INTRODUCTION
Reallife systems almost always show nonlinear dynamical behavior. This behavior complicates the task of finding models that accurately describe these systems. While in a large number of applications a linear model shows already satisfactory results, there are numerous situations where linear models are not accurate enough; especially when we deal with very complex systems or require very high performance. Physical knowledge of the system can be a great aid in finding a nonlinear model. However, this knowledge is not always available. In these cases we have to determine a model from a finite number of measurements of the system's inputs and outputs.
The blockoriented approach to the nonlinear system modeling and identification
assumes that the system consists of nonlinear memoryless and linear dynamic
subsystems (Bendat, 1990). The approach has been used
in many fields (Emerson et al., 1992; Eskinat
et al., 1991; Haber and Unbehauen, 1990;
Huebner et al., 1990; Kalafatis
et al., 1997; Ralston et al., 1997;
ZiQiang, 1993).
Hammerstein system identification is one of the block oriented nonlinear system
identification methods. The Hammerstein model consists of a static nonlinearity
followed by a linear dynamic system and usually the signals between the nonlinear
and linear blocks are inaccessible to measurements. The identification of the
Hammerstein model involves estimating both the nonlinear and linear parts from
the inputoutput measurements (Sjoberg et al., 1995).
The SingleInput SingleOutput (SISO) Hammerstein model has been successfully
used to model (Leonessa and Luo, 2001), control (Haddad
and Chellaboina, 2001). There are wellknown algorithms for identification
of this kind of nonlinear systems (AlDuwaish and Nazmul
Karim, 1997; Van Pelt and Bernstein, 2000).
Pariz et al. (2003) has presented a new approach
for analysis and modeling of nonlinear systems. Pariz et
al. (2003) and Abdollahi (2002) expand this approach
for analyzing and modeling of continuous and discrete nonlinear systems. Modal
series can expressed a nonlinear system in a new form which is more accurate
than the linearized model of system and it expresses many nonlinear effects
of main system. This new modeling structure of nonlinear systems can be used
to model and identify a nonlinear system effectively.
In this study, we present a method to determine a novel modeling and identification
approach for nonlinear systems using modal series state space model from a finite
number of measurements of the inputs and outputs. An identification algorithm
will also be introduced for this model which uses subspace and MIMO Hammerstein
algorithms.
MODAL SERIES
As expressed by Pariz et al. (2003), Abdollahi
(2002) and Modir Shanechi et al. (2003) any
nonlinear system which is in the form of
where, x = [x_{1}, x_{2}, …, x_{n}]^{T} is the state vector, u = [u_{1}, u_{2},…,u_{m}]^{T }is the input vector and g : R^{2} x R^{m} → R^{n} is a smooth vector function which g(0,0)=0, can be modelled by Eq. 2 and 3 called modal series.
Remarks:
• 
Equations 2 are categorized in three classes’ v, w and
z 
• 
Class v is affected by the initial condition and is the zero
input response of the system 
• 
Class w is affected by the input and is the zero state response
of the system 
• 
Class z is affected by both initial condition and input. It
is the interaction between initial condition and input and differs from
zero when both of them do exist 
• 
In linear systems the complete response of a system is equal
to sum of its zero input and zero state responses, but this is not the case
for nonlinear systems, because of the existence of equations class z 
• 
Modal series method provides a solution for the system in
terms of the modes of the system and the input. This can be better seen
if we apply the transformation x=Ty, where T is the matrix of the right
eigenvectors of B_{10}, use modal series approach to yield the solution
and use back transformation y=T^{1}x to obtain the solution of
Eq. 2 
Extension to discrete modal series is straight and it will bring us to similar
equations (Abdollahi, 2002).
where
and
so on.
A MODIFIED REPRESENTATION OF MODAL SERIES MODEL OF NONLINEAR SYSTEMS
Definition: For matrices P and Q with dimensions n_{p}xm_{p} and n_{q}xm_{q}, respectively, the Kronecker product is defined as a (n_{p}n_{q})x(m_{p}m_{q}) matrix:
We define the superscript notation q and (p) for referring to Kronecker product and the repetitive application of the Kronecker product, respectively.
Now we can express that the class v deals with transient states of nonlinear
system which depends on initial conditions. We can neglect transient effects
and assume zero initial condition for class v when we want to identify nonlinear
system. Since class z equations depends on class v and w and class v states
are assumed zero, so class z equations are zero, too. Then we can rewrite the
discrete modal series in the form of Eq. 5. Where A, B_{1},
B_{2} … and u_{1}, u_{2 }… are defined by
Eq. 6 and 7.
It is supposed that the nonlinear system is linear in output equations. It
means
where, K_{m} is the maximum number of modal series terms. So output of each w_{k} equation is as follows:
Looking at Eq. 51, 52,
… implies that inputs (u_{k}(t), k=2,3,…) for w_{k}(t)
(k=2,3,…) state equations, rely on w_{j}(t1) (j=k1,k2,…1)
and u_{1}(t1).
