Biquadratic stability of uncertain linear systems

Abstract

Deals with the problem of stability analysis for linear systems
with uncertain real, possibly time-varying, parameters. A robust
stability approach based on a Lyapunov function which depends
quadratically on the uncertain parameters as well as in the system state
is proposed. This robust stability approach, referred to as biquadratic
stability, is suited to deal with uncertain real parameters with
magnitude and rate of change which are confined to a given convex
region. A linear matrix inequality based sufficient condition for
biquadratic stability is developed. The proposed robust stability
analysis method includes quadratic stability and affine quadratic
stability as particular cases

Full text

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 8, AUGUST 2001 1303
Clearly, ???? ?? ? ???? ? ?? ? ??????? ? ?. Moreover
???
???????? ?? ??
???
???????? ?? ??
?fixed ?
?fixed ??
(45)
Hence, there exists a function of class ??, ?, such that ????? ??? ?
???? ????. By virtue of (43), then
?
?
????? ? ????? ??
?
? ?
?????? ?????
?????????
????
? ??????? ?????
???? ???
(46)
If instead ?????? ???????, (42) yields
?
?
????? ? ???????
?
??
?
???
??????? ??????? ? ????
????
??? ?? ?????
????
????????
????
? ???
????
????
??
?
? ???
????
????
?
? ??? ????
???????
? ????
?
?????? ????????
?
(47)
Hence, if we define ???? ? ??and the ?? function ???? ?? ??
???????????? ??? ????? ????????, then we can write that ? ? ?
implies ??
estimate holds for all ? ?
sion for energy of trajectories outside the circle ?. However, numerical
simulations of (41), augmented with the equation ? ? ? ??? ??, which
was used in order to compute energy, clearly showed that a ?? bound
is enforced also for ? ? ? ? (Fig. 1 (right) shows power as a function of
time for initial conditions in a small neighborhood of the origin).
???? ? ?????? ??, ?? ? ?. It is hard to show that a similar
?since we do not have an explicit expres-
REFERENCES
[1] D. Angeli and E. D. Sontag, “Forward completeness, unboundedness
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[3]
, “Further equivalences and semiglobal versions of integral input to
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[4] A.Isidori,
NonlinearControl
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[5] H. K. Khalil, Nonlinear Systems.
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tive Control Design. New York: Wiley, 1996.
[7] D. Liberzon, E. D. Sontag, and Y. Wang, “On integral input-to-state
stabilization,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp.
1598–1602.
[8] D. Neˇ sic ´, A. R. Teel, and E. D. Sontag, “Formulas relating ?? sta-
bility estimates of discrete-time and sample-data nonlinear systems,”
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[9] L. Praly and Y. Wang, “Stabilization in spite of matched unmodeled dy-
namics and an equivalent definition of input-to-state stability,” Math.
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SystemsII.London,U.K.:
Upper Saddle River, NJ: Prentice-
[10] E.D.Sontag,“Commentsonintegralvariantsofinput-to-statestability,”
Syst. Control Lett., vol. 34, pp. 93–100, 1998.
[11]
, “Smooth stabilization implies coprime factorization,” IEEE
Trans. Automat. Contr., vol. 34, pp. 435–443, Apr. 1989.
[12] E. D. Sontag and Y. Wang, “New characterizations of input to state sta-
bility,”IEEETrans.Automat.Contr.,vol.41,pp.1283–1294,Sept.1996.
[13]
,“Oncharacterizationsoftheinput-to-stateproperty,”Syst.Control
Lett., vol. 24, pp. 351–359, 1995.
[14] E. D. Sontag, “Stability and stabilization: Discontinuities and the ef-
fect of disturbances,” in Nonlinear Analysis, Differential Equations and
Control, F. H. Clarke and R. J. Stern, Eds.
pp. 551–598.
