电子工程师手册（英文版六）：ch104

Cook, G.E., Anderson, K., Barnett, R.J., Wallace, A.K., Spée, R., Sznaier, M., Sánchez Pe?a, R.S. “Industrial Systems” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 104 Industrial Systems 104.1 Welding and Bonding Control System Requirements ? System Parameters ? Welding System ? Sensing ? Modeling ? Control ? Conclusions 104.2 Large Drives Configurations ? Selection and Compatibility ? Principles and Features of Operation ? Control Aspects ? Future Trends 104.3 Robust Systems Robustness and Feedback ? Robust Stability and Performance ? H ∞ Control ? Structured Uncertainty ? Robust Identification 104.1 Welding and Bonding George E. Cook, Kristinn Andersen, and Robert Joel Barnett Most welding processes require the application of heat or pressure, or both, to produce a bond between the parts being joined. The welding control system must include means for controlling the applied heat, pressure, and filler material, if used, to achieve the desired weld microstructure and mechanical properties. Welding usually involves the application or development of localized heat near the intended joint. Welding processes that use an electric arc are the most widely used in industry. Other externally applied heat sources of importance include electron beams, lasers, and exothermic reactions (oxyfuel gas and thermit). For fusion welding processes, a high energy density heat source is normally applied to the prepared edges or surfaces of the members to be joined and is moved along the path of the intended joint. The power and energy density of the heat source must be sufficient to accomplish local melting. Control System Requirements Insight into the control system requirements of the different welding processes can be obtained by consideration of the power density of the heat source, interaction time of the heat source on the material, and effective spot size of the heat source. A heat source power density of approximately 10 3 W/cm 2 is required to melt most metals [Eagar, 1986]. Below this power density the solid metal can be expected to conduct away the heat as fast as it is being introduced. On the other hand, a heat source power density of 10 6 or 10 7 W/cm 2 will cause vaporization of most metals within a few microseconds, so for higher power densities no fusion welding can occur. Thus, it can be concluded that the heat sources for all fusion welding processes lie between approximately 10 3 and 10 6 W/cm 2 heat intensity. Examples of welding processes that are characteristic of the low end of this range include oxyacetylene welding, electroslag welding, and thermit welding. The high end of the power density range of welding is occupied by laser beam welding and electron beam welding. The midrange of heat source power densities is filled in by the various arc welding processes. For pulsed welding, the interaction time of the heat source on the material is determined by the pulse duration, whereas for continuous welding the interaction time is proportional to the spot diameter divided by the travel speed. The minimum interaction time required to produce melting can be estimated from the relation for a planar heat source given by [Eagar, 1986] George E. Cook Vanderbilt University Kristinn Andersen Marel Corporation Robert Joel Barnett Vanderbilt University Alan K. Wallace Oregon State University René Spée Oregon State University Mario Sznaier Pennsylvania State University — University Park Ricardo S. Sánchez Pe?a University of Buenos Aires Argentina ? 2000 by CRC Press LLC ADVANCED WELDING TORCH he concept of variable polarity plasma arc (VPPA) welding employs a variable current waveform that enables the welding system to operate for preset time increments in either of two polarity modes for most effective joining of troublesome light alloys such as aluminum and magnesium. Although the VPPA concept dates back to 1947, it was never fully developed. In the late 1970s, when the Space Shuttle was in early development, NASA recognized that the then- existing welding techniques were inadequate for the job of joining the huge aluminum segments of the Space T ? 2000 by CRC Press LLC t m = [K/p d ] 2 where p d is the heat source density (watts per square centimeter) and K is a function of the thermal conductivity and thermal diffusivity of the material. For steel, Eagar gives K equal to 5000 W/cm 2 /s. Using this value for K, one sees that the minimum interaction time to produce melting for the low power density processes, such as oxyacetylene welding with a power density on the order of 10 3 W/cm 2 , is 25 s, while for the high energy density beam processes, such as laser beam welding with a power density on the order of 10 6 W/cm 2 , is 25 ms. Interaction times for arc welding processes lie somewhere between these extremes. An example of practical process parameters for a continuous gas tungsten arc weld (GTAW) are 100 A, 12 V, and travel speed 10 ipm (4.2 mm/s). The peak power density of a 100-A, 12-V gas tungsten arc with argon shielding gas, 2.4-mm diameter electrode, and 50-degree tip angle has been found to be approximately 8 3 10 3 W/cm 2 . Shuttle External Tank. Marshall Space Flight Center (MSFC) initiated the development of VPPA welding. The B&B Precision Machine Variable Polarity Plasma Arc welding torch. (Photo courtesy of National Aeronautics and Space Administration.) In the course of its development, it became apparent that the technique had broad potential for improving weld reliability and lowering costs. Since there were no suitable commercially available tools for VPPA welding, MSFC expanded its development effort to include a welding torch that would have dual utility, as a component of NASA’s external tank welding system and as a component of derivative systems for com- mercial applications. The torch contract was awarded to B&B Precision Machine, Owens Cross Road, Alabama. B&B, working in cooperation with MSFC’s Materials and Processing Laboratory, developed and patented a shuttle-use torch and continued development of VPPA. A major step in the late 1980s was a program to fully automate the system and eliminate the hand of the welder on the controls. In 1989, a NASA decision to change the material of the external tank triggered a ? 2000 by CRC Press LLC Assuming an estimated spot diameter of 4 mm, the interaction time (taken here as the spot diameter divided by the travel speed) is 0.95 s. At the other extreme, 0.2-mm (0.008-in.) material has been laser welded at 3000 in./min (1270 mm/s) at 6 kW average power. Assuming a spot diameter of 0.5 mm, the interaction time is 3.94 3 10 -4 s. Spot diameters for the high density processes vary typically from 0.2 mm to 1 mm, while the spot diameters for arc welding processes vary from roughly 3 mm to 10 mm or more. Assuming a rule of thumb of 1/10 the spot diameter for positioning accuracy, we conclude that typical positioning accuracy requirements for the high power density processes is on the order of 0.1 mm and for the arc welding processes is on the order of 1 mm. The required control system response time should be on the order of the interaction time and, hence, may vary from seconds to microseconds, depending on the process chosen. With these requirements it can be concluded that the required accuracy and response speed of control systems designed for welding increases as the power new B&B development. The new alloy in some cases required “tack” welds prior to robotic seam welding. Since tack welds are performed by hand, B&B was assigned to develop a smaller version of the torch that would be easier to manipulate and would meet the needs of applications where access was limited. B&B delivered a prototype small torch in 1992. The small torch has the same features and advantages as the original torch, but it fits in approximately half the space. The VPPA welding system and the B&B torch continue to make all the welds in the external tank of the Space Shuttle and they have been selected as the preferred welding approach for the International Space Station. (Courtesy of National Aeronautics and Space Administration.) A small version of the B&B torch is used in commercial sheet metal welding. (Photo courtesy of National Aeronautics and Space Administration.) density of the process increases. Furthermore, it is clear that the high power density processes must be automated because of the human’s inability to react quickly and accurately enough. System Parameters The variables of the welding process are separated here into direct weld parameters (DWP) and indirect weld parameters (IWP) [Cook, 1981]. The DWP are those pertaining to the weld reinforcement and fusion zone geometry, mechanical properties of the completed weld, weld microstructure, and discontinuities. The IWP are those input variables that collectively control the DWP. The IWP are the welding equipment setpoint variables, e.g., voltage, current, travel speed, electrode feed rate, travel angle, electrode extension, focused spot size, and beam power. Welding System The various DWP, or process variables, that we would like to control and the many possible IWP, or equipment variables, that we may set to achieve the desired output are shown in Fig. 104.1. From the standpoint of feedback control, the welding process depicted in Fig. 104.1 presents two principal problems: (1) in most cases the relationships between the IWP and DWP are nonlinear, and (2) the variables are generally highly coupled. With most production welding today, the designer of the welded part specifies the desired weld characteristics (the DWP), including acceptable tolerance windows. The job of the welding engineer then is to determine a set of IWP that will produce the desired DWP. Most automated welding systems today may be expected to have good control over the IWP, including joint tracking for heat source positioning. Therefore, if production floor conditions do not differ too much from the laboratory conditions under which the weld procedures were developed, then the welding operation can be expected to satisfy quality inspection and control procedures. If not, human operators must be depended upon to provide the necessary feedback to make corrective actions in the welding equipment settings. The human involvement in this scenario can be reduced or eliminated by sensing selected DWP, comparing the sensed variables with desired values, and implementing a multivariable controller that will reduce auto- matically the error between the desired and sensed DWP to zero or an acceptably low difference. Dynamic and steady-state process models are required for both design and stable operation of the multivariable feedback control system. However, the models do not need to be as globally accurate as the models required for open- loop control. In exchange for accuracy, the models used for control system purposes must be computable in real time, and generally, it is important that they provide both steady-state and dynamic information of the interrelationships between the coupled variables of the system. It is generally important that these relationships FIGURE 104.1 Input and output variables of welding process. ? 2000 by CRC Press LLC be “tunable” in real time to permit calibrating the multivariable system controller to the actual operating conditions at any given time. Successful implementation of multivariable weld process control involves (1) sensing, (2) modeling, and (3) control. Issues dealing with each of these will be discussed in the following sections. Sensing In recent years, great strides have been made in sensor technology, particularly in the areas of optical sensors, arc sensing, and infrared, acoustic, and ultrasonic sensing. Optical Sensing Optical sensing technology has been developed and used for a number of applications, including joint tracking and fill control, sensing of molten pool width, sensing of weld bead profile, arc length sensing and control, sensing and control of electrode extension in gas metal arc welding (GMAW), and sensing of weld depth or penetration. Yi [1991] and Barnett [1993] have investigated the ability to estimate GTA weld penetration by means of measuring the weld pool vibration frequency. Yi and Barnett used the reflection of the welding arc from the weld pool surface as a means of sensing the weld pool vibration. Digital signal processing was used to estimate the oscillation frequency of the weld pool from the sensed optical signal. References to other work dealing with weld pool vibration sensing and analysis may be found in Yi [1991] and Barnett [1993]. Other potential applications include sensing of proper fusion characteristics at the sidewalls, detection of surface contaminants, and sensing of metal transfer mode in GMAW. Liu [1991] has demonstrated that the droplet rate in GMAW can be extracted from the arc infrared signal by means of power spectral estimation. Liu establishes the relationships between the metal droplet rate and the welding parameters, arc voltage, arc current, wire-feed speed, and the contact tube-to-workpiece distance (CTWD). Liu proposes a PC-based digital control system for controlling the metal droplet rate in GMAW. One of the first real-time optical tracking systems, and certainly one of the more novel approaches, was a coaxial viewing system developed by Richardson [Richardson et al., 1984]. With this approach, the imaging system is integrated into the welding torch. The point of welding is viewed coaxially with the welding electrode from within the welding torch. Advantages reported for this system of viewing include (1) the bright core of the arc is blocked by the electrode/contact tip, (2) the entire weld area can be viewed without obstruction and without distortion by the viewing angle, and (3) the system is nonintrusive into the weld area and is nondi- rectional. A number of optical tracking systems make use of a projected laser strip or a scanned laser beam to provide structured lighting that permits three-dimensional profiling of the joint, typically in front of the heat source. Several such tracking systems are commercially available and offer robust solutions to the joint tracking problem. A viewing system that provides remarkably good images of the electrode and molten pool area has been developed from laser and night imaging technology. The system’s operation is based on the use of a high- intensity pulsed laser or strobe light synchronized with an image intensifier and camera to suppress the arc light and produce a clear view of the arc area. The excellent image obtained with this system offers a great deal of potential for various types of optical process sensing requirements. Arc Sensing Arc sensing (or through-the-arc sensing) has many applications, some, such as automatic voltage control, dating back 30 years or more. The obvious advantage of arc sensing is that use of the arc itself as a sensor means there is not any need for external sensors, with the associated concern for their reliability in the harsh environment of the welding arc. One of the most widely reported recent applications of arc sensing is for purposes of vertical and lateral tracking and width control [Cook, 1983]. For this application, the sensing method is based on the changes in current and/or voltage when the arc is weaved back and forth across the joint. Inventions have been disclosed for both nonconsumable arc welding processes and consumable arc welding processes (see references in Cook et al., 1990). Applications range from pipe welding to robotic arc welding to turbine blade repair. For submerged ? 