1. Overhead Controller Design Project Name Department and University Date Class Name

2. Project Statement System Descriptions: Performance Specifications: Motion less than 2 sec 5% overshoot 2% settling time No steady state error

3. Linearized System Dynamics Assuming small changes of θ, sin θ≈ θ cos θ ≈ 1

4. Laplace Transform Where,

5. Block Diagram GCGC G3G3 G2G2 G1G1 H R ++ - - +

6. System Transfer function

7. Controller Design Transfer function:

8. Characteristic Equation: Where, From characteristic equation, it could be concluded that there will be 5 poles and 3 zeroes.

9. System Parameters M= 2kgK=100N/mm = 0.8kg L = 0.2 m Hence, we change values of K c, α, z, q and r to work towards the performance criteria.

10. Root Locus for Design-1 Set-1: z = 1; q = 3; r = 5, α = 30, K c = 20.1 Poles: -30, ±5.56i, ±10.4i Zeores: -1, -3±5i You may study the loci by choosing several sets of parameters.

11. m= 0.8kg L=1.0mm= 0.8kg L=0.2m Poles: ±5.56i, ±10.4i, -30 Poles: ±3.62i, ±7.16i, -30 Set-2: z = 1; q = 3; r = 5, α = 30, K c = 20.1 Root Locus for Design-2

12. K c = 20.1

13. Comparison of Root Locus Plots,

14. Simulation Results

15. Settling time, T s = 1.41s Very small steady-state error

16. Simulation Results x 1 -x 0 is less than 1.0, hence this system is practically usable.

17. Summary of Performance such as Steady State Error TimeX1X1 1.975000.993551 2.000000.993699 2.500000.999968 3.000001.000413 3.500000.999894 4.000000.999992 4.500001.000010 It can be seen that the steady state error is very small.

18. It could be seen with the increased of length, The poles on the imaginary axis moves towards the real axis pushing the root locus together with it Thus, the frequency will decrease. ……movement decreases, and the period increases. Discussions on Controller Design and Simulation Under Different Weights

19. The decrease in frequency cause the x 0 to response slower. θ fluctuate a longer period of time due to the increase of length thus a longer settling time As a result, it caused the settling time in x 1 to increase. m= 0.8kg, L= 0.2m, T s =1.41s m= 0.8kg, L= 1.0m, T s =2.06s

20. Effect of mass, m Poles: ±5.56i, ±10.4i, -30 Poles: ±4.96i, ±11.22i, -30 m= 0.8kg L=0.2mm= 1.5kg L=0.2m

21. m=0.8kg, L=0.2m Settling time of x 1 = 1.41 sec Percent overshoot = 2.9% m=1.5kg, L=0.2m Settling time of x 1 = 1.50 sec Percent overshoot = 3.5% As m increases, the shapes of the curve remain relatively similar. However, that is not the case in the increase of L. It is because m is not a variable of K P, K D, K I However, L is a variable of K P. Therefore, it alter the system more compared to m.

22. Conclusions Stability of a system can be better defined as the “relative stability” A system can never be perfect. Fast response will lead to a high percent overshoot. Low percent overshoot will lead to slow response -> settling time. Hence, we adjust these variable to meet the performance criteria.