**3.1 Centralized structure**

Centralized control scheme is a full multivariable controller where the controller matrix is not a diagonal one. The decentralized control scheme is preferred over the centralized control scheme mainly because the control system has only n controlling n output variables, and the operator can easily understand the control loops. However, the design methods of such decentralized controllers require first pairing of input-output variables, and tuning of controllers requires trial and error steps. The centralized control system requires n x n controllers for controlling n output variables using n manipulated variables. But if we are calculating the control action using a computer, then this problem of requiring n x n controllers does not exist. The advantage of the centralized controller is easy to tune even with the knowledge of the steady state gain matrix alone, multivariable PI controllers can be easily designed.

For the centralized structure, Internal model control-proportional integral tuning is adopted, based on studies on the studies and recommendations of Reddy et al (1997) on the design of centralized PI controllers for a Multi-stage flash desalination plant using Davison, Maciejowski and Tanttu-Lieslehto methods.

The IMC-PID tuning relations are used in tuning the controller. For a first order system of

the form <sup>1</sup> *Ds <sup>p</sup> k e s* , the PI controller settings are as follows:

$$k\_c = \frac{\pi}{k\_p \mathcal{L}}\tag{3.1}$$

Identification and Control of Multivariable Systems – Role of Relay Feedback 117

amplitude a and period of oscillation P, are correlated with the ultimate gain (ku) and

4 *u <sup>h</sup> <sup>k</sup>* 

> 2 *<sup>u</sup> Pu*

Detuning factor F determines the stability of each loop. The larger the value of F, more stable the system is but set point and load responses are sluggish. This method yields settings that give a reasonable compromise between stability and performance in

The decentralized scheme is more advantageous in the fact that the system remains stable even when one controller goes down and is easier to tune because of the less number of tuning parameters. But however pairing (interaction) analysis needs to be done as n!

This structure has additional elements called decouplers to compensate for the interaction phenomenon. When Relative gain Array shows strong interaction then a decoupler is designed. But however decouplers are designed only for orders less than 3 as the design

The BLT (Luyben 1986) procedure of tuning the decentralized structure follows the generalized way for all n x n systems as mentioned above. The centralized controllers are tuned using the IMC-PI tuning relations which are appropriately selected for first order and

The decoupled structure adopts the various methods like partial, static and dynamic decoupling to procedure the best results. The design equations for a general decoupler for n

> ( ) ... 0 () () () () ; ; ... ( ) ... () () () () 0 ... ( )

> > 1 ... *n*

;

*M*

*M* 

11

*H s*

22

*C GM* (3.9)

*M Du* (3.10)

*nn*

*C*

1 ... *n*

*C*

*C*

 

*H s*

x n systems are conveniently summarized using matrix notations defined as follows:

*G D H Hs*

*M*

Manipulated variable (new) Manipulated variable (old) Output

*<sup>a</sup>* (3.7)

(3.8)

frequency (wu) by the following relationships:

pairings between input/output are possible.

procedure becomes more complex as order increases.

11 1 11 1

*Gs Gs Ds Ds*

*n nn n nn*

For a decoupled multivariable system, output can be written as

*GsGs Ds Ds*

*n n*

Transfer function matrix; Decoupler matrix; Diagonal matrix of decoupler

1 1

multivariable systems.

**3.3 Decoupled structure** 

second order systems.

1 ... *n*

*u*

*u*

 ;

*u*

$$
\boldsymbol{\sigma}\_{i} = \boldsymbol{\sigma} \tag{3.2}
$$

where max 1 /0.7 ,0.2 *D* 

These tuning relations are derived by comparing IMC control with the conventional PID controller and solving to determine the proportional gain and integral time.

#### **3.2 Decentralized structure**

In spite of developments of advanced controller synthesis for multivariable controllers, decentralized controller remain popular in industries because of the following:


The design of a decentralized control system consists of two main steps:

Step 1 is control structure selection and step 2 is the design of a SISO controller for each loop.

In decentralized control of multivariable systems, the system is decomposed into a number of subsystems and individual controllers are designed for each subsystem.

