Kalman Filter-based Estimators for Dual Adaptive Neural Control

A Comparative Analysis of Execution Time and Performance Issues

Simon G. Fabri and Marvin K. Bugeja

Department of Systems and Control Engineering, University of Malta, Msida, Malta

Keywords:

Dual Adaptive Control, Kalman Filter, Neural Networks, Intelligent Control.

Abstract:

The real time implementation of neural network-based dual adaptive control for nonlinear systems can become

signiﬁcantly demanding because of the amount of network parameters requiring estimation. This paper ex-

plores the effect of three different estimation algorithms for dual adaptive control of a class of multiple-input,

multiple-output nonlinear systems in terms of tracking performance and execution time. It is shown that the

Unscented and Square-root Unscented Kalman ﬁlter estimators lead to a signiﬁcant improvement in tracking

performance when compared with the Extended Kalman ﬁlter, but with an appreciable increase in execution

time. Such issues need to be given due consideration when implementing controllers for on-line operation.

1 INTRODUCTION

The use of dual adaptive techniques for neural net-

work closed-loop control of nonlinear systems has

been investigated from different perspectives in the

recent past (Bugeja and Fabri, 2007; Bugeja et al.,

2009;

Simandl et al., 2005). Inspired by the pioneer-

ing work of Fel’dbaum (Fel’dbaum, 1965), these dual

adaptive methods handle joint estimation and control

of nonlinear systems that are subject to functional un-

certainty by characterizing them within a stochastic

framework.

Neural networks are used to learn the unknown

nonlinear functions in real-time, whereas the dual

controller generates system inputs that exhibit two

desirable properties: (a) caution; whereby the un-

certainty of the neural network estimates is taken

into consideration by the controller so as to maintain

good tracking capabilities, and (b) probing; whereby a

component of the input is used to excite the system so

that the neural network training algorithm reduces the

functional uncertainty rapidly and efﬁciently (Fabri

and Kadirkamanathan, 2001; Filatov and Unbehauen,

2004).

The dual adaptive control paradigm is rooted in

the methodology of stochastic estimation and con-

trol. Stochastic algorithms are employed for estima-

tion of the neural network parameters which often ap-

pear in nonlinear form. Such estimators effectively

“train” the neural networks to capture the system’s

nonlinear functions recursively in real-time, based on

measurements of its inputs and outputs. Nonlinear

stochastic estimators which have been proposed in the

dual adaptive control literature include the Extended

Kalman ﬁlter (Fabri and Kadirkamanathan, 1998), the

Unscented Kalman ﬁlter (Bugeja and Fabri, 2009)

and the Gaussian Sum ﬁlter (

Simandl et al., 2005).

The aim of this paper is to investigate and com-

pare the computational demand of a select set of non-

linear estimation algorithms when applied within the

context of neural network, dual adaptive control of a

class of uncertain, nonlinear, multiple-input/multiple-

output dynamic systems. Computational demand

analysis is a crucial consideration for the implemen-

tation of control algorithms which operate on com-

puter hardware in real-time (

Astr

om and Wittenmark,

2011). The discrete-time nature of such digital con-

trol systems entails that the control law, including the

estimation algorithm, executes, calculates and gener-

ates a fresh control signal with minimal delay at ev-

ery sampling instant. As a consequence, the estima-

tion and control algorithms must execute well within

the sampling period at which the discrete-time control

law is operating.

The analysis reported in this work would be of

help for designers to judge the feasibility of imple-

menting a particular estimation algorithm for real-

time, dual adaptive neuro-control. Three nonlinear

estimation techniques are considered and analysed in

this paper - the Extended, Unscented and Square-root

Unscented Kalman Filter algorithms - all three being

variations of the well-known Kalman ﬁlter method-

169

Fabri S. and Bugeja M..

Kalman Filter-based Estimators for Dual Adaptive Neural Control - A Comparative Analysis of Execution Time and Performance Issues.

DOI: 10.5220/0004455601690176

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 169-176

ISBN: 978-989-8565-70-9

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

ology for the nonlinear case. These algorithms are

evaluated by testing their effect on the execution time

and on the capability of the dual adaptive controller to

meet the control system’s performance requirements

when embedded within the closed loop control sys-

tem.

The rest of the paper is organized as follows: Sec-

tion 2 presents the background of the dual adaptive

control problem for nonlinear systems and describes

the control and estimation algorithms. This is fol-

lowed by Section 3 where the performance and com-

putational demands of the different estimation algo-

rithms are tested and the results are analysed. Con-

clusions are provided in Section 4.

