SchoolofAutomation,XJTU取Lyapunov候选函数如下1P(t) = pT(t) ≥ 0V(t) =2x(t)P(t)x(t),12CAIYUANLI
School of Automation, XJTU © CAI YUANLI 12 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 取 Lyapunov 候选函数如下： 𝑉(𝑡) = 1 2 𝑥 𝑇(𝑡)𝑃(𝑡)𝑥(𝑡), 𝑃(𝑡) = 𝑃 𝑇(𝑡) ≥ 0
School of Automation,XJTU7V(t) =亏xT(t)P(t)x(t) +亏(t)P(t)x(t) +=xT(t)P(t)x(t)xi22.11[Ax + Bu]TPx +?2xTpx+:=xTP[Ax + Bu]12211111xTPx+TBTPx+2xATPx+xT PAx +PBuu!-21222.11T[ATP+ PA + P]x+uT BTPx +xTPBu-2121[ATP + PA + P]x +[u + R-1BT Px]'BT Px213CAIYUANLI
School of Automation, XJTU © CAI YUANLI 13 𝑉̇(𝑡) = 1 2 𝑥̇ 𝑇(𝑡)𝑃(𝑡)𝑥(𝑡) + 1 2 𝑥 𝑇(𝑡)𝑃̇(𝑡)𝑥(𝑡) + 1 2 𝑥 𝑇(𝑡)𝑃(𝑡)𝑥̇(𝑡) = 1 2 [𝐴𝑥 + 𝐵𝑢] 𝑇𝑃𝑥 + 1 2 𝑥 𝑇𝑃̇𝑥 + 1 2 𝑥 𝑇𝑃[𝐴𝑥 + 𝐵𝑢] = 1 2 𝑥 𝑇𝐴 𝑇𝑃𝑥 + 1 2 𝑥 𝑇𝑃𝐴𝑥 + 1 2 𝑥 𝑇𝑃̇𝑥 + 1 2 𝑢 𝑇𝐵 𝑇𝑃𝑥 + 1 2 𝑥 𝑇𝑃𝐵𝑢 = 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇]𝑥 + 1 2 𝑢 𝑇𝐵 𝑇𝑃𝑥 + 1 2 𝑥 𝑇𝑃𝐵𝑢 = 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇]𝑥 + 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝐵 𝑇𝑃𝑥
School ofAutomation,XJTU11xpTBR-1BTPxxTPB[u + R-1BTPx]21xTPBR-1BTPx21[ATP + PA + P- PBR-1BTP]x2t1=[u + R-1BTPx]T R[u + R-1BTPx]121[u + R-1BTPx]'Ru+xPB[u + R-1BTPx]14CAIYUANLI
School of Automation, XJTU © CAI YUANLI 14 + 1 2 𝑥 𝑇𝑃𝐵[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] − 1 2 𝑥 𝑇𝑃 𝑇𝐵𝑅 −1𝐵 𝑇𝑃𝑥 − 1 2 𝑥 𝑇𝑃𝐵𝑅 −1𝐵 𝑇𝑃𝑥 = 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇ − 𝑃𝐵𝑅 −1𝐵 𝑇𝑃]𝑥 + 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝑅[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] − 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝑅𝑢 + 1 2 𝑥 𝑇𝑃𝐵[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥]
School of Automation,XJTUTpTBR-1BTPxY21[ATP+PA + P-PBR-1BTPlxt11[u + R-1BTPx]"R[u +R-1BTPx] -Zu'Ru21xT[ATP+PA + P- PBR-1BTP+Qlx211[u + R-1BTPx]TR[u + R-1BTPx]Ru4215CAIYUANLI
School of Automation, XJTU © CAI YUANLI 15 − 1 2 𝑥 𝑇𝑃 𝑇𝐵𝑅 −1𝐵 𝑇𝑃𝑥 = 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇ − 𝑃𝐵𝑅 −1𝐵 𝑇𝑃]𝑥 + 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝑅[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] − 1 2 𝑢 𝑇𝑅𝑢 = 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇ − 𝑃𝐵𝑅 −1𝐵 𝑇𝑃 + 𝑄]𝑥 + 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝑅[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] − 1 2 𝑢 𝑇𝑅𝑢 − 1 2 𝑥 𝑇𝑄𝑥
School of Automation, XJTU为了使系统稳定(V(t)≤0)，要求112u'Ru+2x"Qx1≥2x[ATP+PA+p- PBR-1BTP+Q]x12[u + R-1BTPxITR[u+ R-1BTPx]+：即CAIYUANLI16
School of Automation, XJTU © CAI YUANLI 16 为了使系统稳定（𝑉̇(𝑡) ≤ 0），要求 1 2 𝑢 𝑇𝑅𝑢 + 1 2 𝑥 𝑇𝑄𝑥 ≥ 1 2 𝑥 𝑇 [𝐴 𝑇𝑃 + 𝑃𝐴 + 𝑃̇ − 𝑃𝐵𝑅 −1𝐵 𝑇𝑃 + 𝑄]𝑥 + 1 2 [𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 𝑇𝑅[𝑢 + 𝑅 −1𝐵 𝑇𝑃𝑥] 即