Dawson - Asymptotic Robust Adaptive Tracking of Nonlinear Systems

8/6/2019 Dawson - Asymptotic Robust Adaptive Tracking of Nonlinear Systems

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of relative degree two in the presence of input and output disturbances, which guaranteesthe tracking error is ultimately bounded and small in the mean square sense. A robustadaptive backstepping controller with a leakage-based adaptation law was designed in [15]for nth-order, SISO, nonlinear parametric strict-feedback systems with function uncertain-ties (including external disturbances) satisfying a triangular bound. The proposed design

ensures global uniform ultimate boundedness of the system state. A similar class of systemswas considered in [4] in the development of two robust adaptive control methods using thetuning function design and the modular design of [10]. Both methods give L/L2 estimateson the effect of the uncertainties/disturbances on the tracking error. In [6], an adaptive back-stepping controller with tuning functions for linear systems with output and multiplicativedisturbances was designed with a switching -modification. The controller gives a trackingerror proportional to the size of the perturbations. In [11], a class of SISO nonlinear systemsaffected by unknown, time-varying bounded parameters and additive disturbances was con-sidered, and a robust adaptive tracking controller was presented that achieves boundednessof all signals and arbitrary disturbance attenuation. The work in [14], which studied a classof systems similar to the one in [11], proposed a robust adaptive controller that ensures theL2 norm of the tracking error is bounded. The tracking control problem for SISO nonlinearsystems with unknown control coefficients and time-varying disturbances was recently stud-ied in [5]. The robust adaptive controller proposed in [5] was shown to guarantee the globaluniform boundedness of the tracking error.

In this paper, we consider multi-input/multi-output (MIMO) nonlinear systems sub-jected to bounded additive disturbances that are twice continuously differentiable and havebounded time derivatives. For these systems, we present a continuous robust adaptive con-trol construction that guarantees asymptotic tracking. The proposed construction is basedon the nonlinear robust control technique of [18], which was originally used to compensatefor unstructured uncertainties. Here, we use it as a robustifying mechanism for adaptive con-

trollers. That is, adaptation is used to compensate for structured (parametric) uncertaintieswhile the robust mechanism compensates for disturbances, hence recovering the disturbance-free, asymptotic tracking property of the adaptive controller. We address the aforementionedproblem for two classes of nonlinear systems. First, we consider a class of nth-order non-linear parametric strict-feedback systems with matched, unknown, time-varying, additivedisturbances and unmatched, uncertain, constant parameters. The standard adaptive back-stepping design [10] is judiciously modified to allow the use of the robust control techniqueof [18]. Also instrumental to our new construction is the use of the sufficiently smoothprojection-based adaptation law recently introduced in [3]. This allows the adaptive stabi-lizing functions of the backstepping design to be differentiable as many times as necessary.A Lyapunov-type stability analysis is used to prove the proposed robust adaptive controller

yields semi-global asymptotic tracking. In the second part of the paper, we consider a classofflat1 nth-order nonlinear systems with matched and unmatched, unknown, time-varying,additive disturbances. A judicious formulation of the error system allows us to construct arobust adaptive controller that ensures semi-global asymptotic tracking.

1 A flat system is one where there exist special outputs, equal in number to the inputs, that are functionsof the state vector and of a finite number of its derivatives. That is, a system is flat if its relative degree isthe same as the system degree [13].

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The remaining of the paper is organized as follows. In Sections II. and III., we presentthe robust adaptive control design for parametric strict-feedback systems and flat systems,respectively. Numerical examples are included in each section to illustrate the control per-formance. Section IV. summarizes the contributions of this work.

II. Parametric Strict-Feedback Systems

A. Problem Statement

We consider a class of parametric strict-feedback systems of the form

x1 = |

1 (x1) + x2 (1a)

x2 = |

2 (x1, x2) + x3 (1b)...

xi = |

i (x1,...,xi) + xi+1 (1c)...

xn = |

n (x1, x2,...,xn) + d + u (1d)

y = x1 (1e)

where xi(t) Rm, i = 1,...,n are the system states, i Rpm, i = 1,...,n are knownnonlinearities, Rp is an uncertain constant parameter vector, d(t) Rm is an unknownadditive disturbance, u(t) Rm is the control input, and y(t) Rm is the system output.We make the following assumptions regarding the system:

A1. i Cn+1i, i = 1,...,n.

A2. d C2

and d(t), d(t), d(t) L.A3. The parameter vector belongs to a compact convex set := { : kk 0} where 0

is a known positive constant.

Let the output tracking error be defined as

e := y yr (2)where the Cn+1 reference trajectory yr(t) Rm is such that

y(i)r (t) L, i = 0,...,n + 1, (3)

and ()(i) (t) denotes the ith derivative with respect to time. Our goal is to construct a statefeedback control u(x1, x2,...,xn) that ensures e(t) 0 as t and the boundedness of allclosed-loop signals. The following notation will be used throughout this section: i (t) Rp,i = 1,...,n 1 are parameter estimates2;

i(t) := i(t), i = 1,...,n 1 (4)2 Despite the existence of the single parameter vector , the control law to be proposed will contain n 1

parameter estimates; i.e., the control law will be overparameterized.

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denote the corresponding parameter estimation errors; Proj(i, i) Rp, i = 1,...,n 1,i(t) Rp denote Cni1 projection operators used to ensure i(t) L independent ofthe stability analysis (see Appendix A for details); i Rpp, i = 1,...,n 1 are constant,diagonal, positive-definite matrices; and ci, i = 1,...,n + 1 are positive constants. To reducethe notational complexity and facilitate the readability of this section, the control construc-

tion that follows is presented for the case where m = 1. Note, however, that the main resultis readily applicable to the MIMO case.

