Stability and robustness analysis

Before embarking on the stability and robustness analysis for the adaptive IMC schemes just proposed, we ®rst introduce some de®nitions [9, 4] and state and prove two lemmas which play a pivotal role in the subsequent analysis.

De®nition 4.1 For any signalx:‰0; 1† !Rn,xtdenotes the truncation ofx to the interval‰0;tŠand is de®ned as

xt…† ˆ x…† if t

0 otherwise

…1:21†

De®nition 4.2 For any signalx:‰0; 1† !Rn, and for any0,t0,kxtk2 is de®ned as

kxtk2ˆD Z t

0 e …t †‰xT…†x…†Šd

12

…1:22†

The k…†tk2 represents the exponentially weighted L2 norm of the signal truncated to ‰0;tŠ. When ˆ0 and tˆ 1, k…:†tk2 becomes the usual L2 norm and will be denoted by k:k2. It can be shown that k:k2 satis®es the usual properties of the vector norm.

De®nition 4.3 Consider the signalsx:‰0;1† !Rn,y:‰0;1† !R‡ and the set

S…y† ˆ x:‰0;1† !Rnj Z t‡T

t xT…†x…†d

Z t‡T

t y…†d‡c

for somec0 and8t; T 0. We say thatxisy-small in the mean ifx2S…y†.

Lemma 4.1 In each of the adaptive IMC schemes presented in the last section, the degree ofQ^d…s;t†in Step 3 of the certainty equivalence design can be made

time invariant. Furthermore, for the adaptiveH2 andH1 designs, this can be done using a single ®xedF…s†.

Proof The proof of this lemma is relatively straightforward except in the case of adaptiveH2optimal control. Accordingly, we ®rst discuss the simpler cases before giving a detailed treatment of the more involved one.

For adaptive partial pole placement, the time invariance of the degree of Q^d…s;t† follows trivially from the fact that the IMC parameter in this case is time invariant. For model reference adaptive control, the fact that the leading coecient ofZ^0…s;t†is not allowed to pass through zero guarantees that the degree ofQ^d…s;t†is time invariant. Finally, for adaptiveH1 optimal control, the result follows from the fact that the leading coecient of Z^0…s;t† is not allowed to pass through zero.

We now present the detailed proof for the case of adaptive H2 optimal control. Let nr;mr be the degrees of the denominator and numerator polynomials respectively ofR…s†. Then, in the expression forQ…s;^ t†in (1.19), it is clear that P^M1…s;t† ˆnth order polynomial

lth order polynomial while RM1…s† ˆ nrth order polynomial

mrth order polynomial. Also B^P1…s;t†RM…s†

ˆ…n 1†th order polynomial

nth order polynomial where nnr, strict inequality being attained when some of the poles of RM…s† coincide with some of the stable zeros of B^P1…s;t†. Moreover, in any case, thenth order denominator polynomial of ‰B^P1…s;t†RM…s†Š is a factor of thenrth order numerator polynomial of RM1…s†. Thus for the Q…s;^ t† given in (1.19), if we disregard F…s†, then the degree of the numerator polynomial is n‡nr 1 while that of the denominator polynomial is l‡mr4n‡nr 1.

Hence, the degree ofQ^d…s;t†in Step 3 of the certainty equivalence design can be kept ®xed at…n‡nr 1†, and this can be achieved witha single ®xed F…s†of relative degree n l‡nr mr 1, provided that the leading coecient of Z^0…s;t†is appropriately constrained.

Remark 4.1 Lemma 4.1 tells us that the degree of each of the certainty equivalence controllers presented in the last section can be made time invariant. This is important because, as we will see, it makes it possible to carry out the analysis using standard state-space results on slowly time-varying systems.

Lemma 4.2 At any ®xed timet, the coecients ofQ^d…s;t†,Q^n…s;t†, and hence the vectorsqd…t†,qn…t†, are continuous functions of the estimate…t†.

Proof Once again, the proof of this lemma is relatively straightforward except in the case of adaptive H2 optimal control. Accordingly, we ®rst discuss the simpler cases before giving a detailed treatment of the more involved one.

For the case of adaptive partial pole placement control, the continuity follows trivially from the fact that the IMC parameter is independent of…t†.

