Robust digital design of continuous-time nonlinear control systems using adaptive
prediction and random-local-optimal NARMAX model
Zhi-Ren Tsai
Department of Computer Science & Information Engineering, Asia University, Taiwan
Graduate Institute of Biostatistics, China Medical University, Taiwan
ren@asia.edu.tw
Abstract
In this paper the time-delay and uncertainty of continuous-time (CT) systems are
considered, and it is suggested that input and output of a discrete-time (DT) Neural Plant
Model (NPM) and recursive neural controller have scaling factors which limit the value
zone of measured data from a system. Adapted scaling factors cause the tuned parameters
to converge to obtain a robust control performance. However, the proposed Random-LocalOptimization (RLO) design for a model/controller uses off-line initialization to obtain a
near global optimal model/controller. Other important issues are the considerations of cost,
greater flexibility, and highly reliable digital products for these control problems. This issue
of DT control design for CT plant is more difficult than that of CT control design for CT
plant, because of the need to process the modeling error between the CT plant and DT model.
The input-delay, uncertainty, and sampling distortion of a CT nonlinear power system need
to be solved by developing a digital model-based controller. Here, this is called the DT
tracking control design of CT systems (DT-CT).
Therefore, the DT structure of the adaptive controller for the CT nonlinear power
system should be designed as a kind of feed-forward-Recursive-Predictive controller (FRP).
First, due to the problem of delays, a digital neural controller with feed-forward of the
1
reference signal and a Nonlinear Auto-Regressive Moving Average eXogenous (NARMAX)
neural model design is adopted to reduce this difficulty. The most important contribution
is that the more reasonable and systematic two-stage control design, the CT nonlinear
delayed system to be controlled is modeled using a NARMAX technique with the first-stage
(off-line) method by the proposed global optimal network algorithm and second-stage (online) adaptive steps. Second, the dynamic response of the system is controlled by an
adaptive NARMAX neural controller via a sensitivity function. A theorizing method is then
proposed to replace the sensitivity calculation, which reduces the calculation of Jacobin
matrices of the BP method. Finally, the feed-forward input of reference signals helps the
digital neural controller to improve the control performance, and the technique works to
control the CT systems precisely.
Keywords:
Random-local-optimization
algorithm,
controller, DT-CT design
2
NARMAX
model-based
neural
1. Introduction
During the past decade optimal control [1, 2] has attracted great attention from both the
academic and industrial communities, and there have been many successful applications.
Despite this success, it has become evident that many basic and important issues [3] remain
to be further addressed. Of these, stability analysis and systematic designs are among the
most important issues for optimal control systems [4] and robust control theories [1, 5-8],
and there has been significant research on these issues (see [4, 9, 10]). In addition, a neural
controller has been suggested as an alternative approach to conventional PID control
techniques [11] for complex control systems [1]. Moreover, Neural-Network (NN) based
modeling has become an active research field because of its unique merits in solving
complex nonlinear system identification and control problems [10]. Neural networks (NNs)
or NARMAX neural networks [12] are composed of simple elements operating in parallel,
inspired by biological nervous systems. A neural network can be trained to represent a
particular function by adjusting the weights between elements. Due to discrete-time (DT)
controllers being cheaper and more flexible than continuous-time (CT) controllers, the DT
control problem for CT plant is worth studying. In modern control engineering, controllers
are commonly implemented directly by the hardware or software of digital computers.
However, one important issue has to be faced; that is, the new design (DT-CT design)
problem effects a new type of application, and an adaptive NN-model-based design method
has not yet been developed to adjust the parameters of a discrete-time (DT) adaptive neural
controller such that the original continuous-time (CT) system, with time delays and
uncertainties, is uniformly ultimately bounded (UUB) stable.
The study of CT control of CT time-delay systems has received considerable attention
in recent years since delay is a major cause of poor performance in many important
3
engineering systems [13-15]. Hence, the future direction of CT time-delay control systems
needs to involve the DT control problem. The amount of delay has different impacts on the
various approaches [15-19]. As is known, the delay control problem is an important and
complex factor in the stability performance of CT nonlinear systems. In general, a delay
signal happens in a signal’s long-distance or heat translation.