Remark 1 Solutions to some special cases of the above-described problem were presentedin preliminary versions of this paper. In [1], we considered the case where n = 2 in (1). In[2], we addressed the regulation problem, i.e., yr 0 in (2), for the general nth-order system(1).

B. Construction of Robust Adaptive Control Law

Step 1

We begin by differentiating (2) and substituting from (1a) to obtain

e = |1(y) + x2 yr. (5)

After adding and subtracting the term |1(yr) to (5), we write the error system in thefollowing form

e = |1(yr) + x2 yr + w1. (6)where

w1 = (|

1(y) |1(yr)) (7)

Remark 2 Due to assumption A1 and (140), we can use the Mean Value theorem to show

kw1k 11 (kek) ktanh(e)k (8)

where 11() R0 is some globally invertible, nondecreasing function.

Let

2 :=1

c1(x2 1) (9)

where 1 is a stabilizing function yet to be designed. To facilitate the notation, let

1 := e, (10)

and define

V1 = ln (cosh(1)) +1

2|

111 1. (11)

Differentiating (11) along (6) gives

V1 = tanh(1) (|

1(yr) yr + 1 + w1 + c12) |

111

.

1 . (12)

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Based on (12), we design the stabilizing function and parameter update law as follows

1 = c1 tanh(1) |1(yr)1 + yr (13).

1 = 1Proj(1, 1), 1 = 1(yr) tanh(1). (14)

Substituting (13) and (14) into (12) gives

V1 = c1 tanh2 (1) + tanh (1) (c12 + w1) + |

1

1 11

1

c1 tanh2 (1) + tanh (1) (c12 + w1) (15)

where property P2 of the projection operator was used (see Appendix A).

Remark 3 The final step of the control design will require that.

i (t) L, i = 1,...,n 1independent of the stability analysis. This motivates the use of the term ln(cosh(i)) in

the Lyapunov function candidate of thefi

rst n 1 steps of the backstepping procedure. Inparticular, due to property P3 of the projection operator (see Appendix A), we know

.

i (t) L ifi(t) L. The boundedness ofi is facilitated by the fact that ln(cosh(i)) / i =tanh(i).

Remark 4 Using (9) and (13), the state x2 can be decomposed into

x2 = c12 c1 tanh(1) | {z } |1(yr)1 + yr | {z } (16)1 b1

where the term 1

is a function of1and

2, and the term

b1is bounded. The usefulness of

this decomposition will become apparent in the next step.

Step 2

Let

3 :=1

c2(x3 2) (17)

where 2 is the second stabilizing function. Using (9) and (1b), we obtain

2 =1

c1(|2 + 2 1 + c23) . (18)

After differentiating 1 of (13), we obtain

1 =1

x1x1 +

1

1

.

1 +1

yryr +

1

ydyr

=1

x1|

1 +

1

x1x2 +

1

1

.

1 +1

yryr +

1

yryr

| {z } (19)

1(x1, x2, 1, yr, yr, yr)

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where 1 is known. Using (19) in (18) produces

2 =1

c1

|2 1x1|1

| {z }

1 + 2 + c23

(20)

w2(x1, x2, 1)

Adding and subtracting the term wb2/c1, where wb2 := w2(yr, b1, 1), to the right-hand sideof (20) yields

2 =1

c1(wb2 1 + 2 + c23 + w2) (21)

wherew2 = (w2 wb2) . (22)

Remark 5 Using assumption A1, (16), (139), and (140), we can use the Mean Value theorem

to showkw2k c1 (21 (k2k) ktanh (1)k + 22 (k2k) ktanh (2)k) (23)

where 2j () R0, j = 1, 2 are some globally invertible, nondecreasing functions and2 := (1, 2)

|. Note that the calculation of (22) is facilitated by the fact that x2 b1 = 1(see Remark 4).

Now, define

V2 = V1 + ln (cosh(2)) +1

2|

212 2, (24)

and differentiate (24) along (21) to obtain

V2 c1 tanh2(1) + tanh(1) (c12 + w1)+ tanh(2)

1

c1(wb2 1 + 2 + c23 + w2)

|

212

.

2 (25)

where (15) was used. Based on (25), we design the second stabilizing function and parameterupdate law as follows

2 = c2 tanh(2) wb22 +1 (26).

2 = 2Proj(2, 2), 2 =1

c1w|b2 tanh(2). (27)

Substituting (26) and (27) into (25) and making use of property P2 of the projection operatoryields

V2 c1 tanh2(1) c2c1

tanh2 (2) + tanh (1) (c12 + w1) +1

c1tanh(2) (c23 + w2) . (28)

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Remark 6 Using (17) and (26), the state x3 can be written as

x3 = c23 c2 tanh(2) wb22 +1= c23 c2 tanh(2) +1 b1

| {z } +b1 wb22

| { z } (29)

2 b2

where b1 := 1(yr, b1, 1, yr, yr, yr), 2 is a function of1, 2, and 3, and b2 is bounded.

Step i (3 i n 1)Let

i+1 :=1

ci(xi+1 i) (30)

where i is a stabilizing function, and differentiate i :=1

ci1(xi i1) to obtain

i =

1

ci1 |i + cii+1 + i i1 (31)where (1c) was used. The derivative ofi1 can be written as

i1 =i1Xj=1

i1

xjxj +

i1Xj=1

i1

j

.

j +iX

j=1

i1

y(j1)r

y(j)r

=i1Xj=1

i1

xj|j +

i1Xj=1

i1

xjxj+1 +

i1

j

.

j

!+

iXj=1

i1

y(j1)r

y(j)r

| {z } (32)

i1(x

1,...,xi,

1, ..., i

1, yr,...,y

(i)

r

)

where i1 is known. Using (32), we can rewrite (31) as

i =1

ci1

|i

i1Xj=1

i1

xjTj

! | {z }

i1 + i + cii+1

(33)

wi(x1,...,xi, 1, ..., i1)

Adding and subtracting the term wbi/ci1 to (33) yields

i = 1ci1

wbi i1 + i + cii+1 + wi (34)

where wbi := wi(yr, b1,..., b(i1), 1,..., i1) and

wi = (wi wbi) . (35)

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Remark 7 Using assumption A1, (139), and (140), we can show

kwik ci1iX

j=1

ij (kik)

tanhj

(36)

where ij () R0, j = 1,...,i are some globally invertible, nondecreasing functions andi := (1,...,i)

|.