2000 by CRC Press LLC arc welding (SAW), for example, current variations of approximately 10% at the sidewalls have been observed while welding in a joint consisting of a 45-degree included angle with a 5-mm root opening. With a nominal current of 580 A at the center of the joint, the current at the sidewalls is approximately 640 A. Variations of this magnitude may be used to implement robust control algorithms for joint tracking and width control. Shepard [1991] presents a thorough treatment of the mechanisms that establish and influence self-regulation in GMAW. Components of a dynamic GMAW process model are identified, including the power source, joule heating in the electrode, electrode burn-off rate, and arc voltage. A numerical simulation of the nonlinear dynamic model for self-regulation is implemented, computing current I and electrode extension in response to CTWD, voltage, and feed rate. The I/CTWD response is shown to be frequency dependent, increasing significantly at higher frequencies. The frequency at which the response increases is shown to be primarily dependent on electrode current density, occurring at lower frequencies for lower current densities. A linearized closed-form model for the I/CTWD frequency response is derived from the simulation equations and is shown to provide accurate results. The closed-form model clearly indicates the relationships between the model parameters that establish the observed characteristics of self-regulation dynamics. Initial implementations of through-the-arc seam tracking methods use simple current levels to identify the lateral limits of the weld joint, adjusting the torch centerline to maintain symmetry. The dynamic model developed by Shepard provides a basis to infer actual joint geometry from position and current information acquired during cross-seam oscil- lation. The relationships developed by Shepard also refine the basis for selection of welding procedures in GMAW applications, particularly for through-the-arc sensing applications. The models define the relationships to generate surfaces to facilitate selection of electrode diameter, feed rate, voltage, electrode extension, and CTWD to optimize desired characteristics such as low-frequency sensitivity, high-frequency sensitivity, and transition frequency subject to requirements on heat input and deposition rate. These interrelationships may be used as extensions to existing expert systems for selection of welding procedures. Arc sensing has been proposed as a means of sensing GTA pool motion after excitation from pulsations in the current. The concept of using weld pool motion as a pool geometry sensing method is based on the fluid dynamics of the constrained weld pool, which depend on the properties of the molten pool material, the surface tension, and the shape of the pool. Another potential application of arc sensing is detection of the metal-transfer mode in GMAW. The droplet transfer mode in the GMAW process has a large effect on weld pool metallurgy, influencing penetration, solidification, heat flow, and mass input. Researchers have attempted to correlate perturbations in the electrical arc signals with droplet transfer. This work has demonstrated the ability to detect the detachment of individual droplets and to distinguish among the three transfer modes: globular, spray, and streaming, as defined by Lancaster [1986]. Measurements of the incremental arc resistance by Shepard [1991] suggest that the metal-transfer mode of the gas metal arc may also be detected by the rapid transitions of the incremental arc resistance at the transition regions of metal transfer (particularly at the spray-to-streaming transition). The incremental resistance was obtained by perturbing the voltage with a 1-V p-p , 15-Hz sinusoidal variation. In the arc resistance measurements, CTWD and electrode extension (and hence arc length) were held constant and data were taken over a wide range of current. A nominal CTWD of 25.4 mm was used, with a 15-mm electrode extension. Feed rate was varied from the globular/spray transition point to the upper ranges imposed by equipment limitations. A small (1 V p – p ), “high-frequency” (15 Hz) sinusoidal perturbation was superimposed on the power source voltage to allow measurement of the incremental resistance at each operating point. The frequency was sufficiently high that the electrode extension did not vary significantly. For each data point, an 8-s record was acquired at 1- kHz sample rate. The frequency response function (FRF) was used to compute the incremental resistance by calculating the current produced in response to the sinusoidal voltage perturbation. The FRF gives the magni- tude and phase angle of a linear model of the arc V-I characteristic about the given operating point, making up the total resistance of arc plus electrode. Results of the incremental arc resistance measurements were plotted as a function of current. The most significant feature of these data was the large peak in incremental resistance in the region of the projected/streaming transition. The height of the peak is roughly twice the nominal resistance at higher currents. The incremental arc resistance increases sharply at the upper end of projected transfer mode, peaking just after the transition to decline to a relatively steady level through the upper end of the streaming transfer range. ? 2000 by CRC Press LLC For weld procedures that include cross-seam oscillation, or weaving, of the heat source, arc sensing provides a reliable indicator of sidewall/adjacent bead fusion. As the sidewall or adjacent bead is approached in the weave cycle, the electrical signals change in response to the change in CTWD for GMAW or arc gap for GTAW. This change is, of course, the signal used for tracking control in through-the-arc tracking; however, it provides a useful indicator of proper penetration into the sidewall or adjacent bead independently of whether arc sensing is used for tracking purposes. Andersen et al. [1989] have reported the use of arc signal parameters as a potential control means for GMAW, short-circuiting transfer. Digital signal processing was used to extract from the electrical signals various features, including average and peak values of voltage and current, short- circuiting frequency, arc period, shorting period, and the ratio of the arcing to shorting period. Additionally, a joule heating model was derived that accurately predicted the melt-back distance during each short. The ratio of the arc period to short period was found to be a good indicator for monitoring and control of stable arc conditions. Any change in the arcing voltage, for a given power circuit condition, leads to corresponding changes in the arcing/shorting time ratio. Such changes in arcing voltage may occur with change in the shielding gas, in the surface condition (in the form of contaminates) of the electrode wire and work, and in their composition, such as the presence of rare earths, in the wire electrode or work materials that affect the arc characteristics. Andersen et al. [1989] show that if the average arc current may be assumed nominally constant because of constant electrode feed, then the arcing/shorting time ratio serves as a sensitive index of the operation of the GMAW short-circuiting system. The arcing/shorting time ratio can be used to control the short-circuiting gas metal arc in a feedback loop by adjusting the open circuit voltage to compensate for variations in the arcing voltage. Finally, the electrical arc signals vary as a function of contaminants on the workpiece and/or electrode, and these variations may be sensed and correlated with the changes observed in surface conditions. Infrared Sensing Infrared sensing has inherent appeal for weld sensing. Potential applications include cooling rate measurements, discontinuity sensing, penetration estimation, seam tracking, and weld pool geometry measurement. Acoustical Sensing The acoustical signals generated by the welding arc are a principal source of feedback for manual welders. Recently, acoustical signals have been studied as a sensing means for automated welding as well. Sound generated by the electric arc of a gas tungsten arc weld has been used for arc length control. With this system the current is pulsed a small amount at an audible rate to generate an audible tone at the arc. The intensity of the arc-generated tone has been shown to be proportional to the arc length and, hence, can be suitably processed to provide a feedback signal for arc length control. Acoustical signals generated by gas metal arcs have been correlated with the detachment of individual droplets from the filler wire. Research has demonstrated the ability to detect the detachment of individual droplets and to distinguish among transfer modes: globular, spray, and streaming transfer. This may lead to a means of closed-loop control of the heat and mass input during both pulsed and nonpulsed GMAW. Acoustical signals have also been reported as a means of plasma monitoring in laser beam welding (LBW). Specifically, experiments have been conducted to characterize the interaction between the incident laser light, the plasma formation, and the target material during pulse welding with an Nd:YAG laser. In the experiments, the acoustical signal, picked up by a microphone, was used to signal plasma initiations and propagation. A correlation was observed between the number of plasmas generated and the weld pool penetration in a target. Acoustic emission has been used for monitoring LBW in real time. The acoustic sensor has been reported to detect laser misfiring, loss of power, improper focus, and excess root opening. Ultrasonic Sensing The use of ultrasonics for weld process sensing has the potential to detect weld pool geometry and discontinuities in real time. However, to be useful in realistic production systems, a means must be developed for injecting the ultrasound and receiving it with noncontacting sensors. Lasers have been proposed as a sound source, and electromagnetic acoustic transducers (EMATs) have been proposed for ultrasound reception. With this proposed ? 2000 by CRC Press LLC approach, the pulsed laser is directed to impinge on the molten pool, setting up stress waves that are transmitted through the workpiece and picked up by the EMAT receiver. Modeling Weld process models intended for control purposes are characterized by the need to be computable in real time. This rules out many of the more exact numerical models that have been developed for finite element and finite difference methods. However, these computationally intensive numerical models may be quite useful in developing simpler models that can be used in the control of multivariable weld feedback control systems. Another important aspect of process models used for control purposes is that they generally need to provide both static and dynamic information. Analytical Models Since the 1940s considerable research has been focused on developing steady-state models that would predict DWP, given a set of IWP. Easily computed analytical models, based solely on conductive heat transfer, are reasonably accurate but primarily are of value in establishing approximate relationships. Improvements to these early analytical models have been proposed that permit obtaining a better match to actual conditions and that may be calibrated in real time; however, accuracy remains limited in the absence of modeling extensions that require computationally intensive numerical solution. Empirical and Statistical Models Other approaches taken to developing steady-state weld process models include: empirically derived relation- ships between the IWP and DWP, with coefficients chosen to match experimental data and statistically derived relationships. Both of these approaches have proven to possess only a limited range of applicability, and they do not lend themselves to real-time “tuning” in a multi-variable control system application. Artificial Neural Network Models A promising method based on an artificial neural network (ANN) has been studied and found to be accurate and computationally fast in the application mode. Furthermore, the ANN can be refined at any time with the addition of new training data and thus promises a method of continuously adapting to the actual welding conditions. Andersen [1992] has reported the application of an ANN to mapping between the IWP’s arc current, travel speed, arc length, and plate thickness and the DWP’s bead width and penetration for GTAW. A back-propa- gation network, using 10 nodes in a single hidden layer (Fig. 104.2), was used for the modeling. A variety of different network configurations were initially evaluated for this purpose. Generally, it was found that one hidden layer was sufficient for weld modeling, and the best training rate was obtained with on the order of 5 to 20 nodes in the hidden layer. The same plate material was assumed throughout the experiment, which eliminated the need for specifying any of the material parameters. Otherwise, additional input parameters might have included thermal conductivity, diffusivity, etc. A total of 72 welds, produced on two material thick- nesses of 3.175 and 6.350 mm, were used for the purpose of training and testing the network for modeling purposes. Weld current values of 80, 100, 120, and 140 A, travel speeds of 2.12, 2.75, and 3.39 mm/s, and arc lengths of 1.52, 2.03, and 2.54 mm were used. Eight of the welds, which were randomly selected, were not used in the training phase but were reserved for testing the model. With a learning rate parameter of 0.6 and a momentum term of 0.9, the network was trained for 200,000 iterations. FIGURE 104.2A neural network used for weld modeling. ? 