Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 146–149___________________________________________________________________________________________________________Weighted Multiple Model Adaptive Control for a Categoryof Systems with Colored NoiseYuzhen Zhang

School of Automation and Electrical Engineering, University of Science and Technology Beijing, 30 Xueyuan Road
Haidian District, Beijing, China
Weicun Zhang

School of Automation and Electrical Engineering, University of Science and Technology Beijing, 30 Xueyuan Road
Haidian District, Beijing, China
Qing Li

School of Automation and Electrical Engineering, University of Science and Technology Beijing, 30 Xueyuan Road
Haidian District, Beijing, China
E-mail: zhangyz_ustb@163.com, weicunzhang@ustb.edu.cn, liqing@ies.ustb.edu.cn
www.ustb.edu.cn

Abstract
The multiple model adaptive control (MMAC) of discrete-time stochastic system with colored noise is consideredin this paper. Model set contains one adaptive identification and some fixed models. Based on the output errors ofthe models, a simple weighting algorithm is adopted with guaranteed convergence. The proofs of the stability andconvergence of system are presented. Besides, the influence of initial value of adaptive model on system
performance is described. Finally, computer simulation results can verify the theoretical results.
Keywords: discrete-time; colored noise; multiple model adaptive control; stability; convergence.

time. In Refs. 4 and 6, the stability and convergence areproved based on the virtual equivalent system (VES)
theory. But, its model set just includes fixed models. If
the real model isn’t in the model set, the system is notgood.
Considering the above problems, in this paper, afinite number of fixed and one adaptive identification
models are designed to cover the discrete-time
stochastic system with colored noise.

1. Introduction
As the largely change of system parameters makes thedynamical system become complicated. The classicaladaptive control can’t meet the requirements. Therefore,
MMAC developed. The most suitable controller is used
to control the system1.
The weighted multiple model adaptive control(WMMAC) was researched in 1960’s2. In that system,
Kalman filter and posterior possibility evaluator (PPE)
algorithm are important3. Later, K. S. Narendra put
forward the concept of switching multiple model
adaptive control4, 5. To some extent, WMMAC can be
regarded as a kind of slow switching4, 5 . Therefore, thescheme can alleviate the vibration when switching.
However, the system stability analysis of theWMMAC has been plaguing the researchers all the2. Description of WMMAC
For analyzing system clearly, a block diagram of ageneral discrete-time WMMAC system is shown inFig.1.
The meanings of symbols are shown: P is theuncertainty plant; Ci is a local controller corresponded

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 146–149___________________________________________________________________________________________________________

recursive extended least square is used to estimatedparameter7. That isθˆ(k )= θˆ(k − 1) + K (k )[ y (k ) − ϕˆ T (k )θˆ(k − 1)]P(k − 1)ϕˆ (k )(6)
 K (k ) =1 + ϕˆ T (k ) P(k − 1)ϕˆ (k ) P(k ) =[I − K (k )ϕˆ T (k )]P(k − 1)
to the fixed model; Cθ is the local controller
corresponded to the adaptive identification model;pi (k ) is weighted value for its local controller; pθ (k )is the weighted value for the adaptive identification
controller. Through the algorithm, the weighted values
are known. Then, the u (k ) can be obtained by weighted
sum.
C1
yr(k) +-.
.
.

Ci.
.
.

where
ϕˆ (k ) = [− y (k − 1), ⋅⋅⋅, − y (k − na ), u (k − d ), ⋅⋅⋅, u (k − d − nb )

p1(k)
ω(k)

pi(k)

∑u(k)

P


, ωˆ (k − 1), ⋅⋅⋅, ωˆ (k − nc )]Tˆθ [aˆ1 , ⋅⋅⋅, aˆna , bˆ0 , ⋅⋅⋅, bˆnb , cˆ1 , ⋅⋅⋅, cˆnc ]T=T ˆ
ωˆ (k ) = y (k ) − yˆ (k ) = y (k ) − ϕˆ (k ) θy(k)pn(k)

Cnpθ(k)Cθ

As initial values will affect the system performance,
we will discuss that specifically in simulation studies.
For each local model, its output yi (k ) is used todefine the output error ei (k ) . As shown in Fig.1, theglobal control u (k ) is obtained byFig. 1. Concise block diagram of a general WMMAC system

Consider the following discrete-time dynamicalplant, the mathematical model is=A( z −1 ) y (k ) z − d B( z −1 )u (k ) + C ( z −1 )ω (k )= z − d B( z −1 )u (k ) + u (k )N +1u (k ) = ∑ pi (k )ui (k )(1)

In this paper, the algorithm4 to calculate weight is

p i=(0) li=(0)y (k ) = ϕ (k − d )θ + ω ( k )ω (k − 1), ⋅⋅⋅, ω (k − nc )]T ∈ R ( na + nb +1+ nc )×1(9)where α is 0.001, avoiding li′(k ) = 0 .

