New sufficient conditions for robust stability of discrete-time systems subject to bounded rate parameters

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF ...

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G-2e-15-2

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P .R. China

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF DISCRETE-TIME SYSTEMS SUBJECT TO BOUNDED RATE PARAMETERS F . Atnato·, M~ Mattei t , A . Pironti* '" [Tniversita degli Stlldi di Napoli Federico If, Diportimento di Informatica S'iste1nistr;ca via Claudio 21 - 80125 Napoli, 11L1LY~ Tel: +8.9 (081) 7683513 J Fax.~ + 39 (081) 7683186, e-rrl.ail: {amato,pironti} @di8na. dis. unina. it

e

t [Iniversita degli Studi di Reggio Calabria, Dipartimento di Infor1natica~ Afatelnaticu) ElettronicQ. e 11~asporti

via Graziella, Localild Fea di ~fito - 89100 Reggio Calabria) €-'m,ail: [email protected]. unina. it

ITAI~Y.

Abstrac.t. In this paper we consider a linear ~ discrete-tilne system vv'hose dynamic matrix depends nlultiaffinely on uncertain, reaL time-varying paralneters; a bound on the rate of variat.ion of such parameters is assumed to be known. By using pa.rameter dependent, piece\vise constant Lyapunov functions, "VC can take into account the rate of variation of the paralneters and obtain sufficient. c.onditjons for stability which are less restrictive than those imposed by the classical parameter independent Lyapunov functions used in the quadratic stability theory. C'opyright © j 999 IFA C Key Words. paranleters.

Uncertain linear systellls, discrete-tilne systems, robust stability, bounded rate

1. INTRODlTC'TION In the past years, two major approaches have en1erged to investigate the stability of discretetinle linear systellls containing; uncertain parallleters. T'he first one aSSUIIles that the nn~ertain­ t.ies are time-invariant; it establishes, via t.he use of the Jury Criterion (see Jury, 1974: C;h. 2) and the discrete-tiITle couterparts of Kharitonov The~ orenl (see RoBot and Bartlett., 1996) for the one pararnetAr and the multj-parameter case respectively, whether or not the eigenvalues of the systenl dynamic Inatrix are in the unit disk for ail the al1u\
F'or tilne-varying parameters, a.nother approach? called the quadratic stability approach can be u tilized; see (Kolla et al, 1989; G arofalo et ai, 1990; _~mato et al, 1998). In this approach one looks for a quadratic Lyapunov function which is independent of the uncertainties and which guarantees stability for all the allo·w·abJe values of the param~ eters. Roughly speaking, because the Lyapunov function is independent of the uncertainties, this approach guarantees uniform asYfilptoLic stability when the parameters arc tilue-varying; also there is no restriction on how fast the paralneters may vary_

The main result of this paper i~ a novel sufficient condition to prove t.he robust stability of a di.screte-time linear systen1 depending on tinlCvarying paranleters \vith a known bound on their J~Qt€ of variation. This resul~ does not suffer of the robust stab-ility above-mentioned limitation,

j

3547

Copyright 1999 IFAC

ISBN: 0 08 043248 4

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF ...

14th World Congress of IFAC

and turns out to be less conservative than the quadratic stability approach \.vhich: as previonsly said, guarantees stability in the presence of unbounded rates of variation of the paranleter~ (see the Examples 1-2 in Section 5). ~ote t.hat, in real \vorld situations, the paralneters often exhibit a bounded rate of variation (sec for exalnplc Aluato and Mat.tei) 1998). The condition gu aranteeing unjform asynlptotic st.a.bilit.y is found with the aid of param.efer dependent piccewise constant Lyapunov fUllctions. 8hov.,r how to convert the problem of finding such a parameter dependent Lyapul10v fUl1ction r ~which enables one to prove the stability of the uncerta.in discrete-time system under consideration to a feasibility problen1 involving Linear h1atrix Inequalities (L1\11s),

",re

l

It is important to reulark that the novel approach proposed in this pa.per can also be applied in t.he liU'lit case of parameters with an -unbounded rate of variation; again our approach obtains lcs~ conservative results than those obtainable ",,~ith the quadratic stability approach (see Example 3 in Section 5). T'he paper is organized as [ollo\vs: in Section

