Question

Q2) The input-output relationship of a linear system is given by y(t) + 250y = 50u(t) a) Simulate the step response of the above system using Simulink. Comment on the figure obtained. (5 marks) b) Give a state space representation of the above system. (5 marks) c) Give a discrete-time representation of the above system using a zero-order hold. (15 marks) d) Analyse the stability and controllability of the discrete time model and comment on the result obtained. (5 marks) e) Assuming T=0.15, design a state feedback of the form u(n)= x(n)+v(n) so that the closed-loop system is stable. Confirm your result using Simulink. (10 marks) f) For T=0.15, calculate the value of v(n) such that the value of y(n) converges to 5 in steady state. Validate your result using Simulink. (10 marks)

The simulink part can be neglected as I can manage. Just need the mathematical solution for all the parts (specially part f)

Answer 1

sol Given input-output relationship of linear sis as- 4(e) +250 ° = 500W olp; &ltilp. 24 yee Given relation. I ara arte | qte)+ 250 - 50000 Now to solve this since Lti sls, we can we Laplace transform. 2 so v icc s²yls) node at y(t) Syo two ut 1. N o Therefore SYIS) + 2503 YO) = 50. s Y(S) = 5 50 ( 5 +250) 250 52 (5+250) VIS: S+ 250-s s26+250) 5 y = x * - o ] 9 40) = 3 - Hotel Aero) » Y = Set mese [to share] P40 zis - ja toist + eso es -0

Taking capbre inverse of equation, a = 0.2r(t)- I .UCt) 0 1250 -250t & l e ult) 1250 ml -250t Tylt) = 0.2 &lt) + I if esot 1 ult OV 1250 L a sls, State space Representation of above sls. for this first we have find out Tf of given na ) (+850yet) = 50 VLE s²yrs) +250sy (5) = 50 UCS) VIS) 03 50 € vis) ?(5250) on UO Sote - [ct 4 D= 2 (0) Đ. (1) - 1,14) TX, (t) To Tales To TULE [200] 10 -250/2016) 50] A matrix B matrix yo -250 spate, modell percinta ry 3611 cmatrix

continous. © Discrete time representation of above system with zelo order hold eg 2() At ten) Switch. Note: Firet diffence in discrete First derivatice in and so on. I So first finding out first diffence. On y (t) = yeni-y (n-1) 0 Now second diffenece - ÝCu = First' difforce of [yen- yen-] - Lyon) -y (n-1)] - [yenu)- y(n-1)] ý (t) & yen-ay(n-1) + (n -2) 3 using ② & ③ Pyan)-2ycoat)- yen-2)] + 250 [yen) -y (4)] = 50 vt 5251 yen) - 259yent) + y(n-2) = 50 ucne discrete sepresentation of above system with zesu os der hold. so the discrete system- Analysis of stability of Given' discrete sls 251 yen) - 252 y (n-1) + yen-2) = 5oven) - ©

Taking Thertore - transform of equation 4 .: yuth 2-transt by (2) 351 462) – 252 244(z+ ?y(z) = 50 UCZ) H (2) = 4(2) 50 GA U(2) 2225227 +951 H (2) = 50 (21-251) (2-1) = Hlz) = 250 - z- (2-251) (27) 251) (2011 03 51 2-1 5X251 251 Poles are H(2) ase 1 and I 251 Note: For Stability of discrete ss, boles wol location should he inside the unit carile. and as per magninide of ppoles as mentioned aleone, system is stable.

Nole since the sls is not ip, but also previous If any variation in actual opp will vary. only dependent on outputs also old olp's takes place control over ilf » we can say syslem will be that no fall uncontrollable.