Based on the timer of the micro-controller or Digital Signal Process (DSP) chip, the effect
of delay in neural system identification can be approximated by many tape-delay terms.
This reduces the difficulty of delay identification. The DT NARMAX model is general
sufficient to approximate an unknown, nonlinear, dynamical and delayed CT system by
selecting an appropriate sampling time.
DT control design for DT plant [20] and CT control design for CT plant are two kinds
of well-known problems. [20] has inspired consideration of the more difficult problem of
DT control design for CT plant, because of the need to process the modeling error between
the CT plant and DT model, except for proposing the novel adaptive control law. The
modeling and controlling performance can be guaranteed by the appropriate sampler and
some theories. The modeling performance of plant is important for this research. The
stability of results and the robust control design are related to this precise plant model.
Hence, a two-stage training scheme is needed to guarantee a well-behaved model, referred
to as a predictive controller. Based on this correct model, the control parameters can be
updated by the BP method. That is one contribution of this paper. Another contribution is
the proposal of a theorizing method to replace the sensitivity calculation or reduce the
calculation of Jacobin matrices of the BP method. That is why the adaptive prediction control
method is used to improve the DT control performance of the proposed DT-CT design by
tuning the parameters of the model and controller.
4
The feed-forward term in [20] is derived indirectly by assuming many constraints, and
due to the over-fitting and local optimal problems of NN modeling, the method [20] is not
suitable for on-line applications because of the need for a lengthy convergence time.
Therefore, to satisfy the on-line working requirements for accurate modeling of the plant,
the NARMAX plant and control models are trained by initially using off-line methods.
On the other hand, these neural techniques [21-23] have usually been demonstrated
under nonlinear control due to their powerful nonlinear modeling capability [24] and
adaptability. However, they must exhibit the optimal problem of falling into the local
minimum easily by using the Back-Propagation (BP) or Levenberg-Marquardt BP algorithm
(LMBP) [25] method. Hence, the RLO algorithm is proposed to improve this drawback. It
not only guarantees the gradient decent method [26] against the local optimal solution, but
also speeds up the convergence of the Particle Swarm Optimization (PSO) [31, 32].
Inspired by the DT neural controller of [20] only for a DT system, a digital neural control
design for a CT system is proposed and an approximate inverse of the delayed plant
dynamics is used to act as the NARMAX neural controller. The adaptive controller and
NARMAX models are easier to converge than [21-23] by the proposed two-stage scheme.
Moreover, the modeling error between the model and physical system is considered in the
theorems by Lyapunov functions [27]. This paper concludes with a simulation example and
experimental data to demonstrate these techniques.
2. System description
First, consider a general nonlinear system with delays; described as follows:
P:
x (t )  fCT ( x, u, t , t   , )
and
y (t )  g ( x) ,
where P shown in Fig. 1 is a controlled plant; the bounded uncertainties
5
(1)
 (t ) create the
dynamic quality of the system parameters which refer to electrical elements of the power
system; the control input u (t ) ;  is the time delay; g () is the relational function of the
state x(t ) and system output y (t ) . Then, f CT is discretized by setting the appropriate
sampling time or sampling period Ts (sec) of DTC-CTP design, and t  k  Ts , where k is
a positive integer, to the following DT nonlinear system f DT :
x(( k  1)  Ts )  f D T ( x(k  Ts ), u (k  Ts ), Ts ), y(k  Ts )  g ( x(k  Ts ))
or
x(k  1)  f D T ( x(k ), u (k ), Ts ), y (k )  g ( x(k )) ,
(2)
where k indicates the signal sequence, the DT state vector is x(k  Ts ) , and the DT control
input is u (k  Ts ) . The zero-order-hold control input u (t )  u (k  Ts )  u (k ) , where k is also
the index of the discrete result u (k ) of u (t ) referring to the NN model P̂ (see Fig.1) of
(1).
In this control structure in Fig. 1, the NN plant model is designed to approximate this
nonlinear system Eq.(1). This NN plant model, or control model is built following the
subsequent mathematical equations.
An NN plant model or control model with L layers each having N l ( r  1l ,2l ,..., N l ;
l  1,2,..., L ) neurons is established to approximate a DT nonlinear system
f DT Eq.(2).
Superscripts are used to distinguish between these layers. Specifically, the number of the
layer is appended as a superscript to the name of the variable. Thus, the weight matrix for
the l -th ( l  1,2,..., L ) layer is written as W l and {WPl ,WCl }  W l . Moreover, it is assumed
that vr ( r  1l ,2l ,..., N l ) is the net input and that all the transfer functions
Tr (vr ) of units
in the NN system are described by the following function:


2
Tr (vr )    
 1,
 1  exp( vr / q) 
for
6
r  1l ,2l ,..., N l
and l  L ;
Tr (vr )  vr ,
where q
and
r  1L ,2 L ,..., N L ;
for
 are positive parameters associated with the sigmoid function. The
transfer function vector of the l -th layer is defined as:
 l (vr )  [Tr (v1 ), Tr (v2 ),..., Tr (vN )]T , l  1,2,..., L ,
l
l
l
where Tr (vr ) ( r  1l ,2l ,..., N l ) is a transfer function of the r -th neuron. The final outputs of
the NN plant model P̂
and control model C F can then be inferred as follows:
respectively:
yˆ (k )   L (W L L 1 (W L 1 L  2 (... 2 (W 2 1 (W 1Z (Ts )))...)))
 Pˆ ( y(k  1), y(k  2),..., y(k  n), u(k ), u(k  1),..., u(k  p), WP (k ), Ts ) , and
u F  C F (u (k  1), u (k  2),..., u (k  cu ), r (k ), r (k  1),..., r (k  c y ), WC (k ), Ts ) ,
where r (k ) is a reference input,
Z T (Ts )  [ g ( x(( k  1)  Ts )), g ( x(( k  2)  Ts )),..., u (k  Ts ), u (( k  1)  Ts ),...,1] ,
the adaptive parameter WP (k )  [WP1 ,WP2 ,...] or
WC (k )  [WC1 ,WC2 ,...] of neural weights’ and
biases’ refers to the iteration k and the proposed adaptive laws for the controller and plant
models are as follows:
WC (k  1)  WC (k )  WC (k ) ,
and WP (k  1)  WP (k )  WP (k ) .
Although WP (k ) and WC (k ) are the proposed adaptive laws of plant model and
control model, respectively, where:
W P (k )   P  ( yˆ (k )  y (k ))
but implementing
dyˆ (k )
,
dW P (k )
WC (k ) T   C  ( yˆ (k )  r (k ))
dyˆ (k )
,
dWC (k )
dyˆ (k )
needs too many Jacobin matrices’ calculations, so the following
dWC (k )
T
adaptive prediction control law is used WC (k )   X u X
7
du (k )
to replace the above
dWC (k )
adaptive laws to reduce computing time,
where  P , C , X are learning rates; u X  C X (e(k ))  [ K 1 (e1 (k )), K 2 (e2 (k )), K 3 (e3 (k )),...] T
is the predictor output, where the tracking error is e(k )  r (k )  y (k ) , and
e(k )  [e1 (k ), e2 (k ), e3 (k ),...]T , K1 (), K 2 (), K 3 (),... are defined by the user.
A composite controller, u(k )  u F  s  u X , where s  {0,1} is proposed in the next section.
3. Control architecture, neural-model-based controller design and control scheme
3.1 Adaptive digital neural controller design through neural plant model
In this paper an adaptive prediction control structure is proposed, as shown in Fig. 1,
where the FRP controller CF is designed as follows:
u(k )  u( z (k ))  C F ( z (k ), WC (k ), Ts )  s  C X (e(k ))  u F  s  u X ,
0,
where e(k )  r (k )  y (k ) , the switch index s  
1,
~
e (k  1)  0,
And,
~
e (k  1)  0.
u( zˆ(k ))  C F ( zˆ(k ), WC (k ), Ts ),
where
uP  uF  u X
,
(3)
yˆ 1 (k  1)  Pˆ (u P (k  1))
eˆ1 (k  1)  r (k  1)  yˆ1 (k  1) , eˆ2 (k  1)  r (k  1)  yˆ 2 (k  1) ,
(4)
,
yˆ 2 (k  1)  Pˆ (u F (k  1))
,
eˆ1 (k  1)  eˆ2 (k  1)  ~
e (k  1) ,
as shown in Fig. 1. The feed-forward terms are reference signals [r (k ), r (k  1),..., r (k  p)] ,
and recursive terms are control signals [u (k  1), u (k  2),..., u (k  q)] . The off-line training
input of controller is:
zˆ(k )  [ y (k ), y (k  1),..., y (k  p), u (k  1), u (k  2),..., u (k  q)] ,
The on-line recursive input of controller is:
z (k )  [r (k ), r (k  1),..., r (k  p), u (k  1), u (k  2),..., u (k  q)] . The controller has two working
phases: z (k ) is the data vector of the testing phase, and zˆ ( k ) is the data vector of the training
8
phase. The tuned parameter vector of the controller is: WC (k ) of (3).
The proposed on-line digital neural controller
[r (k ), r (k  1),..., r (k  p)] and recursive structure
u (k )
has feed-forward terms
[u (k  1), u (k  2),..., u (k  q)] . Hence, it
uses a NARMAX neural model or inverse of the plant dynamics to aid control precision in
the face of a delayed plant with uncertainties. Adapting the neural controller can suppress
the uncertainty of the plant P shown in Fig. 1. Although the structure of the neural controller
is chosen as (3), the neural controller has not been designed because the parameter vector
WC (k ) is not specified.   Ts is the chosen tape-delay time,  is a positive integer. The idea
of the inverse-model-based neural controller is proposed by the following simplified
relation:
If y (k )  Pˆ (u (k )) , u (k )  Pˆ 1 (r (k ))  C F (r (k )) , then y (k )  r (k ) ,
(5)
where Pˆ () is the adaptive NARMAX neural model of plant; CF () is the adaptive
NARMAX neural controller; r (k ) is the desired output. According to the idea of Eq.(5), the
recursive structure Pˆ () can be designed with tape delays as follows:
y(k )  yˆ (k )  Pˆ ( yˆ (k  1), yˆ (k  2),..., yˆ (k  n), u(k ), u(k  1),..., u(k  p),WP (k ), Ts ) ,
(6)
ˆ u , respectively.
where n, p  1 are the amount of tape delays of y,
But, due to the parameters of the recursive structure are converged much harder, the
weights and biases WP (k ) of this model are trained by the feed-forward structure as
follows:
yˆ (k )  Pˆ ( y(k  1), y(k  2),..., y(k  n), u(k ), u(k  1),..., u(k  p),WP (k ), Ts ) .
(7)
The plant output is compared with the desired output to create a tracking error signal
e(k )  r (k )  y (k ) . The system errors eˆ(k )  r (k )  yˆ (k ) and e(k )
are used by the
adaptation algorithm to update the parameters of P̂ and CF . Next, the performance index
9
for minimizing the tracking error is designed, as follows:
1
1
1
J (k )  e(k )T e(k )  (r (k )  y (k )) T (r (k )  y (k ))  ( y (k )  r (k )) T ( y (k )  r (k )) ,
2
2
2
(8)
is a simple cost function to be minimized by the proposed algorithm. Then, the on-line BP
algorithm adapts the control parameter matrix WC (k ) . That is, the change in control
parameters WC (k ) is calculated as
dJ (k )
d ( y (k )  r (k ))T ( y (k )  r (k ))
WC (k )  C (k )
 C (k )
dWC (k )
2  dWC (k )
T
  C (k )( y (k )  r (k ))
dy (k )
dy (k ) du (k )
,
  C (k )( y (k )  r (k ))
dWC (k )
du (k ) dWC (k )
(9)
where the small positive C (k ) can be selected as a stable learning rate via the following
theorems.
Theorem 1: If the number of neurons and tape-delay terms of the neural model is sufficient,
and the appropriate sampling time Ts is selected to let
y (k )  y(k )   and the following
condition
0   P (k ) 
2
dyˆ (k )
dWP (k )
2
 P ,
(10)
is satisfied, where
py
pu
dyˆ (k )
yˆ (k )
yˆ (k ) du (k  i )
yˆ (k ) dyˆ (k  i )



