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Y. M. Cho and R. Rajamani Abstract A reliable model of elevator vertical motion is of tremendous value in many aspects of elevator design, installation and service. The challenges in developing and validating a dynamic model for an elevator arise from the large size of the dynamic system involved, its position-dependent or time-varying nature and from the limited number of variables available for measurement. In this paper, a physics-based dynamic model of elevator vertical motion, scalable to varying rises, is first derived. Then, extensive experimental data is obtained from two elevator systems with rises over 100 m and 250 m. The corresponding parameters of the two elevator systems are identified via modal analysis and a numerical mode-matching procedure so that the model-predicted transfer functions may best match the experimentally estimated ones. The scalability of the model is subsequently examined to extend the validity of the model to untested elevator systems. Finally the experimentally validated model is successfully used in predicting the performance indices of high-rise elevators. |
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M. Greminger and R. Rajamani Abstract Gyroscopes are typically designed to measure angular rate of rotation. A measurement of the angle itself is useful in many applications but cannot be obtained by integrating the angular rate due to the presence of bias errors which cause a drift. This paper develops an innovative design for a vibrating gyroscope that can directly measure both angle and angular rate. The design relies on a feedback control system that maintains a constant level of vibration energy in a suspended mass. The stability of the vibrating system is shown analytically. Simulation studies indicate that the gyroscope can measure both angle and angular rate accurately for low bandwidth applications. |
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S. Larparisudthi and R. Rajamani Abstract The design of vibration isolation mounts that reduce transmission of vibrations from a machine to the structure on which it is mounted is considered. The trade-offs between relative displacement, machine acceleration and structure acceleration that limit the performance of passive isolation systems are well known. The performance of active isolation systems is investigated. Two configurations – disturbance input being a force and disturbance input being a displacement – are considered. In each case, performance is limited by the presence of invariant points in the relative deflection and acceleration transfer functions. In the case of the force disturbance input, the invariant point occurs at the structural resonant frequency. In the case of the displacement disturbance input, the invariant point does not coincide with any resonant frequency and therefore good broadband performance can be obtained. An active control law that optimizes all three transfer functions is obtained. A semi-active version of the control law is implemented on a vibration test rig using a magneto-rheological fluid damper as an actuator. Experimental results confirm the predicted performance characteristics of the semi-active vibration isolation system. |
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TURBOCHARGED DIESEL ENGINES R. Rajamani Abstract This paper addresses the problem of controlling a turbocharged diesel engine so as to minimize NOx and smoke emissions while ensuring that the driver’s torque demands are met. A diesel engine equipped with a variable geometry turbocharger (VGT) and an exhaust gas recirculation (EGR) valve is considered. The technical challenges in the control design task include the multivariable nonlinear dynamics of the system and the unavailability of key states for feedback. A control strategy based on nonlinear control synthesis is developed and shown to accurately control the air-to-fuel ratio AFR and the burned-gas-fraction in the intake manifold to desired values in the presence of changing operating conditions. The variables and AFR are shown to be crucial for feedback. Since neither of these variables can be measured, an observer based on flow and pressure sensor measurements is developed for their real-time estimation. Lyapunov theory is used to show that the developed observer is asymptotically stable. Simulation results confirm the performance of the observer and the observer-based feedback controller. The importance of the developed observer extends beyond the application discussed in this paper. It could be useful for a wide variety of different control and diagnostic applications in diesel engines. |
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J.O. Hahn, R. Rajamani and L. Alexander Abstract Vehicle control systems such as collision avoidance, adaptive cruise control and automated lane-keeping systems as well as ABS and stability control systems can benefit significantly from being made "road-adaptive". The estimation of tire-road friction coefficient at the wheels allows the control algorithm in such systems to adapt to external driving conditions. This paper develops a new tire-road friction coefficient estimation algorithm based on measurements related to the lateral dynamics of the vehicle. A lateral tire force model parameterized as a function of slip angle, friction coefficient, normal force and cornering stiffness is used. A real-time parameter identification algorithm that utilizes measurements from a differential GPS system and a gyroscope is used to identify the tire-road friction coefficient and cornering stiffness parameters of the tire. The advantage of the developed algorithm is that it does not require large longitudinal slip in order to provide reliable friction estimates. Simulation studies indicate that a parameter convergence rate of one second can be obtained. Experiments conducted on both dry and slippery road indicate that the algorithm can work very effectively in identifying a slippery road. |
