Where step response approximated by ( G(s) = \fraca e^-Ls1+Ts ). Time domain : [ \dot\mathbfx = A\mathbfx + B\mathbfu, \quad \mathbfy = C\mathbfx + D\mathbfu ]
:
(for unity feedback, ( H(s)=1 )):
| Controller | ( K_p ) | ( \tau_i ) | ( \tau_d ) | |------------|----------|--------------|--------------| | P | ( 1/a ) | – | – | | PI | ( 0.9/a ) | ( 3.33 L ) | – | | PID | ( 1.2/a ) | ( 2 L ) | ( 0.5 L ) |
(rank of ( \mathcalC = [B \ AB \ A^2B \dots] ) = n) Observability (rank of ( \mathcalO = [C; CA; CA^2; \dots] ) = n)
: [ G_c(s) = K_c \left(1 + \frac1\tau_i s\right)(\tau_d s + 1) ]
: [ G(s) = C(sI - A)^-1B + D ]
Where step response approximated by ( G(s) = \fraca e^-Ls1+Ts ). Time domain : [ \dot\mathbfx = A\mathbfx + B\mathbfu, \quad \mathbfy = C\mathbfx + D\mathbfu ]
:
(for unity feedback, ( H(s)=1 )):
| Controller | ( K_p ) | ( \tau_i ) | ( \tau_d ) | |------------|----------|--------------|--------------| | P | ( 1/a ) | – | – | | PI | ( 0.9/a ) | ( 3.33 L ) | – | | PID | ( 1.2/a ) | ( 2 L ) | ( 0.5 L ) | control systems engineering exam reference manual
(rank of ( \mathcalC = [B \ AB \ A^2B \dots] ) = n) Observability (rank of ( \mathcalO = [C; CA; CA^2; \dots] ) = n) Where step response approximated by ( G(s) =
: [ G_c(s) = K_c \left(1 + \frac1\tau_i s\right)(\tau_d s + 1) ] control systems engineering exam reference manual
: [ G(s) = C(sI - A)^-1B + D ]
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