;
dWP (k ) WP (k ) i 0 u (k  i ) dWP (k ) i 1 yˆ (k  i ) dWP (k )
y (k ) is the output of the optimal model, then the trajectories yˆ (k ) converging to plant
output y (k ) is a uniformly ultimately bounded (UUB) approximation on the bounded
error yˆ (k )  y (k ) .
10
3.2 Proof of Theorem 1
First, consider the following ideal Lyapunov candidate [27] for the model part,
1
1
V1 (k )  ( yˆ (k )  y (k )) T ( yˆ (k )  y (k ))  yˆ (k )  y (k )  y (k )  y (k )
2
2

where
V2 (k ) 
1
yˆ (k )  y (k )
2
2
2
1
2
yˆ (k )  y (k )   (k )  V2 (k )   (k ) ,
2
(11)
is an actual Lyapunov candidate of reachable and
assumptive trajectory y (k ) , the bounded approximation error
 (k ) 
1
2
y (k )  y (k )  ( yˆ (k )  y (k )) T ( y (k )  y (k )) ,
2
and the number of neurons of the neural model is sufficient and the appropriate sampling
time Ts is selected to let y (k )  y (k ) . The next task is to train this neural model such that
V2 (k ) is minimized,
WP (k )
dyˆ (k )
 ( yˆ (k )  y (k ))
 P (k )
dWP (k )

dV2 (k )
d ( yˆ (k )  y (k ))
dyˆ (k )
 ( yˆ (k )  y (k ))
 ( yˆ (k )  y (k ))
.
dWP (k )
dWP (k )
dWP (k )
(12)
Then, the following Lyapunov candidate for the controller is designed:
V3 (k ) 
1
1
2
( yˆ (k )  r (k ))T ( yˆ (k )  r (k ))  yˆ (k )  r (k ) ,
2
2
(13)
thus the change in the Lyapunov function is obtained by:
V3 (k  1)  V3 (k ) 
1
2
2
( yˆ (k  1)  r (k  1)  yˆ (k )  r (k ) ) .
2
(14)
Finally, the update law of the control parameters of the controller is obtained as follows:
WC (k )T
dV (k )
dyˆ (k )
 3
 ( yˆ (k )  r (k ))
.
C (k )
dWC (k )
dWC (k )
11
(15)
This study develops some convergence theorems to select appropriate stable learning rates.
First, the difference of modeling error eP (k )  yˆ (k )  y(k ) can be represented by
T
  de (k ) T
 deP (k )  
deP (k ) 
de (k ) 
eP (k  1)  eP (k )  
 P (k )eP (k )
 eP (k )1   P
 P (k ) P




  dWP (k ) 
dWP (k ) 
dWP (k ) 
 dWP (k )  


  dyˆ (k ) T
dyˆ (k ) 
 eP (k )1  

(
k
)
,
  dWP (k )  P
dWP (k ) 


(16)
thus the change in the Lyapunov function is obtained by:
2


  dyˆ (k )  T

ˆ
1
1
d
y
(
k
)
2
2
2


V2 (k  1)  V2 (k )  ( eP (k  1)  eP (k ) )   eP (k ) 1  
 (k )
 eP (k ) 
  dWP (k )  P dWP (k ) 
2
2





2
T





ˆ
ˆ
1
d
y
(
k
)
d
y
(
k
)
2 


 eP ( k ) 1  
 (k )
 1 .
  dWP (k )  P

2
dWP (k ) 



Hence,
if
  dyˆ (k )  T
dyˆ (k ) 
 1  1  
 P (k )
1

  dWP (k ) 
dWP (k ) 


and
y (k )  y (k )  
,
then
V2 (k  1)  V2 (k ) , that is V2 (k )  0 or yˆ (k )  y(k ) , makes the UUB approximation of this
model on the bounded yˆ (k )  y (k ) . The proof is thereby completed.
Furthermore, the following theorem for the convergence of the controller is obtained by the
same procedure as the above proof.
Theorem 2: If Theorem 1 in Eq.(10) is satisfied, the function
to let the following condition,
12
dyˆ (k )
in Eq.(15) is computed
dWC (k )
0  C (k ) 
2
dyˆ (k )
dWC (k )
2
 C ,
(17)
be satisfied.
Where
with
py
pu
dyˆ (k )
yˆ (k ) du (k  i )
yˆ (k ) dyˆ (k  i )
,


dWC (k ) i 0 u (k  i ) dWC (k ) i 1 yˆ (k  i ) dWC (k )
cu
du (k )
u (k )
u (k ) du (k  i)


,
dWC (k ) WC (k ) i1 u (k  i ) dWC (k )
then the nonlinear systems (1) in Fig. 1c are UUB stable, and the tracking errors
e(k )  r (k )  y (k ) are bounded via the controller.
Hence, the dynamic response of the system P can be controlled using CF , as shown
in Fig. 1. This CF needs the plant model P̂ to adjust control parameters via sensitivity
function
yˆ (k )
.
u (k  i )
The digital feedback controller includes a delay block D, as shown in Fig. 1. Here, the
error e~ (k  1) is used to estimate
u X , and the proposed predictor of the delayed system
can let us cancel some complex computations, such as
yˆ (k  1) yˆ (k  1)  yˆ (k ) yˆ (k  1) yˆ (k  1)



,
u (k  1)
u (k  1)
u (k  1)
uX
of sensitivity function
yˆ (k )
in the BP algorithm. Hence, the following theorem is
u (k  i )
proposed to update the control parameters of FRP under the assumption of providing a
model which applies a lower prediction error, and a more correct
eˆ(k  1)  r (k  1)  yˆ (k  1)
u X . The prediction error
is bounded, due to the previous eˆ(k )  r (k )  yˆ (k )
being
bounded at any time. Hence, the prediction error eˆ( k  1) will be bounded by using
Theorem 1-2. Furthermore, the following theorem is obtained for the convergence of the
13
adaptive prediction controller by the same procedure as Theorem 1.
Theorem 3: If Theorem 1 in Eq.(10) is satisfied, the predictive function
du (k  1)
is
dWC (k  1)
computed to let the following condition,
0   X (k  1) 
2
du (k  1)
dWC (k  1)
2
X ,
(18)
be satisfied, then the nonlinear systems (1) in Fig. 1a-1b are UUB stable, and the tracking
errors
e( k )  r ( k )  y ( k )
are
bounded
via
the
predictive
controller
u(k  1)  uF (k  1)  s  u X .
The tracking error is e(k )  r (k )  y (k ) , and e(k )  [e1 (k ), e2 (k ), e3 (k ),...]T , but the parameters
of adaptive control C F are updated by using the predictive offset
u(k  1)  u p  u(k )  uX  CX (e(k ))  [ K1 (e1 (k )), K2 (e2 (k )), K3 (e3 (k )),...]T ,
of the prediction control input u p  u (k  1) , where
replaces
yˆ (k  1) yˆ (k  1)  yˆ (k ) yˆ (k  1)


