西安交通大学：《智能控制理论与方法》课程教学资源（参考文献）Optimal Design of CMAC Neural-Network Controller for Robot Manipulators

EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Optimal Design of CMAc Neural-Network Controller for robot manipulators Young h. Kim and Frank L. Lewis, Fellow, IEEE Abstract-This paper is concerned with the application neural-network based, closed-loop control can be found [12 of quadratic optimization for motion control to feedback For indirect or identification-based, robotic-system control, sev control of robotic systems using cerebellar model arithmetic eral neural network and learning schemes c can be found in the lit computer (CMAC) neural networks. Explicit solutions to the I control erature. Most of these approaches consider neural networks as of robotic systems are found by solving an algebraic Riccati equa- very general computational models. Although a pure neural-net- tion. It is shown how the CMAC's can cope with nonlinearities work approach without a knowledge of robot dynamics may be through optimization with no preliminary of -line learning phase promising, it is important to note that this approach will not be punov stability analysis, so that both system-tracking stability and very practical due to high dimensionality of input-output space error convergence can be guaranteed in the closed-loop system. In this way, the training or off-line learning process by pure con- The filtered-tracking error or critic gain and the Lyapunov nectionist models would require a neural network of impractical function for the nonlinear analysis are derived from the user input size and unreasonable number of repetition cycles. The pure in terms of a specified quadratic-performance index Simulation connectionist approach has poor generalization properties results from a two-link robot manipulator show the satisfactory In this paper, we propose a ne performance of the proposed control schemes even in the presence that integrates linear optimal-control techniques and CMAC neural-network learning methods. The linear optimal control Index Terms--CMAC neural network, optimal control, robotic has an inherent robustness against a certain range of model uncertainties [9]. However, nonlinear dynamics cannot be taken nto consideration in linear optimal-control design. We use L. INTRODUCTION the Cmac neural networks to adaptively estimate nonlinear I hERE has been some work related to applying optimal ncertainties, yielding a controller that can tolerate a wider control techniques to the nonlinear robotic manipulator. range of uncertainties. The salient feature of this H-J-B control These approaches often combine feedback linearization and op- design is that we can use a priori knowledge of the plant timal-control techniques Johansson [6] showed explicit solu- dynamics as the system equation in the corresponding linear tions to the Hamilton-Jacobi-Bellman(H-J-B)equation for optimal-control design. The neural network is used to improve optimal control of robot motion and how optimal control and performance in the face of unknown nonlinearities by adding adaptive control may act in concert in the case of unknown nonlinear effects to the linear optimal controller. or uncertain system parameters. Dawson et al. [5] used a gen- The paper is organized as follows In Section II, we will re- ral-control law known as modified computed-torque control view some fundamentals of the CMAC neural networks.In Sec MCTC)and quadratic optimal-control theory to derive a pa- tion I, we give a new control design for rigid robot systems rameterized proportional-derivative(PD)form for an auxiliary using the H-J-B equation In Section IV, a CMAC controller oput to the controller. However, in actual situations, the robot combined with the optimal-control signal is proposed In Sec- dynamics is rarely known completely, and thus, it is difficult to tion v, a two-link robot controller is designed and simulated in express real robot dynamics in exact mathematical equations or to linearize the dynamics with respect to the operating point Neural networks have been used for approximation of non linear systems, for classification of signals, and for associative memory For control engineers, the approximation capability of Let R denote the real numbers, n the real n-vectors, and tification-based control. More work is now appearing on the a∈as础l-=√+…+ and the norm of a matrix use of neural networks in direct, closed-loop controllers that AE mxn as (A!l=vAma[AT A] where Ama[ 1 and Amin[l yield guaranteed performance [13]. The robotic application of are the largest and smallest eigenvalues of amatrix. The absolute ue is denoted as· aii and B∈我m×n, the Frobenius norm is supported by NSF Grant ECS-952167. defined by非1=tr(44)=∑吗 with tr() as the trace 种如需分+mmmm ykim50@hotmail.com;flewis@arri.uta.edu) Publisher Item Identifier S 1094-6977(00)00364-3 Ar2≤‖Arl