2000 by CRC Press LLC Once the network had been trained with the 64 training welds, the remaining 8 welds were applied to test the modeling network. The root mean square (RMS) values of the errors were calculated separately for the bead width and penetration, resulting in about 5% and 18% RMS errors, respectively. These results agree with other similar experiments reported by Andersen, in that modeling accuracy is typically on the order of 10-20%. Weld modeling studies have also been carried out on the variable polarity plasma arc welding (VPPAW) process. Modeling of the crown and root width in the keyhole welding mode was of specific interest, and the model inputs were the forward and reverse current values, the torch standoff distance, and the travel speed. The crown and root width errors of the model were generally determined to be on the order of 10–20% or better. An observation relating to the weld modeling experiments should be noted here. The precision of the bead measurements was 0.1 mm, which corresponds to 2 and 7% precision for the average bead width and pene- tration, respectively. Furthermore, inaccuracies in measurements of the data, which were used to train the neural network model, tend to degrade the general performance of the model. Width measurements are generally more reliable than penetration measurements, as they are made in several locations along the top of the bead. A penetration measurement is usually made on a single cross section, and it requires chemical etching, which results in a relatively blurred boundary between the bead and the surrounding base metal. This difference is reflected in the consistently lower accuracy of the penetration modeling, compared with the width modeling. A back-propagation network was also constructed by Andersen [1992] to model the inverse relations, i.e., the DWP-IWP relations, of the weld sample set used in the forward modeling study. A number of neural network configurations were initially used in attempting to train networks to determine the necessary current, travel speed, and arc length for desired bead width and penetration. Preliminary attempts did not result in acceptable training convergence. Closer examination revealed that welds which resulted in full or almost full penetration yielded very irregular bead measurements. It was hypothesized that these irregularities might contribute to the poor training performance. These welds (total of five), which represented the largest pool dimensions on the 3.175-mm test plate, were removed from the training data, and to maintain an equally large data set for the 6.350-mm plate, the five largest welds were ignored there as well. Six welds were randomly selected from the remaining data for each plate thickness for testing only. Using the revised data set, a network of 50 nodes in a single hidden layer was successfully trained. The learning rate was 0.6, the momentum term was 0.9, and the network was trained for 300,000 iterations. The equipment parameters, or IWP, suggested by the neural network were compared with the actual parameters used to produce the test welds. The RMS deviations between these were current, 9.7%; travel speed, 23.9%; and arc length, 25.5%. Although these deviations between the IWP used to produce the original training set and the IWP suggested by the ANN are rather large, the results are not unexpected because of the nonuniqueness of the inverse problem. The results do not imply that the resulting bead geometries would be accordingly erroneous, because a given width-penetration pair may be attained through multiple nonunique combinations of equipment parameters. For example, an arc current increase may be largely offset by a corresponding increase in travel speed. To assess the reliability of the ANN for equipment parameter selection, the parameters suggested by the inverse model were used to produce a new set of welds, and bead width and penetration measurements were carried out as before. These widths and penetrations were compared with the original data set. The RMS errors were width, 5.5%, and penetration, 19.9%. These differences between the new geometry parameters and the original ones are approximately the same as the errors observed from the weld model. Again, it is suggested that uncertainty in bead measurements contributes significantly to these errors. When compared to other control modeling methodologies, neural networks have certain drawbacks as well as advantages. Of the drawbacks, the most notable is the lack of comprehension of the physics of the process. Relating the qualitative effects of the network structure or parameters to the process parameters is usually impossible. On the other hand, other control modeling methods resort to substantial simplifications of either the physical process or more exact numerical models and therefore also trade computability for comprehensi- bility. The advantages of neural models include relative accuracy and generality. If the training data for a neural network is general enough, spanning the entire ranges of process parameters, the resulting model will capture the complexion of the process, including nonlinearities and parameter cross couplings, over the same ranges. Model development is much simpler than for most other models. Instead of theoretical analysis and develop- ment for a new model, the neural network tailors itself to the training data. The network can be refined at any ? 2000 by CRC Press LLC time with addition of new training data. Finally, the neural network can calculate its results relatively quickly, as the input data are only propagated once through the network in the application mode. The reader is referred to Andersen [1992] for a more thorough discussion of the neural network approach to weld process modeling. Andersen also presents a detailed comparison of neural network modeling to two analytical models and a statistically based multidimensional parameter interpolation approach. Control Practical Considerations The easiest approach to controlling multiple weld process parameters can be realized if input variables can be found that affect only a single output quantity. If the output variable is affected by another input variable as well, then one may be the primary variable while the other may constitute a secondary feedback loop that is capable of controlling the output quantity by a relatively small amount with respect to the basic level set by the primary variable. For example, high-frequency pulsation of the current in GTAW may provide a means of controlling the depth of penetration over a small range without affecting the width of the weld bead. In this case the heat input, as determined by the voltage, current, and travel speed, would be the primary input variable controlling the width and penetration, while the high-frequency pulsation would be the secondary variable capable of producing small corrections to the basic penetration depth. Even for single-variable weld process control, nonlinearities in the process may call for an adaptive system to automatically adjust the parameters of the controller when the process parameters and disturbances are unknown or change with time. For example, Bjorgvinsson [1992] shows that a simple automatic voltage control (AVC) system may be unstable over a wide range of current settings because of the variation of the arc sensitivity (voltage change per unit change of arc length) with current. A simplified schematic of an AVC system is shown in Fig. 104.3. The arc voltage (proportional to the arc length) is compared with a reference voltage in a simple position servo. If an error exists between the reference voltage and the arc voltage, the servo motor moves the welding torch up or down to reduce the error to zero. If K a is the gain of the AVC motor drive system and K s is the arc sensitivity (K s = dV arc /dL arc ), then the overall loop gain K is given by K = K a K s . The closed-loop stability of the position control system is dependent on the loop gain and will obviously vary from its design setting if K s changes. Bjorgvinsson shows that for helium shielding gas, the arc sensitivity may vary by approximately a 5:1 ratio over a current range of 15 to 150 A. In this case, for a standard proportional controller, the overshoot to a step input at 15 A is approximately 40% if the controller gain K a is fixed and set for optimum response at 150 A. Bjorgvinsson proposes a gain-setting adaptive controller (see Fig. 104.4) to vary the controller gain in such a manner as to compensate for the changing arc sensitivity for all levels of welding current. Knowing the arc current, the adaptive controller uses information stored in a look-up table or computed from a mathematical FIGURE 104.3 Simplified gas tungsten arc welding setup. ? 2000 by CRC Press LLC model of the arc to adjust K a in response to changes in K s such that the product K a K s = K is maintained constant independent of the current. The result is uniform closed-loop stability characteristics of the AVC system throughout the complete weld. This includes the up-slope period, when the current is varied from the low arc- initiation value to the nominal welding current, which is maintained until the down-slope period, when the current is brought back to a low value for termination of the arc. General Approaches to Multivariable and Adaptive Weld Process Control The welding process is generally nonlinear, and the different variables are normally coupled. If we can assume localized linearity, then adaptive control techniques can be used to change the controller characteristics in response to changes in the operating domain. To handle the multivariable control problem, we attempt to decouple the process input–output variables by appropriate controller design in order to reduce the system to a set of essentially noninteracting loops. Controller design can then be carried out using single-loop techniques. Necessary and sufficient conditions have been derived for decoupling a multivariable system. Unfortunately, the conditions are, in general, unlikely to be satisfied in practice because of model approximations, measurement uncertainties, parameter perturbations, and other causes. Therefore, system decoupleability may be inhibited by constant compensation techniques. In these situations, it is more appropriate to decouple the system in real time using an adaptive controller. It has been shown that such an adaptive controller can be expected to eventually achieve exact decoupling after the system parameters have converged. A general multivariable adaptive direct weld process control system is shown in Fig. 104.5. It will frequently be the case that not all of the DWP that we wish to control can be directly sensed with available sensors. In this case, we may estimate the DWP(s) that we cannot measure and use the estimated values for feedback information. Control of these parameters will obviously not be any better than the model used to estimate them. However, the model may be continuously tuned, i.e., calibrated, from both the IWP and those DWP that are directly sensed. Cook et al. [1991] have described a multivariable weld process control system that makes use of a model to estimate one of two DWP(s) controlled. The system, shown in Fig. 104.6, was configured to accept weld bead width and weld penetration as its two inputs. The system used width sensing, but penetration was only available as an estimate from the forward process model acting in parallel to the actual process. Conventional time-based up-sloping/down-sloping was used for weld initiation and termination, so an inverse process model was used to provide initial weld IWP(s) (following up-slope) to the weld start sequencer. Referring to Fig. 104.6, the desired bead width and penetration are specified by the user as W o and P o , respectively. These parameters, as well as the workpiece thickness H, are routed to a neural network setpoint selector (inverse process model), FIGURE 104.4 Gain-setting adaptive automatic voltage control. ? 2000 by CRC Press LLC which produces the nominal travel speed, current, and arc length (v o , I o , and L o , respectively). Arc initiation and stabilization are controlled in an open-loop fashion by the weld start sequencer. Given the desired equip- ment parameters, the arc is typically initiated and established at a relatively low current, with the other equipment parameters set at some nominal values. Once the arc has been established, the equipment parameters are ramped to the setpoint values specified by the neural network. When the setpoint values have been reached, at time t = T, the closed-loop process control is enacted. As stated previously, the bead width from the process was monitored in real time, while a real-time penetration sensor was not used. Therefore, a second neural network (forward process model) is run in parallel with the process to yield estimates of the penetration. The measured bead width and the estimated penetration are subtracted from the respective reference values, processed through proportional-plus-integral controllers, and added to the final values obtained from the setpoint sequencer. When a workpiece thickness variation is encountered in the process, the system adjusts the current and the arc length accordingly to maintain constant bead geometry. To demonstrate the multivariable weld process control system Cook et al. report an experiment using mild steel for the workpiece material. Plates of two thicknesses, 3.175 and 6.35 mm, were joined together, and a bead-on-plate weld using the nominal parameters (I = 100 A, L arc = 2.54 mm, v = 2.54 mm/s) was made across the boundary between the plates, from the thicker section to the thinner one. The bead width and penetration FIGURE 104.5 Multivariable adaptive weld process control system. FIGURE 104.6 Closed-loop weld process control system. ? 2000 by CRC Press LLC were 3.6 and 0.9 mm, respectively, on the thicker plate. With the controller disabled (equipment parameters maintained constant), the bead width increased to 4.0 mm and the penetration increased to 1.2 mm when the weld pool entered the thinner plate. With the controller enabled, the width and penetration were maintained the same on the thin plate as they were on the thick plate with only a slightly discernible transient. Intelligent Control Practical weld process control implementation, particularly with multivariable and adaptive control, involves a substantial body of heuristic knowledge concerning the weld process and the numerous constraints that are involved in its control. The role that intelligent control concepts can play is to provide a systematic approach to dealing with these constraints. For example, for a given set of material parameters, one may wish to control several geometrical parameters plus cooling rate for the GMAW process, while maintaining operation in the spray transfer mode of the process. Because of the close coupling among the equipment, material, and geometric parameters, and because of the small latitude of permissible variation of one parameter once the others are specified, tight constraints on the control system will be necessary to achieve the desired process quality. It will be desirable to specify degrees of control permitted over the various parameters in terms of a hierarchy of parameter importance. For example, while the wire feed rate has an influence on bead width in the GTAW process, it would not be desirable to allow the wire feed rate to be varied excessively as a means of controlling bead width. Further, the allowable variation of a given parameter, or parameters, may not be symmetrical about the desired set point. Again, for the GTAW process, an increase in current may be partially offset by an increase in travel speed, whereas a reduction in both parameters would tend to more rapidly force the geometrical parameters outside the desired range. Consideration of the process dynamics is also necessary, particularly for successful control during the initiation and termination phases of the overall welding operation. In addition to the hierarchical considerations referred to above, the time sequence and rate of change of each parameter should be considered. Intelligent control concepts may be used to handle these practical control issues in a formal and logical manner. Conclusions Rapid advances have occurred in the development of sensors and in the development of both steady-state and dynamic models suitable for real-time weld process control applications. In combination with multivariable, adaptive control theory methods, the tools are becoming available for significant progress in multivariable, direct weld process control. Long-range efforts will focus on combining process modeling and microstructural evolution modeling for eventual control of both macro and micro parameters. Defining Terms Direct weld parameters (DWP): A collection of parameters that characterize the weld in terms of the weld reinforcement and fusion zone geometry, mechanical properties, weld microstructure, and discontinui- ties. Electron beam welding: A welding process that produces coalescence of metals with the heat obtained from a concentrated beam composed primarily of high-velocity electrons impinging on the surfaces to be joined. Electroslag welding: A welding process that produces coalescence of metals with molten slag that melts the filler metal and the surfaces of the parts to be joined. Gas metal arc welding (GMAW): A welding process that produces coalescence of metals by heating them with an arc between a consumable filler metal electrode and the parts to be joined. The process is used with shielding gas and without the application of pressure. Gas tungsten arc welding (GTAW): A welding process that produces coalescence of metals by heating them with an arc between a nonconsumable tungsten electrode and the parts to be joined. The process is used with shielding gas and without the application of pressure. Filler metal may or may not be used. ? 