Learning Mechanisms (Parameter adaptation) Coordination Mechanisms. Adjustable Model Compensation. Robust Control Law

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1 A Proposal Submitted to The National Science Foundation Faculty Early Career Development (CAREER) Program Title: CAREER: Engineering Synthesis of High Performance Adaptive Robust Controllers For Mechanical Systems and Manufacturing Processes Principal Investigator: Bin Yao School of Mechanical Engineering Purdue University 1288 Mechanical Engineering Building West Lafayette, IN (317) FAX: (317)

2 Project Description: Career Development Plan I. Introduction Modern mechanical systems such as machine tools, microelectronics manufacturing equipment, robot manipulators, and automatic inspection machines are often required to operate at high speeds to yield high productivity. At the same time, precision/accuracy requirements are becoming more and more stringent; two examples for such requirements are the reduced size of components in modern mechanical devices or microelectronics products and high quality surface nishing requirements in machining. Advanced controls play a signicant role in meeting these challenges. Several limiting factors in designing high performance control algorithms are: (a) nonlinearities inherent in the system dynamics, (b) uncertainties associated with the physical system, and (c) complexity due to the involvement of large quantity of dierent elements (e.g., machines, material, process, environment). In practice, though we may apply physical laws to model the system, parameters of the system (e.g., the inertia parameters of an object grasped by a robot) may depend on operational conditions and may not be precisely known in advance. Because of factors such as aging eects, the parameters may also be slowly time-varying. These types of uncertainties are called parametric uncertainties and may cause the controlled system, designed on the nominal model, to be unstable or its performance may degrade. In mechanical systems, nonlinearities such as nonlinear friction force and backlash as well as external disturbances cannot be modeled exactly. These types of uncertainties can be classied as uncertain nonlinearities. On the whole, the system may be subjected to both parametric uncertainties and uncertain nonlinearities. High performance control of uncertain dynamics is, thus, essential for successful operations of modern mechanical systems, which is the subject of the proposed research. II. Research Plan To meet the industrial need for high-performance robust controllers, the proposed research proceeds in the following ways: (i) the development of a general theoretical framework for the design of such controllers, (ii) the integration of the framework with actual sensor properties and physical properties in solving practical issues in implementation, and (iii) the application of the integrated design approach to signicant industrial processes to have an immediate impact. In this proposal, two important applications, electro-hydraulic servo systems and linear-motor driven high-speed/highaccuracy positioning systems, will be investigated since both drive systems have great potentials for widespread use in industry and are not well studied. Overall layout of the proposed research and benets is graphically shown in Appendix 1. The theory development focuses on issues universal to all physical systems and processes. Specifically, two major diculties, nonlinearities and uncertainties, will be thoroughly treated. Signicant universal practical problems such as control input saturation will be considered. Past experiences have shown that, to have a successful implementation, beyond applying any well-developed theoretical framework, particular properties of an application have to be incorporated into the problem formulation and the controller design stage to customize and ne tune the controller to satisfy the special needs of an application. Thus the second part of the proposed research focuses on the integration of the general framework with actual sensor properties and physical properties. Two general means will be emphasized: one is to judiciously select the design 1

3 model needed in the general framework based on the particular properties of a system (i.e., controloriented modeling), and the other is to actively utilize those particular properties to facilitate the controller design and the gain tuning process. In general, specic techniques for system integration are process or system dependent. Thus, the system integration portion of the proposed research will be embedded in the integrated controller design for the proposed two applications. II.1 Theory Development As mentioned before, the theory development focuses on the control of a general uncertain nonlinear system, which is organized as follows. The motivation and summary of research achievements are given in subsection II.1.a. A general framework for the proposed adaptive robust control (ARC) is then proposed in subsection II.1.b. Under the proposed general framework, some of the future research directions and solving strategies for the construction of simple and practical ARC controllers are given in subsection II.1.c. II.1.a Motivation and Summary of Research Achievements Control of uncertain nonlinear dynamics has been one of the main areas of focus in control community during the past twenty years. Two approaches have been popular: adaptive control [1, 2, 3, 39, 5] and deterministic robust control (DRC) [6, 7, 8, 9]. The PI has been doing research in the elds of both adaptive control and DRC during the past several years and has published a substantial number of journal and conference papers on the subject of robust motion and force control of robot manipulators in various operations: by using DRC in [10, 11, 12, 13, 29] and by using adaptive control in [14, 15, 16]. Those theoretical and practical experiences have helped the PI to develop a deep understanding of the limitations and advantages of both adaptive control and DRC. Recently, a breakthrough was made to preserve the advantages of both adaptive control and DRC while overcoming their limitations for a reasonably large class of nonlinear systems. Those results are briey summarized in the following paragraphs. In [17], a systematic way to combine adaptive control with sliding mode control (SMC) [30, 31, 9] for trajectory tracking of robot manipulators was developed. This is the rst theoretical result in the literature that proves the following qualitative statement: under the standard assumption of DRC, 1 one can achieve a guaranteed transient performance and a guaranteed nal tracking accuracy 2 in the presence of both parametric uncertainties and uncertain nonlinearities. At the same time, asymptotic tracking is achieved in the presence of parametric uncertainties by using a control law which is free of discontinuous control terms and innite feedback gains. In other words, we have the advantages of both adaptive control and DRC while overcoming their major practical limitations-the poor transient performance and non-robustness to uncertain nonlinearities of adaptive control and the conservative design of DRC. In [18, 19, 20], combinations of dierent adaptive control schemes and DRC schemes were developed for trajectory tracking of robot manipulators. All algorithms, as well as two benchmark adaptive control schemes [32, 33], are implemented and compared on a two-link direct-drive robot. Comparative experimental results show that compared to either 1 The bounds of the parametric uncertainties and uncertain nonlinearities are known, which is a reasonable and practical assumption. 2 the tracking error exponentially converges to a region with exponential convergence rate and the size of the region controlled by adjusting, in a known way, certain controller parameters 2