For tuning the controller, Biggest Log Modulus Tuning (BLT) method (Lubed 1986) is used, which is an extension of the Multivariable Nyquist Criterion and gives a satisfactory response. A detuning factor F (typical values are said to vary between 2 and 5) is chosen so that closed-loop log modulus, Lcmmax >= 2n,

$$L\_{cm} = 20\log\left|\frac{w}{1+w}\right|\tag{3.3}$$

$$w = -1 + \det\left(I + G\_p G\_c\right) \tag{3.4}$$

where Gc is an n x n diagonal matrix of PI controller transfer functions, Gp is an n x n matrix containing the process transfer functions relating the n controlled variables to n manipulated variables.

Now the PI controller parameters are given as,

$$k\_{ci} = \prescript{k\_{ciZ-N}}{}{\textstyle F} \tag{3.5}$$

$$
\pi\_{li} = \mathsf{F}\pi\_{liZ-N} \tag{3.6}
$$

where *kciZ N* and *IiZ N* are Zeigler-Nichols tuning parameters which are calculated from the system perturbed in closed loop by a relay of amplitude h, reaches a limit cycle whose amplitude a and period of oscillation P, are correlated with the ultimate gain (ku) and frequency (wu) by the following relationships:

$$k\_u = \frac{4h}{\pi a} \tag{3.7}$$

$$
\rho\_u = \frac{2\pi}{P\_u} \tag{3.8}
$$

Detuning factor F determines the stability of each loop. The larger the value of F, more stable the system is but set point and load responses are sluggish. This method yields settings that give a reasonable compromise between stability and performance in multivariable systems.

The decentralized scheme is more advantageous in the fact that the system remains stable even when one controller goes down and is easier to tune because of the less number of tuning parameters. But however pairing (interaction) analysis needs to be done as n! pairings between input/output are possible.

#### **3.3 Decoupled structure**

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

*p*

(3.1)

*<sup>w</sup>* (3.3)

*<sup>k</sup> <sup>k</sup> <sup>F</sup>* (3.5)

(3.6)

*w I GG* 1 det *<sup>p</sup> <sup>c</sup>* (3.4)

(3.2)

*k* 

*c*

*i* 

These tuning relations are derived by comparing IMC control with the conventional PID

In spite of developments of advanced controller synthesis for multivariable controllers,

3. The operators can easily retune the controllers to take into account the change in

4. Some manipulated variables may fail. Tolerances to such failures are more easily incorporated into the design of decentralized controllers than full controllers. 5. The control system can be bought gradually into service during process start up and

Step 1 is control structure selection and step 2 is the design of a SISO controller for each

In decentralized control of multivariable systems, the system is decomposed into a number

For tuning the controller, Biggest Log Modulus Tuning (BLT) method (Lubed 1986) is used, which is an extension of the Multivariable Nyquist Criterion and gives a satisfactory response. A detuning factor F (typical values are said to vary between 2 and 5) is chosen so

> 20log <sup>1</sup> *cm <sup>w</sup> <sup>L</sup>*

where Gc is an n x n diagonal matrix of PI controller transfer functions, Gp is an n x n matrix containing the process transfer functions relating the n controlled variables to n

*ciZ N*

*IiZ N* are Zeigler-Nichols tuning parameters which are calculated from

 *Ii IiZ N F*

the system perturbed in closed loop by a relay of amplitude h, reaches a limit cycle whose

*ci*

*k*

controller and solving to determine the proportional gain and integral time.

decentralized controller remain popular in industries because of the following:

where

loop.

max 1 /0.7 ,0.2 *D*

**3.2 Decentralized structure** 

process conditions.

manipulated variables.

where *kciZ N* and

1. Decentralized controllers are easy to implement. 2. They are easy for operators to understand.

taken gradually out of service during shut down.

that closed-loop log modulus, Lcmmax >= 2n,

Now the PI controller parameters are given as,

The design of a decentralized control system consists of two main steps:

of subsystems and individual controllers are designed for each subsystem.