2000 by CRC Press LLC Indirect weld parameters (IWP): A collection of parameters that establish the welding equipment setpoint values. Examples include voltage, current, travel speed, electrode feed rate, travel angle, electrode geom- etry, focused spot size, and beam power. Laser beam welding (LBW): A welding process that produces coalescence of materials with the heat obtained from the application of a concentrated coherent light beam impinging on the surfaces to be joined. Oxyacetylene welding: An oxyfuel gas welding process that produces coalescence of metals by heating them with a gas flame obtained from the combustion of acetylene with oxygen. The process may be used with or without the application of pressure and with or without the use of filler metal. Thermit welding: A welding process that produces coalescence of metals by heating them with superheated liquid metal from a chemical reaction between a metal oxide and aluminum, with or without the application of pressure. Variable polarity plasma arc welding (VPPAW): A welding process that produces coalescence of metals by heating them with a constricted variable polarity arc between an electrode and the parts to be joined (transferred arc) or between the electrode and the constricting nozzle (nontransferred arc). Shielding is obtained from the hot, ionized gas issuing from the torch as well as from a normally employed auxiliary shielding gas source. Pressure is not applied, and filler metal may or may not be added. Related Topics 56.1 Introduction ? 66.1 Generators References K. Andersen, Studies and Implementation of Stationary Models of the Gas Tungsten Arc Welding Process, M.S. Thesis, Vanderbilt University, 1992. K. Andersen, G. E. Cook, Y. Liu, D. S. Mathews, and M. D. Randall, “Modeling and control parameters for GMAW, short circuiting transfer,” in Advances in Manufacturing Systems Integration and Processes, D. A. Dornfeld, Ed., Dearborn, Mich.: Society of Manufacturing Engineers, 1989. R. J. Barnett, Sensor Development for Multi-parameter Control of Gas Tungsten Arc Welding, Ph.D. Thesis, Vanderbilt University, 1993. J. B. Bjorgvinsson, Adaptive Voltage Control in Gas Tungsten Arc Welding, M.S. Thesis, Vanderbilt University, 1992. G. E. Cook, “Feedback and adaptive control in automated arc welding systems,” Metal Construction, vol. 13, no. 9, pp. 551–556, 1981. G. E. Cook, “Robotic arc welding: Research in sensory feedback control,” IEEE Transactions on Industrial Electronics, vol. IE-30, no 3, pp. 252–268, 1983. G. E. Cook, K. Andersen, and R. J. Barnett, “Feedback and adaptive control in welding,” in Recent Trends in Welding Science and Technology, S. A. David and J. M. Vitek, Eds., Metals Park, Ohio: ASM International, 1990, pp. 891–903. G. E. Cook, K. Andersen, R. J. Barnett, and J. F. Springfield, “Intelligent gas tungsten arc welding control,” in Automated Welding Systems in Manufacturing, J. Weston, Ed., Cambridge, England: Abington Publishing, 1991. T. W. Eagar, “The physics and chemistry of welding processes,” in Advances in Welding Science and Technology, S. A. David, Ed., Metals Park, Ohio: ASM International, 1986, pp. 291–298. J. F. Lancaster, The Physics of Welding, New York: Pergamon Press, 1986. Y. Liu, Metal Droplet Rate Control for Gas Metal Arc Welding, Ph.D. Dissertation, Vanderbilt University, 1991. R. W. Richardson, A. Gutow, R. A. Anderson, and D. F. Farson, “Coaxial weld pool viewing for process moni- toring and control,” Welding Journal, vol. 63, no. 3, pp. 43–50, 1984. M. E. Shepard, Modeling of Self-Regulation in Gas-Metal Arc Welding, Ph.D. Dissertation, Vanderbilt University, 1991. Y. C. Yi, Weld Pool Vibration Analysis in Gas Tungsten Arc Welding, M.S. Thesis, Vanderbilt University, 1991. ? 2000 by CRC Press LLC Further Information Other recommended reading on welding technology, welding processes, and welding automation and control includes Welding Handbook, Volume 1—Welding Technology (American Welding Society, Miami, 1987), Welding Handbook, Volume 2—Welding Processes (American Welding Society, Miami, 1991), Advances in Welding Science and Technology (edited by S. A. David ASM International, Metals Park, Ohio, 1986), Recent Trends in Welding Science and Technology (edited by S. A. David and J. M. Vitek, ASM International, Metals Park, Ohio, 1990), Developments in Mechanised and Robotic Welding (edited by G. R. Salter, The Welding Institute, Cambridge, England, 1980), Modeling and Control of Casting and Welding Processes (edited by S. Kou and R. Mehrabian, The Metallurgical Society, Inc., Warrendale, Penn.), Developments in Automated and Robotic Welding (edited by D. N. Waller, The Welding Institute, Cambridge, England, 1987), Developments and Innovations for Improved Welding Production (The Welding Institute, Cambridge, England, 1983), Automated Welding Systems in Man- ufacturing (Abington Publishing, Cambridge, England, 1991), and Robotic Welding (edited by J. Lane, IFS Publications Ltd., Bedford, England, 1987). 104.2 Large Drives Alan K. Wallace and René Spée A drive is a system that converts electrical energy into useful, controlled, mechanical work. As such, it is a vital component in many industrial processes. The adjustable speed and torque of drives, in contrast to the typically uncontrolled values obtainable directly from most electrical motors, have been made possible by the introduc- tion of high-power electronic devices operating in switching modes. Appropriate selection, installation, and operation are essential for the process effectiveness and energy efficiency necessary for industrial competitiveness. Drives may be considered as consisting of three major subsystems: the motor or machine, which converts electrical energy to the required driving torques over specified speed ranges; the converter, which processes the electrical energy, received from the utility at constant voltage and frequency, into the forms required by the motor; and the controller, which adjusts the operation of the converter based on performance requirements and comparison with measured signals of actual performance. These three subsystems are interlinked by a communications subsystem as shown in Fig. 104.7. Although the demarcation between large and small drives is somewhat subjective, in general, devices such as positioning actuators and machine tools are examples of small drives, whereas large drives are applied to loads such as pumps, compressors, bulk material processing in “heavy” industries and mining operations, electric traction, and the forced- and induced-draft fans of fossil fuel power plants. The rating of a large drive is expressed in hundreds or thousands of kilowatts. The supplies for these drives are three-phase power obtained from the utility system at medium or high voltages. An advanced contemporary industrial drive is the result of an integration of several continually evolving technologies. In machines, improvements in the materials for magnetic circuits and electrical insulation enable higher specific ratings (i.e., better rating per unit mass or volume). In converters, the development of higher FIGURE 104.7Typical drive system. ? 2000 by CRC Press LLC power and faster switching semiconductor devices increases ratings and enables more sophisticated operational techniques. In controllers, incorporation of faster, more powerful microprocessors enables the use of adaptive control techniques with such features as self-diagnostics and automatic setup. Many significant developments in these areas are described in compilations of technical papers [Bose, 1981] and appropriate texts [Bose, 1986]. Configurations In contrast to small drives and servosystems in which many diverse forms of both direct current (dc) motors and alter- nating current (ac) motors are found, large drives are domi- nated by only four distinct motor types: separate (or shunt) field dc motors (DCM), cage-rotor induction motors (CRIM), wound-rotor (or slip-ring) induction motors (WRIM), and synchronous (dc field) motors (SM). For adjustable operation the DCM requires a controllable dc source that can be provided by either an ac-to-dc converter, such as a controlled rectifier (CR), or a dc-to- dc converter, known as a chopper. The latter is not common in industrial drives, being more appropriate for vehicle traction, and, consequently, will not be considered here. The three ac machines require ac-to-ac con- verters with frequency adjustability. This is produced by voltage source inverters (VSI), current source inverters (CSI), machine commutated inverters (MCI), and cycloconverters (CYCLO). Although other combinations may be found in some cases, Table 104.1 summarizes the more com-monly used drive configurations. In certain cases the converters do not operate to control the main power supply to the machine but, as described later, perform a slip energy recovery (SER) function. Details of the form and construction of these converters, motors, and drives can be found in appropriate texts [Gyugyi and Pelly, 1976; Sen, 1981; Leonard, 1985]. Selection and Compatibility An appropriately applied drive first must meet the shaft torque range and speed range of its load. From these, the appropriate motor type, number of poles, and (for ac machines) the frequency range can be selected. This selection is based on two basic equations that relate motor armature current (i), supply frequency (f), air gap flux density (B), number of poles (P), angular shaft synchronous speed (v s ), and shaft torque (T): T μ PBI (104.1) (104.2) From the products of torque and speed the motor (output) rating is derived. Large machines have good efficiencies (greater than 95%) and good power factors at rated operating conditions. Consequently, the output ratings of the converters are not substantially higher than those of the motors that they operate. Figure 104.8 shows areas typical of drive system operation; these result from a combination of physical limitations and economic considerations. Figure 104.8 should be interpreted in conjunction with Table 104.1 while noting that SER systems are a special case of WRIM operation. Certain processes may have, in addition, requirements for the response of a drive to follow changes of the torque and/or speed of the load and for the tolerable level of torque pulsations. These requirements may call for special controller functions and detailed knowledge of the interaction of motor and converter. Electrical motors can be made to operate in regenerative modes, i.e., energy is extracted from the load by the drive. This improves the dynamic response and/or reversing performance. This requirement is expressed in terms of operating quadrants as shown in Fig. 104.9. Hence, a single-quadrant drive is required to motor in one direction only. A two-quadrant drive has to motor and brake in one direction. A four-quadrant drive has to be regenerative and reversible. The number of required quadrants is reflected in the complexity of the converter. TABLE 104.1Drive Component Combinations CR VSI CSI MCI CYCLO DCM X CRIM X X WRIM X X X SM X X w s f P μ ? 2000 by CRC Press LLC The power and speed envelopes of Fig. 104.8 show considerable areas of overlap. Drive selection in these cases is generally based on required response, the operational environment, and economic considerations. For example, dc motors are larger, more complex and vulnerable, and more costly than their equivalently rated ac counterparts. Depending upon the operational quadrants required, however, a controlled rectifier is substan- tially cheaper than an inverter. It follows that, in many cases, a dc motor system is more economical than an induction motor equivalent. In damp, dirty, corrosive, or explosive environments, however, the simplicity and robustness of the induction motor makes it preferable for purely practical reasons. FIGURE 104.8 Classification of drives by rating. FIGURE 104.9 Quadrants of operation. ? 2000 by CRC Press LLC The effects of a drive on its environment are significant in the selection and design process. Converters that are called upon to switch very large currents, hundreds or thousands of amps, at frequencies up to several kilohertz produce serious magnetic fields around the devices themselves and their cables or leads. The electro- magnetic compatibility (EMC) issue must be addressed to ensure that other equipment, such as controllers and computers, is not adversely affected by the operation of the drive. In addition, power electronic converters present nonresistive, nonlinear loads to the power supply system. Consequently, the currents drawn can be of poor power factor and high total harmonic distortion (THD), which is defined in terms of the fundamental and harmonic components of current as (104.3) Significant THD can result in financial penalties being imposed on the drive/operator by the supplying authority and cause overheating of adjacent equipment. Moreover, power quality issues are the subject of new standards both in North America (revisions to ANSI-IEEE Standard 519) and in the European Community. In general, the order of priority of drive selection criteria is performance and response; operating environ- ment; power factor and THD; EMC; economics. Principles and Features of Operation Before the introduction of power semiconductors, both induction motors and synchronous motors were effectively fixed-speed machines, except where highly expensive rotary frequency conversion sets could be justified. Under these conditions the DCM was traditionally the basis of adjustable speed drives. Figure 104.10 shows schematically the two major components of a dc motor: the armature (rotating) and the field winding (stationary). In a DCM the armature current reacts with the air gap flux produced by the field to develop torque in accordance with Eq. (104.1). For a given constant field winding current, if the armature current is maintained at the rated value, the motor will develop rated torque at all speeds. However, the applied voltage (V) must overcome the internal voltage of the armature (E), which is given by E μ w r B (104.4) where v r is the actual speed of the motor. Hence V must be increased to increase motor speed. When the limit of the applied voltage is reached, the motor speed can only be increased further by reducing the air gap flux to maintain the armature voltage in accordance with Eq. (104.4). This is done by reduction of the field winding current in the field weakening mode of operation. The result is a decreasing torque in accordance with an approximately constant power curve, as shown in the single-quadrant torque-speed characteristic of Fig. 104.11. The three-phase thyristor bridge converter shown in the schematic of Fig. 104.10 will produce the output voltage waveform shown in Fig. 104.12, which can be shown to produce a mean (dc) voltage of FIGURE 104.10Schematic of dc motor drive. THD= all harmonics harmonic 2 fundamental I I ? ×100% ? 