2010 8th IEEE International Conference on Control and Automation Xiamen, China, June 9-11, 2010
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Conjugate Gradient-based Parameters Identification Shengyu Wang, Hong Mi, Bin Xi, Daniel(Jian) Sun Abstract—In the paper, the conjugate gradient algorithm is introduced in detail and applied in system identification. After the problem of parameter estimation being transformed to a linear system equation, the problem can be solved by conjugate gradient iteration. Relative error of parameter identification is less than 0.1 percent. This simulation result shows its feasibility in parameter identification. Indexing terms:conjugate gradient algorithm, parameter identification, feasibility
I.
INTRODUCTION
A
S we know, system identification is crucial for control, supervision and fault diagnosis, etc. Model structure selection and parameter identification constitute the consecutive steps for system identification. Generally, model structure can be determined in two ways: one is the discretization of continuous plant model; the other is the so called "black box" model. In this paper, the first method is used to determine the model structure. Under the selected model structure, parameter identification can be viewed as a process to find the solution of an optimization problem in some sense. Usually the final form of the optimization result is linear system equation. How to solve this equation is the main objective of this paper. In practice, analytical solutions of the model are not able to be solved. Fortunately, numerical methods which can give us approximate solutions instead of analytical solutions can be used to solve the problem. There is a lot of work where a variety of effective calculation methods are discussed. For example, reference [1] has classified a variety of numerical methods in detail, and made a related presentation. In the past 60 to 70 years, the application of PID controller and digital computers promotes the research and development numerical methods, parameter identification has been a tremendous development because of these development. Reference [1] presents and discusses the emergence of conjugate gradient algorithm and discussed the algorithm. [4] makes presentation on the conjugate gradient algorithm and the convergence of the process. More and more effective numerical methods (such as: the direct search method,
steepest descent method, least squares) are used in various industries of different fields. In this paper, the conjugate gradient algorithm is applied in parameter identification. II.
PARAMETER IDENTIFICATION
Review Many scholars have done a lot of research in the area of recognition of plant model parameters. In this paper, iterative conjugate gradient algorithm would be introduced for identification of plant model parameters. On the other hand, conjugate gradient algorithm can ensure the convergence of iteration, consequently, it is a very good candidate for parameters identification. It has been shown that conjugate gradient algorithm could make iterative solution converge to the exact solution of equations, that is, it would converge to the exact solution of equations in less r +1 iteration steps '
(r=rank( A '⋅ I ), A is matrix in equation (1)).Interested readers can refer to the reference [1] for the proof of this conclusion. Conjugate Gradient Algorithm In some cases, the steepest descent method would converge slowly. In order to overcome this shortcoming, Hestenes and Stiefel contrived so called conjugate gradient algorithm (or CG) in the early 1950s[1].The algorithm does not need any known parameter to find solution of symmetric, positive definite, linear equations. This approach is very popular in solving the large sparse linear equations. Because of its applicability, relevant research has developed rapidly. This causes relevant algorithm and theory become quite mature. Brief summary of the conjugate gradient algorithm is made in the following. The following linear equation is introduced: A' x = b (1) Where, A ' —given symmetric positive definite matrix in n order b —given vector in n dimension x —unknown vector of n dimensions Choosing a suitable direction vector { p0 , p1,⋯ pk , ⋯} , conjugate gradient algorithm makes iterative equation (1) *
Shengyu Wang is with School of Information Science and Technology, Xiamen University, Xiamen 361005, China(e-mail:
[email protected]) Hong Mi is with School of Public Affairs Zhejiang University, Hangzhou 310008, China (phone:13328771835; e-mail:
[email protected]) Bin Xi is with School of Information Science and Technology Xiamen University, Xiamen 361005, China (e-mail:
[email protected]) Daniel(Jian) Sun is with School of Transportation Engineering Tong Ji University, Shanghai 201804, China (e-mail:
[email protected])
978-1-4244-5196-8/10/$26.00 ©2010 IEEE
convergence to true solution— x of equation (1) in finite steps [1].
