ISSN ONLINE(23209801) PRINT (23209798)
Kenshi Matsuo Graduate School of Engineering and Resource Science, Akita University, Akita, Japan 
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In Internet Protocol (IP) networkbased control systems, the control performance is degraded owing to time delays and its jitter over the networks. Hence schemes to overcome the issue are required. In this paper, the author proposes an autotuning scheme based on a partial model matching method that tunes PID controllers for speed control of a small DC motor through IP networks with varying time delays. For confirming the effectiveness, speed control experiments are done after the controller of the system is autotuned by the proposed scheme under several emulated IP network conditions. As a result, the responses obtained by the experiments show that the proposed scheme is effective for the systems.
Keywords 

DC motor, IP network, time delay, PID controller, partial model matching method  
INTRODUCTION 

Recently, many applications using the Internet have been developed in various fields. As one of the applications, the development of Internet Protocol (IP) networkbased control systems gains more attention (e.g., [1], [2]).  
Because the cost of hardware like hubs, routers, network interface cards, etc. is lower, it has been easy to construct IP networkbased control systems connected among actuators, sensors and controllers over the networks. However, the control performance is degraded due to varying time delays such as constant time delay and its jitter over the network. Therefore, schemes to overcome the issue are required. A simple scheme is to tune a proportionalintegralderivative (PID) or PI controller properly and automatically according to network conditions. For example, [1] has proposed a scheme that autotunes a PI controller. Specifically, this scheme is as follows. It is assumed that varying time delays behave like exponential distributions, and the varying delays are characterised. Then the PI controller is tuned automatically according to the present delays, where the PI parameters are decided such as minimizing an evaluation function with respect to each delay beforehand. However, task for the tuning is often much timeconsuming one.  
In this paper, another scheme that autotunes PID controllers according to varying time delays is proposed. The proposed scheme is that first the PID controller is tuned using a partial model matching method by Kitamori [3], which can automatically choose the PID parameters according to time delays, and then the integral parameter is retuned with respect to its jitter. The advantage using the matching method is that proper PID parameters are automatically obtained if the transfer function of the plant model is known. That is, acquiring the value of parameters tuned according to each time delay in advance is not needed. However, the tuning method is still insufficient against jitter. Hence, to overcome the jitter issue, the relationship between the minimal time of varying time delays and the rise time of the step response are examined, where the responses are obtained by control experiment after tuning the controller based on the matching method according to each time delay. Then, based on the relationship to characterize the jitter, a method that retunes the integral parameter is developed. Thus, the author proposes an autotuning scheme including these two methods.  
Finally, an IP networkbased speed control system of an actual small DC motor, which is chosen because of a typical servomotor having a simple structure, is constructed, and after applying the proposed scheme, speed control experiments are carried out. From the step responses obtained by the experiments, it is confirmed whether the proposed scheme is effective against IP networks with varying time delays.  
IP NETWORKBASED CONTROL SYSTEM 

Fig. 1 shows an IP networkbased speed control system for a small DC motor. The rated voltage, current, power and speed of an actual small DC motor (Sanyo Denki R301T011) are 24V, 1.25A, 11W and 3000min1 respectively. The motor is with an inertial load with 2.5 105 kg m2.V(s), r(s), m(s) and e(s) are an applied voltage [V] to the motor, a reference speed [min1], a rotor speed [min1], and the error between the reference and the rotor speed, i.e., e(s) = r(s) m(s), respectively. Gm(s) is the transfer function of the DC motor. A PID controller GPID(s) is denoted as  
where KP, KI and KD are the proportional, integral and derivative parameters respectively. In actual experiments described in later section, the derivative term uses not only KDs, but also KDs/(1+TDs), where and TD=KD/KP. Both sampling time of each computer to operate the controller and the plant in the figure are 0.001s = 1 ms. Each time delay of signals of V(s) and m(s) through the IP network is expressed as L1 [s] and L2 [s] respectively. However, because L1 and L2 are fluctuated due to network traffic conditions, the delays cannot be dealt with as a (constant) dead time in conventional control theory. Hence, in the figure, note that their varying time delays are formally expressed as eL1s and eL2s.  
In this system, the behavior of IP networks between the controller and the plant is emulated by a PC router on which a Linux network emulator netem [5] is installed. Netem can emulate time delays (as mean [ms]), its jitter (as standard deviation [ms]) and distribution (e.g., normal distribution) of varying time delays. A communication protocol used on the network is User Datagram Protocol (UDP). Packets including a control signal may be lost since UDP has no reception confirmation. Moreover a packet sent from the controller earlier may receive at the plant later or vice versa. This issue is packet disorder [6]. In this paper, if a packet has not been received within a sampling period or disorder has occurred, then the receiver employs the latest one obtained at the preceding period. Moreover, it is assumed that the influence of the packets loss can be negligible.  
AN AUTOTUNING OF THE SYSTEM WITH A DEAD TIME BY PARTIAL MODEL MATCHING METHOD 