(2)′ (k ) = min{li′(k )}
lmin

(11)

′ (k )lminli (k − 1)li′(k )

(12)

ili (k )=pi (k ) =(3)l (k )∑i
N +1l (k )j =1 j(13)

The convergence character is presented in Ref. 4.
In the computer simulation, in order to prevent theweight converge to zero, we can limit the thresholdvalue.

whereϕ (k − d ) = [− y (k − 1), ⋅⋅⋅, − y (k − na ), u (k − d ), ⋅⋅⋅, u (k − d − nb ),1
N +1

1 kα + ∑ ei2 ( j ); ei (k ) =li′(k ) =y (k ) − yi (k ) (10)k j =1y (k ) and u (k ) are the output and the input. The ω (k )is a zero-mean white noise. The υ (k ) is colored noise.
The plant also can be written asT(8)i =1where A( z −1 ) = 1 + a1 z −1 + a2 z −2 + ⋅⋅⋅ + ana z − na) b0 + b1 z −1 + b2 z −2 + ⋅⋅⋅ + bnb z − nb B( z −1 =
−n−1−1−2C ( z ) = 1 + c1 z + c2 z + ⋅⋅⋅ + cnc z c−1υ = C ( z )ω (k )

(7)

(4)

=
θ [a1 , ⋅⋅⋅, ana , b0 , ⋅⋅⋅, bnb , c1 , ⋅⋅⋅, cnc ]T ∈ R ( na + nb +1+ nc )×1 (5)3. Stability and Convergence of the System
On the basis of VES4, 6, this section gives the stability
proof of the WMMAC system.

M= {M i , M θ=, i 1, 2, ⋅⋅⋅, N } is the model set. For
every fixed model, the corresponded controller isdesigned by pole placement control. At the same time,
the one adaptive identification model is designed byindirect pole assignment self-tuning control. The

3.1. WMMAC for LTI plant
Theorem 1. If a WMMAC system satisfies:

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 146–149___________________________________________________________________________________________________________The true model of plant, say M j , is included in the
fixed models of the model set Μ ;• Model M j generates with probability one theminimum output error in the sense that∑ k e 2j (r ) <∑ k ei2 (r ), ∀k ≥ d + 1, i ≠ j
 r 1 =r 1=kke 2j (r ) → R j ; 1k ∑ r 1 ei2 (r ) → Ri
 1k ∑ r 1 ==where R j is a constant, Ri may be constant orinfinity, R j < Ri , i ≠ j ;• Each local controller is well defined such that Ci is•

Proof. First, the weighting algorithm is convergent. So,pθ (k ) → 1; pi (k ) → 0, i =1,..., NAs the VES for this WMMAC is similar to the onein Ref. 8. The influence of the parameter identification
process on the system will be included in ei (k ) .
Follow the proof process of Ref. 8, the VES for
WMMAC can be decomposed into three subsystems.
Then, VES theory can be used to verify the stability.
3.2. WMMAC for parameter jump plantstabilizing M i =, i 1, 2, ⋅⋅⋅, N , and the output of theresulting closed-loop system {Ci , M i } ,say yi (k ) , istracking the reference signal yr (k ) in the sense that

To consider the parameter jump plant, we assume thatthe system parameters are piecewise constant functionof time, and time interval of parameter jump is longenough.
According to the assumptions, we know that eachstage is a WMMAC for LTI plant. The overall system isslow switching between all the stages. Therefore,switching system theory9 can be utilized to verify thestability of the WMMAC system.

