The case of arbitrary relative degree

When the relative degree of (3.1) is greater than one, the controller design becomes more complicated than that given in Section 3.3. The main di€erence between the controller design of a relative degree-one system and a system with relative degree greater than one can be described as follows. When (3.1) is relative degree-one, the reference model can be chosen to be strictly positive real (SPR) [19]. Moreover, the control structure and its subsequent analysis of global stability, robustness and tracking performance are much simpler. On the contrary, when the relative degree of (3.1) is greater than one, the reference modelM…s†is no longer SPR so that the controller and the analysis technique in relative degree-one systems cannot be directly applied. In order to use the

similar techniques given in Section 3.3, the adaptive variable structure controller is now designed systematically as follows:

(1) Choose an operatorL1…s† ˆl1…s†. . .l 1…s† ˆ …s‡1†. . .…s‡ 1†such that M…s†L1…s† is SPR and denote Li…s† ˆli…s†. . .l 1…s†;iˆ 2;. . .; 1;L…s† ˆ1.

(2) De®ne augmented signal

ya…t† ˆM…s†L1…s†

u1 1 L1…s†‰upŠ

…t†

and auxiliary errors

ea1…t† ˆe0…t† ‡ya…t† …3:18†

ea2…t† ˆ 1

l1…s†‰u2Š…t† 1

F…s†‰u1Š…t† …3:19†

ea3…t† ˆ 1

l2…s†‰u3Š…t† 1

F…s†‰u2Š…t† …3:20†

...

ea…t† ˆ 1

l 1…s†‰upŠ…t† 1

F…s†‰u 1Š…t† …3:21†

where 1

F…s†‰uiŠ…t†is the average control ofui…t†withF…s† ˆ …s‡1†2, being small enough. In fact,F…s†can be any Hurwitz polynomial in s with degree at least two andF…0† ˆ1. In the literature, 1

F…s†is referred to as anaveraging ®lter, which is obviously a low-pass ®lter whose bandwidth can be arbitrarily enlarged as !0. In other words, if is smaller and smaller, the ®lter 1

F…s†is ¯atter and ¯atter.

(3) Design the control signalsup;ui, and the bounding functionmas follows:

u1…t† ˆX2n

jˆ1

… sgn…ea1j†j…t†j…t†† sgn…ea1†1…t† sgn…ea1†2…t†m…t†

…3:22†

ui…t† ˆ sgn…eai† li 1…s†

F…s† ‰ui 1Š…t†

‡

; iˆ2;. . .; …3:23†

up…t† ˆu…t† …3:24†

with >0 and

…t† ˆ 1

l1…s† 1

l 1…s†‰wŠ…t† ˆ 1

L1…s†‰wŠ…t†

The bounding functionm…t†is designed as the state of the system m…t† ˆ_ 0m…t† ‡1…jup…t†j ‡1†; m…0†>1

0 …3:25†

with0; 1>0 and 0‡2<min…k1;k2; 1;. . .; 1†for some2>0.

(4) Finally, the adaptation law for the control parametersj;jˆ1;. . .;2nand 1; 2 are given as follows:

_j…t† ˆjjea1…t†j…t†j; jˆ1;. . .;2n …3:26†

_1…t† ˆg1jea1…t†j …3:27†

_2…t† ˆg2jea1…t†jm…t† …3:28†

withj…0†>0; j…0†>0 and j>0;gj>0.

In the following discussions, the construction of feedback signals…t†;m…t†and the controller (3.22) (3.23) will be clear.

In order to analyse the proposed adaptive variable structure controller, we

®rst rewrite the error model (3.10) as follows:

e0…t† ˆM…s†‰up 2n1>w‡2n1>wdo‡2n1…s†‰upŠ

‡ …2n1 1†upŠ…t† ‡do…t†

ˆM…s†L1…s†

1

L1…s†‰upŠ 2n1>‡2n1

L1…s†‰>wdo‡2nM 1…s†‰doŠŠ

‡ 2n1

L1…s†‰…s†‰upŠ ‡ …1 2n†upŠ

…t† …3:29†

Now, according to the design of the above auxiliary error (3.18) and error model (3.29), we can readily ®nd thatea1 always satis®es

ea1…t† ˆM…s†L1…s†

u1 2n1>‡ 2n1

L1…s†‰>wdo‡2nM 1…s†‰doŠŠ

‡ 2n1

L1…s†‰…s†‰upŠ ‡ …1 2n†upŠ

…t† …3:30†

It is noted that the auxiliary errorea1 is now explicitly expressed as the output term of a linear system with SPR transfer functionM…s†L1…s†driven by some uncertain signals due to unknown parameters, output disturbances, un- modelled dynamics and unknown high frequency gain sign.

