Title: Method for tuning PID controllers applicable to nonlinear systems
Abstract: A method for tuning a PID controller includes the steps of inducing equivalent relationships between PID gains of the PID controller and parameters of time delay control (TDC), selecting a natural frequency vector and a damping ratio vector so as to acquire a desired error dynamics of the closed PID control loop system, selecting a sampling time of the closed PID control loop system, determining the parameters of TDC on the basis of the natural frequency vector, the damping ratio vector and a closed loop stability condition for TDC, and selecting PID gains of the PID controller on the basis of the equivalent relationships.
Patent Number: 6,937,908 Issued on 08/30/2005 to Chang, et al.
| Inventors:
|
Chang; Pyung Hun (Daejeon, KR);
Jung; Je Hyung (Daejeon, KR)
|
| Assignee:
|
Korea Advanced Institute of Science & Technology (Daejon, KR)
|
| Appl. No.:
|
697816 |
| Filed:
|
October 29, 2003 |
Foreign Application Priority Data
| Feb 03, 2003[KR] | 10-2003-0006597 |
| Current U.S. Class: |
700/37; 700/28; 700/30; 700/31; 700/41; 700/42; 318/609; 318/610 |
| Intern'l Class: |
G05B 013/02 |
| Field of Search: |
700/30- 31,28,37,42,41,32
318/609,610,561
|
References Cited [Referenced By]
U.S. Patent Documents
| 4466054 | Aug., 1984 | Shigemasa et al.
| |
| 4539633 | Sep., 1985 | Shigemasa et al.
| |
| 4754391 | Jun., 1988 | Suzuki.
| |
| 5057993 | Oct., 1991 | Kanda.
| |
| 5229699 | Jul., 1993 | Chu et al.
| |
| 5331541 | Jul., 1994 | Ueda et al.
| |
| 5568377 | Oct., 1996 | Seem et al.
| |
| 5587899 | Dec., 1996 | Ho et al.
| |
| 5742503 | Apr., 1998 | Yu.
| |
| 5818714 | Oct., 1998 | Zou et al.
| |
| 5847952 | Dec., 1998 | Samad.
| |
| 5971579 | Oct., 1999 | Kim.
| |
| 6081751 | Jun., 2000 | Luo et al.
| |
| 6330484 | Dec., 2001 | Qin.
| |
| 6510351 | Jan., 2003 | Blevins et al.
| |
| 6697767 | Feb., 2004 | Wang et al.
| |
| 6847954 | Jan., 2005 | Wojsznis et al.
| |
Other References
Time delay observer: a robust observer for nonlinear plants using time-delayed
signals, Chang, P.H.; Lee, J.W.; American Control Conference, 1995. Proceedings
of the vol. 3, Jun. 21-23, 1995 Page(s): 1638-1642 vol. 3.
A reduced order time-delay control for highly simplified brushless DC motor□□Chang,
P.H.; Lee, J.H.; Park, S.H.; American Control Conference, 1998. Proceedings of
the 1998□□vol. 6, Jun. 24-26, 1998 Page(s):3791-3795 vol. 6.
Time-varying input shaping technique applied to vibration reduction of an industrial
robot,Hyung-Soon Park; Chang, P.H.; Jong-Sung Hur;Intelligent Robots and Systems,
1999. IROS '99. Proceedings. 1999 IEEE/RSJ International Conference on vol. 1, 17-21.
|
Primary Examiner: Patel; Ramesh
Attorney, Agent or Firm: Graybeal Jackson Haley, LLP
Claims
1. A method for tuning a PID controller, wherein the PID controller is comprised
in a closed PID control loop system, the PID control loop receiving an input, the
PID controller being coupled to an object system being controlled, wherein the
object system outputs process variables which is supplied for comparison to the
input, wherein a result of said comparison is supplied to the PID controller, the
method comprising the steps of:
inducing equivalent relationships between PID gains of the PID controller and
parameters of time delay control (TDC);
selecting a natural frequency vector and a damping ratio vector so as to acquire
a desired error dynamics of the closed PID control loop system;
selecting a sampling time of the closed PID control loop system;
determining the parameters of TDC on the basis of the natural frequency vector,
the damping ratio vector and a closed loop stability condition for TDC; and
selecting PID gains of the PID controller on the basis of the equivalent relationships.