Boston, MA: Kluwer, 1999,
Biquadratic Stability of Uncertain Linear Systems
Alexandre Trofino and Carlos E. de Souza
Abstract—Thisnotedealswiththeproblemofstabilityanalysisforlinear
systems with uncertain real, possibly time-varying, parameters. A robust
stability approach based on a Lyapunov function which depends quadrati-
cally on the uncertain parameters as well as in the system state is proposed.
This robust stability approach, referred to as biquadratic stability, is suited
to deal with uncertain real parameters with magnitude and rate of change
which are confined to a given convex region. A linear matrix inequality
(LMI) based sufficient condition for biquadratic stability is developed. The
proposed robust stability analysis method includes quadratic stability and
affine quadratic stability as particular cases.
Index Terms—Quadratic parameter-dependent Lyapunov functions,
real time-varying uncertain parameters, robust stability, uncertain
systems.
I. INTRODUCTION
Stability analysis of linear systems involving uncertain real parame-
ters has attracted a lot of attention in recent years. Research in this area
hasbeenmainly directedtothefollowingthreecases, dependingonthe
time-dependence of the uncertain parameters:
• systems with time-invariant (or constant) parameters;
• systems with very fast time-varying parameters;
• systems with time-varying parameters with bounded rate of vari-
ation.
Many results have been proposed for the first two extreme cases,
where the frameworks of ? analysis (for the first case) and quadratic
stability (for the second case) have been widely used. The notion of
quadratic stability, which consists in seeking a matrix ? ? ?, and in-
dependent of the uncertain parameters, such that a quadratic Lyapunov
function ? ??? ? ???? proves the stability of the system for all ad-
missible parameters, is numerically appealing and originated a body
of important results. However, since the Lyapunov function is param-
Manuscript received June 7, 1999; revised June 13, 2000 and March 20,
2001. Recommended by Associate Editor L. Dai. This work was supported
by “Conselho Nacional de Desenvolvimento Científico e Tecnológico -
CNPq,” Brazil, under Grants 300459/93-9/PQ and 301653/96-8/PQ, and
PRONEX/CNPq 66.2015/98-3.
A. Trofino is with the Department of Automation and Systems, Universi-
dade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil (e-mail:
trofino@lcmi.ufsc.br).
C. E. de Souza is with the Department of Systems and Control, Laboratório
Nacional de Computação Científica - LNCC/MCT, 25651-070 Petrópolis, RJ,
Brazil (e-mail: csouza@lncc.br).
Publisher Item Identifier S 0018-9286(01)07693-0.
0018–9286/01$10.00 © 2001 IEEE

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1304IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 8, AUGUST 2001
eter independent, stability holds even when the parameters change ar-
bitrarily fast, which can be quite conservative in many applications. On
theother hand,thethirdcaseabove,whichisthemostgeneralcase,has
also been studied recently, but to a lesser degree. This case is quite im-
portant in applications as a system may be asymptotically stable if the
parameters are constant but not quadratically stable, and may still re-
mainasymptoticallystableiftheratesofvariationoftheparametersare
restricted to appropriate ranges. In order to reduce the conservatism of
quadratic stability, the notion of parameter-dependent Lyapunov func-
tions was introduced in [1], i.e., quadratic Lyapunov functions where
the Lyapunov matrix depends on the uncertain parameters.
Since the work of [1], a great deal of interest has been devoted to
robust stability analysis and robust control synthesis methods based on
parameter-dependent Lyapunov functions and a number of results has
been proposed in the literature; see, e.g., [3]–[20] and the references
therein. Most of these works deal with uncertain continuous-time sys-
temsandarebasedonaffineparameter-dependentLyapunovfunctions,
i.e., Lyapunov functions which are quadratic on the system state and
depend affinely on the uncertain parameters. The latter methods differ
mainly on the assumptions made regarding the uncertain parameters
andtheirrateofvariation,andontheoverboundingtechniquesadopted.