2 ESTIMATION AND CONTROL

In this work we consider multiple-input, multiple-

output (MIMO) systems with nonlinear dynamics of

the following form:

= f

f (x

k−1

) + G

G(x

k−1

+ ε

(1)

where y

∈ R

is a vector of s outputs,

∈ R

is an s-input control vector, x

k−1

k−n

...y

k−1

k−1−p

...u

k−2

∈ R

s(n+p)

the system state vector (N.B. 0 ≤ p ≤ n and the

u terms vanish from the state vector if p = 0),

vector ﬁeld f

f (x

k−1

) : R

s(n+p)

7→ R

and matrix

G(x

k−1

) : R

s(n+p)

7→ R

s×s

contain the unknown non-

linear functionals governing the system dynamics,

and ε

∈ R

represents an additive output white

noise signal assumed to be zero-mean Gaussian of

covariance R

The control objective is for the output vector y

to track a reference input vector y

∈ R

despite the

uncertainty in the nonlinear functionals comprising f

and G

2.1 Neural Network Estimation Model

Two multilayer perceptron (MLP) neural networks

are used to estimate and approximate the nonlinear

system functionals within an arbitrarily large compact

set χ

χ in which the state vector is known to be con-

tained. The estimates from the neural networks will

be utilized within a dual adaptive control law.

Neural network

f =



···



is used to approx-

imate the functionals in f

f := [ f

··· f

]

, where

ap-

proximates f

. Neural network

g = [ ˆg

··· ˆg

]

is used

to approximate the functionals in G

G, where ˆg

(i−1)s+ j

approximates G

i, j

according to the notation:

G :=







1,1

··· G

1,s

s,1

··· G

s,s







G =







ˆg

··· ˆg

ˆg

−s+1

··· ˆg







Each of the two networks contains one hidden layer of

sigmoidal neurons whose outputs are represented by

vectors φ

,φ

for the

f ,

g networks respectively such

that:

= φ

; i = 1···s

ˆg

= φ

; i = 1···s

(2)

where

denote the synaptic weight vectors of

networks

f and

g respectively. The individual MLP

sigmoidal activation functions are given by



1 + exp(−



−1

; i = 1···L

x =







1 + exp(−



−1

; i = 1···L

x =





with L

denoting the number of neurons in

the hidden layer of the

f ,

g networks respectively,

and

denoting the corresponding hidden layer

weight vectors that shape the sigmoid of the i

acti-

vation function.

Let us group together all the unknown network

weights into one vector

z =

, where

···

and

···

are

the weights associated with the

f and

g networks re-

spectively and which require on-line estimation. Ac-

cording to the Neural Network Universal Approxima-

tion Theorem (Haykin, 1999), there exists a set of

optimal (constant) weights such that an appropriately

sized neural network can approximate any smooth

nonlinear function within a compact subset of its in-

put space, up to any desired degree of accuracy. For

our case, let us denote this optimal set of weights as

∗

, these being the unknown optimal

values of

and

respectively. Let us assume that

the accuracy achieved with this set of optimal weights

is such that the network approximation error can be

considered negligible. In this case, system equation

(1) can be equivalently re-written in terms of the fol-

lowing optimal neural network estimation model:

∗

k+1

= z

∗

+ ρ

f (x

k−1

, p

∗

) +

G(x

k−1

, p

∗

k−1

+ ε

= h

h(x

k−1

∗

) + ε

(3)

where

h(x

k−1

∗

) :=

f (x

k−1

, p

∗

) +

G(x

k−1

, p

∗

k−1

and ρ

is a Gaussian process noise with known covari-

ance Q

, generally set to have very small magnitude.

The latter noise signal was not present in the origi-

nal system equation but is included in the estimation

model because it aids the weight estimation process.

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170

2.2 Parameter Estimation

The optimal neural network parameters vector z

∗

ap-

pearing in model (3) is unknown and needs to be es-

timated recursively in real-time while the control ac-

tions are being executed. In this work, z

∗

is treated as

a random variable having an initial condition that is

assumed to be Gaussian distributed with known mean

and covariance P

. These parameters appear non-

linearly in function h

h(x

k−1

∗

) of the model’s

output equation (3). As a consequence, even with

Gaussian additive noise, the distribution of the param-

eters will no longer remain Gaussian because of the

nonlinear effects of the system dynamics. Thus non-

linear stochastic estimation methods need to be ap-

plied. Three such approaches are considered here:

• the Extended Kalman ﬁlter (EKF),

• the Unscented Kalman ﬁlter (UKF),

• the Square-root Unscented Kalman ﬁlter

(SRUKF).

