APPLICATION OF STATE ESTIMATION AND NEURAL NETWORKS IN
NONLINEAR CONTROL
A. Jadlovská
* Department of Cybernetics and Artificial Intelligence
Faculty of Electrical Engineering and Informatics
Technical University of Košice
Letná 9, 041 20 Košice
Slovak Republic
fax : +421-95-62 535 74 and e-mail : Anna.Jadlovska@tuke.sk
Abstract: This paper considers the use of neural networks for non-linear state estimation,
identification and control of non-linear processes. The non-linear identification is using
feed-forward neural networks as useful mathematical tool to build model between the
input and output of a non-linear process. In this paper is considered the possibility an
on-line state estimation of the actual parameters from off-line trained neural state space
model of the non-linear process using the gain matrix. This linearization technique is
used in the algorithm on-line tuning of the controller parameters based on pole
placement control design for non-linear process.
Keywords: Dynamic neural models, Non-linear State Estimation, Gain Matrix, Nonlinear Control
1 INTRODUCTION
The purpose of this paper is to show how feedforward neural network (Multi Layer Perceptron –
MLP) can be used for modeling and control of the
non-linear processes. When the mathematical model
of the process cannot be derived with an analytical
method, then the only way is using the relationship
between the input and output of the process. Fitting
the model from the data is known as an identification
of the process. For linear processes this technique is
generally well known, (Ljung 1987).
For processes, which are complex or difficult to
model the non-linear identification can use feedforward neural network – MLP as useful
mathematical tool to build the non-parametric model
between the input and the output of the real nonlinear process, (Chen et al. 1990), (Narendra et al.
1990), (Jadlovská 2000), (Jadlovská (a) 2002). We
will consider in this paper the possibility an on-line
state estimation of the actual parameters from off-line
trained neural state space model of the non-linear
process using the gain matrix, introduced later,
(Najvárek 1996).
This linearization technique is used to perform an online tuning of the controller parameters based on pole
placement control design in observer-based control
loop, (Jadlovská (b) 2002). The advantage using
sample-by-sample linearization is that the controller
parameters can be changed in response to process
changes. In fact neural network method also becomes
adaptive, if training of the neural network state space
model as observer is continued on-line.
2 MODELING OF NONLINEAR PROCESS
In this part we will discuss some basic aspects of nonlinear system identification using among numerous
neural networks structures only Multi-Layer
Perceptron–MLP (a feed-forward neural network)
with respect to model based neural control, where the
control law is based upon the neural model.
(tanh) neuron functions. The „1“ shown in Fig.1.
together with the last column in W1 giving the offset
in the network. The net input is represented by the
vector Zin and the net output is represented by the
vector Ẑout . The mismatch between the desired
We know, that a discrete-time non-linear system can
be written on the following general form:
The output from MLP can be written as:
output Z out and Ẑout is the prediction error E.
xk   f  xk  1, uk  1, d k  1
yk   Hx k 
Zˆ out
(1)
in which k is the discrete sample number, x is
a state vector, u is the input vector and y is the
output, d is a vector of disturbances. We will
furthermore include a vector θ containing some set
of parameters sufficient to describe the model at time
k in the description. f and h are given functions
which may be nonlinear, time-varying and multivariable.



Z 
 W2   W1  in 
1 










(3)
From a trained MLP (by Back- Propagation
Algorithm –first-order gradient method, GaussNewton algorithm – second-order gradient method)
which has m0 inputs and m2 outputs can be found
a m2  m0 gain matrix N by differentiating with
respect to the input vector of the network.
We wish to identify a nonlinear (neural network)
mapping F
The gain matrix N can be calculated from (3)
ˆ k   F X
ˆ k  1,U k  1, Dk  1, E k  1
X
N


ˆ k 
Yˆ k   Hx
(2)
E k   Y k   Yˆ k 
based on samples of the system inputs and outputs
uk , yk kN0 (the training set) such that the
prediction error E defined by equations (2) becomes
small. F is chosen to be a Multi-Layer Perceptron
(MLP). We assume, that the output mapping to be
fixed, H  I 0 , where I and 0 are identity and
zero matrices of appropriate dimensions.
We will use in this paper feed-forward neural
network MLP with a single hidden layer. This
structure is shown in matrix notation in Fig.1,
(Najvárek 1996).
Zout
Zin
Y0
W1
1
X1

Y1
1
W2
X2 Ẑ out
+E
-
Fig.1. Matrix block diagram of an MLP
The matrix W1 represents the input weights; the
matrix W2 represents the output weights, 
represents vector function containing the non-linear
d Zˆ out d Zˆ out d Y1 d X 1