For example, in [9] an affine parameter-dependent Lyapunov func-
tion is used and it is assumed that the uncertain parameters and their
time-derivatives are valued in given polytopes. Using multiconvexity
arguments, [9] proposed sufficient conditions for robust stability anal-
ysisbasedonlinearmatrixinequalities(LMIs).Aninterestingstudyon
the connections between affine parameter-dependent Lyapunov func-
tions and thereal ? approachcan be foundin [6],which alsodeveloped
a robust control synthesis method based on such Lyapunov functions.
See also [14] for connections between parameter-dependent Lyapunov
functions and frequency-dependentinput/output scaling for? analysis.
The results in [10], [11], [13] and [16] are also based on affine pa-
rameter-dependent Lyapunov functions. An interesting feature of [11]
and [16] is that they also deal with the robust control problem with
a quadratic performance index. On the other hand, multiaffine Lya-
punov functions have been considered in [3], [7] and [8]. Further, fre-
quency domainapproachesto robust stabilityanalysis forsystems with
slowly time-varying uncertain parameters satisfying the so-called inte-
gral quadratic constraints have been developed in [12], [15] and [17].
This note, which is an extended version of a CDC’99 paper [20],
deals with the problem of robustness analysis for linear systems with
uncertain real, possibly time-varying, parameters. It is assumed that
the parameters values and rates of variation lie in a given polytope. A
new robust stability analysis method is proposed using a parameter-de-
pendent Lyapunov function. In order to try to reduce the conservatism
of the methods based on affine quadratic stability, a robust stability
approach based on a Lyapunov function which depends quadratically
on the uncertain parameters as well as on the system state is adopted.
This stability approach, referred to as biquadratic stability, is suited to
deal with uncertain real time-varying parameters with bounded rate of
change. An LMI based sufficient condition for biquadratic stability is
developed. The proposed stability analysis method includes quadratic
stability and affine quadratic stability as particular cases. Moreover, it
hasanadvantageovermostofexistingstabilitycriteriabasedonparam-
eter-dependent Lyapunov functions as it provides an appropriate ap-
proach for solving problems of robust control synthesis via parameter-
dependent Lyapunov functions, including ?? and ?? control prob-
lems [4], [5].
Notation: ??denotesthe?-dimensionalEuclideanspace,????is
the set of ??? real matrices, ??is the ??? identity matrix, and the
notation ? ? ? (respectively, ? ? ?), for a real matrix ?, means that
? is symmetric and positive–definite (respectively, negative–definite).
II. MAIN RESULTS
Consider the uncertain linear continuous-time system
? ???? ?
???
?
???
????
????
(1)
where ? ? ??is the state, ?? ? ????, ? ? ?? ???? ?, are known
matrices and ??? ? ? ?? ???? ?, are bounded uncertain real, possibly
time-varying, parameters with bounded rates of variation.
It isassumed that the parametervector ? ? ??????????andits time-
derivative?? ? ????????????lie in a given bounded polyhedral domain
? with ? known vertices.
Thispaperisaimedatdeterminingifthesystem(1)isasymptotically
stable for all the admissible parameters ??, ? ? ?? ???? ?, as above. To
this end, the system (1) will be rewritten as follows:
? ???? ? ???? ??????????
(2)
where ? ? ????is a known matrix and ???? ? ????is an uncertain
matrix such that each row depends linearly on the uncertain parameter
?, and the value of ? depends on the choice of ?.
A direct, and natural, choice of ? and ? is
? ? ???
???
????
? ? ?????
???
???????
(3)
Observe that ? ? ?? for ? as above and this is possibly the choice of
? of largest dimensions which is not redundant. It should be noted that
the factorization ?? of
?? ??
?
???
????
is not unique. For example, with ?? of the form
?? ?
?
...
?
?
...
?
???
?
...
?
???
???
????
????
???
????
where ??? ? ? ?? ???? ?, are known scalars, a possible decomposition
??, in addition to that of (3), is
? ? ??
???
?????
? ? ?????
????
???
??????
In addition to (3), alternative factorizations of ?? can be constructed
from factorizations of the matrices ??, namely given factorizations
?? ? ????