4 adaptive control schemes or DRC schemes, the proposed adaptive robust control (ARC) scheme improves tracking performance. In [21, 22, 23, 24], the approach is extended to a class of MIMO nonlinear systems transformable to a semi-strict feedback form through introducing the concept of adaptive robust control (ARC) Lyapunov functions, the backstepping design [2], and the desired trajectory initialization. The form allows coupling and the appearance of unknown parameters in the input channels of each layer, which substantially increases the design diculty. Even in the absence of uncertain nonlinearities, the problem has not been solved. The form includes the constrained motion and force control of robot manipulators in contact with a rigid surface [25], the motion and force tracking control of robot manipulators in contact with unknown stiness surfaces [26], and the motion control of machine tools [27]. In viewing the above theoretical achievements and the improvement of performance in experiments, a great need exists to formalize and generalize the approach to establish a general theoretical framework for the design of high-performance robust controllers. Such a framework is proposed in the following subsection. II.1.b General Philosophy and General Structure of ARC Controllers The proposed general framework unies the previously proposed ARC controllers through their underlining working mechanisms; this will provide a clear guideline in the design of future highperformance robust controllers. In addition, the framework will go beyond the scope of the previously proposed ARC schemes and new means will be sought and incorporated to improve performance. The proposed ARC will seek both means to achieve high performance: (i) robust lter structures will be employed to attenuate the eect of model uncertainties coming from both parametric uncertainties and uncertain nonlinearities as much as possible to guarantee certain transient performance and nal tracking accuracy in general, and (ii) mechanisms which will reduce model uncertainties (e.g., parameter adaptation) will be sought and introduced in the design whenever possible to further improve performance. The rst mean is normally accomplished by robust feedback control design and the second mean is normally done by some learning processes such as parameter adaptation. In general, the two means will interact with each other and the design techniques associated with them may have serious philosophical conicts. Thus, one of the major diculties in designing ARC controllers lies in being able to solve the conicts to integrate the two means. In some situations, some compromises have to be made. The general design philosophy is that nominal robust performance provided by the rst mean should not be lost when introducing learning mechanisms. In other words, the approach takes the viewpoint that a control law has to be robust rst (in the sense of not only stability but also performance). Learning mechanisms are introduced only when their destabilizing eects can be controlled. Such a design philosophy diers fundamentally from other schemes that recently appeared in the literature that use both the adaptation and control terms normally used in DRC. In all those schemes [34, 35, 32, 36, 37], transient performance was not guaranteed. To achieve the objectives described above, an ARC controller needs the following four components: (i) robust control law; (ii) adjustable model compensation; (iii) learning mechanisms; and (iv) coordination mechanisms. The general structure is illustrated in Fig.1. The functionality and the design strategies for each element are briey described in the following (For simplicity, learning mechanisms will be restricted to parameter adaptation). 3