 W 2   X 1  W1 (4)
d Z inT
d Y1T d X 1T d Z inT
where W1  W1
(excl. last column).
Above mentioned the gain matrix N allows an online estimation of the actual model parameters from
off-line trained neural model - MLP of the non-linear
process.
We know from linear identification, that for a linear
process with m inputs, n states and p outputs the
following model is often used
xˆ k  Φ xˆ k  1  Γ uk  1  Ke k  1
y k   H xˆ k   e k 
(5)
where x̂ is the state vector (order n), u is the input
vector (order m), y is the output vector (order p)
and e is the prediction error (order p). For the linear
case Φ , Γ , K and H are constant matrices of
dimension
n  n ,n  m ,n  p
and
pn
respectively. It is assumed that x̂ is not measurable
(incoplete stste information) and the model is named
the Innovation State Space model. The matrix H is
choosen to H  I p , p O p ,n p , where I p , p is p  p


unity matrix and O p ,n p is a p  n  p  zero matrix.
This means, that the first p elements of the state
vector x̂ is filled with ŷ .
With the inspiration from this linear state space
model (5) in our paper we will applicate idea using
dynamic neural network model, where no all inputs
and outputs from the neyworks are measurable. This
type of the model is well-known Nonlinear
Innovation State Space model (NISS), (Norgaard
1997), (Jadlovská 2002).
Y(k)
q-1
X̂ k-1
U(k – 1) Neural
network
q
-1
X̂ ( k )
H
Ŷ ( k )
E(k – 1)
+ E(k)
-
The non-linear Innovation State Space model
(Kalman Predictor) can be defined by equations (5)

ˆ k   F X
ˆ k  1,U k  1, E k  1,θ
X

ˆ k 
Yˆ k   HX
Fig.2. Non-linear State Space Model (NISS)
(6)
Y k   Yˆ k   E k 
where F is non-linear vector function, θ represents
vector of the parameters and E is the prediction
error (order p), U is input vector (order m), X is
state vector (order n) and Y is the output vector
(order p).
The neural NISS model with the input vector Zin k 
and output vector Z out k 
 Xˆ k  1


Z in k   U k  1 
 E k  1 


ˆ k 
Z out k   X
(7)
is shown in Fig.2, which is a recurrent network
After training neural network MLP can be on-line
estimated the actual gain matrix N k  , which is
calculated by (4) and for NISS model by (8):
N k  
The model of non-linear process and the training
method has been considered generally for the
multivariable case in part 2. Next we will think about
control design for non-linear MISO process using
neural state space model NISS (6).
A trained neural NISS model representing a model of
the non-linear process we use for an on-line state
estimation actual process parameters by gain matrix
N k  , (Najvárek 1996), (Norgaard 1997). This
linearization technique allows an on-line tuning of the
controller parameters using pole-placement control
strategy, which is well known from the linear control
theory, (Åström et al. 1990). In this paper the pole
placement method as control concept is formulated in
a way, which allows the neural model found in
section 2 to be used as state observer. An example of
the control structure using estimation process
parameters from NISS model is illustrated on Fig.3.
y(k + 1)
Proces
ˆ k 
ˆ k 
dX
dX
out