22 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 Optimal Design of CMAC Neural-Network Controller for Robot Manipulators Young H. Kim and Frank L. Lewis, Fellow, IEEE Abstract—This paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using cerebellar model arithmetic computer (CMAC) neural networks. Explicit solutions to the Hamilton–Jacobi–Bellman (H–J–B) equation for optimal control of robotic systems are found by solving an algebraic Riccati equa￾tion. It is shown how the CMAC’s can cope with nonlinearities through optimization with no preliminary off-line learning phase required. The adaptive-learning algorithm is derived from Lya￾punov stability analysis, so that both system-tracking stability and error convergence can be guaranteed in the closed-loop system. The filtered-tracking error or critic gain and the Lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic-performance index. Simulation results from a two-link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances. Index Terms—CMAC neural network, optimal control, robotic control. I. INTRODUCTION T HERE has been some work related to applying optimal￾control techniques to the nonlinear robotic manipulator. These approaches often combine feedback linearization and op￾timal-control techniques. Johansson [6] showed explicit solu￾tions to the Hamilton–Jacobi–Bellman (H–J–B) equation for optimal control of robot motion and how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. Dawson et al. [5] used a gen￾eral-control law known as modified computed-torque control (MCTC) and quadratic optimal-control theory to derive a pa￾rameterized proportional-derivative (PD) form for an auxiliary input to the controller. However, in actual situations, the robot dynamics is rarely known completely, and thus, it is difficult to express real robot dynamics in exact mathematical equations or to linearize the dynamics with respect to the operating point. Neural networks have been used for approximation of non￾linear systems, for classification of signals, and for associative memory. For control engineers, the approximation capability of neural networks is usually used for system identification or iden￾tification-based control. More work is now appearing on the use of neural networks in direct, closed-loop controllers that yield guaranteed performance [13]. The robotic application of Manuscript received June 2, 1997; revised June 23, 1999. This research was supported by NSF Grant ECS-9521673. The authors are with the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX 76118-7115 USA (e-mail: ykim50@hotmail.com; flewis@arri.uta.edu). Publisher Item Identifier S 1094-6977(00)00364-3. neural-network based, closed-loop control can be found [12]. For indirect or identification-based, robotic-system control, sev￾eral neural network and learning schemes can be found in the lit￾erature. Most of these approaches consider neural networks as very general computational models. Although a pure neural-net￾work approach without a knowledge of robot dynamics may be promising, it is important to note that this approach will not be very practical due to high dimensionality of input–output space. In this way, the training or off-line learning process by pure con￾nectionist models would require a neural network of impractical size and unreasonable number of repetition cycles. The pure connectionist approach has poor generalization properties. In this paper, we propose a nonlinear optimal-design method that integrates linear optimal-control techniques and CMAC neural-network learning methods. The linear optimal control has an inherent robustness against a certain range of model uncertainties [9]. However, nonlinear dynamics cannot be taken into consideration in linear optimal-control design. We use the CMAC neural networks to adaptively estimate nonlinear uncertainties, yielding a controller that can tolerate a wider range of uncertainties. The salient feature of this H–J–B control design is that we can use a priori knowledge of the plant dynamics as the system equation in the corresponding linear optimal-control design. The neural network is used to improve performance in the face of unknown nonlinearities by adding nonlinear effects to the linear optimal controller. The paper is organized as follows. In Section II, we will re￾view some fundamentals of the CMAC neural networks. In Sec￾tion III, we give a new control design for rigid robot systems using the H–J–B equation. In Section IV, a CMAC controller combined with the optimal-control signal is proposed. In Sec￾tion V, a two-link robot controller is designed and simulated in the face of large uncertainties and external disturbances. II. BACKGROUND Let denote the real numbers, the real -vectors, and the real matrices. We define the norm of a vector as and the norm of a matrix as where and are the largest and smallest eigenvalues of a matrix. The absolute value is denoted as . Given and , the Frobenius norm is defined by with as the trace operator. The associated inner product is . The Frobenius norm is compatible with the two-norm so that with and . 1094–6977/00$10.00 © 2000 IEEE

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER receptive field adjustable Fig. 1. Architecture of a Cmac neural network Fig. I shows the architecture and operation of the CMAC. The 3 =[1 u2. sional Receptive-Field Functions: Given any A. CMAC Neural Networks 2)Multisim nl E 3e", the multidimensional receptive- CMAC can be used to approximate a nonlinear mapping y(r): field functions are defined as Ⅺ→ Y where X" C9 is the application in mensional input space and Y c gm in the application 9i1,2,…,j ,n(x1)·p2,(x2)…pm,n(xn)(3) space. The CMAC algorithm consists of two primary functions for determining the value of a complex function, as shown in Fig 1 1,……, n The output of the CMAc is given by R:X→A P.A→Y v(x)=>m(x),j=1,…,m( where where X continuous n-dimensional input space Wji E s output-layer weight values, a NA-dimensional association space O: continuous, multidimensional receptive-fie Y m-dimensional output space → function; The function p= R()is fixed and maps each point in the NA number of the association point. input space onto the association space A The function P() The effect of receptive-field basis function type and partition computes an output ye Y by projecting the association vector number along each dimension on the Cmac performance has determined by R(r)onto a vector of adjustable weights such not yet been systematically studied The output of the CMAC can be expressed in a vector notation y=P(p)=wp v(r)=w p(a) R()in(1)is the multidimensional receptive field function I)Receptive-Field Fumction: Given x=[T1r2..mI E Re, let Eci mim Si]ev1<i<n be domain of interest w matrix of adjustable weight values For this domain, select integers Ni and strictly increasing par- p(r), vector of receptive-field functions titions Based on the approximation property of the CMAC, there ex ists ideal weight values W, so that the function to be approxi mated can be represented as 丌;=[x;,1x,2…x;,N 1<< f(a=wpr)+e(c) or each component of the input space, the receptive-field basis function can be defined as rectangular [1] or triangular [4] or with e(=r)the"functional reconstructional error"and E()< any continuously bounded function, e.g., Gaussian 31