The partial model matching method by Kitamori [3] can decide PID parameters automatically such that the closed loop transfer function of a provided system including the controller coincides with a function of a reference model, where the response of the reference model is the one that a designer wants to design. In this section, it is explained how the method is applied to a system with a “constant” time delay, i.e., a dead time L as shown in the upper figure of Fig. 2, not “varying” time delays as in Fig. 1. This method with a dead time is provided as follows [4].  
APID controller GPID(s) is denoted as (1). First, the transfer function of the DC motor model in the plant is described as Gm(s) = 1/(g0+g1s+g2s2), and Gmd(s), which is the function of the motor with a dead time L, is  
Let g0 = 4.807 103, g1 = 6.346 104 and g2 = 7.232 108, whose parameters of the motor are empirically derived beforehand. Moreover, let  
Now, in (2), the Maclaurin series expansion of eLs is done, i.e., eLs = 1 + Ls + L2s22! + L3s33! +…. Then, since (2) is equivalent to (3), the following relations (4) are obtained,  
Second, a reference model transfer function W(s) is introduced as  
where let {0, 1, 2, 3, 4} = {1, 1, 17/40, 39/400, 109/7599} [4] and is a time scaling parameter. The response of W(s) in the system shown in the lower figure of Fig. 2 means the desirable one that a designer wants to design in the actual system illustrated in the upper figure. Then PID parameters KP, KI and KD, are determined such that the coefficient of each term from low to high order in s of the closedloop function G(s) = GPID(s)Gp(s)/{1+GPID(s)Gp(s)} matches that of W(s), as illustrated in Fig.2. That is, these parameters are done as following formulas:  
where is a minimum positive solution, and satisfies the following equation  
THE PROPOSED AUTOTUNING SCHEME ACCORDING TO VARYING TIME DELAYS 

