Tham khảo tài liệu 'adaptive control design and analysis part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 62 Chapter 2 Systems Theory asymptotically stable . In this section we present some fundamental tools for analyzing the stability of adaptive control systems the Bellman-Gronwall lemma for feedback structure equivalence a small-gain lemma for feedback stability some operator stability properties for system analysis and the Lefschetz-Kalman-Yakubovich lemma for passivity and system matrices . Bellman-Gronwall Lemma There are different versions of the Bellman-Gronwall lemma 78 151 426 . Two of them are presented here with proofs which are useful in stability analysis for different adaptive control schemes in the next chapters. Lemma Let f t f g t and fc t be continuous and g t 0 k t 0 Vt to o nd for some to 0. If a continuous function z t satisfies z t f t g t i k zfr dr Vi to J to then z t f t g t f k r f r e k dr Vi t0. Jto Proof Introducing w t fỊ0 k r z r dr and using and the fact that A t 0 we obtain w t fc t .z t fc t t Defining A t w t e ftok s deri and using we have à í w t e ỉ ok dơ - w í fc í ỡ í e- cr s T dí7 k t f i e k ơìdơ. Since A to cư to 0 from it follows that A t r fc r r e_ fc ơ ẩ r d dr. to With the definition of A t leads to w t A t e-f o r ff r dơ r k T f r eỉ k -ữ dữ dr J to which with for g t 0 and the definition of w t implies . V Input-Output Stability 63 Figure Feedback system diagram of Bellman-Gronwall lemma. A control system block diagram can be used to illustrate Lemma as shown in Figure where g t e R and k t e R from which we have y t d t gự w t0 k r y r dr f i ỔƠ i k r y r dr Jto for t d t g t w tũ . From similar to we have 11 01 1 01 lỡ 0I r k f i t dr. 0 Lemma Ifk t is continuous and k t 0 Vi to 0 and a continuous function z t satisfies z t c f k r z r dT Vt to J to where c is a constant then z t ce- ofc T dT t t0. Proof Introducing w t c fto kfr z r dr and A t w t e k dơwith A to w to c from we derive w t fc
