Adaptive-pole selection in the Laguerre parametrisation of model predictive control to achieve high performance

Abstract

In this paper, we study an adaptive method to select online the pole value for a Laguerre scheme in Model Predictive Control (MPC) that yields high performance. It has been observed that, while still using a small numbers of decision variables, the location of the pole affects the closed-loop behaviour significantly. In the present paper, an adaptive algorithm is developed to systematically improve the closed-loop performance of the system as well as the volume of the feasible region and robust feasible region in the case of using a small numbers of decision variables. In order to do this, a method to select a pole value that yields high performance for the initial condition of the system is proposed. The method generates a lookup table of the high-performance pole value obtained through off-line computations. Then, the table is used to assign the pole in the online process. Closed-loop stability for the scheme is established using sub-optimality arguments. Simulations illustrate the suggested method's effectiveness to achieve a balance between performance, optimality, and computational load.

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Adaptive-pole selection in the Laguerre
parametrisation of model predictive control to
achieve high performance
Massoud Hemmasian Ettefagh, Jose De Dona, Farzad Towhidkhah & Mahyar
Naraghi
To cite this article: Massoud Hemmasian Ettefagh, Jose De Dona, Farzad Towhidkhah &
Mahyar Naraghi (2021): Adaptive-pole selection in the Laguerre parametrisation of model
predictive control to achieve high performance, International Journal of Systems Science, DOI:
10.1080/00207721.2021.1933252
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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
https://doi.org/10.1080/00207721.2021.1933252
Adaptive-pole selection in the Laguerre parametrisation of model predictive
control to achieve high performance
Massoud Hemmasian Ettefagh a,JoseDeDona b, Farzad Towhidkhah cand Mahyar Naraghi a
aDepartment of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran; bSchool of Electrical Engineering and Computing,
University of Newcastle, Callaghan, Australia; cDepartment of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran
ABSTRACT
In this paper, we study an adaptive method to select online the pole value for a Laguerre scheme
in Model Predictive Control (MPC) that yields high performance. It has been observed that, while
still using a small numbers of decision variables, the location of the pole affects the closed-loop
behaviour significantly. In the present paper, an adaptive algorithm is developed to systematically
improve the closed-loop performance of the system as well as the volume of the feasible region
and robust feasible region in the case of using a small numbers of decision variables. In order to
do this, a method to select a pole value that yields high performance for the initial condition of
the system is proposed. The method generates a lookup table of the high-performance pole value
obtained through off-line computations. Then, the table is used to assign the pole in the online
process. Closed-loop stability for the scheme is established using sub-optimality arguments. Sim-
ulations illustrate the suggested method’s effectiveness to achieve a balance between performance,
optimality, and computational load.
ARTICLE HISTORY
Received 8 November 2018
Accepted 16 May 2021
KEYWORDS
Model predictive control;
Laguerre; adaptive-pole
tuning; suboptimal;
computation time
1. Introduction
Model predictive control (MPC) is the general name
for various control algorithms that predict the behav-
iour of the system in the future with the aid of
a plant model repeatedly at each sampling time
(Mayne et al., 2000).Thefactthatthismethodtreats
constrained-multivariable processes in a systematic
manner constitutes the main reason that the scheme is
well established. However, there still exist some trade
os between the region of attraction, online compu-
tation times and performance. Usually, a large region
of attraction and a satisfactory performance demand
signicant computation times that prevent us from
applying the scheme to fast systems.
These issues have been addressed by dierent
authors from dierent points of view. Using time-
varying control policies, for instance, Limon et al.
(2005) increased the volume of the feasible region.
Explicit MPC (Bemporad et al., 2002; Seron et al.,
2000)anditsextensions(Zeilingeretal.,2008,2011)
exploit multi-parametric optimisation and bi-level
optimisation to reduce computational burden. How-
ever, the computation becomes hardly tractable for
CONTACT Massoud Hemmasian Ettefagh hemmasian@aut.ac.ir
high-order systems as well as for systems with a large
number of constraints and long horizons (Shekhar
&Manzie,2015).
The idea of reducing the total number of decision
variablesinthecontrollawgoesbacktotheearlystages
of MPC. Choosing the control horizon smaller than
the prediction horizon is introduced in Ricker (1985).
Known in the literature as moving block technique,this
method has successfully shown that it is able to reduce
the computational load (Cagienard et al., 2007); how-
ever, specic considerations should be implemented
to maintain performance and the volume of the fea-
sible region (Shekhar & Manzie, 2015). Predictive
Functional Control (PFC) (Richalet et al., 1987)is
another computationally low method that is attrac-
tive in the industry. Providing less conservative stabil-
ity proofs and improving the performance for unsta-
ble, non-minimum phase or highly oscillatory sys-
tems are active research directions in this eld (Izzud-
din et al., 2015,). Employing orthonormal functions
to reduce the total number of decision variables was
introduced in Wang (2004,2010), and has proven to
be eective in enlarging the feasible region, reducing
© 2021 Informa UK Limited, trading as Taylor & FrancisGroup