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x0 ( initialvector ) x k +1 = xk + α k pk ( k = 0,1,⋯)
(2)
Therefore, selection of the direction vector is the key issue
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of conjugate gradient method. [1] deals with this topic. It
α n −1 = −
indicated that the selected vectors { p0 , p1,⋯ pk , ⋯} should be orthogonal each other with respect to Λ . That
( p , Λp ) = 0
is:
i
j
when
i ≠ j
λn =
and
pi ≠ 0 ( i = 0,1,⋯, n − 1) . We then have
(r (r
n
(r (p n
r
, Ap n−1
n −1
, Ap
,r n
, Ar
) n
)
)
n −1
)
( n = 1,2,⋯)
( n = 1,2,⋯)
4, x n +1 = x n + λn p n , go to step 2 So far, conjugate gradient algorithm has been briefly
n −1
x * = x0 + ∑ α i pi
(3)
introduced. In the rest parts of paper, application of CG in
i =0
Now, we can use iteration
system identification will be discussed.
n −1
xk = x0 + ∑ α i pi
(4)
III. EXPERIMENTAL MODEL
i =0
Furthermore,
In the control of bottle blowing machines, temperature
x * = xn
system model of bottles embryo and mold is a typical
(5)
temperature system model. Furthermore, temperature of could be hold. That means the true solution would be
bottles embryo and mold model affect the quality of Blower
reached in finite steps through equation (4).The process of construction of α i , Pi will be detailed in the following
in a large extent. So the control of temperature of bottles
iterative process of conjugate gradient(CG) algorithm. For given linear equation such as (1), iterative process of conjugate gradient(CG) algorithm is detailed as follows: T
1, initialize initial vector x (for example x = [0,0] ) 0
0
embryo and mold is an important element in controls of bottle blowing machine. In order to control temperature adaptively, identification of model parameters is the first problems to be solved. Least-squares method was applied for identification of plant in Reference [6]. For temperature plant, the transfer function is first-order plus time delay plant
2, r n = b − Ax n
model. And mathematic expression in form of discretization
If
is:
rn ≤ ξ
τ U k − K pTs + TY ( k − 1) Ts Y (k ) = T + Ts
Where,
ξ is the threshold of maximum permissible error. Then
(6)
Where, n
Output x and terminate program
K p —gain parameter
else
τ —lag parameter, which is assumed to be known before r
n
≤ξ
identification
T —inertia coefficient
go to step 3
Ts —Sampling period, which could be specified before
r , ( n = 0 ) 3, p n = n n −1 r + α n −1p , ( n = 1,2,⋯) n
identification This is the experimental model, and it was just introduced briefly here. The pre-processing of model parameters and identification of model parameters will be discussed as
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following.
A X = B
IV. PRE-PROCESSING AND IDENTIFICATION
Y ( k i ) ; Y = Y ( k j )
As mentioned above, the model of object in temperature
In order to guarantee positive definite of the weight matrix
control plant can be expressed in mathematical form as
in equation (9), both sides of equation (9) would be left
equation (6). (6) could be rewritten in form of (7):
τ Y ( k ) = AU k − T s
+ BY ( k − 1)
multiplied by W
. So equation (9) is transformed to
equation (10):
(7)
W T WX = W T Y
(10)
After a serie of transformation, parameters A , B can be
Where,
A= For
T
given
K pTs
B=
T + Ts
sequence
identified easily with conjugate gradient algorithm. Finally,
T T + Ts
relationship between A , B and T , K p will be used for
U (1), U (2),⋯, U ( k ),⋯ and
Y (1),Y (2),⋯,Y ( k ),⋯ , one can construct equation group
calculation of parameters T and K p . Before experiment, it
(8) with two samples in different time(i.e. U ( k i ),Y ( k i ) and
is necessary to filter the samples. The average filter would be used here. Before the process of filter will be introduced, we
U ( k j ),Y ( k j ) ; where k i ≠ k j ):
should keep one point in mind—ten samples would be selected for one sample. The process of filter is described as
τ Y k i = AU k i − + BY ( k i − 1) Ts Y k = AU k − τ + BY k − 1 (j ) j ( j) Ts
follow. Firstly, when ten samples are obtained, the average
( )
value would be computed; secondly, any samples would be (8)
eliminated if absolute difference between its value and average value is bigger than given threshold; third, new mean would be computed for the rest of samples. Entire
Equation group (8) could be rewritten in form of matrix,
shown in Fig.1.