5 Learning Mechanisms (Parameter adaptation) Coordination Mechanisms r Adjustable Model Compensation Robust Control Law u Plant x x y Figure 1: General Structure of ARC controllers Plant Characterization: will be considered For simplicity, the following general MIMO nonlinear plant model _x = f(x; ; t)+b(x; ; t)u + D(x; t)(x; ; u; t); y = h(x; t) (1) where y 2 R m and u 2 R m are the output and input vectors respectively, x 2 R n is the state vector, (t) 2 R p is the vector of unknown parameters and could be time-varying, h(x; t);f(x; ; t);b(x; ; t), and D(x; t) are known 3, and (x; ; u; t) 2 R l d represents uncertain nonlinearities such as disturbances and modeling errors. The following practical assumption is normally made: the unknown parameters lie in a known bounded region, and uncertain nonlinearities are bounded by some known functions (x; t) multiplied by some unknown but bounded disturbances d(t), i.e., 2 = f : min << max g and kk (x; t)d(t) where min, max and (x; t) are known. In the case where is time varying, (t) _ will be assumed to be bounded with unknown bounds. Note that these assumptions are less restrictive than those made in the previously proposed ARC controllers. Namely, unknown parameters can be time-varying and the bounds of uncertain nonlinearities do not have to be known. The relaxation of these assumptions will make the approach more applicable without losing the achievable nominal performance. Adjustable Model Compensation: In order to track a reference signal r(t), model compensation is necessary to provide the correct feedforward term. However, since the plant model has parametric uncertainties, exact model compensation is not possible. Unlike DRC design, where xed parameter estimates are used in the model compensation, here, parameter estimates will be adjusted by learning mechanisms to provide an improved model compensation. Robust Control Law: A robust control law is used to attenuate the eect of model uncertainties to guarantee transient performance and nal tracking accuracy in general. The diculty here is that the usual DRC design technique cannot be directly applied to synthesize the robust control law since DRC normally handles xed model compensation only. Thus robust control techniques, which can account for the eect of adjustable model compensation, have to be sought. The techniques used by the PI in previous ARC designs can be generalized to accomplish this task. Learning Mechanisms: The adaptation law used to update the parameter estimates can be synthesized by adaptive control techniques. In previous ARC designs, the PI only used the direct adaptive control approach, which applies to minimum phase systems only. The indirect adaptive control approach, least-squares estimation, will be studied and used for non-minimum 3 Avector or matrix is called known if each of its components is a known function of its variables 4

6 phase systems and compared to the direct adaptive control approach for minimum phase systems. Coordination Mechanisms: Coordination mechanisms are used to solve the inherent con- icts between the robust control law design and adaptive control design. The parameter estimates provided by adaptive control may go unbounded when the plant has uncertain nonlinearities and thus may destabilize the system since no robust control law can attenuate an unbounded model uncertainty. Intelligent utilization of the prior information, such as the bounds of parametric uncertainties [24], is the key to solving the destabilizing eect of parameter adaptation problem while retaining its nominal learning capability. For example, the smooth projection and other modication techniques used by the PI in previous ARC designs can be generalized to accomplish this task. II.1.c Constructing Simple and Practical ARC Controllers Under the above general framework, some future research directions and strategies are proposed in the following for the construction of simple and practical ARC controllers. Discontinuous Projection Based ARC Design For nonlinear systems with "relative degree" of more than one, the underline parameter adaptation laws in the previously proposed ARC controllers [21] and the robust adaptive control designs [40, 41] are based on the tuning function based adaptive backstepping design [2], which needs to incorporate the adaptation law in the design of control functions at each step. As a result, either smooth projections [21, 23, 20] or smooth modications of adaptation law such as the generalized -modication [17, 40] are necessary since the control functions have to be smooth for backstepping design; either method is very technical and is hard to be implemented. The PI proposes to address this problem by strengthening the underline robust control law such that the adaptation functions are not needed in the control law design at each step. As a result, the widely used discontinuous projection method in adaptive systems [39, 4] can be used in solving the conicts between the robust control design and adaptive control design. The resulting controller will be simpler and the parameter adaptation process will become more robust in the presence of uncertain nonlinearities. This strategy has been successfully tested for nonlinear systems with "relative degree" of one in [17, 19, 27]. Desired Compensation ARC Law The desired compensation structure{the regressor in the model compensation and the adaptation law depends on the reference trajectory only{has the following desirable features: (i) The regressor can be calculated o-line and thus on-line computation time can be reduced; (ii) The interaction between the parameter adaptation and the robust control law is minimized, which leads to an almost total separation of the robust control law design and parameter adaptation design; and (iii) The eect of measurement noise is minimized since the regressor does not depend on actual measurements. As a result, a fast adaptation rate can be chosen in implementation to speed up the transient response and to improve overall tracking performance. These claims have been veried by the comparative experiments on the motion control of robot manipulators [19, 20]. The PI proposes to generalize the design procedure in [19] to construct ARC controllers with such a desired compensation structure for a larger class of mechanical systems. ARC in the Presence of Control Input Saturation Performance of all physical systems are limited by the capacity of the actuators, which is normally referred to as the control input saturation. Control input saturation normally causes controller windup [42] or integrator windup [43], especially when a controller uses dynamic feedback 5

