Problem 1 Reconsider the vibration control problem of the SDOF oscillator: x + 2n x + 2nx(t) = 2nu(t) (1) and the PID control u(t) = kP (r(t) x(t)) + kD( x) + kI t 0 (r() x())d (2) where r(t) is the reference input. Note that the derivative feedback does not contain r(t) so that we can account for non-smooth reference inputs such as the step input. 1. Define the state variables as x1 = x and x2 = x(t). Introduce the third variable x3(t) = t 0 (r() x())d. Write the state equations for the system in the extended state space. x(t) = Ax + Bu (3) Define the matrices A and B. Hence, the PID control can be written in the full-state feedback form u(t) = kP x1 kDx2 kI x3 + kP r(t).
Homework #9 Due November 4, Thursday Problem 1 Reconsider the vibration control problem of the SDOF oscillator: + 25Wni + war(t) = w.ult) (1) and the PID control tu(t) = kp(r(t) - 2(t)) +kp(-4) + ky [(-(r) (r)dt (2) where r(t) is the reference input. Note that the derivative feedback does not contain r(t) so that we can account for non-smooth reference inputs such as the step input. 1. Define the state variables as 21 = x and x2 = i(t). Introduce the third variable = X3(t) -- for (r(T) X(T))dt. Write the state equations for the system in the extended state space. x(t) = Ax+ Bu = (3) Define the matrices A and B. Hence, the PID control can be written in the full-state feedback form tu(t) = -kp01 - kdX2 kjx3+kpr(t). = (4) 4. Based on your knowledge learned from the root locus design, you pick three best poles and apply Ackermann's formula to design the feedback gains. Note that for this problem, the gain vector is defined as K = [kp, kd, ki] (6) Use Matlab command K chosen poles. place (A, B, p). Show a closed-loop step response with your 5. Consider a performance index as 1 J= 2 - } } 0, and R > 0. (a) Choose appropriate matrices Q and R, and use Matlab command (K,S, e)=LQR (A,B,Q,R) to define the full state feedback gain K. Note that R is a scalar for this example. Q can be chosen such that x"Qx represents the mechanical energy of the oscillator. (b) Compute the closed-loop poles of the system. The poles are the eigen-values of the closed-loop system matrix A - BK. (c) Simulate the closed-loop response. For two cases. (1) r(t) = 0. (2) r(t) is a step input. Homework #9 Due November 4, Thursday Problem 1 Reconsider the vibration control problem of the SDOF oscillator: + 25Wni + war(t) = w.ult) (1) and the PID control tu(t) = kp(r(t) - 2(t)) +kp(-4) + ky [(-(r) (r)dt (2) where r(t) is the reference input. Note that the derivative feedback does not contain r(t) so that we can account for non-smooth reference inputs such as the step input. 1. Define the state variables as 21 = x and x2 = i(t). Introduce the third variable = X3(t) -- for (r(T) X(T))dt. Write the state equations for the system in the extended state space. x(t) = Ax+ Bu = (3) Define the matrices A and B. Hence, the PID control can be written in the full-state feedback form tu(t) = -kp01 - kdX2 kjx3+kpr(t). = (4) 4. Based on your knowledge learned from the root locus design, you pick three best poles and apply Ackermann's formula to design the feedback gains. Note that for this problem, the gain vector is defined as K = [kp, kd, ki] (6) Use Matlab command K chosen poles. place (A, B, p). Show a closed-loop step response with your 5. Consider a performance index as 1 J= 2 - } } 0, and R > 0. (a) Choose appropriate matrices Q and R, and use Matlab command (K,S, e)=LQR (A,B,Q,R) to define the full state feedback gain K. Note that R is a scalar for this example. Q can be chosen such that x"Qx represents the mechanical energy of the oscillator. (b) Compute the closed-loop poles of the system. The poles are the eigen-values of the closed-loop system matrix A - BK. (c) Simulate the closed-loop response. For two cases. (1) r(t) = 0. (2) r(t) is a step input
