Random delay effect minimization on a hardware-in-the-loop networked control system using optimal fractional order PI controllers ⋆ Varsha Bhambhani ∗ , Yiding Han ∗ , Shayok Mukhopadhyay ∗ , Ying Luo ∗ and YangQuan Chen ∗ ∗

Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322 − 4160, USA

Abstract: Random delays have serious effects in networked control systems, which deteriorate the performance and may even cause instability of the system. Hence a controller which can make the plant stable at large values of delay is always desirable in NCS systems. Our previous work on OFOPI controller showed that fractional order PI controllers have larger jitter margin (maximum value of delay for which system is stable) for lag-dominated systems as compared to traditional PID controllers, whereas integer order PID controllers have larger jitter margin for delay-dominated systems. In this paper, a telepresence controller based on optimal fractional order proportional and integral (OFOPI) tuning rules is used to obtain the maximum value of delay (jitter margin) at which the system will be stable. To illustrate this, an extensive experimental study on the real-time Smart Wheel networked speed control system is performed using hardware-in-the-loop control. For this purpose, the real-time random delay in the world wide network is collected by pinging different locations, and is considered as the delay in our simulation and experimental systems. Comparisons are made with existing integer order PID controller. It is observed that the proposed OFOPI controller is a promising controller and has faster response time than the traditional integer order PID controllers. Also it is verified that since the plant into consideration viz. the Smart Wheel is a delay-dominated system, PID achieves larger jitter margin as compared to OFOPI tuning rules. Simulation results are presented to illustrate the effectiveness of the proposed OFOPI method. keywords Networked control system, optimum fractional order proportional integral controller, Smart Wheel, random delays, hardware-in-the-loop. 1. INTRODUCTION A networked control system is a feedback control system in which the control loop is closed through a communication network [Beldiman, 2001]. In recent years, due to its costeffectiveness and flexible applications, the research on the use of a data network in a control loop has gained increasing attention [Tipsuwan and Chow, 2003, Zhang et al., 2001, Lian et al., 2001]. Normally, the objective of the networked control system is to make use of the finite network capacity to achieve system stability and good closed loop performance, including stability, rise time, overshoot and other design criteria [Beldiman, 2001, Ramaswamy et al., 2008]. Two approaches are commonly used in the design of networked control systems. One approach is to make modifications to the network protocol for a given plant and controller (in this case, the controller is designed in advance without considering network-induced effects). The other method, followed in this paper, is to consider networkinduced effects and set the design criteria as well as network protocol definition in the controller design for a given plant. Random delays in the network are a serious problem, which deteriorate the performance and may even cause instability of the system. A controller which could ensure the stability of system at large values of delay will revolutionize internet based controllers. Such a jitter-robust controller based on fractional order proportional and integral (FOPI) tuning rules, proposed in [Bhambhani et al., 2008],Bhambhani [2008] is used for speed control of stand-alone Smart Wheel platform at CSOIS. For this purpose, the real random delay in the network is taken into account by pinging different locations and then considering it as the delay in our simulation and experimental systems. ⋆ Corresponding author: Professor YangQuan Chen. 1(435)7970148; F: 1(435)7973054; E: [email protected]; http://fractionalcalculus.googlepages.com/

T: W:

Simulation results are presented to illustrate the effectiveness of the proposed OFOPI controller by a comparison between the fractional order controller and the integer order PID controller [Eriksson and Johansson, 2007a]. This paper is organized as follows. Section 2 describes the “Smart Wheel” system and briefly explains its system identification using step response analysis. Section 3 introduces the OFOPI controller design and tuning rules. Section 4 describes the laboratory setup of the networked control system for speed control of the steering axis of the Smart Wheel using hardwarein-the-loop implementation. Section 5 shows the results for the simulated networked control of the Smart Wheel using OFOPI controller at random time delays. This is compared to optimal PID [Eriksson and Johansson, 2007a,b] controller. Section 6 presents the real-time hardware-in-the-loop control. Finally Section 7 summarizes the results and provides concluding remarks. 2. ARCHITECTURE OF THE CSOIS SMART WHEEL The Smart Wheel is a self-contained robotic wheel which has a steering axis, drive axis and z-axis, each of which can be actuated independently [Flann and Moore, 2000]. The CSOIS has a stand-alone Smart Wheel assembly (shown in Fig. 1). It is equipped with steering and drive motors. It also has a linear actuator for z-axis movement. The Smart Wheel also has a power distribution unit, drive circuitry for the motors and actuators, encoders for drive and steering feedback and a microcontroller. Furthermore, the Smart Wheel’s microcontroller is connected to a serial server. Hence anyone with a computer connected to the network can control the Smart Wheel. The microcontroller on the wheel polls the position encoder on the Smart Wheel. The data read is transmitted over the network through its serial