2000 by CRC Press LLC (104.5) in which V L is the rms line voltage of the ac supply and d is the delay angle. At higher output voltages (i.e., small d), the ripple content is small and the armature current is constant dc. This causes virtually rectangular current pulses at the three-phase input terminals of the rectifier, a high THD condition. Increasing d of the rectifier decreases the voltage applied to the armature. In consequence, the conduction periods of the rectifier shift with respect to the ac supply voltages. Thus, at low power levels, in addition to high THD, the displacement power factor is low. When the applied armature voltage (V) is reduced below the internal voltage (E), with the motor in motion, the second quadrant (braking operation) is entered. In order to achieve four-quadrant operation, either a changeover switch (to reverse the polarity of the armature connections to the rectifier output) or a second converter (with thyristors connected in the opposite sense) is required to enable the required current reversal. The operating speed of a CRIM is best adjusted by control of the terminal supply frequency, in accordance with Eq. (104.2) with a slight adjustment for the operating slip (i.e., the small difference between the synchro- nous speed, v s , and the rotor speed, v r ) (104.6) A basic induction machine drive system is shown in Fig. 104.13. The operation of the motor at constant slip over a range of controlled frequencies can be represented by considering operation at a number of discrete frequencies (f 1 to f 6 ) as shown in Fig. 104.14. For each applied frequency the machine assumes operation at the given slip resulting in the operating points (m 1 to m 4 ) for a constant load torque. Except at low speeds (where the resistance predominates), the impedance of the machine is effectively controlled by the inductive reactance, FIGURE 104.11Controlled operation of dc machine. FIGURE 104.12Phase controlled rectifier voltage. VV L = 32 1 p d( sin)- slip= -ww w sr s ? 2000 by CRC Press LLC which is proportional to the applied frequency. Hence, in order to maintain rated motor current, the voltage must be increased following a constant volts per hertz ratio. However, above a certain frequency, the output voltage of the inverter becomes limited by the dc link voltage developed from the input rectifier. Rated motor current cannot be maintained, and the resultant torque is reduced to typical operating points (m 5 and m 6 ). The loci of the operating points form a torque-speed characteristic similar to that shown in Fig. 104.11 for the dc drive. Braking operation of an induction machine drive can be obtained by observing the rotor speed and exciting the machine at a frequency that produces a negative slip, i.e., operating points b 1 to b 6 in Fig. 104.11 result from this strategy. Under these conditions, however, the inverter stage of the converter rectifies the output of the motor. This increases the voltage of the dc link to a level above the normal output of the rectifier stage. If the rectifier has controllable devices, it can be made to invert the energy in the dc link to utility frequency and hence return it to the three-phase supply. Alternatively, if the rectifier stage is an uncontrolled diode bridge, the regenerated energy must be dissipated in the dc link; this is often achieved by switching a resistor across the link in the braking mode. Switching of the inverter stage devices of the converter causes the potential of the dc link to be sequentially applied, removed, and then reverse connected to the motor terminals. At it simplest, this is equivalent to the application of rectangular voltage waves to a machine that is designed for sinusoidal excitation. Although the machine will operate adequately from rectangular, or overlapping, step-wave excitation, the high harmonic content of the resulting currents cause additional losses in the motor, resulting in a performance derating. In very large drives, where line commutated thyristors are needed to handle the power, or in more moderate-sized FIGURE 104.13 Induction motor drive. FIGURE 104.14 Development of induction motor drive torques. ? 2000 by CRC Press LLC drives at high speeds, where the commutation (switching) losses in the semiconductors prevent more sophis- ticated modes of operation, step-wave excitation may be unavoidable. However, increased ratings of gate-turn- off thyristors (GTO) and the development of MOS-controlled thyristors (MCT) have the potential to make voltage modulation techniques available to larger drives in the near future. Unlike a regular thyristor, which requires either a natural or forced current zero for turn-off, the more advanced devices can be controlled by relatively small gate (or firing) pulses. This enables numerous commutations during one period of the funda- mental frequency. Figure 104.15(a) shows the voltage waveform produced by applying the technique known as pulse-width modulation (PWM). Apart from the fundamental, the lowest- order harmonics of this function appear in a sideband around the modulation frequency. The resulting current is much closer to a sinusoid, as shown in Fig. 104.15(b), because high-frequency components are attenuated by the predominantly inductive nature of the motor impedance. Although PWM techniques reduce unnecessary losses in the motors, the higher frequencies may excite mechanical resonances in the audio frequency range. Thus, the motor may become a source of acoustic noise, which, depending on application and existing environment, may be of concern. For reasons of manufacturability and operational efficiency, the largest induction machines are of the wound- rotor (WRIM) type. These can be, and often are, controlled in the same manner as the CRIM just described. However, access to the rotor circuits via slip-rings enables the alternative form of control known as slip-energy recovery, as shown in Fig. 104.16. The advantages of SER are in reduced size and cost of the converter if the required speeds do not extend greatly from the natural synchronous speed of the motor. This is often the case for large drives. For very large drives, the cycloconverter replaces the inverters as the most appropriate converter in either the stator controlled or SER configuration. Cycloconverters develop the adjustable frequencies required by directly forming approximations to ac waveforms from segments of all the phases of the supply. Hence, each phase of the input supply needs to be connectable to every phase of the machine with both positive and negative polarities. Figure 104.17 is a schematic of a single phase of a CYCLO power circuit, and a typical voltage FIGURE 104.15(a) PWM line-to-line voltage. (b) Motor current due to PWM excitation. ? 2000 by CRC Press LLC waveform development is shown in Fig. 104.18. Examination of the voltage waveform illustrates that cyclocon- verters are only appropriate for generating output frequencies that are significantly lower than the input supply frequency (typically, f out,max ? f supply ). The largest of all industrial drives extend up to ratings of 100 MW. At an order of magnitude below this rating the short (~1 mm) air gaps between stator and rotor, needed for efficient induction motor operation, become untenable mechanically. Synchronous machines, with dc rotor fields excited via slip rings, can operate at high efficiencies and with controllable power factors while employing air gaps of several millimeters and are thus the only practical ac machine for very large drives. In addition, large converters cannot be produced without multiple power electronic devices connected in series and/or parallel. A more practical solution is often FIGURE 104.16Slip energy recovery drive. FIGURE 104.17One phase of cycloconverter. FIGURE 104.18Cycloconverter output waveforms. ? 2000 by CRC Press LLC found in the parallel connection of whole converters. If parallel converters are justified, parallel motor windings, arranged in a six-phase configuration, can be useful for purposes of isolation and reduced criticality of controls. A typical very large drive is shown schematically in Fig. 104.19. The operation of this and alternative configu- rations is described in the literature [Stemmler, 1991]. Control Aspects A comprehensive description of control techniques is significantly beyond the scope of this chapter, but detailed coverage is available in the recommended literature. Almost without exception, large drives are both controlled and protected in response to performance measurements which provide signals for control loops. The type of control strategy, the type of control loop, and the relative importance of the particular loops for a given application depend on the performance requirements. For example, where rapid response to changes in the load torque and/or speed is needed, shaft speed will constitute the major feedback signal and vector control can enable an induction motor drive to respond as well as the more traditional dc motor system. For very large drives system inertia is such that dynamic response is not an issue. More likely, the optimization of specific performance parameters, such as efficiency, is of value to the user. For this application the predominant control loops will be based on current and/or power measurements working in self-optimization or other adaptive control strategies. Future Trends The most significant future developments in large drives are likely to result from improvements in, and the application of, more advanced power electronic devices. This will enable converter operation at higher ratings and higher frequencies. The most direct initial evidence of this will be the ever-increasing rating at which inverters replace cycloconverters. The proposed revisions to ANSI/IEEE-519 concerning the tolerable harmonic current pollution levels of the supply will promote control strategies and converter topologies to replace expensive front-end filtering. Advanced control of inverter rectifier stages and new topologies such as matrix converters and resonant converters will be introduced in lieu of inverters and cycloconverters. The present cadre of machine types will remain, although the trend away from dc motor drives will continue. For certain highly specialized applications, requiring extremely high efficiency and specific performance regard- less of cost, synchronous motors using high-coercivity permanent magnet fields will be used. For work in severe environments and where high specific torque is needed, the switched reluctance motor system shows consid- erable promise [Greenhough, 1991]. Defining Terms Cycloconverter: A system of power electronic devices that converts alternating current energy at a constant voltage and constant frequency to an output of adjustable voltage and adjustable frequency. The conver- sion is done directly without the intermediate direct current stage used in a rectifier/inverter combination. FIGURE 104.19Very large synchronous motor drive. ? 2000 by CRC Press LLC Direct current (dc) motor: An electrical to mechanical energy conversion machine usually powered from a direct current source. The stator consists of a field winding system of a number of salient poles connected to produce a stationary pattern of alternate north and south polarity magnetic flux in the air gap between stator and rotor. The windings of the rotor (or armature) are connected to the energy source via a mechanical switching system, known as the commutator. Sliding electrical contact with the commutator is made by carbon brushes. Induction motor: A machine powered only from an alternating current source. The stator windings are a three-phase system symmetrically displaced around the internal periphery. The combination of the physical spatial placement of the phase windings and the time delay or sequence of the currents flowing in them produces a magnetic field pattern of alternate north and south poles that rotates within the air gap. The rotor can take one of two forms: a system of high-current, short-circuited conductors called a squirrel cage or a three-phase winding system with terminals brought out via slip rings and brushes. Inverter: A system of power electronic devices that converts direct current energy to alternating current energy by controlled sequential switching. Various control techniques have been developed to enable control of both the output frequency and output voltage. Rectifier: A system of power electronic devices that converts alternating current energy to direct current energy. Two generic forms is common: the uncontrolled rectifier and the controlled rectifier, the output voltage of which can be adjusted. Most rectifiers contain filtering elements, such as series inductors or parallel capacitors, at their outputs to reduce the ripple of the terminal voltage. Synchronous motor: A machine requiring both direct current and alternating current sources. The stator winding system is three-phase, similar to that of the induction motor. The rotor is a direct current system similar to the stator of the dc motor but with the mechanical freedom to rotate. Access to the rotor field winding is via slip rings and brushes. Related Topic 66.2 Motors References ANSI/IEEE Standard 519-1992, Guide for Harmonic Control and Reactive Compensation of Static Power Con- verters, December 1992. B. K. Bose, Ed., Adjustable Speed AC Drive Systems, New York: IEEE Press (Wiley), 1981. B. K. Bose, Power Electronics and AC Drives, Englewood Cliffs, N.J.: Prentice-Hall, 1986. P. Greenhough, Switched Reluctance Drives for Applications in Hazardous Areas, 5th International Conference on Electrical Machines and Drives (IEE 341), London: IEE, 1991, pp. 11–16. L. Gyugyi and B. R. Pelly, Static Power Frequency Changers, New York: Wiley, 1976. W. Leonard, Control of Electrical Drives, Berlin: Springer-Verlag, 1985. P. C. Sen, Thyristor DC Drives, New York: Wiley, 1981. H. Stemmler, “High power industrial drives”, Proc. IEEE, 82, 1266–1286, 1994. Further Information The Institute of Electrical and Electronics Engineers (IEEE) has three publications reporting on power elec- tronics, electric machines, and drives. The Transactions on Industry Applications is published bimonthly, while the Transactions on Energy Conversion and the Transactions on Power Electronics appear quarterly. The technical IEEE societies associated with these journals also sponsor semiannual or annual conferences. For information, contact IEEE Service Center, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. Other sources of information include the Proceedings of the Institution of Electrical Engineers (IEE) and the European Power Electronics and Drives Journal in Europe as well as the Transactions of the Institute of Electrical Engineers of Japan. ? 2000 by CRC Press LLC 104.3 Robust Systems Mario Sznaier and Ricardo S. Sánchez Pe?a Robustness and Feedback Feedback and control theory are two concepts that are intimately related. In fact, the latter is considered as the theory of feedback systems. Next we explain why the need for feedback is only due to uncertainty. Consider the feedback system of Fig. 104.20, where S represents the physical system to be con- trolled, K(s) and d(s) the mathematical models of the linear controller and the external disturbance at the output of the plant, respectively. Next, add and sub- tract inside the loop a linear mathematical model of the plant G(s), so that the feedback loop remains unchanged. Finally, redefine the connection between the models of the controller and the plant as: C(s) ? = K(s)[I + G(s)K(s)] –1 . The objective of these transfor- mations inside the feedback loop is to leave the feed- back signal f(s) expressed only in terms of its necessary components. This is: (104.7) From the above, we see that the need for a feedback signal is due exclusively to the uncertain elements in the loop: disturbance d(s) and model uncertainty ?. The disturbance is considered as unknown because otherwise, if we knew exactly the type of signal and the time at which it disturbs the loop, another signal could be injected in the loop that counteracts the effect of d(s). In classical and modern control, which generally assume knowledge of the type of signal (step, ramp, sinusoid), there is no certainty in the moment the disturbance will appear. In robust control, the hypotheses are relaxed and the disturbances are assumed to be bounded (in energy, power, or magnitude). Model uncertainty ? represents the fact that a mathematical model does not copy exactly the relevant physical phenomena taking place in the system. The main difference between classical/modern control theories and robust control is the fact that in the latter, uncertainty is explicitly incorporated in the hypothesis of the problem. Therefore, in robust control, the word “model” is not equivalent to “system,” the latter meaning physical system or plant. Specifically, the system is treated mathematically as a family of models or set, represented by a nominal model G(s) (the same one used in classical/modern control) and bounded uncertainty ?. 