Contents overview

This dissertation is organized as follows: the Taylor estimator for different data dropout rates is presented in Chapter 2. Chapter 3 introduces the real-time transition probability estimator and Chapter 4 presents the model-free intelligent message estimator (IME) based on the real-time estimated transition probability. In Chapter 5, a switching system dynamics modeling and its stability analysis are introduced for motion NCS. In Chapter 6, the PDC compensation for the delay in NCS is introduced in motion NCS and its integration with the IME together solve the two major problems in NCS. Finally, conclusions and recommendations for further research are provided in Chapter 7.

Chapter 2

The Taylor message estimator for motion NCS

Although NCS possesses some advantages such as low cost, extensibility, flexibility, and easy maintenance, the unavoidable time-delay effect induced in the network seriously degrades its control performance and also reduces system reliability and stability. Recently, different approaches were proposed for NCS to mainly compensate for the time-delay effect like the queuing methodology (Luck and Ray, 1994), the sampling time scheduling (Kim et al., 1996), the gain scheduling PI control (Tipsuwan and Chow, 2004), scheduling and control co-design (Lu et al., 2004), and robust control (Chen et al., 2006; Jiang and Han, 2006).

A general network system is basically an event-triggered system and the time delay is mainly concerned. However, the real-time motion NCS, which includes the fixed sampling time and an interpolator to conduct provided motion contouring commands, is a typical event/time-triggered system (Hsieh and Hsu, 2005). When the network communication becomes heavy, some network nodes may not properly receive/transmit messages on time and the data dropout may thus occur. In general, the dropout rate of the network is closely related to both the network transmission rate and the specified sampling period. The data dropout not only increases system uncertainty of the NCS, but also it degrades motion accuracy in tracking and contouring (Hsieh et al., 2006). A Markov chain with two states treated as the vacant sampling can be applied to model the data dropout in a stochastic nature (Nilsson, 1998). Moreover, to handle the data dropout, there are two approaches: (1) using the past control signals to estimate the lost data (Ling and Lemmon, 2002) and (2) including the estimator which based on the power spectral density of NCS output signals (Ling and Lemmon, 2002; Ling and Lemmon, 2004).

A message estimator is proposed for the motion NCS to compensate for its dropout effect. Both simulation and experimental results have shown that a message estimator with a 3^rd-order Taylor expansion is effective for estimating missing motion signals. Moreover, the motion control performance on the NCS is satisfactory by including the estimator in the controller. Since the proposed structure leads to a more reliable motion NCS with less uncertainty, it has been successfully integrated with the

feedforward control design to achieve high-precision motion accuracy (Tomizuka, 1978; Yeh and Hsu, 1999). The proposed network control structure has been successfully realized on a DYNA MTYE 1007 CNC machine tool to prove the feasibility of the present motion NCS.

2.1 Data dropout effects

Motion systems with synchronized control on multiple axes are designed mainly to meet specifications of precision accuracy in tracking or contouring. When motion control systems are realized on the NCS, the data bus containing either the command messages or the feedback measurements are transmitted through the network protocol, as shown in Fig. 2.1. The induced time delay in the NCS is unavoidable and the transmitted message may miss the hard real-time deadline, the sampling time T, and it always leads to erroneous motions in precise systems. Thus, the caused data dropout is crucial to motion performance in the real-time NCS. For the controller area network (CAN) bus, Table 2.1 shows all experimental measurements of the dropout rate with different transmission rates and sampling periods. Experimental results indicate that the dropout rate significantly decreases as the sampling period increases. Note that the control performance of the system also decreases as the sampling period increases in NCS (Lian et al., 2002). Therefore, to select a proper sampling time for the NCS, it is a trade-off by concerning between the network transmission and the control performance.