7 (e.g. adaptive controllers). So far, there is no systematic way to deal with the problem, even for linear time-invariant (LTI) systems. For LTI systems, the usual practice is to design the linear controller ignoring control input saturation and add anti-windup bumpless transfer (AWBT) compensation to minimize the adverse eects of the control input saturation on closed loop performance [42, 44, 45, 46]. Contrary to this approach, the PI proposes to treat the control input saturation as a nonlinear element in the design and actively incorporate the saturation bounds into the ARC design to optimize the achievable performance and to have a built-in anti-windup mechanism. Specically, the saturation bounds combined with the prior information on the bounds of the parametric uncertainties and uncertain nonlinearities will be used by the coordination mechanism to decide the extent of the disturbance that the system can handle so that a good anti-windup mechanism is naturally built in the resulting ARC controllers. This point has been used by a recently proposed ARC algorithm [27]. The saturation bounds will also be actively used in the robust control law design to optimize the achievable performance. This is possible since the robust control law used in the ARC design is synthesized based on min-max type nonlinear robust control techniques and the control input saturation nonlinearites could be incorporated in the design as a known active element. ARC Design for Systems with Discontinuous Parameter Variations In many applications, parameters experience discontinuous changes during the system operation. One example in mechanical systems is the Coulomb friction force, which has a discontinuous jump when the direction of velocity changes. Such a discontinuous jump will normally cause the controlled system to have a large transient response. For example, in machining applications, the largest tracking errors are normally caused by the uncompensated Coulomb friction force, which is referred to as "lost motion" or "quadrant glitches" [47, 48]. Because of the robust lter structure used in the proposed ARC design, the inevitable transients caused by these jumps will be attenuated to a large extent. But in some applications such as positioning and machining, the performance requirement is so high that any available means have to be used to reduce the transient errors. It is thus of practical signicance to incorporate new means, beside the continuous parameter adaptation to reduce model uncertainties to further improve performance. The PI proposes to develop techniques to detect the locations and the strength of jumps from the actual tracking error and then design parameter switching algorithms to catch the jumps to reduce the transient. Two possibilities will be studied and compared. The rst one is to use an idea similar to that in the multiple model switching schemes [49, 50], which use the prediction errors of multiple models as indexes to determine the parameter value to switch to. The second approach is to nd mechanisms to detect the locations and the strength of jumps directly from the actual tracking error with the assistance of the prior knowledge of the plant such as the extent of the measurement noise and the sampling time. II.2 High Performance Control of Electro-Hydraulic Servo-systems Hydraulic systems have been used in industry in a wide number of applications by virtue of their small size-to-power ratios and the ability to apply very large forces and torques; examples like electro-hydraulic positioning systems [51], active suspension control [52], and hydraulic robot manipulators [53, 54]. However, hydraulic systems also have a number of characteristics which complicate the development of high performance closed-loop controllers, and advanced control techniques have not been developed to address those issues. Other drive technologies, such as electric servo-motors, although not being able to compete directly in terms of size and power, have succeeded in displacing hydraulic drives from many precision control applications (e.g., machine tools in machining). This leads to the urgent need for advancing hydraulics technologies by combining 6

8 the high power of hydraulic actuation with the versatility of electronic control. II.2.a Issues to be Addressed The dynamics of hydraulic systems are highly nonlinear [55]. Furthermore, the system may be subjected to non-smooth and discontinuous nonlinearities due to control input saturation, directional change of valve opening, friction, and valve overlap. In addition, hydraulic servo-systems also have large extent of model uncertainties consisting of both parametric uncertainties and uncertain nonlinearities; examples of parametric uncertainties include the large changes in load seen by the system in industrial use and the large variations in the parameters (e.g., bulk modulus) that characterize the system due to the change of temperature and component wear [56]; and examples of uncertain nonlinearities are the external disturbances, leakage, and friction. Nonlinear robust control techniques, which can deliver high performance in spite of both parametric uncertainties and uncertain nonlinearities, are essential for successful operations of high-performance hydraulic servo-systems. II.2.b Proposed Solution and Preliminary Results This propose intends to apply the proposed ARC approach to handle the diculties mentioned above. To test the eectiveness of the proposed approach, the robust motion control of a typical one degree-of-freedom (DOF) electro-hydraulic servo-system shown in Fig.2 will be rst considered. The system consists of an inertia load driven by a double rod cylinder regulated by a two-stage servovalve. The goal is to have the inertia load to track any specied motion trajectory as closely as possible; examples like a machine tool axis [27] and injection molding process. P s R = 0 i P1 Q 1 P L = P1 - P 2 P2 Q Q1+ Q Q 2 L = 2 XL P1 P Figure 2: A One DOF Electro-Hydraulic Servo System Based on the physical properties of the system, the control-oriented model can be chosen as [28] mx L = P L A, b _x L, F fc (_x L )+ f(t; ~ x L ; _x L ) V t P_ L =,A _x L, C tm P L + Q L 4 e Q L = C d wx v q Ps,sgn(x v)p L _x v = 1 (,x v + Ki) In (2), the rst equation describes the dynamics of the inertia load, where x L and m represent the displacement and the mass of the load respectively, P L = P 1, P 2 is the load pressure of the cylinder, A is the ram area of the cylinder, b represents the combined coecient of the modeled damping and viscous friction forces on the load and the cylinder rod, F fc represents the modeled 7 (2)