1 The goal of robust control is to compute the least conservative conditions providing certainty on loop stability and performance of an uncertain model (bounded family of models) that represents a physical system. When these properties, stability and performance, refer to the nominal model, they are called nominal. When they refer to the complete family of models or uncertain model, they are called robust. Next, let us imagine ideally, that there is exact knowledge of d(s) and an exact mathematical representation of the system S, i.e., S ≡ G(s). By the arguments in the above paragraphs, without loss of generality, we can assume d(s) ≡ 0 and therefore f(s) ≡ 0. In this case, there is no need for feedback: any desired output could be obtained or, stabilization of G(s) could be achieved by conveniently designing an open-loop controller C(s). 1 If ? is not bounded, the problem becomes ill-posed, since by taking it large enough we could always destabilize the feedback loop. FIGURE 104.20 Feedback and uncertainty. fs ds S Gs us ( ) = ( ) +? ( ) [] ( ) ? 12434 ? 2000 by CRC Press LLC Nevertheless, these assumptions do not include the physical connection between the controller and the system. Through any physical connection (the electrical signals from the D/A of a computer controller to the actuator), there is a possibility of having external disturbances (electrical noise, quantization). If the system is open-loop unstable (e.g., inverted pendulum), there exist disturbances injected at the plant input that could produce an undesirable diverging output. Again, the lack of knowledge of possible disturbances entering the loop at different points makes open-loop control a useless choice. Thus, in any realistic situation there is no way to avoid feedback, due to the existence of uncertainty. The basic objective of both robust control and robust identification is to develop methods that explicitly take into account this uncertainty, leading to the design of robust systems: systems where a desirable property (such as stability or performance) can be guaranteed a priori, even in the presence of uncertainty. Robust Stability and Performance In this section, we address the issues of nominal and robust stability and performance problems in single-input single-output (SISO) systems. The analysis proceeds from stability of the nominal model of the plant to the final objective of robust control: robust performance. Nominal Internal Stability It is well known that a system described by a rational transfer function G(s) is bounded input bounded output (BIBO) stable if and only if it has all its poles in the open left half complex plane Re(s) < 0. However, as illustrated by the following example, this classical input/output stability concept may fail to capture the stability properties of a feedback loop. Consider the loop of Fig. 104.21 and let (104.8) The transfer function from the input u 2 to the output y 1 is given by T y 1 u 2 = , which is stable in the usual sense. However, the transfer function between the input u 1 and the output y 1 is T y 1 u 1 = , which is obviously unstable. As we will see next, this is caused by the cancellation of the unstable plant pole at s = 1 by a zero of the controller. This example shows that there is a difference between the stability of a certain system, considered as a mapping between its input and output 2 which we define as input-output stability, and stability of a feedback loop which will be defined next. In the latter, we must guarantee that all possible input-output pairs are stable, which leads to the concept of internal stability. Definition 104.1 The feedback loop of Fig. 104.21 is internally stable if and only if all transfer functions obtained from all input-output pairs have their poles in H11923 – ? {s: Re(s) < 0} (input-output stable). It is easy to show [9] that to verify internal stability it is sufficient to check the input-output stability of the four transfer functions between the inputs [u 1 (s), u 2 (s)] and the outputs [e 1 (s), e 2 (s)]. Moreover, it is not difficult to prove that the feedback loop in Fig. 104.21 is internally stable if and only if [1 + g(s)k(s)] –1 is stable and there are no right half plane (RHP) pole-zero cancellations between the plant and the controller. Thus, the concept of internal stability formalizes the well-known design rule that no unstable pole/non-minimum phase zero cancellation between plant and controller should be allowed. 2 Even in the MIMO case with several inputs and outputs. gs s ss ks s s ( ) = + ( ) ? ( ) + ( ) ( ) = ? ( ) + ( ) 1 13 1 1 , FIGURE 104.21 Feedback interconnection to evaluate internal stability. 1 4()s + () ()() s s s + +? 1 1 4 ? 2000 by CRC Press LLC Robust Stability Roughly speaking, a given property of a system (such as stability or performance) is robust if it holds for a family of systems that represents (and contains) the nominal plant. In this context, robustness can be quantified by defining a robustness margin in terms of the distance of the nominal model that represents the system, to the nearest model that lacks the property under consideration. Thus, a given robustness margin is related to a specific type of model uncertainty. In classical control theory, this leads to the well-known concepts of phase and gain margins. Both of these measures can be interpreted in terms of the Nyquist plot, as shown in Fig. 104.22. Here φ m and g m represent the “distance” in angle and gain, respectively, to the critical point z = –1. Thus, the feedback loop remains stable even when the nominal plant g o is replaced by g(s) = δg o (s), where δ = c or δ = e jφ and where c and φ are uncertain values contained inside the intervals I c = [1, g m ] and I φ = [0, φ m ], respectively. Note that these definitions implicitly assume that both types of uncertainty (phase and gain) act on the loop one at a time. As a consequence, these margins are effective as analysis tools only when the model of the plant has either phase or gain uncertainty and do not guarantee robust stability for the more realistic situation where both phase and gain are simultaneously affected by uncertainty. For instance, in the system depicted in Fig. 104.22, both φ m and g m have adequate values. Nevertheless, with small simultaneous perturbations in the phase and gain of the loop, the plot will encircle the critical point z = –1. A more realistic uncertainty description, leading to controller designs that perform better in practice, is multiplicative dynamic uncertainty. In this context, the actual physical system is described by the set (104.9) as illustrated in Fig. 104.23. Here, g o and W δ (s) represent the nominal plant and a fixed weighting function containing the frequency distribution of the uncertainty, and the stable transfer function ?(s) represents bounded dynamic uncertainty. 3 A typical function W δ (s) has high-pass characteristics, with small magnitude (i.e., low uncertainty) at low frequencies, increasing at high frequencies. If its magnitude becomes larger than one above a certain crossing frequency ω o (more than 100% uncertainty); in order to guarantee stability of the closed-loop system, the controller must render the nominal loop function g o (jω)k(jω) small enough at frequen- cies above ω o . This guarantees that the Nyquist plot does not encircle the critical point. A condition guaranteeing stability of all elements of the family G, i.e., robust stability of g o (s), is derived next. 3 Without loss of generality, the bound on ? can be taken to be one, since any other value can be absorbed into the weight W δ (s). FIGURE 104.22 Phase and gain margins. G = ( ) ( ) = ( ) + ( ) ( ) [] ( ) ( ) < ? ? ? ? ? ? gs gs g s sW s s j o j : , , 11?? ? δ ω ωstable sup ? 2000 by CRC Press LLC Theorem 104.1 Assume the nominal model g o (s) is (internally) stabilized by a controller k(s). Then all members of the family G will be (internally) stabilized by the same controller if and only if the following condition is satisfied: (104.10) with T(s) ? g o (s)k(s) [1 + g o (s)k(s)] –1 the complementary sensitivity function. Using Fig. 104.24, we can interpret condition (104.10) graphically, in terms of the family of Nyquist plots corresponding to the set of loops. First observe that Eq. (104.10) is equivalent to: (104.11) For a given frequency ω, the locus of all points z(jω) = g o (jω)k(jω) + g o (jω)k(jω)W(jω)?), |?| < 1 is a disk D(ω), centered at g o (jω)k(jω) with radius r = |g o (jω)k(jω)W(jω)|. Since |1 + g o (jω)k(jω)| is the distance between the critical point and the point of the nominal Nyquist plot corresponding to the frequency jω, it follows that condition (104.10) is equivalent to requiring that, for each frequency ω, the uncertainty disk D(ω) excludes the critical point z = –1. Therefore, robust stability for SISO systems can be checked graphically by drawing the envelope of all Nyquist plots formed by the set of circles centered at the nominal plot, with radii |g o (jω)k(jω)W(jω)|, and checking whether or not this envelope encloses the critical point z = –1. In the MIMO case, although an equivalent condition can also be obtained, there is no such graphical interpretation. FIGURE 104.23 Disturbance rejection at the output for a family of models with multiplicative uncertainty. FIGURE 104.24 Set of Nyquist plots of the family of models. TsWs Tj Wj j () () = ()() ∞ ≤ ? sup ω ωω1 1+ ( ) ( ) ≥ ( ) ( ) ( ) ?=gsks gsksWs j oo s ω ? 2000 by CRC Press LLC Nominal Performance In the context of robust control theory, performance is defined on the basis of the ability of the control system to reject a family of disturbances, possibly appearing at different parts of the loop, i.e., sensors, actuators, outputs, etc. In the sequel, for simplicity we consider the case where these disturbances appear at the output of the plant, but the results can be easily generalized to other cases. Definition 104.2 The feedback loop of Fig. 104.25 achieves nominal performance if and only if the weighted output remains bounded by unity, i.e., ||W y (s)y(s)|| 2 ≤ 1, for all disturbances in the set {d ∈ L 2 , ||d|| 2 ≤ 1}. In other words, nominal performance is achieved if for all possible exogenous perturbations d(s) with energy less than one, the energy of the weighted output W·y also remains below one. As before, W d (s) and W y (s) are fixed weighting functions used to give more weight to some regions of the spectrum. Note that checking nominal performance using Definition 104.2 directly requires a search over all bounded energy disturbances, which is clearly not possible. Fortunately, nominal performance can be checked by checking the following frequency domain condition: Theorem 104.2 The feedback system of Fig. 104.25 achieves nominal performance, if and only if: (104.12) A graphical interpretation of the nominal performance condition can be obtained by means of a Nyquist plot (see Fig. 104.26). To this end, define W(s) ? W y (s)·W d (s) and note that Eq. (104.12) is equivalent to: (104.13) Consider, for each frequency jω, a disk D(jω) centered at z = –1, with radius r = |W(jω)|. Then Eq. (104.13) can be interpreted graphically as nominal performance being achieved if and only if, for every frequency jω, the disk D(jω) does not intersect the Nyquist plot of g o (jω)k(jω), the nominal loop. Robust Performance The final goal of robust control is to achieve the performance requirement on all members of the family of models (i.e., robust performance), with a single controller. Next, we will establish a necessary and sufficient condition for robust performance by making use of the conditions for nominal performance and robust stability. Definition 104.3 The feedback loop of Fig. 104.23 achieves robust performance if and only if ||W y (s)y(s)|| 2 ≤ 1, for all possible disturbances in the set {d ∈ L 2 | ||d|| 2 ≤ 1}, and for all models in the set G = {g: g(s) = [1 + W δ (s)?(s)]g o (s), ? stable, |?(jω)| < 1}. FIGURE 104.25 Augmented feedback loop with performance weights. WsSsWs W j gj kj Wj yd y d ( ) ( ) ( ) = ( ) + ( ) ( ) [] ( ) ∞ ≤? sup jω ωωωω1 1 Wj g j kj o ωωωω ( ) ≤+ ( ) ( ) ?1, ? 2000 by CRC Press LLC Applying the conditions for nominal performance and robust stability to all members of the set G leads to the following necessary and sufficient condition for robust performance. Theorem 104.3 A necessary and sufficient condition for robust performance of the family of models in Fig. 104.23 is: (104.14) As before, a graphical interpretation of the robust performance condition can be obtained by means of the Nyquist plot of Fig. 104.27. Notice that Eq. (104.14) is equivalent to: (104.15) FIGURE 104.26 Nyquist plot for disturbance rejection interpretation. FIGURE 104.27 Nyquist plot for robust performance interpretation. WjSj TjWj d ωω ω ω δ ( ) ( ) + ( ) ( ) ≤ ∞ 1 10+ ( ) ? ( ) ( ) + ( ) ( ) ≥lljWjjWj d ωωωω δ ? 2000 by CRC Press LLC From the figure we see that the robust performance requirement combines both the graphical conditions for robust stability of Fig. 104.23 and nominal performance of Fig. 104.22. Robust performance is equivalent to the disk centered at z 1 = –1 with radius r 1 = |W d (jω)| and the disk centered at z 2 = H5129(jω) with radius r 2 = |W δ (jω)H5129(jω)| being disjoint. Clearly, this is more restrictive than achieving the robust stability and nominal performance conditions separately. Moreover, while nominal performance and robust stability can be verified by computing the infinity norm of an appropriately weighted closed-loop transfer function, the condition for robust performance (Eq. (104.14)) cannot be expressed in terms of a single closed-loop weighted infinity norm. Rather, a new measure μ, the structured singular value, must be used. This measure will be defined later, where we will indicate how to compute it, a procedure known as μ-analysis. Extension to MIMO Systems A common practice to extend SISO results to multivariable systems is to use a loop at a time approach, where the SISO tools are applied to each input-output pair of the MIMO system. Unfortunately, as we will show by means of a simple example, this approach can be misleading, overestimating the robustness properties. Example 104.1 Consider the following nominal plant and controller: (104.16) For the above design, the nominal loop L(s) = G(s)K(s) and the complementary sensitivity functions are given by (104.17) Next, open each of the loops, while the other remains closed (see Fig. 104.28). Opening only the first loop leads to u 2 = –y 2 , hence y 1 = –u 1 /s. Similarly, closing the first loop and evaluating the second one, we obtain y 2 = –u 2 /s. Thus, each loop has infinite gain margin and 90° phase margin, and one is tempted to conclude that the system has good robustness properties. However, if the system is affected by multiplicative uncertainty of the form (104.18) FIGURE 104.28 “Loop at a time” analysis. Gs s s s ss s s Ks s s ss ( ) = + ? + ( ) + ( ) + ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( ) = + + ( ) + ( ) ? ? ? ? ? ? ? ? ? ? 