Design and analysis of a new adaptive robust control scheme for a class of nonlinear uncertain systems

Design and analysis of a new adaptive robust control scheme for a class of nonlinear uncertain systems

DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR... 14th World Congress of IFAC E-2c-16-3 Copyright © 1999 IFAC 14th Triennial World Congress, Be...

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DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

14th World Congress of IFAC

E-2c-16-3

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

DESIGN AND ANALYSIS OF A NE'W' ADAPTIVE ROBUST CONTROL SCHEME FOR A CLASS OF

NONLINEAR UNCERTAIN SYSTEMS Jian-Xin Xu *

Qing-~ei

Jia * Tong-Heng Lee*

* Department of Electrical Engineering National University of Singapore 10 Kent Ridge Crescent Singapore 119260

Abstract: This paper presents a new adaptive robust control scheme which is the extension of (Xu et al., 1997) in the sense that more general classes of nonlinear uncertain dynamical systems are under consideration. To reduce the robust control

gain and \viden the application scope of adaptive techniques, the system uncertainties are classified into t,vo different categories: the structured and non-structured uncertainties. The structured uncertainty is estimated witll adaptation and compensated. 1:1eanwhile, the adaptive robust method is applied to deal with the non-structured uncertainty by estimating unknown parameters in the upper bounding function. The t-L-modification scheme is used to cease parameter adaptation in accordance with the adaptive robust control law. The new control scheme guarantees the uniform

boundedness of the system and at the same time, the tracking error enters an arbitrarily designated zone in a finite time. The new control scheme also improves the result of (Xu et al., 1997) in that the unkno\vn input distribution matrix of the system can be state-dependent instead of being a constant matrix. Copyright © _~~9 IFAC Keywords: Adaptive robust control, nonlinear system, uncertain system

14 INTRODUCTION

and Neto, 1995) an adaptive law using a dead-zone

and a hysteresis function is proposed to guarantee Numerous adaptive robust control algorithms for

both the uniform boundedness of all the closed-

systems containing uncertainties have been devel-

loop signals and uniform ultitnate boundedness

oped (Xu et al., 1997; Liao et al., 1990; Brogliato

of the system states. In both control schemes,

and Net 1995; Ioannou and Sun, 1996; Taylor et al., 1989; Sastry and A.Isidori, 1989; Narendra

it is assumed that the system uncertainties are bounded by a bounding function \vhich is a prod-

and Annasvlamy, 1989). In (Liao et al., 1990)

uct of a set of known functions and unknown

"'Rriable structure control ~vith an adaptive law is developed for an uncertain input-output linearizable Ilonlinear systeln, where linearity-inparameter condition for uncertainties is assumed. The unknown gain of the upper bounding function is estimated and updated by adaptation law so that the sliding condition can be met and the error state reaches the sliding surface and stays on it. To deal with a class of nonlinear systems \vith partially kno\vn uncertainties, in (Brogliato

positive constants. The objective of adaptation is to estimate these unknown constants.

°,

In (Xu et aL., 1997), a ne\v adaptive robust control scheme is developed for a class of nonlinear uncertain systems with both parameter uncertainties and exogenous disturbances. Including the categories of system uncertainties in (Liao et ai., 1990) and (Brogliato and Neto, 1995) as its subsets, it is assumed that the disturbances is bounded by

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14th World Congress of IFAC

DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

a known upper bounding function. Furthermore, the input distribution matrix is assumed to be

Assumption A2:

constant but unknown.

For E E Rnxn,

\!XEV

VtE[O)oo)

In this paper we proposed a continuous adaptive robust control scheme y..rhich is the extension of (Xu et al., 1997) in the sense that more general classes of nonlinear uncertain dynamical systems are under consideration. The unknown input distribution matrix of the system input can be statedependent here instead of being a constant matrix in (Xu et al., 1997). To reduce the robust control gain and vviden the application scope of adaptive techniques, the system uncertainties are supposed to be composed of two different categories: the first can be separated and expressed as the product of known function of states and a set of unknown constants, and the other category is not separable but with partially known bounding functions~ It

is further assumed that the bounding function is convex to the set of unkno\vn parameters, Le., the bounding function is no longer linear in parameters. The first category of uncertainties is dealt with by means of the well used adaptive control method. Meanwhile an adaptive robust

lIT

T max

) ~

> -1 (2)

rmin

Assumption A8: The structured uncertainty g(x, p, t) = [91 (x, p~ t), .. ~ ,gn(x, p, t)]T E Rn is a nonlinear function vector which can be expressed as

g(x, p, t) == 8(p )~(x, t)

e=

diag(B!, ... , OJ)

