**6. Stability analysis**

#### **6.1 INA and DNA methods**

Rosenbrock extended the nyquist stability and design concepts to MIMO systems containing significant interaction. The methods are known as the inverse and direct Nyquist array (INA and DNA) methods. As an extension from the SISO nyquist stability and design concepts, these methods use frequency response approach. These techniques are used because of their simplicity, high stability, and low noise sensitivity. In actual applications, there will be a

*ss s*

*ss s*

34.68 46.2 0.87(11.61 1) 8.15 1 10.9 1 (3.89 1)(18.8 1)

*s s ss*

*s s s*

1 2 3

(5.8)

*u u u*

 

2.6 3.5

 

0.66 0.61 0.0049 6.7 1 8.64 1 9.06 1

*ee e sss*

2.36 2.3 0.01 5 1 5 1 7.09 1

*ss s <sup>y</sup> e e s e*

Table 5.1. Analytical expressions for PID controller parameters for standard transfer

 **FOPDT SOPDT IPDT** 

Rosenbrock extended the nyquist stability and design concepts to MIMO systems containing significant interaction. The methods are known as the inverse and direct Nyquist array (INA and DNA) methods. As an extension from the SISO nyquist stability and design concepts, these methods use frequency response approach. These techniques are used because of their simplicity, high stability, and low noise sensitivity. In actual applications, there will be a

max 0.25 ,0.2 *Dp p*

max 0.25 ,0.2 *Dp p*

=DP10

=DP10

*ee e*

1 3 3 1.2

3 9.2 9.4

2

The closed loop response is given in Figure 5.2.

*y*

*y*

functions

**PI** 

**PID** 

Table 5.2.

selection rule

**6. Stability analysis** 

**6.1 INA and DNA methods** 

max 1.7 ,0.2 *Dp p*

max 0.25 ,0.2 *Dp p*

> 

region of uncertainty for interaction, as the process transfer function can be different from what was used in the controller design (due to modeling errors and process variations).

Fig. 5.1. Closed-loop responses (a: Loop-1 and b: Loop-2) to setpoint changes of example (5.1) -processes using PID controller

#### **6.2 Nyquist Stability Theorem**

Suppose that *G s* is an n x n system with a decentralized control system *C s diag c s c s* <sup>1</sup> ,....., *<sup>n</sup>* and that the matrix, 1 *GsCs* , is column diagonally dominant on the nyquist contour, i.e.

$$\left|1 + g\_{ll}(s)c\_{l}(s)\right| > R\_{l}(s)\left|c\_{l}(s)\right|\tag{6.1}$$

Identification and Control of Multivariable Systems – Role of Relay Feedback 129

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

Time

(b)

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

Time

(c)

Fig. 5.2. Closed-loop responses (a: loop-1; b: loop-2 and c: loop-3) to setpoint changes of

0.2

0.1

example 5.2 -processes using PID controller

0.2

0.3

0.4

0.5

Response,y3

0.6

0.7

0.8

0.9

1

0.4

0.6

0.8

Response,y2

1

1.2

1.4

where

$$R\_l(s) = \sum\_{k=1, k \neq 1}^{n} \left| \mathbf{g}\_{kl}(s) \right| \tag{6.2}$$

for *l n* 1,2,........, and for all s on the Nyquist contour

Suppose that *G s* is an n x n system with a decentralized control system *C s diag c s c s* <sup>1</sup> ,....., *<sup>n</sup>* and that the matrix, 1 *GsCs* , is column diagonally

> 1, 1

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

Time

(a)

*n l kl k k R s g s* 

1 *g sc s R s c s ll l l l* (6.1)

(6.2)

**6.2 Nyquist Stability Theorem** 

0.2

0.4

0.6

Response,y1

0.8

1

1.2

1.4

where

dominant on the nyquist contour, i.e.

for *l n* 1,2,........, and for all s on the Nyquist contour

Fig. 5.2. Closed-loop responses (a: loop-1; b: loop-2 and c: loop-3) to setpoint changes of example 5.2 -processes using PID controller

Identification and Control of Multivariable Systems – Role of Relay Feedback 131

<sup>2</sup> (10 1) *<sup>s</sup> e s* = <sup>2</sup> (10 1) *<sup>j</sup> e j* 

<sup>2</sup> (10 1) (10 sin 2 cos2 ) ( 10 cos2 sin 2 ) *<sup>j</sup> e j j*

 

(cos12 60 sin 12 ) (60 cos sin 12 )

 

Recall that the magnitude of a complex number is the square root of the sum of real part

1 1 0 1 *<sup>k</sup>*

Thus in this chapter, it was found that least square and subspace methods have been used to identify process in open loop and sequential identification technique is used to estimate the process in closed loop. And the decentralized controllers are tuned using BLT method results in a stable controller. Finally, all the interaction tools are discussed as well the stability of the MIMO processes. The IMC-PID tuning rule suggested in this article yields

The following step-by-step procedure may be employed to solve a multi-variable control

1. Choose an appropriate pairings of controlled and manipulated variables, by interaction

2. If interaction is modest, one may consider SISO controllers for the multi-variable

3. If interaction is significant, it may be possible to use decouplers to reduce interaction in

4. An alternative to steps 2 and 3 is to use a full multi-variable control technique that

Based on the concept of sequential identification-design, an approach for the automatic tuning of multivariable systems is discussed. Several system identification methods like subspace identification, least squares, relay feedback methods are used to determine

Authors wish to acknowledge the financial support of DST / SR-S3-CE-90-2009 in carrying

dynamic parameters of a specific model structure from plant data (real time).

 

*g j* <sup>12</sup>

squared and the imaginary part squared. Therefore, g12(0) =1

A constant pre-compensator was designed to obtain dominance. This was

 

 

(6.5)

 

 

(6.7)

(6.6)

Consider the g11 element, first replace s with jw which produces:

Using Euler's relation,

Consider w=0, g11(0)=-1

**7. Conclusion** 

problem:

analysis.

system.

fast and robust responses.

**8. Acknowledgement** 

out this research work

To compute the radius, g12(w) is calculated as:

conjunction with PID-type controllers.

inherently compensates for interactions.

## **6.3 INA design methodology**

The following is the design procedure for the INA technique:

