Using Stokes’s theorem, one has or The electromagnetic torque T acts on the infinitesimal current loop in a direction to align the magnetic moment m with the external field B, and if m and B are misaligned by the angle θ , we have The incremental potential energy and work are found as Using the electromagnetic force, we have and Coordinate Systems and Electromagnetic Field The transformation from the inertial coordinates to the permanent-magnet coordinates is We use the transformation matrix If the deflections are small, we have T i dA ∇ rB⋅()B ∇ r×()dA⋅ s ∫ ° –× s ∫ ° idAB× s ∫ ° == T iAB× mB×== TmBqsin= dW dΠ T dq mB q dq and Wsin Π mB qcos– mB⋅–== = == = dW dΠ– F dr⋅∇Πdr⋅–== = F ∇Π– ∇ mB⋅()m ∇⋅()B== = r T r r = q y q z coscos q y q z sincos q y sin– q x q y q z q x q z sincos–cossinsin q x q y q z q x q z sincos+sinsinsin q x q y cossin q x q y q z q x sin q z sin+cossincos q x q y q z q x q z cossin–sinsincos q x q y coscos x y z = r x y z , r x y z == T r = q y q z coscos q y q z sincos q y sin– q x q y q z q x q z sincos–cossinsin q x q y q z q x q z sincos+sinsinsin q x q y cossin q x q y q z q x q z sinsin+cossincos q x q y q z q x q z cossin–sinsincos q x q y coscos T rs 1 q z −q y −q z 1 q x q y q– x 1 = 0066_Frame_C20.fm Page 122 Wednesday, January 9, 2002 1:47 PM ©2002 CRC Press LLC Using Stokes’s theorem, one has or The electromagnetic torque T acts on the infinitesimal current loop in a direction to align the magnetic moment m with the external field B, and if m and B are misaligned by the angle θ , we have The incremental potential energy and work are found as Using the electromagnetic force, we have and Coordinate Systems and Electromagnetic Field The transformation from the inertial coordinates to the permanent-magnet coordinates is We use the transformation matrix If the deflections are small, we have T i dA ∇ rB⋅()B ∇ r×()dA⋅ s ∫ ° –× s ∫ ° idAB× s ∫ ° == T iAB× mB×== TmBqsin= dW dΠ T dq mB q dq and Wsin Π mB qcos– mB⋅–== = == = dW dΠ– F dr⋅∇Πdr⋅–== = F ∇Π– ∇ mB⋅()m ∇⋅()B== = r T r r = q y q z coscos q y q z sincos q y sin– q x q y q z q x q z sincos–cossinsin q x q y q z q x q z sincos+sinsinsin q x q y cossin q x q y q z q x sin q z sin+cossincos q x q y q z q x q z cossin–sinsincos q x q y coscos x y z = r x y z , r x y z == T r = q y q z coscos q y q z sincos q y sin– q x q y q z q x q z sincos–cossinsin q x q y q z q x q z sincos+sinsinsin q x q y cossin q x q y q z q x q z sinsin+cossincos q x q y q z q x q z cossin–sinsincos q x q y coscos T rs 1 q z −q y −q z 1 q x q y q– x 1 = 0066_Frame_C20.fm Page 122 Wednesday, January 9, 2002 1:47 PM ©2002 CRC Press LLC IV Systems and Controls 21 The Role of Controls in Mechatronics Job van Amerongen Introduction • Key Elements of Controlled Mechatronic Systems • Integrated Modeling, Design and Control Implementation • Modern Examples of Mechatronic Systems in Action • Special Requirements of Mechatronics that Differentiate from “Classic” Systems and Control Design 22 The Role of Modeling in Mechatronics Design Jeffrey A. Jalkio Modeling as Part of the Design Process • The Goals of Modeling • Modeling of Systems and Signals 23 Signals and Systems Momoh-Jimoh Eyiomika Salami, Rolf Johansson, Kam Leang, Qingze Zou, Santosh Devasia, and C. Nelson Dorny Continuous and Discrete-Time Signals • z Transform and Digital Systems • Continuous- and Discrete-Time State-Space Models • Transfer Functions and Laplace Transforms 24 State Space Analysis and System Properties Mario E. Salgado and Juan I. Yuz Models: Fundamental Concepts • State Variables: Basic Concepts • State Space Description for Continuous-Time Systems • State Space Description for Discrete-Time and Sampled Data Systems • State Space Models for Interconnected Systems • System Properties • State Observers • State Feedback • Observed State Feedback 25 Response of Dynamic Systems Raymond de Callafon System and Signal Analysis • Dynamic Response • Performance Indicators for Dynamic Systems 26 The Root Locus