Switched affine systems using sampled-data controllers: robust and guaranteed stabilization

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Switched affine systems using sampled-data controllers:

robust and guaranteed stabilization

Pascal Hauroigné, Pierre Riedinger, Claude Iung

To cite this version:

Pascal Hauroigné, Pierre Riedinger, Claude Iung. Switched affine systems using sampled-data

con-trollers: robust and guaranteed stabilization. IEEE Transactions on Automatic Control, Institute

of Electrical and Electronics Engineers, 2011, 56 (12), pp.2929-2935. �10.1109/TAC.2011.2160598�.

�hal-00569453�

guaranteed stabili zation

Pas al Hauroigné,Pierre Riedinger, ClaudeIung

Abstra t

The problemof robust and guaranteed stabilization isaddressed for swit hed ane systems

using sampled state feedba k ontrollers. Based on the existen e of a ontrol Lyapunov

fun tion for a relaxed system, we propose three sampled-data ontrols. Global attra ting

sets, omputedbysolvingasequen eofoptimizationproblems,guaranteepra ti alandglobal

asymptoti stabilizationfor the whole system traje tories. In addition, robust margins with

respe t to parameters un ertainties and non uniform samplingare provided using

input-to-state stability. Finally, a bu k-boost onverter is onsidered to illustratethe ee tiveness of

the proposed approa hes.

Keywords: swit hed ane systems, stabilizationof hybrid systems, input-to-statestability,

robust ontrol

1. Introdu tion

During thepastde ades, hybridsystems haveattra tedalargeinterestfromthes ienti

ommunity. Indeed, a wide range of systems an be modeled in a hybrid ontext: physi al

systems involving impa ts, multi-model approa hes, ele tri al ir uits ontaining swit hing

elements (diodes, transistors,...), et . Here, we study a parti ular lass of hybrid systems:

swit hed ane systems. It onsists in a nite olle tion of ane subsystems whi h are

se-le ted by aswit hing rule. Usually, the ontroldesign of these systems relies on anaveraged

model[1℄. Averaging methods [2, 3℄ are largely used in power ele troni s in order toprovide

feedba k strategies using Pulse Width Modulation (PWM) ontrol. Advan ed methodssu h

as passivity based ontrol [4℄, sliding modes [5, 6℄, optimal [7, 8, 9℄ and predi tive ontrol

a ountthe dis ontinuitiesintrodu ed by swit hings.

Two main problems are related to the stability of swit hed systems: the rst on erns

the stability onditions foran arbitraryswit hing lawand the se ond on erns the swit hing

strategy whi h keeps the system stable. In this paper, we treat the latter problem. Several

surveys are available [15, 16, 17, 18℄ on this topi . Most of the available te hniques for

either analyzing the stability or synthes izing ontrol laws are based on Lyapunov fun tions:

quadrati [19℄, multiple [20, 21, 22℄, pie ewise quadrati [23, 24℄, et . In [17℄, the author

presents a Lie algebra approa h for the study of swit hing systems. The role of dwell time

and the impa tof time-delayhave alsobeen emphasized in[25, 26℄. Even various te hniques

are employed, most of them deal with swit hed systems whose subsystems share a ommon

equilibrium.

Unlikethesestudies,weaddressthe asewhereno ommonequilibrium anbedened. A

large lass of systemshaving apra ti alinterest,su h asDC-DCpower onvertersis overed

by this framework. Based on the existen e of a ommon quadrati Lyapunov fun tion, a

ontinuous time stabilizing swit hing strategy is provided in a re ent paper [27℄. In [28℄, in

a dis rete time framework, a positively invariant set [29℄ formed by the union of bounded

ellipsoids is determined and used in a predi tive ontrol algorithm to steer the state inside.

However, the method uses a LMI formulation to ompute these ellipsoids whi h introdu es

some onservatism in the result. Indeed, LMIs imply that the swit hed system possesses a

swit hing sequen e

S

of a pres ribed length for whi h a property of uniform stability w.r.t. the initial onditionissatised. So, the omputedinvariantsare notparti ularlytightaround

the target.

Inthis paper,basedontheexisten eofaControlLyapunov Fun tion(CLF)forarelaxed

system - obtained by relaxing the ontrol domain to its onvex hull -, robust stability for

sampled swit hed strategies is investigated. In this framework, the referred targets, named

operating points, are dened as the equilibria of the relaxed system. Assuming that a

on-tinuous time CLF is known for the relaxed system, dierent sampledswit hed strategies are

Pre isely, we prove that positive invariant sets an be obtained by solving optimization

problems. Sin enoassumptionismadeontheCLF,thisproblemisingeneralnontrivialand

non-linear. Fortunately,when the targetdenes astable equilibriumof the relaxedsystem, a

quadrati Lyapunov fun tion an beeasilyexhibitedandthe optimizationproblemrevealsto

bea quadrati ally onstrained quadrati program(QCQP)for whi he ient solvers exist.

Therobustnessaspe tsoftheproposedsampledswit hedstrategiesin aseofnonuniform

sampling and parameter un ertainties are also studied and dis ussed. The Input-to-State

Stability (ISS) formulation [30, 31℄ is used inorder to provide stabilitymargins.

The paperisorganizedasfollows. Se tion2givesnotations anddenitionsused

through-out thepaper. Thesystemdes riptionisgivenandtheoperatingpointsare denedinSe tion

3. In Se tion 4, we propose three dierent sampled-data ontrols for the swit hed system,

dedu ed from a known CLF for the relaxed system. Using this CLF, a set of optimization

problems isalsoformulated. InSe tion5,weprovethatthe solutionsof theseproblemsallow

to dene global attra ting sets for the sampled swit hed ane system. Se tion 6 provides

some relations of in lusions between these attra ting sets. An extension of those results in

the aseofparameterun ertainties andnon-uniformsamplingisgiven inSe tion7. The

om-putational aspe ts are addressed inSe tion8. Abu k-boost onverter is usedin Se tion9to

illustrateour results. We show thatthe stabilityis guaranteedeven inpresen e of parameter

un ertainties. To on lude,Se tion10summarizesthe results ofthis paperandtheir interest

in the resear h eld of swit hed ane systems.

