Question

Problem 1 An inverted pendulum driven by a d-c motor is governed by the following differential and algcbraic equa tions: (a) Determine the transfer function of the process. (b) It is proposed to control the process using "proportional control": where yr is a constant reference value. Determine the value Kmir for which the gain K must exceed in order that the closed-loop system be stable. (c) Determine the value of K for which the magnitude of the error is less than 10 percent. (d) It is determined that the performance of the system with the gain set at the value determined in Part (c) is unsatisfactory. Explain why. To improve the performance of the control system for the process of Problem variable" feedback control: lt is decided to use "state- Determine the control gains Ki, K2, and Ko for which the closed loop poles are located at and that the steady-state error is zero, i.e., As an alternative to state-variable feedback proposed in Problem hit is decided to use "proportional plus integral (PI) control" Determine the range in the K3, Ka plane for which the closed-loop system is stable when using this control law For the control law of it is proposed to sct K4 2K3 (a) Find the range of K'3 for which the closed-loop system is stable. (b) Sketch the locus of close-loop poles using this control law, i.e E(s) as K3 varies from 0 to oo, Your sketch should include important features, such as: asymptotes of loci for large values of K3, segments on the real axis, crossings of imaginary axis, etc. IN The control gain is set at In the design of Ks 30 (a) Determine the poles of the closed-loop system using this gain setting. (b) Sketch unit step response of this system. Using this sketch, estimate () the percentage overshoot, (i) the time it will take to reach its maximum value ("peak time)

Answer 1

ct IY2 stq8-5 TIF o0 coytrelas fr) ond 'teat the closed Lecp ystem be stable.

Veh @ д€ eamtolthe trar) s beq t un chona, Ibe prc cess U. Bt D 0

C t Co -o) Y2 *t 1s -, tqS-5 0o valu te

the Syserり芐 to be stable oul the Eleonenk fotbe. ,first columり should be Kp fne *be value o bic the a C. *p

beuaus duie to ls