A Dynamical Robust Adaptive Controller

Copyright «:> IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

A DYNAMICAL ROBUST ADAPTIVE CONTROLLER T. Ahmed-Ali, F.Lamnabhi-Lagarrigue

Laboratoire des Signaux et Systemes, C.N.R .S., Supelec,91192 Gif-sur- Yvette, Cedex, France

Abstract : In this paper, we propose a robust adaptive control schemes for a large class of single-input single-output nonlinear uncertain systems in order to track bounded and smooth trajectories. The method is based on the introduction of a dynamical iterative change of coordinates (DCC) which is a modified version of a dynamical backstepping procedure. The performances of this algorithm are illustrated on a BuckBoost converter. Copyright© 1998 IFAC Keywords: Uncertain nonlinear systems, Robust tracking control, Adaptive control, Disturbance attenuation.

fact, our design is of interest as soon as the set of geometric conditions that we can find in the above mentionned works and which are necessary in order to track a given output using a static feedback only, are not fulfilled .

1. INTRODUCTION

This papers deals with the design of a dynamical robust adaptive feedback control in order to achieve output tracking of uncertain singleinput single-output nonlinear control systems of the form

X = f(x, u) { y=h(x , u)

+ G(x , u)~ + R(x , u)d(t)

More precisely, we here address the following new control problem : how to design a dynamical adaptive controller and an adaptive law in order to track a given smooth and bounded trajectory, Yr (t), t ~ to with bounded derivatives while guaranteeing some robustness properties such as ,

(1)

which are observable and minimum phase . x E Rn, f, G, Rand h are assumed to be smooth functions . ~ is a constant unknown parameter vector which takes any value in a closed ball Q C RP of known radius r(Q) centred at the origin and d(t) is an arbitrary bounded disturbance.

RI : 11 x(t) 11 and 11 O(t) 11 are bounded ,

R 2 : when d(t) = 0 asymptotic tracking achieved , R3 : when d(t) E L 2 , Y - Yr E L2 satisfies

J t

Several methods using a static feedback have been developed in the literature for the robust tracking of a class of uncertain nonlinear systems which satisfies some geometric conditions. One can cite for instance, Marino and Tomei, (Marino and Tomei, 1995a), (Marino and Tomei, 1993) , (Marino and Tomei, 1995b) , Freeman and Kokotovic (Freeman and Kokotovic, 1995), Polycarpou and Ioannou (Freeman and Kokotovic , 1995) .

(y(-r) - yr(-r)f d-r :5 f30

to

IS

J t

+ f31

11 d(-r) 112 d-r(2)

to

where O(t) is the estimate of ~ and where the functions {30 and {31 depend only on the initial conditions of the system (1) . Property R3 is called L 2 -gain disturbance attenuation (Marino and Tomei, 1995b) . The scheme proposed in this paper is based on a dynamical iterative change of coordinates (DCC) which is a modified version of a dynamical back-

The aim of this paper is to design a robust and adapti ve scheme for general systems (1) . In

189

stepping procedure introduced by Rios-Bolivar, Zinober , and Sira-Ramirez (Rios-Bolivar and SiraRamirez , 1995) and whose existence is guaranteed by the observability assumption derived by Fliess (Fliess, 1989) . Our algorithm is an extension of the one introduced by Marino an Tomei (Marino and Tomei, 1995a) combined with a new Lyapunov function inspired from the work of Praly (Praly, 1992) . This scheme need less restrictive geometry conditions than the one described in the recent literature . However, the observability condition HI

ay ay rank{ ox ' OX ' . . . , 'if, E

n

Z2

= hl(x, 8, u , it) + odx , t , u) - y~l) _ Vl(X , 8, u, t).