2. Notations

Let

R

,

N

and

N

∗

denote the set of real, natural and stri tly positive natural numbers, respe tively. Moreover, for any

a ∈ N

, let

N

≤a

denotes the set

{k ∈ N | k ≤ a}

.

k · k

is the Eu lidian norm ofa ve tor and

k · k

∞

the innitenorm of a fun tion. In this paper, systems are of the form

˙x(t) = f (x(t), u(t))

where

f : R

n

_{× R}

m

_{→ R}

n

is lo ally Lips hitz ontinuous.

So, fora given input

u

,there is aunique solution ofthe initialvalue problemand isdenoted

Denition 1 (

N

0

,

K

and

K

∞

−

fun tions). A fun tion

α : R

+

_{→ R}

+

is a

N

0

−

fun tion, if it is ontinuous, nonde reasing and satises

α(0) = 0

. Moreover,

α

is a

K−

fun tion if

α ∈ N

0

and isstri tly in reasing.

α

is a

K

∞

−

fun tion if it is an unbounded

K−

fun tion.

Denition 2 (

KL−

fun tion). A lass

KL−

fun tion is a fun tion

β : R

+

_{× R}

+

_{→ R}

+

su h

that

β(·, t) ∈ K

for ea h xed

t ≥ 0

and

∀r ≥ 0, β(r, t) → 0

as

t → +∞

.

Denition3 (ISS). Asystem oftheform

˙x = f (x, u)

issaidtobeInput-to-StateStable(ISS) if there existsome

β ∈ KL

and

γ ∈ K

su h that

kx(t, x

0

, u)k ≤ β(kx

0

k, t) + γ(kuk

∞

), ∀u ∀x

0

.

Denition 4 (Pra ti al stability). A system of the form

˙x = f (x, p)

where

p

is a xed parameter, is saidto be pra ti ally stable if there exist some

β ∈ KL

and a positive onstant

c(p)

su h that

kx(t, x

0

, u)k ≤ β(kx

0

k, t) + c(p), ∀x

0

.

Denition 5 (0-GAS). A system of the form

˙x = f (x, u)

issaid to be 0-Globally Asymptot-i ally Stable (0-GAS), if there exists some

β ∈ KL

su h that

kx(t, x

0

, 0)k ≤ β(kx

0

k, t), ∀x

0

.

Denition 6 (AG). A system of the form

˙x = f (x, u)

has the Asymptoti Gain property (AG), if there exists some

α ∈ N

0

su h that

lim sup

t→+∞

kx(t, x

0

, u)k ≤ α(kuk

∞

), ∀u ∀x

0

.

Denition 7 (CLF). For a system of theform

˙x = f (x, u)

, a ontrolLyapunov fun tion isa fun tion

V

thatis ontinuous, dierentiable,positive-denite,proper,and su hthatfor all

x

, there exists

u

for whi h the dire tional derivative along the traje tory satises

V (x; u(x)) :=

˙

∂V

∂x

· f (x, u(x)) ≤ −γ(kxk)

where

γ

is a lass

K−

fun tion.

3. System Des ription

A swit hed ane system has the form:

˙x(t) = A

0

x(t) + B

0

+

m

X

i=1

u

i

(t)(A

i

x(t) + B

i

)

(1) where

u

i

_(t)

,

i = 1, . . . , m

are omponent values of the ontrol

u(t) ∈ U = {0, 1}

m

and

x(t) ∈ R

n

represents the state value at time

t

.

A

i

and

B

i

are real matri es of appropriate dimensions. As previously laimed, the most studied ase in the literature is the parti ular

ase where

B

i

are all distin t from

0

and for whi h no ommon equilibrium an be dened, e.g. DC-DC onverters.

System(1)belongstothe lass ofnonsmoothsystemsforwhi hthenotionofsolution an

be properly dened and generalized in the sense given by Fillipov [32, 33℄. The generalized

solutions are dened by onsidering the followingrelaxed system:

˙x(t) = A

0

x(t) + B

0

+

m

X

i=1

u

i

(t)(A

i

x(t) + B

i

)

with

u

i

(t) ∈ [0, 1]

(2)

where the ontrol domain is now the onvex hull o

(U )

of the original one. A link between the solutions of the system (1) and those of the system (2) an be established by a density

theorem ininnite time [34℄:

Theorem 1. If

z

is a global solution of (2) starting from

z

0

and

ε : [0, +∞) → (0, +∞)

is ontinuous, then there exists a solution

x

of (1), starting from

x

0

∈ B(z

0

, ε(0))

su h that

kz(t) − x(t)k < ε(t)

for all

t ∈ [0, +∞)

.

Therefore, swit hinglaws

u ∈ L

∞

([0, +∞), U )

(where

L

∞

denotes theBana hspa eofall

essentiallyboundedmeasurablefun tions)existsu hthatthetraje toryofthesystem(2) an

beapproa hed as loseas desiredby theone of the system(1). Forthis reason,the operating

points set of the swit hed system (1) denoted by

X

ref

, is dened as the set of equilibrium pointsof the system (2):

X

ref

=



x

ref

∈ R

n

: A

0

x

ref

+ B

0

+

m

X

i=1

u

i

ref

(A

i

x

ref

+ B

i

) = 0

,

u

i

ref

∈ [0, 1]



.

(3)

This set denes the ontroltargetsfor the state of the system (1).

