An Idea on Adaptive Optimal Control of SFC
Zhang Guixu

Huang Tuanhua

IMP,the Chinese Academy of Sciences,LANZHOU,730000,CHINA
Abstract
The idea of application of optimal control theory to thecomputer control system of SFC is presented. The process
consists of mathematical modeling of SFC parameter
identifying, designing of the fastest response controller andthe techniques of realizing adaptive control.
1 Introduction
HIRFL (Heavy Ion Research Facility in Lanzhou) is alarge and modern scientific facility in our institute, SFC is
the injector of HIRFL. It is very important to match SFC
with an automatic control system. But the application ofautomatic control theory to the control of SFC has not beenrealized up to now.
This paper presents how I am preparing to employ thecontrol theory in SFC. If a satisfied result is done, I’ll
expand it to the other parts of HIRFL.
2 Designing step
According to computer control theory, the adoptivecontrol system is shown as Fig.1, where G is the controlled
facility, i.e. SFC, D is a controller, X is a observer, and Q isa identifier. The D, X and Q are all programmed incomputer.

values of variables are small incremental changing near thesteady-state operating point, which usually they can belinearized.
The mathematical model derived from the physicalformula can not help much in modeling, because it is toocomplicated to be done. In practical terms, the modelobtained from practical experience is very reliable anduseful.
The first modeling can be done by some instruments or
normal methods. A crude model can be written down when
the wave shapes of input and output are being seen on adouble trace oscillometer or a set of input and output data
are detected. For example, a step function signal is input
into a device, then its output is observed, so that the unittransient response does adequately reflect the qualitative
behavior of the device considered. The main poles andzeros of its transfer function can be estimated, so an
empirical model is obtained. Or we can use a frequency
response detector to obtain a frequency response model.
The model is expressed as :x’(t) = Ax(t) + Bu(t)y(t) = Cx(t)

(1a)
(1b)

The further modeling work for precisely determining thesystem parameters will be done by identification method ofcontrol theory. Basing sequence data of inputs and outputs[u(k),y(k)] recorded by a computer system, a group ofprediction values y$ ( k ) could be computed. To simplify
the identification algorithm for real-time application, arecursive least square parameter identification scheme isused. Its cost function is defined as:
NJ (a, b, c) = ∑ [ y( k ) − y$ ( k )]

2(2)

K =1

Fig.1 The Adaptive Control System
2.1 Mathematical Modeling of SFC
Before solving the optimal control problem, themathematical modeling of SFC must be obtained. The
mathematical model should be able to represent SFC, or G
in Fig.1.
The essential physical principle of SFC is that a charged
particle moves in a magnetic and/or electrical field. The ion
beam qualities can be adjusted by varying the currents ofmagnets and quadrapoles. The beam qualities can bemeasured by diagnostic devices.
In general, the mathematical model for SFC describes therelationship among the beam qualities (outputs) and theadjustable data (inputs) and the states. They are expressedby differential equations or their transformations. The

where a,b,c are the elements of matrices A, B, C
respectively. By means of computer, a group of parameters
a,b,c may be selected when the cost function J achieves aminimum, which is called a curve-fitting method.
A computer control system is a discrete time system. The
computer system samples measured qualities, makecalculations, and transmits adjusting data in time sequence.
Equations (1) can be changed into the following
different equations:x(k+1) = Ax(k) + Bu(k)y(k) = Cx(k)

(3a)
(3b)

where k is the sampling-adjusting ordinal number forvariables u,x,y in Fig.1, the control vector u is ldimensional, the measurement vector y is m dimensional,
the state vector x is n dimensional, here n ≥ l ≥ m, A is an
n×n matrix, B is an n×l matrix, and C is an m×n matrix.