There exist other nonlinear estimation algorithms

such as Particle ﬁlters and Gaussian Sum ﬁlters but

these fall outside the scope of the analysis reported in

this work.

2.2.1 The EKF Estimator

Use of the EKF algorithm as a nonlinear estimator for

the parameters of neural networks is well documented

(Haykin, 2001). The EKF is based on a rather crude,

ﬁrst-order linearization of the system dynamics, upon

which a Kalman ﬁlter is applied. It effectively ignores

the non-Gaussian distribution of the random variables

and propagates the density functions by means of the

standard Kalman ﬁlter equations which would only be

correct in a linear scenario, where Gaussianity is pre-

served. This assumption often leads to a well-known

disadvantage of the EKF: divergence of the parameter

estimates when propagated through time.

Let us denote the estimate to vector z

∗

calculated

at time step k as

. The EKF algorithm, operating in

predictive mode, recursively generates estimates for

the parameters of the neural network controller being

considered in this work, as follows:

Algorithm 1 - EKF:

Initialize with

and P

. Then at every time step k...

1. Calculate ∇

∇

k−1

, the Jacobian of h



k−1

∗



with respect to z

∗

evaluated at

∇

k−1



∇

k−1

∇

k−1



∂

k−1

∂p

∗



∂



k−1



∂p

∗



2. Approximate the covariance of the output mea-

surement’s distribution:

= ∇

∇

k−1

∇

k−1

+ R

3. Calculate the Kalman gain:

= P

∇

k−1

−1

4. Use the current output measurement to generate

the innovations:

= y

−h

h(x

k−1

)

5. Predict the parameter estimate:

k+1

+ K

6. Predict the covariance of the estimate:

k+1

= P

−K

∇

k−1

+ Q

Step 1 of this algorithm highlights a second crucial

disadvantage of the EKF algorithm, namely that it en-

tails a prior, off-line derivation of the Jacobian ma-

trix for calculation in step 1. This derivation can be-

come rather complex in the context of our neural net-

work control schemes that typically demand hundreds

of network parameters (Bugeja, 2011). Space limita-

tions preclude us from showing the derivation and ﬁ-

nal form of this Jacobian matrix. Sufﬁce to say that it

will be of size s ×(L

(s + L

) + L

+ L

)), where

= s(n + p) + 1 denotes the length of vector

2.2.2 The UKF Estimator

The UKF is based on the Unscented Tranformation

which propagates the mean and covariance of the

random variables through appropriate nonlinear

transformations (Julier and Uhlmann, 2004). A

set of so-called sigma points are chosen from the

statistics of the nonlinear transformation so as to

propagate second-order properties of the probability

distribution. As a consequence, the UKF is a more

accurate estimator than the EKF (Wan and van der

Merwe, 2001). In addition, unlike the EKF, it does

not require any derivations or calculation of complex

Jacobian matrices. When applied within the context

of the neural network controller discussed in this

paper, the algorithm takes the following form:

Algorithm 2 - UKF:

Initialize with

and P

. Then at every time step k...

1. Form a matrix

whose columns are the 2N + 1

sigma vectors

generated as follows:

+ γσ

i = 2,...,N + 1

−γσ

i = N + 2,...,2N + 1

KalmanFilter-basedEstimatorsforDualAdaptiveNeuralControl-AComparativeAnalysisofExecutionTimeand

PerformanceIssues

171

where σ

is the i

column of Σ

, the latter de-

noting the lower-triangular Cholesky factoriza-

tion of P

such that Σ

= P

. In practice,

this is obtained by calling a Cholesky factoriza-

tion function in software, denoted in this paper as

chol

{

}

N denotes the length of

(in this case N =

(s + L

)+L



+ L



) and γ =

√

N + λ where

λ = α

(N + κ) −N, α being a constant that de-

termines the spread of the points in the sigma

vectors around

, typically set within the range

1 ≥ α ≥ 10

−4

. κ is a constant scaling parameter

normally set to 3 −N.