It is worth noting that none of the ontrols

u

ref

∈

o

(U )\U

from (3)and orresponding to an equilibrium

x

ref

, is admissible for the swit hed system (1). The out ome is that the swit hed system state

x

annot be maintained on

x

ref

by a ontrol taking its values in

U

(unless the time duration between swit hings tends towards 0). Consequently, if the target

forthe swit hed systemisanoperatingpoint

x

ref

,theasymptoti behaviorofthetraje tories of (1) is hara terizedby:

•

a y le near

x

ref

ifadwell time ondition isappliedonthe swit hings (i.e. alowerlimit

exists for the time durationbetween swit hings);

•

an innite swit hings sequen e with a vanishing time duration between swit hings as

t → ∞

.

Thesefeatures on ernindire tlymostofthe ontroldesignswhi huseanaveragedmodel

[2, 3,35℄.

4. Sampled ontrol strategies

Many ontrol strategies like optimal [7, 11, 9℄, predi tive [13, 36℄, sliding mode [37℄ or

stabilizing ontrols [38, 6, 5℄ an be investigated to steer the state of the system (1) near a

dened operating point

x

ref

∈ X

ref

. As shown in [9℄, some singularities known as singular ar s that render parti ularly hard the optimal ontrol synthes is, appear in the resolution

of optimal ontrol problems for the system (1) or (2). This ertainly explains why dis rete

time predi tiveformulationwith ontrolrestri ted to

U

orLyapunov based approa hes seem to be more tra table. Our aim is not to dis uss the advantages and/or drawba ks of ea h

method. The paper fo uses ontwo pra ti al aspe ts: the use of sampled swit hed laws and

the guarantee of stability margins.

For agiven

x

ref

∈ X

ref

, letusdenethe hangeof oordinates

z = x − x

ref

. The system (1) or(2) an be rewritten inthe form:

˙z = A(u)z + B(u)

(4)

with

A(u) = A

0

+

P

m

i=1

u

i

A

i

and

B(u) = A

0

x

ref

+ B

0

+

P

m

i=1

u

i

(A

i

x

ref

+ B

i

)

. Suppose that the followingassumptionis satised:

Assumption 1. A ontrol Lyapunov fun tion

V

is known for the system (4) with a ontrol domain relaxed to

co(U )

.

Assumption 1impliesthat a ontinuous time state feedba k strategy:

rea hed, i.e. the target

x

ref

in

x−

oordinates, the relation

u

∗

_{= u}

ref

ne essarily holds. From Assumption 1,we an dedu e three sampledswit hed ontrolstrategies:

1. Pulse-Width Modulationstrategyisasimplewaytoapplya ontroldened by(5)

to the swit hed system. For agiven sampleperiod

T

s

, the

i

th

omponent of the ontrol

u

is approximated by:

v

i

(t, T

s

) =

∞

X

k=0

I

_[t

k

, t

k

+u

i

k

T

s

[

(t), ∀t ∈ R

+

(6)

where

I

A

(.)

stands for the indi atorfun tion whi h takesthe value

1

when

t ∈ A

and

0

otherwise,

t

k

= kT

s

and

u

k

= u(t

k

)

.

2. Steepestdes entstrategy onsistsin hoosingattime

t

k

,

k ∈ N

,themostde reasing dire tion of the CLF

V

amongthe nitevalues given by

u ∈ U

:

v(t, T

s

) =

∞

X

k=0

u

k

I

[t

k

, t

k+1

[

(t), ∀t ∈ R

+

(7)

where

u

k

= arg min

u∈U

˙

V (z

k

; u)

(8)

with

V (z

˙

k

; u)

the derivativeof

V

in the dire tiongiven by

A(u)z

k

+ B(u)

.

3. Predi tivestrategyminimizes,overahorizon

N

H

andamonganiteset ofsequen es,

V (z

k+N

H

)

from the urrent position

z

k

:

v(t, T

s

) =

∞

X

k=0

u

k

I

[t

k

, t

k+1

[

(t), ∀t ∈ R

+

(9) where

u

k

= arg

1

min

u

k

,u

k+1

,··· ,u

_{k+NH −1}

∈U

NH

V (z

k+N

H

)

(10) with

arg

1

the rst argument of the optimalsequen e

u

k

, u

k+1

, · · · , u

k+N

H

−1

.

Notethat allthe proposed swit hingstrategies deneexpli itlyorimpli itlyastate feedba k

ontrol law:

tion, onsider the losedloopobtained fromone ofthe threefeedba ks

v

. The resultingexa t dis retization of (4)at time

t

k

,

k ∈ N

, an be writtenas:

z

k+1

= A

s

(κ

s

)z

k

+ B

s

(κ

s

).

(12)

When the hosen strategy is the predi tive or the steepes t des ent,

A

s

(κ

s

) = e

A(u

k

)T

s

and

B

s

(κ

s

) =



R

T

s

0

e

A(u

k

)(T

s

−τ

)

dτ



B(u

k

)

with

u

k

given in (8)or(10). Forthe PWM strategy,the

right side expression is obtained re ursively sin e this strategy denes a pie ewise onstant

ontrol onea hinterval

(t

k

, t

k+1

)

depending onthe value

u

k

, given in(6).

Denition 8 (Level sequen e and sublevel set sequen e). For a sequen e

{z

0

, · · · , z

N

}

of length

N + 1

generated bythe system (12) from an initial ondition

z

0

, let us dene the level sequen e by:

L

k

(z

0

) = V (z

k

), k ∈ N

≤N

,

(13)

and the sublevel set sequen e by:

S

L

k

(z

0

) = {z : V (z) ≤ L

k

(z

0

)}, k ∈ N

≤N

.

(14)

Following the fa t mentioned at the end of the previous se tion that a swit hed system

annot be maintained on

x

ref

, it is lear that non-monotone de reasing sequen es

L

k

(z

0

)

,

k ∈ N

≤N

, may exist for some

z

0

. Intuitively and as one an expe t, y li path is followed

near the operating point

x

ref

. Thus, the notion of pra ti al stability seems onvenient to hara terize anattra ting set w.r.t. the period

T

s

.

