EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Inserting(20), (22), and(A10) into(Al3)gives Whence the application of robot property 2, (17)shows that the matrices K, Aof(31)and(32)solve the algebraic Riccati equa- tion of(A20) i PAl+32(P+Q-PBR- B P)E=0. (A14) KA K AKOn×n K Onxn R Since z PAi=(1/2)2(A P+PA)i, (A14)can be written 1Q1 Qi2 Q (A20) 32(P+AP+PA+Q-PBR-B P)=0.(A15) This completes the proof. REFERENCES e We can summarize by stating that if a matrix Pcan be found J.S.Abus,“A at satisfies(A15)VtE(to, oo), then the value function given ler( CMAC),J. Dynamic Syst, Meas, Contr in(A2)satisfies the HJB equation(A1). In this case, the desired (2)A B T S oSAL ol o.97.no.3. optimal control is given by (AlO). Note that if the matrix Psat- approximation bounds for unction,IEEE Trans. Inform Theory, vol. 39, pp. 930-945 isfies the algebraic Riccati equation (28), then Psatisfies(Al5) This completes the proof. 3] C.-T. Chiang and C -S. Lin, " CMAC with general basis functions, al Networks, vol. 9, no. 7, pp. 1199-1211, 1996 [4]S Communi, F L. Lewis, S.Q. Zhu, and K Liu, "CMAC neural net- works for control of nonlinear dynamical systems, "Proc. Neural, Par. allel and Scientific Computing, vol. 1, pp. 119-124, 1995 APPENDIX B 65] D. Dawson, M. Grabbe, and F. L. Lewis, " Optimal control of a modi- PROOF OF THEOREM I fied computed-torque controller for a robot manipulator, Int J. Robot. automat,vol.6,no.3,pp.161-165,1991 From Lemma 1. it is known that trol, IEEE Trans. Automat Contr, vol 35, pp. 1197-1208, Nov 1990 [7 D. E Koditschek, "Quadratic Lyapunov functions for mechanical sys- tems,Yale Univ, Tech Rep. 703, Mar 1987. 2P(q)= On×nM(q) pp 23-30, Apr. 90>CMman, and J JGelfand, "Theory and deveop [9 F. L. Lewis, C. T Abdallah, and D M. Dawson, Control of Robot Ma- [10] F L. Lewis and V. L. Syrmos, Optimal Control, 2nd ed, New York: lves the HJB equation for K= K, A, solving the matrix [11] K.s. Narendra and A M Annaswamy, "a new ad w for robust quation from the quadratic form daptation without persistent excitation, "IEEE Trans. Automat Contr. ol.AC-32,pp.134-145,Feb.1987 [12] F L. Lewis, A. Yesildirek, and K. Liu, "Multilayer neural-net robot con- 2(PA+A P-PBR P+P+Qz=0.(Al7) (13)M. M Polycarpou asa'polyc igr stable adaptive neural control of scheme for non- 88-399,Mar.196. /hear systems,"IEEE Trans. Automat. Contr, vol 41, pp. 447-451,Mar The optimal-feedback control law that minimizes J(u)is [14]Y-F. Wong and A Sideris, " Learning convergence in the cerebellar model articulation controller. " IEEE Trans. Neural Networks. vol. 3 pp.ll5-121,Jan.1992 [15] Matlab Users Guide, Control System Toolbox. Natick, MA: Math- u(t)=-R-)it). works. 1990 Young Ho Kim was born in Taegu, Korea, in 1960 He received the B.S. degree in physics from Korea Let the weighting matrices be given by (30 demy in 1983, the M.s. degree in nsertion of expressions for matrices A, B in(20)and P(g electrical engineering from the University of Central Florida, Orlando, in 1988, and the Ph. D. degree in in(27)into(A2), we have electrical engineering from the University of Texas From1994to1997, at the Automation and Robotics Research ATK ngton. He has extensively in the fields of feedback control usin K High-Level tworks, dynamic neural networks, fuzzy-logic eal-time adaptive critics for intelligent control of robotics, and nonlinear 1Q1 systems. DI KI ived the Korean Army Overseas Scholarship Sigma Xi Doctoral Research Award in 1997. He is a member of30 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 Inserting (20), (22), and (A10) into (A13) gives (A14) Since , (A14) can be written as (A15) We can summarize by stating that if a matrix can be found that satisfies (A15) , then the value function given in (A2) satisfies the HJB equation (A1). In this case, the desired optimal control