The elements of the state vector x consist of the qualities of
the ion beam everywhere, and some physical quantities
reflected the operating states of SFC, while some of them
are employed for conventional mathematical treatment.
2.2 The Fastest Response Controller DesignThe expected beam-quality vector is expressed asfollows: r1 r 2r =   M   rm x$(k +1) = A⋅ x$(k) +B⋅u(k) +K(k +1)⋅[ y(k +1) −C(A⋅ x$(k) +b⋅u(k))]

K(k +1) = Pk
( +1)⋅C′⋅[CPk( )C′+v(k)]−1Pk( +1) = AGk( )A′+HWk( )H′
where x$ is state estimation of state vector x, that x$ isgradually approaches x, G(k) is the covariance matrix ofstate estimation x$ .

where r is an m-dimension step-function vector, and r1,r2, ..., rm are constants.
The function of the fastest response controller is to bring
the beam qualities from the initial vector y(0) = 0 to theexpected vector y(N) = [r1,r2,...,rm], with the minimal
sample-control times and without the occurrence ofoscillations.
The fastest response controller can be designed asfollows.
Let
F1 = e x p ( A T )T

F2 = ∫ exp( A T )dt ⋅ B
0where T is the sample period. Then u ( 1)  u (2 )   C F N −1 F12 = M   A F1 N − 1 F 2
u ( N ) meaning as those in (2); Two stochastic variables, w and v,
are white noise sequences with an average of zero, and thecovariance matrices of w and v are W and V,respectively.
Then, an on-line Kalman filter can be established asfollows:

C F1 N − 2 F 2
A F1 N − 2 F 2k −1[LLC F2A F2

 r1 
r 
 2 M 0  
 rmB 0  M  0 e ( k ) = r − y ( k ) = r − ∑ C F1k − i −1 F2 u ( i )i=0]

The two response of u(k) and e(k) is transformed into ztransfer function U(Z) and E(Z). Finally, the z-transfer
function matrix representation of the fastest and nonoscillation controller is obtained:D( z ) = U ( z ) ⋅ E −1 ( z )2.3 Observer X
SFC is operated in the presence of strong electromagneticfield. For this discussion, SFC is assumed as a linear
discrete-time stochastic system. The noise spectrum of thestochastic disturbances is assumed as white noise andindependent of any initial states.
Considering stochastic disturbances, the mathematicalmodel of SFC has the following form:x ( k + 1) = A ⋅ x ( k ) + B ⋅ u ( k ) + H ⋅ w ( k )y( k ) = C ⋅ x (k ) + v (k )where the definition of x,u,y,A,B,C and k have the same

The initial estimate of the unknown quantities x$ (0) andcan be approximated by prior knowledge of thecharacteristics of SFC.
2.4 The Realization of Adaptive Control
It should be pointed out that a designed optimal controller
could actually be changed into a non-optimal controllerover a long period of time. This would be due to theparameters of SFC gradually shifting after a long period ofoperation.
To prevent this, we must consider the use of adaptive
control techniques. Recently, a number of adaptive control
algorithms have been proposed. The dynamic parametersof SFC would be updated in every sampling period.
Here An Adaptive Observer and Identifier can beconsidered.
The mathematical model of SFC can be transformed into
the following standard form: x 1′ a1x′a 2 2 x 3′  =  a 3  M  M
 x n′  a n1− r20M0y = [1LLLOL10− r3
M000L1   x 1   b1 0   x 2  b 2    0   x 3  + b3 u   M  M   M − rn   x n   b n 0 ]x

In Fig.1, the output of its adaptive observer X and
identifier Q, the estimate vectors x$ , a$ , b$ , all gradually
approach the real-value vector x,a,b. But the methods only
solve problems for SISO system. For a MIMO System thestate equation of the system model ought to be transformed
into a diagonal form, then these adaptive control methods
could be applied appropriately.
3 ProjectThis engineering adaptive optimal control in SFC is
expected to be started next year. Firstly, get familiar withthe present state of our beam tuning. Secondly, try to set up
the model of SFC. Thirdly, design the controller, observer
and identifier on computer. Fourthly, try and fail, until
success. I hope we may get a satisfied result and expand toother parts of HIRFL. I believe that proper control theory
application would result in significant benefits.