**4. Input-output pairing**

Many control systems are multivariable in nature. In such systems, each manipulated variable (input signal) may affect several controlled variables (output signals) causing interaction between the input/output loops. Due to these interactions, the system becomes more complex as well as the control of multivariable systems is typically much more difficult compared to the single-input single-output case.

#### **4.1 The Relative Gain Array analysis**

The RGA is a matrix of numbers. The *i* jth element in the array is called *ij* . It is the ratio of the steady-state gain between the ith controlled variable and the jth manipulated variable when all other manipulated variables are constant, divided by the steady-state gain between the same two variables when all other controlled variables are constant.

$$\mathcal{B}\_{ij} = \frac{\left[\mathcal{Y}\_i \bigwedge\_{m\_j} \right]\_{\overline{m\_k}}}{\left[\mathcal{Y}\_i \bigwedge\_{\overline{y\_k}} \right]\_{\overline{y\_k}}} \tag{4.1}$$

For example, suppose we have a 2 **X** 2 system with the steady-state gains *pij k*

$$y\_1 = k\_{p11}m\_1 + k\_{p12}m\_2\tag{4.2}$$

$$y\_2 = k\_{p21}m\_1 + k\_{p22}m\_2$$

For this system, the gain between y1 and m1 when m2 constant is

$$
\left[\begin{array}{c}
y\_1\\\nearrow \boldsymbol{m}\_1
\end{array}\right]\_{\overline{\boldsymbol{m}\_2}} = k\_{p11}
$$

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

$$y\_1 = k\_{p11}m\_1 + k\_{p12}m\_2$$

$$0 = k\_{p21}m\_1 + k\_{p22}m\_2$$

$$y\_1 = k\_{p11}m\_1 + k\_{p12} \left[\sum\_{p=21}^{k\_{p21}} \bigwedge\_{p\_{p22}} \right]$$

$$y\_1 = \left[\frac{k\_{p11}k\_{p22} - k\_{p12}k\_{p21}}{k\_{p22}}\right]m\_1\tag{4.3}$$

$$\mathbb{E}\left[\bigvee\_{m\_1}^{y\_1}\bigvee\_{\overline{y\_2}}\right] = \left\lfloor \frac{k\_{p11}k\_{p22} - k\_{p12}k\_{p21}}{k\_{p22}} \right\rfloor \tag{4.4}$$

Therefore the term 11 in RGA is

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

1 22 12

<sup>1</sup> <sup>1</sup> 19.4 18.9

0.15698 0.15293 0.0534 0.1035 *<sup>D</sup>*

3

0.15698 0.15293 16.7 1 21 1 6.6 19.4 0.0534 0.1035

Many control systems are multivariable in nature. In such systems, each manipulated variable (input signal) may affect several controlled variables (output signals) causing interaction between the input/output loops. Due to these interactions, the system becomes more complex as well as the control of multivariable systems is typically much more

the steady-state gain between the ith controlled variable and the jth manipulated variable when all other manipulated variables are constant, divided by the steady-state gain between

*i*

*y m*

*i*

2 21 1 22 2 *p p y km km*

*y m*

*ij*

For example, suppose we have a 2 **X** 2 system with the steady-state gains *pij k*

For this system, the gain between y1 and m1 when m2 constant is

*k*

*<sup>j</sup> <sup>m</sup>*

*<sup>j</sup> <sup>y</sup>*

*k*

7 3

*s s*

*s s*

*e e*

10.9 1 14.4 1

The RGA is a matrix of numbers. The *i* jth element in the array is called

the same two variables when all other controlled variables are constant.

*e e s s*

*D G*

 

*Q GD* or <sup>1</sup> *Q GG* 0 (3.17)

(4.1)

1 11 1 12 2 *p p y km km* (4.2)

*ij* . It is the ratio of

(3.16)

21 22

1 0 0

det 0 0 0

*G g g*

123.58 6.6 12.8

0

The transfer function of the statistically decoupled system is given by

12.8 18.9

*s s <sup>Q</sup>*

difficult compared to the single-input single-output case.

**4.1 The Relative Gain Array analysis** 

**4. Input-output pairing** 

*g g D G*

0

$$\beta\_{11} = \frac{1}{1 - \frac{k\_{p12}k\_{p21}}{k\_{p11}k\_{p22}}} \tag{4.5}$$

**Example**: Calculate 11 element of RGA for the wood and berry column

$$k\_p = \begin{bmatrix} 12.8 & -18.9 \\ 6.6 & -19.4 \end{bmatrix}$$

$$\mathcal{J}\_{11} = \frac{1}{1 - \frac{k\_{p12}k\_{p21}}{k\_{p11}k\_{p22}}} = \frac{1}{1 - \frac{(-18.9)(6.6)}{(12.8)(-19.4)}} = 2.01$$

#### **4.2 Singular Value Decomposition**

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 matrices as follows:

$$k = \mathcal{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tag{4.6}$$

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 "singular values"

Identification and Control of Multivariable Systems – Role of Relay Feedback 123

In 2004, Salgado and Conley investigated the channel interaction by considering controllability and observability gramians so called participation matrix. Similarly, Wittenmark and Salgado (2002) introduced Hankel Interaction Index array. These gramian measures namely HIIA, PM overcome the disadvantages of RGA. One key property of these is that the whole frequency range is taken into account in one single measure. Interaction measures recommend the input-output pairings that result in the largest sum when adding the corresponding elements in the measure. HIIA and PM give appropriate suggestions for

<sup>1</sup>

Since NI is positive, the closed loop system with the specified pairing may be stable.