This leads to

Zl =

-ClZl

+ Z2 + ~~ G(x , u)(f, -

8)

oh + ox R(x , u)d(t) + vdx, 8, u, t) Step k , k=2, .. . ,n-l At this step we choose

oy(n-l) ox } = n

Zk+l

=hk(x , 8, u, ... , u(k») + Ok(X , t , 8, u, . .. , u(k-l») - y~k ) _ Vk(X , 8, u , . .. , u(k-l ), t)

'id(t) E Loo .

\

where

must be satisfied.

(k)

oh k -

l

Note that the DCC has also been used by the authors in (Ahmed-Ali and Lamnabhi-Lagarrigue, 1997b) in order to design a dynamical sliding controller for the same non-parameterized class of systems and in (Ahmed-Ali and LamnabhiLagarrigue, 1997 a) for a parameterized version .

and

This algorithm is applied to the robust control design of a Buck-Boost converter.

OOk-l ) ) ok(x ,t , 8, u, . .. , u (k-l ) )=a;-(f(x , u)+G ( x,u 8

hk(x , 8, u , . .. , u

) = ---a;-(f(x , u)

+ G(x , u)8)

k-l ~h

' " ' ~ (i+l)

+ ~ ou(i)

u

k-2 ~ ' " ' VOk-l

+~~u

2. ADAPTIVE CONTROL DESIGN

i =O

2.1 Dynamical change of coordinates and control law

+ G(x , u)8)

~

+ CkZk + Zk-l

i=O

where Ck is a strictly positive constant. Therefore (3)

Zk

=- Zk-l + [oh k - l ox ohk- l + [ 08 oh k - l + [ 08

CkZk

+ Zk+l

+ OOk-l

OVk-ljG(X, uHf, _ 8) ox OOk-l _ OVk-lj8 + 08 08 OOk-l _ OVk-l ( )d( ) + 08 08 jR x , u t +Vk (8 x , ,u, . . . ,u (k-l) , t)

~~ (f(x, u) + G(x , u)8) + ~~ it

oh + ox G(x, u)(f, - 8) - y~l) . Now let us set

_

ox

where the terms Vk will be designed together with the adaptive law.

+ G(x, u)8)

oh . +-u OU

Step n: At the last step we find the following expression for Zn

and

Ol(X , t,U) =

u)

'"' VVk-l (i+1) - ~ ou(i ) u

This gives

. . oh hdx , 8, u , u) = ox (J(x, u)

OVk-l

~(f(x , k-2

Step 1: Let us consider the tracking error

Zl =

OVk-l

+{ft-{it -

Zl=Y-Yr(t)

vU

OOk-l

Let us consider again the system of the form (1)

(i+1)

ClZl,

Zn= h n-l (8 x, ,u , . . . ,u (n-l»)

where 8 is an estimate of the unknown parameterf, and Cl is a strictly positive constant. Therefore, we introduce the new state Z2 and the function VI which will be precised below,

+ On-l (x,t ,17{} , U, . . . , U (n-2»)

- Yr(n-l) - Vn-l (x,8 ,u , ... , u (n-2) , t) . Now let us introduce the functions

190

(4)

0~:-1 (f(X , U) + G(X , u)B)

hn(x , B, u , . . . , u(n )) =

n-l

+

, the map r l ( Z, u, 8, Yr , Yr , ... , y~n-l ) )

J:l

~ vh n - 1 (i+l) ~ OU(i ) U

b oun d ed·f · arguments z, -u , B, Yr, Yr · ,.·· , Yr(n-l ) 1 a 11 Its are bounded . Note that the existence of 4>-1 IS guaranteed on the set 5 such that . IS

(5)

i=O

and

04>

Q n

+

G(

(x,t, 8 ,u, ... , u (n-l)) = n-2

X, u

)B)

a;--

U sing the new coordinates ZI , . . . , Zn, the system can be written as follows

!:>

J:l

~ VQ n -l

+~

rank ox = n .