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FOR A PLATOON OF AUTOMATED VEHICLES R. Rajamani , S.B. Choi, J.K. Hedrick, B. Law, R. Prohaska and P. Kretz Abstract This paper presents the design and experimental implementation of a longitudinal control system for the operation of automated vehicles in platoons. The control system on each vehicle is designed to have a hierarchical structure and consists of an upper level controller and a lower level controller. The upper controller etermines the desired acceleration for each vehicle in the platoon so as to maintain safe string-stable operation even at very small inter-vehicle spacing. The lower controller utilizes vehicle-specific parameters and determines the throttle and/or brake commands required to track the desired acceleration. A special challenge handled in the design of the lower level controller is low-speed operation that involves gear changes and torque converter dynamics. The paper also presents the design of longitudinal intra-platoon maneuvers that are required in order to allow any car in the platoon to make an exit.
The paper presents extensive experimental results from the public NAHSC demonstration of automated highways
conducted in August 1997 at San Diego, California. The demonstration included an eight-car platoon operating
continuously over several weeks with passenger rides given to over a thousand visitors. The maneuvers
demonstrated included starting the automated vehicles from complete rest, accelerating to cruising speed, allowing
any vehicle to exit from the platoon, allowing new vehicles to join the platoon and bringing the platoon to a
complete stop at the end of the highway.
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LATERAL AND LONGITUDINAL CONTROL FOR THE OPERATION OF AUTOMATED VEHICLES IN PLATOONS R. Rajamani, H.S. Tan, B. Law and W.B. Zhang Abstract This paper presents the design and experimental implementation of an integrated longitudinal and lateral control system for the operation of automated vehicles in platoons. The challenges handled in the design of the longitudinal control system include nonlinear vehicle dynamics, string-stable operation with very small inter-vehicle spacing, operation at all speeds from a complete stop to high-speed cruising, and the execution of longitudinal split and join maneuvers in the presence of communication constraints. The challenges handled in the design of the lateral control system include high speed operation using a purely "look down" sensor system and lane changing without transitional lateral position measurements. The paper also describes the design of an on-board supervisor that utilizes inter-vehicle communication and coordinates the operation of the lateral and longitudinal controllers in order to execute entry and exit maneuvers.
Extensive experimental results are included in the paper from the NAHSC demonstration of automated highways
conducted in August 1997 at San Diego, California. The demonstration included a closely-spaced eight-car platoon
operating continuously over several weeks, with passenger rides being given to visitors. The maneuvers demonstrated
included starting the automated vehicles from complete rest, accelerating to cruising speed,
automated lane keeping, allowing any vehicle to exit from the platoon with an automated lane change, allowing
new vehicles to join the platoon and bringing the platoon to a complete stop at the end of the highway.
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VEHICLE-FOLLOWER CONTROL SYSTEMS R. Rajamani and S.E. Shladover Abstract This paper is a comparative study of the performance of constant-time-gap autonomous control systems and cooperative longitudinal control systems that use inter-vehicle communication. Analytical results show that the minimum time gap that can be achieved in autonomous control is limited by the bandwidth of the internal dynamics of the vehicle. Experimental results from typical sensors and actuators are used to show that in practice it is very difficult to achieve a time gap less than 1 second with autonomous vehicle following. This translates to an inter-vehicle spacing of 30 meters at highway speeds and a theoretical maximum traffic flow of about 3000 vehicles per hour. The quality of radar range and range rate measurements pose limitations on the spacing accuracy and ride quality that can be achieved in autonomous control. Dramatic improvements in the trade-off between ride quality and spacing accuracy can be obtained merely by replacing radar range rate in the autonomous control algorithm with the difference between the measured velocities of the two cars (a rudimentary form of co-operation). As a baseline comparison, the experimental performance of fully cooperative control is presented. An inter-vehicle spacing of 6.5 meters is maintained in a platoon of 8 cooperative vehicles with an excellent ride quality and an accuracy of 20 cm. Extending this to a ten-vehicle platoon makes it possible to achieve theoretical maximum traffic flows of about 6400 vehicles per hour.