To get some insights: for a sequen e

{z

0

, · · · , z

N

}

, generated by the system (12) from an initial ondition

z

0

, one an sear h w.r.t.

z

0

, the highest level

L

N

(z

0

)

at the end of the sequen e that an berea hed fromalowerlevel

L

0

(z

0

)

. Denote this optimizationproblemby

P

N

:

P

N

:

max

z

0

∈R

n

L

N

(z

0

)

(15) s.t.

z

k+1

= A

s

(κ

s

)z

k

+ B

s

(κ

s

), k ∈ N

≤N −1

(16)

L

N

(z

0

) ≥ L

0

(z

0

)

(17)

Remark 1. For all

N ≥ 1

, the onstraints (16) and (17) an be trivially satised with an initial ondition

z

0

= 0

. Therefore

z

0

= 0

is always a feasible argumentfor

P

N

.

If

z

∗

0

is anoptimalargument of

P

N

,then

L

∗

N

= L

N

(z

0

∗

)

denotes the optimum and

S

∗

L

N

=

S

L

N

(z

∗

0

)

the orresponding sublevelset.

Denition 9. The problem

P

N

is said to be bounded if the optimum

L

∗

N

isnite.

From thedenitionof

P

N

,any sequen e

{z

0

, · · · , z

N

}

with

z

0

outside

S

∗

L

_N

learlysatises

L

N

(z

0

) < L

0

(z

0

)

. The next se tion uses this feature, onne ts the asymptoti properties of

the system (12) to the set

S

∗

L

_N

,

N ∈ N

∗

, and proves pra ti alstabilityresults for the system (12).

5. A su ient ondition for global and pra ti al stabilization

Let usbegin by re allingsome denitions:

Denition10. Aset

Ω

issaidto bepositivelyinvariantfor thesystem(12),ifforall

z

0

∈ Ω

, the state sequen e

z

k

∈ Ω, k ∈ N

∗

.

Denition 11. A traje tory is saidto approa h a set

Ω

, ifthe distan e

d(z

k

, Ω) = min

ω∈Ω

kz

k

−

ωk → 0

as

k → ∞

.

Denition 12. A losedpositively invariantset

Ω

issaidtobe aglobalattra ting setof (12), if for all initial onditions

z

0

∈ R

n

, the traje tories approa h

Ω

. Now, some properties on erning the sublevel sets

S

∗

L

N

,

N ∈ N

∗

an be established:

Theorem 2. Under Assumption 1, if the problem

P

N

is bounded, then

S

∗

L

N

is a global at-tra ting set for all traje tories of the system (12).

Proof. Consider a traje tory

(z

k

)

k∈N

of the system (12) obtained from an arbitrary initial ondition

z

0

. First,letusprovethat

S

∗

L

_N

isapositiveinvariantsetforallinnitesubseq uen es

(z

pN

+r

)

p∈N

, r ∈ N

≤N −1

. Suppose that a state

z

r

∈ S

∗

L

N

exists su h that

V (z

r

) ≤ L

∗

N

<

absurd. Consequently, if

V (z

r

) ≤ L

∗

N

then

V (z

pN+r

) ≤ L

∗

N

, for all

p ∈ N

, whi h means that

S

∗

L

N

is apositiveinvariantset for innite subsequen es of the form

(z

pN

+r

)

p∈N

, r ∈ N

≤N −1

. Now, let show that

S

∗

L

N

is a global attra ting set for the system (12). Assume that an index

s ∈ N

≤N −1

exists su h that

z

s

∈ S

/

∗

L

N

,then two ases must be distinguished:

•

either an index

p

0

∈ N

∗

exists su h that

V (z

p

0

N+s

) ≤ L

∗

N

then the positive invarian e property of

S

∗

L

N

implies

∀p ≥ p

0

∈ N

∗

,

z

pN+s

∈ S

∗

L

N

;

•

orthisindex

p

0

doesnotexist,then,followingthedenitionof

L

∗

N

,astri tde reasingof thesequen e

V (z

pN

+s

), ∀p ∈ N

∗

ismandatory. Therelation

V (z

pN

+s

) > V (z

(p+1)N +s

) >

L

∗

N

ne essarily holds,for

p ∈ N

∗

. As

V

isalso ontinuous and thesequen e isbounded, a limit

ℓ

s

exists su h that

lim

p→∞

V (z

pN+s

) = ℓ

s

. Assume

ℓ

s

> L

∗

N

. By ompa tness, a subseq uen e

(z

ϕ

s

(pN +s)

)

p∈N

(with

ϕ

s

: N → N

stri tly in reasing), that onverges to a limit point

y

s

an be extra ted. Moreover,

y

s

ne essarily satises

V (y

s

) = ℓ

s

. Considering

y

+

s

the

N

th

iterate of

y

s

by (12), the relation

V (y

+

s

) = V (y

s

) = ℓ

s

also holds. However,

V (y

+

s

) = V (y

s

)

impliesthat

y

s

isa possible argument for

P

N

whi h is absurd. Therefore the sequen e

(z

pN

+s

)

p∈N

fullls

lim

p→∞

V (z

pN+s

) = L

∗

N

.

Sin eallsubsequen esoftheform

(z

pN

+s

)

p∈N

,

s ∈ N

≤N −1

,followoneofthetwoaforementioned ases, then the whole sequen e

(z

k

)

k∈N

approa hes

S

∗

L

N

i.e.

lim sup

k→∞

V (z

k

) ≤ L

∗

N

. Therefore,

fromDenition12andthefa tthat

S

∗

L

N

isapositivelyinvariantset,

S

∗

L

N

isaglobalattra ting set for the system (12).

Corollary 1. A su ient ondition for the global and pra ti al stabilization of the sampled

swit hed system (

12

) isthat an integer

N

exists su hthat

P

N

is bounded.