~ ~ [';1,

eJ

e2, .. ~ ,';n]T

(3)

where ~ [eil,8i2~-··,8iPJ is an unkno~"'n pararneter vector and == I~il, c;i2, ... ,eipJ is a known function vector. The nODstructured uncertainty 6g(x, p, W, t) [6g 1(x, p, w~ t), 6g2(x, p, w, t), ... , 6g n (x, p, w~ t)]T is bounded such that

1;,;

'it E 'R,+

\Ix E D

'Vp E P

JI6g(x,p,w,t)11 ~ Pd(X, q, t)

in the upper bounding function are estinlated ,vith

This paper is organized as follows. Section 2 describes the class of nonlinear uncertain systems to be controlled. Section 3 gives the design procedure of the adaptive robust control and the stability analysis. Simulation-based studies on the proposed method are given in Section 4.

+ 2E

where ...\ (.) indicates the eigenvalues of " . " ~

method is applied to deal with the second category of uncertainties, where the unknown parameters adaptation.

2. A("2 E

VpEP

where 11 ~ \\ represents the Euclidean norm; 1) is a compact subset of 'Rn in which the solution of (1) uniquely exists with respect to the given desired state trajectory Xd(t). Pd(X, q, t) is an upper bounding function with unknown parameter vector q E P . Here Pd(X~ q, t) is differentiable and convex to q, that is

2. PROBLElVI FORlV1ULATION Consider a class of uncertain dynamical systenl described by

x = f(x, t) + B(p){[I + E(x, p, t)]u(t) + g(x, p, t) (1)

+6g(x i p,w,t)}

nn

where x [Xl,X2,···,x n ]T E is the measurable state vector of the system; u == [Ul, U2,···, un]T E is the control input of the system; p E P is an unknown system parameter vector and P is the set of admissible system

nn

The control objective is to find a suitable control input u for state tracking with the desired state

trajectory Xd E 'Rn, \vhere ously differentiable.

Remark 1:

Assumption Ai: f(x~ t) = [/1 (x, t)~ f2(X~ t), ... ~ fn(x, t)]T E Rn is a known nonlinear function vector. The matrix B(p) is unknown but B(p) > o 'vip E P.

is at least continu-

In assumption Al ,ve assume that

f(x: t) is a kno~vn function vector. If it is unknown, but able to be decomposed into three parts: known, structured uncertain and non-structured uncertain vectors as below

f(x, p} t) == f' (x, t)

parameters; f(x, t) E Rn is a nonlinear function vector; B(p) E nnxn is the input distribution matrix; g(x) p~ t) and Dg(X, p, w, t) are nonlinear uncertain function vectors of the state x, unknown parameter p, time t alO;) well as a set of random variables w. Here we make the following assumptions:

Xd

+ g' (x~ p, t) + 6g' (x, p, w, t) (5)

which are analogous to fex, t)t g(x, p, t) and 6g(x, p, w, t) respectively in (1) and satisfying definitions in Assumption Al and A3, then the plant can be rearranged into

x=:;; f'(x, t) + B(p){(I + E)u + g"(x, p, t) +6g"(x, p, w, t)}

(6)

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14th World Congress ofIFAC

DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

where cl, El, (. are the estimates of q, 8,


,:vhich is exactly in the form of (1) with g"(x, p~ t) ~ [8, B-lel)[~T, ~,T] II~gll (x, p, w, t)11

(11)

= 116g + B- 1 .6.g'j I S Il6g!1 + ltB- 1 L1g'lI

:s Pd(X, q, t) + qOp~(x, q', t)

The control la\v u and the corresponding adaptive la"\\r are chosen to be

vvhere g' = e','~ 116g'll ::; p~(x,q!,t) which is convex to q', and qo = A1Tlax (B-1) is an unknoV\."n constant. Remark 2: .6g is the only system uncertainty taken into account in (Liao et al., 1990) and (Brogliato and Neto, 1995), v,rhich are particular cases of (1). Besides, in (Liao et al., 1990) and (Brogliato and Neto, 1995), Pd is a linear combination of unknown parameters and known functions, namely Pd = 2: qiPi(X, t), "\vhich corresponds to the case of "::::::~' in (4).

