Fal!2001 16.3122-7 EXample: simple system with to=0 and tf= 10sec J=x2(10) o 8 =4o+[=o[801 (t)+ru(t)dt Compute gains using both time-varying P(t)and steady-state value Find state solution a(0)=11 using both sets of gains 1=1h=5 Figure 2: Set q=l, r=1,h=10, Kl 0.73Fall 2001 16.31 22—7 • Example: simple system with t0 = 0 and tf = 10sec. ∙ ∙ 0 1 0 x˙ = 0 −1 ¸ x + 1 ¸ u " # Z 10 ∙ ∙ J = xT (10) 0 0 x(10) + xT (t) q 0 0 0 ¸ x(t) + ru2 (t) ¸ dt 0 h 0 • Compute gains using both time-varying P(t) and steady-state value. • Find state solution x(0) = [1 1]T using both sets of gains q = 1 r = 1 h = 5 1.4 K1 (t) K1 5 4.5 1.2 4 1 3.5 3 0.8 2.5 0.6 2 0.4 1.5 1 0.2 0.5 K2 (t) K2 x 1 x 2 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Time (sec) Dynamic Gains Static Gains 1.4 1.4 1.2 x 1 x 2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Time (sec) States Gains States Gains Figure 2: Set q = 1, r = 1, h = 10, Klqr = [1 0.73]