**3.2. Control of frequency transmission**

An approach to schedule a real time distributed system based upon modifications on frequency transmission of individual components in the system is presented in [4], this shows that scheduling of a distributed system can be accomplished through modifications on transmission frequencies into a region where the system performance is not affected. A linear time invariant model in which the coefficients of the state matrix are the relations between the transmission frequencies of each agent and through a feedback controller to modify transmission frequencies bounded between maximum and minimum values of transmission. This approach drives the frequency transmission based on three parameters: minimum frequency (*fm*), maximum frequency (*fh*) and real frequency (*fr*). Frequency transmission dynamics can be modeled as a linear time-invariant subsystem which state variables are transmission frequencies of the sensor nodes involved on the system. Note that for each primary task of a sensor *Si*, *i* = 1, ..., *m* frequency can be expressed as *fi* = 1/*pi*. There is a relationship between nodes' frequencies an external input frequencies which serves as coefficients of the linear system. Therefore it is possible to control the NCS through the input vector *u* such that the outputs *y* are nodes' frequencies into a region *L* bounded by maximum and minimum transmission frequencies, see Fig. 2.

**Figure 2.** Transmission frequencies bounded by a schedulability region

6 Will-be-set-by-IN-TECH

(Token Bus), FDMA, TDMA (TTP), Switched Ethernet, WLAN (802.11b), and ZigBee (802.15.4) are some types of network supported [2]. The network blocks are mainly configured using blocks dialogs. Some parameters common to all types of networks are bit rate, the minimum frame size, and the network interface delay. For each type of network there are some parameters that specifie the number of nodes, data rate (bits/s), preprocessing delay, minimum frame size, maximum frame size, frame overhead, etc. The network blocks may be used having one kernel block for each node in the network. The tasks into the kernels can send and receive arbitrary Matlab structure arrays over the network using ttSendMsg and ttGetMsg kernel primitives. This way is to quite flexible but requires to program some routines to configure the system. An useful network scheduler viewer shows the network activity for all nodes involved. An overview of all Truetime's primitives can be found on [12].

In this section a formal definition of task model is provided and it gives an overview of the control of frequency transmission in a distributed systems and how it impacts over quality

Chen *et al.* [3] gives a formal definition of system model of a typical distributed systems consisting of a set of processors and a set of tasks. A distributed system are characterized as follows. A set of processors Ω = {*S*1, *S*2, ..., *Sm*} where Ω is the processor set, *Si* is the *i* − *th* processor and *m* is the total number of processors. In this model, all processors are assumed to be identical to assure same execution time for each task on different processors. It is also supposed that enough processors are provided. A set of primary copy of real-time tasks Φ = {*τ*1, *τ*2, ..., *τn*}, where *τ<sup>i</sup>* = {*ci*, *pi*} , *i* = {1, 2, ..., *n*}. Here Φ is the set of tasks, *τ<sup>i</sup>* is the *i* − *th* task, *n* is the number of tasks which are periodic, independent and preemptive, *ci*

An approach to schedule a real time distributed system based upon modifications on frequency transmission of individual components in the system is presented in [4], this shows that scheduling of a distributed system can be accomplished through modifications on transmission frequencies into a region where the system performance is not affected. A linear time invariant model in which the coefficients of the state matrix are the relations between the transmission frequencies of each agent and through a feedback controller to modify transmission frequencies bounded between maximum and minimum values of transmission. This approach drives the frequency transmission based on three parameters: minimum frequency (*fm*), maximum frequency (*fh*) and real frequency (*fr*). Frequency transmission dynamics can be modeled as a linear time-invariant subsystem which state variables are transmission frequencies of the sensor nodes involved on the system. Note that for each primary task of a sensor *Si*, *i* = 1, ..., *m* frequency can be expressed as *fi* = 1/*pi*. There is a relationship between nodes' frequencies an external input frequencies which serves as coefficients of the linear system. Therefore it is possible to control the NCS through the input vector *u* such that the outputs *y* are nodes' frequencies into a region *L* bounded by maximum

**3. Frequency transmission scheduling**

denotes the execution time of *τi*, *pi* denotes the period of task *τi*.

**3.2. Control of frequency transmission**

and minimum transmission frequencies, see Fig. 2.

performance.

**3.1. Task model**

The objective of controlling the frequency is to achieve coordination through the convergence of values. Each sensor *Si* knows its minimum and maximum frequencies based upon messages sent to controller and it could be estimated its own real transmission frequency. Let a NCS with a set Ω = {*S*1, *S*2, ..., *Sl*} nodes that performs a set of task *τ<sup>i</sup>* = {*ci*, *pi*} for *i* = {1, 2, , *n*}, a subset of Ω is sensor nodes subset Ω*<sup>s</sup>* = {*S*1, *S*2, ..., *Sm*}.

