**An Application of Robust Control for Force Communication Systems over Inferior Quality Network**

Tetsuo Shiotsuki *Tokyo Denki University Japan*

## **1. Introduction**

372 Challenges and Paradigms in Applied Robust Control

Cao, Y. and & Zhang, W. (2005). Modified Fuzzy PID Control for Networked Control

Zhang, W. (2001). *Stability Analysis of Networked Control Systems*, PhD Thesis, Case Western

Zurawski, R. (2005). *The Industrial Communication Systems. Handbook*. CRC Press, ISBN:

*Technolo*gy, vol. 12, pp. 520-523

Reserve University

9780849330773

Systems with Random Delays, in: *World Academy of Science, Engineering and* 

The developments of computer and network technologies have provided a virtual reality environment and ubiquitous network systems. Especially audio-visual devices play an important role in communication. For example, voice communication by telephone, audio-visual communication, streaming technology, digital television system and so on. However, we know that the human makes communication not only by audio-visual information but also by using all five-senses (touch, taste, hearing, eyesight, and smell). The realization of the five-senses communication system is one of the prospected technologies.

Especially force communication is a hopeful application in the coming e-world. Several kinds of gimmicks can be considered for transmitting or exchanging the sense of touch, haptic, tactile, force and kinesthetic. In the area of the wearable computing technologies some force-like communication system is realized by using pressure, tension, bending, stress sensors and vibration or pressure actuators, which give the illusion of force communication. On the other hand, robotic researchers have discussed on bilateral tele-operation systems, which realizes remote-manipulations with the sense of reaction forces caused by collision or touching of remote objects and environments. An aim of the technology is that the communication channel between two terminals simulates as if a rigid rod or tight rope. In this article, we consider the bilateral tele-operation systems as a force communication device. It is a well-known that the computer network has inevitable time-delay and jitter in the transmission of the data. And in control engineering deterioration of the stability and performance of the closed loop systems is a well-known fact. Control researchers have proposed several kinds of approaches to overcome the problems. The rest of the chapter is composed as follows. In Section 2, a characterization of the computer network from the view point of transmission delay is discussed. In Section 3, control systems of force sensorless bilateral tele-operation system and the problems caused by transmission delay are examined with a brief historical review. Section 4 presents a procedure how to design a robust control system over the uncertain time-delay network. In section 5 a simulation result is introduced, and some discussions are presented. In section 6 experimental results over the real broadband computer network are introduced. And the results of experiments and investigation are explained in detail. Section 6 concludes the article.

<sup>0</sup> <sup>6</sup> <sup>12</sup> <sup>18</sup> <sup>24</sup> <sup>0</sup>

(right) Histogram (delay vs. frequency)

**3.2 Delay and instability**

a in Fig.2.

time[hour]

contour of the *H*∞ norm of the transfer function

This plot says the following facts.

(1) If *L* = 0 then the gain margin is infinite.

**3.3 Scattering and wave variable method**

time-delay deteriorate as the time-delay grows.

*γ* :=  <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>0</sup>

time delay[sec]

(2)

10

20

number of packets[packet]

Fig. 1. An example of transmission delay of the Internet: (left)time history(time vs. delay),

PE(position error) type is the most simple one. Master and slave exchange the position data each other. Both controllers compensate the deviation of the positions independently. It means that the system is a combination of two position feedback control systems. When the master and the slave equipments have the same characteristics the structure of the system is completely symmetric. In FR(Force reflection) type, master equipment transmits the position data and receives the force data from the slave equipment. On the other hand FRP(Force reflection with passivity) type exchanges the velocity data and the force data respectively.

It is well-known that the time delay in the loop deteriorates the stability, performance and robustness of the feedback systems. Fig.2(b) shows a demonstration of the tarde-off of gain *K* and time-delay *L*. If (*L*, *K*) is chosen in the range of *stable* region the closed loop system depicted in Fig. 2(a) becomes stable, and vice versa. This trade-off curve is identical to the

with *γ* = 1. It is easy to calculate that the transfer function matches to one at the cutting point

(2) The gain margin decreases rapidly as the time-delay grows, that is, the robustness to

According to the considerations the necessity of careful investigation to time-delay is required.

Anderson and Spong (6) introduced a new communication architecture for tele-operation over the network with time-delay. Their method is based on the passivity and scattering representation of the network. Thus the strictly passivity of master and slave systems

<sup>−</sup>*Ls*) ∞

*K <sup>s</sup>*<sup>2</sup> <sup>+</sup> *<sup>K</sup>* (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*

30

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 375

1

round trip time[sec]

2

## **2. Communication network and time delay**

Fusion of computer and tele-communication technologies has provided the revolution of the computer network such as the Internet. Before the revolution usual tele-communication is established in two steps. Firstly, according to the request from the sender the system searches the receiver and establishes a communication channel by reserving network resources exclusively. Secondly, the session starts on the reserved real communication channel. After the end of the session, the reserved resources are released. In this case, the time-delay over the communication channel is so small as can be ignored.

On the other hand, communication on the computer network between two terminal nodes is realized as a set of the exchange process of datagrams (frame, packet, cell, ˛Ac). For example, the information is converted into digital data and divided into datagrams. These datagrams are put on the node and travel along the path while looking for appropriate next node until they reach to the destination. In general the data exchange process includes the huge number of data processing such as encoding, storing and (route) switching. And the length of the processing time depends on the size of datagram and transmitting rate the busyness of the equipments. Especially the network routers are shared by multi-users. Since the practically implemented algorithm is almost trying and error type, the data buffer sometimes overflowed and fails data(packet loss). In order to ensure the reproducibility of the data several kinds of data processing algorithms are implemented according to transmission protocols. TCP/IP(Transmission Control Protocol/Internet Protocol) provides confirming of receiving data (acknowledge), control of window size, and data retransmission and so on. Because of the complexity of the mechanism and sharing of the resource of the network the time-delay is greater than the circuit channel type communications. And the jitter, variation of the time delay occurs frequently.

Fig.1 shows an example of time delay during a day between two campuses(Kumamoto and Fukuoka) in 1998. The left graph (a) indicates the time series from midnight to midnight, and the right(b) is the histogram of the number of packets with respect to transmission delay. It is too difficult to construct the prediction model of time-delay because of the randomness and chaos. Here we adopt a statistical model as a rectangular distribution as follows.

$$0 < L \le L\_{\text{max}} \tag{1}$$

In practice it is possible to set *Lmax* such that 95% of packets are travels in the time interval [0, *Lmax*].

#### **3. Historical review of tele-operation systems**

#### **3.1 Master slave system**

Suppose the situation in which an operator manipulates (push/ pull/ lift/ put on and so on) some object through the communication network. Such a kind of system is called a master slave system or *tele-operation system*. Usually the terminals for the operator and the object are called the *master* and the *slave* equipment each other. Operators motion is converted to the motion data by the master mechanism, transmitted to the slave side and realized as a motion of the slave equipment. If the system can transmit the force information caused in the slave side to the master side it called *bilateral tele-operation systems*. Several kind of mechanisms are proposed for the control of bilateral tele-operation systems.