noise when differentiated to get velocity, hence a proper data length selection is very important. The Smart Wheel reports data in a particular format, hence it is parsed to get the value of steering angle (θ) and time (t) from it. Also an s-function is written which implements the same MATLAB code for communication. The s-function block properly sizes the θ and t vectors so that they are of the same length and are synchronized. Figure 2 shows the steering speed of Smart Wheel corresponding to a control input of 19. X-axis is the time in seconds and Y-axis is the steering speed of Smart Wheel obtained by taking derivative of θ with respect to t in radians/seconds.

Fig. 2. Step Response analysis of Smart Wheel

Fig. 1. Stand-alone Smart Wheel on a mobile rig port. It accepts velocity values through the serial port and converts them to pulse width modulated (PWM) signals. These signals drive the motor. An operator with a computer and access to the internet can install a virtual COM driver and can gain access to the Smart Wheel. Algorithms can be used to read the data from the Smart Wheel and to calculate the wheel’s velocity based on the encoders position data. A controller can thus be designed on a remote computer to send velocity commands to the Smart Wheel. An internet camera (DLink - model DCS5300) [DLink Systems Inc., 2003] located near the Smart Wheel assembly sends streaming audio and video of the wheel’s motion to the remote computer. The audio and video streams are independent of the controller data which uses the serial server. The encoder data can be plotted on the screen of the remote computer to analyze the performance of the closed loop system. 2.1 System Identification of the Smart Wheel The first step is the system identification of the the stand-alone Smart Wheel assembly at the CSOIS. The system identification is also done through a hardware-in-the-loop simulation. For this code was written in MATLAB to enable communication between the remote computer and the Smart Wheel. The first step is to connect to the port on the Smart Wheel and obtain access to it. This is done by sending the REMOTE REQ or ASCII character 47 to the Smart Wheel. The Smart Wheel responds with a REMOTE ACCEPT or ASCII character 33 in its buffer. Connection is verified if this character is received. When connected, data is read in from the buffer on the Smart Wheel. Size of the data available is limited by the size of the buffer. Longer data produces greater computational delay and

Furthermore, it should be noted that the input issued to the motor over the network is in the range of (0 to ±19), with the signs representing the direction of rotation. Also, since the input is not 1 or 5 volts, as is the case in a traditional step response system ID, the value of K obtained for the motor is low. A hardware in loop implementation of the Smart Wheel is done using an s-function block which handles communication between the plant and the remote computer. The sampling interval for the s-function block in Simulink is set equal to the time needed to read data from the buffer to attain synchronization. A system identification is performed by obtaining the step response of the Smart Wheel. From the step response, the transfer function for the FOPDT model of the Smart Wheel is obtained as P (s) =

K 0.1484 −0.592s e−Ls = e Ts + 1 0.045s + 1

(1)

3. OFOPI DESIGN METHODS & PRACTICAL TUNING RULES The past decade has seen an immense amount of research work on fractional order controller design and tuning methods [Oldham and Spanier, 1974, Vinagre and Chen, 2002, Chen et al., 1977, Lubich, 1986, Podlubny and Misanek, 1993, Podlubny, 1994a,b]. This section provides a brief summary of the design method and tuning strategy developed in [Bhambhani et al., 2008]. The motivation for the research was from ideas developed in [Eriksson and Johansson, 2007a,b, Bhaskaran et al., 2007b,a] where tuning rules for PI/PID controllers where developed for a class of systems which can be approximated with a good FOPDT model. The objective was to design an optimal controller such that the jitter margin and system performance are maximized and yet the closed loop feedback system is robust and stable. For this a multi-objective optimization method was used which simultaneously minimizes two objective functions namely the ITAE factor and jitter margin which are functions of controller gain parameters x bounded by