2. Propagate the sigma vectors through the neural

network estimation model’s output equation:

k−1

where

k−1



k−1



;

k−1



k−1



;

i = 1,...,2N + 1; vectors

are the columns

of matrices

which in turn are submatrices

of size



(s + L

)×2N + 1



and



+ L



2N + 1



respectively, taken from the matrix of

sigma vectors

after repartitioning it as follows:





3. Approximate the mean and covariance of the dis-

tribution of the output measurement using the

propagated sigma vectors, calculated by the fol-

lowing weighted summations:

2N+1

∑

i=1

(i)

2N+1

∑

i=1

(i)



−



−



+ R

2N+1

∑

i=1

(i)



−



−



where the so-called unscented transform weights

are given by

(1)

N + λ

, W

(1)

N + λ

+ 1 −α

+ β

(i)

= W

(i)

2(N + λ)

, i = 2,...,2N + 1.

β is a constant parameter that depends upon prior

knowledge of the estimate’s distribution where,

for the Gaussian prior case, β = 2 is optimal.

4. The Kalman gain is calculated as:

= P

−1

5. The innovations are calculated as:

= y

−

6. Predict the parameter estimate:

k+1

+ K

7. Predict the covariance of the estimate:

k+1

= P

−K

+ Q

In its credit, and in contrast with the EKF, the UKF

algorithm does not make use of the Jacobian ma-

trix. Additionally, several studies have been pub-

lished which illustrate the improved performance of

the UKF over the EKF for estimation of variables in

nonlinear systems (Julier and Uhlmann, 2004; Wan

and van der Merwe, 2001).

2.2.3 The SRUKF Estimator

The numerical properties of the UKF algorithm can

be further improved by a square-root implementation,

leading to the SRUKF algorithm (Wan and van der

Merwe, 2001). This guarantees that the covariance

matrices remain positive semi-deﬁnite.

Additionally, whereas the UKF needs to perform

a Cholesky factorization of the covariance matrix

at every time step in order to compute its square

root Σ

(refer to Step 1 in the UKF algorithm), the

SRUKF propagates Σ

directly, avoiding the recursive

Cholesky factorization operations. In the parameter

estimation scenario, this leads to a computational

complexity which has the same order as that of

the EKF in terms of ﬂoating point instructions per

iteration (Wan and van der Merwe, 2001). However

this does not necessarily translate to an equal or a

similar execution time, especially when the amount

of parameters is high, leading to a large number of

sigma points (LaViola, 2003). When applied to the

neural network controller developed in this paper, the

SRUKF algorithm takes the following form:

Algorithm 3 - SRUKF:

Initialize with

and Σ

= chol

{

}

. Then at every

time step k...

1. Form matrix

of 2N + 1 sigma vectors as fol-

lows:

+ γσ

i = 2,...,N + 1

−γσ

i = N + 2,...,2N + 1

where σ

is the i

column of Σ

. The constants

N, γ, λ and α are deﬁned in an identical manner

as for the UKF.

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2. Propagate the sigma vectors through the neural

network output equation:

k−1

where

k−1

are deﬁned in the

same way as for the UKF.

3. Approximate the mean and the Cholesky factor of

the covariance of the output measurement’s dis-

tribution through the following weighted summa-

tions of the propagated sigma vectors:

2N+1

∑

i=1

(i)

= qr

(2:2N+1)



2:2N+1

−



= cholupdate



−

, W

(1)



2N+1

∑

i=1

(i)



−



−



where the unscented transform weights

(i)

, W

(i)

and parameter β are deﬁned as

in the UKF algorithm.

4. The Kalman gain is calculated as:

= (P

)/Σ

5. The innovations are calculated as:

= y

−

6. Predict the parameter estimate:

k+1

+ K

7. Predict the Cholesky factor of the covariance of

the estimate as either:

Option 1:

k+1

= λ

−0.5

RLS

cholupdate



, K

,−1



where λ

RLS

is a constant forgetting factor param-

eter set to a value just less than 1 e.g. 0.9995,

or Option 2:

k+1

= cholupdate



, K

,−1



+ D

where

= −Diag

{

}

Diag

{

}

+ Diag





with function Diag

{

}

denoting the formation of

a diagonal matrix whose diagonal elements are a

copy of the diagonal terms in the matrix of the

function’s argument.

The qr{·}, ·/· and cholupdate{·} symbols used in

steps 3, 4 and 7 of the above algorithm denote func-

tions that execute QR factorization, efﬁcient least

squares and Cholesky factor updating operations re-

spectively. More details on this can be found in (Wan

and van der Merwe, 2001).

