INTRODUCTION
Decentralized control methodology provides an effective solution to control complex processes with multi inputs and multi outputs. Simple tuning procedure for single input and single output controllers and implementation considerations are among the many reasons for its wide use (Takagi and Nishimura, 2003; Tan et al., 2001; Asano and Morari, 1998; Skogestad and Postlethwaite, 2005).
However, a successful decentralized design would require an appropriate inputoutput section a prior (Skogestad and Postlethwaite, 2005). Since the seminal work by Bristol (1966) and presentation of RGA concept for inputoutput pairing, there have been many extensions and modifications to the method (KhakiSedigh and Shahmansoorian, 1996; Conley and Salgado, 2000; Wittemark and Salgado, 2002; Xiong et al., 2005). In spite of the extensive research of the previous decades, the inputoutput pairing of nonlinear multivariable systems remains an open problem and only recently it has been addressed. Glad^{ } (1999) presents an extension of RGA to inputoutput pairing for nonlinear multivariable systems. Where, this approach proposes a two stage static and dynamic inputoutput pairing analysis. Also, Moaveni and KhakiSedigh (2006) proposed a new online estimation of RGA using neural network for linear, nonlinear or uncertain linear multivariable systems.
INPUTOUTPUT PAIRING FOR NONLINEAR MULTIVARIABLE SYSTEMS
Inputoutput pairing for nonlinear multivariable systems is mainly performed by the indirect approach, where the system is linearized around its operating points and any of the approaches for linear multivariable systems are used. Recently, Glad (1999) has presented an extension of RGA to inputoutput pairing for nonlinear multivariable systems.
In the following theorem a mathematical relationship between inputs and outputs of nonlinear multivariable systems is derived and using it a new inputoutput pairing analysis based on relative gain definition will be presented.
Theorem, Consider the class of multivariable nonlinear systems, described in
affine state space model with m inputs and m outputs, given by:
Where, f(x), g_{1}(x),..., g_{m}(x) are smooth vector fields and h_{1}(x),..., h_{m}(x) are smooth functions, defined on an open set of u_{j} shows the jth input of input vector, u and y_{i} shows the ith output of output vector, y (Isidori, 1995).
Inputoutput pairing analysis for nonlinear multivariable systems can be done
using Γ_{nlRGA} as:
Where:
and r≤n is the maximum value of vector relative degree from y_{i}
to u_{i}, where, i, j = 1,..., m.
Proof: As h_{i}(x) are assumed smooth functions, Taylor series
expansion of y_{i}(t) at point t = τ, can be written in
the following form:
Where:
so:
therefore, Eq. 4 can be rewritten as:
Where:
and so:
Now, to propose an interaction measure or inputoutput pairing analysis we
can use the relative gain definition (Bristol, 1966). Relative gain is the ratio
of the process gain in an isolated loop to process gain in that same loop when
all other control loops are closed. Process gain in an isolated loop of process
can be computed as:
Also, process gain in that same loop when all other control loops are closed
is:
So, nonlinear RGA, when, t is close to τ, is defined as:
Where, the operator ⊗ denotes the elementbyelement multiplication
of the two matrices and T denotes the inverse transpose of the matrix. However,
computation of nonlinear RGA using (12) is not simple, since it requires Φ
matrix. So, to propose a practical inputoutput pairing analysis and to reduce
the computational load we can ignore some terms of matrix Φ. These terms
includes high order terms (k>r), where, r is the maximum value of vector
relative degrees from y_{i} to u_{j} (i, j = 1,..., m), because
t is assumed close to τ. The low order terms (k<r), where the effect
of some inputs on some outputs are not apparent, can be ignored. So, matrix
Φ can be approximated as:
Therefore nonlinear RGA and Eq. 12 can be rewritten as:
It is readily seen that the nonlinear RGA in Eq. 14 is time
independent and has similar properties to the conventional RGA in linear multivariable
systems. But Γ_{nlRGA} is not computed in the steady state and
depends on the operating point of the process.
LINEAR INTERPRETATION OF THE NONLINEAR RGA
The above method can be applied to linear multivariable systems. The above
method is applied to linear multivariable systems and an interpretation of the
nonlinear RGA is provided. Suppose that a linear multivariable system is given
by the following state space equations:
Where:
c^{i} 
= 
ith row of matrix C, 
b_{j} 
= 
jth column of matrix B. 
According to Eq. 1 and 15, functions f
(x), g_{j }(x) and h_{i} (x) for the corresponding linear state
space model are:
Equation 13 and 16 are employed to compute
the matrix R and to find the relationship between the inputs and outputs of
the corresponding linear system. In this case, matrix R is as in
Eq. 17 and it is interesting to note that it is similar to the decoupling
matrix (Falb and Wolovich, 1967).
Hence, using Eq. 17 and 14, a new approach
to inputoutput pairing for linear multivariable systems is provided. Where,
it is important to observe that the proposed method takes the full dynamic effects
of the system into account, though, in RGA only the steadystate or the behaviour
at a single frequency is considered. Hence, the Γ_{nlRGA} can
be used as a dynamic interaction measure for linear and nonlinear multivariable
systems, where this method is most effective for nonlinear multivariable systems.
Algorithm: To choose the appropriate inputoutput pair for affine nonlinear
multivariable systems following steps are proposed:
• 
Calculate the vector relative degrees from y_{i}
to u_{j} (i, j = 1 ..., m) and assign the largest relative degree
to r. 