Method Hitay Özbay Introduction • Desired Pole Locations • Root Locus Construction • Complementary Root Locus • Root Locus for Systems with Time Delays • Notes and References 27 Frequency Response Methods Jyh-Jong Sheen Introduction • Bode Plots • Polar Plots • Log-Magnitude Versus Phase plots • Experimental Determination of Transfer Functions • The Nyquist Stability Criterion • Relative Stability 28 Kalman Filters as Dynamic System State Observers Timothy P. Crain II The Discrete-Time Linear Kalman Filter • Other Kalman Filter Formulations • Formulation Summary and Review • Implementation Considerations ©2002 CRC Press LLC In the initial conceptual design phase it has to be decided which problems should be solved mechan- ically and which problems electronically. In this stage decisions about the dominant mechanical properties have to be made, yielding a simple model that can be used for controller design. Also a rough idea about the necessary sensors, actuators, and interfaces has to be available in this stage. When the different partial designs are worked out in some detail, information about these designs can be used for evaluation of the complete system and be exchanged for a more realistic and detailed design of the different parts. Although the word mechatronics is new, mechatronic products have been available for some time. In fact, all electronically controlled mechanical systems are based on the idea of improving the product by adding features realized in another domain. Good mechatronic designs are based on a real systems approach . But mostly, control engineers are confronted with a design in which major parameters are already fixed, often based on static or economic considerations. This prohibits optimization of the system as a whole, even when optimal control is applied. In the last days of gramophones, the more sophisticated designs used tacho feedback in combination with a light turntable to achieve a constant number of revolutions. But a really new design was the compact disc player. Instead of keeping the number of revolutions of the disc constant, it aims for a constant speed of the head along the tracks of the disc. This means that the disc rotates slower when tracks with a greater diameter are read. The bits read from the CD are buffered electronically in a buffer that sends its information to the DA converter, controlled by a quartz crystal. This enables the realization of a very constant bit rate and eliminates all audible speed fluctuations. Such a performance could never be obtained from a pure mechanical device only, even if it were equipped with a good speed control system. In fact, the control loop for the disc speed does not need to have very strict specifications. It should only prevent overflow or underflow of the buffer. The high accuracy is obtained in an open loop mode, steered by a quartz crystal (Fig. 21.3). The flexibility introduced by the combination of precision mechanics and electronic control has allowed the development of CD-ROM players, running at speeds more than 50 times faster than the original audio CDs. A new way of thinking was necessary to come to such a new solution. On the other hand, the CD player is still a sophisticated piece of precision mechanics. No solid-state electronic memory FIGURE 21.1 Optimization of the controller. FIGURE 21.2 Optimization of the all system components simultaneously. ©2002 CRC Press LLC . q z −q y −q z 1 q x q y q– x 1 = 0066_Frame_C20.fm Page 122 Wednesday, January 9, 20 02 1:47 PM 20 02 CRC Press LLC IV Systems and Controls 21 The Role of Controls in Mechatronics Job van Amerongen . q x q y coscos T rs 1 q z −q y −q z 1 q x q y q– x 1 = 0066_Frame_C20.fm Page 122 Wednesday, January 9, 20 02 1:47 PM 20 02 CRC Press LLC Using Stokes’s theorem, one has or The electromagnetic torque. solid-state electronic memory FIGURE 21 .1 Optimization of the controller. FIGURE 21 .2 Optimization of the all system components simultaneously. 20 02 CRC Press LLC