U e

+Uv

(12)

- ~[f -

-Ke - e~

Xd] -

Vd

r~axflucl12e

uv =

(1

+r

7n in)

(r 7nax lie T ucrl

+ cl)

.-.2 Vd;:;;;;;

e

Pd

Pdllelj

+£2

e=rl(e~T -}.LIE» ~ = r 2 (e(f - Xd) T - J.t2~]

it = r 3(11ell ~~ 1..1 -

Remark 3: (Xu et al., 1997) is also a particular case of (1) in the following three aspects: 1. The input distribution matrix B(p) considered in. (Xu et al., 1997) is assumed to be constant, V\.thile in this paper 1ve deal with a more general class of input distribution matrix expressed as B(p)[I + E(x, p, t)J. Obviously B(p) is a subset of B(p)[I +E(x,p, t)]. 2. The bounding function Pd(X, t) in (Xn et al., 1997) is a particular case of Pd(X} q, t) in ·w·hich Pd(X, q~ t) = L: qiPi(X, t) with qi = 1, i.e., the bounding function is fully kno,vn. 3. The function f(x~ p} t) in (5) can have the uncertain term 6g'(x, p, w, t), whereas (Xu et al., 1997) cannot, as the upper bounding function of B- I 6.g' still contains unknown parameters qo and unkno~vn parameter vector q'.

=

U

Uc =

(13)

J1.34)

where cl and E2 are positive constants; K E nnxn is a gain matrix; Pd ~ Pd(X, q, t); r i ) i = 1,2,3 are positive definite matrices chosen to be

ri =

(14)

diag(,it, ri2,·· . ,1'in).

J.ti, i = 1,2,3, which constitute the tt-modification scheme, are defined as

~ { ki(co 0

Pl -

lIell) e E

Eo elsewhere

(15)

,vhere k 1 , k 2 and k 3 are positive constants. Eo where

6"0

~

= {e : Ilell < EO}

(16)

is a positive constant specifying the

desired tracking error bound. 3. ADAPTIVE ROBUST CONTROL \VITH f.L - AI0DIFICATIOf\l The adaptive robust technique is used in this section to develop a controller which guarantees the global boundedness of the system. The design procedures are presented in detail as follo,vs.

A. The Adaptive Robust Control Law Define tIle measured state tracking error vector as e=x -

Xd

ll. Convergence Analysis

For the above adaptive robust controller, vote have the following theorem.

Theorem : By choosing the control gain matrix such that Arnin(K) 2: ~~Cl with Cl > 0, the o proposed adaptive robust control law (12)-(15) ensures that the system trajectory enters the set Eo in a finite time~ 1vloreover, the tracking errors as well as the parameter estimation errors are bounded by the set

(7) D

and the parameter error matrices as

q=q-q

(8)

6=9-E>

(9)

~===-~

(10)

== {e, e, ~, q : e T e + trace{ eT e} + trace{ ~ T 4-} +qT q < k/[~kleotrace{eTEl} 1 T +'21 k2cQ trace{ T ~} + "2k3CQ q q + cl}

(17)

where k' is defined. to be

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DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

k' k

11

==

14th World Congress of IFAC

+1l2trace{ ti> T ~}

1

kif min{Amin(B-l), Amin(ri l 2min{A'lnin(K), k1h, k 2 b, k 3 0}

=

max{..\max(B~

1

~l

),Amax(ri

i

,

)}

= 1, 2, 3.

+l-' l trace{9 T 8} +J.L2trace{~T~} -llell (Pd - Pd) + Jl3Cl T q + £1

A7nax (A) and Arnin(A) indicates the maximum and minimum eigenvalues of the matrix A respectively, and E and 8 are positive values to be defined later.

~

-eT

Ke

+ 11-1trace{ST (El - e)} + cl + 6"2

+tL2trace{ T (cl> - T 8}

DD

Proof:

+ P2 trace{ &T ep } + ,u3qT q + E

- J-L2 trace{ ~ T }

The folloV\ring positive definite function V is selected

- Jl3q T q

1

T

v

= !eTB-le

2

+ .!trace{E>T r1

1

2

-T-

S; -e Ke - "2Jlltrace{8 8} -

e}

+~trace{.pTr21-J.} + ~qTr31q.