2.3 The Control Law

The ideal solution to the dual control problem in-

volves the minimization of a cost function whose so-

lution is practically impossible to implement in most

situations (Fabri and Kadirkamanathan, 2001). Some

adaptive control schemes thus optimize a much more

basic cost function which leads to control laws, such

as Heuristic Certainty Equivalence (HCE) and Cau-

tious control, that lack the desirable effects of dual

control. These lead to an inferior performance typi-

cally exhibiting large overshoots in HCE, or long re-

sponse times in Cautious control. A better alterna-

tive, as used in this work, is to adopt a suboptimal

cost function but which to a certain extent still retains

dual-like properties in its control actions.

This work is based on the suboptimal dual cost

function originally proposed in (Milito et al., 1982)

for linear stochastic systems, but generalized to the

case of nonlinear MIMO systems as follows:

inn

= E



k+1

−y

k+1





k+1

−y

k+1









k+1





where E{·}denotes mathematical expectation over all

random variables, I

is the information state at time-

step k deﬁned as I

{

...y

k−1

...u

}

, and i

is the innovations vector from the estimator. Design

parameters Q

and Q

are (s ×s) positive deﬁnite di-

agonal matrices with real positive elements. Q

an (s ×s) diagonal matrix satisfying −Q

≤ Q

≤ 0

element-wise. Matrix Q

imposes a penalty on track-

ing errors and matrix Q

induces a penalty on large

control inputs. Q

affects the innovations vector. Its

setting determines whether the control law will act in

HCE mode, Cautious mode or alternatively induce the

desired dual adaptive control characteristics.

The optimization of cost function J

inn

subject to

the dynamics of the system as represented by the op-

timal neural network model (3), leads to a dual control

law of the following form:



+ Q

+ N



−1



k+1

− f



−κ



(4)

where f

, G

, N

and κ

are terms which depend

on the variables of the selected estimator. A detailed

derivation and full description of this control law falls

outside the scope of this paper. Further details on

the control aspect of the work may be obtained from

(Bugeja, 2011). N

and κ

depend on the covariance

estimates from the estimator and embody informa-

tion regarding the uncertainty of the estimates at every

time instant k. When matrix Q

is set equal to −Q

KalmanFilter-basedEstimatorsforDualAdaptiveNeuralControl-AComparativeAnalysisofExecutionTimeand

PerformanceIssues

173

the terms N

and κ

will become null and vanish from

control law (4). The controller will thus ignore the

uncertainty of the estimates, leading to an HCE con-

troller which tends to generate aggressive inputs that

may be beneﬁcial for probing the system but at the

expense of excessive overshoot and a low degree of

stability. At the other extreme, with Q

= 0

0, the con-

trol law gives maximum attention to the uncertainty

terms, which leads to Cautious control. Whereas this

does not give rise to excessive overshoot and a low de-

gree of stability, it is typically too weak to make the

system react promptly and generate probing signals

which enhance the estimation process. Both these ex-

tremes are inferior to Dual control in terms of per-

formance. When Q

is set in between these two ex-

tremes, the controller will exhibit dual-like character-

istics obtained by reasonably balancing out the ag-

gressive probing of HCE and the sluggish actions of

Cautious control.

3 TESTING AND RESULTS

The structure of the complete control algorithm can

be subdivided in terms of the following tasks:

TASK A - Measurement of the system outputs y

TASK B - Model Estimation (EKF, UKF, SRUKF):

I. Calculation of Jacobian matrix (for EKF) or

sigma vectors matrix (for UKF, SRUKF), fol-

lowed by the output measurement covariance or

its Cholesky factor (for SRUKF).

II. Calculation of Kalman gain, innovations, pa-

rameter prediction, and parameter covariance

prediction.

TASK C - Dual Control Law Calculations:

I. Calculation of f

, G

, N

, κ

II. Calculation of u

TASK D - Input of u

to the system.