u(k  1)
u p  u (k )
uX
dyˆ (k )
in the following recursive equation:
du (k )
py
pu
dyˆ (k )
dyˆ (k ) du (k )
yˆ (k ) du (k  i )
yˆ (k ) dyˆ (k  i )



,
dWC (k ) du (k ) dWC (k ) i 0 u (k  i ) dWC (k ) i 1 yˆ (k  i ) dWC (k )
du ( k )
or
dWC ( k )
therefore,
only
du ( k  1)
need to be calculated to update WC (k ) or WC (k  1) , respectively,
dWC ( k  1)
and K1 (), K 2 (), K 3 (),... are defined by the user.
3.3. Two-stage scheme
Fig. 1 shows a block-diagram of an adaptive recursive control system. The system to be
14
controlled is labeled as the plant P , which is subject to uncertainties and delays. Due to
gradient-descent based training algorithms, let the model/controller converge to a local
minimum in the solution space. Hence, the two-stage training algorithm is proposed, as
follows.
In the first stage, the measured data is used to train the global optimal NARMAX plant
and neural controller by the training-data-shuffle method. This method shuffles the training
data to avoid most of the local optimal solutions obtained by the off-line training procedure
in next section. The measured data is divided into a training data and other testing data.
This testing data is not used for training the NN. However, the final performance of the NN
is decided by the testing data and the training data.
In the second stage, the global optimal NARMAX plant model and neural controller is
adapted. The two stages are divided into the following five steps:
Step 1: First, the reference signal, r (k ) , is designed. By the white noise of input u (k ) for
plant, output data y (k ) is collected and a training-data-shuffle method is used to
shuffle the input/output pairs’ data. These shuffled data are ready to train the
NARMAX model/controller. Here, the following reasonable conditions need to be
taken into account:
max (r (k ))  max ( y (k )) , min (r (k ))  min ( y (k )) , max (u (k ))  uU , and min (u (k ))  u L ,(19)
k
k
k
k
k
k
need be satisfied, where uU is the upper bound of u (k ) , and u L is the lower bound
of u (k ) . According to Eq.(19), much of the excessive control effort u (k ) can be
avoided. If Eq.(19) is satisfied, then go to Step 2.
Step 2: The feed-forward structure model P̂ is trained/tested off line
yˆ (k )  Pˆ (Su u(k ), Su u (k  1),..., Su u (k  pu ), S y y(k  1), S y y(k  2),..., S y y(k  p y ), WP (k ), Ts )(1 / S y ),
15
(20)
via the shuffled input/output pairs’ data. After system identification P̂
is
performed, and the digital neural controller CF for the CT system can be built by
using this inverse NARMAX plant model Pˆ 1 in the next step.
Step 3: In practice, according to the exchanged output/input pairs’ data from Step 2, the offline stage to train/test the neural controller can be passed through
u (k )  CF ( Su u (k  1), Su u (k  2),..., Su u (k  cu ), S y y (k ), S y y (k  1),..., S y y (k  c y ), WC (k ), Ts )(1 / Su ) .
(21)
If Eq.(20) and Eq.(21) work, go to Step 4.
Step 4: Update the on-line weights and biases WP of the recursive structure model P̂ :
yˆ (k )  Pˆ (Su u(k ), Su u (k  1),..., Su u (k  pu ), S y yˆ (k  1), S y yˆ (k  2),..., S y yˆ (k  p y ), WP (k ), Ts )(1 / S y ),
(22)
to approximate the CT nonlinear system by using Remark 1 and Theorem 1. Due to
the adaption laws for Eq.(20) and Eq.(22), an exchange for both of them can be
designed to switch into the system, as a switching in Fig. 1, when Eq.(22)’s absolute
approximation error is too big. If Eq.(20) and Eq.(22) work, then go to Step 5.
Step 5: Adapt the digital neural controller for the modeling error and tracking error by using
Remark 1 and Theorem 1-2. Finally, update the on-line parameters of the neural
controller C F
u (k )  CF ( Su u (k  1), Su u (k  2),..., Su u (k  cu ), S y r (k ), S y r (k  1),..., S y r (k  c y ), WC (k ), Ts )(1 / Su ) ,
(23)
to minimize the tracking error, and finish the above two stages: the off-line stage and
on-line stage.
16
To make sure of the robustness of the control system, the convergence to the global optimal
solution of parameters of the model/controller has to be guaranteed. Hence, some random
initial weights and biases of the model are designed by Particle Swarm Optimization (PSO)
[31, 32] with the parameters of the controller first. The PSO algorithm consists of the velocity
vi ( j  1)  vi ( j )   1i  ( pi  xi ( j ))   2i  (G  xi ( j )) ,
and position
xi ( j  1)  xi ( j )  vi ( j  1) ,
where i  1,2,..., H is the particle index; j  1,2,..., N is the iteration index; vi is the velocity
of ith particle; xi is the position of ith particle; pi is the best position found by ith
particle (personal best); G is the best position found by the swarm (global best, best of
personal best);  1i ,  2i are the random numbers on the interval [0,1] applied to the ith
particle.
The PSO supplies random initial parameters, hence, it is an initial parameters’
conductor. These initial parameters are then converged locally by the LMBP method and
the best solution for the initial model/controller is chosen. Finally, the global optimal
solution of parameters can be found every time. Hence, this idea has been named the
Random-Local-Optimization (RLO) algorithm. The RLO algorithm is a composite of the
LMBP algorithm and a random initialization procedure of evaluating fitness value
1 /(   0.01) , where     1  (1   )   2 ,   [0,1] . The total of absolute training error 1
is obtained by LMBP via the training data, and  2 is the total of absolute testing error of
the model/controller output via the testing data input. In this paper, off-line RLO is used as
a learning algorithm for the feed-forward structure model Eq.(20) due to the on-line tuning
parameters of the recursive structure of the plant model being not converged. After the offline training stage, in order to tune on-line the parameters of the plant model Eq.(22)
17
recursively,
dyˆ (k )
of Eq.(10) needs to be calculated as follows:
dWP (k )
py
pu
dyˆ (k )
yˆ (k )
yˆ (k ) du (k  i )
yˆ (k ) dyˆ (k  i )
.