6. Relations of in lusion and smallest attra ting set

A natural question on erns how the sets

S

∗

L

N

,

N ∈ N

∗

, are imbri ated.

Theorem 3. Assume problem

P

N

is bounded. Then

∀p ∈ N

∗

, the problem

P

pN

is bounded and the followingin lusions hold:

S

∗

L

pN

⊆ S

∗

L

N

.

Proof. Following Remark 1, the set of andidates is not empty sin e for all

N

,

z

0

= 0

is alwaysaninitial andidate. Consider asequen e

z

k

, k ∈ N

≤pN

,foranypositiveinteger

p

,and suppose that

z

pN

∈ S

/

∗

L

N

. As inthe proof of Theorem 2, the subsequen e

V (z

jN

), j ∈ N

≤p

, is ne essarily stri tly de reasing and then, this sequen e isnot feasible for

P

pN

. It impliesthat the optimalsequen e

z

∗

k

, k ∈ N

≤pN

, for

P

pN

fullls

z

∗

pN

∈ S

∗

L

N

(and a fortiori

z

∗

0

). Then

P

pN

is bounded sin e

P

N

is bounded.

One mightexpe t astri t in lusionbetween the sets

S

∗

L

_N

. This isnot the ase ingeneral and it iseasyto exhibitanexample showingthat a relationas

L

∗

N

≥ L

∗

N+1

, ∀N ∈ N

∗

, annot hold. However, the upperbound

L

∗

1

≥ L

∗

N

,

∀N ∈ N

∗

remainsvalid when

P

1

is bounded.

Corollary 2. Assume thata nonempty set of integers

I

exists(ne essarily innite following Theorem 3) rendering

P

i

,

i ∈ I

, bounded. Then

S

∞

=

T

i∈I

S

∗

L

i

= lim inf

_i→∞

S

∗

L

i

is the smallest attra ting set of the system (12) given by the set of problems

P

.

7. Robust stabilization

In ordertoinvestigatetherobustness ofthe proposed sampled-data ontrollers,an

input-to-stablestabilitypropertywiththesampletimeasinputisgivenhereafter. Then,inthenext

subse tion, a generalization of this result is provided in the ase of parameter un ertainties

and non-uniformsampling.

7.1. Input-to-state stability w.r.t. the sample time

Theorem 4. UnderAssumption1andassumingforevery sampledperiod

T

s

,

0 < T

s

≤ T

s

max

,

thataninteger

N (T

s

)

existssu hthattheproblem

P

N

isbounded,thesystem(12)when

T

s

→ 0

is

0−

GAS.

In order to establishthe proof of this theorem, onsider the followingdenition:

Denition 13 (Supporting hyperplane). A hyperplane

H

of dimension

(n − 1)

is said to support a losed and onvex set

M (⊂ R

n

₎

on point

y ∈



∂M ∩ H



if

M

is ompletely lo ated

a ve tor

λ

is inward-pointing normal to this supporting hyperplane

H

of

M

on point

y

then

λ

T

_{y = inf}

z∈M

λ

T

_z

.

Proof. First, we show that the system (12) is GAS whatever the hosen swit hed strategy

is, when

T

s

→ 0

. For the PWM strategy, sin e

lim

T

s

→

0

v(t, T

s

) = u(t) = κ(z(t))

holds almost

everyw here, the ontrollaw orresponds exa tlyto the feedba k given by the CLF. So, from

Assumption 1,the system is

0−

GAS when

T

s

→ 0

.

Forthe

N

H

−

step predi tivestrategy,the best de reasing value for

V (z

k+N

H

)

from

V (z

k

)

, when

T

s

vanishes, orresponds tothe dire tiongiven by

arg min

u∈U

˙

V (z

k

; u)

whi hpre isely

or-responds to the steepest des ent. A rst order Taylor expansion an be used to prove this

point. So it only remains to prove the fa t that the steepes t des ent strategy is GAS when

T

s

→ 0

.

Noti ethattheinstantaneousswit hinglawfroma urrentposition

z

alongthetraje tory is given by

arg min

u∈U

˙

V (z; u)

. Now, if

min

u∈U

˙

V (z; u) ≤ ˙

V (z; κ(z)) < −γ(kzk)

(18)

where

γ

is a lass

K−

fun tion, the GAS property holds. Note that the existen e of the fun tion

γ

is dedu ed from the denition of aCLF.

In order to prove the left side inequality of (18), observe that along the traje tory, the

derivative of

V

is given by

V (z(t); u) =

˙

∂V

∂z

T

f (z, u)

. For a xed

z

,

f (z, u) = A(u)z + B(u)

is ane w.r.t.

u

. So, the set dened by

n

f (z, u), u ∈

o

(U )

o

mat hes with the set

Λ =

o

n

f (z, u), u ∈ U

o

andisa losedpolyhedron. Let

λ =

∂V

∂z

(z)

and

G

itssupportinghyperplane on

Λ

. Denote

u

∗

= arg min

_u

λ

T

_{f (z, u)}

. Then, on the point

ρ = f (z, u

∗

)

, we have

λ

T

_{ρ =}

min

w∈Λ

λ

T

_w

. Two ases must be distinguished: either

ρ

is single, then

ρ

is a vertex of the polyhedron

Λ

and

u

∗

_{∈ U}

, or

ρ

is non single, then

ρ

belongs to an edge or a fa e of the polyhedron

Λ

. At least one vertex

δ

exists su h as

δ ∈ ∂Λ ∩ G

(Figure 1). So, a ontrol

u

∗

_{∈ U}

always exists su hthat (18) holds.

