The case of relative degree one

WhenP…s†is relative degree one, the reference modelM…s†can be chosen to be strictly positive real (SPR) (Narendra and Annaswamy, 1988). The error model (3.10) can now be rewritten as

e0…t† ˆ M…s†2n1

up >w‡>wdo‡2nM 1…s†‰doŠ ‡…s†‰upŠ

…t†

…3:11†

In the error model (3.11), the terms >w;>wdo‡2nM 1…s†‰doŠ and …s†‰upŠare the uncertainties due to the unknown plant parameters, output disturbance, and unmodelled dynamics, respectively. Let…Am;Bm;Cm†be any minimal realization ofM…s†2n1 which is SPR, then we can get the following state space representation of (3.11) as:

_

e…t† ˆAme…t†‡Bm…up…t† >w…t†‡>wdo…t†‡2nM 1…s†‰doŠ…t†‡…s†‰upŠ…t††

e0…t† ˆCme…t† …3:12†

where the triplet…Am;Bm;Cm†satis®es

PmAm‡A>mPm ˆ 2Qm; PmBm ˆ Cm> …3:13†

for somePmˆP>m>0 and QmˆQ>m>0.

The adaptive variable structure controller for relative degree-one plants is now summarized as follows:

(1) De®ne the regressor signal w…t† ˆ a…s†

…s†‰upŠ…t†;a…s†

…s†‰ypŠ…t†;yp…t†;rm…t†

>

ˆ ‰w1…t†;w2…t†;. . .;w2n…t†Š>

…3:14†

and construct the normalization signal m…t† [15] as the state of the following system:

_

m…t† ˆ 0m…t† ‡1…jup…t†j ‡1†; m…0†>1

0 …3:15†

where0; 1>0 and0‡2<min…k1;k2†for some2>0. The parameter k2>0 is selected such that the roots of…s k2†lie in the open left half complex plane, which is always achievable.

(2) Design the control signalup…t†as up…t† ˆX2n

jˆ1

… sgn…e0wj†j…t†wj…t†† sgn…e0†1…t† sgn…e0†2…t†m…t† …3:16†

sgn…e0† ˆ

1 if e0>0 0 if e0ˆ0 1 if e0<0 8>

<

>:

(3) The adaptation law for the control parameters is given as _j…t† ˆjje0…t†wj…t†j; jˆ1;. . .;2n _1…t† ˆg1je0…t†j

_2…t† ˆg2je0…t†jm…t† …3:17†

wherej;g1;g2>0 are the adaptation gains andj…0†; 1…0†; 2…0†>0 (in general, as large as possible)jˆ1;. . .;2n.

The design concept of the adaptive variable structure controller (3.15) and (3.16) is simply to construct some feedback signals to compensate for the uncertainties because of the following reasons:

. By assumption (A5), it can be easily found that j>wdo…t†‡

2nM…s† 1‰doŠ…t†j 1 for some1>0.

. With the construction ofm, it can be shown [15] that…s†‰upŠ…t† 2m…t†;

8t0 and for some constant2>0.

Now, we are ready to state our results concerning the properties of global stability, robust property, and tracking performance of our new adaptive variable structure scheme with relative degree-one system.

Theorem 3.1 (Global Stability, Robustness and Asymptotic Zero Tracking Performance) Consider the system (3.1) satisfying assumptions (A1)±(A5) with relative degree being one. If the control input is designed as in (3.15), (3.16) and the adaptation law is chosen as in (3.17), then there exists>0 such that for 2 ‰0; Š all signals inside the closed loop system are bounded and the tracking error will converge to zero asymptotically.

Proof: Consider the Lyapunov function Vaˆ12e>Pme‡X2n

jˆ1

1

2j…j jjj†2‡X2

jˆ1

1

2gj…j j†2

wherePm satis®es (3.13). Then, the time derivative ofVa along the trajectory (3.12) (3.17) will be

V_aˆ e>Qme‡e0 up >w‡>wdo‡2nM 1…s†‰doŠ ‡…s†‰upŠ

‡X2n

jˆ1

1

j…j jjj†_j‡X2

jˆ1

1

gj…j j†_j e>Qme X2n

jˆ1

je0wjj…j jjj† je0j…1 1† je0j…2 2†m

‡X2n

jˆ1

1

j…j jjj†_j‡X2

jˆ1

1

gj…j j†_j

qmjej2

for some constant qm>0. This implies that e2L2\L1 and j;jˆ 1;. . .;2n; 1; 2;e02L1 and, hence, all signals inside the closed loop system are bounded owing to Lemma A in the Appendix. On the other hand, it can be concluded thate_2L1by (3.12). Hence,e2L2\L1ande_2L1readily imply thateande0 will at least converge to zero asymptotically by Barbalat's lemma

[19]. Q.E.D.

In Theorem 3.1, suitable integral adaptation laws are given to compensate for the unavailable knowledge of the bounds onjjjandj. Theoretically, the adaptive variable structure controller will stabilize the closed loop system with guaranteed robustness and asymptotic zero tracking performance no matter whatj…0†'s andj…0†'s are. However, according to the following Theorem 3.2, we will expect that positive and large values of j…0†; j…0† should result in better transient response and tracking performance, especially when j…0†>jjj; j…0†> j.

Theorem 3.2 (Finite-Time Zero Tracking Performance with High Gain Design) Consider the system set-up in Theorem 3.1. If j…0†

jjj; j…0† j; then the output tracking error will converge to zero in ®nite time with all signals inside the closed loop system remaining bounded.

Proof Consider the Lyapunov function Vbˆ12e>Pme where Pm satis®es

(3.13). The time derivative ofVb along the trajectory (3.12) becomes V_bˆ e>Qme X2n

jˆ1

je0wjj…j jjj† je0j…1 1† je0j…2 2†m e>Qme

k3Vb

for some k3>0 since j…t† jjj; j…t† j;8t0. This implies that e approaches zero at least exponentially fast. Furthermore, by the fact that

e0e_0ˆe0fCmAme‡CmBm…up >w‡>wdo‡2nM 1…s†‰doŠ ‡…s†‰upŠ†g k4je0jjej X2n

jˆ1

je0wjj…j jjj† je0j…1 1† je0j…2 2†m k4je0jjej je0j X2n

jˆ1

jwjj…j jjj† ‡ …1 1† ‡ …2 2†m

where k4ˆ jCmAmj, and that jej approaches zero at least exponentially fast, there exists a ®nite timeT1 >0 such thate0e_0 k5je0jfor allt>T1and for somek5>0. This implies that the sliding surfacee0 = 0 is guaranteed to be reached in some ®nite timeT2>T1 >0. Q.E.D.

Remark 3.2: Although theoretically only asymptotic zero tracking perform- ance is achieved when the initial control parameters are arbitrarily chosen, it is encouraged to set the adaptation gainsj andgj in (3.17) as large as possible.

This is because the large adaptation gains will provide high adaptation speed and, hence, increase the control parameters to a suitable level of magnitude so as to achieve a satisfactory performance as quickly as possible. These expected results can be observed in the simulation examples.