dWP (k ) WP (k ) i 0 u (k  i ) dWP (k ) i 1 yˆ (k  i ) dWP (k )
(24)
Similarly, in order to tune the on-line parameters of the controller Eq.(23) recursively, and
dyˆ (k )
of Eq.(18) needs to be calculated as follows:
dWC (k )
py
pu
dyˆ (k )
yˆ (k ) du (k  i )
yˆ (k ) dyˆ (k  i )


,
dWC (k ) i 0 u (k  i ) dWC (k ) i 1 yˆ (k  i ) dWC (k )
where
cu
du (k )
u (k )
u (k ) du (k  i)


.
dWC (k ) WC (k ) i1 u (k  i ) dWC (k )
(25)
(26)
Hence, the following algorithm adapts a NARMAX neural controller for a NARMAX neural
model of plant.
Step 1: Back propagate through C F to form
u (k )
u (k )
and
in Eq.(26). If update
WC (k )
u (k  i )
du (k )
du ( k  i )
of Eq.(26), and shift
down in Eq.(25), then go to Step 2.
dWC ( k )
dWC (k )
Step 2: Back propagate through P̂ to form
update
yˆ (k )
yˆ (k )
and
in Eq.(25). If
yˆ (k  i )
u (k  i )
dyˆ (k )
dyˆ ( k  i )
of Eq.(25), and shift
down in Eq.(25), then go to Step 3.
dWC (k )
dWC ( k )
T
Step 3: Compute WC (k )  C (k )( yˆ (k )  r (k ))
dyˆ (k )
. If update weights
dWC (k )
WC (k  1)  WC (k )  WC (k ) , and WP (k  1)  WP (k )  WP (k ) , then go to Step 1.
To clarify this method, in [20], a robust and adaptive method was used to allow learning
to occur on-line, tuning performance as the system runs. But, [20] didn’t consider the
prediction, modeling error, global optimal initialization of control parameters, the problem
18
of lengthy convergence time of on-line control, delayed terms, uncertainties in plant and
DT-CT problems. Moreover, the choice method of initial parameters of the on-line controller
still lacks the ability to overcome the over-fitting problem of the controller. Hence, the offline stage is proposed for a RLO learning algorithm to choose the initial weights and biases
of the on-line neural controller in the simulation example of the power plant, as shown in
the following case study.
4. Cases study
First, the conventional PWM buck converter, by using AM-OTS-DS [28, 29]
methodology, is modeled to the following equivalent circuit plant:
 1 
R  RC 
RL 




 iL (t ) 
L 
R  RC 




R
vC (t ) 
 C  (R  R )
C


R

 1

  VD 
L  ( R  RC )   iL (t )   RM iL (t )  Vin  VD 
 L ,

u
(
t
)

L
 vC (t ) 
1

 0 
0




C  ( R  RC ) 
 R  RC
vo (t )  
 R  RC
R   iL (t ) 
,

R  RC  vC (t )
(27a)
In this paper the robustness of the above control system is emphasized, so uncertainty  ,
and delay  are added to the original control system.
 1 
R  RC 
R 
 iL (t )   L  L R  RC 



R
v
(
t
)
 C  
 C  (R  R )
C


R

 1

  VD 
L  ( R  RC )   iL (t )   RM iL (t )  Vin  VD 
 L ,

u
(
t
)

L


1
 vC (t ) 

 0 
0





C  ( R  RC ) 
 R  RC
vo (t )  
 R  RC
where
R R 
;
R 6
R   iL (t ) 
   vo (t   ),