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 23 Fig. 1. Architecture of a CMAC neural network. A. CMAC Neural Networks Fig. 1 shows the architecture and operation of the CMAC. The CMAC can be used to approximate a nonlinear mapping : where is the application in the -di￾mensional input space and in the application output space. The CMAC algorithm consists of two primary functions for determining the value of a complex function, as shown in Fig. 1 (1) where continuous -dimensional input space; -dimensional association space; -dimensional output space. The function is fixed and maps each point in the input space onto the association space . The function computes an output by projecting the association vector determined by onto a vector of adjustable weights such that (2) in (1) is the multidimensional receptive field function. 1) Receptive-Field Function: Given , let be domain of interest. For this domain, select integers and strictly increasing par￾titions For each component of the input space, the receptive-field basis function can be defined as rectangular [1] or triangular [4] or any continuously bounded function, e.g., Gaussian [3]. 2) Multidimensional Receptive-Field Functions: Given any , the multidimensional receptive￾field functions are defined as (3) with , . The output of the CMAC is given by (4) where output-layer weight values; : continuous, multidimensional receptive-field function; number of the association point. The effect of receptive-field basis function type and partition number along each dimension on the CMAC performance has not yet been systematically studied. The output of the CMAC can be expressed in a vector notation as (5) where matrix of adjustable weight values vector of receptive-field functions. Based on the approximation property of the CMAC, there ex￾ists ideal weight values , so that the function to be approxi￾mated can be represented as (6) with the “functional reconstructional error” and bounded

EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Then, an estimate of f(a)can be given by Property 1--Inertia: The inertia matrix M(g) is uniformly bounded f(r) m1I≤M(q)≤m2Im,m2>0andI∈我n,(l6) where W are estimates of the ideal weight values. The Lya punov method is applied to derive reinforcement adaptive Property 2-Skeww Symmetry: The matrix learning rules for the weight values. Since these adaptive learning rules are formulated from the stability analysis of the N(q立=M(q)-2Vm(g立 controlled system, the system performance can be guaranteed or closed-loop control s skew-symmetric Robot Arm Dynam II. OPTIMAL-COMPUTED TORQUE-CONTROLLER DESIGN The dynamics of an n-link robot manipulator may be ex- A.H-J-B Optimization pressed in the Lagrange form [91 Define the velocity-error dynamics M(q)+Vm(q立+F+fc(①)+gq)+7(t)=T(t et)=-Ae(t q(t)∈我 variable The following augmented system is obtained M(q)∈ inertia Vm(q),q)∈我× Coriolis/centripetal forces = gq∈究 gravitational force L+」=[0nxm-M-v M-1 u(ty diagonal matrix of viscous friction co- or with shorter notation efficients. ∫c(i)∈我 Coulomb friction coefficient x(1)=A(q,q)(t)+B(q)u(t) external disturbances The external control torque to each joint is t(tEgen with A(g立∈我x如,B(q)∈我2n,and(t)∈界nx Given a desired trajectory ga(t)E R", the tracking errors are i(t)is defined as it)T=[etT r(tT].A quadratic perfor e(t)=qa(t-t) and e(t)=qa(t)-i(t) (9 mance index J(u) is as follows and the instantaneous performance measure is defined L(E, u)dt rt=et)+ Aet) (10) with the Lagrangian where AE R xn is the constant-gain matrix or critic (not nec- L(E, u)=22(+Qa(t)+3uT(t)Ru(t) The robot dynamics( 8)may be written as 1Q121「e Qi2 Q,r+2 uTRu(22) M(q(t)=-Vmn(,@)r(t)-T(t)+(x)(11) Given the performance index J(u), the control objective is where the robot nonlinear function is to find the auxiliary control input u(t)that minimizes(21)sub. ject to the differential constraints imposed by (19). The optimal h(r)=M(a(d+ Ae+Vm(a, i a+Ae) control that achieves this objective will be denoted by u (t). It is Fq+f(q)+9(q)+7(t) (12) worth noting for now, that only the part of the control-input-to- robotic-system denoted by u(t) in(14) is penalized. This is rea and. for instance sonable from a practical standpoint, since the gravity, Coriolis, and friction-compensation terms in(12)cannot be modified by (t)=[ereq效] (13)the optimal-design phase A necessary and sufficient condition for u(t) to minimize This key function h(r)captures all the unknown dynamics of (21)subject to(20)is that there exist a function V=v(a, ty the robot arm satisfying the H-J-B equation Now define a control-input torque as o(t+milr(2u一 (2,t) (t)=b(x)-u(t) with u(t)E an auxiliary control input to be optimized later. where the Hamiltionian of optimization is defined as The closed-loop system becomes Mart=-vm(, or(t)+ut (15) H av(E, t) L(2,v)+(t