Define

Vi = Vi1 + ln (cosh (i)) +1

2|

i 1i i (37)

whose derivative is

Vi i1Xj=1

cj

cj1tanh2(j) +

1

cj1tanh

j

cjj+1 + wj

+tanh(i) 1ci1wbi i1 + i + cii+1 + wi |i 1i .i (38)

where c0 = 1. Based on (38), we design

i = ci tanh(i) wbii +i1 (39).

i = iProj(i, i), i =1

ci1w|bi tanh (i) . (40)

Substituting (39) and (40) into (38) gives

Vi iX

j=1

cjcj1 tanh2(j) + 1cj1 tanh j cjj+1 + wj . (41)Remark 8 Using (30) and (39), the state xi+1 can be written as

xi+1 = cii+1 ci tanh (i) wbii +i1= cii+1 ci tanh (i) + b(i1) i1 | {z } b(i1) wbii | {z } (42)

i bi

where b(i1) := i1(yr, b1,..., b(i1), 1, ..., i1, yr,...,y(i)r ), i is a function of1,...,i+1,

and bi is bounded..

Step n

In the last step, we modify the standard backstepping procedure in order to use the newrobust control mechanism of [18] to deal with the additive disturbance in (1d). Let thevariable r(t) be defined as

r :=1

cnn + n (43)

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where

n :=1

cn1(xn n1) . (44)

After differentiating (43), we obtain

r = 1cnn + cnr cnn. (45)

Differentiating n twice produces

n =1

cn1

|n + d + u n1

(46)

where (1d) was used.We first concentrate on the calculation of the term n1 in (46). To that end, we have

n1 =

n1

Xj=1

n1

xj xj +

n1

Xj=1

n1

j

.

j +

n

Xj=1

n1

y(j1)r y

(j)

r

=n1Xj=1

n1

xjxj+1 +

n1Xj=1

n1

j

.

j +nX

j=1

n1

y(j1)r

y(j)r | {z } +

n1Xj=1

n1

xj|j (x1, , xj)

(47)

where is known. Differentiating (47) gives

n1 = +n1

Xj=1n1

xj "j

Xk=1 |j

xk xk+1 + Tk #

+n1Xj=1

"n1Xk=1

2n1

xjxk

xk+1 +

Tk

+2n1

xjk

.

k

!+

nXk=1

2n1

xjy(k1)r

y(k)r

#Tj

= + g1(x1,...,xn, 1, ..., n1, yr,...,y(n)r )

+f1(x1,...,xn, 1,..., n1, yr,...,y(n1)r ) (48)

where

g1 =n1

Xj=1"n1

Xk=12n1

xjxk(xk+1 +

|

k) +2n1

xjkbk

!+

n

Xk=12n1

xjy(k1)r

y(k)r

#Tj

+n1Xj=1

n1

xj

"jX

k=1

|j

xk(xk+1 +

|

k)

#Tj , (49)

f1 =n1Xj=1

n1Xk=1

2n1

xjkk

!|

j , (50)

and property P3 of the projection operator was used.

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We now turn our attention to the calculation of the term |n in (46). Thus, we have

Tn =n1Xj=1

|nxj

xj +|nxn

xn

=

n1Xj=1

|

nxj

|j + xj+1 + |nxn cn1cnr cn1cnn +n1Xj=1

n1xj|j +!

= g2(x1, , xn, 1, , n1, yr,...,y(n)r ) + f2(x1, , xn, 1, , n1, yr,...,y

(n)r ) (51)

where

g2 =|nxn

n1Xj=1

n1

xj|

j +n1

xjxj+1 +

n1

jbj +

n1

y(j1)r

y(j)r

!

+n1

Xj=1|nxj

|j + xj+1

(52)

and

f2 =|nxn

cn1cnr cn1cnn +

n1Xj=1

n1

jj

!. (53)

After substituting (48), (51), and (46) into (45), we obtain

r =1

cn1cn

f2 f1 + d + u + g2 g1

+ cnr cnn. (54)

We now add and subtract the terms gb1 := g1(yr, b1, ...,b(n1), 1,..., n1, yr,...,y(n)r ) and

gb2 := g2(yr, b1, ...,b(n1), 1,..., n1, yr,...,y(n)r ) to the right-hand side of (54) to obtain

r =1

cn1cn

f2 f1 + d + u + g2 gb2 + gb1 g1 | {z } + gb2 gb1| {z }

+ cnr cnn (55)

f3 h

where h(yr, b1,..., b(n1), 1, ..., n1, yr,...,y(n)r ) has the special property that

h(t), h(t) L (56)due to (3) and properties P1 and P3 of the projection operator. Finally, we rewrite (55) as

r =1

cn1cn + h + d + u + f + cnr cnn (57)where f := (f2 f1 + f3) /cn1cn.Remark 9 Using (137), (139), and (140), we can show f has the special property that

kfk f(kxk) kzk (58)where f() R0 is some globally invertible, non-decreasing function, and

x = (1, 2,...,n, r)| z =

tanh(1), tanh(2),..., tanh(n1),n, r

|

. (59)

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Based on (57), we design u as [18]

u = sgn(n) cn1cn(cn + cn+1 + 1)r (60)

where > 0. The actual C0 control input can be written from (60) as follows

u(t) = (t) (0) cn1cn(cn + cn+1 + 1) (n(t) n(0))Zt

0

[cn1cn(cn + cn+1 + 1)n() + sgn(n())] d(61)

where u(0) = 0. After substituting (112) into (55), we obtain the closed-loop system

r = r cnn + f cn+1r +1

cn1cn

h + d sgn(n)

. (62)

C. Main Result

We now state the main result for the robust adaptive controller for the parametric strict-feedback system (1). In proving the main result, we will utilize the technical lemmas inAppendix C.