some non-linear equality and inequality constraints. Expressed mathematically, the two objective functions targeted are: Z ∞ O1 (x) = t|e(t)|dt (2) 0

and

O2 (x) =

1 δmax

Here δmax can be computed from (3) as: 1 + G(jω)C(jω) δmax = min | | (3) ωǫ[0,∞] jωG(jω)C(jω) Hence the multi-objective optimization problem takes the form as in (4) and (5). Minimize O(x) = [O1 , O2 ] (4) at x = [Kp , Ki , α] such that x satisfies the equality and inequality constraints given by ( |D + G(iω)C(iω)|2 ≥ R2 i = 1, · · · , a1 σ = ∂|D + G(iω)C(iω)|2 (5) = 0 i = 1, · · · , a2 ∂ω Here objective function O1 (x) is the ITAE criterion and O2 (x) is the inverse of jitter margin. These values should be minimized while still ensuring robustness of the system. The set of equations defined by σ ensures robustness and stability. The inequality constraint |D + G(iω)C(iω)|2 is the sensitivity constraint and is a function of Kp , Ki , α and ω and must be greater than R2 . Here D and R are the center and radius of the circle which encloses both the Ms and Mp circles and are given by (6) as Ms − Ms Mp − 2Ms Mp2 + Mp2 − 1 D= (6) 2Ms (Mp2 − 1) Ms + Mp − 1 R= 2Ms (Mp2 − 1) Ms and Mp are the maximum absolute values of sensitivity and complementary sensitivity functions respectively. Furthermore, |1 + G(iω)C(iω)|2 = 0 is the stability region of the sensitivity constraint and satisfies the boundary condition at critical point or the point at which D = 1 and R = 0. For more information read [Bhaskaran et al., 2007b,a] The OFOPI tuning rules obtained by the above procedure as derived in [Bhambhani et al., 2008] are stated as: 0.2T Kpo = + 0.16 (7) L 0.25 0.19833 Kio = + + 0.09 TL L o α = τ − 0.04L + 1.2399 L where τ = L+T . Whereas the PID tuning rules for varying time delay systems [Eriksson and Johansson, 2007a,b] are listed as: 0.4T − 0.04 0.16 k= + (8) Kp L Kp −0.11T 3 + 1.5T 2 − 1.5 0.35T 2 + 4T + 50  ki = 0.01 + Kp L2 Kp L 0.4T 2 + 11T  kd = 0.01 Kp It should be noted that the OFOPI tuning rules derived in [Bhambhani et al., 2008] were based on a set of FOPDT systems such that their delay values are L = [1, 2, 3, ... , 9, 10]T , values of the time constant are T = [1, 2, 3, ... , 9, 10]T and

Fig. 3. Laboratory set up and Hardware-in-the-Loop steady state gain K = 1, whereas the Smart Wheel lies out of the range. Efforts were made to simulate another set of FOPDT systems with time delay and time constant values in the range L = [0.1, ... , 1]T and T = [0.1, ... , 1]T . However due to computational limitations and frequency mismatch, this still remains a difficult task and is one of the major challenges. Similar difficulties were encountered for a PID controller using the approach found in [Eriksson and Johansson, 2007a] for the new set of FOPDT systems, although the range of plants considered in that paper were L = [0.1, 1, ... , 10]T , T = [0.1, 1, ... , 10]T and Kp = 1. To see how well the existing OPID/OFOPI tuning rules perform for the Smart Wheel system at different random delays, real-time experiments were conducted. It was noticed that the tuning rules still hold true i.e the networked system is still stable for delays less than jitter margin. However the upper bound of jitter margin increases by 60% to 70%. This was concluded from simulating several systems lying in the new range. ′′ = 1.6 ∗ δmax (9) δmax ′′ where, δmax is the jitter margin for new set of FOPDT systems at the same OFOPI/OPID tuning tuning parameters.

