**3.4 Examples**

#### **3.4.1 Centralized controller**

A first order plus dead time process with 1 *<sup>p</sup> k* , 1 *<sup>p</sup>* and 0.25 *Dp* is chosen for simulation study. The controller is designed with a first order filter with 1.4286 , 0.7 *<sup>c</sup> k* and 1 *<sup>I</sup>* . Closed loop responses with the present controller are obtained. The results are shown below:

Fig. 4. Closed-loop response of example -processes using PID controller

#### **3.4.2 Decentralized controller**

118 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

*GD H* (3.13)

For a 2 x 2 system, equations are derived for decouplers, taking that loop and the other

. Closed loop responses with the present controller are obtained. The results are

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>0</sup>

Time

Fig. 4. Closed-loop response of example -processes using PID controller

simulation study. The controller is designed with a first order filter with 1.4286

*C GDu* (3.11)

*C Hu* (3.12)

<sup>1</sup> *D GH* (3.14)

and 0.25 *Dp* is chosen for

, 0.7 *<sup>c</sup> k*

The equation (3.10) becomes,

The equation (3.11) becomes,

which defines the decoupler

interacting loops into account.

**3.4.1 Centralized controller** 

A first order plus dead time process with 1 *<sup>p</sup> k* , 1 *<sup>p</sup>*

**3.4 Examples** 

and 1 *<sup>I</sup>* 

0.2

0.4

0.6

0.8

Response

1

1.2

shown below:

where,

or

The wood and berry distillation column process whose transfer function

$$
\begin{bmatrix}
\frac{12.8e^{-s}}{16.7s+1} & \frac{-18.9e^{-3s}}{21s+1} \\
\frac{6.6e^{-7s}}{10.9s+1} & \frac{-19.4e^{-3s}}{14.4s+1}
\end{bmatrix}
$$

is chosen for simulation study. The controller is designed using BLT method with F=2.55, <sup>1</sup> 0.375 *<sup>c</sup> k* , <sup>1</sup> 8.29 *<sup>I</sup>* (loop 1 controller settings) and 2 0.075 *<sup>c</sup> k* , <sup>2</sup> 23.6 *<sup>I</sup>* (loop 2 controller settings). With these settings, the closed loop responses are obtained and are shown below.

Fig. 5. Closed-loop response with BLT tuning for WB -Column using PID controller (solid line is loop 1 response and dashed line is loop 2 response)

#### **3.4.3 Decoupled PID controller**

The Wood and Berry binary distillation column is a multivariable system that has been studied extensively. The process has transfer function

$$\begin{vmatrix} \frac{12.8e^{-s}}{16.7s+1} & \frac{-18.9e^{-3s}}{21s+1} \\ \frac{6.6e^{-7s}}{10.9s+1} & \frac{-19.4e^{-3s}}{14.4s+1} \end{vmatrix} . \tag{3.15}$$

The decoupler is given by

Identification and Control of Multivariable Systems – Role of Relay Feedback 121

2

The gain between y1 and m1 when y2 is constant (y2 = 0) is found from solving the equations

1 11 1 12 2 *p p y km km*

21 1 22 2 0 *p p km km*

11 22 12 21 1 1 22 *pp pp p kk kk y m k* 

11 22 12 21 <sup>1</sup> 1 22

*y p kk kk y m k* 

11 element of RGA for the wood and berry column

12.8 18.9 6.6 19.4 *<sup>p</sup> <sup>k</sup>*

1 1 2.01 ( 18.9)(6.6) <sup>1</sup> <sup>1</sup>

(12.8)( 19.4)

*<sup>T</sup> kUV* (4.6)

<sup>1</sup> *<sup>p</sup> <sup>m</sup> <sup>y</sup> <sup>k</sup> m* 

11

21 1

*p p p <sup>p</sup> k m y km k <sup>k</sup>*

*pp pp*

12 21 11 22

1 1 *p p p p k k k k*

22

(4.3)

(4.4)

(4.5)

1

1 11 1 12

2

11

12 21 11 22

SVD is a numerical algorithm developed to minimize computational errors involving large matrix operations. The singular value decomposition of matrix K results in three component

where K is an n x m matrix. U is an n x n orthonormal matrix, the columns of which are called the 'left singular vectors'. V is an m x m orthonormal matrix, the columns of which are called the 'right singular vectors'. is an n x m diagonal matrix of scalars called the

*p p p p k k k k*

Therefore the term

**Example**: Calculate

matrices as follows:

"singular values"

**4.2 Singular Value Decomposition** 

11 in RGA is

11

$$D = G^{-1}(0) = \frac{1}{\det\begin{pmatrix} \mathcal{G}(0) \end{pmatrix} \begin{pmatrix} g\_{22}(0) & -g\_{12}(0) \\ -g\_{21}(0) & g\_{22}(0) \end{pmatrix}} \tag{3.16}$$

$$D = G^{-1}(0) = \frac{1}{-123.58} \begin{pmatrix} -19.4 & -18.9 \\ 6.6 & 12.8 \end{pmatrix}$$

$$D = \begin{pmatrix} 0.15698 & 0.15293 \\ 0.0534 & -0.1035 \end{pmatrix}$$

The transfer function of the statistically decoupled system is given by

$$Q = \text{GD} \quad \text{or} \ Q = \text{GG}^{-1}(0) \tag{3.17}$$

$$Q = \begin{bmatrix} \frac{12.8e^{-s}}{16.7s+1} & \frac{-18.9e^{-3s}}{21s+1} \\ \frac{6.6e^{-7s}}{10.9s+1} & \frac{-19.4e^{-3s}}{14.4s+1} \end{bmatrix} \begin{pmatrix} 0.15698 & 0.15293 \\ 0.0534 & -0.1035 \end{pmatrix}$$