OQ n -l (f( x , u )

ou(i) u

(i+l)

+

VQ n -l

at

ZI

OV n -l

OVn -l (f(

-~-~

+ G( X,U )B)

_

x,u

~ OVn-l

~!:> (i) U i=O vU

= -CI Z l + Z2 + wi(€ -

8)

+ 4>i d(t)

+vJ(x, 8, u, t)

) (i+l)

Let us denote the set of points (X, B, u) by D = Br X Be x Bfl C Rn X n X Rn where u = . ... , u (n-l))T suc h th a t ( u, u, ohn _ ou CB) (x(t),B(t) , u(t))

# 0,

Vt E R

(3 denoting the highest order of the derivatives of u having a nonzero coefficient in the expression (5) for h n . Note that we do not consider here the singular case where this coefficient could vanish at some points . Now given an initial condition

where T _

Wk -

[oh k _ 1 ox

+

OVk-l ]G( ) ox x, u

OQk-I _

ox

and

(XO , 80 , uo) E D and

let us choose the following dynamic controller as the solution of

",T

'I' k

OVk-l]R( ). oB x, u

_

The function Vi(X , t, 8, u , .. . , u(i-l )) will be determined in the next section

hn(x , B, u , .. . , u(n)) = Qn(x , t, B, u , ... , u(n-l ))

+ y~n)

= [ohoBk- 1 + OOk-l oB

_ Zn-l - CnZn

+ vn(x , t , B, u , .. . , u(n-l))

=

2.2 Adaptive controller design

We then set Wi u(i-l), i = 1, . . . , n . The differential equation (6) may be written

In order to design an adaptive controller let us consider the following Lyapunov function inspired from the work of Praly (Praly, 1992)

V

(6)

= W + 0; (€ - 8) + 2~ II€ - 811

2

(8)

where F(z , t,8,u , .. . , u(n ))

= 0

(9)

In the following we assume that the following assumptioms hold (8) may also be written

H2 : The system (6) is bounded-input boundedstate for all BEn, Z being the input and Wi, i = 1, .. . , n being the states. are bounded .

V

=W - ~lloo~ W+ 2~IIB -

Let us now choose {,

H3 : 4> being the coordinate transformation

191

°o~ 112.(10)

0 < { ~ 1 such that

{W ~ W -

z = 4>{x, u, 8, Yr , Yr , . .. , y~n-l )) )

€+ Q

Q

"211

oW oB

11

2

(11)

or equivalently such that f}z ( ::; 1 - 0' 11 f}0 112.

Clearly, if 0' satisfies

It follows

1-(

0' <

II~;W '

.

V::; -c

then V is positive definite. Now let us define f} z B = {( z, u , O, t)/IIf}OIl

R 2n +p+1

.

V::;

V.

With this in mind , let us now compute follows n

' " 2 LCiZi

Zi Vi

+

i=1

L

i=1

z i4>T d(t )

f}W B_ ~ (~ f}0 0'

f}0

Note that

f}W· ~ f}0 0 = L

n

~ Zi

(14)

(15)

In order to prove that x(t) is bounded we need to prove that z(t) is bounded . From (8) we have

i= 1

+ f}W (~ _ 0) _

[12 ~ 2] + -ryld(t)1 2.

B y integrating (15), and from the fact that V(t) > o we derive R 3 .

n

+L

-c

1]

n

i=1

n

2T ~ T 4"1] ~ ~ Zi 4>i 4>i + ~ Zi4>i d(t) .

cz; ::; -V(t) + ~ld(t)12.

It

" T (~-O)+ 'LZiqi " T 0· + 'LZiWi

i=1

-

In the case d(t) = 0, from (14) and LaSalle theorem , we can ~ee that R 2 is fulfilled . Now in the general,~ase , from (14) we obtain

Then from (11) we can see that V is positive definite on B.

. V=-

2]

We obtain

< r} C

where r > 0 and arbitrarily chosen. Moreover let us choose 0' such that 1- ( 0'<[2.

n

[12 ~ ~ Zi

V ::;

_ O)T B

W

1 f}W 2 f}0 11

+ 211

1

1

2

+ (20' + 2)11~ - 011 .