24 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 Then, an estimate of can be given by (7) where are estimates of the ideal weight values. The Lya￾punov method is applied to derive reinforcement adaptive learning rules for the weight values. Since these adaptive learning rules are formulated from the stability analysis of the controlled system, the system performance can be guaranteed for closed-loop control. B. Robot Arm Dynamics and Properties The dynamics of an -link robot manipulator may be ex￾pressed in the Lagrange form [9] (8) with joint variable; inertia; Coriolis/centripetal forces; gravitational forces; diagonal matrix of viscous friction co￾efficients; Coulomb friction coefficients; external disturbances. The external control torque to each joint is . Given a desired trajectory , the tracking errors are and (9) and the instantaneous performance measure is defined as (10) where is the constant-gain matrix or critic (not nec￾essarily symmetric). The robot dynamics (8) may be written as (11) where the robot nonlinear function is (12) and, for instance (13) This key function captures all the unknown dynamics of the robot arm. Now define a control-input torque as (14) with an auxiliary control input to be optimized later. The closed-loop system becomes (15) Property 1—Inertia: The inertia matrix is uniformly bounded and (16) Property 2—Skew Symmetry: The matrix (17) is skew-symmetric. III. OPTIMAL-COMPUTED TORQUE-CONTROLLER DESIGN A. H–J–B Optimization Define the velocity-error dynamics (18) The following augmented system is obtained: (19) or with shorter notation (20) with , , and . is defined as . A quadratic perfor￾mance index is as follows: (21) with the Lagrangian (22) Given the performance index , the control objective is to find the auxiliary control input that minimizes (21) sub￾ject to the differential constraints imposed by (19). The optimal control that achieves this objective will be denoted by . It is worth noting for now, that only the part of the control-input-to￾robotic-system denoted by in (14) is penalized. This is rea￾sonable from a practical standpoint, since the gravity, Coriolis, and friction-compensation terms in (12) cannot be modified by the optimal-design phase. A necessary and sufficient condition for to minimize (21) subject to (20) is that there exist a function satisfying the H–J–B equation [10] (23) where the Hamiltionian of optimization is defined as (24)

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER and V(a, t) is referred to as the value function. It satisfies the where h(er) is given by(12). It is referred to as an optima partial differential equation computed torque controller(OCTC) av(e,t=L(i, u, )+222.(25)B Stability Analysis Theorem 2: Suppose that matrices K and A exist that satisfy The minimum is attained for the optimal control u(t)=w(t), the hypotheses of Lemma 1, and in addition, there exist con- and the Hamiltonian is then given by stants K,i and k2 such that 0 the origin of the error space. It remains to show that dv/dt0≠0 (37) See Appendix A for proof. The time derivative of the lyapunov function is negative defl Theorem 1: Let the symmetric weighting matrices Q, R be nite, and the assertion of the theorem then follows directly from chosen such that the properties of the Lyapunov function [9] Q Qu Q 75 R=Q2(30) IV CMAC NEURAL-CONTROLLER DESIGN with Q12+Q120nXn (31) trol law given in Theorem I plus the CMAC neural-network The nonlinear robot function can be represented by a CMAC eural network h(x)=W2y(x)+E(x)l|=(x)川≤EM(38) with(32)solved for A using Lyapunov equation solvers(e.g MatLab[15D) where p(sr)is a multidimensional receptive-field function for See Appendix B for proof. the CMAc Remarks Then a functional estimate h(r)of h(r)can be written as 1)In order to guarantee positive definiteness of the con- structed matrix Q, the following inequality [7] must be h(a)=w plr) Aim(Q2)>|Ql2/am(Q1).(3) r(t)=Wry(x)-'(t)-v() 2)With the optimal-feedback control law c(t)calculated using Theorem 1, the torques r(t) to apply to the robotic where u(t) is a robustifying vector. Then(11)becomes system are calculated according to the control input Mr(t)=-Vmr(t)+Wp(=)+E(a)+Ta(t) T*(t)=(x)-"(t +u(t)+u(t

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 25 and is referred to as the value function. It satisfies the partial differential equation (25) The minimum is attained for the optimal control , and the Hamiltonian is then given by (26) Lemma 1: The following function composed of , and a positive symmetric matrix satisfies the H–J–B equation: (27) where and in (10) and (27) can be found from the Riccati differential equation (28) The optimal control that minimizes (21) subject to (20) is (29) See Appendix A for proof. Theorem 1: Let the symmetric weighting matrices , be chosen such that (30) with . Then the and required in Lemma 1 can be determined from the following relations: (31) (32) with (32) solved for using Lyapunov equation solvers (e.g., MatLab [15]). See Appendix B for proof. Remarks: 1) In order to guarantee positive definiteness of the con￾structed matrix , the following inequality [7] must be satisfied (33) 2) With the optimal-feedback control law calculated using Theorem 1, the torques to apply to the robotic system are calculated according to the control input (34) where is given by (12). It is referred to as an optimal￾computed torque controller (OCTC). B. Stability Analysis Theorem 2: Suppose that matrices and exist that satisfy the hypotheses of Lemma 1, and in addition, there exist con￾stants and such that , and the spectrum of is bounded in the sense that on . Then using the feedback control in (29) and (20) results in the controlled nonlinear system (35) This is globally exponentially stable (GES) regarding the origin in . Proof: The quadratic function is a suitable Lyapunov function candidate, because it is positive radially, growing with . It is continuous and has a unique minimum at the origin of the error space. It remains to show that for all . From the solution of the H–J–B equation (A12), it follows that (36) Substituting (29) for (31) gives (37) The time derivative of the Lyapunov function is negative defi￾nite, and the assertion of the theorem then follows directly from the properties of the Lyapunov function [9]. IV. CMAC NEURAL-CONTROLLER DESIGN The block diagram in Fig. 2 shows the major components that embody the CMAC neural controller. The external-control torques to the joints are composed of the optimal-feedback con￾trol law given in Theorem 1 plus the CMAC neural-network output components. The nonlinear robot function can be represented by a CMAC neural network (38) where is a multidimensional receptive-field function for the CMAC. Then a functional estimate of can be written as (39) The external torque is given by (40) where is a robustifying vector. Then (11) becomes (41)