This structure has additional elements called decouplers to compensate for the interaction phenomenon. When Relative gain Array shows strong interaction then a decoupler is designed. But however decouplers are designed only for orders less than 3 as the design procedure becomes more complex as order increases.

The BLT (Luyben 1986) procedure of tuning the decentralized structure follows the generalized way for all n x n systems as mentioned above. The centralized controllers are tuned using the IMC-PI tuning relations which are appropriately selected for first order and second order systems.

The decoupled structure adopts the various methods like partial, static and dynamic decoupling to procedure the best results. The design equations for a general decoupler for n x n systems are conveniently summarized using matrix notations defined as follows:

$$\mathbf{G} = \begin{bmatrix} \mathbf{G}\_{11}\text{(s)} & \mathbf{G}\_{1n}\text{(s)}\\ \mathbf{G}\_{n1}\text{(s)} & \mathbf{G}\_{nn}\text{(s)} \end{bmatrix}; \mathbf{D} = \begin{bmatrix} D\_{11}\text{(s)} & D\_{1n}\text{(s)}\\ D\_{n1}\text{(s)} & D\_{nn}\text{(s)} \end{bmatrix}; \mathbf{H} = \begin{bmatrix} H\_{11}\text{(s)} & \dots & \mathbf{0} \\ \dots & H\_{22}\text{(s)} & \dots \\ \mathbf{0} & \dots & H\_{nn}\text{(s)} \end{bmatrix}$$

Transfer function matrix; Decoupler matrix; Diagonal matrix of decoupler

$$\boldsymbol{u} = \begin{bmatrix} \boldsymbol{u}\_1 \\ \dots \\ \boldsymbol{u}\_n \end{bmatrix}; \qquad \qquad \boldsymbol{M} = \begin{bmatrix} \boldsymbol{M}\_1 \\ \dots \\ \boldsymbol{M}\_n \end{bmatrix}; \qquad \qquad \boldsymbol{C} = \begin{bmatrix} \boldsymbol{C}\_1 \\ \dots \\ \boldsymbol{C}\_n \end{bmatrix}$$

 Manipulated variable (new) Manipulated variable (old) Output For a decoupled multivariable system, output can be written as

$$\mathbf{C} = \mathbf{G}M \tag{3.9}$$

$$M = D\mu\tag{3.10}$$

The equation (3.10) becomes,

$$\mathbf{C} = \mathbf{G}Du\tag{3.11}$$

Identification and Control of Multivariable Systems – Role of Relay Feedback 119

12.8 18.9 16.7 1 21 1 6.6 19.4 10.9 1 14.4 1

7 3

*s s*

*s s*

*e e s s e e s s*

is chosen for simulation study. The controller is designed using BLT method with F=2.55,

controller settings). With these settings, the closed loop responses are obtained and are

Fig. 5. Closed-loop response with BLT tuning for WB -Column using PID controller (solid

The Wood and Berry binary distillation column is a multivariable system that has been

*e e s s e e s s*

12.8 18.9 16.7 1 21 1 6.6 19.4 10.9 1 14.4 1

7 3

*s s*

*s s*

3

line is loop 1 response and dashed line is loop 2 response)

studied extensively. The process has transfer function

**3.4.3 Decoupled PID controller** 

The decoupler is given by

(loop 1 controller settings) and 2 0.075 *<sup>c</sup> k* , <sup>2</sup> 23.6 *<sup>I</sup>*

3

. (3.15)

(loop 2

The wood and berry distillation column process whose transfer function

**3.4.2 Decentralized controller** 

<sup>1</sup> 0.375 *<sup>c</sup> k* , <sup>1</sup> 8.29 *<sup>I</sup>* 

shown below.

The equation (3.11) becomes,

$$\mathbf{C} = H u \tag{3.12}$$

where,

$$\mathbf{G}D = \mathbf{H} \tag{3.13}$$

or

$$D = \mathbf{G}^{-1} H \tag{3.14}$$

which defines the decoupler

For a 2 x 2 system, equations are derived for decouplers, taking that loop and the other interacting loops into account.