For model reference adaptive control, the continuity is immediate from (1.18) and the fact that the leading coecient of Z^0…s;t† is not allowed to pass through zero. Finally for adaptiveH1 optimal control, we note that the right half plane zero b^1 of Z^0…s;t† is a continuous function of …t†. This is a consequence of the fact that the degree ofZ^0…s;t†cannot drop since its leading coecient is not allowed to pass through zero. The desired continuity now follows from (1.20).

We now present the detailed proof for theH2optimal control case. Since the leading coecient ofZ^0…s;t†has been constrained so as not to pass through zero then, for any ®xedt, the roots ofZ^0…s;t†are continuous functions of…t†.

Hence, it follows that the coecients of the numerator and denominator polynomials of‰P^M…s;t†Š 1ˆ ‰B^P…s;t†Š‰P…s;^ t†Š 1 are continuous functions of …t†. Moreover, ‰‰B^P…s;t†Š 1RM…s†Š is the sum of the residues of

‰B^P…s;t†Š 1RM…s† at the poles of RM…s†, which clearly depends continuously on…t†(through the factor‰B^P…s;t†Š 1). SinceF…s†is ®xed and independent of , it follows from (1.19) that the coecients of Q^d…s;t†, Q^n…s;t† depend continuously on…t†.

Remark 4.2 Lemma 4.2 is important because it allows one to translate slow variation of the estimated parameter vector …t† to slow variation of the controller parameters. Since the stability and robustness proofs of most adaptive schemes rely on results from the stability of slowly time-varying systems, establishing continuity of the controller parameters as a function of the estimated plant parameters (which are known to vary slowly) is a crucial ingredient of the analysis.

The following theorem describes the stability and robustness properties of the adaptive IMC schemes presented in this chapter.

Theorem 4.1 Consider the plant (1.8) subject to the robust adaptive IMC control law (1.12)±(1.16), (1.17), where (1.17) corresponds to any one of the adaptive IMC schemes considered in the last section and r…t† is a bounded external signal. Then, 9 >0 such that 82 ‰0; †, all the signals in the closed loop system are uniformly bounded and the errory y^2S c22

m2

somec>07 for

7In the rest of this chapter, `c' is the generic symbol for a positive constant. The exact value of such a constant can be determined (for a quantitative robustness result) as in [13, 9]. However, for the qualitative presentation here, the exact values of these constants are not important.

Proof The proof is obtained by combining the properties of the robust adaptive law (1.12)±(1.16) with the properties of the IMC-based controller structure. We ®rst analyse the properties of the adaptive law.

From (1.10), (1.13) and (1.14), we obtain

"ˆ ~T‡

m2 ; ~X …1:23†

Consider the positive de®nite function V…† ˆ~ ~T~

2

Then, along the solution of (1.12), it can be shown that [9]

V_ ~T"

ˆ"‰ "m2‡Š (using (1.23)

1

2"2m2‡1 2

22

m2 (completing the squares) …1:24†

From (1.11), (1.15), (1.16), using Lemma 2.1 (Equation (7)) in [4], it follows that

m2L1. Now, the parameter projection guarantees that V;; ~ 2L1. Hence integrating both sides of (1.24) fromttot‡T, we obtain

"m2S 22 m2

Also from (1.12)

jj _ j"mjjj

m …1:25†

From the de®nition of, it follows using Lemma 2.1 (equation (7)) in [4] that

m2L1, which in turn implies that_2S c22 m2

. This completes the analysis of the properties of the robust adaptive law. To complete the stability proof, we now turn to the properties of the IMC-based controller structure.

The certainty equivalence control law (1.17) can be rewritten as snd

1…s†‰uŠ ‡1…t†snd 1

1…s†‰uŠ ‡. . .‡nd…t† 1

1…s†‰uŠ ˆqTn…t†and…s†

1…s†‰r "m2Š where 1…t†, 2…t†,. . .; nd…t† are the time-varying coecients of Q^d…s;t†.