In the EKF estimator, TASK B(I) is covered in steps

1, 2 and TASK B(II) is implemented in steps 3, 4, 5, 6

of Algorithm 1. In the UKF and SRUKF estimators,

TASK B(I) is implemented in steps 1, 2, 3 and TASK

B(II) is covered in steps 4, 5, 6, 7 of Algorithms 2 and

Whereas TASK B(I) for the EKF involves an ex-

tensive prior off-line effort to derive equations for the

Jacobian matrix ∇

∇

k−1

that can be rather complex in a

neural networks scenario, calculation of the numerical

value of ∇

∇

k−1

and P

during run-time in steps 1 and

2 is rather straightforward. By contrast, the Jacobian-

free UKF and SRUKF estimators require no prior off-

line effort at all for TASK B(I), but instead shift the

effort within steps 1 to 3 during run-time through rel-

atively more intense calculations for Cholesky or QR

factorizations and weighted summations across sigma

vectors. This computational effort is not trivial in a

neural networks scenario where the amount of param-

eters, and consequently also the number of sigma vec-

tors, is large.

A number of simulation trials were performed in

order to validate the effects of the three estimators on

the control system. The objective of these trials is

twofold, namely to quantify across the three estima-

tors: (a) the ability of the control algorithm to track

the reference input vector y

, (b) the time taken for

execution of the complete control algorithm, with spe-

ciﬁc focus on the impact of TASKS B and C.

Simulation trials were performed on MATLAB

using a 2-input, 2-output dynamic MIMO system hav-

ing the form of Equation (1) with n = 2, p = 1 and

nonlinear functions:

f =





0.7x

1+x

+ 0.25x

+ 0.5x

0.5x

sinx

1+x

+ 0.5x

+ 0.3x





G =

cos

0.1

1+3x

0.1x

−5.5

The measurement noise covariance R

= 5 ×10

−4

I denotes the identity matrix) and that of the model

process noise Q

= 1 ×10

−5

I. The two neural net-

works in the estimation model are structured with

= L

= 7 hidden layer neurons. This leads to a

total of N = 140 neural network parameters requir-

ing estimation. The initial condition of the parame-

ter vector

is generated at random from the interval

[−0.1,0.1] and its covariance P

= 0.8I

I. The UKF

and SRUKF parameter α is set to 0.9. 2N + 1 = 281

sigma vectors, each composed of N = 140 elements,

are required by these two algorithms. The control law

was tested under three different settings of Q

corre-

sponding to HCE, Cautious and Dual control modes

= −0.3I

I in the latter case).

Figure 1 shows a sample of results for dual control

with a UKF estimator over a 10s interval. The ref-

erence input signals are shown in black and the two

system outputs in colour. Notice how the proposed

adaptive control system results in good tracking of the

reference inputs following an initial period of “learn-

ing” due to the controller’s lack of knowledge of the

nonlinear system functions in f

f and G

G. The single

trial results of Figure 1 are however not sufﬁcient to

characterize the general performance of the control

system due to variations introduced by the random

effects of the noise and the initial parameter vector.

A Monte Carlo characterization was therefore per-

formed by repeating the experiment over 150 trials,

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

174

0 1 2 3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0.5

1.5

time in seconds

0 1 2 3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0.5

1.5

time in seconds

Time in seconds

output

Figure 1: Tracking results with Dual control using UKF.

each time generating fresh realizations of the noise

signals and the initial neural network parameters. Q

was set to 0.1I

I in all three cases. In each individ-

ual trial, all three control modes (HCE, Cautious and

Dual) and all three estimators (EKF, UKF, SRUKF)

were subjected to the same realization of random sig-

nals so as to ensure a fair comparison. The SRUKF

estimator is tested for both Cholesky factor predic-

tion options in step 7 of the algorithm. The tracking

performance at the end of each trial was quantiﬁed in

terms of an error metric C =

∑

end

k=0

||y

−y

which

captures the squared tracking error over the whole

simulation interval which was set to 5s in these trials.

Lower values of C indicate a better tracking perfor-

mance. The Monte Carlo analysis results are summa-

rized in Table 1, showing the mean value of C and

its variance across all trials for each possible com-

bination of control mode and estimator. The results

clearly show the superior performance of Dual con-

trol over the Cautious and HCE modes in all estima-

tors, with C exhibiting the smallest mean and variance

in each case. Moreover, comparing across estimators,

the inferior control performance arising from the in-

accuracies of the EKF estimator is clearly evident in

all control modes. On the other hand, the UKF and

the two SRUKF options exhibit an error metric hav-

ing means and variances of the same order, reﬂecting

similar tracking performance.

The simulation experiments were also used to

time the main steps of the estimation and control al-

Table 1: Results of Monte Carlo analysis.