2lim 1 ∑ r =1[ yi (r ) − yr (r=)]
R ′, R ≤ R ′ < ∞ .
kk →∞ kthen it is stable.
Proof. First, the weighting algorithm is convergent. So,p j (k ) → 1; pi (k ) → =0, i 1,..., N , i ≠ j; pθ (k ) → 0 (14)where p j (k ) , pθ (k ) index weights of the true modeland the adaptive identification model.
Then, the VES for WMMAC in this case is similar
to the one in Ref. 4. The influence of the adaptive modelon the system will be included in ∆u (k ) . And,ei (k ) = y (k ) − yi (k ) = y (k ) − ϕ T (k − d )θ i

4. Simulation Studies
In this section, the results of computer simulations will
be presented. Consider an uncertain discrete-time plant(15)(1 + a1 z −1 + a2 z −2 ) y (k ) = z −1 (b0 + b1 z −1 )u (k )where ϕ (k − d ) = [− y (k − 1), ⋅⋅⋅, u (k − d ), ⋅⋅⋅, ωˆ (k − 1), ⋅⋅⋅] .
Follow the proof process in Ref. 4, VES theory canbe used to verify the stability.
Theorem 2. If a WMMAC system satisfies:• The true model of plant isn’t included in the fixedmodels of the model set Μ ;• Model M θ generates with probability one theminimum output error in the sense that∑ k eθ2 (r ) <∑ k ei2 (r ), ∀k ≥ d + 1, i =1,..., N r 1 =r 1=
kk1eθ2 (r ) → Rθ ; 1k ∑ r 1 ei2 (r ) → Ri
k ∑ r 1==where Rθ is a constant, Ri may be constant orinfinity, Rθ < Ri ;• The adaptive controller Cθ is stabilizing M θ ,and
the output of the resulting closed-loop system{Cθ , M θ } , say yθ (k ) , is tracking the reference
signal yr (k ) in the sense that

+ (1 + c1 z −1 )ω (k )where ω (k ) is(17)obtained by sqrt (0.01) ∗ randn(1) inMATLAB, c1 = 0.5 . This discrete-time model can beobtained by the simple continuous-time in (18) withsample time ts = 0.5s and the zero order hold.

h(18)s − 3s + 2
For simplicity, four fixed local models are used as
the uncertainty of h , that is, h1 = 0.7 , h2 = 0.8 ,h3 = 1.2 , h4 = 1.3 . The four corresponded controllers
are designed by pole placement control.
The expected closed-loop characteristic polynomial
is corresponds to the characteristic polynomial of thefollowing continuous-time second-order systemωn 2(19)
2s + 2ξωn s + ωn 2
G (s) =2lim 1 ∑ r =1[ yθ (r ) − yr (r=)] R ′, R ≤ R ′ < ∞ .
k(16)k →∞ kwhere ξ is 0.7, ωn is 1.

then it is stable.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 146–149___________________________________________________________________________________________________________Then we can know the four fixed controller easily.
At the same time, the adaptive identification model isdesigned by indirect pole assignment self-tuningcontrol. The recursive extended least square is used toestimated parameter.
The
initialvalueis106 ∗ eye(na + nb + 1) . The calculation in detail isshown in formula (6) and (7).
The true model of plant is not included in the fourfixed model, which h is 1. The initial value of adaptivemodel θ will affect the system performance.
The initial value θ = (θ1 + θ 2 + θ3 + θ 4 ) 4 .The
simulation results are shown in Fig.2 and Fig.3.

In the two cases, the controller 5 designed according
to the adaptive identification model all are chosenrapidly. However, the latter one works better.
5. Conclusions
In this paper, a finite number of fixed and one adaptive
identification models are designed for a category ofsystems with colored noise. This method can effectively
deal with issue when the real model isn’t in the model
set. And, the stability of WMMAC in the case is proven.
At the same time, an adaptive model with real-time
initial value, the parameter value of the fixed model
closest to the real model, can improve the systemperformance.
Acknowledgements
This work was supported by National Natural Science
Foundation of China (61520106010) and National Key
Technologies R&D Program (2013BAB02B07).

Fig. 2. Controller weight signals of case 1

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Fig. 3. Output, reference and control signals of case 1

The initial value is the parameter value of the fixedmodel closest to the real model. The simulation results
are shown in Fig.4 and Fig.5.

Fig. 4. Controller weight signals of case 2

Fig. 5. Output, reference and control signals of case 2149