Remark 4.1 The construction of the adaptive variable structure controller (3.22) is now clear since the following facts hold:

. Since 2n1 L1…s†

h>wdo‡2nM 1…s†‰doŠi

…t†is uniformly bounded due to (A5), we have

2n1 L1…s†

h>wdo‡2nM 1…s†‰doŠi …t†

1 …3:31†

for some1.

. With the design of the bounding functionm…t†(3.25), it can be shown that 2n1

L1…s†

h…s†‰upŠ ‡ …1 2n†up

i…t†

2m…t† …3:32†

for some2>0.

The results described in Remark 4.1 show that the similar techniques for the controller design of a relative degree-one system can now be used for auxiliary errorea1. But what happens to the other auxiliary errorsea2;. . .;ea, especially the real output errore0as concerned? In Theorem 4.1, we summarize the main results of the systematically designed adaptive variable structure controller for plants with relative degree greater than one.

Theorem 4.1 (Global Stability, Robustness and Asymptotic Tracking Performance) Consider the nonlinear time-varying system (3.1) with relative degree >1 satisfying (A1)±(A5). If the controller is designed as in (3.18)±

(3.25) and parameter update laws are chosen as in (3.26)±(3.28), then there exists >0 and >0 such that for all 2 …0; † and 2 …0; †, the following facts hold:

(i) all signals inside the closed-loop system remain uniformly bounded;

(ii) the auxiliary errorea1 converges to zero asymptotically;

(iii) the auxiliary errorseai;iˆ2;. . .; , converge to zero at some ®nite time;

(iv) the output tracking errore0will converge to a residual set asymptotically whose size is a class K function of the design parameter.

Proof The proof consists of three parts.

Part I Prove the boundedness ofeai and1;. . .; 2n; 1; 2.

Step 1 First, consider the auxiliary error ea1 which satis®es (3.30). Since

M…s†L1…s†is SPR, we have the following realization of (3.20) _

e1ˆA1e1‡B1

u1 2n1>‡ 2n1

L1…s†‰>wdo‡2nM 1…s†‰doŠŠ

‡ 2n1

L1…s†‰…s†‰upŠ ‡ …1 2n†upŠ

ea1ˆC1e1 …3:33†

with P1A1‡A>1P1ˆ 2Q1; P1B1ˆC1> for some P1ˆP>1 >0 and Q1 ˆ Q>1 >0. Given a Lyapunov function as follows:

V1 ˆ 12e>1P1e1‡X2n

jˆ1

1

2j j j 2n 2

‡X2

jˆ1

1

2gj…j j†2 …3:34†

it can be shown by using (3.32) and (3.31) that V_1ˆ e>1Q1e1‡ea1

u1 2n1>‡ 2n1

L1…s†‰>wdo‡2nM 1…s†‰doŠŠ

‡ 2n1

L1…s†‰…s†‰upŠ ‡ …1 2n†upŠ

‡X2n

jˆ1

1

j j j 2n

_j‡X2

jˆ1

1

gj…j j†_j e>1Q1e1 X2n

jˆ1

jea1jj j j 2n

jea1j…1 1† jea1j…2 2†m

‡X2n

jˆ1

1

j j j 2n

_j‡X2

jˆ1

1

gj…j j†_j

ˆ e>1Q1e1

q1je1j2

for someq1>0 if the controller in (3.22) and update laws in (3.26)±(3.28) are given. This implies thate1; 1;. . .; 2n; 1; 22L1 andea12L2\L1.

Step 2 From (3.19)±(3.21), we can ®nd thatea2;. . .;ea satisfy _

ea2ˆ 1ea2‡u2 l1…s†

F…s†‰u1Š ...

_

eaˆ 1ea‡u l 1…s†

F…s† ‰u 1Š

Now by the following facts that foriˆ2;. . .; : d

dt 12e2ai

ˆeaie_ai

ˆeai i 1eai‡ui li 1…s†

F…s† ‰ui 1Š

ˆ i 1e2ai‡eai sgn…eai† li 1…s†

F…s† ‰ui 1Š

‡

li 1…s†

F…s†‰ui 1Š

or

d

dtjeaij i 1jeaij …3:35†

whenjeaij 6ˆ0. This implies thateaiwill converge to zero after some ®nite time T>0.

Part II Prove the boundedness of all signals inside the closed loop system.

