Journal of Robotics, Networking and Artificial Life, Vol. 2, No. 2 (September 2015), 69-72

Adaptive Multiple-Model Control of A Class of Nonlinear Systems
Chao Yang
The Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA),
Beijing 100191, China
Yingmin Jia
The Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA),
Beijing 100191, China
E-mails: yangchao_buaa@163.com; ymjia@buaa.edu.cn

Abstract
In this paper, an adaptive multiple-model controller is developed for nonlinear systems in parametric-strict-feedback
form. Unlike the previous results, a switching scheme is not required here to switch the most appropriate model into
the controller design. The new scheme reduces the number of identification models and uses information provided by
all the models more efficiently than previous results by using the convex combination of estimates of parameters. The
method guarantees parameter convergence and global asymptotic stability of the closed-loop system. The global
boundness of closed-loop signals and asymptotic convergence to zero of tracking error are proved. A simulation
example is included to demonstrate the effectiveness of the obtained results.
Keywords: adaptive control, multiple-model design, nonlinear systems, asymptotic tracking

1. Introduction
In recent years, adaptive control has been wellinvestigated1-5. It is proposed to cope with the problemof parametric uncertainties in systems lager than
traditional method can handle. The typical idea ofadaptive control is estimating unknown parameters online and designing controller with the estimates as thereal ones2. Plenty of different stable adaptive controllers
have been designed for linear or nonlinear uncertain
systems7-9. A distinguished problem for adaptive control
is that transient performance of the system may include
unacceptable large peaks, especially when there exist
large initial estimation errors. To overcome thisdrawback and enhance the transient performance,
adaptive control using multiple models for linear
systems was proposed in Refs.7, 8, 10and 11. The basic
idea of Ref.10 was running in parallel multiple models,and design an index of performance based on
identification errors, so that model closest to the realplant can be selected by switching quickly and transient
response can be improved. Latter, the method was
extended to switching and tuning in Refs.12 and 13.
Numerous of simulations and applications havedemonstrated adaptive multiple-model control canenhance transient performance significantly.
However, the methods are mainly concentrated onlinear systems. Adaptive multiple-model control for
nonlinear systems is initially considered in Ref.14. Then,
adaptive multiple-model control method for nonlinear
requiring persistence of excitation is proposed in Ref.15,
where the unknown parameters can be calculated in thefirst place.
In Ref.16, a sufficient condition concerning theparameter convergence is given by constructing a global,
explicit strong Lyapunov function. Inspired by this, wedevelop a novel multiple model adaptive controller for aclass of nonlinear system in parametric-strict-feedback
form. Unlike the usual hypothesis, the restrictive
matching conditions are not required, but thepersistency of excitation condition is assumed. The
control scheme employs adaptive identification models
based on adaptive parameters from a known compact
parameter set and a virtual identification model, of
which the estimation parameter is both adaptive andresettable. The algorithm reduces the number ofidentification models by using convex combination
method to estimate parameter even as the system is in

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Chao Yang, Yingmin Jia

operation. Also, global asymptotic stability of theclosed-loop system is proved.
This paper is organized as follows. The problem
formulation and assumptions are given in Section 2. In
Section 3, backstepping adaptive method and the newadaptive multiple-model method are introduced. In
Section 4, the closed-loop system dynamics is analyzed
including stability and parameter convergence. Section
5 presents a simulation which compares the proposed
method with single model adaptive control.

3. Design of adaptive multiple model controller
3.1 Introduction of backstepping adaptive control [2]The standard backstepping adaptive control andadaptation law for parametric-strict-feedback system aregiven by(4)u   n ( x,ˆ, yr( n ) )ˆ(5)   n  WZ
where

2. Problem formulationθ  [1 , 2 , , p ]T , θ  θ  θConsider the following nonlinear systems inparametric-strict-feedback form [2]:Z  [ z1 , z2 , , zn ]x1  x2  f1T ( x1 )θ

xi  xi 1  f i T ( x1 , x2 ,..., xi )θ,(1)

i  1, 2...n  1xn  u  f ( xn )θ
T
n

(6)

