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Computation of time-optimal switchings for linear
systems with complex poles
Frédéric Grognard, Rodolphe Sepulchre
To cite this version:
Frédéric Grognard, Rodolphe Sepulchre. Computation of time-optimal switchings for linear systems
with complex poles. European Control Conference (ECC03), Sep 2003, Cambridge, United Kingdom.
�hal-01091712�
linear systems with omplex poles F. Grognard y , R. Sepul hre y;z y
Center for Systems Engineering and Applied Me hani s,
Universite atholique de Louvain,
Av. G. Lema^tre, 4. B1348 Louvain-la-Neuve,
Phone: +32-10-472597. Fax +32-10-472180
grognardauto.u l.a .be
z
Institut Monte ore, B28
Universite de Liege,
B4000 Liege Sart-Tilman, Belgium.
r.sepul hreulg.a .be
Submitted as regular paper to the European Control
Conferen e 2003
Abstra t
The minimum-time bounded ontrol of linear systems is
generi- ally bang-bang and the number of swit hings does not ex eed the
dimension of the system if the eigenvalues of the system matrix are
real orif the initial onditionis suÆ iently loseto the target. This
paper extends the method of [8 ℄ for omputing the swit hing times
of time-optimal ontrollers to linear systems with omplexpoles and
demonstratesits appli ation on MPCs hemes.
Keywords: Time-optimal, bounded ontrol, MPC, bang-bang ontrol,
In this paper, we will onsider the problem of steering a solution from an
initial ondition z
0
to the origin forsingle-input linear systems
_
z =Az+bv (1)
subje t to the input onstraint
jvj1
where z 2 IR
n
, v 2 IR , and the pair (A;b) is ontrollable.
The orresponding stabilization problem has long been re ognized as a
signi antnonlinear ontrolproblem,so thatmany solutionshave been
pro-posed: anti-windups hemes, low-gain ontrollaws orModelPredi tive
Con-trol (MPC) s hemes.
The anti-windup s hemes are extensively used in industry but they are
often ad-ho and rarely propose stability proofs (though some an be found
in[10℄and [17℄). Low-gain ontrollawsprovideproofsof semiglobalstability
([11, 12, 16℄), but do so at the expense of performan e. MPC s hemes are
alsowidelyusedinindustry,buttheirappli ationdependsontheexisten eof
fast algorithmsfor the omputationof optimal ontrolproblems. In [2℄ and
[4℄, this problemisavoided by givinganexpli it formofthe MPC ontroller
whi h does not require the online omputation. Su h a ontroller annot
always be omputed, sothatone mustrelyonthe online omputationof the
solutionofoptimal ontrolproblems. Inthispaper, weareinterested insu h
an algorithm,wherethe ost tominimizeis the total time.
Themostnatural ontrolmethodforlinearsystemswith magnitude
on-straintistime-optimal ontrol,whi hiswellknowntobebang-bang,withthe
swit hings o uring onso alled \swit hing urves" in the state spa e. The
omputation of those urves is equivalent to omputing a feedba k ontrol
lawv
(z), and isuntra table forlarge systems.
Thispra ti allimitationimpliesthattheimplementationoftime-optimal
ontrolisbest a hieved throughthe omputationof open-loop ontrol. Also,
due tothe la kofrobustnessof open-loop ontrol, itissuggestedto losethe
loopby nesting this open-loop ontrol in an MPC s heme: every units of
time, a time-optimal ontrol law v
(t) is numeri ally omputed online with
the urrent z(k ) asinitial ondition, and this ontrollaw is appliedduring
units of time;attime (k+1) ,the same ontrolproblemisre omputed,...
It is therefore important to design algorithms that an rapidly solve online
the optimal ontrol problem that is posed every units of time. We fo us
pute the time-optimal ontrol law v
(t) for any given z
0
. Several
gradient-based iterative methods have been proposed. These gradient methods
typ-i ally iterate on the adjoint initial or nal state together with the time of
response (seeforinstan e [5,6,9,13℄, and,forasummaryof thosemethods,
[14℄). Itisknownthatthesemethodsare,ingeneral,sensitivetothe starting
ondition (initial guess) and have poor onvergen e properties.