4. LABORATORY SET UP AND HARDWARE-IN-THE-LOOP This section gives a brief introduction about the implementation of OFOPI control of the steering axis speed of the Smart Wheel over the network loop. Figure 3 shows a simple implementation of a OFOPI controller with the Smart Wheel in the loop. The block labeled ‘smartwheel’ is the actual block which handles the entire process of communicating remotely with the hardware. Operational details of that block are illustrated with help of a flow chart shown in Fig. 4. As inferred from the flowchart, the first step is to connect the Smart Wheel’s port and obtain access to it. The connection process is done at the start of the simulation. A count is maintained and connection is performed only on the first ever operation. In the event of timeout or no receipt of connection code, the simulation shuts down. When connected, data is read in from the buffer on the Smart Wheel. Size of the data available is limited by the size of the buffer, longer data produces greater computational delay and noise when differentiated to get velocity, hence a proper data length selection is very important. The Smart Wheel reports data in a particular format, hence data is parsed to get the value of steering angle (θ) and time(t) from it. The s-function properly sizes the θ and t vectors so they are the same length and are synchronized. The current velocity and the error between the setpoint and the current velocity is obtained next. If the error is zero then the s-function block issues a REM OT E REQ command, ASCII character 92, which issues a stop command and the wheel stops spinning. For non-zero error, the controller produces some output. The steering motor rotates clockwise when it receives ASCII values 70 to 89 (70 - stopped, 89 maximum speed). It rotates anticlockwise when it receives

Table 1. Gain parameters for different PID controllers Aus

PID

China

PID

Kp 1.524 0.8866 1.5015 0.91103

AMIGO OPID AMIGO OPID

Ki 0.43189 4.2004 0.38166 3.684

Kd 0.009323 0.03341 0.00952 0.03341

JM 1.8696 1.5596 2.1424 1.7739

Table 2. Gain parameters for different FOPI controllers Aus

FOPI

China

FOPI

FMIGO OFOPI FMIGO OFOPI

Kp 2.1227 0.1716 2.1079 0.17015

Ki 4.5671 7.524 3.9876 6.5769

α 1.1 2.154 1.1 2.1561

JM 0.2168 0.5550 0.2186 0.6384

parameters, OFOPI and OPID controllers are used to control the steering speed of the Smart Wheel for different delays. Figure 5 and Fig. 6 show the simulated result for random delays obtained by pinging the university in China whereas Fig. 7 and Fig. 8 are for random delays obtained by pinging the Australian university.

Steering speed of Smart Wheel (Radians/Second)

Simulated network control of steering speed of Smart Wheel using OFOPI tuning rules: China 4 L(t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 1.25* JM’’

Fig. 4. Flowchart describing working of Smart Wheel ASCII values 70 to 51 (70 - stopped, 51 maximum speed). The control input lies in the range of [0, 20) for either direction of rotation. As seen in Fig. 3, a saturation block is placed in the loop which modulates the controller output to acceptable values. This necessitates the use of an anti-windup feedback block to the integrator to prevent integrator saturation. The correct values are fed into the s-function block which communicates with the Smart Wheel and sends appropriate commands. In each cycle, the s-function block reads data, parses data, calculates error, takes input from the controller, sends commands to the Smart Wheel and plots current received data to the screen.

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Fig. 5. Simulated Network control of steering speed of Smart Wheel using OFOPI: China Here L(t) denotes the random delay and includes both the network delays and inherent delays due to computation and other miscellaneous factors, JM represents δmax and JM ′′ ′′ represent δmax . It can be said that the OFOPI controller has a faster response as compared to OPID controller. Also the simulation results are as expected.

5. SIMULATION ILLUSTRATION The next step is to compute the controller gain parameters for the two controllers, i.e. OFOPI and OPID based on equations (7) and (8). Two different sets of random time delays are obtained by pinging two different places located far apart from the Smart Wheel platform. These are HuaZhong University of Science and Technology in China and the Australian National University. The gain parameters so obtained for different controllers are listed in Tables 1 and 2, where AMIGO / FMIGO stand for Approximate / Fractional Ms Constrained Integral Gain Optimization. It should be noted that the jitter margin values for OPID are larger as compared to OFOPI controller, which is true as the Smart Wheel is a delay dominated system. Based on the above