Theorem 1 The control law (61) ensures that all system signals are bounded and e(t) 0as t , provided

> kh(t)kL +d(t)

L+

1

cn

h(t)L

+d(t)

L

, (63)

cn > cn1

2cn22

, (64)

and the control gains ci, i = 1,...,n + 1 are selected sufficiently large relative to the systeminitial conditions.

Proof. Let the function P(t) R be defined as follows

P(t) := b Zt0

L()d (65)

where

L =

r

cn1cn h + d sgn(n) (66)b =

1

cn1c2n

h|n(0)| + n(0)

h(0) + d(0)

i. (67)

If is selected to according to (63), it follows from Lemma 1 that P(t) 0.We now define the following function V

V := Vn1 +1

22n +

1

2r2 + P. (68)

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Using (138), we can bound (68) as follows

1 ln (cosh(ksk)) V 2 ksk2 (69)

where

s =

x|, |

1,..., |

n1,P

|

, 1 =12

min

1,min1i

, 2 = max

12max

1i

, 1

,

(70)and x was defined in (59).

After taking the time derivative of (68) and substituting from (43), (62), and (65), weobtain3

V = Vn1 cn2n r2 + rf cn+1r2 +r

cn1cn

h + d sgn(n)

L

= Vn1 cn2n r2 + rf cn+1r2 (71)

upon use of (66). Substituting now from (41) for j = n 1 and (58) gives

V n1Xj=1

cj

cj1tanh2(j) +

1

cj1tanh

j

cjj+1 + wj

cn2n r2 +f(kxk) kzk krk cn+1r2

. (72)

We can upper bound (72) by using (36) as follows

V n1

Xj=1cj

cj1tanh2(j) cn2n r2 + f(kxk) kzk krk cn+1r2

+n1Xj=1

"cj

cj1tanh

jj+1 +

tanh j jXk=1

jk

j ktanh(k)k#

. (73)

Using (139) and (140), we can show that for k = 1,...,n 2ck

ck1tanh (k) k+1 +

tanh k+1 k+1,k k+1 ktanh(k)k ck

ck1 k+1

+ 1

ktanh(k)k

tanh

k+1

+k+1,k

k+1 ktanh (k)ktanh k+1= k+1,k

k+1 ktanh(k)ktanh k+1

(74)

3 Let f(s, t) denote the right-hand side of the closed-loop system on which the stability analysis is beingperformed. Notice from (62) and (66) that f(s, t) has a discontinuity on the set of Lebesgue measure zero{(s, t) :

n= 0}. Since Lemma 2 requires that a solution exist for s = f(s, t), see [18] for a discussion on the

issue of existence of solutions.

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wherek+1,k

k+1 := ckck1 k+1 + 1 + k+1,k k+1 . (75)

Now, let cj = cj1 (kj + n j), j = 1,...,n 2 and cn1 = cn2 (kn1 + 2), and rewrite (73)as

V n1Xj=1

(kj + 1) tanh2(j) cn2n r2 +

n1Xj=1

jjj tanh2 j

+n1Xj=1

j1Xk=1

jk

j ktanh(k)ktanh j tanh2(j)+

cn1cn2

tanhn1

n tanh2

n1

+f(kxk) kzk krk cn+1r2

(76)

where j+1,j = j+1,j and 10 = 0. After completing squares on the bracketed terms of (76),we obtain

V n1Xj=1

tanh2(j) r2 "

cn

cn12cn2

2#2n +

2f(kxk)

4cn+1kzk2

n1Xj=1

"kj jj

j 14j1Xk=1

2jkj

#tanh2

j

. (77)

Let

j j

:= jj j

14

j1

Xk=12jk

j, j = 1,...,n 1, (78)

cn > (cn1/2cn2)2 , (79)

and := min

1, cn

cn12cn2

2, then (77) can be written as

V n1Xj=1

kj j

j tanh2 j 2f(kxk)4cn+1

kzk2 . (80)

It follows from (80) thatV

kzk2 (81)

for

kj > jj , j = 1,...,n 1 and cn+1 > 2f (kxk)4 (82)

where the constant satisfies 0 < < 1.We now apply Lemma 2 by first determining from (69) and (81) that

W1(s) = 1 ln (cosh(ksk)) W2(s) = 2 ksk2 W(s) = kzk2 . (83)

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From (82), we define the sets D and S as follows

D =

s : ksk < min1j (kj),

1f (2

cn+1)

, j = 1,...,n 1 (84)

S =

s D : W2(s) < 1 ln

cosh

min

1j (kj),

1f (2

cn+1)

, j = 1,...,n 1. (85)

We can now invoke Lemma 2 to state that s(t)

L. From (3), we then know x1(t)

L.

From (13), (14), and property P3 of the projection operator, we know 1(t),.

1 (t) L.We can now use (9) to show x2(t) L. From (1a), we know x1(t) L. Continuingwith this procedure, we can show i(t), xi+1(t), L, i = 2,...,n 1. We can then statei(t), i1(t) L, i = 1,...,n 1 by using (33). Using (62) and assumption A2, we canshow r(t) L. From (43) and (47), we know n(t), n1(t) L; hence, from the timederivative of (44), we can conclude xn(t) L. Finally, we can use (1d) to show thatu(t) L.