Fig. 2.1 Networked motion control systems

Table 2.1 The data dropout rate of CAN bus transmission rate +

-

) (t

Remote

y

System

)

(t r

CAN bus (delay )



Controller

Master Slave

Feedback Sensor

Transmission rate Dropout rate ( )

ms T

2

Dropout rate ( )

ms T 1



1M bit/s 0.48% 0.49%

500 bit/s 0.51% 19.97%

250 bit/s 20.21% 42.14%

The data dropout occurs randomly on the network transmission either in command or feedback measurement signals. Actually, the dropout commands are properly estimated since most commands are relatively smooth compared with the measurements (Hsieh and Hsu, 2005; Hsieh et al., 2006). Therefore, this paper focuses on compensating the effect of the measurement data dropout. To model the data dropout in transmitting the feedback message data, Fig. 2.2 shows that dis a binary process with probability distribution of

P

( k

d

[ ] )1 



,

P

( k

d

[ ]0)1



, and the data dropout occurs when

d

[

n

]1 [11-12]. The transmitted feedback signal

y

[n] is modeled as













1 ] [ ,

0 ] [

0 ] [ ], [ ] [

k d if k

y

k d if k y k

y

dropout

 (2.1)

Fig. 2.2 Modeled NCS with data dropout.

Fig. 2.3 Experimental result with data dropout rate = 19.97%

Experimental results with an 1 ms sampling period and a 500K bit/s transmission rate are shown in Fig. 2.3. Results indicate that the data dropout occurred in the feedback data directly affects the system performance. In the present experiments, the missing feedback messages are all treated as 0 values and it makes the designed controller to loss efficacy. To compensate for the dropout data, the designed message estimator

) (z

F

is shown in Fig. 2.4 and the NCS can be expressed as in the following:



Fig. 2.4 NCS with the dropout compensator (Ling and Lemmon, 2004)

The power spectral density of the system output response are as in the following (Ling and Lemmon, 2004):

 , is the unique positive solution to the following equation



where w is the frequency. Moreover, the networked control system shown in Fig. 2.4 can be transformed to an LTI system as shown in Fig. 2.5. The optimal dropout compensator

F

(z) can be thus designed by minimizing the power spectral density of the output response

y

[k] under the noise contamination

n

[k] (Ling and Lemmon, 2002; Ling and Lemmon, 2004).

Fig. 2.5 Equivalent LTI systems (Ling and Lemmon, 2004)

2.2 The Taylor message estimator

Based on the structure shown in Fig. 2.4, the output of a message estimator will estimate the missing message when the dropout happens as in Eq. (2.2). The missing messages can be thus recovered to some extents to improve performance of the motion NCS. For the messages in a relatively low frequency, the improvement of control performance with a simple 1-delay message estimator is acceptable (Ling and Lemmon, 2002). However, as the frequency of the transmitted/received signals increases, the motion NCS owns faster dynamics and the improvement of the 1-delay message estimator is thus limited.

2.2.1 The order of the estimator

In the present paper, a Taylor message estimator is proposed for the motion NCS, because most dynamics of motion commands or motion measurements can be represented by a Taylor expansion with a suitable order except the motion commands

containing significant variation, as shown in Fig. 2.6 with different dynamic natures.

Fig. 2.7 shows that the transmission error decreases when the order of the Taylor estimator increases for smooth commands. However, it also shows that the transmission error increases when the order of the Taylor estimator increases for commands with significant variation. Therefore, the selection of orders of the Taylor estimator is very important in motion NCS. So this paper used the integrated absolute errors (IAE) of transmission errors as a performance index to determine the orders of the Taylor estimator. Results shown in Fig. 2.8 indicate that the 3^rd-order Taylor message estimator is more suitable in real applications by concerning different motion commands.

Fig. 2.6 Motion commands with (a) smooth variation, (b) Significant variation ^(a) (b)

Time(ms) Time(ms)

Amplitude (mm) Amplitude (mm)

Fig. 2.7 Simulation result with the Taylor message estimator (a) smooth variation

(b) significant variation

Fig. 2.8 Analysis of compensation effects with different orders 2.2.2 Coefficients of Taylor message estimator

If the current

k

_th position data

P

(k) is lost, the Taylor expansion is

The estimated value of the current position command can be expressed as ˆ 1

P

. Therefore, the estimated current result from the past messages is obtained. The different order Taylor message estimators F(z) can be simply expressed in the z-transform as

