Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 3 Part 11 potx

where (30.201) (30.202) (30.203) The system poles are s = ±14.1050, s = ±4.5299. Eigenvector analysis shows that the fast instability at s = 14.1050 is primarily associated with the upper (shorter) link, while the slower instability at s = 4.5299 is primarily associated with the lower (longer) link. The system does not possess any natural integrators (i.e., no zero eigenvalues) and, as expected, the singular values σ i [P(j ω )] are flat at low frequencies (see Fig. 30.9). Closed Loop Objectives A controller to be implemented within a negative feedback loop is sought. The closed loop system should exhibit the following properties: (1) closed loop stability, (2) zero steady state error to step reference com- mands, (3) good low frequency reference command following (step commands followed with little overshoot within 3 s), (4) good low frequency disturbance attenuation, (5) good high frequency noise attenuation, (6) good stability robustness margins at the plant output. Each step of the control system design process is now described. A central idea is the formation of a so-called design plant P d from the original plant P. The design plant P d is what is submitted to our H 2 - LQG/LTR design machinery. Step 1: Augment Plant P with Integrators to Get Design Plant Pd == == [A, B, C] In order to guarantee zero steady-state error to step reference commands, we begin by augmenting the plant P = [A p , B p , C p ] with integrators—one in each control channel—to form the design plant P d = [A, B, C]; i.e., P d = P(I 2×2 /s). This is done as follows: (30.204) FIGURE 30.8 Two degree-of-freedom PUMA 560 robotic manipulator. A p 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 31.7613 33.0086– 0.0000 0.0000 56.9381– 187.7089 0.0000 0.0000 = B p 0.0000 0.0000 0.0000 0.0000 1037.7259 3919.6674– 3919.6674– 2030.8306 = C p I 22× 0 22× []= A 0 2×2 0 2×4 B p A p = 0066_Frame_C30 Page 33 Thursday, January 10, 2002 4:44 PM ©2002 CRC Press LLC where (30.201) (30.202) (30.203) The system poles are s = ±14.1050, s = ±4.5299. Eigenvector analysis shows that the fast instability at s = 14.1050 is primarily associated with the upper (shorter) link, while the slower instability at s = 4.5299 is primarily associated with the lower (longer) link. The system does not possess any natural integrators (i.e., no zero eigenvalues) and, as expected, the singular values σ i [P(j ω )] are flat at low frequencies (see Fig. 30.9). Closed Loop Objectives A controller to be implemented within a negative feedback loop is sought. The closed loop system should exhibit the following properties: (1) closed loop stability, (2) zero steady state error to step reference com- mands, (3) good low frequency reference command following (step commands followed with little overshoot within 3 s), (4) good low frequency disturbance attenuation, (5) good high frequency noise attenuation, (6) good stability robustness margins at the plant output. Each step of the control system design process is now described. A central idea is the formation of a so-called design plant P d from the original plant P. The design plant P d is what is submitted to our H 2 - LQG/LTR design machinery. Step 1: Augment Plant P with Integrators to Get Design Plant Pd == == [A, B, C] In order to guarantee zero steady-state error to step reference commands, we begin by augmenting the plant P = [A p , B p , C p ] with integrators—one in each control channel—to form the design plant P d = [A, B, C]; i.e., P d = P(I 2×2 /s). This is done as follows: (30.204) FIGURE 30.8 Two degree-of-freedom PUMA 560 robotic manipulator. A p 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 31.7613 33.0086– 0.0000 0.0000 56.9381– 187.7089 0.0000 0.0000 = B p 0.0000 0.0000 0.0000 0.0000 1037.7259 3919.6674– 3919.6674– 2030.8306 = C p I 22× 0 22× []= A 0 2×2 0 2×4 B p A p = 0066_Frame_C30 Page 33 Thursday, January 10, 2002 4:44 PM ©2002 CRC Press LLC 31 Adaptive and Nonlinear Control Design 31.1 Introduction 31.2 Lyapunov Theory for Time-Invariant Systems 31.3 Lyapunov Theory for Time-Varying Systems 31.4 Adaptive Control Theory Regulation and Tracking Problems • Certainty Equivalence Principle • Direct and Indirect Adaptive Control • Model Reference Adaptive Control (MRAC) • Self-Tuning Controller (STC) 31.5 Nonlinear Adaptive Control Systems 31.6 Spacecraft Adaptive Attitude Regulation Example 31.7 Output Feedback Adaptive Control 31.8 Adaptive Observers and Output Feedback Control 31.9 Concluding Remarks 31.1 Introduction The most important challenge for modern control theory is that it should deliver acceptable performance while dealing with poor models, high nonlinearities, and low-cost sensors under a large