ProblcIll 1 Given systerrl (1) ,vith p(.) being any vector of un.kno·wn tinle-varying paralucters satisfying (3) and (4), establish a sufficient condition for the unifoTIU asymptotic stability of system RClllark 1 If the discrete-time systerIl \ve are

dealing \vith corrles from the discretization of a continuous-t.ime process subject to bounded rate paraU1-eters Pi (t), with !Pi (t) I :; he" i == 1, ... ) r the value of hi in (4) depend:=; on the sampling perjod. Indeed, if the sampling period T.<; is suf~ ficiently small, one has (letting tk+l == tk + T s and, according with the notation used in the paper, k ::::: tk) l

Pi ( k) j 'T S

11

2

3. !vlAIN RESULT In order to prove the Inain result of the paper v..:e state the follovving lenl1ua. Lelnnla 1 :L.et M N Ent'? x n be ttVO positive definite matrices}· then the tnatrix-valued junction 1

AT(p)MA(p) - N is 1?egativ€ definite for all pER :=== ] if and only if

fa], b]] x ... x

[ar.~ b r

AT (p)MA(p) - N

VpE{al,bl}x ..

(1)

< 0,

~x{ar,br}.

(6)

At a first stage, assume that r ::::: J (the one parameter case); in this case the Lemn"la readily follov..~s from the observation that for a fixed x E ill11. X n the function Proof.

where

(2) i

T

,,'there i E [k, k + 1J and the last equality follows from the ylean value
STATEMEN'l~

:=

+ 1) -

(5)

tain system described by the state space equation

A(p)

11

(1).

In this paper we consider the discrete-time uncer-

== A(p)x(k)

(4)

[Pi (i) ITs

in Section 6_

+ 1)

1,2: ... ~ r .

The main objective of the paper i~ to find a solution [or the fol]o,~;,ing problem:

Pi (k

tion 5 we luakc evidence of all the benefits of the proposed result by means of a number of nUIllcrical examples. Finally SOIne conclusions are given

x(k

==

!

the problern we deal with is precisely stated and in Section 3 the main result of the paper is provided, Section 4 is devoted to clarify the application of the lnaiTI result and t.o discus.s the computational aspects of the proposed technique, while in Sec-

2. PROBLEM

i

1l i"2I···. ir ===O

A(·) E IR nxn and p:::: (Pl

is convex (indeed its second derivative is alv.~aYB

r

P2 Pr)T E IR is the vector of the uncertain parameters_ The parameters Pi are assumed to

notlnegativc) , No,"v, corning back to the general multi-paralneter case, if ..v e define n~ == [al: b 1] x ... x [al, bl ], the Lemma can be proven by llleans of the follovving cha.in of equivalent conditions;

b~

time-varying and to satisfy t.he folIo,ving two conditions

i) Each paran1eter belongs to a known interval

(3) ii) 'I"he rate of variation of each isfies the follovving conclitjon

paralTH~ter

sat-

<=?

x T (AT (p)1\.f A(p) - N)~ ;eT (AT (p)M A(p) - N);v

{::}

V ( Pl p,. -1 ) E R·,.-l 1 Pr E {a r 1 br } xT(AT (p)M A(p) - N)x

<

0

Vp E

n

3548

Copyright 1999 IFAC

ISBN: 0 08 043248 4

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF ...

v (Pl

Pr-2) E 'R r -

14th World Congress of IFAC

T'he first difference of v( ') .) along solutions of (1)

2 ,

is given by

E {ar-l' br - 1 } Pr E {at"', br }

Pl'-l

j

'V (

{::}

xT(AT(p)MA(p) - N)3!

x

VPl E {a), b1 } ,P2 E {a21 b2 }) Pr E {aT' br

+ 1), pC k + 1) - l' ( X ( k ) ~ p ( k ) ) [AT (p( k np (p(.!: + 1 »A(p(k» JX (k

_x T F(p(k))x .

.•• ,

(i1)

} .