The controllability Gramian, P, defined for stable time-invariant systems as

0

A stable system will be *state observable* if the observability Gramian, Q, defined as

0

*<sup>T</sup> A TA P e BB e d* 

*<sup>T</sup> A TA Q e CC e d* 

These Gramians can be obtained by solving the following continuous time Lyapunov

*T T AP PA BB A Q QA C C*

Hankel singular values with controllability and observability gramians P and Q is given by

(1) *<sup>G</sup> <sup>H</sup>*

 *ij <sup>H</sup> <sup>H</sup> ij kl kl <sup>H</sup> G G*

 

 

*T T*

 

 

> 0 0

(4.9)

(4.10)

(4.11)

*i n* 1,2,....... (4.12)

*<sup>H</sup>* max *PQ* (4.13)

(4.14)

*p N j pjj*

*Det k k*

**4.4 Gramian based interaction measures** 

decentralized multivariable controller.

If P has full rank, the system is state controllable.

If Q has full rank, the system is state observable

**Hankel interaction index array** 

The normalized version is the HIIA given by

( )*i H i* 

The Hankel norm of the system with the transfer function G is

equations:

(4.8)

12.8 19.4 18.9 6.6 NI 0.498 12.8 19.4

SVD is designed to determine the rank and the condition of a matrix and to show geometrically the strengths and weaknesses of a set of equations so that the errors during computation can be avoided.

#### **4.2.1 Example**

Consider a very simple mixing example, a multivariable process whose gain matrix is as follows:

$$k = \begin{bmatrix} 0.7778 & -0.3889\\ 1.0000 & 1.0000 \end{bmatrix}$$

which decomposes to

$$\begin{aligned} \boldsymbol{U} &= \begin{bmatrix} 0.2758 & -0.9612 \\ 0.9612 & 0.2758 \end{bmatrix} \\\\ \boldsymbol{V} &= \begin{bmatrix} 0.8091 & -0.5877 \\ 0.5877 & 1.0000 \end{bmatrix} \\\\ \boldsymbol{\Sigma} &= \begin{bmatrix} 1.4531 & 0 \\ 0 & 0.8029 \end{bmatrix} \end{aligned} $$

At this point these singular values and vectors are merely numbers; however, consider the relationship between these values and an experimental procedure that could be applied to measure the steady-state process characteristics.

#### **4.3 Niederlinski index**

A fairly useful stability analysis method is the Niederlinski index. It can eliminate unworkable pairings of variables at an early stage in the design. The controller settings need not be known, but it applies only when integral action is used in all the loops. It utilizes only the steady state gains of the process transfer function matrix. The method is necessary but not the sufficient condition for stability of a closed loop system with integral action. If the index is negative, the system will be unstable for any controller settings. If the index is positive, the system may or may not be stable. Further analysis is necessary.

$$\text{Niederlinski index} = \text{NI} = \frac{\text{Det}\left[k\_p\right]}{\prod\_{j=1}^{N} k\_{pj}} \tag{4.7}$$

where, kp is a matrix of steady state gains from the process openloop transfer function

kpjj is the diagonal elements in steady state gain matrix

Example: Calculate the Niederlinski index for the wood and berry column:

$$k\_p = \begin{bmatrix} 12.8 & -18.9 \\ 6.6 & -19.4 \end{bmatrix}$$

$$\text{NI} = \frac{\text{Det}\left[k\_p \right]}{\prod\_{j=1}^{N} k\_{pj}} = \frac{(12.8)(-19.4) - (-18.9)(6.6)}{(12.8)(-19.4)} = 0.498\tag{4.8}$$

Since NI is positive, the closed loop system with the specified pairing may be stable.