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)

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER which is guaranteed negative as long as either(54)or(55) hold F+J(=02+1((58 ≥(∈M+1H成/{m(q)+hm(分}≡B2 (54) where sqn(a) is a signum function The weighting matrices are as follows: WP≥V∈M+n+是WM≡B (55) Q1 where B: and Bw are convergence regions. According to a Q1 standard Lyapunov theory extension[11], this demonstrates uni- formly ultimate boundedness ofet),r(t), and w(t) Q21=Q12 Remarks Q21=Q 1)The OCTC is globally asymptotically stable if h(ar)is fully known, whereas the neural-adaptive controller is Q2=1_「300 UUB. In both cases, there is a convergence of trackinger- rors. UUB is a notion of stability in the practical sense that Solving the matrices K and A using MatLab [15] yields is usually sufficient for the performance of closed-loop systems, provided that the bound on system states is small K=-3 A 6 1897919404(60 2)Robotic manipulators are subjected to structured and/or The motion problem considered is for the robot end-effector unstructured uncertainties in all applications. Structured to track a point on a circle centered at x= y=0.05 m and incertainty is defined as the case of a correct dynamical radius 0.05 m, which turns 1/2 times per second in slow motion model but with parameter uncertainty due to tolerance and two times per second in fast motion. It was pointed out that variations in the manipulator-link properties, unknown control-system performance may be quite different in low-speed loads, and so on Unstructured uncertainty describes the and high-speed motion. Therefore, we carry out our simulation case of unmodeled dynamics that result from the presence for two circular trajectories of high-frequency modes in the manipulator, nonlinear The desired positions in low speed are friction. The adaptive optimizing feature of the proposed neural controller is suitable even without full knowledge x(t)=1+0.05sin(t) of the system dynamics (t)=10+0.05co(mt) (61) 3)From Barron results [2], there exist lower bounds of order (1/NA)2/n on the approximation error EM if only the and the high-speed positions profiles are parameters of a linear combination of basis functions are adjusted. Our stability proof shows that the effect of the xd(t)=1+0.05sin(47 bounds on the approximation error can be alleviated by (t)=1+0.050(4mt) the judicious choice of weighting matrices Q and R 4)It is emphasized that the neural-weight values may be By solving the inverse kinematics, we obtain the desired joint- initialized at zero, and stability will be maintained by the angle trajectory in fast motion optimal controller(t) in the performance-measurement The responses of the OCTC, where all nonlinearities are loop until the neural network learns. This means that there exactly known, are shown in Fig.3 without disturbances and is no off-line learning or trial and error phase, which often friction. The simulation was performed in low speed and high requires a long time in other works speed. After a transient due to error in initial conditions, the 5)The advantage of the CMAc control scheme over other position errors tend asymptotically toward zero existing neural-network architectures is that the number To show the effect of unstructured uncertainties, we dropped of adjustable parameters(i.e, weight values)is signifi- a term(m1 +m2)glici in gravity forces. The simulation re antly less, since only weights in the output layer are to sults are shown in Fig. 4(a) in low speed. Note that there is be adjusted. It is very suitable for closed-loop control. a steady-state error with OCT b)shows the effect of external disturbances and friction forces which is difficult to V. SIMULATION RESULTS model and compensate. This is corrected by adding a CMAC neural network as follows The dynamic equations for an 7-link manipulator can be The CMAC can be characterized by found in [9]. The cost functional to be minimized is number of input spaces:c=[g i J()=2/(2 Qz+u Ru)dt number of partitions for each space: Ni= 3, i · number of association points:NA=3×3×3×3 An external disturbance and frictions are receptive field-basis functions: pi, i (i)= exp[(, T()=[8sin(2)8cx(切) (57) mn,2=0, 1,t=1