2M. HEMMASIAN ETTEFAGH ET AL.
the computational load and keeping performance lev-
els (Khan & Rossiter, 2013). This method, as well
as PFC and the moving block technique, can all be
seen as sub-categorises of a more general framework,
namely aggregation strategies (Li & Xi, 2007). Khan
and Rossiter (2013) used Laguerre and higher-order
parametrisations in MPC and addressed the prob-
lem of determining the pole values for generalised
orthonormal functions parametrisation and the total
number of decision variables with a multi-objective
optimisation approach. An evolutionary algorithm
for a multi-objective optimisation is employed by
Gutiérrez-Urquidez et al. (2015) to systematically tune
the parameters of Laguerre parametsized MPC. The
goal of the multi-objective optimisation was to keep
performance loss and computational load at minimum
levels while trying to maximise the feasible region.
In all of the aforementioned works, a xed pole (or
a set of xed parameters in the case of using Kautz
or higher-order systems) was selected as a result of
an o-line optimisation. In our previous work (Hem-
masian Ettefagh et al., 2018), we established the sta-
bility of suboptimal methods for constrained nonlin-
ear time-varying systems and by virtue of that we
established the general framework of the orthonor-
mal functional parametrisation method to linear time-
varying systems. In the present paper, we focus on
theonlineparametertuningoftheorthonormalfunc-
tional parametrisation method for time-invariant sys-
tems.Inthisrespect,thecontributionofthepresent
paperistwofold.Firstly,wepresentanewstability
proof for Laguerre parametrisation in MPC where
some of the assumptions are relaxed in comparison
with the existing proofs of Wang (2004). This relax-
ation permits us to use a smaller number of decision
variables and to freely choose the prediction horizon.
The second, and main, contribution of this paper is
to recognise that in the Laguerre-parametrised MPC
problem the optimum pole value is related to the given
initial condition, and then to suggest an algorithm for
theonlineselectionoftheLaguerrepoleforthesys-
tem. In the proposed method, the high-performance
pole values for dierent initial conditions are stored in
a lookup table. During the online computations, the
lookuptableisusedtoassignthepolefortheopti-
misation problem which is to be solved. The method
provides a simple yet eective algorithm to increase the
feasible region and performance, compared to the case
of using a single xed value for the pole.
The remainder of this paper is organised as follows:
After providing necessary background about Laguerre
model predictive control in Section 2,wepresenta
less conservative proof for closed-loop stability and
examine inuential parameters on the performance
and optimality of the Laguerre method in Section 3.
An algorithm to determine the high-performance pole
location is presented in Section 4.Illustrativeexam-
ples are given in Section 5to demonstrate the signif-
icant features of the proposed method, followed by
conclusionsinSection6.
2. Laguerre functions and model predictive
control
In this section, we introduce the backgroundand basic
assumptions used in this article.
2.1. Modelling and traditional model predictive
control
Consider a constrained linear time-invariant state-
space model of the form
xa(k+1)=Aaxa(k)+Baua(k)
xa(k)∈Xa,ua(k)∈Ua,(1)
where xa(·)∈Rnis the state vector,ua(·)∈Rmis
the input vector,Aa∈Rn×nis a time-invariant sys-
tem matrix and Ba∈Rn×mis a time-invariant input
matrix. The system is considered to be controllable
and the input and state constraint sets Xaand Ua
are polytopes that contain the origin in their interior.
Orthonormal functions are used here to parametrise
the input vector with less number of decision variables.
Because the particular functions chosen are stable and,
hence, their impulse response decays to zero, the input
vector should be modied in such a way to satisfy this
property. Consequently, in this paper, the system is
augmented with an integrator to change the input vec-
tor uato an incremental value, namely u(k)=ua(k)−
ua(k−1). The augmented system is then:
xa(k+1)
ua(k)
 
x(k+1)
=AaBa
0Im×m
 
A
xa(k)
ua(k−1)
 
x(k)
+Ba
Im×m
 
B
u(k). (2)

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 3
Forsimplicity,thefollowingnotationswillbeused
henceforward to describe the augmented system:
x(k+1)=Ax(k)+Bu(k),(3)
x(k0)=(xa(k0),ua(k0−1)) x0,
x(k)∈X,u(k)∈U,(4)
where XXa×Uadenotes the new constraint set on
variable x(k). In addition to the constraints in (1), we
have added a constraint set Ufor slew-rate constraints
on u(k)=ua(k)−ua(k−1).
In the model predictive control paradigm the fol-
lowing cost function is usually considered as the
performance index that is to be minimised for sys-
tem (3)–(4) at each sampling time k,
J(xk,πk)=
k+Np−1

i=kxi|k2
Q+ui|k2
R+xk+Np|k2
P,
(5)
where x2
Q=xTQx and Q,Rand Pare positive de-
nite state, input and terminal cost weighting matrices,
respectively. πkis the nite sequence of future control
values dened as
πk=uk|k,uk+1|k,...,uk+NP−1|k,(6)
and xi|ksatises:
xi+1|k=Axi|k+Bui|k,fori=k,...,k+Np−1,
with
xk|k=xkx(k).
In this note, we establish stability by employing a ter-
minal constraint set
xk+Np∈XP,(7)
in which XPisapolytopesthatcontainstheoriginin
its interior. The construction of the particular choice
of XPis such that the optimisation of the performance
index J(xk,πk)in (5) for xk∈XPdoes not activate any
constraint; therefore, πkcan be obtained through the
control policy uk+i|k=−Kxk+i|kin which Kand Pare
obtained from the Algebraic Ricatti Equation as
P=ATPA +Q−ATPB BTPB +R−1BTPA, (8)
K=BTPB +R−1BTPA. (9)
For system (1) with the feedback control law ua(k)=
ua(k−1)−Kx(k),thesetXPis chosen as the maxi-
mal output admissible set Gilbert and Tan (1991)
XP=x∈X|(A−BK)ix∈X,
−K(A−BK)ix∈U,∀i≥0. (10)
It is easy to write the ‘prediction vector’ Xas an explicit
function of the future control vector πkand the initial
value xk|k=xkas
Xk=SXxk|k+Sππk, (11)
where Xk,πk,SX,Sπ,Qand R(the latter two, for future
use) are given by
XT
k=xT
kxT
k+1|kxT
k+2|k... xT
k+Np|k, (12)
πT
k=uT
k|kuT
k+1|kuT
k+2|k... uT
k+Np−1|k,
(13)
SXT=IA
T(A2)T... (ANp)T, (14)
Sπ=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
000
B00
AB B 0
A2BAB B
.
.
..
.
..
.
.
.
.
..
.
..
.
.
ANp−1BA
Np−2BA
Np−3B
... ... 0
... ... 0
... ... 0
0... 0
...0
.
.
....0
... ... B
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (15)
Q=blockdiag{Q,Q,...,Q,P}, (16)
R=blockdiag{R,R,...,R}. (17)
Using (11) in (5), we can summarise the resulting
optimisation problem, that we call Traditional MPC
(TMPC) as follows.
Algorithm 2.1 (TMPC): For system (3)–(4) with cost
function (5) and terminal constraint set Xp,theTMPC
algorithmisdenedasfollows.
J(xk,πk)=xT
kF0xk+πT
kHπk+2πkF1xk, (18)

4M. HEMMASIAN ETTEFAGH ET AL.
π∗
k=arg min
πk
J(xk,πk), (19)
Subject to Gπk−g≤0, (20)
where F0=ST
XQSX,H=ST
πQSπ+R, F1=SπQSX,
and G, g are dened appropriately from the constraint
sets U,Xand XP.
We use the rst element of π∗
kas the control input of
the system and then the algorithm proceeds in a reced-
ing horizon fashion. This control scheme achieves
guaranteed closed-loop stability in the case of feasibil-
ity (Mayne et al., 2000).
2.2. Laguerre parametrisation of model predictive
control
The basic idea of parametrisation in MPC is to pro-
duce a controller over the prediction horizon (Np)
with fewer number of decision variables, N<mNp.In
Laguerre parametrisation, the future control vector πk
is parametrised with Laguerre polynomials rather than
using an FIR basis set, hence it allows us to decrease the
number of decision variables. The discrete-time trans-
fer function of Laguerre polynomials in the z−domain
is expressed as
q(z)=√1−a2
z−a1−az
z−aq−1
, (21)
where qis the order of the function. The parameter a
is the pole of the function, which is referred in the lit-
erature as the time scaling factor or Laguerre pole,and
it is chosen to be a∈[0, 1)in order to produce a stable
transfer function. A set of Laguerre functions {q|q=
1, ...,N}is called a Laguerre network of degree N,and
itconstitutesasetoforthonormalbasisfunctions.The
necessary and sucient condition for the complete-
ness of a Laguerre network is (see Oliveira et al. (2011)
for further details)
∞