9 Coulomb friction force, and f(t; ~ xl ; _x L ) represents the external disturbances as well as terms like the unmodeled friction forces. The second equation describes the actuator (or the cylinder) dynamics [55, 57], where V t is the total volume of the cylinder and the hoses between the cylinder and the servovalve, e is the eective bulk modulus, C tm is the coecient of the total leakage of the cylinder due to pressure, and Q L is the load ow. Q L is related to the spool valve displacement of the servovalve, x v,by the third equation, where C d is the discharge coecient, w is the spool valve area gradient, and P s is the supply pressure of the uid. Although the actual servovalve dynamics is third-order, for control-purpose, it can be modeled as a rst-order system described by the last equation of (2) since the bandwidth of the other pair of poles is normally very high [57]. As seen from (2), the system is subjected to parametric uncertainties due to the variations of m, b, F fc, e, C tm, C d, ; and K. However, if one is to adapt all the parameters, the resulting controller will be too complicated and dicult to tune. Here, physical properties of the system can be used to determine the major parameters to be adapted. Based on this consideration, only important parameters like m, e, and d n, the nominal value of the disturbance d, will be adapted rst. Other parametric uncertainties can be dealt with in the same way if necessary. The main diculties in controlling system (2) are: (i) the system has unmatched model uncertainties since parametric uncertainties and uncertain nonlinearities appear in equations that do not contain control input i; this dicult can be overcome by employing backstepping design via ARC Lyapunov function; (ii) the function relating the ow rate Q L and the valve opening x v is nonsmooth since x v appears as a discontinuous sign function sgn(x v ). This non-smooth nonlinearity has been neglected in all previous studies due to either technical requirements of the smoothness of all terms (e.g., the backstepping adaptive design in [52]) or the use of linearization techniques (either around a nominal operational point for linear controller design or feedback linearization), which demand the dierentiability of all terms. In [28], a preliminary ARC controller that can take into account of the particular nonlinearities of the hydraulic dynamics and the above non-smooth nonlinearity has been developed. A 4-DOF robot arm driven by four hydraulic cylinders (a scaled-down version of the CAT excavator arm) is being built as the test-bed for the proposed research. The immediate future research work will be: (i) obtain accurate but tractable nonlinear models of hydraulic components based on the experimental tests. This includes experimental verication and modication of the design model in (2); (ii) simplify the design procedure based on the actual sensor and actuator dynamics; (iii) implement the strategy and modify the design based on the experimental results; and (iv) generalize the integrated design to the control of multi-link electro-hydraulic systems such as industrial excavators. The PI is also applying a similar strategy to the swing control of the new-generation of excavators (see the support letter from Caterpillar Inc.). Simulation results have demonstrated the eectiveness of the proposed approach. Implementation on the actual machine is being carried out. II.3 Linear Motor Driven High-Speed/High-Accuracy Positioning Systems Manufacturing processes such as machining require the high speed movement of a machine tool to increase productivity. At the same time, tracking a desired prole with a high accuracy is required to guarantee a good surface nishing [58, 59, 60]. Other operations such as wire-bonding in semiconductor industry also require high-speed/high accuracy movement. In all these applications, motion is normally provided by a linear X-Y table. High-speed/high-accuracy robust control of such a table is thus essential for successful operations. The goal is to achieve nominal tracking errors near the measurement resolution (normally a few micrometers or nanometers) during the entire operation including transients. Conventionally, the table is driven by rotary motors with lead-screw/ballnut-type gear trains 8

10 to convert, for each axis, the motors' rotary motion into the linear motion of the table. Such mechanical transmissions not only signicantly reduce the linear motion speed and the dynamic response, but also introduce backlash, large frictional and inertial loads, and structural exibility. Backlash and structural exibility physically limit the accuracy that any control system can achieve. As an alternative, direct drive design, which uses linear motors to directly drive the table, has been recently gaining increasing popularity [61, 62, 63, 64, 65] and shows promise for widespread use in high-speed/high-accuracy operations. In general, the linear motor axis design has the following advantages over its rotary-motor counterpart: (a) no backlash and less friction, resulting in very high accuracy; (b) no mechanical limitations on achievable accelerations and velocities; achievable velocities are only limited by the encoder bandwidth or by the power electronics; (c) bandwidths are only limited by encoder resolution, measurement noise, calculation time, and frame stiness; and (d) mechanical simplicity, higher reliability, and longer lifetime. II.3.a Issues to be Addressed The linear motor drive gains high-speed/high-accuracy potential by eliminating mechanical transmission. However, it also loses the advantage of using mechanical transmission{gear reduction which reduces the eect of the table dynamics on the motor and provides the necessary stiness for processes such as machining. As a result, there is a strong dynamic interaction between the table dynamics and the linear motor. Control of the linear motor thus becomes more dicult because of parametric uncertainties in the table dynamics such as the variation of the inertia (e.g., uncertain payloads) and uncertain nonlinearities like disturbances (e.g., cutting forces in machining). Furthermore, the stiness of the axis in the motion direction is entirely determined by the servo controller since a linear motor has zero static stiness; active control is therefore necessary for proper functioning. The linear motor itself may be subject to some nonlinear eects such as force ripple. The force ripple is an electromagnetic eect caused by the variation of the motor constant with position. Since not all magnets in the linear motor are identical, feedforward compensation by measurements [64] may be too sensitive and costly to be useful. In order for a linear motor driven table to function and to deliver its promising high-speed/high-accuracy potential, a controller which can achieve the required high accuracy, in spite of the uncertainties in the table dynamics and the nonlinear eects in the linear motor, has to be employed. II.3.b Proposed Solution and Preliminary Results The ARC approach is proposed to address the control issues associated with the linear motor table because of its high performance and strong robustness to model uncertainties. To verify these claims and to see if the approach can reduce the actual tracking error to the measurement resolution level, a simple ARC controller which only estimates the disturbance is rst constructed and implemented on the X-Y table of a Matsuura 510VSS high-speed vertical machining center. The X-Y table is driven by conventional AC rotary motors and was intensively used as a testbed by Professor Tomizuka and his coworkers for various high-performance robust digital motion controllers synthesized by linear control theory. A combination of friction compensation, a disturbance observer (DOB), a position feedback controller, and a digital feedforward controller was proposed in [58, 66, 67] as a general controller structure for high performance robust motion control. Recently, instead of using the disturbance observer (DOB) in the structure, the PI applied the ARC approach to make the resulting closed-loop system robust to plant model uncertainties [27]. Extensive comparative experimental tests done on the table show the substantial improvement of performance and the excellent disturbance rejection capability of the proposed ARC [27]. For example, for tracking a circle with a radius of 20mm and a feedrate of 7m/min, with a sampling rate 9