6. EXPERIMENT ON THE REAL-TIME SMART WHEEL SPEED CONTROL SYSTEM The final step is the experiment. Figure 9 and Fig. 10 show the real-time results for random delays obtained by pinging the university in China whereas Fig. 11 and Fig. 12 are for random delays obtained by pinging the Australian university. Noise due to sensors and actuators are an inherent part of real-time network control unlike simulation results. Furthermore, oscillations are observed due to delays and physical phenomenon like suspension movements, frictional changes and motor-cogging. Also the motor velocity has a lower bound due to deadzone. As can be seen in above figures, the OFOPI controller has a faster response as compared to the PID controller. Also for

Simulated network control of steering speed of Smart Wheel using OPID tuning rules: Australia 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 1.25 * JM’’ Steering speed of Smart Wheel (Radians/Second)

Steering speed of Smart Wheel (Radians/Second)

Simulated network control of steering speed of Smart Wheel using OPID tuning rules: China 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 1.25* JM’’ 3

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Fig. 6. Simulated Network control of steering speed of Smart Wheel using OPID: China

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Real−time network control of steering speed of Smart Wheel using OFOPI tuning rules: China 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 0.25*JM’’

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Fig. 8. Simulated Network control of steering speed of Smart Wheel using OPID: Australia

Simulated network control of steering speed of Smart Wheel using OFOPI tuning rules: Australia 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 1.25* JM’’ Steering speed of Smart Wheel (Radians/Second)

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Fig. 7. Simulated Network control of steering speed of Smart Wheel using OFOPI: Australia every case considered, the system becomes more and more unstable as the delay is increased and finally reaches instability ′′ at delay greater than δmax . Moreover, the results obtained in real-time are in confirmation with simulation results obtained in Section 5. 7. CONCLUSION This paper presents an intensive study and experimental work on networked control of the steering speed of the stand-alone Smart Wheel platform at CSOIS, Utah State University. Optimal fractional order proportional and integral tuning rules are used to determine the parameters at which the system has maximum delay and yet is stable. An s-function block is used

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Fig. 9. Real-time Network control of steering speed of Smart Wheel using OFOPI: China

in Simulink to implement the Smart Wheel in the network loop. It is noticed that since the Smart Wheel is a delay-dominated system, it will result in higher jitter margin using an OPID controller. This is in confirmation with the results obtained in the work of [Bhambhani et al., 2008]. However, it is seen that for systems with L and T in the range [0.1, · · · , 1], the upper bound on jitter margin is increased manyfold. Also the OFOPI controller has a faster performance as compared to OPID controller. Hence, the work presented in this paper justifies that the OFOPI tuning rules work well in the networked control setting having random delays.

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Real−time network control of steering speed of Smart Wheel using OFOPI tuning rules: Australia 4 L (t) L (t) + JM 3.5 L (t) + JM’’ L(t) + 1.25* JM’’ Steering Speed of Smart Wheel (Radians/Second)

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Fig. 10. Real-time Network control of steering speed of Smart Wheel using OPID: China

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Real−time network control of steering speed of Smart Wheel using OPID tuning rules: Australia 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 0.25* JM’’ Steering speed of Smart Wheel (Radians/Second)

Steering speed of Smart Wheel (Radians/Second)

Real−time network control of steering speed of Smart Wheel using OPID tuning rules: China 4 L (t) L (t) + JM L (t) + JM’’ 3.5 L (t) + 1.25* JM’’

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Fig. 11. Real-time Network control of steering speed of Smart Wheel using OFOPI: Australia ACKNOWLEDGEMENT Ying Luo is supported by the Ministry of Education of the P. R. China and China Scholarship Council(CSC). The authors acknowledge the benefits from the weekly Fractional Calculus Reading Group meeting at CSOIS (http://mechatronics.ece.usu.edu/foc/yan.li/). REFERENCES O. Beldiman. Networked Control Systems. Ph.D. Dissertation, Dept. of Electrical and Computer Eng., Duke University, 2001. V. Bhambhani, Y. Q. Chen, and D. Xue. Optimal fractional order proportional integral controller for varying time-delay systems. 17th IFAC World Congress., July 2008. Varsha Bhambhani. Optimal fractional order proportional integral controller for processes with random time delays. Master’s thesis, Utah State University, 2008.