Corollary 3. Assumingfor every sampledperiod

T

s

,

0 < T

s

≤ T

s

max

, an integer

N (T

s

)

exists su h that problem

P

N

is bounded. Then, the system (12) is input-to-state stable w.r.t. the

ρ

f (z, u

1

₎

f (z, u

2

₎

f (z, u

3

₎

f (z, u

4

_{) = δ}

f (z, u

5

₎

G

λ

Λ

Figure1: Supportinghyperplane

G

on

Λ

.

lass of onstant input

T

s

.

Proof. A lassi alresult[30℄statesthat ISSisequivalentto0-GAS property ( f. Theorem4)

andasymptoti gainproperty(thesolutionsareultimatelybounded)i.e.

lim sup

k→∞

kz

k

(z

0

, T

s

)k ≤

γ(T

s

)

where

γ

isa

N

0

−

fun tion for

0 < T

s

≤ T

s

max

. From the denitionof

S

∞

, denethe as-so iatedlevel

L

∞

= lim inf

i→∞

L

∗

i

. If

L

∞

(T

s

)

isa lass

N

0

−

fun tionfor

0 < T

s

≤ T

s

max

,theresult

is given by taking

γ(T

s

) = L

∞

(T

s

)

. Ifnot, it isalways possible todene a lass

N

0

−

fun tion

γ

by hoosing

γ(T

s

) ≥ sup

0<T ≤T

s

L

∞

(T )

sin e

sup

0<T ≤T

s

L

∞

(T )

is bounded and nonde reasing for

all

T

s

≤ T

s

max

.

Remark 2. Note that this ISS result is given for the lass of onstant input

T

s

.

7.2. Non-uniform sampling and parameter un ertainties

An improvement an be obtained if the lass ofswit hing laws is relaxedin the following

manner: dene a minimum (resp. maximum) dwell time

δ

min

(resp.

δ

max

) as the minimum (resp. maximum)durationbetweentwoswit hings. Letusdenetheswit hingtimesequen e

t

k

, k ∈ N

, with duration onstraints:

δ

min

≤ τ

k

= |t

k+1

− t

k

| ≤ δ

max

,

(19)

orrespondingtothe time instantswhere the system (4)swit hes fromone mode toanother.

Assumealsoboundedparameterun ertainties

θ

onthematri es

A

i

and

B

i

in(4). Without loss of generality,the un ertainties are given inthe form:

We relaxthe problem

P

N

by:

P

N

(δ

min

, δ

max

, θ

max

) :

max

z

0

,θ,τ

k

L

N

(z

0

)

(21) s.t.

z

k+1

= A

s

(⋆)z

k

+ B

s

(⋆), k ∈ N

≤N −1

(22)

δ

min

≤ τ

k

= |t

k+1

− t

k

| ≤ δ

max

(23)

− θ

max

≤ θ ≤ θ

max

(24)

L

N

(z

0

) ≥ L

0

(z

0

)

(25) where

(⋆) = (τ

k

, θ, κ

s

)

.

Remark 3.

κ

s

remains un hanged and based on the unperturbed model (4).

Remark 4. Inthis relaxed problem, the optimization depends on the initial ondition

z

0

, the swit hingtimesequen e

t

k

andtheparameterun ertainties

θ

. Noti ethatthesetof onstraints (22) is now time dependent following the integration duration

τ

k

.

Remark 5. All the results on erning the attra ting sets

S

∗

L

k

remain valid sin e the given proofsdo notdepend onhowthe losedloopsequen e

(z

k

)

k∈N

isobtained from aninitial guess

z

0

.

Property 1. Theoptimal valueof

P

N

(δ

min

, δ

max

, θ

max

)

is non-de reasing w.r.t.

δ

max

or

θ

max

and non-in reasingw.r.t.

δ

min

.

Proof. It is lear that an optimal argument

(z

∗

0

, θ

∗

, τ

k

∗

, k ∈ N

≤N −1

)

for

P

N

(δ

min

, δ

max

, θ

max

)

is also an admissible argument for

P

N

(δ

min

, δ

max

+ δ, θ

max

)

for all

δ ≥ 0

. The rest of the announ ed properties isalso trivially established.

Corollary 4. Assume

(∆

max

, Θ

max

) > 0

exists su h that for all

δ

min

> 0

(

∆

max

≥ δ

max

≥

δ

min

), an integer

N (∆

max

, δ

min

, Θ

max

)

exists su h that the problem

P

N

(δ

min

, ∆

max

, Θ

max

)

is

bounded. Thenthe system (12) with relaxed swit hing laws (19) and parameterun ertainties

(20), isISSwithinput

(τ, θ)

orrespondingto theswit hingdurationsequen e

τ = (τ

0

, τ

1

, · · · )

and the parameter un ertainties

θ

.

Proof. Theproofuses,asinCorollary3,theequivalen ebetweenISSand(0-GAS

+

AG) prop-erties[30℄. Taking

δ

min

≤ δ

max

→ 0

and

θ

max

→ 0

,

0

-GASpropertyexpressedinTheorem4 re-mainsvalidwiththis lassofrelaxedswit hinglawsandboundedun ertainties. TheAG

prop-erty i.e.

lim sup

k→∞

kz

k

(z

0

, θ, τ

i

, i ∈ N

≤k−1

)k ≤ γ(k(τ, θ)k

∞

)

with

k(τ, θ)k

∞

= max (sup

k

τ

k

, θ)

,

follows from the fa t that the fun tion

φ(δ

max

, θ

max

) =

sup

0<δ

min

≤δ

max

L

∞

(δ

min

, δ

max

, θ

max

)

is

bounded and non-de reasing w.r.t.

δ

max

, for all

δ

max

≤ ∆

max

and respe tively

θ

max

, for all

θ

max

≤ Θ

max

. Then, it is always possible to dene a lass

N

0

−

fun tion

γ

by hoosing for

example

γ(s) ≥ φ(s, s)

with

s = k(τ, θ)k

∞

.

