Design of Optimal PID Controller with 𝜺-Routh Stability for Different Processes

Table 1

Proposed optimal PID controllers of plants.1–3 under various control weight matrices.

Plants	Control weight matrix; Q = diag (𝑞1𝑞2𝑞3𝑞4)	Optimal PID controller			ε-Routh stability
		𝑘^∗𝑝	𝑘^∗𝑖	𝑘^∗𝑑
Plant.1	diag (5 5 1 1)	0.13121	0.58138	0.58655	𝜀=0
	diag (5 2 1 1)	0.01633	0.58138	0.51933
	diag (10 2 1 1)	0.23263	0.82219	0.64449
	diag (10 10 1 1)	0.47606	0.82219	0.77901
	diag (10 10 4 2)	0.15570	0.58138	0.64101
Plant.2	diag (0.2 1 1 1)	1.15698	0.52966	0.86781	𝜀=0.5
	diag (0.2 1 1 0.2)	2.16962	0.84500	1.82140
	diag (0.2 2 2 0.5)	1.78668	0.58700	1.59290
	diag (0.1 0.8 0.8 0.2)	1.85901	0.64942	1.62251
	diag (0.4 0.5 5 1)	1.22806	0.62168	1.42095
Plant.3	diag (0.04 27 158 175)	0.32447	0.01537	1.45256	𝜀=0
	diag (0.04 28 120 80)	0.33078	0.01543	1.48167
	diag (0.02 25 150 175)	0.30719	0.01193	1.39141
	diag (0.05 27 120 100)	0.33727	0.01693	1.50359
	diag (0.06 30 158 175)	0.33603	0.01737	1.49509