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 27 which is guaranteed negative as long as either (54) or (55) holds (54) (55) where and are convergence regions. According to a standard Lyapunov theory extension [11], this demonstrates uni￾formly ultimate boundedness of , , and . Remarks: 1) The OCTC is globally asymptotically stable if is fully known, whereas the neural-adaptive controller is UUB. In both cases, there is a convergence of tracking er￾rors. UUB is a notion of stability in the practical sense that is usually sufficient for the performance of closed-loop systems, provided that the bound on system states is small enough. 2) Robotic manipulators are subjected to structured and/or unstructured uncertainties in all applications. Structured uncertainty is defined as the case of a correct dynamical model but with parameter uncertainty due to tolerance variations in the manipulator-link properties, unknown loads, and so on. Unstructured uncertainty describes the case of unmodeled dynamics that result from the presence of high-frequency modes in the manipulator, nonlinear friction. The adaptive optimizing feature of the proposed neural controller is suitable even without full knowledge of the system dynamics. 3) From Barron results [2], there exist lower bounds of order on the approximation error if only the parameters of a linear combination of basis functions are adjusted. Our stability proof shows that the effect of the bounds on the approximation error can be alleviated by the judicious choice of weighting matrices and . 4) It is emphasized that the neural-weight values may be initialized at zero, and stability will be maintained by the optimal controller in the performance-measurement loop until the neural network learns. This means that there is no off-line learning or trial and error phase, which often requires a long time in other works. 5) The advantage of the CMAC control scheme over other existing neural-network architectures is that the number of adjustable parameters (i.e., weight values) is signifi￾cantly less, since only weights in the output layer are to be adjusted. It is very suitable for closed-loop control. V. SIMULATION RESULTS The dynamic equations for an -link manipulator can be found in [9]. The cost functional to be minimized is (56) An external disturbance and frictions are (57) (58) where sqn is a signum function. The weighting matrices are as follows: (59) Solving the matrices and using MatLab [15] yields (60) The motion problem considered is for the robot end-effector to track a point on a circle centered at 0.05 m and radius 0.05 m, which turns 1/2 times per second in slow motion and two times per second in fast motion. It was pointed out that control-system performance may be quite different in low-speed and high-speed motion. Therefore, we carry out our simulation for two circular trajectories. The desired positions in low speed are (61) and the high-speed positions profiles are (62) By solving the inverse kinematics, we obtain the desired joint￾angle trajectory in fast motion. The responses of the OCTC, where all nonlinearities are exactly known, are shown in Fig. 3 without disturbances and friction. The simulation was performed in low speed and high speed. After a transient due to error in initial conditions, the position errors tend asymptotically toward zero. To show the effect of unstructured uncertainties, we dropped a term in gravity forces. The simulation re￾sults are shown in Fig. 4(a) in low speed. Note that there is a steady-state error with OCTC. Fig. 4(b) shows the effect of external disturbances and friction forces, which is difficult to model and compensate. This is corrected by adding a CMAC neural network as follows. The CMAC can be characterized by: • number of input spaces: ; • number of partitions for each space: ; • number of association points: ; • receptive field-basis functions: with , and ;

28 EEE TRANSACTIONS ON SYSTEMS. MAN. AND CYBERNETICS-PART C: APPLICATIONS AND REVIEWS VOL 30. NO. 1. FEBUARY 2000 Fig 3. Performance of OCTC (34): (a) tracking error for slow motion and(b)tracking error for fast motion(solid: joint 1, dotted: joint 2) 051 Time(second Time(second) Fig 4. Performance of OCTC(34): (a) tracking error with modeling error for slow motion and(b) tracking error with disturbance and friction for slow motion Time(second) Fig. 5. Performance of CMAC neural network controller(40)(a)tracking error for slow motion and ( b)tracking error for fast motion(solid: joint 1, dotted Joint 2) learning rate in the weight-tuning law: VI. CONCLUSION We have developed a hierarchical, intelligent control scheme for a robotic manipulator using the HJB optimization process 08000 and the CMac neural network. It has been shown that the entire closed-loop system behavior depends on the user-specified per and=0.0001 formance index Q and R, through the critic-gain matrix A. The simulation time. 20s Lyapunov function for the stability of the overall system is au The results in Figs. 5 and 6 clearly show the ability of the CMac tomatically generated by the weighting matrices. In the deriva neural-network controller to me uncertainties, both struc- tion of the optimal-computed torque controller, it has been as- tured and unstructured. Note that the problem noted umed that nonlinearities in the robotic manipulator are com- with octc does not arise here. as all the nonlinearities pletely known. However, even with the knowledge of nonlinear- sumed unknown to the cmac neural controller ities, it is difficult to achieve the control objective in the pres-

