping Approach”, PTOC. 34th IEEE CDC, New Orleans,

[lo] M. Rios-Bolivar, “Adaptive Backstepping and Sliding Mode Control of Uncertain Nonlinear Systems”, Ph.D. dissertation, University of Sheffield, (1997).

[ l l ] M. Rios-Bolivar, A. S. I. Zinober and H. Sir&Ramirez,“Dynamical Sliding Mode Control via Adaptive Input-Output Linearization: A Backstepping Approach”, in: Robust Control via VaTiable Structure and Lyapunov Techniques (F. Garofalo and L. Glielmo, Eds.), Springer- Verlag. pp. 15-35, (1996).

[12] R. Rothfuss and M. Zeitz, “A toolbox for symbolic nonlinear feedback design”, Proceedings f3th IFAC World Congress, San Francisco (CA), USA, F, pp. 283-288, (1996).

[13] A.S.I. Zinober, J.C. Scarratt, A. Ferrara, L. Giacomini and E.M. =os-Bolivar, “Dynamical adaptive first and second order sliding mode control of nonlineas non-triangular uncertain systems”, Proceedings 7th IEEE Mediterranean Control Conference, Haifa, (1999).

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P a r a m e t e r estimate

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Figure I: Controlled state variables, parameter estimate and control input of a nontriangular system

1 1999 The Instition of Electrid Engineers. mted and published by lhe IEE, Savoy Place, London WCZR OBL, UK. 5/13