design a feedback controller

Question

A mass-spring-damper system is shown. The input is a force, u, acting on the mass block. The output is displacement, y, of the mass block. Using the well-known Newton’s second law, the dynamic equation of the motion for the mass-spring-damper system can be derived: where m is the mass, k is the spring stiffness, and c is the damping coefficient. The three coefficients satisfy: m=1 kg; c=0; k=10 N/m. Design a feedback controller for this type of plant in order to satisfy stability and good dynamic performance.

(a) well written: text, equations and figures. (25 marks).

(b) design control system by using or transient-response method, or root locus method, or frequency response method. (15 marks); design control system by using or state-space method, or digital control method, or nonlinear control method. (25 marks).

(c) plot simulated responses to a step input and give corresponding analyses. (25 marks)

(d) dynamic behavior to changes in the system parameter (mass and spring), plot the vibration amplitude for various system parameter, and give corresponding analyses. (25 marks)

Answer 1

In terms of the Laplace transforms Y(s) and U(s) of the displacement y(t) and input u(t), the ordinary differential equation becomes

(ms² + cs + k)Y(s) = U(s) + my'(0) + (ms + c)y(0)

that is trivial to solve for Y(s), especially if c = 0. Then interpret "feedback controller" as a new factor on U(s) to achieve some constraint on Y(s).