Using the above boundedness statements, it is clear from (83) that W(s(t)) L, whichis a sufficient condition for W(s) being uniformly continuous. It then follows from Lemma2 that kz(t)k2

0 as t

s(0)

S, which implies from (59) that e(t)

0 as t

s(0) S.Note that the region of stability in (85) can be made arbitrarily large to include any

initial conditions by increasing the control gains ci, i = 1,...,n+1 (i.e., a semi-global stabilityresult). Specifically, we can use the second equation in (83) and (85) to calculate the regionof stability as follows

ks(0)k icosh1exp21 ks(0)k2 , i = 1,...,n 1 and

cn+1 >1

42f

cosh1

exp

2

1ks(0)k2

(87)where

ks(0)k =

vuut nXi=1

2i (0) + r2(0) +

n1Xi=1

|

i (0)i(0) + P(0) (88)

Note that the inequalities in (79) and (87) can be satisfied for large enough gains ci, i =1,...,n + 1 because: i) i() is not a function ofci, ii) i(0), r(0), and P(0) are only a functionof 1/ci, and iii) r(0) is not a function of cn+1 since u(0) = 0.

D. Numerical Example

For simulation purposes, we considered the parametric strict-feedback system (1) with thefollowing model

1 (x1) =

" x211

4x31

#, 2 (x1, x2) =

" x211

4x32

#, 3 (x1, x2, x3) =

" x21

4x21x3

#, =

21

.

(89)

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The reference trajectory was selected as

yr = sin t

1 exp

t

3

3

. (90)

The initial conditions of the system were set to x1 (0) = 1, x2 (0) = 0, x3 (0) = 0,1 (0) =2 (0) = (0, 0)

|, while the controller parameters were set to

c1 = 20, c2 = 300, c3 = 30, c4 = 10, = 10, = 1,

= 1, 0 = 3, 1 = diag (20, 200) , 2 = diag (2, 20) . (91)

In the first simulation, the disturbance was set to d (t) = 10 sint (see Figure 1(a)). Thesimulation results are given in Figures 2 to 4(a). Figure 2 shows the output y (t) versus thereference trajectory yr (t) and the tracking error e(t). Figure 3 shows the elements of theparameter estimate vectors 1 (t) and 2 (t) while the control input u (t) is shown in Figure4(a). Two other simulations were conducted to test the robustness of the proposed control

law to disturbances that do not satisfy assumption A2. To that end, the disturbance wasset to the triangle and square waveforms shown in Figures 1(b,c), and the simulation wasrerun without retuning the control gains. In both cases, the results for the tracking error andparameter estimates were nearly identical to the ones for the sinusoidal disturbance shownin Figures 2 and 3, and thus are not reproduced here. The profile of the control law is givenin Figures 4(b,c).

III. Flat Systems

A. Problem Statement

We now consider a class offlat nonlinear systems of the form

x(n) = f(x, x,...,x(n1), ) + G(x, x,...,x(n1), ) (u + d1) + d2 (92)

where x(i)(t) Rm, i = 0,...,n 1 are the system states, f Rm and G Rmm arenonlinear functions, Rp is an unknown constant parameter vector, d1(t), d2(t) Rm areunknown additive disturbances. We make the following assumptions regarding the systemmodel:

A4. G is symmetric positive definite and

m kk2

|

M(x, x,...,x(n1)

, ) m(x, x,...,x(n1)

) kk2

Rm

(93)

where M := G1, m > 0, and m(x, x,...,x(n1)) > 0 is a globally invertible, nonde-creasing function of each variable.

A5. f, G C2 and affine in .A6. di C2 and di(t), di(t), di(t) L, i = 1, 2.

15


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Given a Cn reference trajectory xr(t) Rm satisfying

x(i)r (t) L, i = 0, 1,...,n + 2 (94)

and the tracking error

e1 := xr x, (95)our goal is to construct a state feedback control u(x, x,...,x(n1)) that ensures e(

i)1 (t) 0 as

t , i = 1,...,n 1 and the boundedness of all closed-loop signals.

B. Construction of Robust Adaptive Control Law

We begin the construction by introducing the following auxiliary error signals ei(t) Rm,i = 2, 3,...,n

e2 := e1 + e1 (96a)

e3 := e2 + e2 + e1 (96b)

...

en := en1 + en1 + en2. (96c)

An expression can be derived for ei, i = 3, 4,...,n in terms ofe1 and its derivatives as follows[18]

ei =i1Xj=0

aije(j)1 (97)

where the constants aij are given by

ai0 = B(i) =1

5

1 +

5

2

!i

1 52

!i , i = 2, 3,...,n (98)

aij =i1Xk=1

B(i k j + 1)ak+j1,j1, i = 3, 4,...,n, j = 1, 2,...,i 2 (99)

ai,i1 = 1, i = 1, 2,...,n. (100)

Using assumption A4, we rewrite (92) as follows

Mx(n) = h + u + d1 + Md2 (101)

where h = M f. We now introduce the variable

r := en + en (102)

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where Rmm is diagonal positive definite. After differentiating (102) and multiplyingboth sides of the resulting equation by M, we obtain

Mr = Mn1

Xj=0anje

(j+2)1 + Men

= M

x(n+1)r +

n2Xj=0

anje(j+2)1 + en

!+ Mx(n) h u d1 Md2 Md2

= 12

M r en u + N d1 Md2 Md2 (103)

where the auxiliary function N(x, x,...,x(n), t) is defined as follows

N = M

x(n+1)r +

n2Xj=0

anje(j+2)1 + en

!+ M

x(n) +

1

2r

+ en h. (104)

We now arrange (103) into the following form

Mr = 12

Mr en u + N + Nr + (105)where

N(x, x,...,x(n), t) =

N Md2 M d2

Nr Mrd2 Mrd2

(106)

Nr(t) = N(xr, xr,...,x(n)r , t) (107)

Mr(t) = M(xr, xr,...,x(n1)r , ) (108)

(t) =

d1

Mrd2

Mrd2. (109)

Remark 10 Due to assumptions A5 and A6, we can use the Mean Value theorem to showN (kzk) kzk (110)where z := (e|1, e

|

2,...,e|

n, r|)| and () R0 is some globally invertible, nondecreasing

function. In view of assumption A5, we can also linearly parameterize Nr in the sense that

Nr = Wr(t) (111)

where Wr(t) Rmp is the known regressor, function only of xr and its derivatives.