2. The method as defined in claim 1, wherein the object system being a multi
input multi output system of degree n which is expressed in a sampled data system.
3. The method as defined in claim 2, wherein the number of the parameters of
TDC is three, a first parameter being determined by the natural frequency vector,
a second parameter being determined by the natural frequency vector and damping
ratio vector and a third parameter being determined as a diagonal matrix.
4. The method as defined in claim 3, wherein all of diagonal elements of the
third parameter has one constant value.
5. The method as defined in claim 3, wherein diagonal elements of the third parameter
have constant values different to each other.
Description
This application claims priority from Korean Patent Application No. 10-2003-0006597
filed 03 Feb. 2003, which is herein incorporated by reference.
FIELD OF THE INVENTION
The present invention relates to a method for designing PID controllers and,
more particularly, to a method for tuning PID controllers which are applicable
to nonlinear systems such as robot manipulators.
BACKGROUND OF THE INVENTION
A plurality of patents has disclosed gain tuning methods of PID controllers.
These
patents may be categorized into two types of methods; one is a gain tuning method
of PID controllers using hardware equipments and the other using software algorithms.
Among these patents, the second type of methods, i.e., gain tuning methods
using software algorithms, to which the present invention pertains, can be exemplified
as follows: U.S. Pat. No. 6,081,751 titled "System and method for closed-loop autotuning
of PID controllers", U.S. Pat. No. 5,971,579 titled "Unit and method for determining
gains of a PID controller using a genetic algorithm", U.S. Pat. No. 5,742,503 titled
"Use of saturation relay feedback in PID controller tuning", U.S. Pat. No. 5,331,541
titled "PID control unit", U.S. Pat. No. 5,229,699 titled "Method and an apparatus
for PID controller tuning", U.S. Pat. No. 5,057,993 titled "Method and system for
acquiring parameters in process control", U.S. Pat. No. 4,754,391 titled "Method
of determining PID parameters and an autotuning controller using the method", and
U.S. Pat. No. 4,466,054 titled "Improved proportional integral-derivative control apparatus".
Based on online tuning, the above-mentioned patents are directed to methods
for autotuning of PID controllers using algorithms which measure set point values
and process variables to suggest suitable gain values.
U.S. Pat. No. 6,081,751, as shown in FIG. 1A, discloses a method for calculating
new PID controller parameters either directly through the formulae associated with
the Ziegler-Nichols reaction curve method or through the intermediate step of calculating
an ultimate period and frequency from the time constant and dead time which are
calculated from the period and amplitude of oscillation generated by a relay.
U.S. Pat. No. 5,971,579 is directed to a method for determining gains of a PID
controller utilizing a genetic algorithm unit shown in FIG. 1B.
U.S. Pat. No. 5,742,503 provides a method for autotuning parameters of a PID
controller, wherein parameters of a transfer function are computed through two
steps and precise parameters of the PID controller are determined from the computed
parameters of the transfer function.
U.S. Pat. No. 5,331,541 discloses a PID control device which identifies the
rise characteristics of a controlled system by a step response method on changing
a reference, moves to PID control when idle time and slope successively obtained
on the rise reach a predetermined value, and computes PID control parameters based
on the idle time and slope obtained up to that point.
U.S. Pat. No. 5,229,699 suggests a method for tuning PID controllers, in which
a proportional control gain is increased until a desired oscillation is obtained,
an amplitude and period are measured from the oscillation, an ultimate gain and
an ultimate period is calculated in accordance with the amplitude and period, and
the parameters of the PID controller are set in dependent upon the ultimate gain
and period.
U.S. Pat. No. 5,057,993 introduces a method for acquiring parameters in the
process control. According to this method, a manipulated variable to which an identification
signal from an identification signal generator is added is inputted to a process
to produce a controlled variable output which is then sampled to obtain a dead
time and a maximum gradient, and the initial values of PID control parameters are
calculated on the basis of the dead time and the maximum gradient. The initial
values of the PID control parameters and the like are set in the adaptation section.
