METHODS AND DEVICES FOR THE DISCRETE SELF-ADJUSTING CONTROLLERS

METHODES ET DISPOSITIF POUR UNITE DE COMMANDE AUTOREGLABLE DISCRETE

English Abstract

Methods to adjust the controller gains of discrete controllers are suggested.
The adjustment relies on a set of parameters called the state parameters.
These parameters are combinations of modes, which propagate the
deviation from minimum variance control and contribute to more variability of
the error variable. The state parameters are re-estimated on-line to portray
the real current behavior of the control system, and this re-estimation
establishes the self-adjusting algorithm. The algorithm can be used for both
adaptive and self-tuning controls. In a nonadaptive environment, the state
parameters converge to constant values. In an adaptive environment, the
state parameters adapt to new values.

French Abstract

Des méthodes pour régler les gains de contrôleur de commandes discrètes sont proposées. Le réglage repose sur un ensemble de paramètres appelés les paramètres d'état. Ces paramètres sont des combinaisons de modes, qui propagent la déviation du réglage de la variance minimale et contribuent à une plus grande variabilité de la variable d'erreur. Les paramètres d'état sont estimés de nouveau en ligne pour dresser le portrait du vrai comportement actuel du système de commande et cette nouvelle estimation établit l'algorithme autoréglable. L'algorithme peut être utilisé à la fois pour des commandes adaptatives et autoréglables. Dans un environnement non adaptatif, les paramètres d'état convergent vers des valeurs constantes. Dans un environnement adaptatif, les paramètres d'état s'adaptent aux nouvelles valeurs.

Claims

What I Claim as My Invention Is
1. A method to design and set up variables for the self-adjusting control al-
gorithms of a single-input-single-output feedback, feedforward-feedback
or PID controller, which comprises of the following steps:
(a) accepting, from the design of a control engineer, the con-
troller type of said controllers and its corresponding or-
ders l, m, n and p,
(b) accepting, from the design of a control engineer, the de-
gree of integration d = 1 for integral action and d = 0
otherwise,
(c) accepting, from the design of a control engineer, the
penalty constant .lambda. as a small positive constant for smooth
control actions,
(d) setting up the appropriate matrices and vectors of the
variables, in the beginning and at each control time, for
the said controllers as below
<IMG>
and
x t = [ .gradient.d u t-.function.-1 .multidot. .gradient.d u t-.function.-
m y t-.function.1 .multidot. y t-.function.-n ]
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for a feedback controller,
X t = [~ d U t-f-1 .cndot..cndot..cndot. Y t -f-1 .cndot..cndot..cndot. d t- f-
1-k .cndot..cndot..cndot. d t-f-p-k ]
for a feedforward-feedback controller with k .gtoreq. 0 as zero
or the difference between the dead time of the measured
disturbance and that of the plant dynamics and
X t = [ ~ d U t-f-1 Y t-f-1 Y t-f-2 Y t-f-3 ]
for a PID controller.
2. A method to determine the initial values of the state parameters in c for
the self-adjusting control algorithms of a discrete control system with
a feedback, feedforward-feedback or PID controller, which comprises of
the following steps:
(a) choosing the state parameters in c for the following quan-
tity
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for the
appropriate controller and if the initial controller gains
are not given,
(b) choosing the state parameters in c for the following quan-
tity
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for the
appropriate controller and if the initial controller gain
vector .beta. t i , is given,
(c) accepting the initial state parameters in c if they are
given.
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3. A method to tune the controller gains and calculate the control action
of a discrete feedback controller at the control time N, which comprises
of the following steps:
(a) determining the parameter µ, with the available values
of the state parameters of the last control time in c N-1,
for the sum of squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
feedback controller and the state parameters in C1 and
C2 are set to c = c N1 ¨ µg(c N-1) with g(c N-1) as the
vector of the derivatives of S N(c) evaluated at the value
C N-1,
(b) setting c N = c N-1 if no positive value of µ can be found
in step (a) and setting c N = c N-1 ¨ µg(c N-1) otherwise,
(c) obtaining the controller parameters from the vector given
by
<IMG>
with the state parameters in C1 and C2 obtained from
step (b),
(d) calculating the control action ~ d U N from the following
equation
<IMG>
with a0 as the first element of the vector ~N.
4. A method to adapt the controller gains and calculate the control action
of a discrete feedback controller at the control time N, which comprises
of the following steps:
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(a) determining the state parameter vector c for the sum of
squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
feedback controller,
(b) comparing the two sums of squares S N(c) and S N(c N-1)
and accepting the new values of the state parameters
c N = c if S N(c) < S N(c N-1) or c N = c N-1 otherwise,
(c) obtaining the controller parameters as shown in step (c)
of claim 3,
(d) calculating the control action ~ d U N as shown in step (d)
of claim 3.
5. A method to tune the controller gains and calculate the control action
of a discrete feedforward-feedback controller at the control time N,
which comprises of the following steps:
(a) determining the parameter µ, with the available values
of the state parameters of the last control time in c N-1,
for the sum of squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
feedforward-feedback controller and the state parameters
in C1 and C2 are set to c = c N-1 ¨ µg(c N-1) with g(c N-1)
as the vector of the derivatives of S N(c) evaluated at the
value c N-1,
(b) setting c N = c N-1 if no positive value of µ can be found
in step (a) and setting c N = c N-1 ¨ µg(c N-1) otherwise,