T

z1  y  yr , zi  xi   i 1 , 2  i  n,

W  [1 , 2 , , n ] i 1 ，fk
k 1 xki 1

1  f1 , i  fi  

Where xi  R, i  1, 2...n i s t he state o f s y st e m,

1  c1 z1  ˆT f1 ( x1 )  y r (t ),u  R is the control input , θ  R is a n
unknown parameter vector belonging to a known compact set S  R p and y  R is the output. The functions

 i   z i  ci z i  ˆ  i pT

the reference signal to be tracked. We suppose th
at the full system states are available. The object
is to improve transient performance in the presence oflarge parametric uncertainties, and meanwhile, ensure the stability of the closed-loop system.
It should be noted that in the classical backstepping
adaptive control design, transient performance can be
improved by choosing sufficiently large high-gain
parameters2. However, a possible shortcoming of suchmethod is that the resulting control efforts may be verylarge. Another way to improve transient performance isusing multiple identifications models.
Before the design of adaptive multiple-model
controller, we need necessary assumptions which areuseful in the following analysis.
Assumption 1(A1) There is a known positive realvalued constant B satisfying(2)

Assumption 2(A2) f1 ( yr (t )) satisfies the classicalpersistent excitation condition，that is，for all tR,
there exist positive real-valued constants μ and T such
that
t

t T

f1 ( yr (l )) f1 ( yr (l ))T dl(9) i 1
 i
ˆi 1 
 i 1 ( k )yr )  ( i 1 xk 1  i 1 xk 1 
x
x


k 1
yr( k 1)kk

f i , i  1, 2, , i are known smooth functions. yr  R is

I  

(8)

 1  1 z1 , i   i 1  i zi ,y  x1

max{| yr (t ) | ,| y r (t ) | }  B(7)i 1 
  ( k 1 ) i z k  y r ( t ), 2  i  n
ˆ
k 2

(10)
where     0 is the adaptation gain matrix, yr isthereference
signaltobetrackedandci  0,1  i  n are constants to be designed. The
T

above control design is based on the Lyapunov function
1 n
1Vn ( z1 , z2 , , zn ,  )   zi2  (ˆ   )T  1 (ˆ   )2 i 1
2
The time derivative of Vn , computed with (5)-(10) isn.

Vn ( z1 , z2 , , zn ,  )   ci zi2 .This

impliesthei 1boundness of the state zi ,1  i  n and ˆ , which in turn
indicates

theboundnessofsystemstate

xi ,1  i  n and control u . Under the standard
arguments of adaptive control theory, it follows thatzi (t ),1  i  n tend to zero asymptotically, and thus

lim z1 (t )  lim( y (t )  yr (t ))  0. Then asymptotic
t 

t 

tracking can be achieved.
Remark: In addition to the boundness of the(3)parameter estimation, there are seldom results aboutparameter convergence. Ref.6 provides sufficientcondition for parameter estimates convergence, but thecondition is difficult to verify. With assumptions (2) and(3), Ref.16 constructs an explicit, global, strong
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Adaptive Multiple-Model Control

Lyapunov function, and proves parameter estimation
convergence. See Ref.16 for more details.

3.2 Design of adaptive multiple-model controller
We will run in parallel (p+2) models. N (N=p+1)
models are in the same structure but different initial
parameter values  j (0), j  1, 2..., N . The initial valuesof  j (0), j  1, 2..., N are known and the region of
uncertainty S of the plant parameter vector θ lies in theirconvex hull K (i.e. S  K).The other one model called‘ virtual model ’ is designed with adaptive andresetting mechanism. The N models will run paralleland parameter estimates are evaluated online by acriterion function J(t).Definet

J j (t )   e 2j (t )    et  e 2j ( )d

(11)

0

as the criterion function for the j-th adaptive model,
where   0,   0 are constants to be designed. Jc is thecriterion function for virtual model. At each resetting
time tr, if J c    min{J1 , , J N } , where   (0,1) is aconstant to be designed, reset the estimate parameter
vector e (t ) of virtual model byN(12) (t )   



e

i 1

i i

where
i 

1
J i (t )

N

1

j 1

j

 J (t ) , i  1, 2, , N

.