In the adaptation section, a pulse transfer function of the process is acquired,
and PID control parameters are calculated from the acquired pulse transfer function
by using a partial matching method in a frequency region.
U.S. Pat. No. 4,754,391 discloses a method of determining the PID parameters
for the PID controllers by monitoring a limit cycle generated in a controlled process
to obtain characteristics of the process and determining optimum PID parameters
to be used for succeeding process control on the basis of the results of the limit
cycle monitoring.
U.S. Pat. No. 4,466,054 suggests a process control apparatus which comprises
a nonlinear controller connected in parallel with a PID controller, a circuit for
identifying a dynamic characteristic of a process and a circuit for determining
gains of the PID controller according to the dynamic characteristic, which is illustrated
in FIG. 1C.
All of the prior arts described above suggest methods for autotuning a PID controller
to acquire a desired response during an online control process and realization
of the methods in a hardware system.
Such conventional methods, however, require additional systems, thereby making
the structure of PID controllers to be complicated. Further, these methods suffer
from the disadvantage of requiring a lot of effort for calculating necessary intermediate values.
In addition, the object systems to which these conventional arts can be applied
are limited to linear systems.
SUMMARY OF THE INVENTION
It is, therefore, an object of the present invention to provide a method for
tuning
a PID controller applicable to nonlinear MIMO systems represented in second order
phase variable form without adding any supplementary devices to the PID controller.
In accordance with an aspect of the present invention, there is provided a method
for tuning a PID controller, wherein the PID controller is comprised in a closed
PID control loop system, the PID control loop receiving an input, the PID controller
being coupled to an object system being controlled, wherein the object system outputs
process variables which is supplied for comparison to the input, wherein a result
of said comparison is supplied to the PID controller, the method comprising the
steps of: inducing equivalent relationships between PID gains of the PID controller
and parameters of time delay control (TDC); selecting a natural frequency vector
and a damping ratio vector so as to acquire a desired error dynamics of the closed
PID control loop system; selecting a sampling time of the closed PID control loop
system; determining the parameters of TDC on the basis of the natural frequency
vector, the damping ratio vector and a closed loop stability condition for TDC;
and selecting PID gains of the PID controller on the basis of the equivalent relationships.
BRIEF DESCRIPTION OF THE DRAWINGS
The above and other objects, features and other advantages of the present invention
will be more clearly understood from the following detailed description taken in
conjunction with the accompanying drawings, in which:
FIG. 1A is a flow chart showing a conventional method for tuning a PID controller;
FIG. 1B is a block diagram showing another conventional method for tuning a
PID controller;
FIG. 1C is a block diagram showing further another conventional method for tuning
a PID controller;
FIG. 2 is a schematic view showing the overall configuration of a control system
in accordance with an embodiment of the present invention;
FIG. 3 is a block diagram showing a control object system and a PID controller
in accordance with the present invention;
FIGS. 4A and 4B are flow charts showing the process of designing a PID controller;
FIG. 5 is a perspective view showing a robot manipulator with six degrees of
freedom which is an object system of a PID controller in accordance with the present invention;
FIG. 6A is a graph showing response when a PID controller designed according
to the present invention is applied to an object system subject to a step trajectory;
FIG. 6B is a graph showing control input obtained when a PID controller designed
according to the present invention is applied to an object system subject to a
step trajectory;
FIG. 7A is a graph showing response error when a PID controller designed according
to the present invention is applied to an object system subject to a sinusoidal
type trajectory; and
FIG. 7B a graph showing control input when a PID controller designed according
to the present invention is applied to an object system subject to a sinusoidal
type trajectory.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The application of the preferred embodiments of the present invention is best
understood with reference to the accompanying drawings, wherein like reference
numerals are used for like and corresponding parts, respectively.
FIG. 2 shows a control system structure to which the present invention is applied.
The control system includes a digital device (e.g., computer)
21, an I/O
board
22, and an object system
23. A program for PID control runs
in the digital device
21.
The object system
23 is a nonlinear MIMO system in a second order phase
variable form and may be represented by equation 1.
where, when the order of the system is n, x is an nX1 vector of variables
to be controlled, A(x,{dot over (x)}) is an nX1 nonlinear vector representing dynamic
characteristics such as friction of the system, B(x,{dot over (x)}) is an nXn matrix
representing input distributions, and u is an nX1 input vector of the object system.
