tailieunhanh - PID Control Implementation and Tuning Part 8

Tham khảo tài liệu 'pid control implementation and tuning part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sampled-Data PID Control and Anti-aliasing Filters 133 and feedback signal Zi is equivalent with the following discrete-time problem xp 1 Fpxi gpUi wp 33 Zi d pXP 34 1 N-1 I iHmoEN L xViQ1x1p 2Xp q12ui q2u2 w 35 where Q1 q12 q2 qw h J Fp T MFp T dT M dp d p 0 d hl fp T Mgp T dT 0 hl gp T Mgp T dT A 0 F I 11 Fp t - s cpc pF t - s dsdz dp 10 0 Fp t eAp t Fp Fp h 36 t gp t I eApVbpdv gp gp h 37 0 and wi is a zero mean vector Gaussian noise with E wpwpl Wp and Wp I eApsCpc peApsds. 38 0 Vectors X0 and wp are independent for all i 0. The optimal control law minimizing the performance index 35 for the discrete stochastic system 33 - 34 is a linear function Ui -k x xp 39 where xp denotes the Kalman filter estimate of xp based on available information up to and including i from 47 - 48 .The feedback gain kx q12 Fp Kgp x q2 g pKgp 40 depends on the positive definite solution K of the following algebraic Riccati equation K Q1 Fp KFp - q12 Fp Kgp q12 Fp Kgp q2 g pKgp 134 PID Control Implementation and Tuning Discrete-time Kalman filter Simple instantaneous sampling with sampling period h consists in taking the values of the sampled signal at discrete time instants ti ih i 0 1 . Available measurements Zi are expressed as zi y2 ti . 41 The problem defined by measurement equation Zi z ih and state equation 1 is equivalent to the following discrete-time system Xi 1 Fxi gui Wi 42 Zi d xi 43 where F t eAT F F h 44 g T f e . g g h 45 0 and Wi is a zero mean vector Gaussian noise with E wiW- w and W Ị eAsCC eA sds. 46 0 Vectors x0 and Wị are independent for all i 0. The limiting Kalman filter Anderson Moore 1979 that provides xi i E xi Zi for the discrete-time system in 42 - 43 as i TO has the form xi 1 i 1 xi 1 i k zi 1 - dxi 1 i 47 x i 1 i Fx i i g2i x0 -1 0 48 where kf ỉẾd L W F fl - -d F. 49 a Ld y d Ld MV LQG Control System Assessment Output and control variances for systems with continuous-time filters can be expressed by following formulae éỊj var yi d0V0d0 02 var