Based on (105) and (111), we design u as

u = (K+ Im) r + CSgn (en) + Wr (112)

= W|r r (113)

where K, C Rmm and Rpp are diagonal positive definite, andSgn() := (sgn(1), sgn(2), ..., sgn(m))

| , = (1, 2,..., m)| . (114)

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Since r is not available for measurement, we can generate via

(t) =

Zt0

W|r ()en()dZt0

W|r ()en()d+ W|

r (t)en(t) W|r (0)en(0), (115)

while the actual control input is given by

u(t) = (K+ Im)en(t) (K+ Im)en(0)

+

Zt0

h(K+ Im)en() + CSgn (en ()) + Wr () ()

id. (116)

Finally, after substituting (112) into (105), we obtain the closed-loop system

Mr = 12

M r en (K+ Im)r + Wr CSgn (en) + N + (117)

where = .

C. Main Result

The main result for the robust adaptive control for the flat system (92) is given next.

Theorem 2 The control law (116) and (115) ensures the boundedness of all system signalsand e(i)(t) 0 as t , i = 0,...,n 1, provided

Cii > ki(t)kL +1

ii

i(t)L

, i = 1,...,m, (118)

min() >1

2

, (119)

and the elements ofK are selected sufficiently large relative to the system initial conditions.

Proof. Let L and b in (65) be given by

L = r| ( Csgn(en)) (120)

b =mXi=1

Cii |eni(0)| + e|

n(0)(0). (121)

If Cii is selected to according to (118), it follows from Lemma 1 that P(t) 0.

We now defi

ne the following function

V :=1

2

nXi=1

e|i ei +1

2r|Mr +

1

2|

1 + P, (122)

which satisfies the bounds1 ksk

2 V 2(ksk) ksk2 (123)

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where4

s =

z|, |

,

P|

, 1 =1

2min

1, m,min

1

, 2 = max

1, m(ksk),

1

2max

1

.

(124)

Diff

erentiating (122) along (96), (102), (113), and (117), we obtain

V = n1Xi=1

e|i ei e|nen + e|n1en r|r + r|N r|Kr (125)

upon use of (120). Using (110), (119), and the fact thateTn1en 12 ken1k2 + kenk2, we

obtain

V 3 kzk2 + krk (kzk) kzk min (K) krk2

3 2(kzk)

4min (Ks) kzk2 (126)

where 3 = min12

,min() 12

. From (126), we can state that

V kzk2 for min (K) > 143

2(kzk) (127)

where 0 < < 3.From (123) and (127), Lemma 2 can be invoked using arguments similar to those in the

proof of Theorem 1. That is, we first determine the sets

D = ns : ksk < 1 2p3min (K)o (128)S =

s D : W2(s) < 1

1(2

p3min (K))

2. (129)

Then, from Lemma 2, we know s(t) L. From (102), we know en(t) L. Using (94) and(95), we can show x(i)(t) L, i = 0, 1,...,n. We now know that M(t), f(t) L. Finally,we can use (101) to show u(t) L. We can now establish the uniform continuity of W(s),and then invoke Lemma 2 to show kz(t)k2 0 as t y(0) S. From the definition ofz, this implies ei(t), r(t) 0 as t y(0) S and i = 1, 2,...,n. From (102), we knowen(t) 0 as t y(0) S. Now, we can recursively use (97) to show e(i)1 (t) 0 ast , i = 1, 2,...,n 1 y(0) S. Finally, from the time derivative of (97) with i = n, wecan show e

(n)1 (t) 0 as t y(0) S. Note that the set (129) can be made arbitrarily

large to include any initial conditions by increasing the eigenvalues of K.

4 Using (94), (95), and (96), one can show(x, x,...,x(n1)) (ksk) where () is some positive function.

Thus, m(x, x,...,x(n1)) m(ksk).

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D. Numerical Example

To illustrate the proposed control, we considered the flat system of (92) with n = 1 and thefollowing model

f =

x1x2x22 , G =

2 + cos x1

10

03 + sin x2

2

,

=

12

=

21

, d1 =

cos2tsin2t

, d2 =

sin3tcos3t

(130)

where x = (x1, x2)|. The reference trajectory was selected as

xr =

xr1xr2

=

sin t

1 exp

t

3

5

2sin t

1 exp

t

3

2

. (131)

The initial conditions of the system were set to x (0) = (0.1, 0.2)| and (0) = (0, 0)|, whilethe controller parameters were chosen as = I2, K = 20I2, C = 15I2, and = 10I2. Figure5 shows the state x (t) versus the reference trajectory xr (t) (top plots) and the tracking errore1(t) (bottom plots). Figure 6 shows the parameter estimate (t) and the control input u (t).