11 of 0:4ms, the tracking errors of both X and Y-axes for the entire operation are within a level of 2.5 encoder resolution (1m). Furthermore, since ARC design actively utilizes physical properties (e.g., the inertia matrix is positive denite) and the resulting controller is passive in nature, it has the following advantages over the disturbance observer (DOB) design: (i) ARC results in better handling of the discontinuous disturbances. As a result, time-consuming and costly rigorous identication of friction model can be alleviated and overall tracking performance can be improved; (ii) Large parameter variation, such as the variation of inertia, can be easily accommodated in the ARC design as contrast to the limited range of parameter variation that DOB can handle; and (iii) ARC is better able to deal with the control saturation because the ARC design has a built-in anti-integration windup mechanism. Although the above test is done on the conventional positioning system, the resulting ARC controller structure can be modied to take into account the particular problems associated with linear motor positioning systems. Specically, the following considerations will be taken to ensure a successful implementation: The variations of parameters such as the inertia of the table can be accounted for by introducing corresponding parameter adaptations. To speed up the adaptation loop to account for the short execution time, an ARC with a desired compensation structure will be constructed. To ensure that the controller can provide sucient high servo stiness to disturbances, disturbances have to be estimated at a fast rate and compensated for while the transient eect has to be attenuated as much as possible by the robust feedback control. The possibility of success has been demonstrated by the experimental results described above. Other means such as parameter-switching, discussed in the theory development, can be incorporated to further increase the servo stiness. On-line adaptive feedforward will be added to compensate for the eect of force ripples of linear motors. In summary, an ARC controller which will enable the linear motor table to fulll its highspeed/high- accuracy potential can be constructed and a successful implementation of the controller is expected. III. Education Plan The focus of the education plan is the integration of the proposed research into the current curriculum, and the development of innovative curricula in control of dynamic systems. The aim is to educate engineers in this area to facilitate technology transfer, i.e., reduce the time for controls research results to the wider engineering community. Specically, the plan focuses on the following four major parts: Enhancing undergraduate control curriculum by emphasizing system integration and exposing students in the classroom and laboratory to advanced control approaches through hands-on experiences. Developing research projects for undergraduates and enhancing experimental facilities for both undergraduate and graduate teaching and research. Interacting with industrial users closely and developing continuing education and short courses for practicing engineers for technology transfer. 10

