Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 4 (March 2015), 291-294

Weighted Multiple Model Adaptive Control of Time-varying Systems∗
Weicun Zhang 1 †, Qing Li 2
1 School of Automation and Electrical Engineering

University of Science and Technology Beijing
Beijing 100083, China
E-mail: weicunzhang@ustb.edu.cn
2 School of Automation and Electrical Engineering

University of Science and Technology Beijing
Beijing 100083, China
E-mail: qingli@ies.ustb.edu.cn

Abstract
This paper is concerned with the weighted multiple model adaptive control (WMMAC) of stochastic plantwith time-varying parameters, such as slowly time-varying plant or parameter jump plant. An improved
weighting algorithm is proposed for this kind of control systems. The effectiveness (sharp and correct 0-1convergence) of the weighting algorithm and the robust stability of the resulting closed-loop WMMAC
system has been verified through many simulation results.
Keywords: Stability, Convergence, Weighting algorithm, Multiple model adaptive control.

1.

INTRODUCTION

As we know, weighting algorithm plays an important role in WMMAC systems. According to reference [1], the closed-loop stability of a WMMAC
system mainly depends on three conditions: first,
the model set includes the true model of the plantor a closet one to the plant; second, the weighting
algorithm converges correctly; third, each local controller stabilizes its corresponding model. Besides it
is obvious that the convergence rate of a weighting
algorithm is related to the transient process of theWMMAC system.

In classical weighted multiple model adaptive estimation (WMMAE) and WMMAC [2−8] , as wellas robust multiple model adaptive control (RMMAC) [9−11] , probabilistic weighting algorithm wasadopted, which is based on multiple Kalman filters, dynamic hypothesis testing, and Bayes’ theorem. Some convergence results on the probabilistic
weighting algorithm have been obtained [12−15] . In[10], Fekri, Athans and Pascoal pointed out that under certain ergodicity and stationarity assumptions,one of the posterior probabilities will converge withprobability 1 to unity and will ’identify’ the model
closest to the true plant, i.e. the one with smallest∗ This work was supported by Major State Basic Research Development Program (973 Program) (No. 2012CB821200) and National

High-Tech Research and Development Program of China (863 Program) (No. 2011AA060408).
† Corresponding author.

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Weicun Zhang, Qing Li

Baram Proximity Measure (BPM)[10,12] .
However, since using Kalman filters to drive thehypothesis testing, the classical WMMAC scheme
can suffer from poor performance due to either large
initial state estimate error or inaccurate knowledgeof the disturbance/noise statistics. Additionally, thecomplexity of the supervisor may hinder its application because every candidate controller requires
a Kalman filter and a post posterior evaluation [16] .
Thus, some substitutive methods have been proposed to improve the above-mentioned situation. In[16-19], fuzzy rules based weighting algorithms replaced the probabilistic weighting algorithm. In [1],a new weighting algorithm based on model outputerror, is proposed. As a result, the long-standing issue, i.e, the closed-loop stability of WMMAC, hasbeen addressed for a class of discrete-time stochastic plant under smooth conditions. The breakthroughresult also benefited from the virtual equivalent system (VES) theory [20] .
This paper is intended to further improve theconvergence rate of weighting algorithm, as wellas to adapt the weighting algorithm to time-varying
systems such as parameter jump system and slowly
time-varying system.
2.

DESCRIPTION OF WMMAC

The plant P(k) can be stable or non-stable, minimum
phase or non-minimum phase. Its output y(k) can berewritten asy(k) = φ T (k − d)θ (k) + ω(k)

(2.3)

where
φ T (k − d) =[y(k − 1), ..., y(k − na), u(k − d), ...,u(k − d − nb)]
(2.4)
θ (k) = [−a1 (k), ..., −ana (k), b0 (k), b1 (k), ..., bnb (k)](2.5)
The WMMAC system consists of three components, the model set M = {Mi , i = 1, 2, ..., N}, the’local’ controller set C = {Ci , i = 1, 2, ..., N}, and theweighting algorithm. The ith model, controller, andweight are denoted as Mi , Ci , and pi (k), respectively.
The control objective is to track the bounded reference signal yr (k) < ∞.
’Local’ controllers can be designed according toany existing methods, such as pole assignment, H ∞method, mixed-µ synthesis tools, PID, etc.
Each ’local’ controller Ci generates local controlui (k); the ’global’ control applied to the plant is obtained by weighting of the local controls, i.e.
N