Fig. 1. An example of transmission delay of the Internet: (left)time history(time vs. delay), (right) Histogram (delay vs. frequency)

PE(position error) type is the most simple one. Master and slave exchange the position data each other. Both controllers compensate the deviation of the positions independently. It means that the system is a combination of two position feedback control systems. When the master and the slave equipments have the same characteristics the structure of the system is completely symmetric. In FR(Force reflection) type, master equipment transmits the position data and receives the force data from the slave equipment. On the other hand FRP(Force reflection with passivity) type exchanges the velocity data and the force data respectively.

## **3.2 Delay and instability**

2 Robust control book 3

Fusion of computer and tele-communication technologies has provided the revolution of the computer network such as the Internet. Before the revolution usual tele-communication is established in two steps. Firstly, according to the request from the sender the system searches the receiver and establishes a communication channel by reserving network resources exclusively. Secondly, the session starts on the reserved real communication channel. After the end of the session, the reserved resources are released. In this case, the time-delay over

On the other hand, communication on the computer network between two terminal nodes is realized as a set of the exchange process of datagrams (frame, packet, cell, ˛Ac). For example, the information is converted into digital data and divided into datagrams. These datagrams are put on the node and travel along the path while looking for appropriate next node until they reach to the destination. In general the data exchange process includes the huge number of data processing such as encoding, storing and (route) switching. And the length of the processing time depends on the size of datagram and transmitting rate the busyness of the equipments. Especially the network routers are shared by multi-users. Since the practically implemented algorithm is almost trying and error type, the data buffer sometimes overflowed and fails data(packet loss). In order to ensure the reproducibility of the data several kinds of data processing algorithms are implemented according to transmission protocols. TCP/IP(Transmission Control Protocol/Internet Protocol) provides confirming of receiving data (acknowledge), control of window size, and data retransmission and so on. Because of the complexity of the mechanism and sharing of the resource of the network the time-delay is greater than the circuit channel type communications. And the jitter, variation

Fig.1 shows an example of time delay during a day between two campuses(Kumamoto and Fukuoka) in 1998. The left graph (a) indicates the time series from midnight to midnight, and the right(b) is the histogram of the number of packets with respect to transmission delay. It is too difficult to construct the prediction model of time-delay because of the randomness and

In practice it is possible to set *Lmax* such that 95% of packets are travels in the time interval

Suppose the situation in which an operator manipulates (push/ pull/ lift/ put on and so on) some object through the communication network. Such a kind of system is called a master slave system or *tele-operation system*. Usually the terminals for the operator and the object are called the *master* and the *slave* equipment each other. Operators motion is converted to the motion data by the master mechanism, transmitted to the slave side and realized as a motion of the slave equipment. If the system can transmit the force information caused in the slave side to the master side it called *bilateral tele-operation systems*. Several kind of mechanisms are

0 < *L* ≤ *Lmax* (1)

chaos. Here we adopt a statistical model as a rectangular distribution as follows.

**2. Communication network and time delay**

of the time delay occurs frequently.

**3. Historical review of tele-operation systems**

proposed for the control of bilateral tele-operation systems.

[0, *Lmax*].

**3.1 Master slave system**

the communication channel is so small as can be ignored.

It is well-known that the time delay in the loop deteriorates the stability, performance and robustness of the feedback systems. Fig.2(b) shows a demonstration of the tarde-off of gain *K* and time-delay *L*. If (*L*, *K*) is chosen in the range of *stable* region the closed loop system depicted in Fig. 2(a) becomes stable, and vice versa. This trade-off curve is identical to the contour of the *H*∞ norm of the transfer function

$$\gamma := \left\| \frac{K}{s^2 + K} (1 - e^{-Ls}) \right\|\_{\infty} \tag{2}$$

with *γ* = 1. It is easy to calculate that the transfer function matches to one at the cutting point a in Fig.2.

This plot says the following facts.


According to the considerations the necessity of careful investigation to time-delay is required.

## **3.3 Scattering and wave variable method**

Anderson and Spong (6) introduced a new communication architecture for tele-operation over the network with time-delay. Their method is based on the passivity and scattering representation of the network. Thus the strictly passivity of master and slave systems

where *Lmax* is the upper bound of the estimated time-delay. This means that the uncertainty caused by the variation of time-delay between [0, *Lmax*] can be covered by the weighting

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 377

*WD*(*s*, *Lmax*) = 2.1*<sup>s</sup>*

delay = 0.5

<sup>10</sup>−2 <sup>10</sup>−1 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>3</sup> 10−5

10−4

4 can be replaced as

**4.2 PE type bilateral tele-operation system**

Fig. 3. variation of the gain casued by time delay and inequality in (4)

Fig. 4. correspondence between time-delay and multiplicative uncertainty

*e*

By using the function *WD*(*s*; *Lmax*) and uncertain bounded function Δ (|Δ| < 1) the inequality

Here we introduce a simple PE type bilateral tele-operation system designed with robust control technique. Two joystick mechanisms, corresponds to master and slave, are considered.

<sup>−</sup>*Ls* = 1 + *WD*(*s*; *Lmax*)Δ. (6)

10−3

10−2

10−1

100

101

*s* +

1 *Lmax* . (5)

function *WD*(*s*; *Lmax*) as a high-pass filter with cut-off frequency 1/*Lmax*[rad/sec]

Fig. 2. Trade-off between time-delay and loop gain (delay vs. loop gain)

and stationary time-delay are assumed, which are strong constraints for design. Moreover Niemeyer and Slotine (7) extended their method by using wave variables. Since it is a generalization of Anderson-Spong method, it has the same constraints and difficulties in practice. On the other hand Leung, Francis and Apkarian (8) proposed a controller designed via *μ*−Synthesis. The proposed method based on robust control theory can deal fluctuation of time-delay and has strong practicability. But all the above methods have the same configuration in which the master and slave system exchanges the velocity and the force variables (*v*, *f*) through the network. This means that the position, integral of velocity *v*, of the master and the slave systems are depend on the initial conditions, and the stability is ensured not in the sense of position but velocity. Moreover the necessity of force sensors makes the systems configuration sophisticated. The more simple architecture is prefer for the practical application.

#### **4. Robust control approach**

#### **4.1 Paradigm of robust control**

There are several kind of strategies to overcome the problem of time delay. Assuming the rectangular (uniform) distribution of time delay *H*∞ control theory can be applied as follows. Fig.4 shows the correspondence between time-delay and multiplicative uncertainty. Now let define a 1-st order high-pass filter *WD*(*s*; *L*) as

$$W\_D(s;L) = \frac{As}{s + \frac{1}{L}}\tag{3}$$

where *<sup>A</sup>* <sup>=</sup> 2.102904074495... It is easy to verify that the norm of (*e*−*Ls* <sup>−</sup> <sup>1</sup>) holds the following inequality for any frequency(on the imaginary axis) and any time delay *L* with 0 < *L* ≤ *Lmax*

$$|1 - e^{-j\omega L}| < \mathcal{W}\_D(j\omega; L) \le \mathcal{W}\_D(j\omega; L\_{\text{max}}), \quad \forall j\omega \in j\mathbf{R}, \tag{4}$$