De®ne eai ˆM…s†Li 1…s†‰eaiŠ;iˆ2;. . .; and Eaˆea1‡ea2‡ ‡ea

which is uniformly bounded due to the boundedness of eai. Then, from (3.18)±(3.21), we can derive that

Eaˆe0‡M…s†L1…s†

u1 1 L1…s†‰upŠ

‡M…s†L1…s†

1

l1…s†‰u2Š 1 F…s†‰u1Š

‡M…s†L2…s†

1

l2…s†‰u3Š 1 F…s†‰u2Š

...

‡M…s†L 1…s†

1

l 1…s†‰upŠ 1

F…s†‰u 1Š

ˆe0‡

1 1

F…s†

M…s†L1…s†

u1‡ 1

l1…s†‰u2Š ‡ ‡ 1

l1…s†. . .l 2…s†‰u 1Š

ˆ4 e0‡R …3:36†

Now, since k…ui†tk1k6k…e0†tk1‡k6;iˆ1;. . .; 1 for some k6>0 by Lemma A in the appendix, it can be easily found that

u1‡ 1

l1…s†‰u2Š ‡ ‡ 1

l1…s†. . .l 2…s†‰u 1Š

t

1 k7k…e0†tk1‡k7

for somek7>0. Furthermore, since theH1norm ofk1s…1 F…1s††k1ˆ2 and ksM…s†L1…s†k1ˆk8 for somek8>0, we can conclude that

k…R†tk1 1

s

1 1

F…s†

1

sM…s†L1…s†

1…k7k…e0†tk1‡k7† …k9k…e0†tk1‡k9†

for somek9>0. Now from (3.36) we have

k…e0†tk1 k…Ea†tk1‡ k…R†tk1 k…Ea†tk1‡…k9k…e0†tk1‡k9† which implies that there exists a >0 such that 1 k9>0 and for all 2 …0; †:

k…e0†tk1 k…Ea†tk1‡k9

1 k9 …3:37†

Combining Lemma A and (3.37), we readily conclude that all signals inside the closed loop system remain uniformly bounded.

Part III: Investigate the tracking performance of ea1 ande0.

Since all signals inside the closed loop system are uniformly bounded, we have ea12L2\L1; e_a12L1

Hence, by Barbalat's lemma, ea1 approaches zero asymptotically and Eaˆea1‡ea2‡ ‡ea also approaches zero asymptotically. Now, from the fact of (3.37) andEa approaching zero, it is clear thate0 will converge to a small residual set whose size depends on the design parameter. Q.E.D.

As discussed in Theorem 3.2, if the initial choices of control parameters j…0†; j…0†satisfy the high gain conditionsj…0†

j 2n

andj…0† j, then, by using the same argument given in the proof of Theorem 3.2, we can guarantee the exponential convergent behaviour and ®nite-time tracking performance of all the auxiliary errors eai. Since eai reaches zero in some

®nite time andEaˆea1‡ea2‡ ‡ea, it can be concluded thatEaconverges to zero exponentially and e0 converges to a small residual set whose size depends on the design parameter . We now summarize the results in the following Theorem 4.2.

Theorem 4.2: (Exponential Tracking Performance with High Gain Design) Consider the system set-up in Theorem 4.1. If the initial value of control parameters satisfy the high gain conditionsj…0†

j 2n

andj…0† j, then there exists a and such that for all 2 …0; Š and 2 …0; Š, the following facts hold:

(i) all signals inside the closed loop system remain bounded;

(ii) the auxiliary errorseai;iˆ1;. . .; , converge to zero in ®nite time;

(iii) the output tracking errorse0 will converge to a residual set exponentially whose size depends on the design parameter .

Remark 4.2: It is well known that the chattering behaviour will be observed in the input channel due to variable structure control, which causes the implementation problem in practical design. A remedy to the undesirable phenomenon is to introduce the boundary layer concept. Take the case of relative degree one, for example, the practical redesign of the proposed adaptive variable structure controller by using boundary layer design is now stated as follows:

up…t† ˆX2n

jˆ1

…e0wj†j…t†wj…t†

…e0†1…t† …e0†2…t†m…t† …3:38†

…e0† ˆ sgn…e0† if je0j> "

e0

" if je0j "

8<

:

for some small >0. Note that…e0†is now a continuous function. However, one can expect that the boundary layer design will result in bounded tracking error, i.e. e0 cannot be guaranteed to converge to zero. This causes the parameter drift in parameter adaptation law. Hence, a leakage term is added into the adaptation law as follows:

_j…t† ˆjje0…t†wj…t†j j…t†; jˆ1;. . .;2n _1…t† ˆg1je0…t†j 1…t†

_2…t† ˆg2je0…t†jm…t† 2…t† …3:39†

for some >0.