•

Now let us return to our robust stability Problenl 1 and split each interval [ri' Pi] into l~i sub-intervals of equal length l.-'i

U [Pi ,.1, Pi j+l]

fEz' Pi] ==

X

T

l

~o\.SSUlne that p(k) E Ll,jl x L2j2 >< ... x Lrljr; since p( . ) is subject to tile constraint (4) and inequality (8) holds} we have

p(k

==:

j~l

U

U

+ 1) E

l/i

Ll

j

jl$ll X

..• X Lf',jr-ffilr ,

l.Elt,·.",lrE1r

(12)

(7)

Lilj ,

,vith

j=l

=

where pi, 1 =!!.i a.ncJ PZ,Vi Pi' ~ow, for a given Pi, define ffii to be t.he maximum number of subintervals that the parameter Pi nlay jUlnp in one

Hence it i~ readily seen that the follo,ving condition guarantees negativeness of (11):

discrete-time st.ep (taking into account (4)); then let Si, i =: 1,2, ... r an r-tuple of integers Vilhich satis(y for j == 1, 2" ... , Vi - 'ITI i j

lPi ..1 - Pilj+S~ 1 ~

-i == 1,2, ... , 1~

hi

A'T (p )P(jl ffi {l, J'2 EB 12, ~ •.. , Jr EB IT) A(pJ

(8)

•

-P(jl,j2,." ,jr)

Vp E The follo\\ring is the luain result of the paper.

Theorem 1: System (1) is tJ,niforrnly aSY1nptotically stable for all vector valued functions

= I, . ~ . ,

==

= -Si,

.I, 2,

(14) x

Lrljr

) Vi 1

-1,0:

+ 1, ... , +Si

,r

11

Based on Theoren1 (t '~ole conclude that Problenl 1 is solvable if the following Linear Matrix Inequa.lity feasibility problem admits a solution.

. . . , ir) > 0 , k , i ::::: I} . . . , 'f"

Vi

0

The proof follows by applying Lemma 1.

P (j 1 ~ j 2 ~ k

jk

lk

k,i==1,2~

satisfying conditions (3) and (4), if there exist m·atrices

j

<

Ll,jl X L2,j2 X . . .

ProbleITl 2 Find positive definite Inatrices

such that A

T

(p) P (j 1 P(jl,

$

11 , j

2

E1~ 12 1

i'2: ... ,jT) <

••

~ y jr EB l r ) A (p) ~

0

which satisfy conditions (14).

(9)

Problem 2 is a classical convex feasibility problem involving Ll\1Is and can be solved by one of the algorithrns proposed in Bayd et al (1994).

\:Ip : Pi E {Pi,jk' Pi,jkffil}

== I 2, .. . ,Vi lk == ~si,··'1-110)+1~···,+si k) i == 1} 2, ... , r jk

4. AP_PLICATION OF THE l\1_~IN R.ESlTLT 1n this Section \,ve ,;vill clarify SOUie aspects of the main result provided in t.he prevjous section and will diRCUSS the computational burden of Theorem 1.

J

and the operation ffi is defined as jk

jk tB

111

:==

{

+ 112

if

jk

VIi

if if

+m > jk + rn <

+m

jk

E [1,

Vi]

First of all we note that, in applying Theorenl 1, 'Ne have two degrees of freedom: the number of sub-intervals 1/£1 i:=: 1, ... , r 1 in \vhich we split the hypcr-rectangle

Vi

1

Proof Let us consider the candidate Lyapunoy function v(z;p) ;eT P(p)x where

=

p :p

E Lt

.j1

x I

2 ,j"2

P(jl, j2,' . . jr) jk

==

11

.,

~,Vi

l

x ... x I

rjr

-+ Vi

>0 k}i:=: 1, ..

·lr.

V2

U It,j x U

(10)

j~l

j:=:l

U r

L2j X . . .

x

U

Lr,j )

j::=l

3549

Copyright 1999 IFAC

ISBN: 0 08 043248 4

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF ...

14th World Congress ofIFAC

assunl€ that

to \vhich the parameter vector is assurned to belong, and the integers Si, i == 1, . - . ) r.

jp(t)1 To better understand the Dleaning of the integers Si, let us consider the i-th component of the para.meter vector, namely Pi. The combination of conditions (8) with (4) means that, in one dicretetilne step, Pi maymov€ t.o a sub-interval which is at a distance (n1easured in number of sub-intervals of the interval [ei' Pi]) which is less than or equal to Si-

set

J

Ll,il&ll X . . . X

Lrfjt-.@},- ,

(16)

It is evident that, for a given splitting of the original hyper-rectangle in \vhich the paralnter vector p attains values, the faster the parameters are allowed to vary in time, the larger is the set (16) to which they may jump in one time st.ep or in other vvords the greater are t he integers Si, i == 1, . - . , 'r, \vhich satisfy (8) ~ j

In the application of the main tl'leorew, since the integers Vi and Si (i = 1, ... , r) both increase the diInension of the LIVII problem to he solved one should try to keep thenl as lo\v a.s possihle, compatibly ,v]th the rate of variation of t.he parameters.