? one can choose
? ? ???
???
????
? ? ?????
???
???????
Note that the choice of the dimension ? of ? is closely related to the
conservatism and computation burden of the robust stability method to
bedeveloped.Indeed,itturnsoutthatanincreaseof? islikelytoreduce
the conservatism as it increases the number of decision variables in the
underlying optimization problem, however it will increase the required
computational effort.
Several methods of robust stability analysis for system (1) has been
recently proposed in the literature, where the majority of them are
based on the notion of affine quadratic stability. Specifically, they
use the so-called affine parameter-dependent Lyapunov functions, i.e.,
? ??? ?? ? ???????, where ???? is a symmetric positive definite
matrix which depends affinely on ?. These methods are extensions of
the quadratic stability approach and is always less conservative than

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 8, AUGUST 20011305
the latter whenever the parameter rate of variation is bounded. In this
paper a new robust stability test for system (1) is proposed using a
parameter-dependent Lyapunov function. In order to try to reduce the
conservatism of the methods based on affine quadratic stability, the
following notion of stability is used in this note.
Definition 2.1: Givenadomain? ofadmissible? and??,thesystem
(1)issaidtobe biquadraticallystableifthereexistsasymmetricmatrix
???? ? ????of the form
???? ????? ??? ? ????
? ? ?????
?
?
?
?
?
?
?
?
? ??
??
??
??
?
??
(4)
such that
???? ? ??
??????? ? ?
(5)
?????
??
????? ???????? ? ???????? ??? ? ??
??????? ? ??
(6)
Note that biquadratic stability implies that the system (1) is asymptot-
ically stable for all ?????? ? ?. Indeed, it turns out that
? ??? ?? ?? ???????
(7)
is a parameter-dependent Lyapunov function for the system (1) that
dependsquadraticallyontheuncertainparameters??andonthesystem
state.
Observe that the condition ???? ? ? does not, in general, imply
? ? ?. Further, the inequalities of (5) and (6) are nonconvex in ? and
??, and thus the problem of verifying their feasibility is not tractable, in
general, neither analytically nor numerically.
Remark 2.1: It should be noted that biquadratic stability comprises
quadraticstabilityandaffinequadraticstabilityasspecialcases.Specif-
ically, when the system (1) is biquadratically stable with ?? ? ? and
?? ? ?, it turns out that ???? ? ??, which implies the quadratic sta-
bility of the system (1).
On the other hand, the notion of affine quadratic stability can be
obtained from biquadratic stability by imposingthe restriction ?? ? ?.
Indeed, in this case it follows that (7) becomes
? ??? ?? ? ??
??? ??? ? ????
?
? ? ??
???
?
???
?????
?
for some symmetric matrices???? ? ? ?? ???? ?.
An LMI based sufficient condition for biquadratic stability is pre-
sented below.
Theorem 2.1: Consider the system (1) and let ? be a polytope of
admissible ??????. Then this system is biquadratically stable if there
existasymmetricmatrix? ? ????????????,andmatrices?and? ?
????????such that the following LMIs are satisfied at all the vertices
of ?:
? ? ????? ? ?????????
(8)
??
???? ? ? ?? ???? ? ????? ? ?????????
(9)
where
? ????
?
??
?
??
?? ? ???
????? ? ??
?? ??
(10)
Moreover, under the previous conditions
? ??? ?? ? ??
?
?
?
?
?
?
?
is a biquadratic Lyapunov function for system (1).
Proof: Assume that the conditions of Theorem 2.1 are satisfied.
First note that, since the inequalities of (9) and (8) depend affinely on ?
and??,thentheseinequalitiesaresatisfiedforall?????? ? ? ifandonly
if (9) and (8) are satisfied at all the vertices of ?; see [2] for details.
Let the following biquadratic Lyapunov function candidate for
system (2) [or equivalently, system (1)]:
? ??? ?? ? ??????? ? ??
?
?
?
?
?
?
?