8. Computational aspe ts

This se tion dis usses some omputational aspe ts that an be en ountered when one

solves the optimization problems

P

N

. Sin e no assumption is made about the known CLF andsin ethestatefeedba kisgenerallyadis ontinuousfun tionofthestate,theoptimization

problems

P

N

are non-linearand non-smooth.

Nevertheless , ifthepredi tiveorsteepes tstrategies are onsidered,thefeedba klawleads

to a partition of the state spa e w.r.t. the ontrol values

u(z) ∈ U

. Then, the smoothness requirement an bea hieved if

P

N

issolved for every xed swit hing sequen es. In this on-text, additional onstraintsrelatedtothe hosenswit hingstrategymustbeadded. Pre isely

for a xed sequen e:

•

Steepest strategy: atea h time

t

k

, the ontrol

u

k

∗

of the hosen sequen e has toverify

2

m

_{− 1}

onstraints:

˙

V (z

k

; u

k

∗) ≤ ˙

V (z

k

; u), u ∈ U, u 6= u

k

∗, k ∈ N

≤N −1

.

(26)

Therefore, for a xed sequen e of length

N

,

N (2

m

_{− 1)}

onstraints are added to

P

N

. The problem is learly smooth inthis ase, if the CLF is.

• N

H

−

predi tivestrategy: atea htime

t

k

, the ontrol

u

k

∗

of the hosen sequen e has to

verify the onstraints:

min

u

k

∗,u

k+1

,··· ,u

_{k+NH −1}

∈U

NH

V (z

k+N

H

) ≤

min

u

k

,u

k+1

,··· ,u

_{k+NH −1}

∈U

NH

with

u

k

6= u

k

∗, k ∈ N

≤N −1

. The left minimization is done over

2

m(N

H

−

1)

elements and

the right one over

2

mN

H

− 2

m(N

H

−1)

elementsfor ea h

N

element of the xed sequen e. As the left and right terms are ontinuous but not dierentiable everywhere, a dire t

sear halgorithmisneeded inordertosolvetheproblem(ex eptthe ase

N

H

= 1

: where the smoothnessrequirement isa hieved).

This aution an be avoided if, atea h time

t

k

, the sequen e

u

k

∗, u

k+1

, . . . , u

k+N

H

−

1

in the left term is xed in advan e. This pro edure implies to dene a set of additional

optimizationproblems orresponding toallpossible sequen esat alltime

t

k

. Then,the total numberof optimization problems be omes

2

mN N

H

.

•

PWM strategy: sin ethestate feedba k laws

u(z)

are generallydis ontinuousfun tions of the state, optimization problems are non-smooth. Nevertheless , there are two ases

where

P

N

an be solved without numeri al issues: if

u(z)

is ontinuous or if

u(z) ∈ U

almosteverywhere and allows to dene a partition of the state spa e. In this ase, the

same previous methodology an be applied.

Now, wehave shown that the smoothness requirement an be met. It an be underlined

that, for many pra ti al ases, quadrati Lyapunov fun tion andidates an be exhibited.

For example, in (4) as

B(u

ref

) = 0

, if

A(u

ref

)

is Hurwitz then there exists a quadrati Lyapunov fun tionasso iated tothe system

˙z = A(u

ref

)z

whi h an beused with one ofthe given strategies. Passivity based ontrolis anotherway to get su h quadrati CLF. It means

that the obje tive fun tion and the onstrains are quadrati fun tions. So, a quadrati ally

onstrainedquadrati program(QCQP) an beused. QCQPisawide-studiedprobleminthe

optimizationliteraturehavingalargenumberofappli ations[39℄. RelaxationsofQCQPbased

onsemideniteprogramming(SDP)andthe reformulation-linearizationte hnique (RLT) an

beane ientwaytosolveit. Globaloptimizationsolvers,su hasGloptiPoly[40℄,that solve

non onvex global optimization problem of minimizing a multivariable polynomial fun tion

subje t topolynomialinequality, equality or integer onstraints,are parti ularly e ient for

QCQP. GloptiPoly allows to solve a series of onvex relaxations of in reasing size, whose

Solver Computation Time (s)

L

∗

1

L

∗

4

L

∗

6

NL Matlab 0.98 399.2 4078 Gloptipoly 1.36 3.91 14.9

extremely fast solver. A omparison between the Non-LinearSolverfmin on of Matlab and

the solverGloptiPo ly isperformedontheexamplegiveninthe next se tion. Theresultsare

summarized for the steepes t strategy ase in table (1).

Now, if parameter un ertainties and non uniform sampling are taken into a ount, the

level set omputed mat hes to the worst ase for the dynami s. In this ase, the programis

not a QCQP but a urate polynomial approximations of (22) an be obtained using Taylor

expansion of

e

(A+∆A)(T

s

+δT

s

)

where

∆A

and

δT

s

dene the un ertainties. More a urate, a polytopi approximation of the dynami like in [26℄ ould be another way to deal with this

issue. Ifapolytopi approximationisusedthen theproblembe omes againaQCQP. So,the

proposed solverremainsadapted for both ases.

9. Appli ation 9.1. DC-DC onverter des ription PSfrag repla ements

E

L

C

R

u

_D

v

C

i

L

Figure2: Bu k-boost onverter

Considerabu k-boost onverter(Figure2)whosestateequationin ontinuous ondu tion

mode (the urrent passingthrough the indu tan e never fallsto zero) isgiven by:

where

x = [i

L

, v

C

]

T

and

A

0

=







0

0

0 −

_RC

1







A

1

=







0

1

L

−

1

C

0







B

0

= [

E

L

, 0]

T

B

1

= [−

E

L

, 0]

T

with

R = 50 Ω

,

C = 220 µF

,

L = 20 mH

and

E = 6 V

. Let the target be

x

ref

= [0.24, −6]

T

_{∈ X}

ref

orresponding to

u

ref

= 0.5

. As

A(u

ref

) =

A

ref

= A

0

+ u

ref

A

1

is Hurwitz and as

B(u

ref

) = 0

, the solutions

P = P

T

_{> 0}

of

A

T

ref

P +

P A

ref

+ Q = 0

with

Q = Q

T

_{> 0}

allow to dene quadrati CLFs

V (z) = z

T

_{P z}

for the

system (4). Taking

Q = 180 × Id

,one gets:

P =







91.05 0.04

0.04

1







. In the next two subse tions,

the results of the proposed approa hes are illustrated through the steepes t and predi tive

strategy. The sampletime is

T

s

= 2.5.10

−5

s.