For instance, a robot manipulator may be represented by equation 1, in which case
x is a vector showing the rotation angles for the individual axes of the robot manipulator.
Referring to FIG. 3, the overall system of the present invention is shown
in a block diagram where x
d is a desired trajectory.
First, the relationship between PID control and time delay control will be
explained. Time delay control method is known as a robust control method for a
sampled data system. Based on the relationship, a gain tuning method in accordance
with the present invention will be described in detail.
The sampled data system includes both a discrete time system and a continuous
time system. When the system represented by equation 1 is to be controlled, a digital
device
21 such as a computer may be used as a controller. In this case,
the digital device
21 is a discrete time system while the object system
23 is a continuous time system.
As explained by K. Youcef-Toumi and S.-T. Wu in "Input/output liberalization
using
time delay control-Trans. Of ASME, J. Dyn. Sys., Meas., Contr., vol. 114, pp.10-19
(1992), the time delay control (TDC) is a nonlinear robust control method.
When TDC is applied to the system represented by equation 1, it is expressed
by the form of equation 2, below.
wherein e(t)=x
d(t)-x(t) is an error vector, λ is a time
delay value, K
D and K
P are nXn constant diagonal matrix determining
the overall closed loop system including the TDC and the object system to have
desired error dynamics, and {overscore (B)}
-1 is a parameter selected
to satisfy equation 3, below
##EQU1##
where the subscript i2 indicates an induced matrix 2-norm and
The induced matrix norm is defined as follows: As mXn matrix A of real elements
defines a linear mapping y=Ax from R
m into R
n, the induced
p-norm of A is defined by
##EQU2##
which for p=1, 2, ∞, is given by
##EQU3##
where λ
max(A
T A) is the maximum eigenvalue of A
TA.
In equation 3, β
1 and β
2 are gains of L
2
(see equation 9 below) whose meanings are explained below. First, ε(t)
is defined by equation 6
H and G is defined as H: ε|→e and G:ε|→ė, respectively,
and ∥•∥
T2 is defined as L
n2 norm
of •(t) truncated at time T. Considering the operator H
i:ε
i|→e
i
for each element of the vector ε(t), the transfer function between
ε
i and e
i can be expressed by equation 7:
##EQU4##
Likewise, considering the operator G
i:ε
i|→ė
i,
the transfer function between ε
i and ė
i can be
expressed by equation 8:
##EQU5##
where k
Di and k
Pi are i
th diagonal entries
of K
D and K
P, respectively. Then, the gain of the transfer
function L
2 is defined by equation 9:
##EQU6##
Further, ∥H∥
2=∥G
H∥
i2
and ∥H∥
2=∥G
H∥i2. Also,
##EQU7##
and
##EQU8##
Thus, β
1 and β
2 are given by equation 10:
βhd
1=∥H∥
2
For a sampled data system, the time t is expressed by the combination of the
sampling time (Δt) of the control system with the step number (a=1, 2, .
. . , k), that is, t=a·Δt. Accordingly, where TDC is used in the sampled
data system, the time delay value λ is the sampling time Δt of the
control system. Further, {umlaut over (x)}(t-λ) is expressed by using the
central differential method which is the error-lowest numerical differential methods
while {umlaut over (x)}
d(t) and ė(t) are expressed by using the
backward differential methods, as shown in equation 11:
##EQU9##
After substituting the parameters of equation 2 by equation 11, rearrangement
into the form of the PID controller makes equation 12:
##EQU10##
In result, equation 12 is a TDC form of a sampled data system.
Generally, a PID controller
31 may be expressed in a form of equation 13:
where K is an nXn constant diagonal proportional gain matrix, T
D
is an nXn constant diagonal matrix representing derivative times, and T
I is
a constant diagonal matrix representing a reset or integral time.
However, in order to show the relationship with TDC, a PID controller which
includes a DC component vector will be considered, as illustrated in equation 14:
where the DC component is a constant value determined by the initial error(è(0),e(0)).