q=1
(1−|a|)=∞⇔|a|<1. (22)
For an impulse input, we can readily use (21) to obtain
the time evolution of the state-space representation of
Laguerre network as:
LN(k)=√bkLN(0), (23a)
where
b=1−a2,
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a00
ba0
−ab b a
a2b−ab b
.
.
..
.
..
.
.
(−1)N−2aN−2b(−1)N−3aN−3b...
... 0
... 0
... 0
...0
.
.
.
ba
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
LN(k)=⎡
⎢
⎢
⎢
⎣
l1(k)
l2(k)
.
.
.
lN(k)
⎤
⎥
⎥
⎥
⎦,LN(0)=⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
−a
a2
.
.
.
(−1)N−1aN−1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Note that the decaying rate of Laguerre functions
(lq,q=1, ...,N)is adjusted by the Laguerre pole a.A
smaller ameans a faster convergence to zero while as a
increases to 1, the values of lq(·)need more time steps
to approach zero. Moreover, the number of steps (k)
that are needed to achieve orthonormality of the out-
puts in the state-space representation is tuned by this
factor. For identication purposes, the system’s dom-
inant pole is the optimum place for the Laguerre pole
(Fu & Dumont, 1993).
In the Laguerre method, the elements of the control
sequence (6) for a single-input system are expressed as
uk+i|k=
N

q=1
lq(i)ηq(k)=LN(i)Tη(k). (24)
Using (23a)–(24), we are able to write the future con-
trol vector πkas
πk=η(k), (25)
where vector η(k)[η1(k),...,ηN(k)]denotesthe
new unknown variables vector and matrix =
(a,N,Np)is dened as
=⎡
⎢
⎢
⎢
⎣
l1(0)l2(0)··· lN(0)
l1(1)l2(1)··· lN(1)
.
.
.
l1(Np−1)l2(Np−1)··· lN(Np−1)
⎤
⎥
⎥
⎥
⎦.
(26)

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 5
Using parametrisation (25) in the optimisation prob-
lem (5), we dene the following Laguerre MPC
(LMPC) algorithm to control system (3)–(4).
Algorithm 2.2 (LMPC): For system (3)–(4) with cost
function (5) and terminal constraint set Xp,theLMPC
algorithmisdenedasfollows.
JLMPC(xk,η(k)) =xT
kF0xk+η(k)TTHη(k)
+2η(k)TTF1xk(27)
η∗(k)=argmin
η(k)JLMPC(xk,η(k)), (28)
Subject to Gη(k)−g≤0. (29)
Then, the control input sequence πkis reconstructed as
π∗
k=η∗(k)and its rst element is applied to control
the system.
In Algorithm 2.2, JLMPC denotes the cost function
when the Laguerre parametrisation is employed. It
should be mentioned here that the optimum value of
JLMPC in Algorithm 2.2 diers generally from the opti-
mum value of JTMPC in Algorithm 2.1. Their values are
identical only if the LMPC algorithm is able to produce
the same control sequence as the TMPC algorithm.
This matter is examined in detail in the next section
when we perform a stability analysis of the Laguerre
method.
Proposition 2.3: The optimisation problem (27)–(29)
has a closed-form solution that can be described by
η(k;xk)=−(TH)−1TF1xk+1
2(G)actλact ,
(30)
λact =−2(G)act TH−1(G)T
act−1
×(G)act TH−1TF1xk+gact.
(31)
Proof: For the optimisation problem (27)–(29), the
Lagrangian function Lis dened as
L=η(k)TTHη(k)+2η(k)TTF1xk
+λT(Gη(k)−g).
It is known from the active set method that for a
given xk, there exist reduced matrices (G)act ,gact (i.e.
aftereliminatingtherowscorrespondingtothenon-
active constraints) that describe the active constraints.
Hence, for xkone has (G)actη−gact =0. For this
case, the KKT conditions are (Goodwin et al., 2005):
2THη +2TF1xk+(G)actλact =0,
(G)actη−gact =0
that can be solved to determine η(k)and λact as (30)
and (31). The solution shows that η(k)=η(k;xk)and
we interchangeably use one form of the notation or the
other. The reason of using η(k;xk)is to insist on the
dependency of it on the initial condition xk.
Remark 2.1: It should be noted that the Laguerre
parametrisation has a dual eect on the computational
load. The introduction of into (27)–(29) adds some
computations. However, the Hessian matrix of the
LMPC algorithm (TH)hasalowerdimensioncom-
pared to the Hessian Matrix of the TMPC algorithm
(H). This tends to decrease the computational load of
nding the inverse of Hessian matrix. In addition, it is
arguedinKhanandRossiter(2013)thatthecomputa-
tional load of the LMPC algorithm relates to (Nη)3,i.e.
the summation of Laguerre network dimension N(i)
used to capture the control signal of each input channel
i∈{1, ...,m}(Nη=m
i=1N(i)),whereasthecompu-
tational load of the TMPC algorithm relates to (mNp).
Roughly speaking, in order to keep the computational
time of LMPC algorithm less than TMPC algorithm,
one should choose Nη≤3
mNp. Simulation compar-
isons of computational times of LMPC versus TMPC
have been performed in Section 5below to illustrate
these points via two examples.
3. Stability and optimality of the Laguerre
method
3.1. Stability
The existing stability proofs in the literature of
Laguerre-parametrised MPC (see, e.g. Wang (2004,
2009))arebasedonthefactthattheoptimalLaguerre-
parametrised cost function Jis equal to the original
optimal cost function, or that the optimisation at sam-
ple kcan reuse the optimal trajectory computed at the
previous sample. However, a key feature of our cur-
rent contribution is that the pole of the Laguerre net-
work is to be adapted (see Section 4)soastoimprove
closed-loop performance as well as the volume of the

