EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Long-term Performance User Input Measure: Cost function User Input: Instantaneous Performance CMAC neural ne Measure: na Unknot Fig. 2. CMAC neural controller based on the H-J-B optimization. with the weight-estimation error W=w-W. The state-space Evaluating(47) along the trajectory of(43)yields description of (4 1)can be given by L=z P(g)Az-2 B P(az+32p(g)i 刻()=A()+Bg(+Wx()+x)++ 2PqB{W()+m)+7+y with z, A, and B given in(19)and(20) Inserting the optimal-feedback control law (29)into(42),we Using iP(g)Ai=(1/2)2T(A P(+PaZ,and from the riccati equation(28), we have i(t)=(A-BRB Pi(t) P+PA+是P=一想Q+ PbrBP(49) +BWry(ax)+(x)+7a(+v(t+},(43) Then the time derivative of Lyapunov function becomes Theorem 3: Let the control action u*(t be provided by optimal controller (29), with the robustifying term given by L=-2i Q2-22TP(aBRB P(i+2P(@ v(t)=-k2r(t)/|r(t) B{+7+v+t{W(FW+9BP2)}.(50) with ba and r(t) defined as the instantaneous-perfor- Applying the robustifying term(44)and the adaptive learning mance measure(10). Let the adaptive learning rule for neural- rule(45),we obtain network weights be given by L≤-22{m(Q)+m(R)} W=Fp(x)BPq)2-N洲W +1M+1(Wy1n子),(s) with F=F>Onxn and K>0. Then the errors e(t), r(t), The following inequality is used in the previous derivation and w(t) are"uniformly ultimately bounded. "Moreover, the errors et)and r(t)can be made arbitrarily small by adjusting w(w-w) weighting matrices Proof: Consider the following Lyapunov function =W,W》F-|W≤|WFWM-‖W.(52) Completing the square terms yields L=2xn1m|2+WFW)(46) where K is positive definite and symmetric given by (31). The L≤-2()m(+ha time derivative L of the Lyapunov function becomes +(w-1)-41 L=2P(+2Pq2+(WFW).(7 (53)26 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 Fig. 2. CMAC neural controller based on the H–J–B optimization. with the weight-estimation error . The state-space description of (41) can be given by (42) with , , and given in (19) and (20). Inserting the optimal-feedback control law (29) into (42), we obtain (43) Theorem 3: Let the control action be provided by the optimal controller (29), with the robustifying term given by (44) with and defined as the instantaneous-perfor￾mance measure (10). Let the adaptive learning rule for neural￾network weights be given by (45) with and . Then the errors , , and are “uniformly ultimately bounded.” Moreover, the errors and can be made arbitrarily small by adjusting weighting matrices. Proof: Consider the following Lyapunov function: (46) where is positive definite and symmetric given by (31). The time derivative of the Lyapunov function becomes (47) Evaluating (47) along the trajectory of (43) yields (48) Using , and from the Riccati equation (28), we have (49) Then the time derivative of Lyapunov function becomes (50) Applying the robustifying term (44) and the adaptive learning rule (45), we obtain (51) The following inequality is used in the previous derivation (52) Completing the square terms yields (53)