**5. Tuning of controller**

Consider a process with transfer function <sup>1</sup> *D sp p p p k e G s s* . This transfer function has two parts. One invertible: *Gp* and the other containing non-invertible part *Gp* (time delay or right half plane zero that gives non-minimum phase behaviour). The IMC controller can be

$$\text{ expressed as: } \text{G}\_c^{\text{'} \text{MC}} = \frac{1}{\text{G}\_p^{-}} \text{ where } \text{G}\_p^{-} = \frac{k\_p}{\tau\_p s + 1} \text{ and } \text{G}\_p^{+} = e^{-D\_p s}.$$

Let us consider the desired closed loop response as 1 1 *D sp y Gp e Rs s* which can be

equated to complimentary sensitive function as <sup>1</sup> *true c p true c p y G G <sup>R</sup> G G* . Thus the true controller can be expressed as:

$$\mathbf{G}\_c^{true} = \frac{\mathbf{G}\_c^{\text{'} \text{'} \text{'} \text{'} \text{'}}}{1 - \left(\bigvee\_R \right)\_d \mathbf{G}\_c^{\text{'} \text{'} \text{'} \text{'}}} = \frac{\mathbf{J}\_p^{\text{'}}}{\left(\lambda s + 1\right) - \mathbf{G}\_p^{+}} \tag{5.1}$$

The right hand side of this equation can be written or rearranged to

$$\mathcal{G}\_c^{true} = \frac{\bigwedge^- \mathcal{G}\_p^-}{\left(\lambda s + 1\right) - e^{-D\_p s}} \tag{5.2}$$

In fact, the standard form of a PID controller can be given as

$$\mathbf{G}\_c^{true} = \frac{f(\mathbf{s})}{s} \quad \text{Or} \quad \mathbf{G}\_c^{true} = \frac{(\beta \mathbf{s} + 1)f(\mathbf{s})}{s(\beta \mathbf{s} + 1)} = \frac{\phi(\mathbf{s})}{s(\beta \mathbf{s} + 1)} \quad \text{where} \quad \beta = a \mathbf{r}\_D \tag{5.3}$$

This true controller can be expanded near the vicinity of s=0 using Laurent series as

$$G\_c^{true}(\mathbf{s}) = \frac{1}{s(\beta \mathbf{s} + 1)} \left[ \sum\_{j = -\infty}^{\infty} c\_j(\mathbf{s}) \right]^j = \frac{1}{s(\beta \mathbf{s} + 1)} \left[ \dots + \phi(0) + \phi'(0)\mathbf{s} + \phi''(0)\frac{\mathbf{s}^2}{2!} + \dots \right] \tag{5.4}$$

By comparing the coefficients of s in equation (5.4) with the standard PID controller, we get

$$\begin{aligned} k\_c &= a\_0 = \phi^\circ(0) = f^\circ(0) + \beta f(0) \\ \frac{k\_c}{\tau\_I} &= b\_1 = \phi(0) = f^\circ(0) \\ k\_c \tau\_D &= a\_1 = \frac{\phi^\circ(0)}{2!} = \frac{f^\circ(0) + 2\beta f^\circ(0)}{2} \end{aligned} \tag{5.5}$$

where

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

Hankel norm is the largest singular values. For elementary SISO subsystems with several HSVs it can be argued that a more relevant way of quantifying the interaction is to take into account all of the HSVs, atleast if there are several HSVs that are of magnitudes close to

Gramian based interaction measures are calculated and these values for benchmark 2-by-2

**PROCESS HIIA PM** 

0.1144 0.3362

*p*

*G s*

right half plane zero that gives non-minimum phase behaviour). The IMC controller can be

1 *p*

1 1 *IMC true <sup>p</sup> <sup>c</sup> <sup>c</sup> IMC <sup>p</sup> <sup>c</sup>*

*G G*

*<sup>y</sup> s G <sup>G</sup> <sup>R</sup>* 

*d*

*p k*

 *s* 

*D sp p*

 

 and *D sp G e <sup>p</sup>* .

