Minimizing Energy Cost in Electric Arc Furnace Steel Making by Optimal Control Designs

Table 1

	LP	LQR
s.t.	min𝑢𝑖𝐽LP(𝑢𝑖)=𝑁∑^4𝑖=1𝑝𝑖𝑢𝑖	min𝑢𝑖𝐽LQR𝑢𝑖𝑟1,𝑟2,𝑟3,𝑟4=𝑁∑^4𝑖=1𝑟𝑖𝑢^2𝑖(𝑟1,𝑟2,𝑟3,𝑟4)
	𝑥1[𝑁+1]=𝑁𝜑1𝑢1	𝑥1[𝑁+1]=𝑁𝜑1𝑢1(𝑟𝑖)
	𝑥2[𝑁+1]=𝑁𝜙1𝑢1	𝑥2[𝑁+1]=𝑁𝜙1𝑢1(𝑟𝑖)
	𝑥3[𝑁+1]=𝑁∑𝑖=1𝑎𝑖𝑢𝑖	𝑥3[𝑁+1]=𝑁∑𝑖=1𝑎𝑖𝑢𝑖(𝑟1,𝑟2,𝑟3,𝑟4)
	0<𝐼𝑙≤𝑢1≤𝐼𝑢	0<𝐼𝑙≤𝑢1(𝑟𝑖)≤𝐼𝑢
	0≤𝑢2≤𝑂𝑢	0≤𝑢2(𝑟𝑖)≤𝑂𝑢
	0≤𝑢3≤𝐺𝑢	0≤𝑢3(𝑟𝑖)≤𝐺𝑢
	0≤𝑢4≤𝐶𝑢	0≤𝑢4(𝑟𝑖)≤𝐶𝑢