28 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000 (a) (b) Fig. 3. Performance of OCTC (34): (a) tracking error for slow motion and (b) tracking error for fast motion (solid: joint 1, dotted: joint 2). (a) (b) Fig. 4. Performance of OCTC (34): (a) tracking error with modeling error for slow motion and (b) tracking error with disturbance and friction for slow motion (solid: joint 1, dotted: joint 2). (a) (b) Fig. 5. Performance of CMAC neural network controller (40): (a) tracking error for slow motion and (b) tracking error for fast motion (solid: joint 1, dotted: joint 2). • learning rate in the weight-tuning law: and ; • simulation time: 20 s. The results in Figs. 5 and 6 clearly show the ability of the CMAC neural-network controller to overcome uncertainties, both struc￾tured and unstructured. Note that the problem noted in Fig. 4 with OCTC does not arise here, as all the nonlinearities are as￾sumed unknown to the CMAC neural controller. VI. CONCLUSION We have developed a hierarchical, intelligent control scheme for a robotic manipulator using the HJB optimization process and the CMAC neural network. It has been shown that the entire closed-loop system behavior depends on the user-specified per￾formance index and , through the critic-gain matrix . The Lyapunov function for the stability of the overall system is au￾tomatically generated by the weighting matrices. In the deriva￾tion of the optimal-computed torque controller, it has been as￾sumed that nonlinearities in the robotic manipulator are com￾pletely known. However, even with the knowledge of nonlinear￾ities, it is difficult to achieve the control objective in the pres-

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER Fig. 6. Performance of CMAC neural-network controller (40):(a) tracking error with disturbance and friction for fast motion and(b)tracking error of mass variation(m2; 2.3-4.0 kg at 5 s, m2;4.0- 2.3 kg at 12 s)with disturbance and friction for fast motion(solid: joint 1, dotted: joint 2) ence of modeling uncertainties and frictional forces. The salient matrix whose elements are partial derivatives of the elements of feature of the CMAC neural-HjB design is that the control ob- Pw.r.t. ei jective is obtained with completely unknown nonlinearities in A candidate for the Hamiltonian H(24)is the sum of(A5) the robotic manipulator. The proposed neural-adaptive learning and the Lagrangian(22). Now we are ready to evaluate how H shows both robustness and adaptation to changing system dy- depends on ut)E en. The ut)=u(t) for which H has namics. To that end, a critic signal is incorporated into the adap- its minimum values is obtained from the partial derivative w.r.t tive-learning scheme. The application potential of the proposed u(t) methodology lies in the control design in areas such as robotics Since ut) is unconstrained, ( A3)requires that and flight control and in motion-control analysis(e.g, of biome chanics) aH V(2,t) 0 APPENDIX A PROOF OF LEMMA 1 which gives a candidate for the optimal control The theorem claims that the HJB equation V(,t) aNa2n=min1(2)+V(么,t (A8) (A1) sInce is satisfied for a function BcH =22P@2K Oxn z We know that(A3)is satisfied by u(t), given(A8). Inserting where .6)into(A8)giv mL(,v)+( =H( (A3) u(t=-RB P(g)i (A10) To derive optimal-control law, the partial derivatives of the Notice that the relation function v need to be evaluated Here, we have the time deriva- tive of the function v DB=[D1 XOnxn +O2mxn XM-I(q]=O2nx2n(All) is used The gradient of v with respect to the error state i is A necessary and sufficient condition for optimality is that the chosen value function V satisfies(23). Substituting(24)for(23) ID eld av(, t) av(a, t) 22+1(2,t) aP P where it is understood that the partial derivatives of V in(Al (A6)are being evaluated along the optimal control u(t).Inserting (A4)into(A12), in(A6, Dhas dimension2n×2n,O2n×lisa2n×1 zero vector, and the not aP(g/aei is used to represent the 2n X 2n zP(q)2+2Pq+L(,x)=0.(A13)

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 29 (a) (b) Fig. 6. Performance of CMAC neural-network controller (40): (a) tracking error with disturbance and friction for fast motion and (b) tracking error of mass variation (m ; 2:3 ! 4:0 kg at 5 s, m ; 4:0 ! 2:3 kg at 12 s) with disturbance and friction for fast motion (solid: joint 1, dotted: joint 2). ence of modeling uncertainties and frictional forces. The salient feature of the CMAC neural-HJB design is that the control ob￾jective is obtained with completely unknown nonlinearities in the robotic manipulator. The proposed neural-adaptive learning shows both robustness and adaptation to changing system dy￾namics. To that end, a critic signal is incorporated into the adap￾tive-learning scheme. The application potential of the proposed methodology lies in the control design in areas such as robotics and flight control and in motion-control analysis (e.g., of biome￾chanics). APPENDIX A PROOF OF LEMMA 1 The theorem claims that the HJB equation (A1) is satisfied for a function (A2) where (A3) To derive optimal-control law, the partial derivatives of the function need to be evaluated. Here, we have the time deriva￾tive of the function (A4) The gradient of with respect to the error state is (A5) with (A6) In (A6), has dimension is a zero vector, and the notation is used to represent the matrix whose elements are partial derivatives of the elements of w.r.t. . A candidate for the Hamiltonian (24) is the sum of (A5) and the Lagrangian (22). Now we are ready to evaluate how depends on . The for which has its minimum values is obtained from the partial derivative w.r.t. . Since is unconstrained, (A3) requires that (A7) which gives a candidate for the optimal control (A8) since (A9) We know that (A3) is satisfied by , given (A8). Inserting (A5) and (A6) into (A8) gives (A10) Notice that the relation (A11) is used. A necessary and sufficient condition for optimality is that the chosen value function satisfies (23). Substituting (24) for (23) yields (A12) where it is understood that the partial derivatives of in (A12) are being evaluated along the optimal control . Inserting (A4) into (A12), we obtain (A13)