(11)
where ? is a symmetric matrix satisfying (8) and (9).
Next, note that in view of the definition of ???? we get
????? ? ??? ?? ?
iff ? ?
?
?
??? ?? ??? ? ??? (12)
Hence, considering (8) it follows that ? ??? ?? satisfies
? ??? ?? ? ???? ? ??
?? ?? ?? ? ????
? ??
Ontheother hand, thetime-derivative of ? ??? ??along thetrajectories
of (2) satisfies
?? ??? ?? ? ??
?
?
?
? ????
?
? ? ?? ????
?
?
?? (13)
Now, taking into account (9) it follows that?? ??? ?? is a negative def-
inite function of ? for all ?????? ? ?. Therefore,?? ??? ?? is a bi-
quadratic Lyapunov function for system (1) and thus, this system is
biquadratically stable.
Remark 2.2: Theorem 2.1 provides a sufficient condition for
biquadratic stability of system (1). To our knowledge, the existing
methods for robust stability analysis of uncertain linear system with
time-varying parameters with bounded rates of variation are also
based on sufficient conditions. The results in [19] are also based on
a similar Lyapunov function and may be used to analyze the stability
of a large class of uncertain linear systems. However, the results of
this paper have the advantage that can be used to solve problems of
robust control synthesis via parameter-dependent Lyapunov functions,
including robust ?? and robust ?? control problems. The latter
problems have been tackled in [4] and [5].
Since quadratic stability is a special case of biquadratic stability, and
considering that Theorem 2.1 provides only a sufficient condition for
biquadratic stability, a natural question that arises is if Theorem 2.1
recovers the well known condition for quadratic stability. Indeed, it
happens that in the case where system (1) is quadratically stable the
conditions of Theorem 2.1 are satisfied as shown below.
Corollary 2.1: Suppose that system (1) is quadratically stable and
let ? ??? ? ????? be a suitable Lyapunov function. Then there exists
a sufficiently small scalar ? ? ? such that the conditions of Theorem
2.1 are satisfied with ?? ? ??, ?? ? ?, ?? ? ? and
???
? ?
?
???
? ?
???
??
?
(14)
Proof: The quadratic stability of system (1) is equivalent to
???? ??????? ?????? ??? ? ??
for all admissible ??

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1306IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 8, AUGUST 2001
As ? is bounded, there exists a sufficiently small scalar ? ? ? such that
??? ??????? ???
???? ??????? ?????? ??? ? ??????? ???
With ?? ? ??, ??? ?, ??? ?, and ? and ? as in (14), the left-hand
side of (8) and (9), denoted by ??and ??, respectively, become
(15)
???
??
????
??????
???? ??????? ?????? ???
??
???
???
????
?
Finally,usingSchur’scomplementsandconsidering(15),itresultsthat
??? ? and ??? ?, which concludes the proof.
In the light of Corollary 2.1, it turns out that the conditions of The-
orem 2.1 with ?? ? ? and ?? ? ? are indeed necessary and sufficient
for quadratic stability of system (1).
Itshould be noted that in addition to quadratic stability, Theorem 2.1
canalsobeusedtotesttheasymptoticstabilityofsystem(1)inthecase
where the parameters are time-invariant; such stability notion will be
referred to as robust stability Indeed, a sufficient condition for robust
stability follows directly from Theorem 2.1 by setting?? ? ?.
Observe that the notions of robust and quadratic stability are asso-
ciated with zero and arbitrary fast parameter variations, respectively,
which are two extreme situations that are somehow ideal in practice.
Note that a system may not be asymptotically stable for arbitrary fast
parameter variations and asymptotically stable for large, but bounded,
parameter variations. On the other hand, a system may be asymptot-
ically stable for zero parameter variations and still be asymptotically
stable for small parameter variations. These facts highlight the rele-
vance of Theorem 2.1.
We now present a “dual” version of Theorem 2.1. To this end, (1)
will be rewritten in the following “dual” form:
???
