Question

Consider the unity-feedback system shown below: R(s) E(s) input: r(t), output: y(t) C(s) P(s) error: e() r(t) y(t) closed-loop transfer-function: Hyr(sD t the closed-loop transfer-function be Hyr(s) Y (s) R(s) Let the transfer-function of the plant be P(s) 10 s (s 1) (s 5) The open-loop transfer-function is G(s) P(s) C(s) DESIGN OBJECTIVES: Find a controller C(s) such that the following are satisfied i) The closed-loop system is stable. ii) The steady-state error ess due to a unit-ramp input r(t) = t 1 (t) is equal to zero, iii) The phase-margin of G(s) is at least 40, i.e., P.M. > 40. To satisfy objective (i), the open-loop transfer-function G(s) must be Type-2; note that the given plant is Type-1. So we will use a proportional+integral (PI) controller C(s) = K, + At First show that design objective (ii) is satisfied with a PI controller STEP I: Plot the Bode plot of the uncompensated system P(s). Find the phase-crossover fre quency wp, the gain-crossover frequency wg the gain margin G.M. and the phase margin P.M. of P(s). Is the closed-loop systern stable with C(s) 1, i.e., G(s)-P(s)? STEP II: Determine the gain-crossover frequency u, such that the desired P.M. is achieved on the Bode plot of P(s). Find the values of Kp and Ki such that the gain-crossover frequency of G(s) C(s)P(s) is u; to do this, take the corner-frequeney weKi/Kp of the PI controller a decade below and use Pug)laB20logKp. Plot the Bode plot of G(s) and find the P.M and G.M. Is design objective (iii) satisfied? STEP III: Now fine-tune your design by adjusting unil the P.M. requirement is satisfied. Plot the Bode plot of G(s) and find wp, w P.M., G.M

Answer 1

P (s) = 10/s (s + 1) (s + 5) The open-loop transfer function is G (s) = P (s)C (s) Design objectives For finding a controller C (s) such that closed loop system is stable steady state error E_ss due to a unit ramp Input r (t) = Atu (t) = 0 PM greaterthanorequalto 40 degree for ramp Input r (t) = Atu (t) Here Input = 1 when Input greaterthan type implies E_ss = infinity Input < type implies E_ss = 0 Input = type implies E_ss = A/K_v K_v = lt_s rightarrow 0 s G (s) K_v = lt_s rightarrow 0 s [10/s (s + 1) (s + 5)] = 10/5 therefore K_v = 2 phase margin of P (s) greaterthanorequalto 40 degree steps for finding PM: - calculate gain cross over frequency (omega_gc) = ? \r\n mod (G (s)) = 1 implies mod (1 (j omega)) = 1 mod (1/omega squareroot (omega^2 + 1) squareroot (omega^2 + 25))