IV. Conclusion

This work considered the tracking problem for nth-order MIMO nonlinear systems in thepresence of additive disturbances and parametric uncertainties. The continuous robust adap-tive controller, whose construction is founded on the fusion of an adaptation law and a dy-

namic robust control mechanism, exploits the two times continuous differentiability of thedisturbances. The proposed construction was applied to two classes of nonlinear systems:parametric strict-feedback systems and flat systems. In both cases, the resulting robustadaptive control law guarantees the semi-global asymptotic convergence of the tracking er-ror to zero and boundedness of all closed-loop signals. Numerical simulations illustrated theperformance of the proposed control laws, including the robustness to disturbances that donot satisfy the given assumptions. Since the tuning function method [10] does not seemapplicable to the proposed construction for parametric strict-feedback systems, the robustadaptive control law for this class of systems is overparameterized.

References

[1] Z. Cai, M.S. de Queiroz, B. Xian, and D.M. Dawson, Adaptive Asymptotic Trackingof Parametric Strict-Feedback Systems in the Presence of Additive Disturbance, Proc.IEEE Conf. Decision and Control, pp. 1146-1151, Paradise Island, Bahamas, 2004.

[2] Z. Cai, M.S. de Queiroz, and D.M. Dawson, Asymptotic Adaptive Regulation of Para-metric Strict-Feedback Systems with Additive Disturbance, Proc. American ControlConf., Portland, OR, 2005, to appear.

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[3] Z. Cai, M.S. de Queiroz, and D.M. Dawson, A Sufficiently Smooth Projection Opera-tor, IEEE Trans. Automatic Control, in press.

[4] R.A. Freeman, M. Krstic, and P.V. Kokotovic, Robustness of Adaptive NonlinearControl to Bounded Uncertainties, Automatica, Vol. 34, No. 10, pp. 1227-1230, 1998.

[5] S.S. Ge and J. Wang, Robust Adaptive Tracking for Time-Varying Uncertain NonlinearSystems With Unknown Control Coefficients, IEEE Trans. Automatic Control, Vol. 48,No. 8, pp. 14621469, 2003.

[6] F. Ikhouane and M. Krstic, Robustness of the Tuning Functions Adaptive BacksteppingDesigns for Linear Systems, IEEE Trans. Automatic Control, Vol. 43, No. 3, pp. 431437, 1998.

[7] P.A. Ioannou and P.V. Kokotovic, Adaptive Systems with Reduced Models, Lecture Notesin Control and Information Sciences, Vol. 47, New York, NY: Springer-Verlag, 1983.

[8] P. Ioannou and J. Sun, Robust Adaptive Control, Englewood Cliff

s, NJ: Prentice Hall,1996.

[9] H. Khalil, Nonlinear Systems, New York, NY: Prentice Hall, 2002.

[10] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control De-sign, New York, NY: John Wiley & Sons, 1995.

[11] R. Marino and P. Tomei, Robust Adaptive State-Feedback Tracking for NonlinearSystems, IEEE Trans. Automatic Control, Vol. 43, No. 1, pp. 84-89, 1998.

[12] K.S. Narendra and A.M. Annaswanny, Stable Adaptive Systems, Englewood Cliffs, NJ:

Prentice Hall, 1989.

[13] R. Ortega, A. Loria, P.J. Nicklasson, and H. Sira-Ramirez, Passivity-based Control ofEuler-Lagrange Systems, London: Springer-Verlag, 1998.

[14] Z. Pan and T. Basar, Adaptive Controller Design for Tracking and Disturbance At-tenuation in Parametric Strict-Feedback Nonlinear Systems, IEEE Trans. AutomaticControl, Vol. 43, No. 8, pp. 1066-1083, 1998.

[15] M.M. Polycarpou and P.A. Ioannou, A Robust Adaptive Nonlinear Control Design,Automatica, Vol. 33, No. 3, pp. 423-427, 1996.

[16] J.-B. Pomet and L. Praly, Adaptive Nonlinear Regulation: Estimation from LyapunovEquation, IEEE Trans. Automatic Control, Vol. 37, No. 6, pp. 729-740, 1992.

[17] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness,Englewood Cliff, NJ: Prentice Hall, 1989.

[18] B. Xian, D.M. Dawson, M.S. de Queiroz, and J. Chen, A Continuous AsymptoticTracking Control Strategy for Uncertain Nonlinear Systems, IEEE Trans. AutomaticControl, Vol. 49, No. 7, pp. 1206-1211, 2004.

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[19] Y. Zhang and P.A. Ioannou, A New Class of Nonlinear Robust Adaptive Controllers,Int. J. Control, Vol. 65, No. 5, pp. 745-769, 1996.

A Projection Operator

A smoothened version of the projection operator introduced in [16] was recently proposedin [3]. The projection operator of [3] replaces the Lipschitz continuity property of thePomet/Praly projection [16] with the stronger property of arbitrarily many times contin-uous differentiability, while introducing minor or no modifications to the other projectionproperties. This new projection operator is useful for backstepping-based, robust adaptivecontrollers, such as the one in Section II., which require multiple differentiations of theadaptation law. The projection operator of [3] is given by

Proj(i, i) = i 12p(i)

4 (2 + 20)ni

20, i = 1,...,n 1 (132)

where

p(i) = T

i i 20, (133)

1 =

pni(i) if p(i) > 0

0 otherwise,(134)

2 =1

2p(i)i +

s1

2p(i)i

2+ 2, (135)

i was defined in (40), is the gradient operator, , are arbitrary positive constants, and0 was defined in assumption A3. It can be proven that the above projection operator has

the following properties [3]: If i(0) , thenP1.

i(t) 0 + t 0;P2.

T

i Proj(i, i) T

i i;

P3. Proj(i, i) = i + bi andProj(i, i) kik

"1 +

0 +

0

2#+

0 +

220, where

i = i 1

1

2p(i)i +s

1

2p(i)i

2

+ 2

p(i)4 (2 + 20)

ni20

,

bi = 1p(i)

4 (2 + 20)ni

20,

(136)

and i kik"

1 +

0 +

0

2#kbik

0 +

220; (137)

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P4. Proj(i, i) Cni1.