• 
Compute the matrix R using Eq. 3 (or 17)
for nonlinear (or linear) multivariable systems. 
• 
Compute the nonlinear RGA using Eq. 14. 
• 
Choose the appropriate inputoutput pair using nonlinear RGA analysis. 
SIMULATION RESULTS
Nonlinear and linear multivariable processes are used to show the effectiveness of the proposed methodology. The measure of goodness of inputoutput pairing method is the decentralized control performance (Hovd and Skogestad, 1992). So, in the following examples we use this measure to compare the proposed method with previous methods.
Example 1: Consider the Quadrupletank with nonlinear state space model
as (Johansson, 2000):
Where, h_{1} and h_{2} (water levels of 1st and 2nd tank) are outputs of the Quadrupletank.
Linearization of Eq. 18 gives the following transfer function
matrix:
Where:
Assume that the Quadrupletank has approximately the following physical constants
(Johansson, 2000):
The conventional RGA for the linear model of the Quadrupletank is:
Where,
If, 0<γ_{1} + γ_{2}≤1, the appropriate pair is (u_{1}y_{2}, u_{2}y_{1}) and if 0<γ_{1} + γ_{2}≤2 the appropriate pair is (u_{1}y_{1}, u_{2}y_{2}) (Johansson, 2000).
Now to investigate the properties of the proposed nonlinear RGA, we can apply the nonlinear RGA method to this process and analyze the results.
To compute the nonlinear RGA, for this process r = 2 and matrix R is:
so, nonlinear RGA is:
which, it can be rewritten as follows:
To find the appropriate inputoutput pair from Eq. 24, the
following new variables are defined as:
Hence:
Where, to choose the appropriate inputoutput pair we should analyze the elements
of the nonlinear RGA in Eq. 26. Therefore, the following
two cases are defined:
• 
If α>1α then the appropriate inputoutput pair
is (u_{1}y_{1}, u_{2}y_{2}). 
This result is easily justified as follows:
Where, it shows that the new result is compatible with the result of conventional
linear RGA analysis.
• 
Also, if α<1α then the appropriate inputoutput
pair is (u_{1}y_{1}, u_{2}y_{2}). So: 
Where, Eq. 28a is compatible with the conventional linear
RGA analysis, but, Eq. 28b is additional condition that derived
by nonlinear RGA analysis. Therefore, Eq. 27 and 28
show that the appropriate inputoutput pairs depend on the operating points
of nonlinear process.
To compare the input output pairing analysis according to nonlinear RGA with
the conventional linear RGA analysis, we consider the following conditions for
the Quadrupletank process:
For the above conditions linear RGA proposes (u_{1}y_{2},
u_{2}y_{1}) as an appropriate inputoutput pair (Johansson,
2000). Using the nonlinear RGA, following inequality holds true and (u_{1}y_{1},
u_{2}y_{2}) is proposed as an appropriate pair:
We use the decentralized control structure with PI controllers introduced by
Johansson (2000) to compare the close loop performances of the proposed inputoutput
pairings. In Fig. 1 responses of the Quadrupletank according
to (u_{1}y_{2}, u_{2}y_{1}) are shown and
in Fig. 2 the responses according to (u_{1}y_{1},
u_{2}y_{2}) are shown.
Hence, the simulation confirms that the closeloop responses according to diagonal pairing suggested by nonlinear RGA have more desirable performance than the closeloop responses according to offdiagonal pairing. Where, this result shows the advantage of the proposed inputoutput pairing method.
Example 2: Consider the wellknown Wood and Berry binary distillation
column process as:

Fig. 1: 
Output responses according to linear RGA analysis 

Fig. 2: 
Output responses according to nonlinear RGA analysis 
Where, X_{D} and X_{B} are the overhead and bottom compositions of methanol, respectively, R is the reflux flow rate and S is the steam flow rate to the reboiler (Wood and Berry, 1973).
Using Eq. 17 in the above method to choose the appropriate
inputoutput pair, matrix R for this process is as follows:
and using Eq. 14:
Where, a first order Pade approximation to realize the time delays are used and (u_{1}y_{1}, u_{2}y_{2}) is the appropriate inputoutput pair similar to the conventional linear RGA analysis (Kariwala et al., 2006).
Also, Hankel norm based inputoutput pairing method, using Eq.
34, proposes (u_{1}y_{1}, u_{2}y_{2})
as the appropriate inputoutput pair.
Example 3: Consider a process given by Grosdidier and Morari (1986)
as:
The conventional RGA implies the (u_{1}y_{2}, u_{2}y_{1})
as the appropriate inputoutput pair. But, the nonlinear RGA, using Eq.
17 and 14, is:
Where, it shows that, (u_{1}y_{1}, u_{2}y_{2}) is an appropriate inputoutput pair. This loop pairing decision was obtained by Grosdidier and Morari (1986) through analyzing both magnitude and phase characteristics of the interaction between the two loops and by Xiong et al. (2005) using the Effective Relative Gain Array (ERGA).
CONCLUSION
In this study, a direct method to inputoutput pairing for nonlinear multivariable systems is proposed. Also, the implications of this direct method for the linear multivariable case are studied. This introduces the use of decoupling matrix for inputoutput pairing of linear multivariable systems. The effectiveness of the method for linear and nonlinear multivariable systems is demonstrated by several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, while the proposed inputoutput pairing method provides accurate results. Also, this method shows that inputoutput pairing for nonlinear multivariable systems is dependent on the operating point.