6M. HEMMASIAN ETTEFAGH ET AL.
feasibleregion.Hence,thefactthatthepoleisadapted,
potentially at every sample time, renders the exist-
ing techniques unable to prove stability since there
is no guarantee that the previous optimal trajectories
canbereused.Also,sinceourmainmotivationisto
reduce the number of computations, we want to dis-
pense with the necessity of the Laguerre-parametrised
method achieving the same level of optimality as the
original problem. Thus, in this paper, we study the
stability of the LMPC method via suboptimal meth-
ods. Using the ideas of Scokaert et al. (1999), we show
that the designed controller preserves stability even if
it does not achieve optimality.
We begin this section by recalling a theorem from
Scokaert et al. (1999) on conditions under which a sub-
optimal control method stabilises a constrained sys-
tem, and then we employ it to show that the Laguerre
method (Algorithm 2.2) satises these conditions.
Consider a time-invariant discrete-time nonlinear
system described by
xk+1=f(xk,uk)(32)
where xk∈Rnand uk∈Rmdenote the state and con-
trol vectors at discrete time kand f(·):Rn×Rm→
Rnis assumed to be time-invariant and continuous at
the origin with f(0, 0)=0.
Thegoalistoregulatethestatexto the origin with
the control sequence πkin (6). Let {xk|k,xk+1|k,...,
xk+Np|k}and xk|k=xkdenote the corresponding state
sequence. The control action ukis chosen to be the rst
vector in the sequence πk,i.e.uk=uk|k.
A function α(·), dened on the nonnegative real
numbers, is said to be a K-functionifitiscontinu-
ous, strictly increasing with α(0)=0. For all r≥0,
n≥1wedenethen−dimensional ball with radius
ras Bn
r:={x∈Rn:x≤r}.
Theorem 3.1 (Scokaert et al. (1999)): Consider the
system (32) with the control law uk=uk|kin (6) and
assume that the following conditions are satised:
(1) ThereexistsafunctionV(·):Rn×RNpm→R
continuous at the origin with V(0, 0)=0and a
K-function α(·),suchthatforallx∈Rn,π∈
RNpm,
V(x,π) ≥α(x)(33)
(2) Thereexistsaconstantr>0and a K-function σ(·),
such that every realisation {xk,πk}of the controlled
system with xk∈Bn
rsatises
πk≤σ(xk). (34)
(3) There exists a set F∈Rnthat contains an open
neighbourhood of the origin and a K-function
γ(·),suchthateveryrealisation{xk,πk}of the con-
trolled system with x0∈Fsatises xk∈Ffor all
k≥0and
V(xk+1,πk+1)−V(xk,πk)≤−γ((xk,uk))
(35)
where ukdenotes the rst element of πk,i.e.u
k=
uk|k.
Then, the closed-loop system is asymptotically stable
in F.
Proof: see Scokaert et al. (1999). 
We are now ready to establish the stability of the
MPC method parametrised by Laguerre functions. For
simplicity, we show the stability for a single-input sys-
tem. The extension to multi-input systems is straight-
forward.
Theorem 3.2: SupposetheTMPCalgorithmproduces
an asymptotically stabilising feasible solution for the
constrained system (3)–(4) that satises (35) with
V=J, the cost function in (18).Then,theLMPC
algorithmwiththeclosed-formsolution(30) produces
an asymptotically stabilising MPC controller for a su-
ciently large N.
Proof: We empl o y JLMPC (·,·)in (27) with the solu-
tion (30) as a candidate for the function Vof
Theorem 3.1.
Condition (33) requires positive deniteness of
the function JLMPC(·,·). Accordingly, we notice that
JLMPC(·,·)is a quadratic function with the property
that JLMPC(0, 0)=0, therefore it is easy to nd a K-
function α(xk)such that JLMPC(xk,ηk)≥α(xk),
for instance α(xk)=xT
kQxkisasuitablechoice.
Condition (34) requires boundedness of the control
sequence πk. We need to nd an r>0andaK-function
σ(·)such that for every xk∈Bn
rthe realisation {xk,πk}
satises πk≤σ(xk). First, we dene the norm
of the sequence πkas πk=(πT
kπk)1/2.Usingthe
Laguerre parametrization (25), we write
πk=πT
kπk1/2=η(k)TTη(k)1/2

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 7
≤λ1η(k)Tη(k)1/2(36)
where λ1denotes the positive square root of the maxi-
mum eigenvalue of T. In (29) (equivalently, in (4)),
we assumed that the constraint set Xisapolytopecon-
taining the origin in its interior, so there exists a ρ0for
which Bn
ρ0lies entirely in the interior of X.Itisalso
straightforward to show that there exists a polytope XI
around the origin such that all constraints are inactive
(for example (10)); hence, in XIwe have λact =0. In
a similar manner, we can nd a ρifor which Bn
ρilies
entirely in the interior of XI. The general solution (30)
can be expressed as η(k;xk)=D(xk)xk+d(xk)which
is a continuous piecewise ane function of xk(Bem-
porad et al., 2002). Since for xk∈Bn
ρ0⊂X,D(xk)and
d(xk)have bounded norms, for all xk∈Bn
ρ0we can
dene (λmax(M)denotes the maximum eigenvalue of
matrix M):
λD=max
xk∈X,
k≥k0
λmax(D(xk)TD(xk)), (37)
λd=max
xk∈X,
k≥k0
(d(xk)Td(xk)), (38)
λDd =max
xk∈X,
k≥k0
λmax 2d(xk)TD(xk)T2d(xk)TD(xk)
(39)
Using (36) and η(k)=D(xk)xk+d(xk),itiseasyto
show
πk≤λ1η(k)Tη(k)1/2
≤λ1xT
kD(xk)TD(xk)xk+d(xk)Td(xk)
+2d(xk)TD(xk)xk1/2
≤λ1λDxT
kxk+λd
xT
kxk
ρ2
i+λDdxT
kxk1/2
.
(40)
So, by dening σ(r)as
σ(r)=λ1λDr2+λd
r2
ρ2
i+λDd r(41)
we nd a proper K-function σ(·)and an r=ρ0that
satises condition (34) for all xk∈Bn
ρ0.Notethatwe
have added the term (xT
kxk)/ρ2
ito (40) to preserve the
inequality d(xk)Td(xk)≤λd(xT
kxk)/ρ2
ifor xkinside of
Bn
ρi(when d=0) and outside of it (when xT
kxk≥ρ2
i).
In order to show the monotonic decrease of
JLMPC(·,·)– condition (35) – note that there always
exists an N0≤Np,suchthatguaranteesthemono-
tonic decreasing of the cost function. Such exis-
tence is due to the completeness of the Laguerre
functions, see (22) and Oliveira et al. (2011)for
the details, the fact that the solution of TMPC is
assumed to be feasible, the fact that by increas-
ing Nthe orthonormal functions used in LMPC
algorithm approximate the optimal solution of TMPC
algorithm more accurately and the fact that (18) is
assumedtosatisfy(35).ForthespecialcaseofN=
Npthe orthonormal parametrisation matrix in (26)
is a square matrix with Npindependent rows. Thus,
Equation (25) constitutes a bijective transformation
with an invertible matrix , between the two variables
πkand η(k), and hence the optimal solution of the
LMPC algorithm is the unique optimal solution of the
original TMPC algorithm. This argument shows that
there exist at least one N0such that for N0≤Npthe
solution of the LMPC algorithm produces a monotoni-
cally decreasing cost function. The asymptotic stability
of the scheme then results from Theorem 3.1, since all
the conditions of the theorem are satised. 
Remark 3.1: Along the stability proof of Theorem 3.2
we did not use the orthonormality property of the
Laguerre network which suggests us that we do not
need to preserve this property. Therefore, unlike Khan
and Rossiter (2013); Wang (2004), there might be no
need to select long prediction horizons. This relaxation
in the stability criteria is helpful for the computa-
tion times since it allows us to use shorter prediction
horizons. Furthermore, it makes computations numer-
ically more stable since the growth of the condition
number of TH(Hessian Matrix) is directly related
to Np(Wang, 2001).
3.2. Selection of the degree of the Laguerre network
In Theorem 3.2, we have proven the existence of N0,
for which the suboptimal method is stabilising. How-
ever, we have not discussed yet how to determine such
N0to ensure stability and feasibility. A simple analogy
between the proposed method and traditional MPC
reveals that Nplays the role of the control horizon
(number of control actions that are determined by

