New Comparisons and Overview of Interpolation Methods for the
uncertain case
Yihang Ding and J.Anthony Rossiter
Abstract— A recent paper [17] proposed a new interpolation
method for predictive control which formulated a control law
that systematically combines two existing approaches. The
key benefit is an algorithm that requires only a quadratic
programming optimisation while giving good feasibility and
potential for robustness. This paper first shows how to extend
this algorithm to the uncertain case. In addition, a detailed
overview of many existing interpolations methods precedes
a full comparison among them. Furthermore, the numeric
comparisons include higher order examples whereas earlier
papers have tended to focus on lower order models.
Keywords: constraints, interpolation, feasibility, computational,
efficiency, uncertain, LPV
I. INTRODUCTION
Model Predictive Control (MPC) [3], [11], is one of the
most important advanced control techniques to have had
a significant and widespread impact on industrial process
control. Within this approach, a common objective in the
MPC community is to guarantee asymptotic stability and
recursive constraint satisfaction for a set of initial states
that is as large as possible while simultaneously obtaining a
minimal control cost and computational load. Typically there
is a conflict existing between the computational efficiency,
which depends on the number of d.o.f., the volume of the fea-
sible region and the performance. Interpolation techniques,
the topic of this paper, [10], [1] provide one favorable
means of trading off between computation and feasibility.
Nevertheless, a recent paper [17] explored the potential of a
combination approach which forms a middle ground between
interpolation and conventional MPC. Interpolation can be an
effective means of augmenting feasibility volumes, but not
necessarily in all state directions, whereas MPC tends to have
limited but omni-directional benefits.
MPC algorithms are often specified for a linear or nominal
case and it is necessary to assume that either, the inherent
robustness of the approach or some form of back off, will
negate the effects of uncertainty. Hence, there is much inter-
est in how to extend MPC to cater explicitly for parameter
uncertainty. This paper focuses on uncertainty modeled by
linear parameter varying (LPV) system. The predominant
papers in the literature use ellipsoidal invariance as a key
tool in establishing the stability of LPV system; one can
use linear matrix inequalities (LMI) to set up conditions
for feasibility, stability and convergence and LMIs give rise
to convex optimizations. However, LMI optimizations can
be computationally demanding and ellipsoidal regions may
be conservative compared to polyhedrals and hence in this
paper, LMIs are used solely to establish convergence but not
for feasibility.
Y. Ding and J.A. Rossiter are with Department of Automatic Control
and Systems Engineering, University of Sheffield, UK. Email: dingyi-
hang2000@hotmail.com
It has been shown (e.g. [8]) that a simple algorithm can be
used to define the the maximal admissible set (MAS) [4] for
the uncertain case (the main downside is the set complexity).
Thus, it is straightforward to extend MPC techniques and
constraint handling [12] to cater for LPV systems. Therefore,
one minor contribution of this paper is to establish that the
interpolation proposed in [17] can indeed use recent insights
to be extended to the uncertain case. Having established this,
the next key question is whether the proposed interpolation
has any value, that is, how does it compare to existing
techniques [18], [16], especially how much improvement
is there compared to a conventional MPC algorithm? This
paper demonstrates the potential through several numerical
examples, and moreover, uses higher order examples than
adopted in many previous papers the literature. A different
form of presentation of the results is used to illustrate the
benefits for high order systems.
In summary, section II will give some background to
MPC, the modeling assumptions and a quick review how
to extend the original interpolation algorithms for LPV
systems. Section III proposes a means of combining the new
interpolation method [17] with a LPV system. Section IV
gives three numerical examples and the paper finishes with
conclusions and suggestions for future work.
II. BACKGROUND
This section introduces standard material from the existing
literature on MPC, invariant sets and some basic interpolation
schemes for uncertain systems. For completeness a brief de-
scription is given of the following interpolation schemes: (i)
ONEDOF (the most basic form); (ii) GIMPC; (iii) GIMPC2
(the most powerful form and development of GIMPC); (iv)
GIMPC2 (a compromise between ONEDOF and GIMPC2);
(v) OMPC (a typical MPC algorithm).
A. Model, objective and constraints
This paper considers uncertain systems of the form
+1
= ()
+ ()
, =0,..., ∞ (1)
where ((),()) ∈ (
1
,
1
),..., (
,
)
and subject to constraints (other constraints possible).
() ∈≡{ : ≤ ≤ }, =0,..., ∞, (2a)
() ∈≡{ : ≤ ≤ }, =0,..., ∞. (2b)
() ∈ ℝ
and () ∈ ℝ
denote state and input vectors
at discrete time with
and
respectively denoting the
number of states and inputs of the system.
An underlying aim is to minimise an upper bound on a
predicted cost of the form:
=
∞
=0
(()
T
()+ ()
T
()) (3)
2010 8th IEEE International Conference on
Control and Automation
Xiamen, China, June 9-11, 2010
FrB1.1
978-1-4244-5196-8/10/$26.00 ©2010 IEEE 1635