Therefore we can easily see that from (11)

11

T· Ziqi 0

f}W 2 f}0 11

::;

2 ;(1- ()W

and

i=1

This suggests the choice of the following adaptive law .

0= O'Proj

[

f}W

~ ZiWi + ( f}0 n

f

]

(12)

Now let us set C

where Proj is the smoothed projection introduced in ****** in the case where n is a closed ball

Pmj(8, 1)

if IIOT 011 ::; r if IIOTOII ~ rand otherwise

~ {;

2 0'

= c'[1 + -(1 - ()] .

It follows

OT[::5 0

.

[2 1+ 2)1I~ 1 - 811 ] + ;(1- ())W + (20' 2

V::; -c' (1 1

1

n

2

+ c'(20' + 2)11~ - 011 + -ryld(t)1

where with

0> O.

and

With this choice we guarantee the boundedness of

V::;

o and the following inequality

-c'V

+ c'( ~ + ~)II~ - 011 2+ ~ld(t)12 . (17) 20'

2

1]

Therefore both V and W - %11 °o'!' 11 are bounded . Now from (11) , we can see that W is bounded , this leads to the boundedness of z . By using H 2 , H3 and the fact that Yr(t) and its derivatives are bounded, we derive that x(t) is bounded and R1 is guaranteed .

Let us set C = 2 min {Ci} , if we now choose the following Vi , i 1, ... , n

= Vi

2

= -4"1] Zi 4>i T 4>i,

then the dynamical controller takes the form

Summarizing, we have proved the following result

192

!I = (JJ(I - u)( -(J2(I - U)Xl - (J3 X2) - y~2) + (Cl + ~u2)((Jl(I- U)X2 + (J4U - Yr(t».O .OI- XC) * xb)

Theorem: The tracking control problem with the robustness properties Rl , R2 and R3 for the nonlinear system (1) can be solved by using the adaptive law (12) and the dynamic controller (13) in the feasability region D n B n S, provided that the assumptions H l , H2 and H3 are fulfilled.

Then

electronics systems or robotics, equation (12) will be linear in (J and therefore, the implementation of the adaptive law will be easier than other approaches like (Marino and Tomei, 1995a) or (Ahmed-Ali and Lamnabhi-Lagarrigue, 1997 a) . Moreover it is worth noting that the equation for the dynamic controller (13) does not contain iJ, therefore the implementation of the dynamical controller will also be easier.

Z2 = ((J4 - (J1X2 T

+!I + W2 (€ .

TJ

- (Jl(l- U)2Xl

and

= =

oW

o(J (€ - (J)

1

+ 20' II€ -

2

(JII .

(J3 = q3 (J 4_- b2 a 11 - a2 bl an a 11 - a2l a 12

X:w = -5V .

where

Zl = Y - Yr(t)

all = 1 - 0'(1 - U)2U 2 a12 = -0'(1 - U)X2U a2l = -0'(1 - U)X2U

The second variable is defined by

+ (J4U -

+

(J2 = q2

Zl = 6(1- U)X2 +€4U + ud(t) - Yr(t) .

Z2 = (Jl(I - U)X2

+ ~u2)uJ .

q'l = O'[zl(I - U)X2 + Z2C1X2(I - u)] q'2 = -0'[Z2(Jl(I- u)2X2J { q'3 = -0'[Z2(Jl (1 - u)xd q4 = O'[Zl U + Z2Cl U],

The desired output is Yr(t) = 30(1- e- 4t ). The tracking error is Then

(Cl

From this we can derive the following update law

In the following we will consider the numerical values 6 3.6103, 6 5.5103, 6 2.25010 3, €4 52.810 3, Cl 3, C2 6, TJ 1, 0' 0.01, (To 1.5, M = 104 and the initial conditions X10 = 15A

- (JJ(I- U)X2

and

Xl = €l (1 - U)X2 + €4U + ud(t) X2 = -6(1 - U)Xl - 6X2 + x 2d(t) . { Y = Xl Our control objective is that the input inductor current Xl should follow a desired output reference signal Yr (t ) .

