Chapter 3 The Real-time transition probability estimator
3.4 Illustrative example
j k i for i j R D
j
k
i ( ) PrPr OW(k) | ( ) , ,
ˆ,
(3.2)Under the definition, the local transition probability,
ˆi,j(k
) can be efficiently estimated because all previous network signals are known. Moreover, the average of a) ˆi,j(
k
with a relatively long duration can represent the transition probability,
i,j.3.4 Illustrative example
By applying the proposed real-time estimator of the local transition probability to the two signals with the same overall transition probability
D,D 0.5, as shown in Fig. 3.5 (a) and (b), respectively, their distributions are significantly different andthe estimation results for two signals at the 15th instance are
ˆD,D(15)0 and% 60 ) 15 ˆD,D(
, respectively. Therefore, the index of local transition probability )ˆi,j(
k
is more appropriate to imply the distribution of the dropout data in motion NCS than the dropout rate only.(a)
(b)
Fig. 3.5 Real-time calculation of local transition probability with the same overall transition probability
D,D 0.5 .Furthermore, the proposed real-time transition probability estimator is verified and simulation results are presented in Fig. 3.6 (a)-(b). In Fig. 3.6(a), the real-time transition probability estimator can efficiently measure different transition probabilities in network signals and its measuring time is less than 0.1 sec in Fig. 3.7.
0 1000 2000 3000 4000 5000 6000
0 0.2 0.4 0.6 0.8 1
time (ms)
Value of transition probability
Estimated local transition probability True transition probability
(a)
0 1000 2000 3000 4000 5000 6000
0 0.2 0.4 0.6 0.8 1
time (ms)
Value of transition probability
Estimated local transition probability True transition probability
(b)
Fig. 3.6 Transition probabilities of (a) average of ˆD,D(k) vs D,D and (b)
R R R
R, (k) vs , ˆ
of
average
1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 0
0.2 0.4 0.6 0.8 1
time (ms)
Value of transition probability
Estimated local transition probability True transition probability
Fig. 3.7 Measuring time of the real-time transition probability estimator.
3.5 Summary
The proposed real-time transition probability estimator is designed with a real-time view of the communication quality because of the short-window signals are applied. The estimated results can be applied to on-line monitor QoS of network and results are summarized as follows:
(1) The transition probability matrix can be used to analyze the data dropout rate by its eigenvector and the expected number of consecutive dropout states can be estimated by obtaining the transition probability.
(2) The transition probability (
ˆD,D ) can efficiently and rapidly monitor the real-time data dropout distribution of motion NCS with a straightforward algorithm by triggering those sampled points as the singles are missing.(3) The 50-points average of the local transition probability,
ˆD,D and
ˆR,R, can suitably represent the transition probability of overall network communication.In simulation, the variation of network traffic load can be rapidly measured by applying the real-time transition probability estimator.
Chapter 4
The Intelligent message estimator
4.1 The structure of the multi-axis motion NCS
The general NCS architecture Fig. 4.1(a) shows that when the number of motion axes increases, the network traffic of the architecture also increases more seriously due to the commands and feedback messages that must be transmitted or received on time within a system sampling period. Therefore, it leads to the development of a faster network infrastructure to meet the requirement of synchronization such as Ethernet and EtherCAD. However, this also leads to higher-cost implementation given the modifications.
(a)
(b)
Fig. 4.1 (a) General NCS architecture and (b) practical motion NCS architecture
In real control applications, practical motion NCS architecture is generally modified as shown in Fig. 1(b), and only the command messages are transmitted from the master to the controller. Thus, the transmission can meet the hard real-time requirement within a sampling period to avoid possible heavy traffic over networks.
Moreover, the feedback messages are transmitted according to certain monitoring functions by the event-triggering approach without occupying the network.
Nevertheless, in motion NCS with multiple axes, when the network delay is longer than the sampling period, the missing message of motion NCS becomes unavoidable.