Consider the following discrete-time stochastic
plant with single input and single output (SISO)

u(k) = ∑ pi (k)ui (k)P(k) : A(k, q−1 )y(k) = q−d B(k, q−1 )u(k) + ω(k)(2.1)
whereA simple and direct weighting algorithm is proposed in [1], which depends only on the model output errors. It gets rid of the inconvenience of Kalmanfilters and off-line membership functions. Details
will be given in the next section.
For each model Mi ∈ M, its output is given byA(k, q−1 ) = 1 + a1 (k)q−1 + · · · + ana (k)q−na
B(k, q−1 ) = b0 (k) + b1 (k)q−1 + · · · + bnb (k)q−nb

yi (k) = φ T (k − d)θid > 1; na > 1; nb > 1
where y(k), u(k) and ω(k) are the output, input andzero-mean white noise of the system, respectively,y(k) = 0, u(k) = 0, ω(k) = 0 for k < 0, and that
1 nlim ∑ [ω(i)]2 = R < ∞n→∞ ni=1(2.2)(2.6)

i=1(2.7)

where θi is the parameter vector of model Mi . Further, define the output error of each model Mi , i.e.
ei (k) = y(k) − yi (k) = y(k) − φ T (k − d)θi(2.8)

As we will see in the next section, ei (k) is used tocalculate pi (k).

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Weighted Multiple Model Adaptive Control

3.

WEIGHTING ALGORITHMS WITHCONVERGENCE ANALYSIS(3.1)

To adapt the above mentioned algorithms toslowly time-varying or parameter jump systems,
there are two methods to avoid the zero lock-in condition of pi from occurring. One method is to set
lower bounds for weights pi , i = 1, 2, ..., N as in [21];The other is to reset each weight according to (3.6)
when the parameters change is detected. In the simulations of this paper, we adopted the latter method.
In detail, the numerical value order of li (k) is used
to detect the parameter changes. The related programme is as follows.

ei (p)

(3.2)

..
.

lmin (k) = min li0 (k)(3.3)

A = [l10 (k)l20 (k)l30 (k)l40 (k)...];(3.4)[lmin (k), index(k)] = min(A);First, we review a weighting algorithm put forwarded in [1]. Then, a modified weighting algorithm will
be proposed in order to improve the convergence
rate, as well as to be suitable for time-varying plant.
Weighting algorithm 1 [1] :li (0) =1; pi (0) = li (0)Nk1
li0 (k) = 1 +∑k p=12ili (k) = li (k − 1)pi (k) =lmin (k)li0 (k)li (k)
N∑i=1 li (k)calculates li (k) according to Weighting Algorithm 2(3.5)

if l1 (k) = 0||l2 (k) = 0||l3 (k) = 0||l4 (k) = 0 · · ·Based on Weighting Algorithm 1, a modified
version is proposed as follows.

if index(k) = index(k − 1)
Weighting algorithm 2:li (0) =resets Weighting Algorithm 2;

1; pi (0) = li (0)N(3.6)

1 k∑ ei (p)2
k p=1(3.7)

li0 (k) = 1 +end

lmin (k) = min li0 (k)ili (k) =(li (k − 1)li (k − 1)[βi (k)]pi (k) =(3.9)
if βi (k) = 1

ceil( 1−β2 (k) )
i..
.

(3.8)

lmin (k)li0 (k)βi (k) =end

if βi (k) < 1(3.10)li (k)
N∑i=1 li (k)(3.11)

where ceil(x) is the ceiling function that generates
the smallest integer not less than x, i.e.,ceil(x) = min{n ∈ Z|x 6 n}

4.

SIMULATION RESULTS

To verify the effectiveness of the proposed weighting algorithm and the performance of the resulting
closed-loop WMMAC system, a lot of simulation
results have been conducted with MATLABr version 6.5. Owing to the space limitation, the corresponding photographes are omitted here. In a word,
the weighting algorithm has very good convergence
performance under various situations, and the corresponding WMMAC systems have satisfactory stability and tracking performance.

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5.

CONCLUSIONS AND FUTURE WORKThe stability and convergence of the closed-loop
WMMAC system depend only on three conditions, i.e., the stabilizing characteristic of each ’local’controller; the effective model set to cover the uncertainty of plant to be controlled; and the appropriate convergence of the weighting algorithm. Thus,
WMMAC provide us with more confidence and flexibility in robust adaptive control practices.
The future research will be focused on two directions. One is to analyze the stability and convergence of WMMAC systems of non-linear timevarying plants. The other is to further relax the convergence conditions of weighting algorithms.
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