+

(se ~ 3)

< <

0.3

(21a)

0.01

(~lb)

ExaJTIple 2 Consider no,v the follo\ving uncertain discrete-time system depending on t\~/o tinlevarying parameters x(k

-0.01)

0.9979 (( 0.01

+ 1)

-0.01) 1

+(Se;3)pdkHO

1)

+ (Be ~ 4) P2 (k) (1

0 )] :z: ( k ) (22)

1

p( k)

(0

1 )] ;Il(k)( 18)

~"hich comes out from the discretization~ via Eulcr>s nlethod (see Astrom and vVittenmark, 1984), of the continuous time process

:r. = [(-in ~l) +

'p(k)f !~(p(k)) !

In the above exa.mple> if the pair of parameters (v == GO, :; == 1) would have not given good results, one might have chosen the pair of paranlcter,t; (v ::::::: 120, S == 2), (// == 180) s ::= 3) and so OIL

5. EXA11:PLES Exal"nple 1 Consider the one paralueter dependent uncertain syst.ern

0.9979 [( 0.01

.in the continous-tin1e dornain which. according to Remark 1, nleans

l

1

+ 1)

(20b)

No\-v, as said, since in our technique t.he distance covered by the paran1eters in one tinle step is n1.easured by the nUlnber Si ~ the larger is the choice of the sub-jntervals length ~ or equivalently the SIn all er is Vi] the smaller are the integers Si satisfying (8). For this reason is preferable to keep the values of Vi as lov.r as possible to limit the dinl.ension of the LMIs feasibility Problen1. 2; however, since our stability condition is only sufficient trying to increase the parameter v at the cost of an heavier computational problem has not to be discarded. Indeed a higher nUlnber of sub-intervals ITleanS a wider class of candidate Lyapunov function for proving stability.

l r El r

""rhere

x(k

(20a)

1

\Ve split the interval [-0.3, 0.3J into 60 subintervals of eqnal length (v == 60) and, acc:ording to (8), Vie let s = l~ \\-"it.h these values of 1.1 and s Problem 2 adlnits a feasible solution; this proves uniform asyrnptotic stability of Rysterrl (18) for all realizations of the time-varying parameter p( . ) subject to the constaints- (21). 11

at the tinle step k + 1, by virtue of the above n1entioned conditions, it will belong to the following

U

0.3

in the discrete-tin1.e donlain.

rI'herefore, if the parameter vector p belongs to the following hypcr-interval at the time step k

<1 El1 •.. ·

<

lpet.) 1 <

(°0

8

) Pl(t){O

1)]

·v·{hich COlnes out frOIn the discretization, via Eufilethod with L :::::: O.Ol~ of the continuous time process

ler~s

x

:e{t)

(19)

=

21

[( -i

-;1) + (°0

+ (O.~8) P2(t)

with sampling period T.r; == 0.01. In this case for IP~ S 0.248 system (18) is quadratically stable;

8

(1

) Pl(t)( 0

0)] ;Il(t)

1)

(23)

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

NEW SUFFICIENT CONDITIONS FOR ROBUST STABILITY OF ...

In this case for jp·d ::; 0.125, i

= 1,2, system

14th World Congress ofIFAC

(22)

is quadratically stable.

1

Now assume that the continuous-tinlc parameter p(t) in (23) is subject, for all t, to the following constraints

Ipl(t)1 IpI (t)!