In consequence, designing the PID controller
31 means selecting K, T
D,
T
I, and DC.
Application of the Laplace transform to equation 14 gives equation 15:
##EQU11##
Multiplication of s to both sides and rearrangement change equation
15 into equation 16:
sU(
s)=
K(
s+TDs2+TI-1)
E(
s)+
DC-sKTDe(0) (16)
The inverse Laplace transform of equation 16 gives equation 17:
where e(0), è(0) and u(0) are values of e(t), è(t) and u(t) at t=t
0,
respectively, and δ(t) is a Dirac delta function.
In equation 17, DC is obtained by equation 18:
where u(0)=u
TDC(0).
Accordingly, equation (17) is simplified to equation 19:
Then, equation 19 may be rewritten to equation 20:
{dot over (x)}(t), {umlaut over (x)}(t), {dot over (x)}
d(t), {umlaut
over (x)}
d(t) are obtained by the backward method, as shown in equation
21.
##EQU12##
In a sampled data system, a PID controller can be expressed as equation 22:
When equations 20 and 21 are substituted into equation 22, equation 23 is obtained:
##EQU13##
Therefore, for a sampled data system, the PID controller can be expressed
as equation 23.
Comparison of equation 12 with equation 23 shows that the PID controller
and the TDC take the same form in a sampled data system. Hence, Relationship
1
can be given between the gain of the PID controller and the parameter of the TDC:
##EQU14##
Also, the DC component can be obtained from equation 24:
In Relationship
1, Δt is the sampling time of the control system,
K
D and K
P are parameters which define the error dynamics
of the overall closed loop system and are determined by the intention of the designer.
Thus, if only {overscore (B)}
-1 is determined, all of the gains of the
PID controller can be automatically determined.
As mentioned above, {overscore (B)}
-1 is selected to satisfy equation
3 and, in general, determined by the following two methods:
(1) {overscore (B)}
-1=α (α is a constant, I is a unit matrix)
(2) {overscore (B)}
-1 is a constant diagonal matrix (({overscore (B)}
-1)
ii=α
i(i=1,
. . . , n)).
A detailed design process of the PID controller based on Relationship
1
will be explained with reference to FIGS. 4A and 4B.
First, an MIMO type object system represented as a second order variable form
is selected (Step S
1000).
Relationships between parameters of PID and TDC are derived in connection
with the object system (Step S
2000).
Then, a desired error dynamics of the object system is determined (Step S
3000).
The error dynamics is determined by the designer by selecting the natural frequency
vector (ω) and the damping ratio vector (ç) K
D and K
P
are determined on the basis of the natural frequency vector and damping ratio
vector selected in Step S
3000 (S
4000). When i
th terms
of the natural frequency vector ω
n and the damping ration vector
ç are ω
ni and ç
i, respectively, i
th
diagonal entries of the matrixes K
D and K
P can be obtained
through the relations k
Di=2ç
iω
ni and
k
Pi=ω
ni2. Afterwards, the sampling time
of the control system is determined (Step S
5000). The sampling time is preferably
set to be a small value, but influenced by the CPU speed of the digital device
210.
In Steps S
3100 and S
3200, {overscore (B)}
-1 is determined.
By one of the two methods mentioned above, {overscore (B)}
-1 is set
to satisfy equation 3. Finally, gains of the PID controller are selected on the
basis of Relationship
1 and equation 24 in Step S
3300.
EXAMPLE
An example will be described, in which the method of the present invention was
applied to a virtual object system on computer.
A Puma type robot manipulator having six degrees of freedom, like that shown
in
FIG. 5, was selected as the object system. This robot is a model depicted in the
article "The explicit dynamic model and inertial parameters of the PUMA 560 arm",
IEEE Int. Conference on Robotics and Automations, pp. 510-518 (B. Armstrong, O.
Khatib, and J. Burdick (1986)). The dynamics of the robot manipulator is expressed
by equation 25:
where M(θ) is a 6×6 inertia matrix, θ is a 6×1 vector
representing the rotation angles with respect of six axes, V(θ{dot over (θ)})
is a vector representing Coriolis force and centrifugal force, G(θ) is a
gravity vector, F(θ,{dot over (θ)}) is a 6×1 vector representing
forces which are not included in system modeling such as frictional force or disturbance
and τ is a 6×1 torque vector applied on joints. Comparing equation 1
with equation 25, one can get B(x,x)→M(θ)
-1.
