IC
C C ’2 0 0 9
SOME REMARKS TO CONTROL BY MASV METHOD
Milan KONECNY
Lachia University of Each Age,
Brušperk, Czech Republic,
[email protected]
Abstract: The paper is focused to problems with finite and infinite models, time optimization of
dynamic parameters in the MASV method, called Method Aggregate State Variables.
Keywords: Nonlinear Control, Optimization, Aggregate State Variables.
1 Introduction
The MASV method, called Method
Aggregate State Variables, and its applications
consist of four parts:

mathematical model

control algorithm

simulation control

application in industry
This paper describes some remarks to models
and algorithms. Classical formulation of the
MASV method does not solve the control in the
finite time and construction algorithm uses the
control in the infinite time. Time optimization
cannot use this formulation. There are not
dynamical parameters T, D .
In the paper we show various ways how to
solve these problems.
x  x1 , x2 ,, xn T ,
u  u1 , u2 ,, um  ,

Let a mathematical model of the nominal
nonlinear subsystem be considered
(1)
dim x  n
dim u  m
T

f  x2 ,, f r , xr 2 ,, f r , xr 2 ,, f n , dim f  n
T
1
1
 0


gr1

 0
G

gr 1
 0



 g n1
1
2
2 Mathematical model of control
system
x0  x 0
x  f (x, t )  G(x, t ) u ,
2
0



gr 2

0



1
2
gr 2 
2
0



g n2

dim G  n, m ,
0 


grm 

0 
,

gr m 
0 


g nm 
1
2
rj  rj 1  n j , r0  0; j  1,2,, m
n  rm 
e  x w  x, dim e  n,
m
nj
j 1
where
x - the vector of the state variables,
u - the vector of the control variables,
f - continuous vector functions,
G - the matrix of the continuous functions gi,j ,
n - the number of the state variables (the order
of the nonlinear subsystem),
nj - the partial order,
m - the number of the control variables,
T - the transpose symbol,
dim - the dimension of a vector or
the order of a matrix.
The condition of controllability of the
nominal nonlinear subsystem (1) must hold
[Zítek & Víteček 1999]
rank Gx, t   m
2 Control algorithms design – MASV method
The task of the optimal tracking
control design is the determination of the
feedback control
u  u(x, x , t )
w
(5)
for the controllable nominal standard nonlinear
subsystem (1), which for a given state trajectory
{xw(t)} ensures its tracking by a real state
trajectory {x(t)} so that the value of the
quadratic objective functional
J 
 e D De  e D T De dt
0
T
T
T
T
2
t 
is minimal,
where
e - the error vector,
D - the constant nonnegative aggregation matrix
[dim D = (m,n), rank (DG) = m],
T - the diagonal matrix of positive time
constants Tj of the order m, i.e.
T  diagT1 , T2 , , Tm 
(8)
By the method of the aggregation of the
state variables, it is possible to obtain the
optimal feedback control (Zítek & Víteček
1999)



u  DGx, t  T1De  D x w  f x, t 
1
(9)
which causes the aggregated optimal closedloop control system
De  T 1De  0,
(3)
It is supposed that i = 1, 2,…, n; j = 1, 2,…,
m and strictly not distinguished between a
subsystem (system) and a model in the entire the
following text.

lim et   lim e t   0 (7)
t 
e0  e 0
(10)
and minimal value of the quadratic objective
functional (6)
J *  eT0 DT TDe0
(11)
If the elements dji of the aggregation matrix
D will be chosen in accordance with the
formulas
d ji  0
for
i  rj 1
d ji  0
for
rj 1  i  rj
or
d jr  1
j
i  rj 