Since it is possible to construct ζ_{k}(t) and then the input
vector u_{k} of every w_{k} equation for every sample time,
it is clear that Eq. 51, 52 … are
linear. w_{k} is in fact the linear model of nonlinear system and all
other terms try to model nonlinear dynamics of the main nonlinear system.
Now we can express that there are a set of equations in the form of (10) which can approximate the main nonlinear system.
where, D_{1} = D and D_{k }= 0 for k = 2,3,…. And the
main state vector can be computed by Eq. 5.
In attention to Eq. 5 to 10, one can proposed
a model structure as shown in Fig. 1. Figure
1 expresses a common linear block and several nonlinear blocks. It also
presents r the Hammerstein structure of subsystems.
In an identification problem for nonlinear systems, we can identify the structure proposed in Fig. 1 as an approximation of nonlinear systems. This would be very flexible method, since we can choose arbitrary number of modal series terms (w_{k}) to be identified.
Because of the state space form of this model of nonlinear systems, we proposed to use a modified version of a subspace algorithm as presented next.
A NOVEL ALGORITHM FOR IDENTIFICATION OF A NONLINEAR SYSTEM USING THE PROPOSED MODEL
Hammerstein systems are known structure for nonlinear system identification.
There are lots of literatures devoted to identification problem of Hammerstein
systems (Gomez and Baeyens, 2000). Identification of
MIMO Hammerstein systems are still an active field but there are well practiced
works around (Michel and Westwick, 1996). A straight
and interesting approach to identification of MIMO systems are subspace identification
algorithms (Van Overschee and de Moor, 1996). Linear subspace
algorithms have shown their ability in linear system identification and considering
the structure introduced earlier we can proposed an algorithm for MIMO nonlinear
system identification.
In order to be able to identify each subsystem of the proposed structure we
need to know input and output of each subsystem. Inputs are distinct, since
u_{1} is defined by Eq. 61 and inputs of ith subsystem
are u_{1}(t1) and states of all previously identified subsystems (w_{j}(t1),
j=1, …, i1).
To produce outputs for each subsystem, consider following output equation expansion for a K_{m} modal series terms representation.
So we proposed the following output equation (10k) for each subsystem when we want to identify the kth subsystem:
Now, the following algorithm is proposed in order to identify such a model.
Algorithm:
• 
Let y_{1}(t) = y(t), u_{1}(t) = u(t), k =
1 and choose an arbitrary number of modal series terms (K_{m}) 
• 
Identify linear subsystem of proposed structure (Fig.
1) using one of the MIMO linear system identification algorithms (preferably
linear subspace algorithms). And produce w_{1}(t) and A, B_{1},
C 
• 
Determine w_{i}(t), i = 1, 2, …, k 
• 
Let k = k+1 and use (12k) to produce y_{k}(t) 
• 
Use one of the MIMO Hammerstein system identification algorithms
while you know that the linear block of this subsystem has been identified
in step 2 and the nonlinear block structure has been defined in proposition
A1. And determine (k1)th MIMO Hammerstein subsystem (Fig.
1) 
• 
if k = Km go to step 7 else go to step 3 
The algorithm uses subspace methods to identify each subsystem (Van
Overschee and De Moor, 1996). We can apply N4SID, CVA, MOESP algorithms.
The proposed algorithm has been applied to identify the example expressed in
the next section.

Fig. 1: 
Presentation of a nonlinear system as proposed by modal series
model (u_{1}=u_{1}(t1), w_{1}=w_{1}(t1),
w_{2}=w_{2}(t1)) 

Fig. 2: 
Van der Pol Oscillator: the solid line represents the nonlinear
simulation output and dashed line represents the predicted output 
SIMULATIONS
In this simulation example we use N4SID subspace identification for linear
system identification and the method proposed by Michel
and Westwick (1996) to identify the MIMO Hammerstein subsystems.
The identification algorithm presented in the previous section is now tested using a sampled data Van der Pol Oscillator with nonlinear input.
System equations presented in 13:
where, w_{1}, w_{2}, v_{1} and v_{2} are white
noise signals, with covariance 0.001, 0.0002, 0.02 and 0.03, respectively.
The chosen input signal u for identification process was a zeromean, white noise sequence of length N = 1000 with a sampling period T = 0.05 sec uniformly distributed between 1 and 1.
In order to validate the proposed model structure and identification algorithm, additional 1000 samples are obtained from the simulation and the result is compared with the predicted output in Fig. 1.
In Fig. 2, the first 50 sec of data are used for identification and the remaining 50 sec of data are used for validation. The solid line represents the nonlinear simulation output and the dashed line represents the predicted output.
As it can be seen in Fig. 2, the algorithm could have done the identification of a severely nonlinear system in a satisfactory manner.
CONCLUSION
In this study, we proposed a new modeling approach for nonlinear systems based on modal series. New model contains linear and Hammerstein subsystems which are common in their linear blocks and the structure of nonlinear blocks are distinct. A MIMO nonlinear system identification algorithm based on subspace identification, Hammerstein modeling and proposed model has been presented. The proposed model and identification algorithm can find many applications in the context of nonlinear system identification and control.
ACKNOWLEDGMENT
This study is a side product of the project financially supported by Islamic Azad University, Kazerun Branch, Iran.