? ???? ? ??? ??
?????? ????
(16)
where ?? ? ????is a known matrix and ????? ? ????is an un-
certain matrix such that each row depends linearly on the uncertain
parameter ?.
Applying Theorem 2.1 to the adjoint system of (16) one obtains the
following result.
Theorem 2.2: Consider the system (1) and let ? be a polytope of
admissible ??????. Suppose that there exist a symmetric matrix ? ?
????????????, and matrices ? and ? ? ????????such that the fol-
lowing LMIs are satisfied at all the vertices of ?:
? ? ?????? ? ??
????????
(17)
?? ???? ? ? ???
?
???? ? ?????? ? ??
????????
(18)
where
???
??
????
?? ????
?
??
????
?
????? ? ???
?? ??
(19)
Then (1) is exponentially stable for all ?????? ? ? and ? ??? ?? ?
?????????, where
???? ?
?
?
?
?
?
?
is a parameter-dependent Lyapunov function for that system.
Similarly to Theorem 2.1, we have the following corollary, whose
proof is similar to that of Corollary 2.1.
Corollary 2.2: Suppose that system (1) is quadratically stable and
let ? ??? ? ????? be a suitable Lyapunov function. Then there exists
a sufficiently small scalar ? ? ? such that the conditions of Theorem
2.2 are satisfied with ?? ? ???
? , ?? ? ?, ?? ? ? and
? ?
?
???
? ?
???
? ??
??
?
(20)
In view of Corollaries 2.1 and 2.2, notice that Theorems 2.1 and 2.2
are equivalent in the case where ???? is independent of ?, i.e., when
quadratic stability is used. However, this is no longer true for a param-
eter-dependent matrix ????. In this case, both Theorems 2.1 and 2.2
shouldbeusedandthebestobtainedresult,intermsofstabilitymargin,
should be adopted.
III. AN EXAMPLE
Consider the following uncertain system, which as borrowed from
[2, p. 73]:
? ???? ?
???
??????
???? ? ??
????
????
?
??
????
(21)
where ??is an uncertain real, possibly time-varying, parameter.
Although system (21) is asymptotically stable for any fixed ?? in
the range ? ? ?? ? ?, it is not quadratically stable for ?? in that
range. However, using Theorem 2.1 it is obtained that system (21) is
asymptotically stable for the same range of values of ?? but with the
rate of variation restricted to ????? ? ?????. On the other hand, by the
method of [9], which is based on an affine parameter-dependent Lya-
punov function, stability is guaranteed only for ????? ? ??????. Note
that using Theorem 2.1 with the constraint that the Lyapunov function
is affine on the parameters, i.e., with ?? ? ?, stability is guaranteed
for ????? ? ??????.
When the range of ?? is increased to ???? ? ?? ? ?, it follows
from Theorem 2.1 that asymptotic stability is ensured as long as ????? ?
?????.Ontheotherhand,theaffinestabilitycriteriaof[9]andTheorem
2.1with ?? ? ? ensureasymptotic stability for ????? ? ???? and ????? ?
?????, respectively.
Now, supposing that ??is constant, the range of value of ??obtained
from Theorem 2.1 ensures stability is ???????? ? ?? ? ????????,
which is the same as provided by the affine stability criteria of [9] and
Theorem 2.1 with ?? ? ?. Note that, the actual range of values ??for
asymptotic stability is ?????? ? ?? ? ???? ????????????, i.e.,
?????????? ? ?? ? ?????????.
Thus, for this example, the stability test based on a biquadratic
Lyapunov function of this paper is less conservative than the affine
quadratic stability method of [9], in the sense that the former provides
a larger bound for the parameter rate of variation which ensures
asymptotic stability.