8M. HEMMASIAN ETTEFAGH ET AL.
solving the optimisation problem (18)–(20)). There-
fore, the eect on stability is rather straightforward,
thelargertheNis, the more likely it is to ensure sta-
bility. In addition, the value of Nisafundamental
parameter in determining the size of the set F.For
N=Npin the LMPC algorithm, the volume of F,
where the controller is stabilising as well as feasible,
is equal to the volume of the same set in the TMPC
algorithm with the same horizon. The volume reduces
with lower values of N.ItisclearthatFmust be com-
puted as a subset of the feasible region of the proposed
method (27)–(29). We dene the feasible region for the
Laguerre method as follows:
Denition 3.3 (Feasible region): In the LMPC
algorithm (27)–(29), the feasible region FNp
Nfor sys-
tem (1) with constraints (4), (7) is dened as the
projection of the set
⎧
⎨
⎩xk∈X∃{η1,η2,...,ηN}such that Axk+i|k
+B
N

q=1
lq(i)ηq∈X,
N

q=1
lq(i)ηq∈U,
i=0, 1, ...,Np−1, Axk+Np−1|k
+B
N

q=1
lq(Np−1)ηq∈XP⎫
⎬
⎭(42)
on Xa.
3.3. Selection of the Laguerre pole
As it will be illustrated in Section 5,thevalueof
Laguerre pole achanges the size of FNp
N.Moreover,
the level of sub-optimality (or the performance loss)
in the LMPC algorithm is considerably inuenced by
a. Therefore, in order to minimise the level of sub-
optimality and obtain the maximum possible feasible
region for a selected N,itisimportanttoselectthe
value of aappropriately. The standard method for
determining the optimum value of ain the LMPC
algorithm would be to solve ∂
∂aJLMPC =0. This leads
us, through the optimisation (27)–(29), to the deter-
mination of the Laguerre network coecients ηand
Laguerre pole asimultaneously and the resulting cost
function is the closest one to the optimum cost func-
tion of the original problem (18)–(20) in the TMPC
algorithm.
Here in order to obtain some insights into the opti-
misation with respect to a, we study this optimisation
for the simple case of an unconstrained system. Since
the optimal value of η(k;xk)in LMPC can be obtained
for a xed value of a, the solution of (28) in the case of
omitting (29) is
η∗(k;xk)=−TH−1TF1xk. (43)
Substituting (43) in (27) and then taking the partial
derivative with respect to a,wehave
∂
∂aJLMPC #xk,η∗(k;xk)$
=−xT
kFT
1
∂
∂aTH−1TF1xk=0,
(44)
fromwherewecanndtheoptimalvalueofa.If
N=Npthen becomes an invertible square matrix
and hence all s in (44) cancel with each other. In
this case, the solution of (44) becomes trivial. For
the case N<Np, however, (44) becomes a high-order
polynomial with respect to ‘a’ that is generally nonlin-
ear and non-convex. This equation can be considered
as an adaptation law because it provides a ne tun-
ing over one of the controller’s parameters. However,
we should keep in mind that solving this equation
online to nd the optimal ais much more cumber-
some and time-consuming than solving the original
problem (18)–(20) of the TMPC algorithm, so we have
to devise a method to avoid this problem. Instead of
solving the adaptation law (44) online, we present an
algorithminthenextsectiontotacklethisissue.As
a middle ground solution, this algorithm allows to
use values of athat yield good performance for the
real-time optimisation of the LMPC algorithm.
It is also worth mentioning here that the optimal
value of a(solution of (44)) is related to the initial con-
dition xk.ThisfactrevealsthatforaselectedN≤Np
thereisnotaglobaloptimumfortheentirefeasible
region. Moreover, note from (44) that scaling the initial
state xkdoes not change the optimal value of a. Hence,
it is the direction of xkthat is relevant for the optimi-
sation. In addition, according to Theorem 3.2, the cost
function JLMPC(xk,η∗(k;xk)) is bounded and posi-
tive denite with respect to xk. Therefore, for any given
xk= 0 the cost function has a minimum in the given
interval for ‘a’. These observations will be exploited in

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 9
the next section to devise an algorithm for the adaptive
selection of the Laguerre pole.
4. Adaptive-pole selection method
It was mentioned in the previous section that for a
selected number of decision variables Nin the LMPC
algorithm, the optimum Laguerre pole depends on the
initial condition xk.Inthissection,weintroducea
systematic strategy to select the Laguerre pole based
on the current initial state. Consider a system, rep-
resented by (3)–(4), that is to be controlled by the
LMPC algorithm with a prescribed network dimen-
sion N. The goal is to determine the best Laguerre pole
ato minimise the cost function for any given xkin
FNp
N. As it was mentioned in the previous section, the
optimisation problem (27)– (29) with respect to ais
nonlinear and non-convex. Therefore, by considering
aas a decision variable of the online optimisation, the
computational load increases considerably.
In this section, an algorithm is proposed to obtain
the value of the Laguerre pole to achieve high perfor-
mance for the given initial condition. The algorithm
consists of two parts: o-line and online. In the o-
line part, an optimisation over ais performed for every
direction of initial conditions, then a lookup table is
built in which the high-performance value of ais deter-
mined for every direction of initial conditions. The
table is used in the online part to assign the value
of ato the measured initial condition, and then the
online optimisation of the LMPC algorithm is per-
formed with the assigned a.Thelookuptableisused
at every sampling time to assign the value of athat
corresponds to the given initial condition.
Algorithm 4.1 (Adaptive LMPC): O-line computa-
tions:
(i) Grid the surface of the origin-centred unit sphere
Bn
1:={x∈Rn|x=1}by p equispaced points.
(ii) For each point xi,i=1, ...,p, solve the optimisa-
tion problem (44) numerically and, in case of mul-
tiple minima, choose the global minimum a(|a|<
1).
(iii) For i =1, ...,pconstructalookuptableforx
i,axi.
Online computations (perform at each sampling
time):
(i) Forthedirectionofthemeasuredinitialstatex
k
interpolate the appropriate akfrom the lookup
table and give it to Algorithm 2.2 (LMPC)
(ii) Use Algorithm 2.2 (LMPC) to determine the future
control values πk.
The proposed algorithm produces a feasible region
thatcanbeapproximatedastheunionofallfeasible
regions for each axi,i.e.
%
FNp
N≈
p
&
i=1
FNp
N(axi), (45)
in which the delity of the approximation is related to
the number of equispaced points pused to grid the
unit sphere in Algorithm 4.1. The sucient number of
equispaced points pvaries from system to system and
can be easily determined with o-line computations in
which sensitivity analysis is employed to monitor the
increase in cost function when interpolated values for
theLaguerrepole‘a’areused.
Remark 4.1: Algorithm 4.1 is based on an optimi-
sation over parameter ‘a’ for the unconstrained case.
Using (30) and (31) in (44), it can be shown that for
the case of constrained systems, the optimum value
of adeviates from the value of Algorithm 4.1. Adding
the eect of constraints to Algorithm 4.1 converts the
lookup table to a pointwise lookup table for every
initial condition, which is computationally intractable
and inecient.
Using the results of the unconstrained system in
Algorithm 4.1 provides a middle ground solution
between two extremes: the high performance and
computationally complex solution of the optimum ‘a’
withrespecttotheconstrainedcaseandthelowper-
formance but computationally simple solution of the
xed-pole LMPC. It is noteworthy to mention that
the unconstrained optimisation in the o-line part
of Algorithm 4.1 is exclusively designed to nd the
Laguerre pole based on the initial condition. This opti-
misation should not be confused with the constrained
optimisation problem of the online part in which all
state and input constraints are involved.
Remark 4.2: Forthecaseofmultivariablesystems
with ‘m’inputs,item(ii) of the o-line section of
Algorithm 4.1 produces a vector of ‘m’ dierent
elements a={a(1),...,a(m)}astheresultofoptimi-
sation (44). Then, the vector axiis assigned to each