(12)
then the characteristic polynomial of the
aggregated optimal closed-loop control system
(10) will be written in the form
N s    N j s  ,
m
j 1
(6)
1

p r
N j s     s   d jp s
 Tj
 p  r 1


rj
j 1
j 1
1
(13)
where
s - the complex variable in the Laplace
transform,
Nj - the characteristic polynomial of the j-th
autonomous control subsystem of the partial
order nj.
It is obvious that in this case the optimal
closed-loop control system consists of
m
autonomous linear control subsystems whose
desired dynamic behaviour can be ensured by a
suitable choice of the time constants Tj and
coefficients
dji
of their characteristic
polynomials (13), i.e. by a suitable choice of the
matrix T and D. It is very important that the
quadratic objective functional (6) has only an
auxiliary purpose.
The feedback control (9) demands
knowledge of the exact mathematical model of
the nominal nonlinear dynamic subsystem (1)
The control u is non-robust and is often called
the equivalent control. It ensures the aggregated
optimal closed-loop control system (10) from
which after the completion with the equations
ei  ei 1, i  r j
(14)


model with dynamic parameters T, D,
optimal control in the time using
dynamic parameters T, D
4 Remarks to open problems
4.1 Models with infinite and finite final time
and with error in final time
We will rate models regarding final time
and error in final time
Model MASV( ∞, 0) – classical MASV
method.
J 
0
lim et   lim e t   0
t 
t 
(16)
(17)
Infinite final time and error is in infinity
equal to 0.
The infinite final time is not realistic.
Model MASV (tf,0) – basic version with
infinite final time, zero error
the full optimal closed-loop control system can
be obtained in the form
e  Ae ,
T T
T T 2
 e D De  e D T De dt

J
T T
T T 2
 e D De  e D T De dt (18)
tf
0
(15)
   
e t f  e t f  0
(19)
which has the characteristic polynomial (13).
This model is realistic, but it is not usually
solvable.
3 Formulation some open question in
the MASV method
Model MASV (tf, error) - version with the
finite final time tf and with limited value of
e t f , e t f
The MASV method was formulated
considering

infinite final time

constant value of parameters D, T

energy optimization as a method is
used.
These properties are basic for development the
MASV method.
We will make some remarks to

finite and infinite models,

control in the finite time,

optimal control in the time

control with error in the finite time,
  
J
T T
T T 2
 e D De  e D T De dt (20)
tf
0
 
e tf
0
 
 erroro , e t f
1
 error1 (21)
In this model the final time tf is finite and e
is less that constants error.
4.1 Control algorithm MASV(tf , error)
The goal is finding tf such that e in the
final time are less than error.
One type of solution is the next algorithm.
There is the main idea – transforming problem
on the method MASV.
Control algorithm :

find control by using method MASP.

compile system of differential
equations for errors

find a priori estimate for solution

from a priori estimate and calculate
the value of tf
4.2 Time eps - optimization
method MASV(tf , error)
using the
Problem statement
Find the minimal time tf , which meets the
MASV Model (tf, eps).
Design control algorithm MASV (tf, error),
choose a priori estimate for solution of this
problem.
4.3 Energy optimization of T, D
One of the possible ways how to optimize
T,D is minimizing the functional
J *  eT0 DT TDe0
Discreet algorithm: Set the discrete time
sequence and calculate the optimal value of T,
D in the every time step during control
algorithm.
Continuous algorithm: Calculate the optimal
value of D,T in every time step.
Conclusions
The paper describes a concept for using the
MASV method in the non-standard condition
models and algorithms, where the final time is
finite and optimization, dynamic parameters of
T,D have to be taken into account.
These ideas will be developed and employed
on realistic problems in the next author’s paper.
References
KHALIL, H.K. 1996. Nonlinear Systems. Second
Ed. Upper Saddle River: Prentice–Hall,
1996.
ZÍTEK, P.  VÍTEČEK, A. 1999. Control
Design
of
Anisochronic
and
Nonlinear Subsystems (in Czech).
Prague: CTU in Prague, 1999.
Find such values of D, T that the
functional J* becomes minimal.
Lachia University of Each Age
4.5 Time eps - optimization the value of T, D
Problem statement
Find such T,D that tf(T,D) froms method
MASV( tf , eps) is minimal.
One of the possible ways how to solve this
problem is optimizing variables using a priori
estimate. We find approximation of solution.
4.6 Dynamic parameters T, D
Dynamism parameters T, D passes from
time optimization with variables T, D.
Open adaptive system of education
and cognitive activities for Lachia
with centre in Brušperk
We prepare miniconference IOCA2009
Optimization,
optimal control and applications.
Brušperk - September 2009
Information - [email protected]