B Hyperbolic Function Inequalities

It can be shown that the following inequalities hold

R

q

1

2tanh2 (kk) ln(cosh(kk))

qXi=1

ln(cosh(i)) kk2 (138)

tanh(kk) kTanh ()k (139)kk

tanh (kk) kk + 1 (140)

where Tanh() :=

tanh(1) , tanh(2) , ..., tanhq

|

.

C Technical Lemmas

The following lemmas are used in establishing the main results of the paper.

Lemma 1 Let z, : R0 Rm, (t), (t) L, andr = cz+ Az (141)

L = r| ( BSgn(z)) (142)where c > 0, A, B Rmm are diagonal positive definite, and

Sgn(z) := (sgn(z1), sgn(z2),..., sgn(zm))| . (143)

If the elements of B are selected to satisfy the following sufficient condition

Bii > ki(t)kL +c

Aii

i(t)L

, i = 1,...,m, (144)

then Zt0

L()d b (145)

where the positive constant b is given by

b = c

mXi=1

Bii |zi(0)| + z|(0)(0)

!. (146)

Proof. After substituting (141) into (142) and then integrating in time, we obtain

Zt0

L()d =

Zt0

(cz() + Az())| (() BSgn(z())) d

=

Zt0

z|()A (() BSgn(z())) d+ cZt0

dz|()

d()d

cZt0

dz|()

dBSgn(z())d. (147)

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After integrating by parts the second integral on the right-hand side of (147), we haveZt0

L()d =

Zt0

z|()A (() BSgn(z())) d+ cz|()()|t0 cZt0

z|()d()

dd

cmXi=1

Bii |zi()||

t

0

=

Ztt0

z|()A

() cA1d()

d BSgn(z())

d

+c (z|(t)(t) z|(0)(0)) cmXi=1

Bii (|zi(t)| |zi(0)|) . (148)

We now upper bound the right-hand side of (148) as follows

Zt

0

L()d Z

t

0

m

Xi=1

|zi()| Aii|i()| + cAii di()

d Bii d+c

mXi=1

|zi(t)| (|i(t)| Bii) + c

mXi=1

Bii |zi(0)| z|(0)(0)!

. (149)

It follows from (149) that if B is chosen according to (144), then (145) holds.

Lemma 2 Consider that a solution exists for the system

= f(, t), f : Rq R0Rq. (150)

Let the set D be defi

ned as D := { : kk < s} where s > 0, and let V: D R0 R0 bea C1 function satisfying

W1() V(, t) W2(), t 0, D (151)

whose derivative along the trajectories of (150) satisfy

V(, t) W(), t 0, D (152)

where W1(), W2() are C0 positive definite functions and W() is a differentiable, positive

semi-definite function. If(0) S, where

S := { D : W2() } , 0 < < minkk=s

W1(), (153)

then (t) is bounded. Furthermore, if W() is uniformly continuous, then

W((t)) 0 as t . (154)

Proof. See proof of Theorem 8.4 in [9].

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0 1 2 3 4 5 6 7 8 9 10

10

5

0

5

10

Disturbance

(a) Sinusoid

0 1 2 3 4 5 6 7 8 9 10

10

5

0

5

10

Disturbance

(b) Triangle

0 1 2 3 4 5 6 7 8 9 10

10

5

0

5

10

Time (sec)

Disturbance

(c) Square

Figure 1: (Parametric strict-feedback system) Disturbance d(t).

0 5 10 15 20 251.5

1

0.5

0

0.5

1

1.5

Time (sec)

yr

y

0 5 10 15 20 250.2

0

0.2

0.4

0.6

0.8

Time (sec)

Error

Figure 2: (Parametric strict-feedback system) Top plot: output y(t) versus reference trajec-tory yr(t). Bottom plot: tracking error e(t).

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0 5 10 15 20 252

1.5

1

0.50

0.5

1

1.5

2

2.5

Time (sec)

Parame

terEstimates

11

hat

12

hat

0 5 10 15 20 251

0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

ParameterEstimates

21

hat

22 hat

Figure 3: (Parametric strict-feedback system) Parameter estimates 1(t) and 2(t).

0 5 10 15 20 2560

40

200

20

40

ControlInput

(a) Sinusoidal Disturbance

0 5 10 15 20 2560

40

20

0

20

40

ControlInput

(a) Triangle Disturbance

0 5 10 15 20 2560

40

20

0

20

40

Time (sec)

Control

Input

(a) Square Disturbance

Figure 4: (Parametric strict-feedback system) Control input u(t).

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0 1 2 3 4 5 6 7 8 991.5

1

0.5

0

0.5

1

1.5

Time (sec)

x1

xr1

0 1 2 3 4 5 6 7 8 90.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

e11

0 1 2 3 4 5 6 7 8 991.5

1

0.5

0

0.5

1

1.5

Time (sec)

x2

xr2

0 1 2 3 4 5 6 7 8 990.25

0.2

0.15

0.1

0.05

0

0.05

0.1

Time (sec)

e12

Figure 5: (Flat system) Top plots: state x(t) versus reference trajectory xr(t). Bottom plots:tracking error e(t).

0 1 2 3 4 5 6 7 8 9

0

0.1

0.2

0.3

0.4

0.5

ParameterEstimate

s

Time (sec)

1

hat

2

hat

0 1 2 3 4 5 6 7 8 94

3

2

1

0

1

2

3

ControlInputs

Time (sec)

u1

u2

Figure 6: (Flat system) Parameter estimate (t) and control input u(t).

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Dawson - Asymptotic Robust Adaptive Tracking of Nonlinear Systems