In[8℄, wehavepresentedanalgorithmbasedonanotherapproa h: ituses
the bang-bang propertyof the time-optimal ontroller. Thealgorithmis
de-signed tooperatewhen the numberof swit hingsis less orequalton 1. It
sees the omputation of the time-optimal ontrol asthe omputationof the
optimalsequen eofswit hingtimes0=t
0 <t 1 ;<t n =T or,equivalently,
the optimal sequen e of time intervals x
1 = t 1 t 0 ;x 2 = t 2 t 1 ;:::;x n = t n t n 1
. In this paper, we onstru t ontinuous time-systems x_ = f(x)
whi h `produ e' the optimal sequen e x = (x
1 ;:::;x
n )
T
, in the sense that
they possess an isolated equilibrium at x = x and that this equilibrium is
asymptoti allystable. The main resultof [8℄shows that, when the
eigenval-ues of A are real, the time-optimal ontroller presents n 1 swit hings or
less, and underpropertime-s alede omposition,the semiglobal onvergen e
of solutions tothe desired equilibriumx=x an be enfor ed.
This paper will on entrate on the ase where the eigenvalues of A are
omplex. In Se tion 2, we indi ate a ase where the number of swit hings
of the time-optimal ontroller is n 1 or less. The algorithmand the main
onvergen e results are then given in Se tion 3. Finally we implement an
MPC s heme for a hange of orbit for a nonlinear model of a satellite in
Se tion 4. Con lusions are given in Se tion5.
2 Swit hings in time-optimal ontrollers
The solution of the time optimal ontrol problem
T =minT s.t. z_ =Az+bv z(0)=z 0 z(T)=0 jv(t)j1 (TO)
has long been hara terized asa ni e appli ation of the Maximum Prin iple
[15℄. The time-optimal ontrolisbang-bang and the swit hing times are the
rootsof (t) T b,where (t)=e At 0
istheadjointresponseofthesystemfor
asuitableve tor
0
swit hing times orrespond tothe roots of some (t) b istime-optimal(the
Maximum Prin ipleisne essary and suÆ ient [1℄). Theorem 1 employs this
propertyandProposition1to hara terizeasetofinitial onditionsthat an
besteered tothe originwithabang-bang ontrolthatinvolvesatmost n 1
swit hings.
Notation 1 Wewilldenote!
max
themaximumofthe imaginaryparts
of theeigenvaluesof A. When !
max =0, r !max denotes+1(forr>0). Let T 2 IR + . The set C IR n
is the set of initial onditions z
0 that
are null- ontrollable. The set C(T) C isthe set of initial onditions
z 0
that are null- ontrollable in time t T.
Proposition 1 [18℄ Let A 2 IR
nn
, b; 2 IR
n
with the pair (A;b)
on-trollable and 6= 0. The number N of roots of the exponential polynomial
P(t)=
T e
At
b inside the interval [0;T℄ satis es
N n 1+
T!
max
(2)
Proposition 1 thenresults in the followingtheorem:
Theorem 1 For any z
0
2 C(
!max
), there exists a unique bang-bang
on-troller whi h steers z(t) from z
0
to the origin with n 1 swit hings or less
andatotaltimeinferiororequalto
!max
. Moreover,thisbang-bang ontroller
is the solution of TO.
Proof: The fa t that z
0
2 C(
! max
) implies that there exists a solution to
TO with T ! max
. This ontroller is unique, bang-bang and we willshow
that itswit hes atmostn 1times. Thisresultsfromthe oin iden eofthe
swit hing times of the optimal ontroller with the roots of
(0) T e At b and
from the bound on the number of roots of an exponential polynomial given
by Proposition1: (a) If T < ! max
, Proposition 1 indi ates that the number of roots of
P(t)= (0) T e At
b inside the interval [0;T
℄ is inferior to areal
num-ber belonging to the interval [n 1;n 2). Be ause the number of
roots is an integer, the a tual upper bound is equal to n 1, so that
the numberof swit hings of v
(t) is inferior orequal ton 1. (b) If T = !max
, the number of roots of P(t) inside the interval [0;T
℄ is
numberof swit hings isalsobounded by n 1,orthe numberof roots
equals n. Inthis latter ase,one rootmustbeequalto0and oneother
equal toT
. Otherwise, one ould nd a smaller interval ontaining n
roots of P(t), whi h is in ontradi tion with (2). This indi ates that
only n 2 roots lie in the interior of the interval [0;T
℄, so that only
n 2a tual swit hings takepla e.