 1^st-order Taylor estimator

 2^nd-message Taylor estimator

2.3 Simulation results

2.3.1 Noise command signals

In the simulation analysis, the NCS structure shown in Fig. 2.4 was built on Matlab. The dynamic model of the DYNA CNC machine tool obtained from the system identification procedure was adopted as

-5

Moreover, three different message estimators were implemented for verifying the noise reduction and control performance as: (1) the 1-delay estimator,

F

(

z

) z^1, (2) the optimal estimator (Ling and Lemmon, 2004), and (3) the proposed 3^rd-Taylor estimator. The dropout rate is chosen as



[0,0.6]. For the cases where the input command

r

[k] is the white noise with zero mean, Fig. 2.9 indicate that based on the index of the auto-correlation value

R , the optimal dropout compensator results in

_yy the best noise reduction to suppress the noise contamination effect up to a 20% data dropout rate. On the other hand, the 3^rd–order Taylor message estimator performs the worst for noise reduction. Note that the Taylor message estimator mainly estimates the missing message from the past data but the noise signals are unpredictable.

Therefore, the obtained results also imply that the Taylor message estimator is not suitable for the highly noise-contaminated NCS.

Fig. 2.9 Output PSD with white noise input.

2.3.2 Circular motion command

In real applications, motion commands in general are simple signals like G01, G02 and G03 in CNC codes as linear, clockwise and counter clockwise circular motions, respectively. Basically, a third-order Taylor expansion is suitable to represent most the basic CNC motion commands. Here, a sinusoidal wave in a single axis with the magnitude 50 mm under the feedrate 3000 mm/min as input

r

[k] is adopted to verify the circular motion performance of NCS. Results of three different message estimators under different dropout rates are shown in Fig. 2.10. Simulation results indicate that by applying the optimal dropout compensator, it leads to the worst control accuracy and its dropout rate is limited to 20% only. Theoretically, the optimal dropout compensator is designed to minimize the power spectral density of the output signals due to the noise input and it is not suitable for the cases with contouring commands. On the other hand, the proposed 3^rd-order Taylor message estimator results in the best control performance when the dropout rate is as high as to 50%.

Fig. 2.10 Tracking accuracy of different message estimators

2.3.3 NURBS motion commands

The circle and butterfly contours are selected as the test control commands produced by applying the non-uniform rational B-Spline (NURBS) curve interpolator (Piegl and Tiller, 1995; Gopi and Manohar, 1997). The NURBS interpolator can create free-form curves easily by manipulating the values of control points, weight and knot vectors. The mathematical formulation of NURBS curve can be described as follows: order B-spline basis function;

R

_i,k(

p

) is the rational basis function. With the circular

commands and the butterfly commands (Yeh and Hsu, 1999), Fig. 2.11 and Fig. 2.12 show that the proposed motion estimator can significantly improve control performance as data dropout occurs.

Fig. 2.11 NURBS simulation result as  19.97%.

Fig. 2.12 NURBS simulation result as  19.97%.

2.4 Experimental results

2.4.1 Experimental setup with the CAN bus

The proposed approach was also verified on a CNC machine tool driven by the AC servo motor. The message estimator together with the controller were implemented on the TI TMS320F2812 DSP microcontroller and its internal CAN protocol was used to transmit/receive messages of the position commands and feedback measurements. The transmitted messages missing the deadline of the fixed sampling time were counted as the data dropout in a time base. Without applying the message estimator, a sinusoidal command was provided with a CAN transmission rate at 250K bit/s. Its missing message transmission error at every sampling period 1 ms was recorded as shown in Fig. 2.13. Experimental results shown in Fig. 2.14 indicate that the proposed 3^rd-order Taylor message estimator effectively reduces the network transmission errors around 1

100.

Fig. 2.13 The transmission error without message estimator

Fig. 2.14 The transmission error with the Taylor message estimator

2.4.2 CNC applications

Furthermore, the proposed 3^rd–order Taylor message estimator and the controller were applied to the DYNA MTYE 1007 CNC machine in a NCS structure, as shown in Fig. 2.15. The sinusoidal command message with the position amplitude 50 mm under the feedrate 3000 mm/min are shown on Fig. 2.16. Experimental results indicate that without the message estimator, the significant tracking error of NCS on CNC leads to a relatively unstable system, as shown in Fig. 2.17. By applying the proposed Taylor message estimator, the motion NCS not only becomes more stable but also greatly reduces the tracking error.

Fig. 2.15 Experimental setup.

CAN bus

DSP control board (F2812)

DYNA CNC machine

DSP control board (F2812)

0 1000 2000 3000 4000 5000 6000 7000 -50

-40 -30 -20 -10 0 10 20 30 40 50

Time (ms)

Amplitude (mm)

Command message

Fig. 2.16 Command message

Fig. 2.17 Experimental results with/without the message estimator.

2.4.3 Feedforward control on NCS

The feedforward control has been successfully applied to motion systems by canceling poles and zeros of the plant model to improve tracking accuracy.

Apparently, the model-based feedforward design is not suitable for general NCS because the dynamic model of a general NCS is usually uncertain due to both the time-delay effect and the data dropout. Since the proposed message estimator may

recover the missing message in the NCS to render a more reliable NCS model, the feedforward control structure as shown in the Fig. 2.18 integrates with the Taylor message estimator becomes feasible.

Fig. 2.18 The motion NCS with the message estimator and the feedforward controller.

Fig. 2.19 The basic structure of ZPETC.

The basic feedforward structure of the zero phase error tracking control (ZPETC) shown in the Fig. 2.19 cancels all removable poles and zeros in the position control loop (Tomizuka, 1978). For those unstable zeros, their conjugate zeros are added to compensate for their phase error through the entire frequency range. If the transfer function of the original position loop is

   

     

 

¹

1 1

1 1 1













  

 



A z

z B z B z z

A z B z z

T

^{a u}

d d

(2.14)

the transfer function of the ZPETC can be expressed as

     

Thus, ZPETC leads to zero phase error in all frequency range. Besides, the DC gain is unity at the zero frequency, as in Eq. (2.16).

Based on the measured results of the CAN bus shown in Table 1, simulation results shown in Figs. 2.20-2.22 indicate that both the 1-delay message estimator and the optimal dropout compensator present unsatisfactory performance as the dropout rate increases. The tracking accuracy of the controller combining the Taylor estimator and the ZPETC leads to significant improvement in motion accuracy as shown in Fig.

2.22. Experimental results on the NCS of CNC shown in Fig. 2.23 also indicate that the tracking error is significantly reduced by applying the proposed message estimator.

Fig. 2.20 Tracking errors of ZPETC and 1-delay estimator with dropout rate (a) 0.49%, (b) 19.97%, (c) 42.14%.

Fig. 2.21 Tracking errors of ZPETC and the optimal dropout compensator with dropout rate (a) 0.49%, (b) 19.97%, (c) 42.14%.

Fig. 2.22 Tracking errors of ZPETC and the 3^rd –order Taylor estimator with dropout rate (a) 0.49%, (b) 19.97%, (c) 42.14%.

Fig. 2.23 Experimental result with the ZPETC and the Taylor message estimator

2.5 Summary

The dropout rate of the CAN bus increases rapidly even when the transmission rate slightly decreases, as shown in the Table 2.1. Therefore, the dropout effect of the NCS causes the serious motion error in precise motion systems. Basic motion control commands (CNC G-code RS-273-A, RS-274-B) can be properly described in both the position and the velocity. From both analytical and experimental results, the proposed

3^rd-order message estimator can be suitably applied to the motion NCS to satisfactorily compensate for the missing commands or measurements. The novel control structure containing a 3^rd-order Taylor message estimator is successfully applied to the motion NCS to improve the control performance significantly. Both simulation and experimental results are summarized as in the following:

(1) Fig. 2.8 indicates that the 3^rd-order Taylor message estimator is more suitable in real applications by considering different motion commands (both smooth and abruptly-changed). Moreover, simulation results indicate that the proposed 3^rd-order Taylor estimator is effective in both the high and the low noise-contaminated signals.

(2) In real applications of the present message estimator, the dropout data must be smooth and predictable, like position commands or velocity commands. Because the velocity, the acceleration and the jerk which are the first, second, and third derivatives of the position, the present 3^rd-order Taylor message estimator is applicable to motion NCS to cover all information of the missing messages of the commands or measurements.