4 Robust control book 3

(a) (b)

and stationary time-delay are assumed, which are strong constraints for design. Moreover Niemeyer and Slotine (7) extended their method by using wave variables. Since it is a generalization of Anderson-Spong method, it has the same constraints and difficulties in practice. On the other hand Leung, Francis and Apkarian (8) proposed a controller designed via *μ*−Synthesis. The proposed method based on robust control theory can deal fluctuation of time-delay and has strong practicability. But all the above methods have the same configuration in which the master and slave system exchanges the velocity and the force variables (*v*, *f*) through the network. This means that the position, integral of velocity *v*, of the master and the slave systems are depend on the initial conditions, and the stability is ensured not in the sense of position but velocity. Moreover the necessity of force sensors makes the systems configuration sophisticated. The more simple architecture is prefer for the practical

There are several kind of strategies to overcome the problem of time delay. Assuming the rectangular (uniform) distribution of time delay *H*∞ control theory can be applied as follows. Fig.4 shows the correspondence between time-delay and multiplicative uncertainty. Now let

*WD*(*s*; *<sup>L</sup>*) = *As*

where *<sup>A</sup>* <sup>=</sup> 2.102904074495... It is easy to verify that the norm of (*e*−*Ls* <sup>−</sup> <sup>1</sup>) holds the following inequality for any frequency(on the imaginary axis) and any time delay *L* with 0 < *L* ≤ *Lmax*

*s* + 1 *L*

<sup>−</sup>*jωL*<sup>|</sup> <sup>&</sup>lt; *WD*(*jω*; *<sup>L</sup>*) <sup>≤</sup> *WD*(*jω*; *Lmax*), <sup>∀</sup>*j<sup>ω</sup>* <sup>∈</sup> *<sup>j</sup>***R**, (4)

stable

K gain

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

unstable region

L delay[sec]

(3)

2 1 *s*

<sup>−</sup> *K*

application.

**4. Robust control approach**

**4.1 Paradigm of robust control**

define a 1-st order high-pass filter *WD*(*s*; *L*) as


1 *Ls e*<sup>−</sup> −

a

−

Fig. 2. Trade-off between time-delay and loop gain (delay vs. loop gain)

where *Lmax* is the upper bound of the estimated time-delay. This means that the uncertainty caused by the variation of time-delay between [0, *Lmax*] can be covered by the weighting function *WD*(*s*; *Lmax*) as a high-pass filter with cut-off frequency 1/*Lmax*[rad/sec]

*s* +

1

. (5)

*WD*(*s*, *Lmax*) = 2.1*<sup>s</sup>*

Fig. 3. variation of the gain casued by time delay and inequality in (4)

Fig. 4. correspondence between time-delay and multiplicative uncertainty

By using the function *WD*(*s*; *Lmax*) and uncertain bounded function Δ (|Δ| < 1) the inequality 4 can be replaced as

$$e^{-Ls} = 1 + \mathcal{W}\_D(\mathbf{s}; L\_{\text{max}})\Delta. \tag{6}$$

#### **4.2 PE type bilateral tele-operation system**

Here we introduce a simple PE type bilateral tele-operation system designed with robust control technique. Two joystick mechanisms, corresponds to master and slave, are considered.

time-delay *L* > 0 exists between master and slave controllers symmetrically. The evaluated

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 379

In the rest of the chapter *em*0,*es*0(*emL*,*esL*) are called as errors in ideal ( computed ) deviation.

= *Xm*(*s*) − *Xs*(*s*) + *Xs*(*s*) − *e*

This means that the minimization of computed deviations (*EmL*(*s*), *EsL*(*s*)) is acomplished by the simultaneous minimization of *Em*0(*s*)(= <sup>−</sup>*Es*0(*s*)), (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*Ls*)*Xs*(*s*) and (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*Ls*)*Xm*(*s*)

As mensioned in previous section the time delay *L* includes uncertainty. But if the upper bound of *L* is obtained as *Lmax* according to 4and 5 the minimization problem can be

Let us consider the two joystick mechanisms as a system with two inputs and two outputs

*Pm*(*s*) 0 0 *Ps*(*s*)

*Cmm*(*s*) *Cms*(*s*) *Csm*(*s*) *Css*(*s*)

acomplished by the minimization of *Em*0(*s*),*WD*(*s*; *Lmax*)*Xs*(*s*) and *WD*(*s*; *Lmax*)*Xm*(*s*). *H*∞ theory gives a design method to obtain an appropriate feedback gain to keep stability and

≤ |*Em*0(*jω*)| + |*WD*(*jω*; *L*)*Xs*(*jω*)|.

= *Em*0(*s*)+(1 − *e*

*EsL*(*s*) = *Es*0(*s*)+(1 − *e*


<sup>−</sup>*LsXs*(*s*)

*emL*(*t*) = *xm*(*t*) − *xs*(*t* − *L*) (11) *esL*(*t*) = *xs*(*t*) − *xm*(*t* − *L*) (12)

<sup>−</sup>*LsXs*(*s*)

<sup>−</sup>*jωL*)*Xs*(*jω*)<sup>|</sup>


 *Um*(*s*) *Us*(*s*)

, *Ps*(*s*) = *Ks*

*s*(*Jss* + *Ds*)

 *Xm*(*s*) *Xs*(*s*)

. (17)

. (18)

(19)

≤ |*Em*0(*jω*)| + |*WD*(*jω*; *Lmax*)*Xs*(*jω*)|. (15)

<sup>−</sup>*Ls*)*Xs*(*s*). (13)

<sup>−</sup>*Ls*)*Xm*(*s*). (14)

deviations (9)(10) at each controller might be computed as follows.

The Laplace transform of computed deviation *emL* (11) is written as

*EmL*(*s*) = *Xm*(*s*) − *e*

In the same way the Laplace transform of *esL* can be written as

robustness against the type of model uncertainty.

The purpose is the design of a controller

 *Xm*(*s*) *Xs*(*s*)

 *Um*(*s*) *Us*(*s*)

 = 

*s*(*Jms* + *Dm*)

*Pm*(*s*) = *Km*

 = 

from the inequality as

plant

**4.3 Plant model** 

where,

Each joystick has a DC-servo motor for torque generation and a position sensor for measurement of the angle of the joy-stick. Force sensors attached to the joysticks are not for use of the control but for the performance evaluation of force communications. They are controlled by computers which are connected to the computer network (see Fig.5). These joysticks are assumed to be modeled as

$$J\_m \ddot{\mathbf{x}}\_m(t) + D\_m \dot{\mathbf{x}}\_m(t) = K\_m u\_m(t) + f\_m(t) \tag{7}$$

$$D\_s \ddot{\mathbf{x}}\_s(t) + D\_s \dot{\mathbf{x}}\_s(t) = \mathbf{K}\_s \boldsymbol{\mu}\_s(t) - f\_s(t) \tag{8}$$

where *x* , *f* and *u* indicate the variables of position of the joystick, external force and input voltage for motor torque generator each other. *J* and *D* indicate the physical parameters of inertia and friction each other. The suffixes *m*,*s* indicate the master and the slave respectively.