#### **4.4 Gramian based interaction measures**

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

SVD is designed to determine the rank and the condition of a matrix and to show geometrically the strengths and weaknesses of a set of equations so that the errors during

Consider a very simple mixing example, a multivariable process whose gain matrix is as

0.7778 0.3889 1.0000 1.0000 *<sup>k</sup>*

0.2758 0.9612 0.9612 0.2758 *<sup>U</sup>*

0.8091 0.5877

0.5877 1.0000 *<sup>V</sup>*

> 1.4531 0 0 0.8029

At this point these singular values and vectors are merely numbers; however, consider the relationship between these values and an experimental procedure that could be applied to

A fairly useful stability analysis method is the Niederlinski index. It can eliminate unworkable pairings of variables at an early stage in the design. The controller settings need not be known, but it applies only when integral action is used in all the loops. It utilizes only the steady state gains of the process transfer function matrix. The method is necessary but not the sufficient condition for stability of a closed loop system with integral action. If the index is negative, the system will be unstable for any controller settings. If the index is

Niederlinski index NI *<sup>p</sup>*

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

where, kp is a matrix of steady state gains from the process openloop transfer function

positive, the system may or may not be stable. Further analysis is necessary.

Example: Calculate the Niederlinski index for the wood and berry column:

kpjj is the diagonal elements in steady state gain matrix

0.8091

1

(4.7)

*N j pjj*

*Det k k*

computation can be avoided.

**4.2.1 Example** 

which decomposes to

**4.3 Niederlinski index** 

measure the steady-state process characteristics.

follows:

In 2004, Salgado and Conley investigated the channel interaction by considering controllability and observability gramians so called participation matrix. Similarly, Wittenmark and Salgado (2002) introduced Hankel Interaction Index array. These gramian measures namely HIIA, PM overcome the disadvantages of RGA. One key property of these is that the whole frequency range is taken into account in one single measure. Interaction measures recommend the input-output pairings that result in the largest sum when adding the corresponding elements in the measure. HIIA and PM give appropriate suggestions for decentralized multivariable controller.

The controllability Gramian, P, defined for stable time-invariant systems as

$$P = \bigcap\_{0}^{\circ} e^{A\tau} B B^T e^{A^T \tau} d\tau \tag{4.9}$$

If P has full rank, the system is state controllable.

A stable system will be *state observable* if the observability Gramian, Q, defined as

$$Q = \bigcap\_{0}^{\approx} e^{A\tau} \mathbf{C} \mathbf{C}^{T} e^{A^{\top} \tau} d\tau \tag{4.10}$$

If Q has full rank, the system is state observable

These Gramians can be obtained by solving the following continuous time Lyapunov equations:

$$\begin{cases} AP + PA^T + BB^T = 0\\ A^T Q + QA + \mathcal{C}^T \mathcal{C} = 0 \end{cases} \tag{4.11}$$

Hankel singular values with controllability and observability gramians P and Q is given by

$$
\sigma\_H^{(i)} \triangleq \sqrt{\mathcal{Z}\_i} \tag{4.12}
\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\tag{4.12}
$$

The Hankel norm of the system with the transfer function G is

$$\left\|\mathbb{G}\right\|\_{H}\triangleq\sigma\_{H}^{(1)}=\sqrt{\mathbb{A}\_{\text{max}}\left(PQ\right)}\tag{4.13}$$

#### **Hankel interaction index array**

The normalized version is the HIIA given by

$$\left\lVert \boldsymbol{\Sigma}\_{H} \right\rVert\_{ij} = \frac{\left\lVert \boldsymbol{G}\_{ij} \right\rVert\_{H}}{\sum\_{kl} \left\lVert \boldsymbol{G}\_{kl} \right\rVert\_{H}} \tag{4.14}$$

#### **Participation matrix**

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 maximum HSV.

Each element in PM is defined by

$$\left[\left[\phi\right]\right]\_{ij} = \frac{\operatorname{tr}\left(P\_j Q\_i\right)}{\operatorname{tr}\left(PQ\right)}\tag{4.15}$$

Identification and Control of Multivariable Systems – Role of Relay Feedback 125

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

' '

*ka f f*

*f f k a*

 

 

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

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

0 0

 

*<sup>s</sup> G s c s <sup>s</sup>*

*ss ss*

1

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

*j*

*s fs s <sup>G</sup>*

*true p c D s G*

*s e*

1

1 *<sup>p</sup>*

 

(5.4)

1 1

<sup>2</sup> 1 1 ' '' ... 0 0 0 ... 1 1 2!

 

0 02 0

 

'' ' '

1 1

3

*s*

*s s s fs*

The method described in earlier section is applied to some standard transfer functions and

7 3

*s s*

*s s*

*e e s s e e s s*

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

2! 2

0 0 (0)

 

 

(5.2)

 

 

(5.5)

(5.6)

. (5.7)

is given in Table 5.2.

*<sup>D</sup>* (5.3)

where

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

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

*j*

*c c I*

*c D*

*s s s s*

0

1

1

*c*

the comprehensive results are presented in Table 5.1 and selection of

has been studied extensively. The process has transfer function

The closed loop response is given in Figure 5.1.

*G s*

*<sup>k</sup> b f*

*true*

*f s*

*c j*

*c*

*G*

*true*

where

*G*

*<sup>s</sup>* Or

*true c*

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

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

Gramian based interaction measures are calculated and these values for benchmark 2-by-2 MIMO process is given in table 4.1.


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