Under such circumstances, a message estimator is thus required to estimate the missing commands and to compensate for their effect. This research is crucial for NCS and is still pursued by industries to promote NCS applications. Therefore, various message estimators with different advantages have been proposed to cope with the dropout effect for motion NCS under different conditions. For example, the one-delay message estimator is easily implemented because of its simple algorithm of replacing the missing message with previously received data (Ling and Lemmon, 2002). However, it cannot predict the variation of messages properly. On the other hand, the non-linear NCS was modeled as a Markovian jump linear system, and the finite loss history estimator (FLHE) was proposed to improve data dropout effects
when the dropout rate is accurately known (Smith and Seiler, 2004). Nevertheless, these methods generally require the accurate plant/network model. On the other hand, model-free strategies for control packet dropout compensators, such as the proportional plus derivative (PD) predictors with a different order of derivatives, are proposed to estimate dropout data and to compensate for their effect (Tian and Levy, 2008). Recently, the Taylor estimator was proposed to significantly improve the control performance in motion NCS (Hsieh et al., 2006; Hsieh and Hsu, 2008). All these reported methods are effective due to their design structures, which are commonly based on the assumption that the dropout data over the network are evenly distributed. Once the missing data occur in a continuous format, this will generally lead to a more serious maximum contouring error. Therefore, intelligent message estimator (IME) is proposed to compensate effects of data dropout based on real-time transition probability estimator.
Moreover, in multi-axis motion NCS, data dropout will lead to the problem of asynchronization among different axes. By applying the proposed IME, synchronization among different axes is also greatly improved. Both simulation and experimental results with the non-uniform rational B-spline (NURBS) motion commands have been verified. With different motion message estimators, the results indicate that the present IME maintains the lowest transmission error as well as the least motion contouring error when the transition probability (
D,D) increases. The CAN-based two-axis AC servo motor control system was also successfully implemented with the proposed IME.In motion NCS, the control messages for each motion axis must be transmitted on time through the network protocol to meet the control design specifications, as shown in Fig. 4.2. Since the time delay exists in stochastic and time-varying natures, the transmitted messages may miss the hard real-time deadline because the network bandwidth is saturated. Generally, it causes data dropout, as the network-induced time delay is longer than the system sampling time TD, as shown in the timing diagram in Fig. 4.3.
Fig. 4.2 Multi-axis motion NCS
Fig. 4.3 Timeline analysis in a control period
4.2 The Least-square estimator
In motion NCS, the real position commands are smooth curves in most practical cases, but their curvatures may vary significantly along the contour.
Practically, missing data with higher transition probability (
D,D) will cause a more serious contouring error around the higher curvature. To estimate the missing messages in NCS, the one-delay estimator simply adopts the last received message as the current missing message, and the Taylor estimator estimates the current missing message from past received signals. If the past signal is also missing, the message obtained from the estimators also becomes unreliable. In this dissertation, the IME is proposed based on the integration of the least-square estimators with different orders based on the online measured local transition probability (
ˆD,D). As the messages are serious dropouts, estimation based on the previous data is no longer reliable, and theone-delay estimator is then included in the proposed IME.
Since the online estimation is time consuming, all parameters of the real-time least-square estimation (LSE) can be obtained in advance. Thus, to achieve an online estimation and compensation algorithm for the missing command in motion NCS, the IME is proposed based on past messages within a short window by applying the least-square approach. For a general time sequence
x
[0],x
[1],x
[M
], a polynomial sequence can be suitably described asN
The normal equation from the least-square approach can be applied to the data to obtain coefficient vector c as
x A A A
c
( T )1 T (4.4)Thus, the missing value for the current missing message can be predicted as
and the estimator matrix
LSE
(M
,N
) can thus be pre-calculated for real-time implementation.M indicates the data number to be counted, and N is the order of
polynomial functions.To achieve an online estimation for NCS, parameters should be determined in advance. Therefore, the order and the data number of the least-square estimator should be determined with practical concerns. For example, the NURBS signal can be approximated by a third-order polynomial equation obtained from the LSE (Sorenson, 1970). Therefore, the length of OW can be properly chosen to as large as five to suitably estimate the NURBS and other curves. Three useful
LSE
(M
,N
) are pre-calculated for real-time applications as follows:
LSE
(5,3)3.2z
12.8z
20.8z
32.2z
40.8z
5 (4.6)
LSE
(3,2)3z
13z
2 z
3 (4.7)
LSE
(2,1)2z
1z
2 (4.8)4.3 Analysis of LSE with different orders
Fig. 4.4 shows the transmission errors obtained by separately applying the estimators of LSE(5,3) and LSE(2,1) to motion NCS in true transition probability
P
D,D0.2 . Simulation results show that LSE(5,3) renders a better compensation effect as compared to LSE(2,1), which should be applied in a more serious data dropout case. However, as PD,D 0.4, which implies that there are about two missing messages among the five transmitted messages, the transmission errorincreases, and LSE(5,3) is not suitable anymore. Fig. 4.5 shows that the compensation results applying LSE(2,1) render better performance.