Fig. 5. A view of experimental equipments: Two joysticks controlled by computers connected to the network


Table 1. Parameters of master and slave joysticks

If there is no time delay between master and slave sides the deviations of the joysticks are evaluated as

$$
\mathbf{x}\_{m0}(t) = \mathbf{x}\_m(t) - \mathbf{x}\_s(t) \tag{9}
$$

$$
\varepsilon\_{\rm s0}(t) = \chi\_{\rm s}(t) - \chi\_{\rm m}(t) = -\varepsilon\_{\rm m0}(t). \tag{10}
$$

When the master and the slave controllers exchange the information through the network, as stated in the previous section, the time-delay must be considered. Let us assume that the time-delay *L* > 0 exists between master and slave controllers symmetrically. The evaluated deviations (9)(10) at each controller might be computed as follows.

$$
\varepsilon\_{mL}(t) = \mathbf{x}\_m(t) - \mathbf{x}\_s(t-L) \tag{11}
$$

$$
\epsilon\_{sL}(t) = \mathbf{x}\_s(t) - \mathbf{x}\_m(t-L) \tag{12}
$$

In the rest of the chapter *em*0,*es*0(*emL*,*esL*) are called as errors in ideal ( computed ) deviation. The Laplace transform of computed deviation *emL* (11) is written as

$$\begin{split} E\_{\mathfrak{m}L}(\mathbf{s}) &= X\_{\mathfrak{m}}(\mathbf{s}) - e^{-Ls}X\_{\mathbf{s}}(\mathbf{s}) \\ &= X\_{\mathfrak{m}}(\mathbf{s}) - X\_{\mathbf{s}}(\mathbf{s}) + X\_{\mathbf{s}}(\mathbf{s}) - e^{-Ls}X\_{\mathbf{s}}(\mathbf{s}) \\ &= E\_{\mathfrak{m}0}(\mathbf{s}) + (1 - e^{-Ls})X\_{\mathbf{s}}(\mathbf{s}). \end{split} \tag{13}$$

In the same way the Laplace transform of *esL* can be written as

$$E\_{\rm sL}(\mathbf{s}) = E\_{\rm s0}(\mathbf{s}) + (1 - e^{-\rm Ls})X\_{\rm m}(\mathbf{s}).\tag{14}$$

This means that the minimization of computed deviations (*EmL*(*s*), *EsL*(*s*)) is acomplished by the simultaneous minimization of *Em*0(*s*)(= <sup>−</sup>*Es*0(*s*)), (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*Ls*)*Xs*(*s*) and (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*Ls*)*Xm*(*s*) from the inequality as

$$\begin{split} |E\_{mL}(j\omega)| &\leq |E\_{m0}(j\omega)| + |(1 - e^{-j\omega L})X\_s(j\omega)| \\ &\leq |E\_{m0}(j\omega)| + |\mathcal{W}\_D(j\omega; L)X\_s(j\omega)|. \\ &\leq |E\_{m0}(j\omega)| + |\mathcal{W}\_D(j\omega; L\_{\text{max}})X\_s(j\omega)|. \end{split} \tag{15}$$

$$|E\_{\rm sL}(j\omega)| \le |E\_{\rm s0}(j\omega)| + |\mathcal{W}\_{\rm D}(j\omega; L\_{\rm max})X\_{\rm m}(j\omega)|.\tag{16}$$

As mensioned in previous section the time delay *L* includes uncertainty. But if the upper bound of *L* is obtained as *Lmax* according to 4and 5 the minimization problem can be acomplished by the minimization of *Em*0(*s*),*WD*(*s*; *Lmax*)*Xs*(*s*) and *WD*(*s*; *Lmax*)*Xm*(*s*).

*H*∞ theory gives a design method to obtain an appropriate feedback gain to keep stability and robustness against the type of model uncertainty.

## **4.3 Plant model**

Let us consider the two joystick mechanisms as a system with two inputs and two outputs plant

$$
\begin{bmatrix} X\_m(s) \\ X\_s(s) \end{bmatrix} = \begin{bmatrix} P\_m(s) & 0 \\ 0 & P\_s(s) \end{bmatrix} \begin{bmatrix} \mathcal{U}\_m(s) \\ \mathcal{U}\_s(s) \end{bmatrix}. \tag{17}
$$

where,

6 Robust control book 3

Each joystick has a DC-servo motor for torque generation and a position sensor for measurement of the angle of the joy-stick. Force sensors attached to the joysticks are not for use of the control but for the performance evaluation of force communications. They are controlled by computers which are connected to the computer network (see Fig.5). These

where *x* , *f* and *u* indicate the variables of position of the joystick, external force and input voltage for motor torque generator each other. *J* and *D* indicate the physical parameters of inertia and friction each other. The suffixes *m*,*s* indicate the master and the slave respectively.

Fig. 5. A view of experimental equipments: Two joysticks controlled by computers connected

*Jm* 0.0140 [Kgm2] *Js* 0.0379 [Kgm2] *Dm* 0.0110 [Nms] *Ds* 0.0250 [Nms] *Km* 0.2557 [Nm/V] *Ks* 0.2557 [Nm/V]

If there is no time delay between master and slave sides the deviations of the joysticks are

When the master and the slave controllers exchange the information through the network, as stated in the previous section, the time-delay must be considered. Let us assume that the

*em*0(*t*) = *xm*(*t*) − *xs*(*t*) (9) *es*0(*t*) = *xs*(*t*) − *xm*(*t*) = −*em*0(*t*). (10)

*Jmx*¨*m*(*t*) + *Dmx*˙*m*(*t*) = *Kmum*(*t*) + *fm*(*t*) (7) *Jsx*¨*s*(*t*) + *Dsx*˙*s*(*t*) = *Ksus*(*t*) − *fs*(*t*) (8)

joysticks are assumed to be modeled as

to the network

evaluated as

Table 1. Parameters of master and slave joysticks

$$P\_m(s) = \frac{K\_m}{s(J\_ms + D\_m)}, \quad P\_s(s) = \frac{K\_s}{s(J\_ms + D\_s)}.\tag{18}$$

The purpose is the design of a controller

$$
\begin{bmatrix} \mathbf{U}\_m(\mathbf{s}) \\ \mathbf{U}\_s(\mathbf{s}) \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{mm}(\mathbf{s}) \ \mathbf{C}\_{ms}(\mathbf{s}) \\ \mathbf{C}\_{sm}(\mathbf{s}) \ \mathbf{C}\_{ss}(\mathbf{s}) \end{bmatrix} \begin{bmatrix} \mathbf{X}\_m(\mathbf{s}) \\ \mathbf{X}\_s(\mathbf{s}) \end{bmatrix} \tag{19}
$$

with two more variables *w*3, *w*<sup>4</sup> which come from the uncertainty (6) as

*w* = �

*z* = �

� *z X* � = �

*w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup> *w*<sup>4</sup>

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 381

*U* = �

*X* = �

*z*<sup>1</sup> *z*<sup>2</sup> *z*<sup>3</sup> *z*<sup>4</sup> *z*<sup>5</sup> *z*<sup>6</sup> *z*<sup>7</sup>

*Um Us*

*Xm Xs*

*G*<sup>11</sup> *G*<sup>12</sup> *<sup>G</sup>*<sup>21</sup> *<sup>G</sup>*<sup>22</sup> � � *<sup>w</sup>*

*U* �

*W*11*Pm* −*W*11*Ps W*<sup>11</sup> −*W*<sup>11</sup> *W*11*Pm* −*W*11*Ps W*12*Pm* 0 *W*<sup>12</sup> 0 *W*12*Pm* 0

0 *W*13*Ps* 0 *W*<sup>13</sup> 0 *W*13*Ps W*<sup>21</sup> 0 00 *W*<sup>21</sup> 0 0 *W*<sup>22</sup> 0 0 0 *W*<sup>22</sup>

0 *WD*2*Ps* 0 0 0 *WD*2*Ps Pm* 0 00 *Pm* 0 0 *Ps* 0 0 0 *Ps*

*Gzw*(*s*) = *<sup>G</sup>*<sup>11</sup> <sup>+</sup> *<sup>G</sup>*12(*<sup>I</sup>* <sup>−</sup> *CG*22)−1*CG*<sup>21</sup> (31)

�*Gzw*(*s*)�<sup>∞</sup> < *γ* (32)

*WD*1*Pm* 0 00 *WD*1*Pm* 0

By applying the controller (19) to the above system a transfer function matrix from *w* to *z*

can be obtained. By using the design procedure based on the *H*∞ control theory a controller is

Let's define the exogenous input *w* and evaluated output *z* as

the generalized plant is obtained as follows ( see Fig.7).