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 of

30 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

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER Frank L. Lewis(S78-M81-SM'86-F94)was he Bs legree in physics and electrical engineering and ngineering at rice niversity, Houston, Tx, in 1971. He received the niversity of West Florida, Pensacola, in 1977. He received the Ph. D. degree from Georgia Institute of Technology, Atlanta, in 1981 trical Engineering with the University of Texas, Ar- gton. He spent six years in the United States Navy, serving as Navigator board the frigate USS Trippe(FF-1075) and Executive Officer and Acti Commanding Officer aboard USS Salinan(ATF-161) He has studied the ge netric, analytic, and structural properties of dynamical systems and feedback control automation. His current interests include robotics, intelligent control ural and fuzzy systems, nonlinear systems, and manufacturing process con- rol. He is the author/coauthor of two U.S. patents, 12 ters and encyclopedia articles, 210 refereed conference papers, and 7 bo tered Professional engineer in the State of Texas and was selected to the Editorial Boards of International Journal of Control Neural Computing and Applications, and International Journal of Intelligent Control Systems. He is the recipient of an NSF Research Initiation Grant and has been continuously funded by NSF since 1982. Since 1991, he has received $1. 8 mil- lion in funding from NSF and upwards of SI million in SBIR/industry/state funding. He was awarded the Moncrief-O Donnell Endowed Chair in 1990 at he Automation and Robotics Research Institute. Arlington. TX. He received a Fulbright Research Award, the American Society of Engineering Education F. E. Terman Award three Sigma Xi Research Awards, the UTA Halliburton En- g Research Award, the UTA University- the ARRI Patent Award, various Best Paper Awards, the IEEE Control Systems Society Best Chapter Award, and the National Sigma Xi Award for Outstanding Chapter(as President). He was selected as Engineer of the year in 1994 by the Ft. Worth, TX, IEEE Section. He was appointed to the NAE Committee on Space Station in 1995 and to the IEEE Control Systems Society Board of Governors in 1996. In 1998, he was selected as an IEEE Control Sys. ems Society Distinguished Lecturer. He is a Founding Member of the board of Governors of the mediterranean Control Association

KIM AND LEWIS: OPTIMAL DESIGN OF NEURAL-NETWORK CONTROLLER 31 Frank L. Lewis (S’78–M’81–SM’86–F’94) was born in Wuzburg, Germany. He received the B.S. degree in physics and electrical engineering and the M.S. degree in electrical engineering at Rice University, Houston, TX, in 1971. He received the M.S. degree in aeronautical engineering from the University of West Florida, Pensacola, in 1977. He received the Ph.D. degree from Georgia Institute of Technology, Atlanta, in 1981. In 1981, he was employed as a Professor of Elec￾trical Engineering with the University of Texas, Ar￾lington. He spent six years in the United States Navy, serving as Navigator aboard the frigate USS Trippe (FF-1075) and Executive Officer and Acting Commanding Officer aboard USS Salinan (ATF-161). He has studied the geo￾metric, analytic, and structural properties of dynamical systems and feedback control automation. His current interests include robotics, intelligent control, neural and fuzzy systems, nonlinear systems, and manufacturing process con￾trol. He is the author/coauthor of two U.S. patents, 124 journal papers, 20 chap￾ters and encyclopedia articles, 210 refereed conference papers, and 7 books. Dr. Lewis is a registered Professional Engineer in the State of Texas and was selected to the Editorial Boards of International Journal of Control, Neural Computing and Applications, and International Journal of Intelligent Control Systems. He is the recipient of an NSF Research Initiation Grant and has been continuously funded by NSF since 1982. Since 1991, he has received $1.8 mil￾lion in funding from NSF and upwards of $1 million in SBIR/industry/state funding. He was awarded the Moncrief-O’Donnell Endowed Chair in 1990 at the Automation and Robotics Research Institute, Arlington, TX. He received a Fulbright Research Award, the American Society of Engineering Education F. E. Terman Award, three Sigma Xi Research Awards, the UTA Halliburton En￾gineering Research Award, the UTA University-Wide Distinguished Research Award, the ARRI Patent Award, various Best Paper Awards, the IEEE Control Systems Society Best Chapter Award, and the National Sigma Xi Award for Outstanding Chapter (as President). He was selected as Engineer of the year in 1994 by the Ft. Worth, TX, IEEE Section. He was appointed to the NAE Committee on Space Station in 1995 and to the IEEE Control Systems Society Board of Governors in 1996. In 1998, he was selected as an IEEE Control Sys￾tems Society Distinguished Lecturer. He is a Founding Member of the Board of Governors of the Mediterranean Control Association