12 Developing courses on nonlinear adaptive and robust controls with emphasis on both theoretical background and experimental skills. The latter is achieved through system integration and extensive laboratory experiences. III.1 Enhancing Undergraduate Control Curriculum At Purdue there are 36,000 undergraduate and graduate students enrolled at the West Lafayette Campus. The School of Mechanical Engineering has more than 50 faculty members teaching more than 800 undergraduate and 300 graduate students. The School is highly regarded in terms of its graduate and undergraduate programs nationally. The PI has lectured the undergraduate core course in control (ME375). The current course follows the standard layout of classical control theory, in which only the conventional feedback structure was taught. This setting tends to narrow undergraduates' understanding about the control and does not reect the recent rapid developments in the control community. In collaboration with other faculty members in the control area, the PI has introduced some of the simple but very useful concepts in modern control theories such as the feedforward compensation into these core undergraduate courses. Introducing these new concepts will let them have a better understanding about the eld and motivate them to continue graduate study in advanced controls. Undergraduate teaching can be summarized by the saying that "tell them - they'll forget, show them - they'll remember, involve them - they'll understand". Thus, extensive hands-on experiences and laboratory demonstration are the key for undergraduates to master any kind of advanced control techniques. The proposed two experiments will be incorporated into existing laboratory courses to give students further exposure for these modern techniques. Furthermore, the two experiments cover two major elds that mechanical students have to work on: electro-mechanical systems and hydraulic systems. Working on the two experiments will also help them in mastering the knowledge in these elds. System integration is another key element for successful implementation of any control algorithm. This point has been greatly emphasized by the PI in lecturing undergraduate control courses through the integration of modeling, analysis, and the controller design based on dierent sensor information and assumptions. It corresponds well to the School of Mechanical Engineering's emphasis on concurrent engineering practice. In fact, a dual level undergraduate/graduate course on mechatronics is being developed to strengthen the link between control and mechanical design. As a controls faculty member, the PI is expected to work with other faculty members to improve the core undergraduate control courses. The PI will take an active role in strengthening the overall undergraduate program in measurements and control, ensuring a coherent and complete treatment of the important aspects of modern control and its application to industrially relevant problems. In addition, all the innovative designs and the descriptions of the laboratory set-ups that the PI will develop will be put on the Web to be disseminated to the entire control community. Direct mailing and presenting papers on the education-related conferences (e.g, ASEE meeting) will also be used to enhance the impact of the work. III.2 Developing Research Projects for Undergraduate Students Part of my education plan is to actively involve undergraduate students in my research program, which aims to provide research experience to undergraduates through direct involvement in individual or team projects. An additional goal is to increase the interaction between undergraduate and graduate students working in related projects; such an interaction should be benecial to undergraduates as they learn new ideas, and graduate students as they learn to teach and impart knowledge. Currently, the School of Mechanical Engineering has two undergraduate elective courses (ME498/499) dedicated to the individual research projects. The PI will actively contribute 11

13 his current research topics to the courses to attract undergraduates interested in advanced controls. Immediate eorts would be to involve undergraduates in the process of setting up and testing the experiments needed for this research and the courses that the PI is going to develop; this will provide them with hands-on experiences in setting up and testing a control system. III.3 Interacting with Industrial Users and Developing Continuing Education and Short Courses The methodology proposed in this work improves the performance of controlled uncertain systems and has wide applications in industries. One of the most eective ways for transferring the methodology into industrial use is through collaborative research. The Engineering School at Purdue has a long tradition of involving industry in university research. The PI has taken this excellent opportunitytointeract closely with industrial users and bring in new research topics. The University was recently awarded a center for electro-hydraulic control by Caterpillar, the largest earth-moving equipment manufacturer with an annual sale of more than 16 billion dollars. The School of Mechanical Engineering plays a major role in this interdisciplinary center and the PI has been actively involved in the center. In fact, in order to ensure a fast and successful technology transfer, the PI is actually working at one Caterpillar plant this summer and is implementing the preliminary version of the proposed ARC approach on the actual machine. The aim is to help Caterpillar to solve one of the major problems they face in the design of the new-generation of high-performance hydraulic machines (see the support letter from Caterpillar Inc.). The PI also intends to develop continuing education and short courses for engineers in industry to facilitate the technology transfer. The proposed two experiments and the laboratory facilities developed for the ME curriculum will be used to provide hands-on experiences during these short courses, including the practical applications of the methodologies taught in the courses. III.4 Developing Innovative Courses on Nonlinear Adaptive and Robust Controls In viewing the important role that advanced controls play in applications such asmanufacturing processes, the School of Mechanical Engineering at Purdue University has decided to build a strong teaching and research program in control to enhance the current mechanical engineering curriculum. Adaptive and nonlinear robust controls, as the mainstream research activities and having wide applications, have naturally become two of the focal areas. However, there are no courses on adaptive control in the current engineering curriculum even within the University and the School lacks advanced control courses emphasizing time-domain analysis and synthesis. As such, there is a demand for courses on adaptive and robust controls; this has received strong support throughout the school (see the departmental support letter). To educate engineering students on advanced controls to meet the current industrial needs, over the next ve years, two courses will be developed and integrated into the current curriculum. The rst course, an introductory undergraduate senior/graduate course entitled "Adaptive Control", will be used to introduce students to the eld of adaptive control and equip them with adequate theoretical fundamentals and hands-on experiences in using the time-domain analysis and synthesis tools. Adaptive control has a rich literature full of dierent techniques for design, analysis, and applications. There are a number of books [38, 3, 39, 5] available on the subject of adaptive control of linear systems and a newly published book [2] summarizing the recent advancements on nonlinear adaptive control. Those books provide a relatively complete technical reference on what has been achieved in the eld. Despite the rich literature on the subject, the eld still appears to an outsider as a collection of unrelated tricks and modications. Students are often overwhelmed and sometimes confused by the vast number of seemingly unrelated designs and analytical methods. Thus, it is important for students to have a broad picture of the eld while maintaining a grasp 12