### **3.3. Network scheduling based on frequency transmission**

An approach that modifies the frequency transmission uses *fm*, *fr*, *fx* frequencies. RTDS dynamics, is modeled as a linear time-invariant system, whose state variables are frequencies transmission rates of the *n* nodes that compose the RTDS [4]. Frequency rates of a node are affected by some external input frequency rates, minimal frequencies of all nodes and particular ratios serve as coefficients of the linear system. So, it is possible to control the NCS using the input vector *u*, such that the output vector *y* contains the frequency rates of all nodes within a nonlinear region *L*, bounded by the maximum and minimum transmission frequency rates. Let we assume that there is a relationship amongst real frequencies *f* <sup>1</sup> *<sup>r</sup>* , *f* <sup>2</sup> *<sup>r</sup>* , ..., *f <sup>m</sup> <sup>r</sup>* and external input frequencies *u*1, *u*2, ..., *um* which serve as coefficients of the linear system:

$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + Bu\_{k\prime} \\ y\_k &= \mathbf{C}\mathbf{x}\_k. \end{aligned} \tag{1}$$

*A*, *B*, *C* are matrices with appropriate dimensions. *A* is the matrix of relationships between frequencies of sensor nodes, *B* is the scale frequencies matrix, *C* is the matrix with frequencies ordered, *xk*<sup>+</sup><sup>1</sup> is a real frequencies vector in time *t* = *k* + 1, *yk* is the vector of output frequencies. The input *uk* is a function of reference frequencies and real frequencies of the nodes in the distributed system, hence *u* = *K*(*fm* − *fr*) where *fm*, *fr* are vectors. Then, the state vector in equation (1) can be written as:

$$\begin{array}{l} \mathbf{x}\_{k+1} = A\mathbf{x}\_{k} + Bu\_{k\prime} \\ \mathbf{x}\_{k+1} = Af\_{rk} + B\left(K(f\_{mk} - f\_{rk})\right) \end{array} \tag{2}$$

#### 8 Will-be-set-by-IN-TECH 56 Numerical Simulation – From Theory to Industry Issues on Communication Network Control System Based Upon Scheduling Strategy Using Numerical Simulations <sup>9</sup>

Note that *K* is the control gain defined as the basics of a LQR algorithm. Matrices *A*, *B* and *C* has the proper dimensions and matrix *A* include the following restriction [11]:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*xc* is a real execution time and *x<sup>r</sup>*

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *x*1 *k*+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Λ *f* 1 *m f* 2 *m f* 1 *m f* 3 *m f* 1 *m f* 4 *m f* 1 *m* 0

*f* 1 *m f* 2 *m* Λ *f* 2 *m f* 3 *m f* 2 *m f* 4 *m f* 2 *m* 0

*f* 1 *m f* 3 *m f* 2 *m f* 3 *m* Λ *f* 3 *m f* 4 *m f* 3 *m* 0

*f* 1 *m f* 4 *m f* 2 *m f* 4 *m f* 3 *m f* 4 *m* Λ *f* 4 *m* 0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *c*<sup>1</sup> *c*<sup>2</sup> *c*<sup>3</sup> *c*<sup>4</sup> 1

*K*<sup>1</sup> *f* <sup>1</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>1</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>1</sup> *r* )

*K*<sup>2</sup> *f* <sup>2</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>2</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>2</sup> *r* )

*K*<sup>3</sup> *f* <sup>3</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>3</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>3</sup> *r* )

*K*<sup>4</sup> *f* <sup>4</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>4</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>4</sup> *r* )

*<sup>r</sup>* ) + *K*2(*f* <sup>2</sup>

*<sup>r</sup>* ) + *K*4(*f* <sup>4</sup>

*<sup>c</sup>* − *xr*)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f* 4 *r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .

*<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>2</sup> *<sup>r</sup>* )+

*<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>4</sup> *<sup>r</sup>* )+

*<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>1</sup>

*<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>3</sup>

*Kc*(*x<sup>r</sup>*

=

+

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *y*1 *k y*2 *k y*3 *k y*4 *k y*5 *k*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*K*1(*f* <sup>1</sup>

*K*3(*f* <sup>3</sup>

*x*2 *k*+1

*x*3 *k*+1

*x*4 *k*+1

*x*5 *k*+1

Thus

*y*1 *k y*2 *k y*3 *k y*4 *k y*5 *k*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*<sup>c</sup>* is a reference execution time.

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*f* 2 *r*

*f* 3 *r f n r*

*xc*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f* 4 *r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .

Issues on Communication Network Control System Based Upon Scheduling Strategy Using Numerical Simulations

57

$$U = \sum\_{i=1}^{n} c\_i / p\_i \le 1,$$

as a new state of the system (1). For simplicity, frequency transition model will be described using a sensor node subset Ω*<sup>s</sup>* = {*S*1, *S*2, *S*3, *S*4}. The elements of the matrices for system 2 are defined as follows:

$$
\overline{a}\_{ij} = \begin{cases}
\frac{\Lambda\left(f\_m^1 f\_m^2 f\_m^3 f\_m^4\right)}{f\_m^1} & i = j \\
\\ \frac{f\_m^j}{f\_m^i} & i \neq j
\end{cases}
$$

$$
\overline{b}\_{ij} = \begin{cases}
f\_{li}^j & i = j \\
0 & i \neq j
\end{cases}
$$

$$
\overline{c}\_{ij} = \begin{cases}
1 & i = j \\
0 & i \neq j
\end{cases}
$$

Note that Λ � *f* 1 *<sup>m</sup>*, *f* <sup>2</sup> *<sup>m</sup>*, *f* <sup>3</sup> *<sup>m</sup>*, *f* <sup>4</sup> *m* � is the greatest common divisor of minimum frequencies, that is the planning cycle Γ expressed in terms of frequencies of the backup task, for shortening, it will be written only as Λ. *fm*, *fh*, and *fr* are vectors of respective frequencies.

Considering the execution time *ci* of each task executing in respective sensor nodes in Ω*<sup>s</sup>* as an additional state, we can rewrite (2) as follows:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *x*1 *k*+1 *x*2 *k*+1 *x*3 *k*+1 *x*4 *k*+1 *x*5 *k*+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Λ *f* 1 *m f* 2 *m f* 1 *m f* 3 *m f* 1 *m f* 4 *m f* 1 *m* 0 *f* 1 *m f* 2 *m* Λ *f* 2 *m f* 3 *m f* 2 *m f* 4 *m f* 2 *m* 0 *f* 1 *m f* 3 *m f* 2 *m f* 3 *m* Λ *f* 3 *m f* 4 *m f* 3 *m* 0 *f* 1 *m f* 4 *m f* 2 *m f* 4 *m f* 3 *m f* 4 *m* Λ *f* 4 *m* 0 *c*<sup>1</sup> *c*<sup>2</sup> *c*<sup>3</sup> *c*<sup>4</sup> 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f n r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *<sup>h</sup>* 0 0 00 0 *f* <sup>2</sup> *<sup>h</sup>* 0 00 0 0 *f* <sup>3</sup> *<sup>h</sup>* 0 0 000 *f <sup>n</sup> <sup>h</sup>* 0 1 1 1 11 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ · ⎛ ⎜⎜⎜⎜⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *K*<sup>1</sup> 0000 0 *K*<sup>2</sup> 000 0 0 *K*<sup>3</sup> 0 0 000 *K*<sup>4</sup> 0 0000 *Kc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛ ⎜⎜⎜⎜⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *m f* 2 *m f* 3 *m f* 4 *m xr c* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f n r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎠ ,

56 Numerical Simulation – From Theory to Industry Issues on Communication Network Control System Based Upon Scheduling Strategy Using Numerical Simulations <sup>9</sup> 57 Issues on Communication Network Control System Based Upon Scheduling Strategy Using Numerical Simulations

,

$$
\begin{bmatrix} y\_k^1 \\ y\_k^2 \\ y\_k^3 \\ y\_k^4 \\ y\_k^4 \\ y\_k^5 \end{bmatrix} = \begin{bmatrix} 1 \ 0 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \ 0 \\ 0 \ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 1 \end{bmatrix} \begin{bmatrix} f\_r^1 \\ f\_r^2 \\ f\_r^3 \\ f\_r^4 \\ x\_c \end{bmatrix}.$$

*xc* is a real execution time and *x<sup>r</sup> <sup>c</sup>* is a reference execution time. Thus

> ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

8 Will-be-set-by-IN-TECH

Note that *K* is the control gain defined as the basics of a LQR algorithm. Matrices *A*, *B* and *C*

as a new state of the system (1). For simplicity, frequency transition model will be described using a sensor node subset Ω*<sup>s</sup>* = {*S*1, *S*2, *S*3, *S*4}. The elements of the matrices for system 2

> *f i m*

> *f j m f i m*

*f i <sup>h</sup> i* = *j*

0 *i* �= *j*

1 *i* = *j*

0 *i* �= *j*

the planning cycle Γ expressed in terms of frequencies of the backup task, for shortening, it

Considering the execution time *ci* of each task executing in respective sensor nodes in Ω*<sup>s</sup>* as