21 21 12 1 2 1 2 21 2 01 2 , Ls s s s Ts s ( ) = + ? ? ? ? ? ? ? ? ( ) = + ? ? ? ? ? ? 1 10 1 1 1 1 10 11 , ?= ? ? ? ? ? ? δδ 12 00 ? 2000 by CRC Press LLC then the corresponding closed loop characteristic polynomial is given by (104.19) and it is easy to verify that the following uncertainty destabilizes the closed loop: (104.20) Thus, a perturbation with “size” (given in terms of the euclidian norm) ||? o || = is destabilizing. In order to motivate the approach that we will follow to generalize the SISO tools to the multivariable case, consider the robust tracking problem shown in Fig. 104.28, where the objective is to synthesize a controller such that, for all elements of the family of plants described by the model: the resulting closed-loop system is internally stable and tracks a reference signal of the form {r(s), ||r(s)|| 2 ≤ 1} with tracking error bounded by 1, i.e., ||?e(s)|| 2 ≤ 1. In Fig. 104.29, this problem is recast into the interconnection of an upper block ? (representing model uncertainty) and a nominal plant M(s) (that includes the nominal model, the controller, and the uncertainty and performance weights) with the following representation: (104.21) where S o (s) = [I + G o (s)K(s)] –1 and T o (s) = G o (s)K(s)[I + G o (s)K(s)] –1 denote the output sensitivity and its complement, respectively. This interconnection is a special case of a Linear Fractional Transformation (LFT), a general structure used in modern robust control theory both for analysis and synthesis. While a complete analysis of the properties of this interconnection is beyond the scope of this chapter (see for example, [9, 10]), we quote below the MIMO equivalent of the SISO robust stability and nominal performance conditions covered above. Assume that ?(s) is stable and that ||?|| ∞ ? sup jω σ [?(jω)] < 1, where σ(·) denotes the maximum singular value; then robust stability and nominal performance of the interconnection of Fig. 104.30 are equivalent to: FIGURE 104.29 Robust tracking problem with sensor uncertainty. ss 2 12 12 210+++ ( ) +++ ( ) =δδ δδ ? o = ?? ? ? ? ? ? ? 1 2 1 2 00 ? o = 2 2 GI sWsGs s o =+ ( ) ( ) [] ( ) ( ) < {} ∞ ??? ? , stable, 1 Ms Ms Ms Ms Ms WsTs WsTs WsSs WsSs oo eo eo ( ) = ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? = ? ( ) ( ) ( ) ( ) ? ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? 11 12 21 22 ?? ? 2000 by CRC Press LLC ? H14067M 11 H14067 ∞ ≤ 1 ? H14067M 22 H14067 ∞ ≤ 1 (104.22) + ? max {H14067M 11 H14067 ∞ , H14067M 22 H14067 ∞ } ≤ 1 As before, robust performance cannot be expressed in terms of the norm of a single transfer function and requires the use of the tools briefly mentioned later. It follows that a controller that achieves robust stability or nominal performance can be found by considering the interconnection shown in Fig. 104.31 and designing K so that ||T zw || ∞ ≤ 1, where T zw denotes the closed-loop transfer function between the input w and the output z. This is the well-known H ∞ control problem addressed in the next section. H ∞ Control As shown in the previous sections, a large number of robust control problems can be described using the block diagram shown in Fig. 104.30. Here, the goal is to synthesize an internally stabilizing controller K(s) such that the worst-case output energy ||z|| 2 due to exogenous disturbances w with unit energy is kept below a given threshold. Since for Linear Time Invariant (LTI) stable systems the L 2 to L 2 induced norm coincides with the H ∞ norm of the transfer matrix [9], this problem is known as the H ∞ (sub)optimal control problem. While a complete analysis of this problem is beyond the scope of this chapter, in the sequel we briefly describe a solution, developed in the early 1990s [4, 5, 8], based on the use of Linear Matrix Inequalities (LMIs). For simplicity, we will assume that the plant is strictly proper. 4 4 This assumption can always be removed through a Loop Shifting transformation [9]. FIGURE 104.30 Statement of the problem as an LFT. FIGURE 104.31 Lower fractional interconnection F H5129 [P(s), K(s)]. Robust stability Nominal performance Nominal performance Robust stability ? 2000 by CRC Press LLC Theorem 104.4 Consider a finite dimensional LTI plant G of McMillan degree n with a minimal realization: (104.23) where the pairs (A, B 2 ) and (A, C 2 ) are stabilizable and detectable, respectively, and where A ∈ R n×n ; D 11 ∈ R n 1 ×m 1 , D 12 ∈ R n 1 ×m 2 and D 21 ∈ R n 2 ×m 1 . Then there exists an internally stabilizing controller K(s) with McMillan degree k that renders the closed-loop transfer function ||T zw || ∞ < 1 if and only if the following Linear Matrix Inequalities in the variables R and S are feasible: (104.24) (104.25) (104.26) where N R and N S are any matrices whose columns form bases of the null spaces of [B T 2 D T 12 ] and [C 2 D 21 ], respectively. Moreover, the set of suboptimal controllers of order k is nonempty if and only if Eq. (104.24–104.26) hold for some R, S satisfying the rank constraint (104.27) Once the matrices R and S have been found, a suitable controller K can be constructed as follows: 1. Form a matrix 2. Solve the following LMI in the variable Θ: (104.28) where (104.29) z y AB B CD D CD w u ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 12 11112 221 0 N I AR RA RC B CR I D BDI N I R T TT TT R 0 0 0 0 0 11 111 1 ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? < N I A S SA SB C BS I D CDI N I S T TT S 0 0 0 0 0 1 111 111 ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? < RI IS ? ? ? ? ? ? ≥0 rank I RS k? ( ) ≤ X SN NI NSR T = ? ? ? ? ? ? =? ( ) ? , 1 1 2 ΨΘ Θ x TT cl cl T ++<QPPQ0 Θ= ? ? ? ? ? ? ? ? AB CD kk kk ? 2000 by CRC Press LLC contains all the controller parameters and where (104.30) In the special case where D 11 = 0, D 22 = 0 and the following conditions hold: (A1) (A, B 2 ) is stabilizable and (C 2 , A) is detectable. (A2) (A, B 1 ) is stabilizable and (C 1 , A) is detectable. (A3) C T 1 D 12 = 0 and B 1 D T 21 =0. (A4) D 12 has full column rank with D T 12 D 12 = I and D 21 has full row rank with D 21 D T 21 = I. The result above reduces to the following theorem, first stated in [3]. Theorem 104.4 Under assumptions (A1)–(A4) there exists an internally stabilizing controller K(s) that renders ||T zw || ∞ < 1 if and only if the following two Riccati equations: (104.31) have positive semidefinite stabilizing solutions X ≥ 0, Y ≥ 0 such that ρ(XY) < 1, where ρ(·) denotes the spectral radius. In this case, a suitable controller is given by (104.32) where (104.33) Ψ ? ?? x o T ooo T o TT o cl TT k o o o AX XA XB C BX I D CDI Xq A A B B CC = + ? ? ? ? ? ? ? ? ? ? ? ? = [] = [] = ? ? ? ? ? ? ? ? = ? ? ? ? ? ? = [] = 11 11 12 21 11 00 0 00 0 0 P DCDB B ; ; ; ; 0 0 0 0 0 0 2 2 12 12 21 21 B I I C D D k k ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? = [] = ? ? ? ? ? ? ? ? ; ; C DD AX XA XBB BB CC AY Y A Y C C C C Y B B TTT TTT T ++ ? ( ) += ++ ? ( ) += 11 22 1 1 11 22 11 0 0 K A F ZL central ≡ ? ? ? ? ? ? ? ? ? ∞ ∞ ∞∞ 0 AABBXBFZLC FBX LYC ZIYX T T T ∞∞∞∞ ∞∞ ∞∞ ∞∞∞ ? =+ + + =? =? =? ( ) 11 2 2 2 2 1 ? 2000 by CRC Press LLC Structured Uncertainty So far, we have considered only global dynamic uncertainty. This name comes from the fact that the uncertainty is attributable to all the systems and describes the unknown higher order dynamics of the plant. However, in many practical applications, this type of uncertainty description covers in a very conservative way the real uncertainty of the model. This is the case when more structured information on the plant is available. Hence, a nonconservative analysis or synthesis procedure should take advantage of this extra information. Next we mention some of these situations. Some excellent references for this area are ([1, 2, 9, 10]). ? Consider the total system as composed of individual subsystems, each with its own dynamic uncertainty description. Take for example the actuators, the system itself, and the sensors described as the following individual sets of models: (104.34) with H5129 = 1, 2, 3 for the actuator, plant, and sensor, respectively. The series interconnection yields, S(s) = G 3 (s) · G 2 (s) · G 1 (s), which can be transformed to an LFT connection between a nominal model ? G(s) and an uncertainty block in the set ? struct ? {diag [? 1 ? 2 ? 3 ], ? H5129 ∈ H11923 k H5129 ×k H5129, ||? H5129 || < 1, H5129 = 1, 2, 3}. This type of uncertainty is called structured dynamic. If, on the other hand, the uncertainty of the plant is described as global dynamic, i.e., {F u [ ? G(s), ?], ? ∈ H11923 n×n , ||?|| < 1} (n = k 1 + k 2 + k 3 ), disregarding the structural information will add more unnecessary models to the set. Hence, the robustness analysis of a closed-loop system with such an uncertainty description will, in general, be conservative. ? In many cases, the plant has a well-known mathematical model usually derived from physical equations. This is the case of some applications from mechanical, aeronautical, and astronautical engineering, where the rigid body model based on Newton-Euler equations provides a good enough description for mild performance specifications. However, the parameters of these models may not be known exactly. Rather, their values are estimated either by classical parameter identification ([6]) procedures or by set mem- bership identification methods ([9]) to within some uncertainty bounds. When these bounds are deter- ministic worst-case bounds, this leads to a plant representation in terms of a family of models with a mathematical fixed structure and parameters that may take values within certain specified sets. This type of uncertainty description is called parametric uncertainty. Take for example the following set of models that represents a plant with uncertain parameters p ? [z ω n ξ] T : Parametric uncertainty can be present in both state space or transfer matrix representations. In the latter case, the uncertain parameters are located in the upper uncertainty block ? of a standard LFT structure, or in the coefficients of the characteristic polynomial of the closed-loop system. In the previous example: (104.35) where k is a constant (nominal stabilizing controller). ? In general, both parametric and structured dynamic uncertainty appear simultaneously. For example, large flexible space structures have a well-known low-frequency model represented by several second- order modes with natural frequencies and damping coefficients within real intervals, i.e., parametric GG lllll l ll sIWs s kk ( ) =+ ( ) [] ( ) ∈< {} × ???; , H11923 1 G sp sz ss zzz nn n ,, , , , ( ) = + ++ ∈ [] ∈ [] ∈ [] ? ? ? ? ? ? ? ? ? ? 22 12 1 2 12 2ωξ ω ωωωξξξ Psp s k s kz n cp n cp o , ( ) =++ ( ) ++ ( ) () () 22 2 1 ωξ ω 12434 12434 ? 2000 by CRC Press LLC uncertainty. The higher-order dynamics can be represented more naturally as dynamic uncertainty. This is the so-called mixed type uncertainty. Recall that stability or performance robustness margins are directly related to the type of uncertainty present in the plant. As special cases, we have mentioned the classical phase and gain margins. For structured uncertainty, the same concept holds; therefore, a general definition of a robustness margin should be made. Stability Margin Characteristic Polynomial Framework A natural way to state the problem in cases of parametric uncertainty is in terms of the closed-loop characteristic polynomial (CLCP): (104.36) where {c i (·), i = 0, …, n – 1} are real functions of the uncertainty vector p and H represents the m-dimensional hypercube of parameters p i ∈ [a i , b i ], i = 1, …, m. Here, a nominal internally stabilizing controller is assumed, i.e., P(s, p 0 ) = 0 has all its roots in H11923 – ? {s ∈ H11923; H11938e(s) < 0}. Robust stability is equivalent to P(s, p) ≠ 0, ? p ∈ H, ?s ∈ H11923 + , where H11923 + ? {s ∈ H11923; H11938e(s) ≥ 0}. Checking this condition requires computing the roots of P(s, p) for all possible values of p ∈ H. On the other hand (since the nominal system is stable), a robust stability margin only needs to indicate at which point the roots of P(s, p) cross over from H11923 – to H11923 + as the uncertainty around the nominal set of parameters p 0 is “increased.” Under certain continuity conditions defined by the Boundary Crossing theorem, as poles move from H11923 – to H11923 + , the first unstable ones reach the jω axis before entering the interior of H11923 + . Hence, the measure of stability, defined as the multivariable stability margin k m is: (104.37) where BH is the unitary m-dimensional hypercube H. Hence, the necessary and sufficient condition for robust stability is k m (jω) ≥ 1, ?ω. As mentioned before, when considering only parametric uncertainty, this is a general framework that includes the LFT formulation as a special case. The complexity of the aforementioned functions of the uncertain parameters c i (p) determines the computational complexity of the solution. In addition, the type of functions considered leads to two clearly different research approaches. This is illustrated in Table 104.2 which classifies the different tools according to these functions. Here, ν represents the set of vertices of H and co(·) is the convex hull. The first row considers the case of an independent set of uncertain coefficients. Kharitonov’s Theorem states that in this case, stability of the complete set of polynomials is equivalent to stability of only four distinguished CLCP. The second row considers the coefficients as affine functions of the uncertain parameters. The Edge theorem states that the stability of the polynomials along the edge of the set of parameters ensures the stability of the whole family of CLCPs. The case of multilinear dependence of the coefficients with the parameters (3rd row) establishes a boundary between two different research approaches. The first one is similar to the previous cases and seeks to compute TABLE 104.2 Coefficient Functions, LFT Structure, Value Sets, and Analysis Results CLCP Framework c i (p) LFT Framework ? p Structure Value Set Result c i = p i M(s) rank 1 Rectangle Kharitonov Affine M(s) rank 1 Polytope Edge Theorem Multilinear Independent δ i ’s M(s) general Non-convex P(ω, H) ? co[P(ω, ν)] Analytical, Computational Polynomial Repeated δ i ’s M(s) general Non-convex P(ω, H) ? co[P(ω, ν)] Computational Psp s c ps c ps c p p n n n o ,, ( ) =+ ( ) ++ ( ) + ( ) ∈ ? ? 1 1 1 L H kj k Pjkp p m ωω ( ) = ∈∞ ( ) ( ) =∈ {} ? inf ,,, 00BH ? 2000 by CRC Press LLC the stability margin for a particular multilinear structure by considering only a smaller number of distinguished models. The second approach starts directly from the general multilinear dependence case and generalizes to poly- nomial functions c i (p). Many of these methods are based on the Mapping theorem and have a clear computa- tional basis. These algorithms are based on a branch and bound method over the two-dimensional value sets in the complex plane for each frequency ω. The general parametric analysis is NP-hard; therefore, the procedures that are able to compute the stability margin exactly (or with guaranteed bounds) have exponential time complexity. LFT Framework When dynamic uncertainty is involved, it is convenient to structure the uncertainty as an LFT, due to the fact that in these cases the model order is not fixed. In the parametric uncertainty case, the LFT setup includes only c i (p) functions that are polynomial in the parameters. Therefore, the previous analysis based on the characteristic polynomial P(s, p) would be more general. The uncertainty structures ? that can be used are as follows: (104.38) for structured dynamic uncertainty, or ? i ∈ H11938 for the parametric uncertainty set ? p , or combinations of both for mixed uncertainty descriptions ? M . The stability margin for these types of uncertainty descriptions is as follows: Definition 104.4 The structured singular value μ ? is defined as: or otherwise μ ? (jω) = 0 if det[I – M(jω)?] ≠ 0 for all ? ∈ ?. Here, the set ? may be any of the previously defined uncertainty structures and M(s) is the lower block of the LFT. From the previous definition in Eq. (104.37), it is clear that k m [M(jω)] =μ ? –1 [M(jω)] when the uncertainty structure ? is the same. The necessary and sufficient conditions for the robust stability of the family of closed-loop systems {F u [T(s), ?], ? ∈ B?