14 on the essential techniques. To achieve this objective, the course will focus on the underlining mechanisms that dierent designs are based on and connect them together by their functionality. The purpose is to let students know when and how to use the designs, not the lengthy and often dicult mathematical derivations. Extensive comparative simulation and laboratory practice will be emphasized. This emphasis is necessary to give students hands-on experiences on the advantages and limitations of dierent designs and to motivate them to search for the addressing solutions in their future work. New perspectives from the PI's research on adaptive robust control will be integrated into the course to help students understand some of the fundamental problems associated with the dierent designs and know which directions to go to address the problems. The advanced course "Engineering Synthesis of High Performance Adaptive Robust Controllers" will provide graduate students with a strong theoretical background together with the experimental skills necessary to design and implement high performance robust control algorithms needed by industry. The course material will be mainly based on the proposed research on adaptive robust control. The course will not only teach graduate students specic skills on how to integrate adaptive control methods with nonlinear robust control methods, but also on how to incorporate particular properties of dierent systems into controller design processes to achieve high performance. To achieve the above objectives, the following specic measures will be taken: Comparative simulation studies will be used to demonstrate the theoretical advantages and limitations of dierent algorithms. Comparative experimental studies on dierent types of mechanical systems will be emphasized and carried out to illustrate the eect of practical limitations on the algorithms. This comparative experimental study process will not only provide students with a deep understanding of dierent control algorithms but also improve their abilities to use their physical intuition in solving the practical problems they will encounter in their future work. Controller implementation, often missed in advanced control courses, will be considered as an integrate part of the course. The proposed two experiments plus the motion control of robot manipulators will provide the necessary laboratory set-ups for the course. The course description is briey outlined in the following. The course will start with the control of linear motor driven positioning systems described in the research plan. The system is a typical example of linear systems subjected to uncertainties and bounded nonlinear eects such as the friction force and force ripples in the linear motors. The system is thus an example where linear feedback control designs do work well but may not be good enough in meeting the high performance requirement. The students will be asked to try various techniques ranging from linear feedback control, adaptive control, and robust control such as DOB mentioned before. This will give them hands-on experiences using dierent control designs and knowing their actual limitations. The proposed ARC design is then naturally introduced, which eectively integrates the digital feedforward control, adaptive control, and robust control. The course will then shift the emphasis to high performance robust control of uncertain nonlinear mechanical systems. A perfect project tting this need is the motion control of robot manipulators. A robot manipulator is constructed to simulate a human being's arm to accomplish a variety of tasks and has been widely used in industry to increase exibility and productivity. Thus, high performance control of robots is of practical signicance. Since robot dynamics are described by a set of highly coupled multivariable nonlinear dierential equations, control of such a system 13

15 is challenging, and has been extensively studied in nonlinear control community during the past decade. In this sense, a robot manipulator is also an ideal testbed for dierent nonlinear robust control algorithms. The PI's extensive research in this area will benet students directly. For example, the PI's experimental practice in [17] can be used to teach students on how to ne tune nonlinear controllers to optimize their actual performance by combining global nonlinear synthesis and local linear analysis during implementation. The course will then focus on the general ARC theory development described in the research plan, through which students can greatly enforce their analytical and synthesis abilities; essential for their future success. The control of electro-hydraulic servo-systems will be used as a practical example of the introduced new design tools. Finally, other applications such as the motion and force control of robot manipulators in various operations [26, 25] will be mentioned. The students will be encouraged to apply the general theory to the specic problems that they are working on, from which immediate benets may be seen and new problems may be generated for future study. IV. Impact of the Proposed Work Today's industry demands high productivity while being able to produce high quality products. High performance advanced controls play a signicant role in meeting these challenges. The proposed research will provide a solid foundation for designing a new generation of high performance robust controllers, and thus will have a notable impact on industry (see the support letter from Caterpillar Inc.). The proposed research also oers new insights into existing elds such as adaptive control and robust control in solving their long-standing practical problems and overcoming their limitations. The high-performance control of electro-hydraulic servo systems and linear motor driven high-speed/high-accuracy positioning systems have the potential for widespread use in manufacturing and other industrial applications, resulting in increases in productivity and improvement in product quality. The proposed research will help such systems to realize their potential. The PI is collaborating with Caterpillar to apply the proposed research to the control of electrohydraulic systems to solve their practical problems. Funding for this work will enhance the relationship with industry and facilitate the associated technology transfer. The proposed education plan will equip engineering students and practicing engineers with a strong theoretical background and the experimental skills necessary to design and implement the high performance robust control algorithms needed by industry. Departmental Endorsement The ocial eective date of the applicant's full time tenure-track appointment: 08/12/96 I have read and endorse this Career Development Plan. Frank P. Incropera, Head School of Mechanical Engineering Date: 14