=

U)X2

Let

V = W

=

.

U)X2(Jl

with

Consider the average Buck-Boost converter model (Rios-Bolivar and Sira-Ramirez, 1995) defined on the input inductor current Xl and the output capacitor voltage X2

= =

+ (1 -

+ [U(Cl + 4"u) + (Jl(I- u)x2Jd(t) + V2

3. APPLICATION TO THE BUCK-BOOST CONVERTER

=

(J)

+ U(J4

wI = [(Cl + ~u2)(I -

= =

~Zl u)u

an = 1 - O'u 2

Yr(t)

+ C1Zl

bl = 0'(1 - u)x2(cdxl - Yr) - Yrd b2 = 20'xdCl (Xl - yr) - yr1) + q4

- Vl

with

+ ql

Then Zl = -C1Zl with

+ Z2 + wT (€ -

(J)

and the following dynamical controller

+ ud(t) + Vl

U

wT = [(1 -

U)X2

0

0

uJ. with

Let us set

193

= (-Zl ((J4 -

C2 Z2 -!I + V2) (J1X2 - ~Zl u)

~-:"----:.:o.....:_~_-:-,-,-

V2 = -~Z2[{U{Cl

Freeman , R. A. and P. V. Kokotovic (1995) . Robust integral control for a class of uncertain nonlinear systems. In: Proc. IEEE CDC. New Orleans Louisana. pp. 2245-2250 . Marino, R. and P. Tomei (1993) . Robust stab ilization of linearisable time-vaying uncertain nonlinear systems. Automatica 29 , 181-189. Marino, R. and P. Tomei (1995a) . Adaptive tracking with disturbance attenuation for a class of non linear systems. In: Proc. IFAC-Nolcos . pp. 125-131. Marino, R. and P . Tomei (1995b). Nonlinear Control Design. Prentice Hall. Praly, 1. (1992) . Adaptive regulation:lyapunov design with a growth condtion. International Journal Of Adaptive Control and Signal Processing 6', ~29-351. Rios-Bolivar , M::j;l.nd A. Zinober H. Sira-Ramirez (1995) . Adaptive sliding modes output tracking via backstepping for uncertain nonlinear systems. In : Proc. European Control Conference. Rome . pp. 699-704 .

+ ~U2) + ih{l- U)X2W .

The simulations are shown in figure 1 without disturbance and in figure 2 with a disturbance.

x,

tell 3.55 3.548 3.546

50~----~~----~

t9.a

5.5.---------------,

2.2004

5.5

2.2002

5.5

.~

3.544 0

t~

.~

22 0

0

0.5 u

52.88,.----------------.

0.5 ·

o

00

0.5

0.5

Fig. 1. Without disturbance

::D x1

'0

5

o

t9.-8

'

.

tet1

tet2

:::~ :::~ 3.54~ 5.5~ 3.53 0

t9.tl

'

5.5 0

0u5

,

':~:B~[J o

disluCfuSance

'

52.7

0

0.5

,

0 0

0.5

,

~B o

0.5

,

Fig. 2. with disturbance

4. REFERENCES Ahmed-Ali , T . and F . Lamnabhi-Lagarrigue ( 1997 a). Dynamical adaptive sliding modes controller. In: European Control Conference. Ahmed-Ali, T. and F. Lamnabhi-Lagarrigue (1997b). Tracking control of nonlinear systems with disturbance attenuation. Comptes rendus de l 'Acadmie des Sciences de Paris

t.325,329-338 . Fliess, M. (1989). Nonlinear control theory and differential algebra. In : Modeling and Adaptive Control (C. I. Byrnes and A. Khurzansky, Eds.) . Number 105 In : Lecture Notes in Control and Information Sciences. SpringerVerlag.

194