Estimator Mode Mean of C Var of C

HCE 15384.0 1.90×10

EKF Cautious 4562.0 3.03 ×10

Dual 43.4 4.32 ×10

HCE 99.8 6.29 ×10

UKF Cautious 42.7 464.8

Dual 35.6 318.5

HCE 90.8 2.18 ×10

SRUKF

Cautious 44.0 485.2

option 1

Dual 36.2 336.9

HCE 101.0 7.90 ×10

SRUKF

Cautious 43.9 611.4

option 2

Dual 35.6 319.8

gorithm across all estimator types. The results, shown

in Table 2, show the mean execution time in mil-

liseconds (ms) calculated over 250 iterations for the

following operations: one complete iteration, Task

B(I), Task B(II) and Task C. Figure 2 is a graphi-

cal depiction of the same results, but expressed as

a factor of the EKF iteration time so as to interpret

the measurements independently of any absolute time

values which would vary according to the hardware

on which the algorithms are executed. Such normal-

ized timings for the separate tasks are shown in dif-

ferent colours as indicated in the legend on Figure

2. The results demonstrate that the execution time

of the UKF/SRUKF increases by a factor of approxi-

mately 8 to 9 with respect to the EKF. This is mainly

due to TASK B(I) in the UKF/SRUKF algorithm, par-

ticularly step 2 of the algorithms which propagates

the 281 sigma vectors through the neural network.

This step takes around 7.5 times the total EKF iter-

ation time, notwithstanding that the UKF/SRUKF al-

gorithms were carefully coded without slow for/next

loops in order to maintain efﬁcient timing. The

SRUKF’s total execution time is of the same order as

that of the UKF, albeit slightly longer, especially with

Option 2. In fact, TASK B(I) in the SRUKF takes

a longer duration than the UKF because of the QR

factorization and Cholesky factor update operations

in step 3 which takes approximately thrice the time

of the corresponding step in the UKF. Although this

is somewhat compensated by the shorter duration of

step 1 in the SRUKF, due to the absence of Cholesky

factorization, this reduction is minimal when com-

Table 2: Execution time in ms (MATLAB running on a 3GHz Pentium 4 CPU with 2GB RAM).

One complete iteration TASK B(I) TASK B(II) TASK C

EKF 10.9 1.8 1.9 7.0

UKF 89.7 85.7 0.6 0.8

SRUKF option 1 91.3 88.3 1.3 0.7

SRUKF option 2 95.5 88.3 5.3 0.7

KalmanFilter-basedEstimatorsforDualAdaptiveNeuralControl-AComparativeAnalysisofExecutionTimeand

PerformanceIssues

175

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,-." /-." 01/-."23425#" 01/-."23425"$"

Factor'of'EKF'itera-on'-me'

Execu-on2-me'graph''

expressed'as'a'factor'of'EKF'itera-on'-me'

678" 9" :;<<=" :;<="

Figure 2: Execution time as a factor of the EKF case.

pared with the increased duration of step 3. TASK

B(II) in the two SRUKF cases is also signiﬁcantly

longer than that of the UKF, particularly with option

2. The dominant component here is the Cholesky fac-

tor update operation in step 7 and, for the case of op-

tion 2, the additional operations required for calcu-

lation of D

. The execution time of the EKF algo-

rithm is dominated by TASK C, which is around 9

times longer than the corresponding task in the other

estimators. However this duration is still much less

pronounced than the contribution of TASK B(I) in the

UKF/SRUKF cases.

4 CONCLUSIONS

The results of the previous section clearly illustrate

the general tracking superiority of Dual control over

Cautious and HCE control. In addition, use of the

UKF or SRUKF estimators leads to even better track-

ing results than the EKF - C is reduced by circa 18%

on average in the Dual control case. The SRUKF of-

fers no signiﬁcant advantages in terms of tracking per-

formance with respect to the UKF, neither with option

1 nor option 2.

However, this improvement in tracking perfor-

mance comes at the cost of signiﬁcantly increased

execution time. The UKF and SRUKF with option

1 respectively take 8.2 and 8.4 times longer than the

EKF-based controller, while the SRUKF with option

2 takes around 8.8 times longer.

One therefore concludes that Dual control with a

UKF-based estimator should be used for best track-

ing performance, provided that the hardware is able to

execute the estimation and control algorithm within a

small percentage of the sampling interval. If this con-

dition is not satisﬁed, the faster EKF-based Dual con-

troller could be implemented instead but with a com-

promise; namely an inferior tracking performance.

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