IV. CONCLUSION
This note addressed the problem of robust stability analysis linear
systems with uncertain real time-varying parameters. The uncertain
parameters enter affinely in the state matrix of the system state-space
model and the admissible values of the parameters and their rates of
variation are assumed to belong to a given polytope. The note pro-
posed an LMI method of robust stability analysis which is based on
a parameter-dependent Lyapunov function with quadratic dependence
on the uncertain parameters. The proposed method includes quadratic
stability and affine quadratic stability as particular cases and has the
advantage that it provides an appropriate approach for robust control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 8, AUGUST 20011307
synthesis via parameter-dependent Lyapunov functions, including ro-
bust ??control and robust ??control.
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Philadelphia, PA: SIAM,
control of
Efficient Active Set Optimization in Triple Mode MPC
Mark Cannon, Basil Kouvaritakis, and J. Anthony Rossiter
Abstract—An active set algorithm tailored to quadratically constrained
quadratic programming in model predictive control (MPC) is presented.
It enables efficient use of augmented ellipsoidal invariant sets in conjunc-
tion with polytopic constraints in triple mode MPC. The algorithm gives
improved optimality and larger stabilizable initial condition sets than con-
ventional quadratic programming MPC algorithms of comparable online
computational burden.
Index Terms—Constrained control, optimization, predictive control,
quadratic programming.
I. INTRODUCTION
Early dual mode model predictive control (MPC) algorithms [1], [2]
optimized predicted performance online over a finite number (say ?)
of future control inputs. To guarantee closed-loop stability, the ?-step
ahead predicted plant state was constrained to lie within a set which is
feasible and invariant under a postulated fixed-term feedback law. For
the case of linear dynamics with input and/or linear state constraints
and a quadratic performance index, the associated online optimization
can be formulated as a linearly constrained quadratic program (QP).
However, forhigh order systems or stringent constraints a large ? may
be needed to ensure initial feasibility, and this can make the online
computational burden prohibitive.
An efficient dual mode strategy proposed in [3], [4] was based on
feasible invariant ellipsoidal sets in the space of the plant state aug-
mented by the degrees of freedom (dof) in predicted input trajectories.
These sets are computed offline and guarantee closed-loop stability
when invoked as constraints on the augmented state at the beginning
of the prediction horizon. The online optimization was thus reduced to
a 2-norm minimization subject to a single quadratic constraint which is
solved extremely efficiently. A drawback of the approach is that ellip-
soidal feasible invariant sets are conservative with respect to the orig-
inal input/state constraints. Although the associated suboptimality can
be reduced to an insignificant level by scaling with minimal additional
computational effort [5], the set of stabilizable initial conditions is nec-
essarily smaller than for a QP-based MPC algorithm employing the
same parameterization of dof in predictions.
Recent work [6], [7] improved the feasibility properties of MPC
based on augmented ellipsoidal sets by using a triple mode strategy
which splits the prediction horizon into three intervals: 1) ? samples
over which predictions are subject to the input/state constraints speci-
fied by the plant model and control problem; 2) ? samples over which
input/state constraints are approximated by an augmented ellipsoidal
set; and 3) the remaining infinite interval on which predictions revert
to a fixed-term feedback law.
The dof associated with mode 2) allow for a larger stabilizable set
than is obtained using modes 1) and 3) alone, but a direct implementa-
tion involves both linear and quadratic constraints. To avoid this, [6],
[7]proposedmodifiedoptimizationproblemswhichreducedtheonline
ManuscriptreceivedNovember20,2000.RecommendedbyAssociateEditor
V. Balakrishnan. This work was supported by the U.K. EPSRC.
M. Cannon and B. Kouvaritakis are with the Department of Engi-
neering Science, Oxford University, Oxford, OX1 3PJ, U.K. (e-mail:
mark.cannon@eng.ox.ac.uk; basil.kouvaritakis@eng.ox.ac.uk).
J. A. Rossiter is with the Department of Automatic Control and Sys-
tems Engineering, Sheffield University, Sheffield, S1 3JD, U.K. (e-mail:
j.a.rossiter@sheffield.ac.uk).
Publisher Item Identifier S 0018-9286(01)07692-9.
0018–9286/01$10.00 © 2001 IEEE