and equation (9) would be gotten:
WX = Y
process of is showed once again in form of flow chart. This is
(9)
Where
τ U ki − ,Y ( ki − 1) Ts W = ; τ U k − ,Y ( k − 1) j j Ts
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Fig.1. flow chart for filter process
V. EXPERIMENT
Fig.2.exprment result for two different models. Initial value for
Now, identification of model parameters would be
A and B are both equate to one and ξ = le − 5
in two different experiments.
simulated in environment of matlab7.8.0 (R2009a). Two
a b
first-order plus time delay (FOPDT) plant models are supposed. Their transfer functions are expressed as follow:
What can we get from Figure 2 is two curves appear at the
Transfer function for model (a)
beginning of convergence process for either parameter A
K p = 2;τ = 5;T = 10
or B . Why? The reason is that iterative process for conjugate
Transfer function for model (b)
gradient algorithm is an oscillating and convergent process.
K p = 5;τ = 10;T = 30
With relationship between A , B and T , K p ; parameters
Parameter identification results are showed respectively in
T , K p can be computed. The results are: model (a) K p
Fig.2 . Fig.2.a is the result for model(a) and Fig.2.b if the result for model(b).
=2.00 T =10.00 model (b) K p =5.0 T =30.00. The results above show that identification of plant parameters with conjugate gradient algorithm is feasible. Relative error is
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smaller than 0.1 percent. VI. CONCLUSION This paper has discussed one method which is used for parameter identification. And result of simulated experiment shows that Relative error is within 0.1%. While r n ≤ 10 −5 and true parameters K p =5, T =30, it will reach the accuracy within 450 steps. While
r n ≤ 10 −5 and true
parameters K p =2, T =10, it will reach the accuracy within 250 steps. The results gave us another piece of information: it would take more time for large coefficient than small coefficient to identify parameter; in the other way, it would take more steps for large coefficient than small coefficient to convergence to the same accuracy. At last, what would be showed here is conjugate gradient algorithm provides a feasible way in online parameter identification. The result being displayed in paper is a reference for application of conjugate gradient algorithm in industrial. REFERENCES [1]
Qingyang Li, Dayi Yi, Chaoneng Wang. Modern Numerical Analysis (first edition). BeijingÈHigher Education Press, 1995 , pp. 240-270.
[2]
Guobo Xiang. Optimal Control of Time-delay Systems. Beijing, China Electric Power press, 2008, pp. 21-75
[3]
Bin Xu. Process Control. Beijing, China Machine Press, 2004, pp. 12-21
[4]
“Conjugate
Gradient
[5]
Cunbo Jiang, Li Liu,
Algorithm
and
Basic
Characters”.
Available: http://class.htu.cn/nla/chat5/sect5_2.htm Cong Wang, Fang Zhong. “Identification
Method and Simulation for Temperature Object Parameter”, Science Technology and Engineering, Vol.9 No.11 2009, pp. 3105-3108 [6]
Chunfeng Song, Yuanbin Hou, Junbiao Jiang, Dongyang Zhao. “Parameters Identifying In the Temperature Control System of TRP Laser Gyros”. Journal of Xi’an University of Science and Technology, Vol.25 No.2 2005, pp. 236-239
[7]
Bing Tan. “Numerieal Simulation and Parameter Identification of the Temperature Field In the Nonlinear System of the Sea Ice”. Dalian University of Technology, 2006
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