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

=

where *γ* is a design parameter chosen as small as possible (5).

Moreover defining the following vectors

**4.4 Construction of generalized plant** 

�

obtained such that

*G*<sup>11</sup> *G*<sup>12</sup> *<sup>G</sup>*<sup>21</sup> *<sup>G</sup>*<sup>22</sup> � *w*<sup>3</sup> = Δ*z*<sup>6</sup> , *w*<sup>4</sup> = Δ*z*7. (24)

�*<sup>T</sup>* (25)

�*<sup>T</sup>* (27)

�*<sup>T</sup>* (28)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (29)

(30)

�*<sup>T</sup>* . (26)

which satisfies the requirements specified as follows. The schematic diagram is depicted in Fig.6

Fig. 6. scheme of PE type master-slave system (a)ideal (non time-delay) scheme,(b) practical(implementable ) scheme

#### **tracking performance**

The closed loop system is a kind of regulator which makes the deviation *e* = *Xm* − *Xs* → 0 as time goes. For the robust control systems design the deviation *e* is generalized to a criteria for tracking performance as

$$z\_1 = W\_{11}e\_\prime \; \; \; e = \mathbf{x}\_\mathcal{m} - \mathbf{x}\_\mathcal{s}.\tag{20}$$

In general *W*<sup>11</sup> has to be chosen as high gain at low frequency and low gain at high frequency.

#### **stability augmentation**

More over in order to obtain a adequate local feedback gain which improves the stability and robustness of closed loop system, the criteria for stability is formulated as

$$z\_2 = W\_{12} \mathbf{x}\_{m\prime} \ z\_3 = W\_{13} \mathbf{x}\_{s\prime} \tag{21}$$

#### **properness of controller**

In order to ensure the properness of the controller *C*(*s*) in (19) the input variables for the plant are added to the criteria for design as

$$z\_4 = \mathcal{W}\_{21}(w\_1 + u\_1), \ z\_5 = \mathcal{W}\_{22}(w\_2 + u\_2) \tag{22}$$

where *w*1, *w*<sup>2</sup> are exogenous inputs or exerted external forces(torques) as in Fig.7.

#### **robust stability against time-delay**

As mentioned in the previous section the robustness corresponds to the minimization of *WD*(*s*; *Lmax*)*Xs*(*s*) and *WD*(*s*; *Lmax*)*Xm*(*s*). Thus we introduce new two output variables

$$z\_6 = W\_\mathcal{D} \mathbf{x}\_\mathcal{W} \; ; \; z\_7 = W\_\mathcal{D} \mathbf{x}\_\mathcal{s} . \tag{23}$$

with two more variables *w*3, *w*<sup>4</sup> which come from the uncertainty (6) as

$$
\Delta w\_3 = \Delta z\_6 \, , \, w\_4 = \Delta z\_7 \,. \tag{24}
$$

#### **4.4 Construction of generalized plant**

8 Robust control book 3

which satisfies the requirements specified as follows. The schematic diagram is depicted in

(a) (b)

The closed loop system is a kind of regulator which makes the deviation *e* = *Xm* − *Xs* → 0 as time goes. For the robust control systems design the deviation *e* is generalized to a criteria for

In general *W*<sup>11</sup> has to be chosen as high gain at low frequency and low gain at high frequency.

More over in order to obtain a adequate local feedback gain which improves the stability and

In order to ensure the properness of the controller *C*(*s*) in (19) the input variables for the plant

As mentioned in the previous section the robustness corresponds to the minimization of *WD*(*s*; *Lmax*)*Xs*(*s*) and *WD*(*s*; *Lmax*)*Xm*(*s*). Thus we introduce new two output variables

where *w*1, *w*<sup>2</sup> are exogenous inputs or exerted external forces(torques) as in Fig.7.

robustness of closed loop system, the criteria for stability is formulated as

*z*<sup>1</sup> = *W*11*e*, *e* = *xm* − *xs*. (20)

*z*<sup>2</sup> = *W*12*xm*, *z*<sup>3</sup> = *W*13*xs*. (21)

*z*<sup>4</sup> = *W*21(*w*<sup>1</sup> + *u*1), *z*<sup>5</sup> = *W*22(*w*<sup>2</sup> + *u*2) (22)

*z*<sup>6</sup> = *WDxm*, *z*<sup>7</sup> = *WDxs*. (23)

Fig. 6. scheme of PE type master-slave system (a)ideal (non time-delay) scheme,(b)

Fig.6

practical(implementable ) scheme

**tracking performance**

tracking performance as

**stability augmentation**

**properness of controller**

are added to the criteria for design as

**robust stability against time-delay**

Let's define the exogenous input *w* and evaluated output *z* as

$$\begin{array}{l} w = \begin{bmatrix} w\_1 \ w\_2 \ w\_3 \ w\_4 \end{bmatrix}^T \end{array} \tag{25}$$

$$z = \begin{bmatrix} z\_1 \ z\_2 \ z\_3 \ z\_4 \ z\_5 \ z\_6 \ z\_7 \end{bmatrix}^T. \tag{26}$$

Moreover defining the following vectors

�

$$\mathcal{U} = \begin{bmatrix} \mathcal{U}\_m \ \mathcal{U}\_s \end{bmatrix}^T \tag{27}$$

$$X = \begin{bmatrix} X\_{\mathfrak{m}} \ X\_{\mathfrak{s}} \end{bmatrix}^{T} \tag{28}$$

the generalized plant is obtained as follows ( see Fig.7).

$$
\begin{bmatrix} z \\ X \\ X \end{bmatrix} = \begin{bmatrix} G\_{11} \ G\_{12} \\ G\_{21} \ G\_{22} \end{bmatrix} \begin{bmatrix} w \\ U \end{bmatrix} \tag{29}
$$

$$
\begin{bmatrix} W\_{11}P\_{m} & -W\_{11}P\_{s} \ W\_{11} & W\_{11} - W\_{11} \end{bmatrix} W\_{11}P\_{m} & -W\_{11}P\_{s} \\
\begin{bmatrix} W\_{12}P\_{m} & 0 & W\_{12} & 0 & W\_{12}P\_{m} & 0 \\ 0 & W\_{13}P\_{s} & 0 & W\_{13} & 0 & W\_{13}P\_{s} \\ 0 & W\_{21} & 0 & 0 & W\_{21} & 0 \\ 0 & W\_{22} & 0 & 0 & 0 & W\_{22} \\ W\_{D1}P\_{m} & 0 & 0 & 0 & W\_{D1}P\_{m} & 0 \\ 0 & W\_{D2}P\_{s} & 0 & 0 & 0 & W\_{D2}P\_{s} \\ \hline P\_{m} & 0 & 0 & 0 & P\_{m} & 0 \\ 0 & P\_{s} & 0 & 0 & 0 & P\_{s} \end{bmatrix} \tag{30}
$$