10 M. HEMMASIAN ETTEFAGH ET AL.
direction xiin the lookup table of item (iii).Inasimilar
manner to the case of single input systems, the online
section of the algorithm uses the given initial condi-
tion to determine the appropriate afrom the table.
This shows that the computational complexity of the
online section increases only slightly with the number
of inputs.
In a similar manner, the look-up table of
Algorithm 4.1 occupies m×pmemory cells. This
shows the linearity of the required storage with the
number of inputs and the number of equispaced
points.
Remark 4.3: The previous remark shows that the o-
line computation for determining the optimum pole
value becomes demanding as the number of inputs
grows. However, the computational complexity of the
online section depends mainly on the dimension of
the state space rather than the number of inputs. In
particular, our algorithm uses the given initial condi-
tion (more specically, its direction) and then employs
an interpolation to assign the values of the Laguerre’s
poles. For high-dimensional systems, the computa-
tional time of the search algorithm could increase
drastically even more than the computational time of
the original optimisation problem. This increase could
pose a limitation on the application of the algorithm
for high-dimensional systems.
5. Illustrative examples
This section presents two numerical illustrations to
compare the eciency of the adaptive-pole location
algorithmwithxed-pole(i.e.withoutadaptation)
LMPC as well as with TMPC. In particular, our focus is
on the comparison of the volume of the feasible region
and the level of sub-optimality (performance loss) in
relation to the pole value and the number of decision
variables N.
5.1. Example 1
Consider a discrete-time state-space model with one
input and two states given by:
xa(k+1)=1.24 −0.25
0.45 0.95 xa(k)+0.11
0.02ua(k)
(46)
withweightsandconstraintsgivenby
Q=1.5 0
01.5
,R=0.6, |ua|≤3,
|ua|≤1.5, x∞≤3. (47)
Forthissystemthemaximumfeasibleregionis
achieved by choosing Np=37 and the terminal
set (10) in the TMPC algorithm1. Figure 1and Table 1
compare the volume of the region under which each
algorithm is feasible. The maximum feasible region is
shown by the grey area. The red regions with dashed-
green borders indicate the feasible region of the LMPC
algorithm for the dierent pole values. The feasible
region of the adaptive LMPC algorithm can be approx-
imatedbytheusingofthosesets(see(45)).Inorder
to show the advantage of the LMPC algorithm, the
blue polytope shows the feasible region of the TMPC
algorithm with the same number of decision vari-
ables (Np=2). The cyan polytope at the centre of the
gure (XP) is the maximal output admissible set for
the system (46)–(47) and is chosen as the terminal
constraint. Comparing the numerical values, Table 1
reveals that the Adaptive LMPC Algorithm enlarges
the feasible region 43 percent in comparison with
the TMPC Algorithm with the same number of deci-
sion variables. Additionally, the adaptive-pole location
method, presented in Algorithm 4.1, produces a rel-
atively larger feasible region compared to the case of
using a single xed pole. In the current example, this
method increases the feasible region by 8.5 percent,
compared with the largest feasible region for a xed
pole.
The high-performance values of the Laguerre pole
for dierent initial conditions in Algorithm 4.1 are
presented in Figure 2. The gure shows that the high-
performance values of the Laguerre pole dier sig-
nicantly from the pragmatic value introduced by
Wang ( 2009).Thismeansthatwearenotableto
achieve high performance for the entire feasible region
by using a static pole, no matter what the value is. How-
ever, the adaptive-pole location method selects the
high-performance pole value with respect to the mea-
sured initial condition and provides the best solution
in which the performance loss is minimal.
Figure 3compares the performance of the TMPC
algorithm (NP=37) with LMPC (in two versions:
using xed pole and adaptive-pole) for initial states
x(1)
k=[0.4 −1.5]Tand x(2)
k=[1.05 0.6]T.Notethat

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 11
Figure 1. Visual comparison of volumes for Example 1. The feasible region of LMPC, fixed pole or adaptive, is larger than the feasible
region of TMPC with the same decision variable (N=2).
Tab le 1. Numerical comparison of volumes and performance for
Example 1.
TMPC,
NP=37
TMPC,
NP=2,
Fixed LMPC,
N=2, a=0.8
Adaptive
LMPC, N=2
Feasible region
volume
9.55 3.95 5.20 5.65
J(x(1)
k)†0% ∞152% 12%
J(x(2)
k)†0% ∞36% 31%
Javg 0% – 42% 13%
T(x(1)
k)‡[ms] 92 – 77 88
T(x(2)
k)‡[ms] 95 – 91 91
†Performance loss. See (48) for the definition.
‡Computation time.
we are not able to compare them with the TMPC
algorithm with Np=2 because the selected initial val-
ues do not belong to its feasible region. This gure
shows clearly that for both initial conditions the tra-
jectories resulting from Algorithm 4.1 (black line with
square marker) capture the ‘optimal solutions’ (blue
line with circle marker) more accurately than the tra-
jectories of the xed-pole LMPC (red line with star
marker). Figure 4compares the state and input val-
ues of the methods for the initial condition x(1)
kas time
trajectories.
In order to further evaluate the performance of
the LMPC algorithm, a comparison of the closed-
loop performances under dierent control algorithms
is reported in Table 1.Inthistabletheclosed-loopper-
formance loss J(xk)is dened as (Cagienard et al.,
2007)
J(xk)=J(xk)−J∗
TMPC(xk)
J∗
TMPC(xk)×100%, (48)
where J∗
TMPC(·)is the optimal cost value for the mini-
mum Npwhich results in the maximum feasible region
(inthisexample,Np=37). The value of 0% means the
performance of the selected method, indicated by J(·),