is given by (A10). Note that if the matrix sat￾isfies the algebraic Riccati equation (28), then satisfies (A15). This completes the proof. APPENDIX B PROOF OF THEOREM 1 From Lemma 1, it is known that (A16) solves the HJB equation for , solving the matrix equation from the quadratic form (A17) The optimal-feedback control law that minimizes is (A18) Let the weighting matrices be given by (30). Insertion of expressions for matrices in (20) and in (27) into (A2), we have (A19) Whence the application of robot property 2, (17) shows that the matrices of (31) and (32) solve the algebraic Riccati equa￾tion of (A20) (A20) This completes the proof. REFERENCES [1] J. S. Albus, “A new approach to manipulator control: The cerebellar model articulation controller (CMAC),” J. Dynamic Syst., Meas., Contr., vol. 97, no. 3, pp. 220–227, 1975. [2] A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, vol. 39, pp. 930–945, Mar. 1993. [3] C.-T. Chiang and C.-S. Lin, “CMAC with general basis functions,” Neural Networks, vol. 9, no. 7, pp. 1199–1211, 1996. [4] S. Commuri, F. L. Lewis, S. Q. Zhu, and K. Liu, “CMAC neural net￾works for control of nonlinear dynamical systems,” Proc. Neural, Par￾allel and Scientific Computing, vol. 1, pp. 119–124, 1995. [5] D. Dawson, M. Grabbe, and F. L. Lewis, “Optimal control of a modi￾fied computed-torque controller for a robot manipulator,” Int. J. Robot. Automat., vol. 6, no. 3, pp. 161–165, 1991. [6] R. Johansson, “Quadratic optimization of motion coordination and con￾trol,” IEEE Trans. Automat. Contr., vol. 35, pp. 1197–1208, Nov. 1990. [7] D. E. Koditschek, “Quadratic Lyapunov functions for mechanical sys￾tems,” Yale Univ., Tech. Rep. 703, Mar. 1987. [8] S. H. Lane, D. A. Handelman, and J. J. Gelfand, “Theory and develop￾ment of higher-order CMAC neural networks,” IEEE Contr. Syst. Mag., pp. 23–30, Apr. 1992. [9] F. L. Lewis, C. T. Abdallah, and D. M. Dawson, Control of Robot Ma￾nipulators, New York: Macmillan, 1993. [10] F. L. Lewis and V. L. Syrmos, Optimal Control, 2nd ed, New York: Wiley, 1995. [11] K. S. Narendra and A. M. Annaswamy, “A new adaptive law for robust adaptation without persistent excitation,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 134–145, Feb. 1987. [12] F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer neural-net robot con￾troller with guaranteed tracking performance,” IEEE Trans. Neural Net￾works, vol. 7, pp. 388–399, Mar. 1996. [13] M. M. Polycarpou, “Stable adaptive neural control of scheme for non￾linear systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 447–451, Mar. 1996. [14] Y.-F. Wong and A. Sideris, “Learning convergence in the cerebellar model articulation controller,” IEEE Trans. Neural Networks, vol. 3, pp. 115–121, Jan. 1992. [15] MatLab Users Guide, Control System Toolbox. Natick, MA: Math￾works, 1990. Young Ho Kim was born in Taegu, Korea, in 1960. He received the B.S. degree in physics from Korea Military Academy in 1983, the M.S. degree in electrical engineering from the University of Central Florida, Orlando, in 1988, and the Ph.D. degree in electrical engineering from the University of Texas at Arlington, Fort Worth, in 1997. From 1994 to 1997, he was a Research Assistant at the Automation and Robotics Research Institute, University of Texas, Arlington. He has published extensively in the fields of feedback control using neural networks and fuzzy systems. He authored the book High-Level Feedback Control with Neural Networks. His research interests include optimal control, neural networks, dynamic recurrent neural networks, fuzzy-logic systems, real-time adaptive critics for intelligent control of robotics, and nonlinear systems. Dr. Kim received the Korean Army Overseas Scholarship. He received the Sigma Xi Doctoral Research Award in 1997. He is a member of Sigma Xi