By applying the controller (19) to the above system a transfer function matrix from *w* to *z*

$$\mathbf{G}\_{2w}(s) = \mathbf{G}\_{11} + \mathbf{G}\_{12}(I - \mathbf{C}\mathbf{G}\_{22})^{-1}\mathbf{C}\mathbf{G}\_{21} \tag{31}$$

can be obtained. By using the design procedure based on the *H*∞ control theory a controller is obtained such that

$$\|\|G\_{\rm 2w}(\mathbf{s})\|\|\_{\infty} < \gamma \tag{32}$$

where *γ* is a design parameter chosen as small as possible (5).

*L*[sec] stability tracking

Xm<red> Xs<blue>

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 383

Xm<red> Xs<blue>

Xm<red> Xs<blue>

X[rad]

X[rad]

X[rad]

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time sec[s]

Master and Slave Position

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time sec[s]

Master and Slave Position

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time sec[s]

Master and Slave Position

Xm<red> Xs<blue>

Xm<red> Xs<blue>

Xm<red> Xs<blue>

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −3

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −3

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −3

Time sec[s]

Table 2. Analysis of stability and tracking performance w.r.t. time-delay

Time sec[s]

Master and Slave Position (No force)

Time sec[s]

Master and Slave Position (No force)

Master and Slave Position (No force)

0.00

0.10

0.50

X[rad]

X[rad]

X[rad]

Fig. 7. generalized plant for master-slave systems

#### **5. Simulation and estimates of robustness**

In order to demonstrait the robustness w.r.t. time delay computer simulations on the stability and tracking analysis are shown here. The time-delay is assumed as *Lmax* = 0.1[sec]. According to the specifications discussed above weighting functions {*Wi*}, *i* = 1, 2, 3 are set as follows.

$$W\_{D1}(\mathbf{s}) = W\_{D2}(\mathbf{s}) = \frac{2.1 \mathbf{s}}{\mathbf{s} + 10} \tag{33}$$

$$W\_{11}(\mathbf{s}) = \frac{1.0 \times 10^5 \mathbf{s} + 2.0 \times 10^5}{1.0 \times 10^4 \mathbf{s} + 1.0} \tag{34}$$

$$W\_{12}(\mathbf{s}) = W\_{13}(\mathbf{s}) = \frac{0.01s + 1}{0.1s + 1} \tag{35}$$

$$W\_{21}(\mathbf{s}) = W\_{22}(\mathbf{s}) = 10\tag{36}$$

Fig.8 shows the bode diagrams of the transfer functions.

Table 2 shows simulation results for analysis of stability and tracking performance.

The values in the left column indicate actual time-delay *Lact* in simulation. When the actual time-delay is not greater than the assumed maximum one, that is *Lact* ≤ *Lmax*, the stability and tracking performance are kept well. On the other hand if the time-delay exceeds estimated value , *Lact* > *Lmax*, the performance of the system becomes worse. It points that the importance of the estimate of maximum time delay *Lmax*.

10 Robust control book 3

In order to demonstrait the robustness w.r.t. time delay computer simulations on the stability and tracking analysis are shown here. The time-delay is assumed as *Lmax* = 0.1[sec]. According to the specifications discussed above weighting functions {*Wi*}, *i* = 1, 2, 3 are set

*WD*1(*s*) = *WD*2(*s*) = 2.1*<sup>s</sup>*

*<sup>W</sup>*11(*s*) = 1.0 <sup>×</sup> <sup>105</sup>*<sup>s</sup>* <sup>+</sup> 2.0 <sup>×</sup> <sup>105</sup>

*<sup>W</sup>*12(*s*) = *<sup>W</sup>*13(*s*) = 0.01*<sup>s</sup>* <sup>+</sup> <sup>1</sup>

The values in the left column indicate actual time-delay *Lact* in simulation. When the actual time-delay is not greater than the assumed maximum one, that is *Lact* ≤ *Lmax*, the stability and tracking performance are kept well. On the other hand if the time-delay exceeds estimated value , *Lact* > *Lmax*, the performance of the system becomes worse. It points that the

Table 2 shows simulation results for analysis of stability and tracking performance.

+

+

+

Fig. 7. generalized plant for master-slave systems

**5. Simulation and estimates of robustness**

Fig.8 shows the bode diagrams of the transfer functions.

importance of the estimate of maximum time delay *Lmax*.

as follows.

+

+ −

+

*<sup>s</sup>* <sup>+</sup> <sup>10</sup> (33)

0.1*<sup>s</sup>* <sup>+</sup> <sup>1</sup> (35)

1.0 <sup>×</sup> <sup>10</sup>4*<sup>s</sup>* <sup>+</sup> 1.0 (34)

*W*21(*s*) = *W*22(*s*) = 10 (36)

+

+

+

Table 2. Analysis of stability and tracking performance w.r.t. time-delay

<sup>10</sup>−5 <sup>100</sup> <sup>105</sup> −100

<sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> −100

<sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> −100

<sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> −100

<sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> <sup>50</sup>

w(rad/sec)

w(rad/sec)

Controller(Xs−>Xs) bode

Controller(Xs−>Xm) bode

−50 0 50

100 150 200

phase(deg)

gain(dB)

phase(deg)

gain(dB)

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 385

<sup>10</sup>−5 <sup>100</sup> <sup>105</sup> <sup>50</sup>

<sup>10</sup>−5 <sup>100</sup> <sup>105</sup> −100

<sup>10</sup>−5 <sup>100</sup> <sup>105</sup> −100

Fig. 9. Bode diagram of the controller *C*(*s*)

w(rad/sec)

w(rad/sec)

Controller(Xm−>Xs) bode

Controller(Xm−>Xm) bode

100 150 200

−50 0 50

phase(deg)

gain(dB)

phase(deg)

gain(dB)

Fig. 8. Bode (gain ) diagrams (a) *W*<sup>11</sup> ,(b) *W*12, *W*13, (c) *W*21, *W*22, (d) *WD*1, *WD*<sup>2</sup>

### **Analysis of designed controller**

Fig. 9 shows bode diagrams of controller *C*(*s*). It can be observed that roughly seeing of the controller is a kind of integrator, but it works constant gain in the middle range 1 ∼ 103[rad/sec].

#### **5.1 Analysis via hybrid matrix**

In order to investigate the force communication ability and transperency of master-slave system hybrid matrix is defined as

$$
\begin{bmatrix} F\_s \\ X\_m \end{bmatrix} = \begin{bmatrix} h\_{11} \ h\_{12} \\ h\_{21} \ h\_{22} \end{bmatrix} \begin{bmatrix} X\_s \\ F\_m \end{bmatrix} . \tag{37}
$$

Table 10 shows the bode daigram of hybrid matrix. *h*<sup>12</sup> = *Fs*/*Fm* and *h*<sup>21</sup> = *Xm*/*Xs* indicate that the tracking ability of position and force communication are expected to work in the range from DC upto 1 rad/sec.