Detailed design process will be described according to the process of FIGS.
4A and 4B.
For convenience, assume that poles of error dynamics of all six axes are located
as multiple roots at 5. In this case, the natural frequency ω
ni=5
(i=1, . . . , 6), damping ratio ç
i=1(i=1, . . . , 6). Here, k
Di=10
and k
Pi=25 (i=1, . . . , 6) in all of the six axes. The sampling time
of the controller is Δt=0.001 sec. Puma 560 has an arm part and a wrist part.
Since the two parts greatly differ in inertia due to the difference in mass and
structure between themselves, two values (α
1 and α
2)
were used in which α
1 is for axes 1, 2 and 3
and α
2 for axes 4, 5 and 6.
##EQU15##
From Relationship 1 and equation 24,
##EQU16##
Eighteen gains of the PID controller were selected.
Two types of trajectories shown in FIG. 3 were applied as x
d(t). One
was a step trajectory having initial errors and the other was a sinusoidal type trajectory.
Case 1: Step Trajectory
The step trajectory of equation 26 was applied to each axis.
The control results based on the values of the PID controller designed according
to the method of the present invention are given in FIGS. 6A and 6B, in which response
x(t) (FIG. 6A) and input torque T (FIG. 6B) are plotted versus time. In the upper
graphs of FIGS. 6A and 6B, axes 1 and 4 are represented by solid
lines, axes 2 and 5 by dotted lines, and axes 3 and 6
by dashed lines. Likewise, the lower graphs of FIGS. 6A and 6B have solid lines
for 1 and 4, dotted lines for axes 2 and 5, and dashed
lines for axes 3 and 6. In all of the graphs, only solid lines are
shown since the lines overlap. As apparent from the plots, all of the six axes
were satisfactorily controlled when they were under the rule of the PID controller
designed according to the present invention.
Case 2: Sinusoidal Type Trajectory
The sinusoidal trajectory of equation 27 was applied to each axis.
The control results based on the values of the PID controller designed according
to the method of the present invention are given in FIGS. 7A and 7B, in which the
response error e(t) (FIG. 7A) and input torque τ (FIG. 7B) are plotted versus
time. In the upper graphs of FIGS. 7A and 7B, axes 1 and 4 are represented
by solid lines, axes 2 and 5 by dotted lines, and axes 3 and
6 by dashed lines. Likewise, the lower graphs of FIGS. 7A and 7B have solid
lines for axes 1 and 4, dotted lines for axes 2 and 5,
and dashed lines for axes 3 and 6. In all of the graphs, only solid
lines are seen since the lines overlap. As apparent from the plots, all of the
six axes were satisfactorily controlled when they were under the rule of the PID
controller designed according to the present invention.
As described hereinbefore, the present invention provides a gain selection method
of PID controllers by which a great number of gains of PID controllers can be selected
with a small number of parameters. As exemplified in the examples, the total number
of PID controller gains to be selected in the six-axis robot manipulator is 18.
However, only two parameters (α
1 and α
2) suffice
the selection of as many as 18 gains.
Conventionally, the number of PID controller gains in an n-degree
system totals 3n. However, the method of the present invention can reduce the parameters
into one or a number less than n. In practice, when tuning a PID controller, all
of 3n gains can be suitably tuned by tuning only one to n-1 gains even though every
gain is not tuned one by one.
Additionally, in comparison to conventional methods, the present invention
can be applied to linear and nonlinear systems and even to MIMO systems. Furthermore,
the present invention enjoys the advantage of selecting and tuning gains of PID
controllers only by the program run in the digital device 21 without additional equipment.
The present invention has been described in an illustrative manner, and it is
to be understood that the terminology used is intended to be in the nature of description
rather than of limitation. Many modifications and variations of the present invention
are possible in light of the above teachings. Therefore, it is to be understood
that within the scope of the appended claims, the invention may be practiced otherwise
than as specifically described.
*