Furthermore, the least-square approach with a different M applied to a different
P
ˆD,D shows that applying LSE(5,3) to compensate the missing data obtains the best motion accuracy as 0P
ˆD,D 0.2, but it becomes the worst as ˆ 0.2,D
P
D .Moreover, LSE(3,2) is more suitable for the situation, as 0.2
P
ˆD,D 0.4. Moreover,LSE(2,1) is most suitable for the situation as
0.4P
ˆD,D 0.6. In addition, the one-delay estimator possesses the best compensation effect as 0.6P
ˆD,D 1. These simulation results are shown in Fig. 4.6.0 100 200 300 400 500 600 700
-6 -5 -4 -3 -2 -1 0 1 2 3 4x 10-3
Time (ms)
Transmission errors (mm)
(a)
0 100 200 300 400 500 600 700 -6
-5 -4 -3 -2 -1 0 1 2 3 4x 10-3
Time (ms)
Transmission errors (mm)
(b)
Fig. 4.4 Transmission errors with (a) LSE(5,3) and (b) LSE(2,1) as PD,D0.2.
0 100 200 300 400 500 600 700
-50 -40 -30 -20 -10 0 10 20 30 40
Time (ms)
Transmission errors (mm)
(a)
0 100 200 300 400 500 600 700 -2
-1.5 -1 -0.5 0 0.5 1 1.5 2
Time (ms)
Transmission errors (mm)
(b)
Fig. 4.5 Transmission errors with (a) LSE(5,3) and (b) LSE(2,1) with D,D0.4.
Fig. 4.6 Analysis of compensation effects with different true transition probability PD,D
4.4 The IME architecture
It was discussed in the preceding section that intelligent Message Estimator (IME) adopts four useful message estimators for real-time applications as follows:
LSE
(5,3) for low-data dropout casesIn this transmission case, all data within observation window length 5 are properly received, or at most, only one missing data is estimated among the four received data within the window.
LSE
(5,3) is chosen to estimate a cubic-curve motion command with the order of 3 by using all five previous data, which may include estimated data at most. In other words, the third-order LSE(5,3) can properly estimate the motion trajectory concerning its velocity, acceleration, and even the change of acceleration as the jerk, and the parameters are obtained from Eq. (4.6).
LSE
(3,2) for medium-data dropout casesIn this case, the medium data dropout condition occurs, and the missing data within0.2
P
ˆD,D(k
)0.4. In other words, only three reliable data are accountable within the window to correctly estimate the missing data. Therefore,LSE
(3,2)is chosen to suitably estimate the quadric-curve trajectory with the order of 2 by using three previous data through considering both its velocity and acceleration from Eq.(4.7)
LSE
(2,1) for heavy-dropout case casesIn this situation, the missing data within 0.4PˆD,D(k)0.6, and
LSE
(2,1) is chosen to estimate the motion trajectory concerning its velocity only by applying previous two data, either received or estimated. The one-delay estimator is adopted for serious-data dropout cases.