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Fig. 12. Real-time Network control of steering speed of Smart Wheel using OPID: Australia T. Bhaskaran, Y.Q. Chen, and G. Bohannan. Practical tuning of fractional order proportional and integral controller (II): Experiments. Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, September 2007a. T. Bhaskaran, Y.Q. Chen, and D. Xue. Practical tuning of fractional order proportional and integral controller (I): Tuning rule development. Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, September 2007b. C. F. Chen, Y. T. Tsay, and T. T. Wu. Walsh operational matrices for fractional calculus and their application to distributed system. J. of Franklin Institute, 303:267–284, 1977. DLink Systems Inc. DLink Securicam DCS 5300 Internet Camera Datasheet,. 2003. L. M. Eriksson and M. Johansson. PID controller tuning rules for varying timedelay systems. Proceedings of the 2007 American Control Conference, July 2007a. L. M. Eriksson and M. Johansson. Simple PID tuning rules for varying timedelay systems. Proceedings of the 46th IEEE Conference on Decision and Control, December 2007b. N. Flann and K. L. Moore. A six-wheeled omnidirectional autonomous mobile robot. IEEE Control Systems Magazine, 2000. F. L. Lian, J. Moyne, and D. Tilbury. Anaysis and modelling of networked control systems: MIMO case with multiple time delays. Proceedings of American Control Conference, June 2001. C. H. Lubich. Discretized fractional calculus. SIAM J. Math. Anal., 17(3): 704–719, May 1986. K. B. Oldham and J. Spanier. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974. I. Podlubny. Numerical solution of initial value problems for ordinary fractional-order differential equations. In W. F. Ames, editor, Proceedings of the 14th World Congress on Computation and Applied Mathematics, pages 107–111, Atlanta, Georgia, USA, July 1994a. Late Papers volume. I. Podlubny. Numerical methods of the fractional calculus. Transactions of the Technical University of Kosice, 4(3-4):200–208, 1994b. I. Podlubny and J. Misanek. The use of fractional derivatives for modelling the motion of a large thin plate in a viscous fluid. In Proceedings of the 9th Conference on Process Control, pages 274–278, STU Bratislava, May 1993. Tatransk´e Matliare. B. Ramaswamy, Y. Q. Chen, and K. L. Moore. Omni-directional robotic wheel - a mobile real-time control systems laboratory. International Journal of Engineering Education (SmarWheel web: http://www.neng.usu.edu/ ece/csois/people/smartwheel/), 24(1), Jan 2008. Y. Tipsuwan and M. Y. Chow. Control methodologies in networked control systems. Control Engineering Practice, 11, 2003. Blas M. Vinagre and YangQuan Chen. Lecture Note on Fractional Calculus Applications in Automatic Control and Robotics. The 41st IEEE CDC2002 Tutorial Workshop # 2, http://mechatronics.ece.usu.edu /foc/cc02tw/cdrom/lectures/book.pdf Las Vegas, NE, USA., December 2002. W. Zhang, M. S. Branicky, and S.M. Phillips. Stability of networked control systems. IEEE Control Systems Magazine, pages 84–99, February 2001.

Random delay effect minimization on a hardware-in-the ... - CiteSeerX

Science and Technology in China and the Australian National. University. The gain .... Ying Luo is supported by the Ministry of Education of the P. R. China and China ... systems. IEEE Control Systems Magazine, pages 84–99, February 2001.

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Confidence Sets for the Aumann Mean of a Random ... - CiteSeerX
Almost the same technique can be used to obtain a confidence region for a set ..... Hajivassiliou V., McFadden D.L., Ruud P.: Simulation of multivariate normal rectangle ... Ihaka R., Gentleman R.: R: a language for data analysis and graphics.

A Random Field Model for Improved Feature Extraction ... - CiteSeerX
Institute of Automation, Chinese Academy of Science. Beijing, China, 100080 ... They lead to improved tracking performance in situation of low object/distractor ...

Peer review delay and selectivity in ecology journals - CiteSeerX
Reviewers are keen to do a prompt review job for these relatively few and well regarded ... their manuscripts if the journal has a good reputation, due to the widespread feeling that it is important to .... IEEE Computer, 23, 46–51. Tobin, M. J. ..

Random-effect models for longitudinal data
No requirement for balance in data (good for longitudinal study). Explicit analysis of ... Definition (Stage 2) bi are i.i.d. distributed as N(0,D), independent of ei .