Multiple Models1)

WANG Xin1LI Shao-Yuan1YUE Heng2

1(Institute of Automation,Shanghai Jiaotong University,Shanghai200030)

2(Research Center of Automation,Northeastern University,Shenyang110004)

(E-mail:wangxin26@sjtu.edu.cn)

Abstract To solve the problem such as too many models,long computing time and so on,a

hierarchical multiple models direct adaptive decoupling controller is designed.It consists of multiple

levels.In the upper level,the best model is chosen according to the switching index.Then multiple

fixed models are constructed on line to cover the region which the above chosenfixed model lies in.

In the last level,one free-running and one re-initialized adaptive model are added to guarantee the

stability and improve the transient response.By selection of the weighting polynomial matrix,it not

only eliminates the steady output error and places the poles of the closed loop system arbitrarily,but

also decouples the system dynamically.At last,for this multiple models switching system,global

convergence is obtained under common assumptions.Compared with the conventional multiple

models adaptive controller,it reduces the number of thefixed models greatly.If the same number

of thefixed models is used,the system transient response and decoupling result are improved.The

simulation example illustrates the power of the derived controller.

Key words Multiple models,hierarchical,direct adaptive control,decoupling,pole placement

1Introduction

It is known that a system with unknown time invariant or slowly time varying parameters can get good performance by using an adaptive controller.But when the system boundary condition changes,the subsystem fails,or large external disturbance presents,the parameters of the system will change abruptly and large parameter errors will generally result in poor transient response[1].To solve these problems,multiple models adaptive controller(MMAC)was proposed.For a continuous time system,Narendra et al.used multiple adaptive models to identify the unknown system parameters simultaneously[2].These models were of different initial values and were used to cover the region where the system parameters changed.At any instant one best model was chosen according to the switching index and the corresponding controller was designed.However,after some instants multiple adaptive models would converge to a neighborhood and lose the power when the system parameters changed abruptly again.Thus,multiplefixed models with two adaptive models were used to overcome this problem[3].In[4]the above results were extended to the discrete-time system.But all of these were dealt with the single input single output(SISO)system and adopted the indirect adaptive algorithm, which not only needed to solve equations on line but also degraded the robustness of the algorithm[1].To solve this problem,multiple models direct adaptive decoupling controller(MMDADC)was proposed[5]. The interactions of the system variables were viewed as measurable disturbance and were eliminated using a feedforward strategy[6].

In an MMAC,to improve the transient response,a large number offixed models are needed to cover the region where the system parameters change.In the simulation example in[7],when only one parameter changed,hundreds offixed models must be used.This increased the computation,raised the system expense,even affected the choice of the sampling period of the discrete-time system.So how to reduce the number of the models is an important issue in MMAC,which blocks its practical use in the industrial process[8].Zhivoglyadov et al.presented a localization method.At each instant,incorrect models were discarded on line to guarantee the global stability[8].In[9]a method called Moving Bank was proposed.It tuned the center of the parameter set dynamically to cover the optimal estimation224ACTA AUTOMATICA SINICA Vol.31

No.2W ANG Xin et al.:Multivariable Adaptive Decoupling Controller Using (225)

low,p k

p k low

low+h×226ACTA AUTOMATICA SINICA Vol.31

1+X(t−k)T X(t−k)

[y f i(t)−X(t−k)Tˆθi(t−1)](21)

where y f i=t ii(z−1)y i(t)is the auxiliary system output,X(t)=[y(t)T,···;u(t)T,···,v(t+k−k2)T,···,1]T is the data vector,Θ=[θ1,···,θn]is the controller parameter matrix andθi=[g0i1,···,g0in; g1in,···;h0i1,···,h0in;h1i1,···,h1in,···]T,i=1,2,···,n.The scalar a(t)is designed to avoid the singu-larity problem when estimatingˆH(0).IfˆH(0)is singular,let a(t)equal another constant value in the intervalσIn the i th level,(i=1,2,l+1),the switching index is as follows

J i,s= e f i,s(t) 2

1+X(t−k)T X(t−k)

,s=1,···,m i(22)No.2W ANG Xin et al.:Multivariable Adaptive Decoupling Controller Using (227)

1+X(t−k)T X(t−k)

=0(24) Then there must exist an instant t s,when t>t s

e

f

l+1,2i

(t)2

1+X(t−k)T X(t−k)

i=1,···,n(25)

which means that after instant t s,nofixed controller models can be chosen as the controller.The controller is selected between the free-running adaptive controller model2and the re-initialized adaptive controller model3.According to the switching index(22),it follows that

0 e f i(t)2

1+X(t−k)T X(t−k)

(26)

So

lim t→∞

e i(t)2228ACTA AUTOMATICA SINICA Vol.31

No.2W ANG Xin et al.:Multivariable Adaptive Decoupling Controller Using (229)230ACTA AUTOMATICA SINICA Vol.31下载本文