In this situation, network communication presents such a heavy data dropout rate; the missing data within 0.6
P
ˆD,D(k
)1. Therefore, the estimation resultsbased on the mentioned least-square approach is not reliable anymore, and the one-delay estimator is chosen to estimate the position only by directly adopting the previous data as
estimator 1
delay
-1 z (4.9)
All switching laws according to Eq. (4.10a or 4.10b) based on the estimated
D
P
ˆD, thus agree with both the simulation results and the theoretical analysis, as shown in Fig. 4.7. The proposed IME switching law based on the index ofP
ˆD,Dcan thus be applied suitably to estimate and recover the missing data for both centralized and distributed missing messages in motion NCS. Although the Taylor estimator has been proven to render more accurate results than the one-delay estimator, Fig. 4.7 further indicates that the proposed IME presents a much better performance under a different transition probability
P
D,D, especially as the missing data becomes more serious. In summary, different LSEs are applied to different real-timeP
ˆD,D, as shown
in Fig. 4.8.Fig. 4.7 Simulation results with different estimators
Fig. 4.8 The system architecture of the proposed IME
4.5 Simulation results
Applications of the present IME based on real-time transition probability
D
P
ˆD, have been applied to the two-axis motion NCS, as shown in Fig. 4.1. The NURBS commands and the system response withP
D,D 0.2 are shown in Fig. 4.9.The results show that the Taylor estimator can reduce the effects of data dropout at a lower
P
D,D. However, Fig. 4.10 also shows that the contouring error obtained by applying IME is significantly reduced to achieve better contouring accuracy. Fig. 4.11shows that the contouring accuracy of the present IME even renders a much better contouring accuracy when
P
D,D increases to 0.5. Furthermore, when the value ofD
P
D, increases to 0.6, the Taylor estimator will lead to an unstable motion as shown in Fig. 4.12. Nevertheless, the proposed IME still results in a stable motion and maintains the contouring error as the least from the simulation results.Fig. 4.9 Contours of motion NCS without/with the Taylor estimator (PD,D 0.2)
0 100 200 300 400 500 600 700
-4 -3 -2 -1 0 1 2 3 4
(a)
Contouring errors (mm)
time (ms)
0 100 200 300 400 500 600 700 -4
-3 -2 -1 0 1 2 3 4
(b)
Fig. 4.10 Contouring errors with (a) the Taylor estimator (b) IME (PD,D0.2)
0 100 200 300 400 500 600 700
-4 -3 -2 -1 0 1 2 3 4
(a)
Contouring errors (mm)
time (ms)
Contouring errors (mm)
time (ms)
0 100 200 300 400 500 600 700 -4
-3 -2 -1 0 1 2 3 4
(b)
Fig. 4.11 Contouring errors with (a) the Taylor estimator and (b) IME (PD,D 0.5)
0 100 200 300 400 500 600 700
-20 -15 -10 -5 0 5 10 15 20 25
(a)
Contouring errors (mm)
time (ms)
Contouring errors (mm)
time (ms)
0 100 200 300 400 500 600 700 -20
-15 -10 -5 0 5 10 15 20 25
(b)
Fig. 4.12 Contouring errors with (a) the Taylor estimator and (b) IME (PD,D 0.6)
4.6 Experimental results
The proposed IME was applied to the CAN based two-axis AC servo motor control systems, as shown in Fig. 4.13. The butterfly NURBS profile for both the X-axis and Y-axis position amplitudes is 30 mm under the feed rate 3,000 mm/min, which is the same as in. Furthermore,
P
D,D is measured as 0.32 and 0.54, respectively, for the present CAN-bus implementation with the baud rate 1 M bit/sec under different sampling periods as 0.5 ms and 0.4 ms, respectively. The results indicate that increasing the sampling rate will result in more serious missing data due to the saturation of network bandwidth. Fig. 4.14 shows the contouring error when32 .
,D 0
P
D . The first-order differential results of the measured contouring error with less oscillation are also shown in Fig. 4.15. All results indicate that the proposed IME renders a more stable and reliable motion than the Taylor estimator. By observing the contouring error as shown in Fig. 4.16 with a more serious data dropout, the results also show that the proposed IME is more effective in reducing the asynchronization effect than the Taylor estimator in rendering a more accurate motion. Similar resultsContouring errors (mm)
time (ms)
provided by the circular NURBS profile for the motion NCS obtained as shown in Figs. 4.17-4.18 also indicate the applicability of the proposed IME to different motion profiles.
