Question

(30) 3. Determine the feedback gain k>1 that minimizes the performance index J = " [x²(t)+ u?(t)] dt %3D for x(0)= v2 x(t) = x(t) + u(t), where u(t) = -k x(t). Show all steps. What is the value of Jmin ?

Answer 1

Ja to Y226+) + UP () ]dt. n°С+) = x 445 406+) Ko} = US UCt) = -K2C+) 7°C+) = X(+) – k 26+) on Touring Lahalace terrains form on 6041 Licles. Sycs) -Y 10-) = X($) (1–) ©x () [s + CX-10 7 = vē, Yls) - Vo = S+(K-1) 20144) = ve e- (K-1) 4: = x+). 24) -- VE - CM-vt. J = 500 2P (4) + k Parcu J = 0 xP (4) CI+HP). It four finewing Join . fiuest fineling en le er cent will give that value of ks which gives Jmin. Ja po olva e-(k-1) + JP CHP) ot. - Som pe -PCK-Ut) Citre) alt = echip). epek-ut you ->(N=13.10 = 2.614kB) J= [-1). Is -0 como deteta (x4) (0+xx) – CI+WP) [t) O = 212_213 - RPM -PCN-1) [

Dell JJ = We PK- roots of dJ zo WP_Ant=0 K = 20414, 00 418 kyl, so [R= 20414 Putting n=2.414 in ean ☺ to get Jmin = 172.4147 20414-1 = 140898] Anti Jimin