(13)

Otherwise, the estimate parameters remainunchanged and the process repeats. An obvious problem
is that whether the resetting is finite or infinite. In thefollowing section, we will prove that only a finite
number of resetting can occur.

4. Stability analysis
Theorem Suppose assumptions (2),(3) hold, and theadaptive multiple-model controller (4) and adaption law(5) in the paper are applied to system (1). Then, for all
initial conditions only a finite number of resetting canoccur, all closed-loop states and  , j  1, 2,, n are bounded,furthermore asymptotic tracking is achieved ,i.e.,lim( y (t )  yr (t ))  0 as t→∞.
t 
j

t

] , where T is the resetting interval, andresetting N r  [ maxT[·] means the floor function. Thus finite number ofresetting is proved. Further, after every resetting time tr ,all adaptive models and the resetting model operate
under respective control input and adaptation law, thenall closed-loop state is bounded. Once t  tmax , thesystem will operate under the classical backstepping
scheme,sowecanfurther
point
out
lim(
y
(
t
)

y
(
t
))

0
r
.This completes the whole proof.
that t 

5. Simulation result
The nonlinear system in parametric-strict-feedback
form is as follows:x1  x2   x1
x2  u   cos( x1 x2 )y  x1
where   [0.1, 5.5] is an unknown parameter. The
output y  x1 is to asymptotically track the reference
signal yr  sin t .
In simulation, the controller and adaptation law is
B  1 , (2) holds.
developed as (4), (5). For all constantt
2
For any 0  (0,  ) , t  2 yr (l ) dl   0 follows for
all t  R , then let   0 , T  2 , and (3) holds. The
unknown parameter is   1 , c1  1.2, c2  1 , the adaptive
gain   1 , initial state x1 (0)  x2 (0)  0 ,and the initial
estimates of parameter for classical adaptive control andadaptive
multiple-mode
controlare1(0)  2.8,2 (0)  0.1,3 (0)  5.5, (0)  2.8

As seen from Figure 1, the state x1 of the system isbounded, and asymptotic tracking is achieved. Adaptive
multiple-model control (d) performs better than single
model (a-c) especially when initial parameter estimation
error is large (b and c). In Figure 2, all parameter
estimations of θ asymptotically converge to its true
value while the estimation in adaptive multiple-model
control (d) is the fastest. Figure 3 illustrates
the control input u is bounded in the two different
control schemes. Thus the scheme proposed in thispaper is feasible and effective.

Proof.
First, we will prove that the number ofresetting is finite. After the above analysis, we j (t )  L [0, ), j  1, 2,, n
have
,
i

1,
2,

,
n
,

ò

0,

t

[0,

)
for
, such that | i (t ) | ò, t  ti .
iLet

tmax=max{t1,t2,…,tn}.
For
Based on the resettingt  tmax ,| i (t ) | ò, i  1, 2, , N .
nmechanism,
Then theresetting adaptive model will estimate parameters withzero error. So, in the worst situation, the amount of e (t ) j 1

 j  j (t )  max{ j (t ) , j  1, 2,..., N}  ò.

Fig.1. Output and reference signal: (a), (b), (c) single modelwith  (0)  0.1,5.5, 2.8 respectively, and (d) retting modelwith  (0)  2.8

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Chao Yang, Yingmin Jia

Fig.2. Parameter estimation: (a), (b), (c) single modelwith  (0)  0.1, 5.5, 2.8 , (d) retting model with  (0)  2.8

Fig.3. control input u: dashed line, dash-dot line and dottedline for single model with  (0)  0.1,5.5, 2.8 , solid line forvirtual model with  (0)  2.8

6. Conclusion
In this paper, an adaptive multiple-model controller
is developed for a class of nonlinear systems inparametric-strict-feedback form. Unlike previous
results ， a switching scheme is not necessary toguarantee the model closest to the real plant to be
switched into the controller design. Global asymptotic
stability of the closed-loop system, globally uniformly
bounded of all the closed-loop signals and convergenceof parameter estimates are proved. The simulation
results illustrate the feasibility and effectiveness of the
proposed method.

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Acknowledgements
This work was supported by the National Basic Research
Program
of
China
(973
Program:
2012CB821200, 2012CB821201) and the NSFC
(61134005, 60921001, 61327807).

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