tailieunhanh - Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanics Part 3

Tham khảo tài liệu 'recent advances in robust control theory and applications in robotics and electromechanics part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 49 holds the condition 37 is satisfied. In fact 39 implies p f0 x Aj x fj x 0 Vx e S. 40 j 1 Moreover if L x A e E then 39 holds from Lemma 1 and hence 37 holds. These facts are summarized in the following lemma. Lemma 3. If the following i or ii holds the condition 37 holds. i There exists A x e R x p such that L x A 0 Vx e Rn . ii There exists A x e R x p such that L x A e E. 4. Proposed method Theorem 2 implies that the state feedback 20 will stabilize the closed-loop system and 25 is satisfied if we can obtain a positive definite symmetric matrix P satisfying 19 . In this section we propose an SOS optimization method to find such P. To this end let us introduce sufficiently small e 0 and define M x P M x P - eI 41 0 x P P-1 x TM x P P-1 x . 42 Then it is easy to see M x P h 0 Vx e Rn M x P z 0 Vx e Rn 43 0 x P 0 Vx e Rn f x P 0 Vx 0 e Rn. 44 Hence for obtaining the feedback 20 it suffices to find P 0 such that Ị x P 0 Vx e Rn. 45 From 42 the condition 45 can be written as hTM x P h 0 V x h e R2n such that h P-1 x 46 and moreover this can be written as hTM x P h 0 V x h e S 47 where S x h e R2n I x - Ph 0 . 48 By this the condition 45 is represented as the condition 47 including the equality constraint x - Ph 0. For the condition 47 we define a generalized Lagrange function as in Section as follows 50 Recent Advances in Robust Control - Theory and Applications in Robotics and Electromechanics L x h K P hTM x P h - KT x h x - Ph 49 where K x h e R x h n. Then from Lemma 3 47 is satisfied if there exit K and P 0 which satisfies L x h K P 0 V x h e R2n 50 Here suppose the degree of K is given say m then K can be written as K x h Hvm x h where vm x h is a vector of size 2n mCm which contains all monomials in x and h whose degrees are less than or equal to m and H is an n X 2n mCm real matrix. From this 50 is reduced to Lm x h H P hTM x P h - vm x h HT x - Ph 0 V x h e R2n 51 and the .