R  RC  vC (t )
; Vin  30
is a DC voltage source;
19
(27b)
C  202.5  10 6
;
  0.52 sin( 2  t / 3) ;   0.1 ;
  102 (1  sin( 10t )) ; L  98.58 106 ; RL  48.5 103 and
RC  162  103 are the parasitic resistances of the inductor and capacitor, respectively. The
element RM  0.27 is the static drain to source resistance of the power MOSFET and
VD  0.82 is the forward voltage of the power diode. u (t ) is the duty ratio of conventional
PWM buck converter. x(t )  x  [iL (t ), vC (t )]T is the state of the system, and the output of
this power system is y(t )  vo (t ) .
The nonlinear, uncertain, hotter circuit’s components, time-delay, and digital control
problems of PWM buck converter CT system renders a tracking control problem difficult to
analyze. A simulation system in Eq.(27) is built with uncertainty. In this study, it is assumed
that the parameters of the circuit’s components are not ideal, and the capacity of the digital
controller is limited by using a lower-cost chip. Here, the sampling period Ts  104 is
designed for this power system Eq.(27b). Hence, the delay  is very large for this system.
Referring to Fig. 1, and the above sections, it can be seen how to model the plant
dynamics by considering the modeling error, and how to use the neural model of plant to
adapt a neural controller. To compare with other methods, the following cases are
introduced:
Case 1: This case is in [29], and its digital controller is a kind of T-S fuzzy controller with
integral term.
Case 2: This case is in [31], and its controller is a kind of PID, with PSO to compare with
Case 3.
Case 3: This is the control method presented here, and the proposed neural-modelbased neural controller is adaptive and globally optimal.
The detail designs of the Case 1−Case 3 are as follows:
20
Case 1 is a LMI control method of original example for this power plant. The control
parameters of Case 1 are as follows:
K11  K 21  [0.0476, 0.9348]T , K12  K22  -426.4969 , and this T-S fuzzy controller of
Case 1 is designed as:
u (k )  h1  ( K11x(k )  K12  e(k )Ts )  h2  ( K 21x(k )  K 22  e(k )Ts ) .
k
k
Case 2 is an optimal control method. This PID controller is designed as:
u (k )  u (k  1)  K p  (e(k )  e(k  1))  K i
Ts
1
(e(k )  e(k  1))  K d (e(k )  2e(k  1)  e(k  2)) .
2
Ts
And, the parameters H  30, N  50 of PSO of Case 2.
Case 3 is also an optimal control method, but its NARMAX neural control design
method is very different from PSO of Case 2. The predictive controller of Case 3 is
u X  K1 (e(k )) , where K1  0.1 .
T
First, the initial state is set to x(0)  [0,0] , and the reference signal r (k )  12 sin( 30k  Ts ) .
Case 3 uses NN structure 5-8-1 of the NARMAX plant model, it has 5 inputs,
[u (k ), u (k  1), u (k  3), yˆ (k  1), yˆ (k  2)] , 8 tansig(·) neurons in the hidden layer, 1 purelin(·)
neuron in the output yˆ (k ) layer. Also, Case 3 uses NN structure 5-8-1 of the NARMAX
controller, it has 5 inputs, [r (k ), r (k  1), r (k  3), u (k  1), u (k  2)] , 6 tansig(·) neurons in the
hidden layer, and 1 purelin(·) neuron in the output u (k ) layer.
The weights and biases, WP , are trained as follows by selecting two suitable scaling
factors Su  1 and S y 
1
of the plant model, whose summation WPS of WP
24
updated as shown in Fig. 2a. The summation WCS
is
of the neural control parameters WC
is updated as shown in Fig.2a. The tracking control performance of Case 3 is shown in Fig.2b.
21
The optimal parameters K p  -0.003241, Ki  34.280925, Kd  3.4004 10-6 of the PID
controller of Case 2 are obtained by using PSO. The tracking control performance of Case 2
is shown in Fig.3. The tracking control performance of Case 1 is shown in Fig.4.
Finally, the control performances of Case 1−Case 3 are compared, as shown in Fig. 5.
Fig. 2 shows the precise neural control performance of Case 3. Fig. 3 shows the digital
PID with PSO control performance of Case 2. Fig. 4 shows the LMI control performance of
Case 1.
It is clear that the proposed two-stage scheme, Case 3, has excellent tracking
performance when compared with Case 1 and Case 2.
Conclusions
The proposed two-stage adaptive prediction control converges very quickly, works
highly effectively, and precisely. It works for nonlinear delayed plants with uncertainty. The
recursive and feed-forward control scheme is partitioned into two stages that can be
independently optimized. First, an off-line neural model of a continuous-time (CT)
nonlinear power plant is made; second, a constrained off-line digital neural controller is
generated; then, an adaptive plant model is made, and an adaptive NARMAX neural
controller with predictor is generated; finally, all processes may continue concurrently, and
robustness and DT-CT problems for a power plant are solved.
Acknowledgment
The author would like to thank the National Science Council for it support under
contracts: NSC-97-2218-E-468-009, NSC-98-2221-E-468-022, NSC-99-2628-E-468-023, NSC100-2628-E-468-001, NSC-101-2221-E-468-024 and Asia University under contract 98-ASIA22
02, 100-asia-35 and 101-asia-29.
References
[1] H. Dong, Z. Wang, D.W.C. Ho, H. Gao, “Robust H∞ neural output-feedback control with
multiple probabilistic delays and multiple missing measurements,” IEEE Trans. Neural
Syst., vol. 18, pp. 712-725, 2010.
[2] L.X. Wang, A Course in Neural Systems and Control, Prentice-Hall, New Jersey, 1997.
[3] Z. Wang, Y. Liu, G. Wei, X. Liu, “A note on control of a class of discrete-time stochastic
systems with distributed delays and nonlinear disturbances,” Automatica, vol. 46, pp. 543548, 2010.