12 M. HEMMASIAN ETTEFAGH ET AL.
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
Figure 2. Values of the high-performance pole in Exam-
ple 1 against the direction of the initial condition
θ=arctan(x2/x1),r=high-performance pole.
The grid size of the unit sphere for the high-performance pole in
Algorithm 4.1 is p=360.
Figure 3. Performance comparison of the adaptive-pole selec-
tion algorithm with TMPC and LMPC with fixed pole in Example 1.
The feasible region of adaptive LMPC is the union of all green poly-
gons and is larger than the feasible region of each of the fixed-pole
LMPC which is just one of the green polygons.
and that of J∗
TMPC(·)are equal and the selected method
does not cause any performance loss. We say the per-
formance loss is ∞if the selected initial condition is
outside of the feasible region.
0 5 10 15 20 25 30 35 40
-1
0
1
0 5 10 15 20 25 30 35 40
-1
0
1
0 5 10 15 20 25 30 35 40
-4
-2
0
2
4
Figure 4. Comparison of state and input trajectories in Example
1 for the initial condition x(1)
k.
Table 1shows that, for the selected initial condi-
tions x(1)
k,x(2)
k, the adaptive LMPC algorithm improves
the level of sub-optimality by decreasing the perfor-
manceloss.Theaverageperformanceloss(Javg )is
calculated for initial states that belong to the feasible
region of the xed-pole LMPC algorithm as well as the
LMPC algorithm with adaptive-pole value (red region
in Figure 1). As can be seen in Figure 3,theselected
initial conditions do not belong to the feasible region
of the TMPC algorithm with NP=2. Therefore, no
solutions exist for these points, and the correspond-
ing performance losses are innity in the table. The
comparison of average performance losses, reported
in Table 1, shows the superiority of the adaptive-pole
LMPC method over the xed-pole LMPC algorithm in
the sense of sup-optimality.
The computational times of the dierent algorithms
forthetwoselectedinitialconditionsx(1)
kand x(2)
kare
reported in Figure 5. The simulations were done on
MATLAB with an active set method for the required
optimizations. The gure shows that the LMPC algo-
rithms,eitherxedoradaptive,arecomputationally
less demanding that the TMPC algorithm with Np=
37. The consistent decrease in computational time
in Adaptive LMPC in comparison with the TMPC
algorithm suggests that our method is successful in
the reduction of computation time. The gure also
shows that the calculation times of the adaptive LMPC

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 13
Figure 5. Comparison of computational times in Example 1 for two different initial conditions x(1)
kand x(2)
k. The consistent decrease in
computational time in Adaptive LMPC when compared to the TMPC is noticeable.
algorithm are slightly larger than the calculation times
of the xed-pole LMPC algorithm. At larger times,
the constraints become inactive, hence the calculation
times of all methods decrease to some constants.
Figure 6. Visual comparison of feasible sets for Example 2. The
feasible set of LMPC, fixed pole or adaptive, is larger than the
feasible set of TMPC with the same number of decision variables
(N=3). Also, the feasible set of adaptive LMPC is the union of
all red polytopes and is larger than the feasible set of each of the
fixed-pole LMPC which is just one of the red polytopes.
5.2. Example 2
Next we consider a higher dimensional and more real-
istic example which has been proposed in Cavallo
et al. (1992) for the longitudinal stability of an agile
aircraft with naturally unstable dynamics. The goal of
using this model is to evaluate the eciency of the pro-
posed algorithm on a more practical application. The
discrete-time state-space model of the aircraft can be
expressed as
xa(k+1)=⎡
⎣
0.983 0.502 2.360
0.002 0.996 −0.295
0 0 0.869 ⎤
⎦xa(k)
+⎡
⎣−2.527
−0.024
0.130 ⎤
⎦ua(k)(49)
Tab le 2. Numerical comparison of volumes and performance for
Example 2.
TMPC,
NP=72
TMPC,
NP=3
Fixed LMPC,
N=3, a=0.82
Adaptive
LMPC, N=3
Feasible region
volume
49.55 5.20 29.7 33.94
J(x(1)
k)†0% ∞15.20% 7.14%
J(x(2)
k)†0% ∞0.005% 0.001%
Javg 0% – 3.41% 1.72%
T(x(1)
k)‡[ms] 390 – 264 257
T(x(2)
k)‡[ms] 324 – 112 115
†Performance loss. See (48) for the definition.
‡Computation time.

14 M. HEMMASIAN ETTEFAGH ET AL.
Figure 7. Values of the high-performance pole for Example 2.
θ=arctan(x2/x1),ψ=arccot(x3/x2
1+x2
2). The high-
performance pole values are displayed by colour intensity on the
unit ball. The grid size of the unit ball for the high-performance
pole in Algorithm 4.1 is p=36000.
in which the weights and constraints are
Q=⎡
⎣
200
0 1000 0
004
⎤
⎦,R=1,
x∞≤5, |u|≤0.3. (50)
The terminal constraint set (10) is dened in accor-
dance with the corresponding unconstrained LQR
controller. The minimum prediction horizon to achi-
evethemaximumfeasiblesetwiththegiventermi-
nal constraint set is Np=72. For this example, the
number of decision variables is selected in accor-
dance with Remark 2.1, i.e. N=3≤3
√72. Figure 6
and Table 2comparethevolumeofthefeasibleset
for each method. The outer white set in the gure
is the maximum feasible set, which corresponds to
TMPC with Np=72. The union of the red sets rep-
resents the feasible set associated with the adaptive
LMPC algorithm. The yellow set is the feasible set
of the TMPC algorithm with Npequal to the num-
ber of decision variables of the LMPC algorithm. The
cyan set at the centre of all other sets is the ter-
minal constraint set. Like in the previous example,
Figure 6and Table 2show the ability of the adap-
tive LMPC method to increase the volume of the
feasiblesetwhilstusingthesamenumberofdeci-
sion variables. For this example, the feasible set of the
adaptive LMPC method is about 14% larger than the
largest feasible set corresponding to a xed value of the
Laguerre pole.
5 101520025303540
-3
-2
-1
0
1
2
3
5 10152025303540
-0.05
0
0.05
0.1
0.15
0.2
5 10152025303540
-0.1
0
0.1
5 10152025303540
-0.5
0
0.5
1
Figure 8. Comparison of state and input trajectories in Example 2 for the given initial condition x(1)
k.

