**6.4 Example**

Johansson and Koivo designed a multivariable controller for a boiler subsystem where the boiler was a 1.6MW water boiler using solid fuel. Significant interaction was present between the loops in the subsystem, which consisted of the boiler underpressure and flue gas oxygen content as outputs with damper position and motor speed of the secondary

blower as associated inputs. The output vector is 1 2 *T yyy* where y1 is the normalized boiler underpressure and y2 is the percentage flue gas oxygen content. The input vector is 1 2 *T uuu* where u1 is the damper position (%) and u2 is secondary blower speed (rpm). The dynamics of the subsystem were determined from step response experiments. First order plus dead time responses were obtained, which produced the transfer function matrix:

$$G(s) = \begin{bmatrix} \frac{e^{-2s}}{(10s+1)} & \frac{-1}{(10s+1)}\\\\ 0 & \frac{e^{-10s}}{(60s+1)} \end{bmatrix} \tag{6.3}$$

The response of the flue gas oxygen content to step change in damper position was very slow and small in amplitude; therefore g21(s) was taken as zero. However, the secondary blower speed, u2, affects both outputs.

The inverse of G can be written immediately as:

$$\mathbf{G}^{-1}(\mathbf{s}) = \begin{bmatrix} -e^{2s} \text{(10s+1)} & e^{12s} \text{(60s+1)}\\ \mathbf{0} & e^{10s} \text{(60s+1)} \end{bmatrix} \tag{6.4}$$

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

$$-e^{2s}(10s+1) = -e^{2/\alpha}(10j\alpha+1)\tag{6.5}$$

Using Euler's relation,

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

.

3. Obtain the inverse nyquist array, which is the <sup>2</sup> *m* nyquist diagrams of the elements of

5. To verify dominance, calculate the appropriate gershgorin circles for the diagonal elements of the INA at various frequencies. The size of the gershgorin circles measures the importance of off-diagonal (interacting) elements relative to diagonal

6. The INA and gershgorin bands provide the amount of gain that may be applied to each

Johansson and Koivo designed a multivariable controller for a boiler subsystem where the boiler was a 1.6MW water boiler using solid fuel. Significant interaction was present between the loops in the subsystem, which consisted of the boiler underpressure and flue gas oxygen content as outputs with damper position and motor speed of the secondary

boiler underpressure and y2 is the percentage flue gas oxygen content. The input vector is

*uuu* where u1 is the damper position (%) and u2 is secondary blower speed (rpm). The dynamics of the subsystem were determined from step response experiments. First order plus dead time responses were obtained, which produced the transfer function

The response of the flue gas oxygen content to step change in damper position was very slow and small in amplitude; therefore g21(s) was taken as zero. However, the secondary

2 12

(10 1) (60 1) ( ) 0 (60 1) *s s*

*es e s*

(10 1) 10 1

*s s*

10

*e s*

10

*s*

*e s*

*s*

(60 1)

1

2

*e*

*s*

0

 

*<sup>c</sup>* , where

*T*

*yyy* where y1 is the normalized

(6.3)

(6.4)

*<sup>c</sup>* is the frequency

to a diagonally

above which the response is certain to become and remain negligible.

4. Design compensators, which transform the non dominant *G s*

of the loops without violating the stability requirement.

blower as associated inputs. The output vector is 1 2

*G s*

1

*G s*

blower speed, u2, affects both outputs.

The inverse of G can be written immediately as:

**6.3 INA design methodology** 

*G s* 

.

dominant.

elements.

**6.4 Example** 

 1 2 *T*

matrix:

The following is the design procedure for the INA technique:

2. Select an appropriate frequency range; usually 0

1. Obtain *G s* and calculate its inverse, *G s*

$$(-e^{2/\alpha}(10jo+1) = (10o\sin 2o - \cos 2o) + j(-10o\cos 2o - \sin 2o) \tag{6.6}$$

Consider w=0, g11(0)=-1 To compute the radius, g12(w) is calculated as:

$$g\_{12}(\alpha) = -\left[ (\cos 12\alpha - 60\alpha \sin 12\alpha) - j(60\alpha \cos \alpha + \sin 12\alpha) \right]$$

Recall that the magnitude of a complex number is the square root of the sum of real part squared and the imaginary part squared. Therefore, g12(0) =1

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

$$k = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \tag{6.7}$$