+ 1l3qT q + Cl

::::-eTKe- AP~llell2 +Pdllell Pdllell + 6"2

)}

1 ~T-'2fl2trace{4>
(18)

1

1

By choosing K such that

have

e}

+ (I + E)(u c + u v ) + .6.g + If>(f(x, t) ->Cd)] - trace{8 T (e{T - f-L1 6 )} -trace {ri» T(e(f - Xd)T -IL 2 c1»} - qTr;l q = e T[9t; + ~(f - }cd) + (1 + E)u v + Eu c -K e - vd + 6g] - e e~ - e -I>(f - Xd) +J.tltrace{EH3 T } + 1l2trace{tf..tI>T} - qT r 3 1 q :5 ~e T Ke - eT Vd + 116gll·llel! 2 _eT (1 + E) r~a", Iluc l1 e (1 + Tmin)(rmaxlleTucll +cl) +rrnaxll eT ucH + J.Lltrace{8Tf)} +J.t2trace{ 4> T 4>} - q T r;14 (19)

= e T[8{

T~

In above derivation the following property of trace is used: trace{QT vw T}

:=

VT

Qw

V :::;

_eT Ke

::;

-Cl,

'i e E nn - E o.(23)

(24) where Eb ~ {e: Ilell < E~} is a subset of Eo· Noting the relation C Eo, (24) implies that the system will enter the set Eo in: a finite time.

Eo

\"Vnen e E

Eb, it is obvious that

(20)

i

(25) we obtain

~E + ~ET 1 + rml n

+e

co

Define 8 = eo -

+

(22)

co

Note that, in terms of the adaptive robust control law (12)-(15), V is a continuous function. We can show that there exists a constant 0 < < co such that (See Appendix A)

U sing the fact that

I

+

E Cl - - 2- ,

where Cl is an arbitrary positive constant, then from (15) ~re have

where v E nnxl, w E ~R}Xl, and Q E R 1Lxl •

eT(I+E)e~eT(I+

(21)

+ C2.

A7nin(K) 2::

-trace{Tr21~} - q T r3"lq

T~

8}

1

(12)-(13),

e T B~le - trace{e T r 1 1

T

+ '2f.t 2 trace{ q. T ep} + 2'/-L3QT q + c. "'There e = El

v=

T

+ 2"J.t1trace{e

Taking the derivative of V along the trajectory of the dynamic system (1) with the control law Vle

1

"2 P3 Q. q

E+E T 2 )e

·

T

V ~ -e Ke -

c~

"21 k16 trace{8-T""8}

l~T--

-

"21 k3b Cl T q

1 T + 2'k 1 c O trace{EJ 8}

1 TIT +'2k2EO trace{ tI> cl>} + 2"k 3 eo q q

it follows that ~

. (25)

> 0, then from (15), (21) and

-2'k 2 8 trace{ tI> q>}

2: 1,

= 1, 2, 3

+ e-

1 T } -k,V,+ "2k1EO trace{8 e 1 TIT +'2k2CO trace{ cl> tI>} + "2k3EO q q

+E

(26)

By solving (26) we can establish that

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DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

The simulation results are given in Fig. 1-2. It can be seen that the control scheme has achieved a good tracking performance in the sense that the system enters the set Eo after a short transient VttTlrich inlplies that e, 6, and q converge exponentially to the residual set (17). DD

4. SI1vIULATION STUDY Consider an uncertain system given as

x = f(x, t) + B[(I + E(x, p, t)u + g(x~ t) +6g(x~ t)]

(27)

where

f(x, t) =

[Si~(:)X~] ,

g(x, t) =

[:~~~]

=

O.5sin(xl) ]

[as~(t) 1

+ t2

In this paper, an adaptive robust control scheme is developed for a class of uncertain systems

'

b o( cos

)] Xl

are as below

bI1 = b22

1.0,

O2

==

1.0,

== 1.0,

bl"2

=

b21

;;;=

b = -0.15,

a::::::: 0.4,

simulation results VttThen the design parameters are Cl = E2 = 0.05 and co = 0.08. It is shown that the system responses and control signals are quite silnilar to, but not as smooth a..'; the previous case with larger magnitudffi.

5. CONCLUSION

The real values of the plant chosen for simulation (}t

Remark 4: The control input is highly dependent on the choice of the design parameters co, €1 and C2. The greater the parameters Cl and C2, the smoother the input. Notice that if Cl and C2 are set to zero, the control scheme is discontinuous. From (22), we can also see that EO has great influence on the control gain K, and a smaller cO results in a larger control gain. Fig. 3-4 give the

[(}2:21S:~Xl] ,

6g(x, t) = [ O.8sin(5t)x2

E(x, p, t) =

[~~~

B =

period, and the control signal is quite smooth.