(c) obtaining the controller parameters and from the vector
given by
<IMG>
with the state parameters in C1 and C2 obtained from
step (b),
(d) calculating the control action ~ d U N from the following
equation
<IMG>
with a0 as the first element of the vector ~ N.
6. A method to adapt the controller gains and calculate the control action
of a discrete feedforward-feedback controller at the control time N,
which comprises of the following steps:
(a) determining the state parameter vector c for the sum of
squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
feedforward-feedback controller,
(b) comparing the two sums of squares S N(c) and S N(c N-1)
and accepting the new values of the state parameters
c N = c if S N(c) < S N(c N-1) or c N = c N-1 otherwise,
(c) obtaining the controller parameters as shown in step (c)
of claim 5,
(d) calculating the control action ~ d U N as shown in step (d)
of claim 5.
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7. A method to tune the controller gains and calculate the control action
of a discrete PID controller at the control time N , which comprises of
the following steps:
(a) determining the parameter p, with the available values
of the state parameters of the last control time cN_1, for
the sum of squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
PID controller and the state parameters in C1 and C2 are
set to c = c N-1 ¨ µg(c N-1) with g(c N-1) as the vector
of the derivatives of S N(c) evaluated at the value c N-1,
(b) setting c N = c N-1 if no positive value of µ can be found
in step (a) and setting c N = c N-1 ¨ µg(c N-1) otherwise,
(c) obtaining the gain vector given by the following equation
<IMG>
and calculating the controller gains as below
<IMG>
with the state parameters in C1 and C2 obtained from
step (b),
(d) calculating the control action ~ d U N from the following
equation
~ d U N = (k p, N + k i, N + K d, N)Y N ¨ (k p, N + 2k d, N) Y N-1 + k d, NYN-
2.
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8. A method to adapt the controller gains and calculate the control action
of a discrete PID controller at the control time N, which comprises of
the following steps:
(a) determining the state parameter vector c for the sum of
squares given by the following equation
<IMG>
to have the minimal value where all the matrices and
vectors are set up as shown in step (d) of claim 1 for a
PID controller,
(b) comparing the two sums of squares S N(c) and S N(c N-1)
and accepting the new values of the state parameters
c N = c if S N(c) < S N(c N-1) or c N = c N-1 otherwise,
(c) obtaining the controller gains as shown in step (c) of claim
7,
(d) calculating the control action .gradient.d u N as shown in step (d)
of claim 7 with the controller gains obtained from the last
step (c).
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Description