> *true c p true c p*

*y G G <sup>R</sup> G G*

1

*D sp y Gp e Rs s* 

 

 

*p k e*

*s*

and the other containing non-invertible part *Gp*

*ij*

*<sup>j</sup> <sup>i</sup> tr P Q* is the sum of squared HSVs of the subsystems with input and output.

**WB** 0.2218 0.3276

*p*

Let us consider the desired closed loop response as 1 1

*G*

Table 4.1. HIIA and PM for benchmark 2-by-2 MIMO process

Consider a process with transfer function <sup>1</sup>

*p*

*<sup>G</sup>* where

equated to complimentary sensitive function as <sup>1</sup>

*G*

 *j i*

(4.15)

0.1741 0.3796 0.463 0.4000

. This transfer function has two

(time delay or

which can be

. Thus the true controller

(5.1)

*tr P Q tr PQ*

**Participation matrix** 

maximum HSV.

Each element in PM is defined by

*tr PQ* equals the sum of all *tr P Q <sup>j</sup> <sup>i</sup>*

MIMO process is given in table 4.1.

**5. Tuning of controller** 

parts. One invertible: *Gp*

expressed as: *IMC* <sup>1</sup> *<sup>c</sup>*

can be expressed as:

*G*

**2X2 MIMO** 

$$\begin{aligned} G\_c(s) &= \frac{\phi(s)}{\left(\beta s + 1\right)}\\ \phi(s) &= \left(\beta s + 1\right) f(s) \end{aligned} \tag{5.6}$$

The method described in earlier section is applied to some standard transfer functions and the comprehensive results are presented in Table 5.1 and selection of is given in Table 5.2. Detailed analysis on synthesis of PID tuning rules can be seen in Panda (2008 & 2009). Example 5.1: The wood and berry binary distillation column is a multivariable system that

$$
\begin{bmatrix}
\frac{12.8e^{-s}}{16.7s+1} & \frac{-18.9e^{-3s}}{21s+1} \\
\frac{6.6e^{-7s}}{10.9s+1} & \frac{-19.4e^{-3s}}{14.4s+1}
\end{bmatrix}.
\tag{5.7}
$$

The closed loop response is given in Figure 5.1.

has been studied extensively. The process has transfer function

Example 5.2: The transfer function of multiproduct plant distillation column for the separation of binary mixture of ethanol-water (Ogunnaike-Ray (OR) column) is given by

Identification and Control of Multivariable Systems – Role of Relay Feedback 127

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).

<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

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

(a)

0.1

0.1

(5.1) -processes using PID controller

0.2

0.3

0.4

0.5

Response,y2

0.6

0.7

0.8

0.9

1

0.2

0.3

0.4

0.5

Response,y1

0.6

0.7

0.8

0.9

1

$$
\begin{bmatrix} y\_1 \\ y\_2 \\ y\_3 \end{bmatrix} = \begin{bmatrix} \frac{0.66e^{-2.6s}}{6.7s+1} & \frac{-0.61e^{-3.5s}}{8.64s+1} & \frac{-0.0049e^{-s}}{9.06s+1} \\ \frac{-2.36e^{-3s}}{5s+1} & \frac{-2.3e^{-3s}}{5s+1} & \frac{-0.01e^{-1.2s}}{7.09s+1} \\ \frac{-34.68e^{-9.2s}}{8.15s+1} & \frac{46.2e^{-9.4s}}{10.9s+1} & \frac{0.87(11.61s+1)e^{-s}}{(3.89s+1)(18.8s+1)} \end{bmatrix} \begin{bmatrix} u\_1 \\ u\_2 \\ u\_3 \end{bmatrix} \tag{5.8}
$$

The closed loop response is given in Figure 5.2.


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


Table 5.2. selection rule