(3) In practice, all motion commands and paths of CNC or robots are predictable and the proposed message estimator in NCS is suitable. Experimental results of the CNC machine tool indicate that by applying the proposed 3^rd-order Taylor message estimator, the maximum tracking error is reduced from 12 mm to 2.4

mm.

(4) The present 3^rd-order Taylor message estimator not only reduces the tracking error, but also degrades the NCS model uncertainty to achieve reliable motion.

By integrating the feedforward control together with the message estimator, the present NCS model uncertainty is reduced and results shown in Fig. 2.23 indicate that the tracking error further decreases from 2.4 mm to 0.08 mm.

(5) The communication delay basically is in a stochastic nature. If it is less than one sampling period T, its delay effect on the degradation of NCS performance is negligible. However, as the delay becomes more serious, say several times than the sampling interval T like in the Ethernet, the data dropout will also become more serious and special design should be considered, like applying the Smith predictor.

By applying the proposed message estimator, advanced control design can be further employed for the motion NCS design to render satisfactory precision and responses.

However, the proposed approach is feasible only as the data dropout rate is small and the missing message can be estimated in a deterministic approach. As the dropout rate increases to 50%, other statistical approaches to determine the stochastic model of the missing messages and to take proper actions may thus be required (Hansen and Yu, 2001).

(a) (b)

Chapter 3

The Real-time transition probability estimator

3.1 The Distribution effect of data dropout

Traditionally, the data dropout rate  is recognized as the quality of service (QoS) for NCS. However, in motion NCS, compared with evenly distributed missing data, continuous missing data will cause even more serious motion error. Fig. 3.1 (a)-(b) show two signals with the same data dropout rate 20% applied to the butterfly profile for fifth-order NURBS commands shown in Fig. 3.2 (Hsieh et al., 2006). By applying the same 3^rd-order Taylor estimator for compensating the missing motion commands, simulation results show that the transmission error become more significant when the data dropout is more centralized, as shown in Fig. 3.3(b).

Therefore, these results indicate that both data dropout and its distribution play crucial roles in NCS motion accuracy.

0 100 200 300 400 500 600 700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 100 200 300 400 500 600 700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 3.1 (a) Distributed and (b) centralized dropout signals with the same dropout rate 20%

(a) (b)

-15 -10 -5 0 5 10

-5 0 5

X-axis (mm)

Y-axis (mm)

Fig. 3.2 NURBS position command

0 100 200 300 400 500 600 700

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

Time (ms)

Transmission error (mm)

0 100 200 300 400 500 600 700

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

Time (ms)

Transmission errors (mm)

Fig. 3.3 Y-axis racking errors of (a) distributed and (b) centralized missing data (20% dropout rate)

3.2 Two-state Markov chain network model

In the Bernoulli model, the data dropout rate



describes the probability of both the received and dropout status (Adas, 1997). The Bernoulli model cannot capture the bursty behavior because the received signal or dropout signal at any instant is independent of all other outcomes. However, transmissions in NCS commonly exhibit bursty behavior. To capture bursty network losses, a two-state Markov chain was used (Gilbert, 1960; Wang and Moayeri, 1995) with both theoretical and practical complexity in modeling. In a Markov chain, the outcome for the future state that the system will transit to depends only on the present state and not any previous ones; the system is also memoryless. In this work, two parameters describe the distribution of packet dropouts, the dropout state D and the received R

In the Bernoulli model, the data dropout rate



describes the probability of both the received and dropout status (Adas, 1997). The Bernoulli model cannot capture the bursty behavior because the received signal or dropout signal at any instant is independent of all other outcomes. However, transmissions in NCS commonly exhibit bursty behavior. To capture bursty network losses, a two-state Markov chain was used (Gilbert, 1960; Wang and Moayeri, 1995) with both theoretical and practical complexity in modeling. In a Markov chain, the outcome for the future state that the system will transit to depends only on the present state and not any previous ones; the system is also memoryless. In this work, two parameters describe the distribution of packet dropouts, the dropout state D and the received R

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