CA 02656235 2012-02-01
Field of the Invention
This invention relates to adaptive and self-tuning controllers for the control
of processes, machines and systems. The invention presents algorithms to
adjust the controller gains of a discrete controller. The controllers are
called
self-adjusting controllers because they can be, depending on the environment
of the control system, either an adaptive controller or a self-tuning
controller.
Background of the Invention
The control of a stochastic regulating control is a difficult problem with
no satisfactory solution yet. The difficulty in the problem is the model of
the disturbance must be modeled in real-time and on-line. An adaptive or
self-tuning controller is the correct approach to the solution. The existing
algorithms are, however, not adequate because of poor vision. From the
first paper in self-tuning control of Astrom, K.J. and Wittenmark, B. ("On
Self-tuning Regulators", Automatica, No. 9, 185-189, 1973) to the relatively
recent paper of Vu, K. et al ("Recursive Least Determinant Self-tuning Regu-
lator", IEE Proceedings - Control Theory and Applications, Vol. 147, No. 3,
285-292, 2000), the algorithms are poor. In the domain of intellectual prop-
erty, we see that neither the patent An Improved Self-tuning Controller of
the FoxBoro Company (Canadian Patent CA 2,122,472, US Patent 5,587,896)
nor the patent Self-tuning process control system of West Instr Limited (UK
Patent GB2279777A) is a good self-tuning algorithm. The reason for these
poor performances is the poor choice of the self-tuning parameters.
It turns out that not all the parameters of a controller at one time have
clear relations with those at the immediate past. Only a selective set of pa-
rameters, called the state parameters, can have these simple relations, which
establish the self-tuning algorithm. The rest of the parameters are calculated
from these parameters. These parameters are called the state parameters be-
cause they represent the state of the dynamics of a process or machine. If
these parameters do not change after some tuning period, we have a non-
adaptive environment and the algorithm is a self-tuning control algorithm.
On the contrary, if the state parameters constantly change, we have an adap-
tive environment and the algorithm is an adaptive control algorithm. For a
common name, the algorithm is called a self-adjusting control algorithm.
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CA 02656235 2012-02-01
Summary of the Invention
It is the object of this invention to introduce an effective self-adjusting
control
algorithm to adjust the controller gains of a discrete controller for
corrective
control of a feedback control system.
It is a further object of this invention to introduce an effective self-
adjusting control algorithm to adjust the controller gains of a discrete con-
troller for corrective control of a feedforward-feedback control system.
Brief Description of the Drawings
Figure I. Block diagram of a self-adjusting discrete feedforward-feedback
control system.
Figure 2. Block diagram of the controller part of a digital control chip.
Description of the Preferred Embodiment
The possibility to express the output or controlled variable as a function of
the controller parameters is the idea behind a self-tuning control algorithm.
In the following, the approach to adjust the parameters of a controller is dis-
cussed and the device to implement the self-adjusting algorithm is described.
Method
Consider the control system depicted in Figure I. With dt = 0 and yr = 0,
the control system is a stochastic feedback control system and is described
by the equation
yt =
w(z1) 0(z-1)
___________________ ut-f-i + __ at,
w(z1)
___________________ llt_f_i 11)(z-1) at
or
w(z-1)0(z-1)0(z-1)11t + 6(z-1)7(z-1)Yt
Yt+ f +1 =
ti(z-1)6)(z-1) 0(z-1)at+f +1,
a(z-1)ut b(z-1)yt
c(z1) __ + et+ f+11
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CA 02656235 2012-02-01
[ut ut-1 = ' ' Yt Yt-1 " *JO ,
'T et+ f +1)
4
c(z1) .
X f+1
1 + + = = = +
+ et+ f +1 =
CLZ-"
When the set point variable yr is not zero, the variable yt can be defined
as the deviation of the output variable Yt from the set point. A nonzero dis-
turbance (it will give a feedforward-feedback control system and the variable
vector Xt+f+i will contain this variable in addition to the other variables ut
and yt.
The equation
0
t+ f +1N
is the equation of the minimum variance controller. Such a controller can
be used for an industrial process to control the quality of a product. An
illustrative example is the control of a paper machine in the paper industry.
The variable yt in Xt+f+1 can be the deviation of the paper sheet moisture
from its set point, and the control variable ut is the steam pressure to the