[4] H.O. Wang, K. Tanaka, M.F. Griffin, “An approach to neural control of nonlinear systems:
stability and design issues,” IEEE Trans. Neural Syst., vol. 4, pp. 14-23, 1996.
[6] B. Shen, Z. Wang, H. Shu, G. Wei, “Robust H∞ finite-horizon filtering with randomly
occurred nonlinearities and quantization effects,” Automatica, vol. 46, pp. 1743-1751, 2010.
[9] B.S. Chen, C.S. Tseng, H.J. Uang, “Mixed H2 /H∞ neural output feedback control design
for nonlinear dynamic systems: an LMI approach,” IEEE Trans. Neural Syst., vol. 8, pp.
249-265, 2000.
[10] K. Tanaka, “An approach to stability criteria of neural-network control systems,” IEEE
Trans. Neural Networks, vol. 7, pp. 629-643, 1996.
[12] H.T. Siegelmann, B.B. Horne, C.L. Giles, “Computational capabilities of recursive
NARX neural networks,” IEEE Trans. Syst., Man, Cybern. B, vol.27, pp.208-215, 1997.
[13] K.R. Lee, J.H. Kim, E.T. Jeung, H.B. Park, “Output feedback robust H∞ control of
uncertain neural dynamic systems with time-varying delay,” IEEE Trans. on Neural
Syst., vol.8, pp.657-664, 2000.
23
[14] Y.Y. Cao, P.M. Frank, “Analysis and synthesis of nonlinear time-delay systems via
neural control approach,” IEEE Trans. on Neural Syst., vol.8, pp.200-211, 2000.
[15] C. Lin, Q.G. Wang, T.H. Lee, “Delay-dependent LMI conditions for stability and
stabilization of T-S neural systems with bounded time-delay,” Neural Sets and Syst.,
vol.157, pp.1229-1247, 2006.
[16] B. Chen, X. Liu, “Delay-dependent robust H∞ control for T-S neural systems with time
delay,” IEEE Trans. on Neural Syst., vol.13, pp.544-556, 2005.
[17] J. Yoneyama, “Design of H∞ control for neural time-delay systems,” Neural Sets and Syst.,
vol.151, pp.167-190, 2005.
[18] S. Hu, Y. Liu, “Robust H∞ control of multiple time-delay uncertain nonlinear system
using neural model and adaptive neural network,” Neural Sets and Syst., vol.146,
pp.403-420, 2004.
[19] H.J. Lee, J.B. Park, Y.H. Joo, “Robust control for uncertain Takagi-Sugeno neural
systems with time-varying input delay,” Journal of Dynamic Syst., Measurement and
Control, Trans. of the ASME, vol.127, pp.302-306, 2005.
[20] G.L. Plett, “Adaptive inverse control of linear and nonlinear systems using dynamic
neural networks,” IEEE Trans. on Neural Networks, vol.14, pp.360-376, 2003.
[21] C.T. Chen, S.T. Peng, “Intelligent process control using neural techniques,” Journal of
Process Control, vol.16, pp.493-503, 1999.
[22] F.J. Lin, W.J. Hwang, R.J. Wai, “A neural network control system for tracking periodic
inputs,” IEEE Trans. Neural Syst., vol.7, pp.41-52, 1999.
[23] Y.C. Chen, C.C. Teng, “A model reference control structure using a neural network,”
Neural Sets and Syst., vol.73, pp.291-312, 1995.
[24] C. Li, K.H. Cheng, “Recursive neural hybrid-learning approach to accurate system
24
modeling,” Neural Sets and Syst., vol.158, pp.194-212, 2007.
[25] M.T. Han, H.B. Demuth, M. Beale, Neural Network Design, PWS, 1996.
[26] J.S.R. Jang, C.T. Sun, E. Mizutani, Neural and Soft Computing, Prentice-Hall, 1997.
[27] J.P. LaSalle, “Some extensions of Lyapunov’s second method,” IRE Trans. Circuit Theory,
pp. 520-527, 1960.
[28] J. Sun and H. Grotstollen, “Averaged modeling of switching power converters:
reformulation and theoretical basis,” IEEE Conf. Syst., pp. 1165-1172, 1992.
[29] C.Y. Huang, “T-S neural controller design for DC-DC power converter,” Master Thesis,
Chung Yuan Christian University, Taiwan, 2002.
[30] Z.R. Tsai, Y.Z. Chang, J.D. Hwang, J. Lee, “Robust neural stabilization of dithered
chaotic systems using IROA,” Information Sciences, vol. 178, pp.1171-1188, 2008.
[31] W.D. Chang, “PID control for chaotic synchronization using particle swarm
optimization,” Chaos, Solitons and Fractals, vol. 39, pp.910-917, 2009.
[32] W.D. Chang, S.P. Shih, “PID controller design of nonlinear systems using an improved
particle swarm optimization approach,” Commun Nonlinear Sci Numer Simulat, vol. 15,
pp. 3632-3639, 2010.
25
Predictor
RLO
off-line
FRP
Plant
Sw
Sampler
ZOH
on-line
off-line
NPM
Switching
(a)
Sampler
NARMAX
neural
D
NPM
ZOH
Plant
|∙|
NPM
|∙|
(b)
26
on-line
RLO
off-line
FRP
Plant
Sampler
ZOH
on-line
off-line
NPM
on-line
Switching
(c)
Figure 1. (a) The proposed two-stage adaptive prediction structure of DT-CT control system.
(b) On-line adaptive prediction block diagram for Theorem 1 and Theorem 3. (c) On-line
adaptive block diagram for Theorem 1 and Theorem 2.
27
43.535
50.045
43.53
50.04
43.525
W PS
W CS
50.05
50.035
43.52
50.03
43.515
43.51
50.025
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.35
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t = k T (sec)
t = k T (sec)
s
s
(a)
20
r
10
k
y
0
k
-10
-20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1
rk - yk
0
-0.1
-0.2
-0.3
t = k T (sec)
s
(b)
Figure 2. (a) The learning curve of the summation of WP , the learning curve of the
summation of WC , and (b) the tracking control performance of Case 3.
28
20
r
10
k
y
0
k
-10
-20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1.5
rk - y k
1
0.5
0
-0.5
t = k T (sec)
s
Figure 3. The tracking control performance of Case 2.
29
15
r
k
10
y
k
5
0
-5
-10
-15
0
0.05
0.1
0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.2
0.25
0.3
0.35
1
0.8
0.6
rk - yk
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
t = k T (sec)
s
Figure 4. The LMI control performance of Case 1.
30
1.2
Case 1
1
Case 2
Case 3
0.8
0.6
rk - yk
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t = k T (sec)
s
Figure 5. Comparison of the control performances for Case 1 to Case 3.
31