Fig. 4.13 Experimental setup
(a)
(b)
Fig. 4.14 Contouring errors with the sampling period = 0.5 ms (a) Taylor estimator (b) IME (PD,D 0.32)
0 100 200 300 400 500 600 700
-200 -150 -100 -50 0 50 100 150
Time (ms)
First order differential of contouring errors
(a)
0 100 200 300 400 500 600 700 -200
-150 -100 -50 0 50 100 150
Time (ms)
First order differential of contouring errors
(b)
Fig. 4.15 First-order differential of contouring errors of (a) the Taylor estimator (b) IME (PD,D0.32)
0 100 200 300 400 500 600 700
-1 -0.5 0 0.5 1 1.5 2
Time (ms)
Contouring error (mm)
(a)
0 100 200 300 400 500 600 700 -1
-0.5 0 0.5 1 1.5 2
Time (ms)
Contouring errors (mm)
(b)
Fig. 4.16 Contouring errors with the sampling period = 0.4 ms (a) Taylor estimator (b) IME (PD,D 0.54)
-5 0 5 10 15 20
-10 -8 -6 -4 -2 0 2 4 6 8 10
X-axis (mm)
Y-axis (mm)
(a)
-5 0 5 10 15 20 -10
-8 -6 -4 -2 0 2 4 6 8 10
X-axix (mm)
Y-axis (mm)
(b)
Fig. 4.17 Circle responses of (a) the Taylor estimator and (b) IME with the sampling period = 0.5 ms (PD,D 0.32)
-5 0 5 10 15 20
-10 -8 -6 -4 -2 0 2 4 6 8 10
X-axis (mm)
Y-axis (mm)
(a)
-5 0 5 10 15 20 -10
-8 -6 -4 -2 0 2 4 6 8 10
X-axis (mm)
Y-axis (mm)
(b)
Fig. 4.18 Circle responses of (a) Taylor estimator and (b) IME with the sampling period = 0.4 ms (PD,D0.54)
4.7 Summary
Based on the on-line estimated transition probability PˆD,D from Chapter 3, a suitable order of the least square estimators under different network communication conditions can be thus determined. Both simulation and experimental results have verified estimation performance by applying different orders of the least square estimators based on different communication quality PD,D. Results indicate that applying the proposed IME, the missing commands can be properly estimated and the data dropout effect can be thus effectively reduced to improve contouring accuracy of motion NCS. Results are summarized as follows:
(1) By applying tracking performance analysis, the switching laws of the proposed IME can be adopted based on estimated transition probability. Furthermore, the IME leads to the lowest contouring error under conditions of different data dropout distribution.
(2) By applying the proposed IME to the CAN based two-axis AC servo motor control systems, contouring accuracy can be maintained well even under severe
missing commands. Moreover, the proposed IME renders the best performance as compared to the one-delay or the Taylor estimators in motion NCS.
Chapter 5
Stability analysis for NCS with the message estimator
Although physical dynamic systems operate in the continuous-time domain, the plant models in NCS are suitably represented in discrete-time domain with the sampling time over the network. This modeling paradigm is well suited for representing motion NCS as shown in Fig. 5.1. With digital computation and communication, digital control design applies the feedback sampled periodically for controller to provide actuator actions periodically. In addition, the network communication is constructed naturally with discrete-time analysis in our framework where transmissions are set to occur periodically. By modeling the continuous plant dynamics in discrete time through integration of the states over the sample period, the model of the NCS is shown in Fig. 5.1.
Fig. 5.1 Structure of the motion network control systems
Traditional state space representations of a continuous time plant given by
)
where is the output vector. This conversion assumes that the continuous time input
u
(t) is held constant for the duration of the sample periodT .
sThrough this work, we usually assume the availability of full-state feedback from plants, and full-state feedback controllers are then obtained. The controller frequently used in this work is the discrete-time infinite time horizon linear quadratic (LQ) controller. The steady state LQ control problem can be posed (Gupta et al., 2005;
Sinopoli et al., 2005; Ogata, 1995): For the linear discrete plant described by
)
Determine the control
) ( )
(
k Kx k
u
(5.5)to minimizes the quadratic cost function
where
Q is positive semi-definite and R
is positive definite. The solution to his problem is well-known (Ji et al., 1991) and it is computed by solving the steady-state Riccati equation for P as.The system models and the controller design for this approach are adopted throughout the remainder of this chapter.
5.1 Modeling NCS dynamics as a switched system
By using the hybrid control system framework with the time-triggered sensing/actuation, and the event-triggered communications shown in Fig. 5.1, the behavior of motion NCS can be naturally fitted into the format of a switched system with two discrete dynamic modes of operation (Kawka and Alleyne, 2009). The closed loop mode (CL) describes the system dynamics evolution over a sample period when the entire round trip communication of feedback and control was successful in the previous sample period. The open loop mode (OL) describes the system dynamics over a period when a new control packet was not received in the preceding sample period. This flow of information can be represented schematically by the diagrams as in Fig. 5.2, and the dropout messages occur mainly due to the increase of the time delay.
Fig. 5.2 Timing diagram of the network and system states
These two dynamics modes of operation can be described mathematically by
)
) 10 . 5 (
where m characterizes the time instant of the last successfully received message.
In the OL mode, although the most recent feedback information ))
(
y k
u k
is not available, information from the last successfully completed round trip transmission (y
(k
m
),y
(k
m
1),u
(k
m
),u
(k
m
1),) can be still used in the NCS applied by the actuator. That is why with this modeling strategy, the NCS naturally fits into the framework of a switched system with two discrete dynamics modes of operation, one for open loop and one for closed loop. For the(