tailieunhanh - Vibration Control Part 8

Tham khảo tài liệu 'vibration control part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 164 Vibration Control Z -1 Zi - P-1Zi-1 - Z -2 5m Derivative of Vi . i-1 V -X P Z Z _Z -2 X - a -1 5n j 1 Stabilizing function íth step a X 1 -P Z - Z -1 a -1 5o where Pi 0 are positive constants. The corresponding notation is shown in figure 1 in which it is illustrated a generic process state space representation corrected with stabilizing functions by means of Backstepping procedure. In order to avoid static control errors for systems type 0 an integrator must be added in series with the plant model. Then at the end of the design procedure this integrator is moved from the plant model to the control equation Hassan K. Khalil 2002 Krstic M. I. Kanellakopoulos P. Kokotovic 1995 . An alternative to such generalized technique consists in adding a cascade Backstepping algorithm following the setpoint generated by a feedback conventional controller. Fig. 1. General process representation stabilized by means of the Backstepping procedure Assumption 1. Assummg the system x f x g x u f 0 0 6 where x e Rn is the state and u e Rn is the control ínput the there exísts a continuous dífferentíable feedback control law u a x a 0 0 7 and a smooth posítíve defimte radíally unbounded function V Rn R such that dV rf x g x a x -W x 0 Vx e R dx 8 where W Rn R is posítíve semí-de tníte. AVC Using a Backstepping Design Technique 165 By means of this assumption the control 7 applied to the system 6 guarantees global boundedness of x t . Let the system 6 be augmented by an integrator according the expression x I x g x ỉ 9 t u and suppose that satisfies the above assumption with ệ as its control variable. Lemma 1 Being W x positive definite then 1 r- -2 Va x ệ V x 1 j-a x 10 is a CLF for the complete system 9 . That means there exists a feedback control u aa x ệ which renders x ệ 0 0 the GAS equilibrium of 9 . Such control results in da dV u -c ệ - a x - f x g x - g x -c y 0 dx dx 11 Up to now it has been introduced the methodology and tools to design a stabilizing control law and .

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