= 0,5

,

c= 0.6.

The control and adapting parameters are chosen

to be

with both unknown parameters and system disturbances. The uncertainties are assumed to be composed of tv..I'O categories= the structured category and the non-structured category with partially kno~vn bounding functions. The structured uncertainty is estimated with adaptive method. Meanwllile, the adaptive robust method is applied to deal with the non-structured uncertainty, where the unknown parameters in the upper bounding function are estimated with adaptation. It is shown that the control scheme developed here can guarantee the uniform boundedness of the system and assure that the tracking error enter the arbitra.rily designated zone in a finite time. The

K

= diag[80, 80],

El =: C2

= k 2 = k 3 = 10.0,

r 1 ::::: r 2 ::::: r3 =

= 0.5,

Pd

k1

=

V + q2X~ ql

VO.25sin2(xl)

both theoretical analysis and simulation studies.

diag(l, 1)

6. REFE,RENCES •

Brogliato, B. and A. Trofino Neto (1995). Practical stabilization of a class of nonlinear systems with partially known uncertainties. A utomatica 31, 145-~150. Ioannou, P.A. and Jing Sun (1996). Robust Adapti~'e Control. Prentice-Hall) Inc. Liao, Teh-Lu, Li-Chen Fu and Chen-Fa Hsu (1990). ~t\.daptive robust tracking of nonlinear systems and ~vith an application to a robotic manipulator. Systems & Control Letters 15, 339-348.

Note that the actual upper bound of 6g is

Il.6.gll =

effectiveness of the control scheme is verified by

+ O.64sin2(5t)x~

5 JO.25 + O.64x~ which is a subset of the given PdThe given trajectory is

(28)

Narendra, K.S. and A.M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, Engle-

and the desired tracking accuracy is co = 0.1 for both el == Xl - Xld and e2 = X2 - X2d. The initial states are Xo == (-2.0, 3.0]T, and the iIlitial values of the parameter estimates ~, ~ and q are zeros.

wood Cliffs, NJ. Sastry, S.S. and A.Isidori (1989). Adaptive control of linearizable systems. IEEE trons. on automatic control 34, 1123-1131.

Xd

= [5sint, 5cost] T

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DESIGN AND ANALYSIS OF A NEW ADAPTIVE ROBUST CONTR...

14th World Congress ofIFAC

Taylor, D.G., P.V. Kokotovic, R.Marino and I~Kanellakopoulos (1989). Adaptive regula-

,

EO

tion of Donlinear systems with unmodeled dynamics. IEEE trans~ on. automatic control 34, 405-412. Xu, Jian-Xin, TOIlg-Heng Lee and Qing-'''~ei Jia (1997). An adaptive robust control scheme for a class of nonlinear uncertain systems. International Journal of Systems Science 28, 429434.

2

-b + Jb 2a

=

+ 4ac

(33)

> O.

DO

(..) . . . 1

~~:[. :: : : : ' : j ~o

2

4

6

8

10

12

14



18

.20

12

1~

18

'6

2';J

1im.(se~)

APPENDIX A: The solution of c~

From (21), it can be seen that

V<

0 if

Fig~

~

then EO can be easily determined by solving the following equation

e + Cl --2-Ij e I1 2 Co

1. Tracking Error

()

1

TIT

= -J-l1trace{e 8} + 2

1 T +2"fl-3q q

-P2trace{}

2

2

4

6

6

10 Yirn8(aeo)

+c

Substituting J.1.1, J-t2 and JL3 in terms of (15) and

letting

Ilell =

c~ yields

Fig. 2. Control Input

~~L

Denote £

o

+ Cl

a=--2Co

b= C

~(kl trace{


~

1 . T } = 2'k 1c otrace{ C{l T ~} + '2k2Eotrace{ 'It W

1

T

+c

(31)

,

--e<:lo

=~

then equation (30) can be transformed to the following aEo,2

+ b! Co

- c

= 0.

o

-b ± y'b 2 2a

+ 4ac

J5

B

10 tinwo(aec;)

12

1...

18

1S

20

2

...

6

8

10

12

14

16

'8

20

12

'4

16

18

20

tlrTl8($liil l; )

:2

...

£>

B

10 tl,...... (s..c)

Fig. 4. Control Input

The solutions of above equation are E~=-----­

4

Fig. 3. Tracking Error

2 1

+"2k3COQ q

.2

' :~~' : ::I

(32)

It is obvious that the solutions E R. Notice that EO is positive, hence the desired solution is

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