number of op- erating conditions. The difficulties encountered are not peculiar to any single class of systems and they appear in virtually every industrial application. Invariably, these systems contain such a large amount of model and parameter uncertainty that “fixed”controllers can no longer meet the stability and performance requirements. Any reasonable solution for such problems must be a suitable amalgamation between nonlinear control theory, adaptive elements, and information processing. Such are the factors behind the birth and evolution of the field of adaptive control theory, strongly motivated by several practical applications such as chemical process control and design of autopilots for high-performance aircraft, which operate with proven stability over a wide variety of speeds and altitudes. A commonly accepted definition for an adaptive system is that it is any physical system that is designed from an adaptive standpoint! 1 All existing stability and convergence results, in the field of adaptive control theory, hinge on the crucial assumption that the unknown parameters must occur linearly within the plant containing known nonlinearities. Conceptually, the overall process makes the parameter estimates themselves as state variables, thus enlarging the dimension of the state space for the original system. By nature, adaptive control solutions for both linear and nonlinear dynamical systems lead to nonlinear time-varying formulations wherein the estimates of the unknown parameters are updated using input–output data. A parameter adaptation mechanism (typically nonlinear) is used to update the param- eters within the control law. Given the nonlinearity due to adaptive feedback, there is the need to ensure that the closed-loop stability is preserved. It is thus an unmistakable fact that the fields of adaptive control and nonlinear system stability are intrinsically related to one another and any new insights gained in one Maruthi R. Akella The University of Texas at Austin ©2002 CRC Press LLC 32 Neural Networks and Fuzzy Systems 32.1 Neural Networks and Fuzzy Systems 32.2 Neuron Cell 32.3 Feedforward Neural Networks 32.4 Learning Algorithms for Neural Networks Hebbian Learning Rule • Correlation Learning Rule • Instar Learning Rule • Winner Takes All (WTA) • Outstar Learning Rule • Widrow–Hoff LMS Learning Rule • Linear Regression • Delta Learning Rule • Error Backpropagation Learning 32.5 Special Feedforward Networks Functional Link Network • Feedforward Version of the Counterpropagation Network • WTA Architecture • Cascade Correlation Architecture • Radial Basis Function Networks 32.6 Recurrent Neural Networks Hopfield Network • Autoassociative Memory • Bidirectional Associative Memories (BAM) 32.7 Fuzzy Systems Fuzzification • Rule Evaluation • Defuzzification • Design Example 32.8 Genetic Algorithms Coding and Initialization • Selection and Reproduction • Reproduction • Mutation 32.1 Neural Networks and Fuzzy Systems New and better electronic devices have inspired researchers to build intelligent machines operating in a fashion similar to the human nervous system. Fascination with this goal started when McCulloch and Pitts (1943) developed their model of an elementary computing neuron and when Hebb (1949) intro- duced his learning rules . A decade later Rosenblatt (1958) introduced the perceptron concept. In the early 1960s Widrow and Holf (1960, 1962) developed intelligent systems such as ADALINE and MADALINE. Nillson (1965) in his book Learning Machines summarized many developments of that time. The pub- lication of the Mynsky and Paper (1969) book, with some discouraging results, stopped for some time the fascination with artificial neural networks, and achievements in the mathematical foundation of the back- propagation algorithm by Werbos (1974) went unnoticed. The current rapid growth in the area of neural networks started with the Hopfield (1982, 1984) recurrent network, Kohonen (1982) unsupervised training algorithms, and a description of the backpropagation algorithm by Rumelhart et al. (1986). Bogdan M. Wilamowski University of Wyoming ©2002 CRC Press LLC . 0.0000 1 037 .7259 39 19.6674– 39 19.6674– 2 030 . 830 6 = C p I 22× 0 22× []= A 0 2×2 0 2×4 B p A p = 0066_Frame_C30 Page 33 Thursday, January 10, 2002 4:44 PM 2002 CRC Press LLC where (30 .201) (30 .202) (30 .2 03) The. 0.0000 0.0000 0.0000 1 037 .7259 39 19.6674– 39 19.6674– 2 030 . 830 6 = C p I 22× 0 22× []= A 0 2×2 0 2×4 B p A p = 0066_Frame_C30 Page 33 Thursday, January 10, 2002 4:44 PM 2002 CRC Press LLC 31 Adaptive. Texas at Austin 2002 CRC Press LLC 32 Neural Networks and Fuzzy Systems 32 .1 Neural Networks and Fuzzy Systems 32 .2 Neuron Cell 32 .3 Feedforward Neural Networks 32 .4 Learning Algorithms