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 15
1234567891011121314
Ti me Instan t
0
50
100
150
200
250
300
350
400
Computat ion T imes [msec]
Compar ison of Computat ion T imes, x
(1)
k
TMPC
Fixed LMPC
Adapti ve LMPC
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Ti me Instan t
0
50
100
150
200
250
300
350
400
Computat ion T imes [msec]
Compar ison of Computat ion T imes, x
(2)
k
TMPC
Fixed LMPC
Adapti ve LMPC
Figure 9. Comparison of computational times in Example 2 for the given initial conditions x(1)
kand x(2)
k. The consistent decrease in
computational time in Adaptive LMPC when compared to the TMPC is noticeable.
Figure 7presents the high-performance values of
the Laguerre pole for initial conditions uniformly dis-
tributed on the surface of the unit ball. The pragmatic
method of Wang (2009)givesa=0.06 which is sig-
nicantly dierent from the high-performance values
showninthisgure.
The performance evaluation of the xed-pole and
adaptive-pole LMPC methods for Example 2 is out-
lined in Table 2. Using (48), the maximum and mini-
mum performance loss indices, correspond to the ini-
tial conditions x(1)
k=[2.2 0.14 −0.13]Tand x(2)
k=
[−2.9 1.14 0.19]T,respectively.Asintherstexam-
ple, the performance loss index for the selected initial
conditions and the average performance loss Javg ,
calculated over 1000 uniformly distributed initial con-
ditions in the feasible region, show that adaptive-pole
LMPC is able to improve the level of sub-optimality
with respect to xed-pole LMPC. To give a graphical
comparison of the performance of the algorithms, the
input and state trajectories are presented in Figure 8.
Like in the previous example, the computational
times of the dierent algorithms for system (49) are
given in Figure 9. In comparison with the TMPC
algorithm, this gure shows that the LMPC algorithm
reduces the computational load in a consistent way.
6. Conclusion
In this paper, an improved method for selecting the
pole value in the Laguerre parametrisation of model
predictive control was presented. The paper employed
a sub-optimality approach to present a novel stabil-
ity proof for the Laguerre parametrisation of MPC
which allows a reduction in the number of deci-
sion variables, then it exploited the results to pro-
pose an adaptive method to assign the Laguerre’s
pole. The method uses the current state (initial
condition of each receding horizon MPC optimisa-
tion) to assign the pole of the Laguerre parametri-
sation used in the optimisation problem which is
solved online. In comparison with the conventional
Laguerre xed-pole selection (regardless of the xed-
pole value), one advantage of the proposed scheme
istheimprovementinthevolumeofthefeasible
region. Additionally, as the pole location is a result
of an optimisation, the method improves the level of
sub-optimality.
In employing the adaptive-pole location method,
one should notice the comments made in remarks 1
and 5 regarding potential sources of increase in the
computation time. Untuned parameters could lead to a

16 M. HEMMASIAN ETTEFAGH ET AL.
computation time larger than the computation time of
the original problem. In order to avoid this problem,
the paper provides a procedure in which the feasible
region, level of sub-optimality, and computation time
are examined. Future work will explore the advantages
of the adaptation of the Laguerre pole in improving
the robust feasible region and the region of recursive
feasibility.
Notes
1. Thereasonfortheexistenceofanitenumbertoachieve
the maximum feasible region and the method of nding it
are articulated in set theories pertinent to MPC, e.g. page
27 of Kouvaritakis and Cannon (2015).
Disclosure statement
No potential conict of interest was reported by the
author(s).
Notes on contributors
Massoud Hemmasian Ettefagh received
his B.Sc. degree from Isfahan University of
Technology, Iran in 2008, and M.Sc. and
P.h.D. degrees from Amirkabir University
of Technology, Iran in 2011 and 2018, all
in mechanical engineering. During 2016-
2019 he was a visiting fellow at University
of Newcastle, Australia and Universidad
Carlos III de Madrid, Spain. Then he joined Huazhong Uni-
versity of Science and Technology (HUST), China as a post-
doctoral fellowship. His research interests include real-time
control, constrained control, model predictive control, time-
varying systems, dierential geometric methods in nonlinear
systems, and design and control of nanopositioning platforms
for AFMs.
Jose De Dona receivedtheB.Eng.degreefrom
Universidad Nacional del Comahue, Argentina in
1989, and a PhD from the University
of Newcastle, Australia in 2000. During
2000 he held a postdoctoral position at the
Universities of Liege and Gent, Belgium.
Since 2001 he has held various positions at
the School of Electrical Engineering and
Computing,The UniversityofNewcastle,Australia,where
he is currently an Associate Professor. During 2008/2009 he
held a Visiting Academic position at Ecole des Mines de Paris,
France. His research interests include constrained control and
estimation, model predictive control, nonlinear control, and
fault tolerant control systems.
Prof. Farzad Towhidkhah was born in
Kermanshah, Iran in 1959. He earned
his Ph.D degree in biomedical engineer-
ingfromUniversityofSaskatchewan,
where he worked on “model predictive
impedance control of joint movement.”
He obtained his bachelor and MSc degrees
in electrical engineering from Amirkabir
University of Technology in Iran. He is a faculty member of
the biomedical engineering faculty of Amirkabir University of
Technology from 1996. His research areas of interest are in
biological system control, motor control, intelligent control,
biological system modeling, system Identication, and bio-
instruments. He was the corresponding author of many papers
in these elds and two books titled “practical usage of infor-
mation technology in medicine” and “human motor control
system.” He is currently leading dierent projects in the eld
of the computational modeling of the attentional control sys-
tem, motor control system, transcranial direct-current stimula-
tion eects, and sensory integration mechanism in his “Cyber-
netics and Modelling of Biological System” laboratory. Prof.
Towhidkhah served as the Dean of Biomedical Eng. Faculty at
Amirkabir University of Tech. for 11 years.
Mahyar Naraghi received his B.Sc. from
University of Minnesota, Minneapolis,
USA, in 1981, and his M.Sc. from Tar-
biat Modarres University, Tehran, Iran,
in 1989, and his Ph.D. degree from Uni-
versity of Ottawa, Canada, in 1996, all
In Mechanical Engineering. He has been
with the Department of Mechanical Engi-
neering at Amirkabir University of Technology since 1996. His
research interests are in robotics and control including both
theoretical and experimental studies.
ORCID
Massoud Hemmasian Ettefagh http://orcid.org/0000-0002-
2123-5218
Jose De Dona http://orcid.org/0000-0002-6880-3227
Farzad Towhidkhah http://orcid.org/0000-0003-3135-9609
Mahyar Naraghi http://orcid.org/0000-0002-0125-848X
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