J. Cent. South Univ. Technol. (2007)05−0701−07
DOI: 10.1007/s11771−007−0134−9
Robustly stable model predictive control based on
parallel support vector machines with linear kernel
BAO Zhe-jing(包哲静)
1
, ZHONG Wei-min(钟伟民)
2
, PI Dao-ying(皮道映)
1
, SUN You-xian(孙优贤)
1
(1. State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China;
2. State Key Laboratory of Chemical Engineering, East China University of Science and Technology,
Shanghai 200237, China)
Abstract：Robustly stable multi-step-ahead model predictive control (MPC) based on parallel support vector machines (SVMs) with
linear kernel was proposed. First, an analytical solution of optimal control laws of parallel SVMs based MPC was derived, and then
the necessary and sufficient stability condition for MPC closed loop was given according to SVM model, and finally a method of
judging the discrepancy between SVM model and the actual plant was presented, and consequently the constraint sets, which can
guarantee that the stability condition is still robust for model/plant mismatch within some given bounds, were obtained by applying
small-gain theorem. Simulation experiments show the proposed stability condition and robust constraint sets can provide a
convenient way of adjusting controller parameters to ensure a closed-loop with larger stable margin.
Key words: parallel support vector machines; model predictive control; stability; robustness
1 Introduction
The use of support vector machine (SVM) in model
predictive control (MPC) has received considerable
attention. Several papers have appeared on SVM based
MPC
[1−5]
, all of which concentrate only on the issue
related to constructing SVM model, combining SVM
model with MPC and deriving the optimal solution
[6]
.
Some papers applied the gradient descent based method
to search the control laws, which cannot be ensured to be
globally optimal
[1−3]
, and others achieved an analytical
solution of control laws
[4−5]
. However no attention is paid
to the stability of the closed-loop system.
In MPC, while the performance of the plant is
optimized over the prediction horizon repeatedly, each
optimization does not care about what happens beyond
the prediction horizon, and so the plant can be put into a
state that it will eventually be impossible to stabilize,
even though the controlled plant itself is stable
[7−8]
.
In this paper, the structure block diagram of parallel
SVMs based MPC closed loop was obtained. Due to
linear kernel, classical theoretical results can be applied
to analyze the stability of SVMs-MPC, and then
according to SVM model the stability condition was
derived. A method of judging the discrepancy between
SVM model and the actual plant was proposed, and
further the constraint sets for the discrepancy were given
to ensure that the real closed-loop system is robustly
stable.
2 Parallel SVMs with linear kernel based
MPC algorithm
The structure of parallel SVMs based
multi-step-ahead MPC is shown in Fig.1, in which H
p
(prediction horizon) parallel SVMs predictive models
constitute multi-step-ahead predictor, y
s
(k+j) is the
desired set-point value of output y at time k+j,
p
ˆ
(|)
y kjk+
is the j-step-ahead predictive output at time
k after feedback correction,
p p
ˆ
(| )
y kk H−
is the H
p
-step-
ahead predictive output at time k−H
p
after feedback
correction, z
−i
and z
−j
are the delay operators. All
sub-models do not influence each other, and then an
accumulation of prediction errors can not occur.
Assume an unknown process can be described as
follows:
(1)
(1) [(),,( 1),yk f yk yk n+= −+L
(),( 1), ,( 1)]uk uk uk m∆− −+L
(1)
where n
≥
1 and m
≥
1 are maximum lags in the outputs
and inputs, respectively, and
() () ( 1)uk uk uk∆= −−
.
According to Eqn.(1) and
() () ( 1)uk uk uk∆= −−
,
Foundation item: Project(2002CB312200) supported by the National Key Fundamental Research and Development Program of China; project(60574019)
supported by the National Natural Science Foundation of China
Received date: 2006−12−20; Accepted date: 2007−03−23
Corresponding author: BAO Zhe-jing, Doctoral candidate; Tel: +86-571-87953761; E-mail: zjbao@iipc.zju.edu.cn