dryer of the paper machine.
The parameters ci's are the state parameters. Each state parameter is
a combination of the modes that propagate the deviation from minimum
variance control and contribute to more variation of the variable ,yt. The
controller parameter vector 0 and the state parameters cz's are obtained by
minimizing a finite sum of squares of the residual et. The minimization with
respect to the controller parameter vector 0 gives, for a finite control
horizon
at the time t, the following equation
CT ¨ tX XT1
Min lim. yT 1 +C2CT2 ______ [C1 C2]Yt (1)
t oT
C -2
with E as a small positive constant indicating closeness of optimality and the
controller parameter vector
Ot _ Fr 0 0 XT
1+ [ t ](C2CT2)-1[Xt C]-1
0 I CT
CT
XT
(C2C2T)-1[C1 Cbtt.
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CA 02656235 2012-02-01
The parameter matrices and vectors in these equations are defined as
below
CI = = ' 1
= = 1=
C1 C1 = = = 1
c1= , c2=
=
= =
=
0 =
= =
Ci = = = 1
Xto Yto¨t
xto+1
Cl
= Yto
C = , Xt == Yt =
_ci_
= =
xt Yt
and
xt = [ Veint_f_i = " f -7n Yt- f -1 = " lit- f -n
for a feedback controller,
xt = [Vdutf -1 = Yt- f -1 ' ' ' dt- f -1- k ' dt-f-p-k
for a feedforward-feedback controller with k > 0 as zero or the difference
between the dead time of the measured disturbance and that of the plant
dynamics and
xt = [ Vdnt-f-i Yt-f- 1 Yt- f -2 Yt- f -3
for a three-mode PID (Proportional, Integral and Derivative) controller.
The integer parameters 1, m, n, p and d are the appropriate orders of the
controller.
In the conventional approach of self-tuning and adaptive controls, the
controller parameter vector at a particular time N, 13N (optimal value of
ON), is obtained as a function of the controller parameter vector 13N-1. In
this invention, it is the vector of the state parameters c = [c1 c2 = = r at
the
time N, which is obtained as a function of that at the time N ¨ 1.
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CA 02656235 2012-02-01
The controller for the control criterion
Al in E { yt2+ f +1 AVditt2lyt, Yt-1, = ' A > 0
vrdut
gives the controller equation
T
A I
,
f +113t Ecivdut_i =0
cto iO
with ao as the first element of the vector f3t.
The state parameters in c are obtained from Eq. (1) and the controller
gains are calculated from these two sets of parameters as given by the last
equation. The initial values of these state parameters can be obtained
directly
from a block of data in the beginning of a control period, from an initial
estimate of the controller gains or an estimate for their values. With the
arrival of new data, the state parameters in c are re-estimated from Eq. (1).
The methods to adjust the parameters and the controller equation are set
forth in the claims. The device to implement the self-adjusting algorithms is
described next.
Device
The self-adjusting controllers are implemented as a digital chip. On this
chip, the controller part of the chip consists of an analog-to-digital
converter
(ADC), a multiplexer, some read-only memory (ROM) and some random-
access memory (RAM). The execution program of the self-adjusting algo-
rithms resides in the ROM of the chip. The controller parameters and the
variables reside in the RAM. The variable to be controlled along with
other variables such as the set point yr and measured disturbance dt, must
be fed through the ADC for discretization. The control variable Vdut is out-
putted as the width of a pulse. The orders of the controller and the sizes
of the arrays for the variables in the self-adjusting algorithms are specified
together with the code for the execution program when it is loaded into the
ROM of the chip. The configuration of the controller part of the chip is
depicted in Figure 2. The operation of the chip is described as follows.
At the time of control, the operating system software of the chip reads
the signals provided by the variables Yt, dt if any, and yr. These signals are

CA 02656235 2012-02-01
then multiplexed to the ADC. The execution program of the self-adjusting
controller is then invoked to read in these signals and store them in arrays
in
RAM. The self-adjusting control algorithms are then invoked to operate on
these data in RAM to produce a new set of values for the state parameters
ci's. The controller gain vector :ji\I is then calculated from these state
param-
eters. The control variable is then computed from these sets of parameters.
The control action, Vdut - the realization of the control variable, is given
by
the width of a pulse.
In applications with larger apparatus, the whole controller part can be
replaced by a function subroutine in the software of the control computer.
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Representative Drawing