T
T
T
ˆ k  1U k  1 E T k  1
d Z in k  d X


ˆ k 
ˆ k 
ˆ k  
 X
X
X


T
T
ˆT
  X k  1  U k  1  E k  1 

3 NONLINEAR CONTROL USING STATE
ESTIMATION

ˆ k  , Γˆ k  , K
ˆ k 
 Φ
Observer
-L
(8)
For a non-linear process the actual linearized
ˆ k , Γˆ k , K
ˆ k  are not constant
parameters Φ
matrices, but depend on the actual values of the input,
state and output vectors.
Because neural state space NISS model on Fig.2
contains feedback loops around MLP we will apply
for training this recurrent network a second order
Recursive Prediction Error Method (RPEM) using
Gauss-Newton search direction, (Ljung 1987), (Chen
et al. 1990), (Jadlovská (b) 2002).
x̂ k 
ˆ k  Γˆ k 
Φ
Place poles
Fig.3. The principle of pole placement control using a
state observer
The aim is to design the feedback gain L in a way so
that the closed loop has a set of prescribed
eigenvalues. Because the linearization model will
change from sample to sample, it is necessary to
recalculate L accordingly. If we have trained a
simulation model – observer network NISS we can
design the control law. Since full state information is
not available, the control law is chosen as a linear
feedback from the state observer by equation (9).
ˆ k 
uk    Lk x
(9)
where L is state feedback matrix which places the
eigenvalues of the closed loop system.
4 SIMULATION RESULTS
The idea and results of the estimation process
parameters from an off-line trained neural NISS
model as an observer and its using for tuning
parameters of controller designed by pole-placement
strategy (non-linear system control) are presented for
non-linear test process – two tanks system, (Jadlovská
2001).
We consider NISS model with 6 inputs and 8 neurons
in the hidden layer. The activation function in the
hidden layer is „tanh“ function and in the output
layer is selected a linear function. The actual gain
matrix N k  can be calculated by (4) and actual
values of estimated parameters can be obtained from
N k  by (8).
Presentation of results of non-linear control designed
by pole placement method using on-line parameter
estimation from an off-line trained neural state model
is illustrated in Fig. 4 at the change of one parameter
of the non-linear process (10).
y(t)
0.6
0.4
The mathematical model of this system can be
described as
1
h1 ( t )   q1,1 ( t )  q2 ,1 ( t )
A
1
h2 ( t )   q1,2 ( t )  q2 ,2 ( t )
A
0.2
0
0
500
1000
1500
2000
2500
3000
3500
(10)
0.6
0.5
u1(t)
0.4
where
0.3
h1( t ) , h2 ( t ) [m] denotes water levels in
two tanks, which are in interaction,
0.2
0.1
u2(t)
0
0
500
1000
1500
2000
2500
3000
3500
q1 ,1 ( t ) , q1,2 ( t ) [m3 s 1 ] denotes input flows
of the tanks,
q2 ,1 ( t ) , q2 ,2 ( t ) [m3 s 1 ] denotes outflows of
Fig.4 State controller based on actual parameter
estimation from neural NISS model
the tanks.
q2 ,1 ( t )  q1,2 ( t )   .z1 ( t ). 2.g .h1 ( t ) ,
q2 ,2 ( t )   .z2 ( t ). 2.g .h2 ( t ) ,
where
z1( t ) and z2 ( t ) [m] are the rises of output
outlets of two tanks.
The other parameters of two tanks are:
the cross section area of the tanks A  1m 2 ,
the parameter of the output outlet of the
tank   0.5m , the constant of gravity
g  9 ,81 ms2 .
The inputs of the system are:
the input flow of the first tank u1 t   q1,1 t ,
the rise of output outlet of the second tank
u2 t   z2 t  ,
the rise of output outlet of the first tank
z1 t  , d t   z1 t .
The output of the system is water level in the
second tank yt   h2 t .
Simulation study shows that the non-linear observerbased control loop with the desired closed-loop poles
giving a critically damped response of the output
tracking.
This example shows real power of the neural
modeling using the structure NISS and the possibility
to apply the method pole-placement known from the
linear control theory for control of non-linear MISO
process.
5 CONCLUSION
In this paper is trained neural NISS model as Kalman
predictor for non-linear MISO process. After training
this NISS model can be used in closed control loop to
an on-line state estimation of the process parameters,
which allow tuning of the controller parameters by
pole-placement method. A practical simulations by
language Matlab/Simulink, Neural Toolbox and
NNSID Toolbox ilustrate, that this control strategy
using linearization technique by the gain matrix from
neural state model produces excelent performance for
control of non-linear MISO process. But this
controller design can be applied for only non-linear
process, which does not contain hard nelinearities.
ACKNOWLEDGMENTS
The work has been supported by the grant agency
VEGA 1/9032/02. This support is very gratefully
acknowledged.
REFERENCES
Åström, K., J. and B. Wittenmark (1990). Computer
Controlled Systems, Theory and design, PrenticeHall, second edition
Chen, S., S. Billings and P. Grant (1990). System
Identification using Neural Networks. In:
International Journal of Control, Vol.51, No.6,
pp. 1191-1214
Jadlovská A. (2000). An Optimal Tracking NeuroController for Nonlinear Dynamic Systems, In:
Control System Design, A Proceedings volume
from IFAC Conference Bratislava, Slovak
Republic, 18-20 June, Published for IFAC by
Pergamon – an Imprint of Elsiever Science,
pp.493-499, ISBN 00-08 043 546 7
Jadlovská, A. (a) (2002). Neural networks Based
Predictive Control of Non-linear System, Acta
Electrotechnica et Informatica, No.4, Vol.2,
Košice, pp.35-38, Slovak Republic, ISSN 13358243
Jadlovská, A. (b) (2002). Non-linear Control Using
Parameter Estimation from Forward Neural
Model, Journal of Electrical Engineering –
Elektrotechnický časopis, No.11-12, Vol.53,
pp.324-327, Slovak Centre of IEE, FEI STU,
Bratislava, ISSN 1335-3632
Leontaritis I., J. (1985). Input-output parametric
models for non-linear systems, part 1 and 2., In:
International Journal of Control, Vol.41, No.2,
pp.303-344
Ljung, L. (1987). System Identification: Theory for
the User, (T. Kailath, Ed.). Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey
Najvárek, J. (1996). Matlab and neural networks,
FEI VUT, Brno
Narendra, K.S. and A.U. Levin (1990). Identification
and control of dynamical systems using neural
networks, In: IEEE Transaction on Neural
Networks, Vol.1, No.1, pp. 4-27
Norgaard, M.(1997). NNSID Toolbox, Version 1.1,
Technical Report, Department of Automation,
DTU