The autotuning method using the partial model matching method described in the previous section is discussed in the case that a time delay in the system is constant as shown in Fig. 2. However, because time delays vary over IP networks as shown in Fig. 1, a tuning scheme according to fluctuating delays, i.e., the jitter, needs to be developed.  
A. The Relationship Between Minimum Delay Time and Rise Time  
It is important to discuss how varying time delays are characterised for developing the autotuning scheme to overcome the varying ones. Now let varying time delays be divided into a constant time delay, i.e., a dead time and its jitter. A dead time L is determined to apply the matching method in the previous section. Let L = Lmean, where Lmean is the mean of round trip times (RTTs), each of which is a time from the controller back to the one via the plant, because varying time delays occur in actual networks, and then the matching method are done.  
However, the jitter still affects the step responses obtained when tuning based only on a dead time Lmean and controlling the system. Therefore, the jitter must be characterized for tuning properly under timevarying network conditions. Now it is explained that the relationship between the minimum time Lmin of RTTs and the rise one Tr of the step response. First, Lstd and Tstd standardizing Lmin and Tr are defined as  
respectively. In (8), Trconst is the rise time of a step response obtained when controlling the system with a dead time L = Lmean, which is shown in the upper figure of Fig. 2, not timevarying. The relationship of Lstd and Tstd can be utilized for characterizing varying delays as described in next subsection.  
For examining the relationship, in the system shown in Fig. 1, under each network condition set by netem, speed control experiments are carried out after autotuning PID parameters by the matching method based on Lmean of RTTs. Netem can set a network condition of oneway trip times (OTTs). Let r(s) = 1500min1. Experiments are done ten times under each condition of all combinations of these netem parameters as shown in Table 1, where the parameters settings are same in the both direction from the controller to the plant and vice versa in Fig.1.Under the netem settings in the cases that the delay is 50, 100, and 150 ms, Trconst are 290.2, 450.4 and 589.0 ms respectively, where Trconst is derived empirically in advance. The distributions set to exponential one, approximated in [1], and normal one.  
The results are shown in Fig. 3. As shown in the figure, it seems that there is a linear relationship between Lstd and Tstd, not depending on the distributions. Hence the relationship f(Lstd) = 1.5 0.572Lstd is obtained by a least squares method.  
B. The Proposed Scheme  
The author proposes an autotuning scheme based on the partial model matching considering varying time delays for the system shown in Fig. 1. The scheme is shown in Fig. 4, which is composed of tuning by a partial model matching method based a dead time and retuning done on the characterization of the jitter by the relationship described in previous subsection. Specifically, first, delay time estimator derives 1000 RTTs, and Lmean and Lstd from Lmean and Lmin are computed. Second, the partial model matching of the gain tuner outputs the PID parameters KP, KI, and KD from Lmean. Third, the integral gain tuner modifies KI in accordance with varying time delays, i.e., outputs KI’ = KI f(Lstd), based on the relationship described in the previous subsection, from Lstd, where the formula modifying KI is empirically adjusted. Finally, the controller GPID(s) is updated by KP, KI’, and KD.  
EXPERIMENT 

For confirming the effectiveness of the proposed scheme, in the system shown in Fig. 4, the speed control experiments are done under network conditions set by netem. Now the netem settings in the both direction between the controller and the plant are as follows:  
1) delay, jitter and distribution are 83ms, 23ms and normal one,  
2) delay, jitter and distribution are 112ms, 37ms and exponential one.  
Let r(s) = 1500min1. Under the above network conditions, following experiments are done. First, each PID controller is tuned by  
A) the proposed scheme, whose output parameters are KP, KI’, and KD,  
B) the only partial model matching method, whose output parameters are KP, KI, and KD,  
C) for comparing C) with A) and B) and showing that A) is more effective than B), the only partial model matching method in the system with a dead time L = Lmean.  
The control system in the case C) is a conventional one with a dead time as shown in Fig. 2, not IP networkbased one as in Fig. 4. The response obtained in the experiment of this system is regard as one in the case without jitter, whereas A) and B) with jitter. Hence, by comparing C) with A) and B), it can be examined whether the proposed scheme is effective against the jitter.  
Next, after the tuning, speed control experiments are carried out. As a result, the obtained step responses are shown in Fig. 5, and the parameters in Table II, where is a standard deviation of the measured RTTs. In the case C), L is set to the time close to Lmean in A) and B) for comparing them as shown in the table. The results show that, in the both cases of 1) and 2), the responses of A) can obtain the similar ones of C) in the overshoot and the rise time. Meanwhile, the ones of B) are obviously affected by the jitter in the overshoot. Therefore, it is found that A) can work well according to varying time delays, especially the jitter.  
CONCLUSION 

An autotuning scheme of PID controller has been proposed considering timevarying network conditions based on a partial model matching method. For confirming the effectiveness of the proposed scheme, speed control experiments have been done in an IP networkbased control system of an actual small DC motor. The experimental result has been that the step response obtained using the scheme has been similar to one done after tuning the system with the constant delay time and has been improved against the jitter. Therefore it has been found that the scheme has worked well against time delays and its jitter. As a result, it has been confirmed that the scheme has been effective.  
ACKNOWLEDGMENT 

The author would like to thank K. Shiroyama for the experimental assistant.  
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