Wehavenowshowntheexisten eofabang-bang ontrollerwithn 1
swit h-ings or less and T
!max
. Uniqueness is proven by showing that any su h
bang-bang ontroller is the unique solution of TO: let v(t) (t 2 [0;T℄) be
a bang-bang ontrol law that steers z(t) from z
0
to the origin with n 1
swit hings orless. Let t=t
j
(j =1;;N n 1) be the swit hing times.
(A) LetT <
! max
. IfN <n 1,then omplementthelistoft
j
withn 1 N
distin tvalueslargerorequaltoT (andsmallerthan
!max ,if!
max 6=0).
One an nd anontrivial
0 su h that T 0 e At j b=0(j =1;;n 1).
This means that the t
j
are the n 1 roots of
T 0 e At b = 0 inside the interval [0;t n 1 ℄ of length inferior to !max . From Proposition 1, we
know that no other root an be found inside this interval [0;t
n 1
℄, so
that v(t) and sign(
T
0 e
At j
b) have exa tly the same swit hing times.
It is then suÆ ient to pi k (0) =
0
or
0
to ensure that v(t) and
sign((0)
T e
At j
b) are identi al. As a onsequen e, v(t) is maximal,
whi h is suÆ ient for v(t) tobe optimalin the ase of TO. Therefore
v(t) is equalto the unique v
(t).
(B) Let T =
!max
. We will ompare v(t) and v
(t) (whi h produ es the
solution z
(t)). Let t
1
be the rst swit hing time of v
(t) and ~ t 1 = min(t 1 ;t 1
). Two ases then arise: either v(t) =v
(t) or v(t) = v (t) inthe interval[0; ~ t 1 ℄. Ifv(t)=v
(t)inthe interval,thenz(
~ t 1 )=z ( ~ t 1 ). The ontrol v(t)(t 2 [ ~ t 1 ; ! max
℄) is then a bang-bang ontroller steering
z(t) from z(
~ t
1
) to the origin in a time smaller than
! max
, and with
n 1 swit hings or less. It is therefore optimal (see point (A)). By
optimality of subtraje tories of an optimal solution, the same an be
saidof v (t),sothatv(t)=v (t)fort 2[ ~ t 1 ; ! max ℄. Finally,v(t)=v (t) for t 2 [0; ! max
℄, so that v(t) is solution of TO. In the ase where
v(t) = v (t) in the interval [0; ~ t 1 ℄, it is lear that T < ! max (in the ase where T = ! max , v
(t) and v(t) would be two di erent optimal
solutions, whi h is impossible). The result of (A) implies that v(t)
(t 2 [;
! max
℄)istime-optimalfromz()with anoptimaltime
! max
.
As ! 0, this optimal time tends to
! max and z() tends to z 0 . By
optimal time from z 0 should then be !max , whi h is in ontradi tion
with the observation that wasmade (T
<
!max ).
Wehavethen shown that any bang-bang ontroller withn 1swit hings or
less andT
!max
thatsteers z(t)fromz
0
totheoriginistheunique solution
of TO. Su h a ontroller istherefore unique. 2
Asa onsequen eof thistheorem, wewillmakethe followingassumption
throughout this paper:
Assumption 1 Suppose that z
0
2 C(
!max ).
ItiseasilyseenthatC(
! max
)isa ompa tsetwiththeorigininitsinterior,
and whoseborder isthe minimum iso hrone orresponding tothe time T
=
! max
. The set C(T) monotoni allyin reases as afun tion of T and tends to
C as T grows unbounded, whi h is alsothe ase of C(
! max ) as ! max goes to
0. Inthe limit,were overthe lassi alresult that thetime-optimal solution
involvesat most n 1 swit hings when all the eigenvalues of A are real.
Theorem 1 justi es the approa h that is taken in this paper: instead
of looking for a time-optimal ontroller, or for the initial ondition
(0) of
the adjoint system as previous algorithmsdid, we look for a ontroller that
swit hesatmostn 1times. Ifthealgorithm onverges, Theorem1indi ates
that optimality an betested as follows:
OptimalityTest: If v(t) (t 2 [0;T℄) is a bang-bang ontroller that steers
z(t) from z
0
to 0 with n 1 swit hingsor less, and if T
! max
, thenv(t)is
the time-optimal solution of TO.
