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Systems, Structure and Control 2012 Part 13

Tham khảo tài liệu 'systems, structure and control 2012 part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A Sampled-data Regulator using Sliding Modes and Exponential Holder for Linear Systems 233 Since we are concerned with a discrete controller the discretization of the continuous system 1 - 3 can be described by xk 1 Adxk Bduk Pdwk wk 1 Sdwk ek Cxk - Rwk where i Ad e0A t A i 0 i Bd f esABds Y A- B 0 i 0 i A Sd eỗs t s i 0 i 3 J i 1 Cd C Rd R Pd f esAPds Y- -P d f Y i 1 where Pi can be computed iteratively from Po P Pi AP-1 PS1 i 1 2 . The classical Robust Regulator Problem with Measurement of the Output for system 1 3 consists in finding a dynamic controller ỉ t Fỉ t Ge t u Heỉ t such that the following requirements hold S The equilibrium point x 0 0 of the closed loop system without disturbances x t Ax t BH t ị t Fệ t GCx t is exponentially stable. 234 Systems Structure and Control R For each initial condition xo W0 ị0 the dynamics of the system x t Ax t BHị t Pw t ị t Fị t G Cx t Rw t w t sw t satisfy that lim e t 0. t A solution to this problem can be found in 1 . This solution is stated in terms of the existence of mappings xss n w ịss Xw satisfying the Francis equations ns An BHex P xs FX 4 for all admissible values of the systems parameters. More precisely the solution can be stated in terms of the existence of mappings Xss nw Uss rw solving the equations ns An Br P 5 0 cn R 6 from which we reckon r A rs x rs q 1 a0r-ars . a rsq 1 0 1 q 1 where the polynomial sq aq_ 1sq 1 . a1s a0 0 is the characteristic polynomial of s. The mapping x nw represents the steady state zero output subspace and Uss r it is the steady-state input which make invariant that subspace. This steady-state input can be generated independently of the values of the parameters of the system and thanks to the Cayley-Hamilton Theorem by the linear dynamical system A Sampled-data Regulator using Sliding Modes and Exponential Holder for Linear Systems 235 n on uss Hn 7a 7b where o diag o 1 .o m H diag Hi . Hm and f o 1 o Ì o o 1 o 0 o o o 1 V ao a a2 a. 1 q 1J Hi 1 o o 1xq. Defining the .