9.2. Attra ting set estimations for the sampled strategies

Figure3shows asystemtraje toryusingthesteepes tdes entfeedba klawandattra ting

sets determined by

P

N

for

N = 1

(red dashedline) and

N = 2

(magentasolid line). Clearly,

S

∗

L

2

isana urateapproximationoftheallsystemtraje tories. UsingGlotipolysoftware,the omputation times are respe tively 1.36 s and 0.97 s.

−0.1

−0.05

0

0.05

0.1

0.15

−1.5

−1

−0.5

0

0.5

1

1.5

PSfrag repla ements

S

∗

L

1

S

∗

L

2

Figure3: Traje toryinthestate-spa eandattra tingsets for

N

= 1

and

N

= 2

For a re eding horizon

N

H

xed to

N

H

= 2

, Figure 4 shows the ase of the predi tive strategy. The bla k solid ellipse is the estimation for a sequen e of length

N = 1

. The

estimationisstilllarge omparingtothelimit y le. Forasequen eof length

N = 8

,abetter approximation isobtained. The omputationtimes, stillusing Glotipolyare respe tively 1.6

s and 29.

10

3

s.

−0.2

−0.1

0

0.1

0.2

−2

−1

0

1

2

PSfrag repla ements

S

∗

L

1

S

∗

L

8

Figure4: Traje toryinthestate-spa eandattra tingsets for

N

= 1

and

N

= 8

Figure 5 represents the evolution of

L

∗

N

in fun tion of

N

for the steepest (solid blue line) and the predi tive strategies (dashed red line). Observe that the relations

S

∗

L

pN

⊆

S

∗

L

N

, ∀p ∈ N

∗

hold as stated inTheorem 3. Whilethe relation

L

∗

N

≥ L

∗

N

+1

does not hold in general. Figure 5 also shows that the above given approximations of the attra ting sets are

a urate although in the ase of predi tive ontrol, this estimation appears not parti ularly

tight aroundthe y le. This an be learly justied by the fa t that there exists at least one

sequen estartinginsidethe sublevelset thatrea hesthe level. This sequen eisobviouslythe

solution of

P

N

. In viewof the evolution of the urve inFigure5, anin rease of

N

seems not to lead toa better estimationof

S

∞

.

In Figure6, the evolution of

L

∗

2

w.r.t.

T

s

isdrawn for both strategies. This gure learly illustrates the ISSproperty of the system. Finally,Figure 7shows the exponential growth of

the omputationtime for the twostrategies.

9.3. Robust attra ting set estimations for sampled strategies

Supposenowthatallparameters

L

,

R

,

E

,

C

areknown with5%ofun ertaintiesand that the sample time

T

s

is time dependent with variation of 5% around its nominal value. The problem

P

N

(δ

min

, δ

max

, θ

max

)

gives the attra ting set in the worst ase. Figure 8 shows in solid linethe level sets orresponding to

L

∗

1

2

3

4

5

6

7

8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

PSfrag repla ements Predi tive strategy Steepest strategy Figure5:

L

∗

N

versus

N

0

1

2

3

4

x 10

−3

0

100

200

300

400

500

600

700

800

900

PSfrag repla ements Predi tive strategy Steepest strategy Figure6:

L

∗

2

versus

T

s

∈ [2.5, 4500]µ

s

in dashed linethe respe tive levelsets for the system withoutun ertainties. This gure also

shows two traje tories, simulated with a uniformly distributed random law for the sample

time variations, and twoparameters sets inside the 5% of un ertainties.

It isworthynoti ingthat,asexpe ted,the attra tingsetsforthesystem withparameters

variationsare biggerthanthe ones forthe nominalsystem. However, the boundedness ofthe

optimizationproblemguarantees the stability of the perturbed system.

10. Con lusion

Inthispaper,robuststabilityforthe lassofswit hedanesystemshasbeeninvestigated.

Basedonthe existen eofaCLF fortherelaxed ontrolproblem,sampledswit hed strategies

1

2

3

4

5

6

7

8

10

−1

10

0

10

1

10

2

10

3

10

4

10

5

PSfrag repla ements Predi tive strategy Steepest strategy

Figure 7: Computationtime(inse onds)versus

N

−0.2

−0.1

0

0.1

0.2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure8: Traje toriesinthestate-spa eandattra tingsets for

N

= 1

and

N

= 8

in aseofun ertainties

The proposed frameworkallowsto ompute tightglobalattra tingsets forthe whole

sys-tem traje tories, by solving a set of onstrained optimization problems. Numeri al aspe ts

have been dis ussed and it has been shown that pra ti ally, the optimization problems

re-veal tobeQCQP or non onvex polynomial optimizationproblems for whi h e ient global

optimization solvers exist. In addition, ISS results with respe t to the sample time and the

parameter un ertainties are formulated. In doing so,some stability marginsare guaranted.

The numeri al illustration given on a bu k-boost onverter shows that quadrati CLF

an be easilydesigned for DC-DC onverters. Applying the steepest or predi tive strategies,

numeri alresultsalsoshowed that itisnot ne essary to onsider a high orderin

P

N

toget a gooda ura y in the over-approximationof

S

∞

.

Asfuturework,a omparisonbetweenoptimal ontrolandthegivenswit hinglawswould

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