3 An algorithm for the omputation of
bang-bang steering ontrols
Des ription of the algorithm
In the set C(
!max
), the sear h for the optimal ontrol an be restri ted to
the steering ontrols that are de ned by asequen e of n time intervalsx
i , t i t i 1
and the orresponding sequen e of onstant ontrolvalues u
i
. This
lassofpie ewise onstant ontrolsis hara terizedbyapairofve tors(x;u),
where x denotes the ve tor of time intervals and u denotes the ve tor of
ontrol values. The time-optimal solution is then de ned by (x;u), with
ju i
0 z(t)=e At z(0)+ Z t 0 e A bv( )d ;
it is seen that a ontrolde ned by the pair (x;u) will steer z
0
toz =0 if it
satis es the `steering equation'
(x)u = z
0
(3)
where the i-th olumnof the matrix is
(:;i) (x), Z t i t i 1 e A bd = Z P i k =1 x k P i 1 k =1 x k e A bd The equation (x)u = z 0
is the nonlinear equation to be solved to
deter-mine the optimal ontrol. In ontrast, (3)is linear in u and is easilysolved
for a given x. Denoting the open positive orthant O
+
n
, it an be seen that
(x) is regular inside the set K = fx 2 O
+ n j P n i=1 x i !max g, so that a
unique solution u(x) =
1 (x)z
0
of (3) exists for any x in K . A natural
lass of iterativemethodsthus onsistsin updatingthe timeintervalsve tor
x su has toenfor e onvergen e ofthe orresponding ontrolve tor u(x) to
a bang-bang sequen e of magnitudeju
i
j=1.
The heuristi s onsidered in [4℄ and [8℄ are the \de entralized"
adapta-tion of the ve tor x: if ju
i
(x)j is larger than one, in rease the length of the
orresponding time interval x
i
; if ju
i
(x)j is smaller than one, de rease the
length of the orresponding time intervalx
i .
In ontinuous-time, these heuristi s yieldthe de entralized adaptation
_ x i =f i (ju i (x)j 1)x i ; i=1;:::;n (4) where f i
shouldbea (smooth) s alarfun tion withitsimageinthe rst and
third quadrant and should only vanish at zero. x
i
multiplies f
i
in order to
guarantee the positive invarian e of the open positive orthant.
Convergen e
In[8℄,wehaveonly onsideredthe asewhere!
max
=0(onlyrealeigenvalues
for A)andprovided aglobalanalysisof the ontinuous-time system(4)with
thefun tionsf
i
sele tedassaturatedlinearfun tions,yieldingthealgorithm:
i _ x i = sat M (ju i (x)j 1)x i ; i 2 f1;;ng; x i (0)>0 (5) With 0 < n << n 1 << << 1
, a time-s ale separation an be
en-for ed between thedi erentx
i
dynami s,andthedi erent ontrolvaluesju
i j
n
theorem of [8℄ an begeneralized tothe ase wherethe eigenvalues of A are
omplex and z 0 2 C( !max ). Theorem 2 If z 0 2 C( !max
), then the equilibrium set of
8 > < > : _ x 1 = sat(ju i (x)j 1)x i . . . n _ x n = sat(ju n (x)j 1)x n (6) inside K = fx 2 O + n j P n i=1 x i !max
g is non empty and is asymptoti ally
stable. It is exponentially stable if is a singleton.
Moreover,ifAonlyhas realnonpositiveeigenvalues,theregionof
attra -tionof in thepositiveorthantisenlargedatwillinO
+ n byproperseparation of the time-s ales n = t n ;; i = t i
InTheorem2,numeri alsimulationssuggestthatthe regionofattra tion
of in ludesthe entire set K . However, atheoreti al hara terizationof the
basin of attra tion seems not immediate in the proof in [8℄. Extension of
the regionof attra tionofbeyond Kisnot feasiblebe auseofthe possible
singularity of (x) and the possible existen eof other equilibria outside K .
A natural way of initializingthe algorithm onsistsin taking allthe
ele-ments of x(0) very small. This almost ensures that x(0) belongs to K , and
that onvergen e to the desired equilibrium takes pla e. However,
onver-gen e tothe time-optimalsolution an onlybe he ked a posterioriby using
the Optimalitytest ofSe tion 2.
Implementation
We illustrate on Figure 1 the implementation of the algorithmon the
on-trolled harmoni os illator:
_ z 1 = z 2 _ z 2 = z 1 +v jvj1 with z 0 = (1 1) T
, an initial ondition su h that the time-optimal solution
only presents one swit hing (x=(0:93051:5709)
T ).