Another issue of importance addressed in the paper is the need to accommodate malfunctions in
radar (ranging sensor) measurements. Measurement errors can occur due to hardware malfunctions as well as
due to road curves, grades and the highway environment in the case of large inter-vehicle spacing. The ability of
a cooperative control system to monitor the health of the radar and correct for such errors and malfunctions
is demonstrated experimentally.
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FOR AUTOMATED VEHICLES R. Rajamani, C. Chen, A. Howell, J.K. Hedrick and M. Tomizuka Abstract A "complete" fault diagnostic system is developed for automated vehicles operating as a platoon on an automated highway system. The diagnostic system is designed to monitor the complete set of sensors and actuators used by the lateral and longitudinal controllers of the vehicle, including radar sensors, magnetometers and inter-vehicle communication systems. A fault in any of the twelve sensors and three actuators is identified without requiring any additional hardware redundancy. The diagnostic system uses parity equations and several reduced-order nonlinear observers constructed from a simplified dynamic model of the vehicle. Nonlinear observer design techniques are used to guarantee asymptotically stable convergence of estimates for the nonlinear dynamic system. Different combinations of the observer estimates and the available sensor measurements are then processed to construct a bank of residues. The paper analytically shows that a fault in any of the sensors or actuators creates a unique subset of these residues to grow so as to enable exact identification of the faulty component.
Both simulation and experimental results are presented to demonstrate the
effectiveness of the fault diagnostic system in the presence of various faults.
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R. Rajamani Abstract This paper presents some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix. The correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented. These conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix. The eigenvalues have to be located sufficiently far out into the left-half plane, and the eigenvectors also have to be sufficiently well-conditioned for ensuring asymptotic stability. Based on these results, a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability. |
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RELATION TO DISTANCE TO UNOBSERVABILITY R. Rajamani and Y.M. Cho Abstract This paper presents a systematic design methodology and some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The existence of an asymptotically stable observer is shown to be guaranteed when the distance to unobservability is larger than the Lipschitz constant of the nonlinear system. An analytical solution for the observer is provided in this case. A methodology for the use of a co-ordinate transformation is then developed so as to reduce the Lipschitz constant and increase the distance to unobservability in the new co-ordinates. The methodology is directly applicable to the important class of feedback linearizable systems. The developed theory is used successfully in the design of an observer for a flexible joint robotic system. |
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Y.M. Cho and R. Rajamani Abstract Geometric techniques of controller design for nonlinear systems have enjoyed great success. A serious shortcoming, however, has been the need for access to full-state feedback. This paper addresses the issue of state estimation from limited sensor measurements in the presence of parameter uncertainity. An adaptive nonlinear observer is suggested for Lipschitz nonlinear systems, and the stability of this observer is shown to be related to finding solutions to a quadratic inequality invovling two variables. A co-ordinate transformation is used to reformulate this inequality as a linear matrix inequality. A systematic algorithm is presented, which checks for feasibility of a solution to the quadratic inequality and yields an observer whenever the solution is feasible. The state estimation errors then are guaranteed to converge to zero asymptotically. The convergence of the parameters, however, is determined by a persistence-of-excitation-type constraint. |
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S. Srinivasan and R. Rajamani Abstract This paper presents a viable modal approach to the modeling of structural-acoustic vibrations in a paneled box. The active control of broadband structural-acoustic vibrations is a topic of current research. For a robust practical solution to this problem, a thorough understanding of the physics of the problem is needed. The model needs to be flexible to allow changes in the types, number and locations of sensors and actuators. The modal approach to modeling offers an efficient solution, since it enables quick construction of the model once the basic modes and the structural-acoustic coupling have been determined. The issue of model size is also addressed. Standard model reduction techniques in the case of very large systems suffer from numerical conditioning problems and are also highly affected by the modal damping frequency values, so that the higher modes tend to be dropped out in the reduced model. To make the model reduction insensitive to damping (which is never correctly known), a model reduction procedure is presented which uses a modal measure to determine the contribution of the different modes to the input-output map. The proposed measures are computed in a straightforward manner and do not suffer from numerical conditioning problems. The proposed model reduction procedure is found to work extremely well in reducing the model size accurately. Further, the model reduction procedure offers insight into possible control solutions to the broadband active control problem. |
