tailieunhanh - Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanics Part 5

Tham khảo tài liệu 'recent advances in robust control theory and applications in robotics and electromechanics part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Optimizing the Tracking Performance in Robust Control Systems 109 phase zeros are located near the origin for which large peaks appear in the sensitivity and complementary sensitivity response curves. We first derive an expression for the tracking delay. To simplify notation let . ft . mB mBT 3 and note that 0 a ft 1. Using the approximation WtT 1 we have 1 lj 1 . . AT AWtjr . Therefore using the straight line approximation Tjr 1 Wt Mt Mt 1 jft ỰÃT 1 jft -ỰMĨ for ft n 1 ỰM 4 and Te AWt .r n arctan .r fty M arctan ftỵ At. 5 Next we derive an expression for the peak steady state error. Since WpS 1 we have 1 j 1 IWj AS AWp . Therefore Sjr Wp As am 1 jr 1 i _ j mA m a mM 1 hJ ĩ j m ms for m On the other hand E s S s R s so that at the steady state we also have Re Ar am. 6 Therefore by equating 1 and 6 and using 4 we obtain am ự 1 M2T 2Mt cos wrTe. 7 110 Recent Advances in Robust Control - Theory and Applications in Robotics and Electromechanics An expression relating 5 to 7 can now be derived noting that ZWp jwr tt m a a A arctan arctan mfMs mfAs 8 Since W AS jwr ARftjWr AS jWr ZWp jWr 9 from 2 4 8 and 9 we obtain Mt sin Wr T r . a a 1 arctan ----A--------- tt m arctan . arctan . 10 V Mt cos Wr Te rnAs VMS Expressions 5 6 7 and 10 are the basic expressions to be used in the selection of the weighting functions. In order to gain insight into the relationships among various parameters involved in these equations we make further simplifications by noting that As At a and ft are small positive numbers. Thus by neglecting appropriate terms these equations reduce to om where WB Mt wBt Ms tt cos Wr Te 2m Wr WB Itan mn YI arctan sin2 Wr Te y mn 2 Mt sin Wr Te 1 Mt cos Wr Te 11 12 13 14 15 M WF Y . Note that Wb Wr so that 12 is well defined. For 13 we have used the trigonometric identity tan x y tan x tan y 1 tan x tan y to obtain the quadratic equation _ Wr Te r. I- . . _ ĩ Wr Te _ ậ Mt At tan - J ft y At y Mt J ft tan - J 0 and then have set At tt 0. .