In order to implement the algorithm, we need to dis retize it. The
sep-aration of the time-s ales results in a very sti set of di erential equations,
whose behavior an only be reprodu ed in dis rete time by taking a very
smalldis retization step. This results inslow onvergen e.
However, wehaveobserved thatthe algorithmisrobust toaredu tion of
the time-s ales separation (see [7℄). It tolerates that we take
i
taking pla e,but thephase-planeismodi ed( ompare thesolid lines,where
1
=1 and
2
=0:1, with the dotted lines, where
1
=
2
=1).
Withoutthe time-s alesseparation,the di erentialequationsarenotsti
anymore,sothatasimplelarge-stepEulerdis retizationgivesagood
approx-imation of the behavior of the ontinuous system ( ompare the dotted and
dash-dotted lines),and a very fast onvergen e (inthe example,the
equilib-rium isrea hedinless than ten stepsfor thefourinitial onditionsof Figure
1). The a tualalgorithmis then
x i (k+1)=x i (k)+Æ sat M (ju i (x(k))j 1)x i (k) fori 2 f1;;ng
where Æ is the dis retization step. We have shown in [7℄ that Æ needs to be
smaller than 1 to ensure invarian e of the positiveorthant. In the example,
we have taken =0:5.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
PSfrag repla ements x 1 x 2Figure 1: Phase plane of the evolution of the algorithm for the ontrolled
harmoni os illator with z
0
=(1 1)
T
. The ontinuous algorithm with
time-s ales separation (solid line), without time-s ales separation (dotted-line),
and the dis retealgorithmwithouttime-s alesseparation (dash-dottedline)
are illustrated. The initial onditions for thealgorithmwhi hare illustrated
are: x 0 =(0:10:1) T ; x 0 =(0:12) T ; x 0 =(20:1) T ; x 0 =(22) T .
From the omments on the initializationand the dis retization of our
algo-rithm,wesuggestthatx(0)bepi ked losetotheorigin,andthatalarge-step
Euler dis retization be employed. After several steps of the algorithm, if it
is onverging, the optimality test des ribed in Se tion 2 should be used to
verify if the result of our algorithm is a time-optimal solution. This test is
only suÆ ient for optimality so that, if the answer of the test is negative,
it does not totaly rule out the fa t that the result of the algorithm is the
time-optimal ontroller.
4 Time-optimal ontrol in a re eding horizon
s heme
In this se tion, the appli ation of re eding horizon based on time-optimal
ontrol and saturated linear ontrol applied to a nonlinear model of an
or-biting satellite are ompared.
Letus onsidertheorbitaltransferproblemforasatellitehavinga ir ular
orbit aroundthe earth. We onsider that the targetisa geostationaryorbit.
It evolves 36000km above the earth, and itsrevolution takes 24hours. The
mass of the satellite isestimated at2000 kg and the maximal thrust(in the
dire tion of the tangent to the orbit) amounts to 2N. We suppose that the
satellite startsits journey 400 kmbelowthe targetgeostationary orbit. The
dynami s of this satellite are:
r = ! 2 r k r 2 _ ! = 2!r_ r + v mr
where r is the distan e of the satellite to the enter of the earth, ! is its
angular velo ity, and v is the tangential thrust [3℄. The onstant m is the
mass of the satellite and k = 3:9851:10
14 m
3 =s
2
is known. The equilibrium
of motion of a geostationary satellite satis es ! =
2
86400
= 7:272 10
5 rad=s
and r=42238km (radius of the earth+36000 km). In order to apply
time-optimal ontrol,we omputethelinearizationofthesystemaroundthetarget
equilibrium of motion and hose the variables like in [3℄: (z
1 ;z 2 ;z 3 ) =(r
r;r;_ (! !) r). This resultsin the linearized system
_ z = 0 0 1 0 3! 2 0 2! 0 2! 0 0 1 A z+ 0 0 0 1 m 1 A v
whi hhasitspolein0and!i. Wehaveshownthatatime-optimalsolution
that takesless than T =
!
is time-optimal). Our algorithm an ompute a bang-bang orbital transfer
for the linear model if T 12h: the ontrol value +2 is applied during
x 1
= 13953 se onds, followed by 2 during x
2
= 14405 se onds and +2
during x
3
=14475 se onds. The transfer takes 42833 se onds, that is lose
to, but smaller than 12 hours. The Optimality test of Se tion 2 indi ates
that this bang-bang ontrol is time-optimal for the linearized model. If we
apply this strategy on the nonlinear model in open-loop, the nonlinearities
prevent the transferfrom being exa tly a hieved.