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R. Rajamani and J.K. Hedrick Abstract An adaptive observer is developed for a class of nonlinear systems. Conditions for convergence of state and parameter estimates are presented. The developed theory is used for observer-based parameter identifiction in the active suspension system of an automobile. A realistic model of the suspension system incorporating the dynamics of the hydraulic actuator is used. The observer is used to adapt on dry friction which is usually present in significant magnitudes in ydraulic actuators. The observer can also be used to adapt on spring stiffness, viscous damping and hydraulic bulk modulus. A special adaptive observer is proposed for identification of the sprung mass of the autmobile. Since the sprung mass depends on the number of passengers and the load on the automobile, it needs to be regularly updated.
The adaptive observers use measurements from two acceleromters and an LVDT. They
yield good experimental performance when implemented on a half-car suspension test
rig.
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J.K. Hedrick, R. Rajamani and K. Yi Abstract This paper proposes a new methodology for designing observers for automotive suspensions. Automotive suspensions are disturbance-affected dynamic systems. Semi- active suspensions are bilinear while active suspensions with hydraulic actuators are nonlinear. The proposed methodology guarantees exponentially convergent state estimation for both these systems. It uses easily accessible and inexpensive measurements. The fact that sprung mass absolute velocity cannot be estimated in an exponentially stable manner with such measurements is also demonstrated.
Controllers using estimated states are implemented experimentally on the Berkeley
Active Suspension Test Rig. Experimental results for two cases are presented : use
of observer states to improve ride quality in an active suspension and use of observer
states to reduce dynamic tire loading in a semi-active heavy vehicle suspension.
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R. Rajamani and J.K. Hedrick Abstract A full-scale half-car suspension test rig is used to evaluate the performance of switchable dampers in semi-active control. A theoretical model is developed for the test rig and its parameters are identified. Two types of switchable shocks are used. A low bandwidth tri-state damper originally designed for manual setting changes is shown to be too slow for real-time ride quality improvement. Semi-active control is successfully demonstrated with a high bandwidth multi-state damper. Conventional ON-OFF, optimal ON-OFF, optimal multi-state control and a robust form of multi-state control are implemented and compared. Semi-active control is shown to give a performance as good as the best of all available passive states at every frequency. Trends in experimental results are accurately predicted by the theoretical model. |
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R. Rajamani, C. Srinivas and K.N. Seetharamu Abstract The finite element method is used to analyse convective heat transfer in a porous medium. Convection past a vertical surface embedded in the medium and convection in a confined porous medium enclosure are analyzed using the above method. The results are compred with those available in the literature and the agreement is found to be good. The method is applicable for two-dimensional analysis in a porous body of any arbitrary shape. The restriction of the boundary layer assumption is relaxed. |
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R. Rajamani, C. Srinivas, P. Nithiarasu and K.N. Seetharamu Abstract A finite element method employing Galerkin's approach is developed to analyze free convection heat transfer in axisymmetric fluid saturated porous bodies. The method is used to study the effect of aspect ratio and radius ratio on Nusselt number in the case of a porous cylindrical annulus. Two cases of isothermal and convective boundary conditions are considered. The Nusselt number is always found to increase with radius ratio and Rayleigh number. It exhibits a maximum when the aspect ratio is around unity; maximum shifts towards lesser aspect ratios as Rayleigh number increases. Results are compared with those in the literature, wherever available, and the agreement is found to be good. |
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