De®ning x1ˆ 1

1…s†‰uŠ;x2ˆ s

1…s†‰uŠ;. . .;xnd ˆsnd 1

1…s†‰uŠ;XX‰x1;x2;. . .;xndŠT,

the above equation can be rewritten as

X_ ˆA…t†X‡BqTn…t†and…s†

1…s†‰r "m2Š …1:26†

where

A…t† ˆD

0 1 0 0

0 0 1 0 0

nd…t† nd 1…t† 1…t†

2 66 66 66 4

3 77 77 77 5

BˆD 0 0 0 1 2 66 66 66 4

3 77 77 77 5

Since the time-varying polynomialQ^d…s;t†is pointwise Hurwitz, it follows that for any®xed t, the eigenvalues ofA…t†are in the open left half plane. Moreover, since the coecients of Q^d…s;t†are continuous functions of…t†(Lemma 4.2) and…t† 2 C, a compact set, it follows that 9s > 0 such that

Refi…A…t††g s8t0 and iˆ1;2;. . .;nd

The continuity of the elements of A…t† with respect to …t†and the fact that _2 S c22

m2

together imply that A…t† 2 S_ c22 m2

. Hence, using the fact that

m2L1, it follows from Lemma 3.1 in [9] that 91 > 0 such that 82 ‰0; 1†, the equilibrium state xeˆ0 of x_ˆA…t†x is exponentially stable, i.e. there exist c0;p0 > 0 such that the state transition matrix…t; † corresponding to the homogeneous part of (1.26) satis®es

k…t; †k c0e p0…t †8t …1:27†

From the identityuˆ1…s†

1…s†‰uŠ, it is easy to see that the control inputucan be rewritten as

uˆvT…t†X‡qTn…t†and…s†

1…s†‰r "m2Š …1:28†

where

v…t† ˆ ‰nd nd…t†; nd 1 nd 1…t†; ; 1 1…t†ŠT and 1…s† ˆsnd‡1snd 1‡. . .‡nd

Also, using (1.28) in the plant equation (1.8), we obtain yˆZ0…s†

R0…s†‰1‡m…s†Š vT…t†X‡qTn…t†and…s†

1…s†‰r "m2Š

…1:29†

Now let 2 …0;min‰0;p0Š† be chosen such that R0…s†, 1…s† are analytic in Re‰sŠ

2, and de®ne the ®ctitious normalizing signal mf…t†by

mf…t† ˆ1:0‡ kutk2‡ kytk2 …1:30†

As in [9], we take truncated exponentially weighted norms on both sides of (1.28), (1.29) and make use of Lemma 3.3 in [9] and Lemma 2.1 (equation (6)) in [4], while observing thatv…t†,qn…t†,r…t† 2L1, to obtain

kutk2c‡ck…"m2†tk2 …1:31†

kytk2c‡ck…"m2†tk2 …1:32†

which together with (1.30) imply that

mf…t† c‡ck…"m2†tk2 …1:33†

Now squaring both sides of (1.33) we obtain m2f…t† c‡c

Z t

0 e …t †"2m2m2f…†d (sincem…t† mf…t†

)m2f…t† c‡c Z t

0 e …t s†"2…s†m2…s† ecRt s"2m2d

ds

(using the Bellman-Gronwall lemma [8]) Since"m2S 22

m2

and

mis bounded, it follows using Lemma 2.2 in [4] that 92 …0; 1† such that 82 ‰0; †, mf 2L1, which in turn implies that m2L1. Since

m,

m are bounded, it follows that , 2L1. Thus

"m2 ˆ ~T‡ is also bounded so that from (1.26), we obtain X2L1. From (1.28), (1.29), we can now conclude thatu,y2L1. This establishes the boundedness of all the closed loop signals in the adaptive IMC scheme. Since y y^ˆ"m2and"m2S 22

m2

,m2L1, it follows thaty ^y2S c22 m2

as claimed and, therefore, the proof is complete.

Remark 4.3 The robust adaptive IMC schemes of this chapter recover the performance properties of the ideal case if the modelling error disappears, i.e.

we can show that if ˆ0 then y ^y!0 as t! 1. This can be established using standard arguments from the robust adaptive control literature, and is a

consequence of the use of parameter projection as the robustifying modi®ca- tion in the adaptive law [9]. An alternative robustifying modi®cation which can guarantee a similar property is the switching-modi®cation [14].