} is μ ? [T 11 (jω)] ≤ 1 for all ω ∈ H11938 (or equivalently k m [T 11 (jω)] ≥ 1 for all ω ∈ H11938), where T 11 (s) is the upper block of T(s) connected to the uncertainty ?. As in the CLCP statement, the robustness condition is tested over the imaginary axis only. In the sequel, we briefly describe a procedure based on the use of upper and lower bounds to compute μ ? . For simplicity, we restrict ourselves to the uncertainty set ? d in Eq. (104.38). It can be shown that μ ? has the following properties: ? μ(MU) = μ(M) for U ∈ U ? {U ∈ ? d , U i unitary; i = 1, …, m} ? μ(DMD –1 ) = μ(M) for D ∈ D ? ? (104.39) Note that the first equality in Eq. (104.39) leads to a non-convex optimization problem. On the other hand, the right-hand side inequality leads to a convex optimization that can be solved in polynomial time. However, ? ? ??? dmi rr ii im = … [] ∈=… {} × diag 1 1H11923 , , , μ ( ) []= ( ) ? ( ) [] = {} ? ? ? ? ? ? ∈ ? ? ?? ? ??Mj I Mjωσ ωinf det 0 1 dI dI Ir d mm ii i 11 0 0 0 O ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? > : identity in max inf Uu D MU M DMD m ∈∈ ? ( ) ( ) ( ) [] =μ ≤ = ≤ρσ D 1 3if ? 2000 by CRC Press LLC it is tight only for structures having no more than three uncertainty blocks, with very recent results indicating that the gap can be arbitrarily large as m grows (but growing no faster than ). Nevertheless, this inequality is used as a standard tool for robustness analysis. For the parametric and mixed uncertainty types ? p and ? M , respectively, computation is not as straightfor- ward. This is due to the fact that the calculation of the exact stability margin (or even an approximation with guaranteed a priori bounds) of an uncertain model with a general parametric uncertainty structure is an NP- hard problem; therefore, there are two research directions. First, looking for exact (or approximate with guaranteed bounds) polynomial time analysis of parametric uncertainty structures that may not be general, but can accommodate relevant practical applications. Second, studying approximate methods (branch and bound, heuristics) that can bound in polynomial time the stability margin for general cases, and although there may not have guaranteed a priori error bounds may work reasonably well in practical situations. For the computation of μ in all these cases, we refer the reader to [9, 10]. Robust Identification A basic point in the development of robust theory are the methods by which a set of models that represents a particular physical process can be obtained. Before the appearance of systematic methods, the family of models was obtained by ad hoc procedures. At the end of the 1980s, the first algorithmic strategies were introduced, based on approximation techniques that provide a uniform error bound. Classical identification procedures [6] rely on stochastic methods to identify a set of parameters of a fixed mathematical structure and thus are more suited for adaptive control applications than for robust control, since the latter relies on a deterministic worst-case approach, with no previous assumption on the order of the system. Moreover, even if families of models with parametric uncertainty could be obtained in this way, there is a limited design machinery for robust analysis and synthesis of this class of uncertain plants, due to the fact that these are NP-hard problems. These considerations led to the development of new deterministic identification procedures, called robust identification, based not only on the experimental data (a posteriori information), but also on the a priori assumptions on the class of systems to be identified. The algorithms produce a nominal model based on the experimental information and a worst-case bound over the set of models defined by the a priori information. A recent survey of the area of robust identification can be found in [7, 9], which include an extensive list of references. Input Data The outcome of a robust identification procedure is a family of models that should include the real physical plant. This family is specified by a nominal model and an uncertainty error measured in a certain norm. The input data to a robust identification algorithm is composed of the class of candidate models S and measurement noise H5114 called a priori information and the experimental data y, called a posteriori information. The a posteriori information is a vector y ∈ H11923 M of experimental data corrupted by noise η ∈ H11923 M . The data can be frequency and/or time noisy samples of the system to be identified. For a model g and a given noise vector η, the experiment can be defined in terms of the operator y = E(g, η), which is linear with respect to both variables. Note that this is not an injective operator because the same outcome y can be produced by different combinations of model and noise. This is a restatement of the fact that the information provided by y is incomplete (M samples) and corrupt (noise η). Therefore, the operator is not invertible and no direct operation over y will provide model g. Instead, a type of set inversion will be attempted. FIGURE 104.32 General statement of the problem. m ? 2000 by CRC Press LLC Consistency Consistency is a concept that can be easily understood if we first define the set of all possible models that could have produced the a posteriori information y, in accordance with the class of measurement noise: (104.40) Therefore, S(y) ? S is the smallest set of models, according to all the available input data (a priori and a posteriori), that are indistinguishable from the point of view of the input information. This means that with the knowledge of (y, S, H5114) there is no way to select a smaller set of candidate models. The “size” of set S(y) places a lower bound on the identification error, which cannot be decreased unless we add some extra infor- mation to the problem. This lower bound on the uncertainty error holds for any identification algorithm and represents a type of uncertainty principle of identification theory. The a priori and a posteriori information are consistent if and only if the set S(y) is non-empty; otherwise, there is no model in S that could have possibly generated the experimental output. Identification Error The a priori knowledge of the real system and measurement noise present in the experiment y is stated in terms of sets S and H5114. The statement of the problem does not assign probabilities to particular models or noise; therefore, it is deterministic in nature. In addition, the modeling error should be valid no matter which model g ∈ S is the real plant (or η ∈ H5114 the real noise vector) that induces a worst-case approach. In this deterministic worst-case framework, the identification error should “cover” all models g ∈ S that combined with all possible noise vectors η ∈ H5114, are consistent with the experiments, i.e., S(y). In practice, however, the family of models conservatively covers this “tight” uncertainty set. Hence, it provides an upper bound for the distance from a model to the real plant. In this framework, the worst-case error is defined as follows: (104.41) where m(·,·) is a specific metric. The identification algorithm A maps both a priori and a posteriori information to a candidate nominal model. In this case, the algorithm is said to be tuned to the a priori information; otherwise, if it only depends on the experimental data, it is called untuned. Almost all classical parameter identification algorithms ([6]) belong to the latter class. The identification error (Eq. (104.41)) can be considered as a priori, in the sense that it takes into account all possible experimental outcomes consistent with the classes H5114 and S before the actual experiment is per- formed. Since it considers all possible experimental data y, it is called a global identification error. A local error that applies only to a specific experiment y can be defined as follows: (104.42) Clearly, we always have e(A, y) ≤ e(A). To decrease the local error more experiments need to be performed, whereas to decrease the global error new types of experiments, compatible with new a priori classes, should be performed, for example, reducing the experimental noise and changing H5114 accordingly. Convergence Now, what happens with the family of models when the amount of information increases? It is desirable to produce a “smaller” set of models as input data increases, i.e., model uncertainty should decrease. The set of models are expected to tend to the real system when the uncertainty of the input information goes to zero. Hence, an identification algorithm A is said to be convergent when its worst-case global identification error SSyy ( ) = ∈= ( ) ∈ {} ? gEg,, ηηH5114 dggS gS mAA, ( ) = ( ) []{} ∈∈ ? sup , , , ,η η H5114 H5114E egS gS mAA, sup , , ,yy y ( ) = ( ) [] ∈ () H5114 ? 2000 by CRC Press LLC e(A) in Eq. (104.41) goes to zero as the input information tends to be “completed.” The latter means that the “partialness” and “corruption” of the available information, both a priori and a posteriori, tend to zero simul- taneously. Input information is corrupted by measurement noise. Thus, “corruption” tends to zero when the set H5114 is a singleton H5114 = {0}. On the other hand, partialness of information can disappear in two different ways. By a priori assumptions when the set S tends to have only one element (the real system) or a posterior measurements when the amount of experimental information is completed by the remaining (usually infinite) data points. This can be unified as follows. The available information (a priori and a posteriori) is completed when the consistency set S(y) tends to only one element: the real system. Hence, an identification algorithm A converges if and only if (104.43) Note that as the consistency set S(y) reduces to a single element, the experiment operator tends to be invertible. Since the identification error is defined in a worst-case sense, its convergence is uniform with respect to the a priori sets H5114 and S. Algorithms and Further Research Topics There are robust identification algorithms that consider frequency domain experiments, called H ∞ –identifica- tion, this being the norm that measures the identification error. The two main ones are the two-stage and the interpolation algorithms. From time-domain measurements, several H5129 1 -identification procedures are available. Due to the fact that robust identification is a currently active research area, there are yet many theoretical and computational aspects that have not been fully developed. Among others, there are problems related to identifying unstable plants and nonuniformly spaced experimental samples. Also, sample complexity is a recent research direction, as well as the mixture of time and frequency experiments and parametric and nonparametric models. A complete description of both frequency (H ∞ ) and time (H5129 1 ) domain identification algorithms and a discussion of the issues mentioned above can be found, for example in [9]. Defining Terms BIBO stable: A system is Bounded Input Bounded Output stable if for all bounded inputs and zero initial conditions, the corresponding output is also bounded. In the case of finite-dimensional linear time invariant systems, this definition is equivalent to having all the poles of the system in the open left half plane Re(s) < 0. Control oriented identification: A deterministic identification procedure that starting from experimental data generates a model consistent with both this data and some a priori assumptions on the class of systems under consideration. Robust stability and performance: A given property of a system (such as stability or performance) is robust if it holds for a family of systems that represents (and contains) the nominal plant. Robustness margin: A quantitative measure of stability, given by the distance from the nominal model representing the system, to the nearest model lacking the property under consideration. Examples are the classical gain and phase margins. References 1. Barmish, B.R., New Tools for Robustness Analysis, Macmillan, 1994. 2. Bhattacharyya, S.P., Chapellat, H., Keel, L.H., Robust Control: The Parametric Approach, Prentice-Hall, 1995. 3. Doyle, J.C., Glover, K., Khargonekar, P., Francis, B., State-space solutions to standard H 2 and H ∞ control problems, IEEE Transactions on Automatic Control, Vol. 34, 1989. lim size S e y ()[] → ( ) = 0 0A ? 2000 by CRC Press LLC 4. Gahinet, P., Apkarian, P., A linear matrix inequality approach to H ∞ control, International Journal on Robust and Nonlinear Control, 4, 421–448, 1994. 5. Iwasaki, T., Skelton, R., A complete solution to the general H ∞ control problem: LMI existence conditions and state-space formulas, Automatica, 1994. 6. Ljung, L., System Identification: Theory for the User, Prentice-Hall, 1987. 7. M?kil?, P.M., Partington, J.R., Gustafsson, T.K., Worst-case control-relevant identification, Automatica, 31, 1799–1819, 1995. 8. Scherer, C., The Riccati Inequality and State-space H ∞ Optimal Control, Ph.D. Dissertation, Universitat Wurzburg, Germany, 1990. 9. Sánchez Pe?a, R., Sznaier, M., Robust Systems Theory and Applications, John Wiley & Sons, 1998. 10. Zhou, K., Doyle, J.C., Glover, K., Robust and Optimal Control, Prentice-Hall, 1996. Further Information Classical Identification: Ljung, L., System Identification: Theory for the User, Prentice-Hall, 1987. S?derstr?m, T., Stoica, P., System Identification, Prentice-Hall, 1989. H5129 1 Optimal Control: Dahleh, M.A., Díaz-Bobillo, I.J., Control of Uncertain Systems: A Linear Programming Approach, Prentice-Hall, 1995. LQG Optimal Control: Dorato, P., Abdallah, C., Cerone, V., Linear-Quadratic Control: An Introduction, Prentice-Hall, 1995. Kwakernaak, H., Sivan, R., Linear Optimal Control Systems, Wiley Interscience, 1972. Anderson, B.D.O., Moore, J.B., Optimal Control: Linear Quadratic Methods, Prentice-Hall, 1990. Robust Control: Doyle, J.C., Francis, B., Tannembaum, A., Feedback Control Theory, Maxwell Macmillan, 1992. Green M., Limebeer, D., Linear Robust Control, Prentice-Hall, 1995. Morari, M., Zafirou, E., Robust Process Control, Prentice-Hall, 1989. Sánchez Pe?a, R., Sznaier, M., Robust Systems Theory and Applications, John Wiley & Sons, 1998. Zhou, K., Doyle, J.C., Glover, K., Robust and Optimal Control, Prentice-Hall, 1996. Parametric Uncertainty: Ackermann, J., Robust Control: Systems with Uncertain Physical Parameters, Springer-Verlag, 1993. Barmish, B.R., New Tools for Robustness Analysis, Macmillan, 1994. Bhattacharyya, S.P., Chapellat, H., Keel, L.H., Robust Control: The Parametric Approach, Prentice-Hall, 1995. Software Packages: Balas, G., Doyle, J.C., Glover, K., Parkard, A., Smith R., μ-Analysis and Synthesis Toolbox, The MathWorks Inc., Musyn Inc., 1991. Gahinet, P., Nemirovski, A., Laub, A., Chilali, M., LMI Control Toolbox, The MathWorks Inc., Natick, MA, 1995. Safonov, M., Chiang, R., Robust Control Toolbox, The MathWorks Inc., 1988. ? 2000 by CRC Press LLC