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*f* 2 *r*

*f* 3 *r f n r*

+

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1

0 *f* <sup>2</sup>

0 0 *f* <sup>3</sup>

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f n r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞

⎞

⎟⎟⎟⎟⎠ ,

⎟⎟⎟⎟⎠

000 *f <sup>n</sup>*

1 1 1 11

*<sup>h</sup>* 0 0 00

*<sup>h</sup>* 0 00

*<sup>h</sup>* 0 0

*<sup>h</sup>* 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

*xc*

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *m f* 2 *m f* 3 *m f* 4 *m xr c*

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ −

⎜⎜⎜⎜⎝

⎧ ⎨ ⎩

> ⎧ ⎨ ⎩

*i* = *j*

,

is the greatest common divisor of minimum frequencies, that is

*i* �= *j*

,

.

*ci*/*pi* ≤ 1,

*n* ∑ *i*=1

has the proper dimensions and matrix *A* include the following restriction [11]:

*U* =

*aij* =

⎧ ⎪⎪⎨ Λ(*f* <sup>1</sup> *<sup>m</sup>*, *f* <sup>2</sup> *<sup>m</sup>*, *f* <sup>3</sup> *<sup>m</sup>*, *f* <sup>4</sup> *m*)

⎪⎪⎩

*bij* =

*cij* =

will be written only as Λ. *fm*, *fh*, and *fr* are vectors of respective frequencies.

are defined as follows:

Note that Λ �

*f* 1 *<sup>m</sup>*, *f* <sup>2</sup> *<sup>m</sup>*, *f* <sup>3</sup> *<sup>m</sup>*, *f* <sup>4</sup> *m* �

an additional state, we can rewrite (2) as follows:

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Λ *f* 1 *m f* 2 *m f* 1 *m f* 3 *m f* 1 *m f* 4 *m f* 1 *m* 0

*f* 1 *m f* 2 *m* Λ *f* 2 *m f* 3 *m f* 2 *m f* 4 *m f* 2 *m* 0

*f* 1 *m f* 3 *m f* 2 *m f* 3 *m* Λ *f* 3 *m f* 4 *m f* 3 *m* 0

*f* 1 *m f* 4 *m f* 2 *m f* 4 *m f* 3 *m f* 4 *m* Λ *f* 4 *m* 0

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

*K*<sup>1</sup> 0000 0 *K*<sup>2</sup> 000 0 0 *K*<sup>3</sup> 0 0 000 *K*<sup>4</sup> 0 0000 *Kc*

=

·

⎛

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎜⎜⎜⎜⎝

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *x*1 *k*+1

*x*2 *k*+1

*x*3 *k*+1

*x*4 *k*+1

*x*5 *k*+1

*x*1 *k*+1 *x*2 *k*+1 *x*3 *k*+1 *x*4 *k*+1 *x*5 *k*+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Λ *f* 1 *m f* 2 *m f* 1 *m f* 3 *m f* 1 *m f* 4 *m f* 1 *m* 0 *f* 1 *m f* 2 *m* Λ *f* 2 *m f* 3 *m f* 2 *m f* 4 *m f* 2 *m* 0 *f* 1 *m f* 3 *m f* 2 *m f* 3 *m* Λ *f* 3 *m f* 4 *m f* 3 *m* 0 *f* 1 *m f* 4 *m f* 2 *m f* 4 *m f* 3 *m f* 4 *m* Λ *f* 4 *m* 0 *c*<sup>1</sup> *c*<sup>2</sup> *c*<sup>3</sup> *c*<sup>4</sup> 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *f* 1 *r f* 2 *r f* 3 *r f n r xc* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *K*<sup>1</sup> *f* <sup>1</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>1</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>1</sup> *r* ) *K*<sup>2</sup> *f* <sup>2</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>2</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>2</sup> *r* ) *K*<sup>3</sup> *f* <sup>3</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>3</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>3</sup> *r* ) *K*<sup>4</sup> *f* <sup>4</sup> *<sup>h</sup>* (*<sup>f</sup>* <sup>4</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>4</sup> *r* ) *K*1(*f* <sup>1</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>1</sup> *<sup>r</sup>* ) + *K*2(*f* <sup>2</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>2</sup> *<sup>r</sup>* )+ *K*3(*f* <sup>3</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>3</sup> *<sup>r</sup>* ) + *K*4(*f* <sup>4</sup> *<sup>m</sup>* <sup>−</sup> *<sup>f</sup>* <sup>4</sup> *<sup>r</sup>* )+ *Kc*(*x<sup>r</sup> <sup>c</sup>* − *xr*) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ *y*1 *k* ⎤ ⎥


*k*