In order to ompensate for the nonlinearities,a re eding horizon s heme
an beused: thetime-optimalstrategy(basedonthe linearmodel)is
re om-puted every ten minutes. However, the omputed ontrol lawis not applied
to the system as is. Indeed, on e the rst time-interval is elapsed, the
so-lution x of the time-optimal ontrol problem ontains one value x
i
, whi h
is very small. Due to the nonlinearities, this value x
i
is not exa tly zero.
Moreover, it an o ur that i = 1, that is the solution of the time-optimal
ontrolproblemstarts with u=+2for a very short time,and then swit hes
to u = 2 for a long time. As this phenomenon an o ur at ea h step of
the Re eding Horizon S heme, the ontrol law will present uselessly many
swit hings. We have eliminated this problem by ignoring the time intervals
that are smaller than ten minutes, so that, if x
1
<600s, the orresponding
ontrol is not applied. It is apparent on Figure 2 that this strategy leads
to an exa t transfer from one orbit to the other. This transfer takes 44400
se onds, that is a littlebit more than twelve hours. It presents more than
twoswit hingsbe ause the\errors"introdu edbythe nonlinearitiesneed to
be ompensated for along the way. Basi ally, the ontrol law is lose to a
stri tbang-bang ontrolwithtwoswit hings: the ontrolvalue+2isapplied
during 13800 se onds, followed by 2 during 16200 se onds and +2 during
14400se onds. However,the ompensationofthenonlinearitiesimpliesthree
o urren es of u =+2 during the se ond time interval, and one o urren e
of u= 2 duringthe third interval.
Asaturated linear ontroller isbuiltfor omparison. We hoose toapply
the design presented in [16℄: a family of Ri ati-based ontrollers is built,
and a ontroller that does not saturate along the solution is hosen, so that
onvergen etotheoriginisnotpreventedbythesaturation. Inordertohave
a balan ed onvergen e to the origin, we res ale the variables of the linear
systems. Indeed, we have z
1 (0)= 400000 and z 3 (0)=44:1555. Therefore, we de ne w 1 = z 1 =400000, w 3 = z 2 =44:1555, and w 2 = z 3 =60 (based on
the observation made onthe time-optimalsolution). Su h anapproa hwith
0
1
2
3
4
5
x 10
4
3.54
3.55
3.56
3.57
3.58
3.59
3.6
x 10
4
0
1
2
3
4
5
x 10
4
−20
0
20
40
60
80
0
1
2
3
4
5
x 10
4
7.2
7.25
7.3
7.35
7.4
x 10
−5
0
1
2
3
4
5
x 10
4
−2
−1
0
1
2
PSfrag repla ements r r_ ! vFigure 2: Orbital transfer using a re eding horizon strategy (solid line) ora
saturated linear ontroller (dash-dottedline)
following ontroller, whi h does not saturate along the solution
u= sat(2:180510 5 z 1 +0:0474z 2 +0:1677z 3 ) (7)
Byessen e, this ontroldesignleads to ontrollerswith in nitegain-margin.
Therefore, we an repla e(7)by
u= sat(k(1:85310 5 z 1 +0:0341z 2 +0:1409z 3 )) (8)
with k > 1. This will make better use of the available a tuation, and still
ensure stability in approximately the same region (we have taken k = 10).
On Figure 2, it appears that the linear ontroller leads to a mu h slower
onvergen e than the time-optimal one. It does not su eed in reprodu ing
the two swit hings. The rst one is present (though early), but the se ond
one is smoothed out.
Not surprisingly, the in lusion of the time-optimal ontroller inside an
MPCloopyieldsimproveperforman ewith respe t towhat isobtainedwith
In this paper, we have proposed an algorithm that omputes time-optimal
swit hingsforlinearsystemswith omplexpoles. Theanalysisextends
previ-ous resultsrestri ted tothe ase ofreal poles. Fastalgorithmsthat ompute
boundedsteering ontrolsareofinterestfortheonline al ulationofbounded
stabilizingfeedba ks. The utilizationof our algorithmin are eding horizon
ontrolimplementation has been illustrated onasatellite example.
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