#### **6. Experiments over the network with time-delay**

In order to demonstrate the robustness of the proposed control systems a networked control system is constructed as in Figure 11.

Master and slave mechanisms and their local controllers (*Cm*, *Cs*) are located on the same cite (at Kumamoto city), and another computer (*CT*) is located beyond the network (at Kitakyushu city, 150km far from Kumamoto city). These three computers are connected to the network JGN , which was Japanese broadband network as an experimental testbed administrated by TAO 1. The controllers *Cm* and *Cs* can communicate each other by way of relay computer *CT*, but not admitted to communicate directly. The transmission capacity of the network is about 100Mbps. The control period at *Cm* and *Cs* is 5[msec] and that of communication period between *Cm* and *Cs* is 10[msec]. The communication protocol UDP/IP is adopted.

<sup>1</sup> Telecommunications Advancement Organization of Japan; reorganized to NICT(National Institute of Information and Communications Technology) in 2004 (http://www.nict.go.jp/)

Fig. 9. Bode diagram of the controller *C*(*s*)

12 Robust control book 3

Fig. 9 shows bode diagrams of controller *C*(*s*). It can be observed that roughly seeing of the controller is a kind of integrator, but it works constant gain in the middle range 1 ∼

In order to investigate the force communication ability and transperency of master-slave

Table 10 shows the bode daigram of hybrid matrix. *h*<sup>12</sup> = *Fs*/*Fm* and *h*<sup>21</sup> = *Xm*/*Xs* indicate that the tracking ability of position and force communication are expected to work in the range

In order to demonstrate the robustness of the proposed control systems a networked control

Master and slave mechanisms and their local controllers (*Cm*, *Cs*) are located on the same cite (at Kumamoto city), and another computer (*CT*) is located beyond the network (at Kitakyushu city, 150km far from Kumamoto city). These three computers are connected to the network JGN , which was Japanese broadband network as an experimental testbed administrated by TAO 1. The controllers *Cm* and *Cs* can communicate each other by way of relay computer *CT*, but not admitted to communicate directly. The transmission capacity of the network is about 100Mbps. The control period at *Cm* and *Cs* is 5[msec] and that of communication period

<sup>1</sup> Telecommunications Advancement Organization of Japan; reorganized to NICT(National Institute of

between *Cm* and *Cs* is 10[msec]. The communication protocol UDP/IP is adopted.

Information and Communications Technology) in 2004 (http://www.nict.go.jp/)

 *Xs Fm*  . (37)

**Analysis of designed controller**

**5.1 Analysis via hybrid matrix**

from DC upto 1 rad/sec.

system hybrid matrix is defined as

system is constructed as in Figure 11.

103[rad/sec].

Fig. 8. Bode (gain ) diagrams (a) *W*<sup>11</sup> ,(b) *W*12, *W*13, (c) *W*21, *W*22, (d) *WD*1, *WD*<sup>2</sup>

 *Fs Xm* = *h*<sup>11</sup> *h*<sup>12</sup> *h*<sup>21</sup> *h*<sup>22</sup>

**6. Experiments over the network with time-delay**

communication

*Cs*

**design parameters**

Network simulator

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 387

*CT*

Network simulator

JGN UDP/IP/ATM non-realtime

**communication prog.** /non-realtime

**control program** /hard-realtime

**Mechatronics/** DC-servo motor Current Amp. Rotary Encoder Inertia and friction Slave side Master side

*Ps Pm*

*<sup>W</sup>*<sup>11</sup> <sup>=</sup> 0.3*<sup>s</sup>* <sup>+</sup> <sup>20</sup> <sup>×</sup> <sup>107</sup>

1 *Lmax*

By specifying the allowable time-delay *Lmax* and upper bound of *H*<sup>∞</sup> norm *γ* in (32) the

*<sup>W</sup>*<sup>12</sup> <sup>=</sup> *<sup>W</sup>*<sup>13</sup> <sup>=</sup> <sup>1</sup>

*WD* <sup>=</sup> 2.1*<sup>s</sup>* <sup>+</sup> *� s* +

communication

**network simulator** /non-realtime(quasi-realtime) timde-delay, packet loss, packet order

control

*Cm*

(41)

physical susyems

3.9 <sup>×</sup> <sup>106</sup>*<sup>s</sup>* <sup>+</sup> <sup>10</sup><sup>5</sup> , (38)

*W*<sup>21</sup> = *W*<sup>22</sup> = 0.2, (40)

1.5 , (39)

control

physical susyems

controller *C*(*s*) is obtained by using MATLAB 2.

<sup>2</sup> MATLAB is a product of The MathWorks, Inc.

Fig. 11. experimental network system for robust control on JGN

Weighting functions in generalized plant (29) are specified as

Fig. 10. Bode (gain) diagram of hybrid matrix (37)

Fig. 11. experimental network system for robust control on JGN

#### **design parameters**

14 Robust control book 3

w(rad/sec) <sup>10</sup>−5 <sup>100</sup> <sup>10</sup><sup>5</sup> −250

w(rad/sec) <sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> −200

−150

−100

gain(dB)

−50

0

50

−200

−150

−100

gain(dB)

−50

0

50

h12

w(rad/sec)

h22

w(rad/sec)

<sup>10</sup>−5 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>5</sup> −50

h21

10−5 <sup>100</sup> <sup>10</sup><sup>5</sup> −250

Fig. 10. Bode (gain) diagram of hybrid matrix (37)

h11

0

−200

−150

−100

gain(dB)

−50

0

50

50

100

gain(dB)

150

200

Weighting functions in generalized plant (29) are specified as

$$W\_{11} = \frac{0.3 \text{s} + 20 \times 10^7}{3.9 \times 10^6 \text{s} + 10^5} \,\text{/}\tag{38}$$

$$W\_{12} = W\_{13} = \frac{1}{1.5} \,\text{\AA} \tag{39}$$

$$W\_{21} = W\_{22} = 0.2 \,\text{,}\tag{40}$$

$$W\_D = \frac{2.1s + \varepsilon}{s + \frac{1}{L\_{\text{max}}}} \tag{41}$$

By specifying the allowable time-delay *Lmax* and upper bound of *H*<sup>∞</sup> norm *γ* in (32) the controller *C*(*s*) is obtained by using MATLAB 2.

<sup>2</sup> MATLAB is a product of The MathWorks, Inc.

**experimental result**


**7. Summary**

**8. References**

(2002)

Company

*H*∞ control systems theory.

0

0.5

position(rad)

1

1.5

Fig. 13 indicates an experimental result for evaluation of tracking performance. In the first half master( solid line ) leads to the slave(dashed line). And last half slave leads to the master.

