Bishop, Robert H. - The Mechatronics Handbook [CRC Press 2002] Part 14 pptx

for all w. This, in turn, implies that the associated sensitivity singular values satisfy (30.156) for all w, where (30.157) • From the above sensitivity singular value relationship, we obtain the follow celebrated KBF loop margins: infinite upward gain margin, at least (6 dB) downward gain margin, at least ±60° phase margin. The above gain margins apply to simultaneous and independent gain perturbations when the loop is broken at the output. The same holds for the above phase margins. The above margins are NOT guaranteed for simultaneous gain and phase perturbations. It should be noted that these margins can be easily motivated using elementary SISO Nyquist stability arguments [2,8]. • From the above sensitivity singular value relations, we obtain the following complementary sensitivity singular value relationship: (30.158) for all w, where (30.159) 2. Recovery of Target Loop L o Using Model-Based Compensator. The second step is to use a model- based compensator K opt = [ A − BG c − H f C, H f , G c ] where G c is found by solving the CARE with A, B, M = C, R = with ρ a small positive scalar. Since r is small, we call this a cheap control problem. • If the plant P = [A, B, C] is minimum phase, then it can be shown that (30.160) (30.161) for some orthonormal W (i.e., W T W = WW T = I ) (30.162) In such a case, PK opt ≈ L o for small ρ and hence PK opt will possess stability margins that are close to those of L o (at the plant output)—whatever method was used to design L o . It must be noted that the minimum phase condition on the plant P is a suficient condition. It is not necessary. Moreover, G c need not be computed using a CARE. In fact, any G c which (1) satisfies a limiting condition for some invertible matrix W and which (2) ensures that A − BG c is stable (for small r), will result in LTR at the plant output. This result s max S KF jw()[] 1 s min S KF jw()[] −1 1 0dB()≤= S KF jw() IG KF jw()+[] −1 = 1 2 s max T KF jw()[]s max IS KF jw()–[]= 1 s max S KF jw()[]26dB()≤+≤ T KF IS KF G KF =–= 1 G KF +[] −1 rI n u n u × X r 0 + → lim 0= rG c r 0 + → lim WC= PK opt r 0 + → lim L o = rG c r 0 + → lim WC= 0066_Frame_C30 Page 28 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 33 Advanced Control of an Electrohydraulic Axis 33.1 Introduction 33.2 Generalities Concerning ROBI_3, a Cartesian Robot with Three Electrohydraulic Axes 33.3 Mathematical Model and Simulation of Electrohydraulic Axes The Extended Mathematical Model • Nonlinear Mathematical Model of the Servovalve • Nonlinear Mathematical Model of Linear Hydraulic Motor 33.4 Conventional Controllers Used to Control the Electrohydraulic Axis PID, PI, PD with Filtering • Observer • Simulation Results of Electrohydraulic Axis with Conventional Controllers 33.5 Control of Electrohydraulic Axis with Fuzzy Controllers 33.6 Neural Techniques Used to Control the Electrohydraulic Axis Neural Control Techniques 33.7 Neuro-Fuzzy Techniques Used to Control the Electrohydraulic Axis C ontrol Structure 33.8 Software Considerations 33.9 Conclusions 33.1 Introduction Due to the development of technology in the last few years, robots are seen as advanced mechatronic systems which require knowledge from mechanics, actuators, and control in order to perform very complex tasks. Different kinds of servo-systems, especially electrohydraulic, could be met at the executive level of the robots. Taking into account the most advanced control approaches, this paper deals with the implementation of advanced controllers besides conventional ones which are used in an electrohydraulic system. The considered electrohydraulic system is one of the axes of a robot. These robots possess three or more electrohydraulic axes, which are identical with the axis studied in this chapter. An electrohydraulic axis whose mathematical model (MM) is described in this chapter presents a multitude of nonlinearities. Conventional controllers are becoming increasingly inappropriate to control the systems with an imprecise model where many nonlinearities are manifested. Therefore, advanced techniques such as neural networks and fuzzy algorithms are deeply involved in the control of such systems. Neural networks, initially proposed by McCulloch and Pitts, Rosenblatt, Widrow, had several Florin Ionescu University of Applied Sciences Crina Vlad Politeknica University of Bucharest Dragos Arotaritei Aalborg University Esbjerg ©2002 CRC Press LLC . least ±60° phase margin. The above gain margins apply to simultaneous and independent gain perturbations when the loop is broken at the output. The same holds for the above phase margins. The above. 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. 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