An Application of Robust Control for Force Communication Systems over Inferior Quality Network 389

master and slave posiiton

slave position[rad] C2M

C2S

mastar position[rad]

C3

<sup>64</sup> <sup>66</sup> <sup>68</sup> <sup>70</sup> <sup>72</sup> <sup>74</sup> <sup>76</sup> <sup>78</sup> <sup>80</sup> -1

In this chapter we discussed on an application of robust control for force communication systems over inferior quality network. According to the investigation of the experiments the effectiveness of bilateral tele-operation system for force communication is confirmed. Especially the most important problem of the robustness w.r.t. time-delay is improved by

[1] Shiotsuki,T., Force communication over the computer network, *3rd IFAC Symposium on*

[2] Shiotsuki,T., Nasu,T., A case study of tele-operation system with time-delay, *SICE02*

[3] Kim,J., Kim,H., Tay,B.K.,Muniyandi, M., Srinivasan,M.A., Jordan, J., Mortensen,J., Oliveira,M. and Slater,M., Transatlantic Touch: A Study of Haptic Collaboration over

[4] Yashiro,D., Tian,D., and Ohnishi,K., Centralized Controller based Multilateral Control with Communication Delay, *Proceeding of The IEEE International Conference on*

[5] Doyle , Francis and Tannenbaum(1992), *Feedback Control Theory*, Macmillan Publish

*Mechatronic Systems, Manly Beach, Sydney, Australia*, (2004)353-358

Long Distance, *PRESENCE*, 13-3, 328-337 (2004)

*Mechatronics*, ICM 2011, Istanbul, Turkey, 13th(2011)

Fig. 13. experimental result of position tracking: master (- slid line), slave(– dashed line)

time(sec)

This means that the master-slave tele-operation system works symmetricaly well.

## **implementation issues**

The controllers are implemented in personal computers. The algorithms are coded by C-language with RT-Linux formats and embedded as a kernel modules of Linux system. The control period is set at 5 [msec] and data exchange rate is set at 10 [msec]. Thus the data processing sequences must be synchronized.

Fig.12 shows a sequence diagram. The time goes from left to right and the datagram travels from master side(top ) to slave side(bottom ) through the network(middle). Because of the control period is a half of the communication period a copied value of the oposit side is used once every two control calculation. The right half part of the diagram assumes the case of long time-delay. In this case copied value is used over and over again until the new datagram reaches again.

Fig. 12. Sequence diagram of data exchanges between master and slave


Table 3. Parameters of master and slave controllers

## **emulation of network with poor quality**

*CT* is a computer located beyond the network to emulate various kind of qualities. It can emulate various kind of probability distribution of transmission delay, packet loss, packet shuffling and so on. Here we set the maximum time delay 1.0[sec] and the jitter in the Pareto distribution . The design parameter for robustness w.r.t. time-delay is set at *Lmax* = 1.5[sec] and <sup>=</sup> 6.6 <sup>×</sup> <sup>10</sup>−<sup>3</sup> Here the <sup>&</sup>gt; 0 is selected to reduce the effects of integrator.

## **experimental result**

16 Robust control book 3

The controllers are implemented in personal computers. The algorithms are coded by C-language with RT-Linux formats and embedded as a kernel modules of Linux system. The control period is set at 5 [msec] and data exchange rate is set at 10 [msec]. Thus the data

Fig.12 shows a sequence diagram. The time goes from left to right and the datagram travels from master side(top ) to slave side(bottom ) through the network(middle). Because of the control period is a half of the communication period a copied value of the oposit side is used once every two control calculation. The right half part of the diagram assumes the case of long time-delay. In this case copied value is used over and over again until the new datagram

**Large time delay**

**time**

**implementation issues**

reaches again.

processing sequences must be synchronized.

**Time delay**

Plant **Control period 5[msec]**

Table 3. Parameters of master and slave controllers

**emulation of network with poor quality**

Fig. 12. Sequence diagram of data exchanges between master and slave

MASTER CPU AMD Duron 600MHz NIC 100/10 Base T

SLAVE CPU Pentium 75+ 166MHz NIC 100/10 Base T

*CT* is a computer located beyond the network to emulate various kind of qualities. It can emulate various kind of probability distribution of transmission delay, packet loss, packet shuffling and so on. Here we set the maximum time delay 1.0[sec] and the jitter in the Pareto distribution . The design parameter for robustness w.r.t. time-delay is set at *Lmax* = 1.5[sec]

RELAY CPU Pentium 600MHz NIC 100/10 Base T OS Linux 2.2.14

and <sup>=</sup> 6.6 <sup>×</sup> <sup>10</sup>−<sup>3</sup> Here the <sup>&</sup>gt; 0 is selected to reduce the effects of integrator.

OS RT-Linux 3.1 on Linux 2.2.19

OS RT-Linux 2.2 on Linux 2.2.14

Master

Slave Network Interface

Slave controller

Slave

**10[msec] Data exchange rate** Fig. 13 indicates an experimental result for evaluation of tracking performance. In the first half master( solid line ) leads to the slave(dashed line). And last half slave leads to the master. This means that the master-slave tele-operation system works symmetricaly well.

Fig. 13. experimental result of position tracking: master (- slid line), slave(– dashed line)

## **7. Summary**

In this chapter we discussed on an application of robust control for force communication systems over inferior quality network. According to the investigation of the experiments the effectiveness of bilateral tele-operation system for force communication is confirmed. Especially the most important problem of the robustness w.r.t. time-delay is improved by *H*∞ control systems theory.

## **8. References**


*USA* 

**Allocation Systems** 

**Robust Control for Single Unit Resource** 

*University of Akron, Southern Illinois University Edwardsville, and Purdue University* 

Supervisory control for deadlock-free resource allocation has been an active area of manufacturing systems research. Most work, however, assumes that allocated resources do not fail. Little research has addressed allocating resources that may fail. Automated manufacturing systems have many types of components that may fail unexpectedly. We develop robust controllers for single unit resource allocation systems with unreliable resources (Chew et al., 2008; Chew et al., 2011; Chew & Lawley, 2006; Lawley, 2002; Lawley & Sulistyono, 2002; Wang et al., 2008; Wang et al., 2009). These controllers guarantee that when unreliable resources fail, parts requiring failed resources do not block the production of parts not requiring failed resources. Further, while resources are down, the system is controlled so that when repair events occur, the system is in a safe and admissible state. There is little manufacturing research literature on robust supervision. Reveliotis (1999) considers the case where parts requiring a failed resource can be re-routed or removed from the system through human intervention. Park & Lim (1999) address existence questions for robust supervisors. Hsieh (2004) develops methods that determine the feasibility of production given a set of resource failures modelled as the extraction of tokens from a Petri net. In contrast, our work models the failure of the workstation server while assuming that buffer space remains accessible after the failure event. We assume that when the server of a workstation fails, we can continue allocating its buffer space up to capacity, but that none of the waiting parts can be processed and thus cannot proceed along their routes until the server is repaired. We further assume that server failure does not prevent finished parts occupying the workstation's buffer space from being moved away from the workstation and proceeding along their routes. Finally, we assume that server failure does not damage or destroy the part being processed and that failure can only occur when the server is working. The last two assumptions are made for notational efficiency and presentation clarity. They can be easily relaxed by adding appropriate events and state variables to our treatment. Our objective is to control the system so that failure of an unreliable resource does not prevent processing of parts not requiring the failed resource. When a resource fails, all parts in the system requiring the failed resource for future processing are unable to complete until the failed resource is repaired. Because these parts occupy buffer space, they can block production of parts not requiring the failed resource. Thus, we want to assure that, when unreliable resources fail, the buffer space allocation can evolve under normal operation so

**1. Introduction** 

Shengyong Wang, Song Foh Chew and Mark Lawley