< <

6. CONCLlfSIONS In this paper, an uncerta.in discrete time) linear ~ystem depending on unknO\Vll, real, time-varying parameters has b~en considered; it is assulned that a bound on the rate of variation of parameters is known. VVe have proposed an approach which, rnaking use of parauleter dependent) piece'~lise constant I.. yapunov functions, obtains less conservative results than the classica.l quadratic st.abjlity approach. SOlne exanlptcs show the benefits of the

0.23,

10,

Ip2(t) I ~ 0.2:3

Ip2 (t ) r $ 10 -

(24a) (24b)

According to (5), the discrete-time parameters

proposed technique.

satisfy

< 0.23, IL.\ (Pl (k )) I :s 0.10 (2 5a) < 0.23, !.d(p2{A~)! ~ 0.10. (25b)

(Pl(k)] jp2(k)(

In this case Problem 2 admits a feasihle solution obtai.ned h.y spliiting the hyper-rectangle [-0.23, 0.23J x [-0.23, 0.23] into 4 x 4 (VI == V2 ~ 4) sub-hyper-rectangles equally spaced; nloreover wc choose SI == 32 == 1.

\Vit.h these vaJue8 of v and s Problem 2 adnl-its a feasible solution; this proves uniforrfi aSYIllpt.otic stability of systen1 (22) for all time-varying realizations of the paranletcr vector p( .) subject to the constaints (25) _

Exalllple 3 Consider again the system of Example 1. As sa.id l for Ipt :S 0.248 the systclll is quadratically stable. This means that if p(k) E [-p~ p] == [-0.248,0.248], the system is uniformly asymptot.ically stable, regardless the rate of variation of the parameter. On the other hand it is possihle to show that Problenl 2 admits a solution with

==

3

f

s == 1, S :::::: 2,

== 4

1

S ::::

v:::::. v

2,

3

l

P := 0.250 P = 0.251 15 == 0.252

REFERE~CES

Aluato, F., Celentano, G. and Garofalo, t-,. (1993) . New sufficient conditions for the stability of slowly varying li.near systems. IEEE Trans. Auto. Control 38, 1409-1411. -".\.mato~ F., and l\1attei~ lvI. (1998). Robust Control of a plaSlll.a \·\rind Tunnel. Proceedings oJ the 1998 Conference on. Control Applications, 'lriestc, Italy. Arnato: F., Mat.tei, :rvI. and Pironti A. (1998) . .t\. note on quadratic stability of uncert.ain linear discrete-time systenls. IEEE lrans. ~4v.to. Contr., 43) 227-229. Astrom) K. J. and Wittenmark, B. (1984). Cornputer ContTolled Systems. Prentice-Hall, Engle~vood Cliffs (N J)_ Boyd, S.) El Ghaoui, L., Feron, E. and Balakrishnall , \T. (] 994). Lin.ear J1fatrix Inequalities in S'yst€Jn and Control Theory. SIAf\,1 Press. Desocr, C. it. (1970). Slo~yly varying discrete systems Xi+l == AiXi. Electron. Lctt., 6, 330340. Gahinet, P., A4Nelnirowski, A.J .Laub and :r-.1.Chilali (1995). I~MI Control Toolbox. The Mathworks~ Natick (lvIA) , Garofalo~ F., Glieln1o, L. and Bevilacqua,1 L. (1990). Schur-Stability of a class of paranleter-dependent matrices. Proceedings of the 28th Alterton Conference 1 Urbana (lL). RoBot, C. V. and Bartlett, C. L. (1986). Sonle discrete-time counterparts to Kharitonov's stability criterion for uncert.ain systenls. lEEE lrans. __4 tdo. Cont-rol, 31, 355-356. I(olla, S. R' J Ycdavalli, R. l{. and l~arison, J. B. (1989). Robust stability bounds on timevarying perturbations for stat.e-space Inodels of linear discrete-tinl€: systelns. Int. J. Contr., 50, 151--159. Jury, E. I. (1974). Inners and /~tability of Dynarnical Systems. \Viley-Interscience, Ne\v York. Qiu, L.~ Bcrnhardsson, B., Rantzer, A.~ Davisoll) E . .1., Young, P.l\1. and Doylc r J. Cl (1995). j\ forlnula. for cornputation of the real stability radius . .A .utomaticQ, 31, 879-890. t

The last example iH devoted to show how the proposed technique is an a.nalysis tool less conservative than the qua.dratic stability approach also wh~n no hound on the rate of variation of parameters is explicitly available. The key consideration is that, in the discrete-tim.e dOH~ain, if the paranleters are assu111ed to belong to a compact s~t~ the rate of variation rnu.sl be necessarily bounded.

v

7.

(26) (27) (28)

In all the three cases above no liJIlitation on the ra.te of variation of the parameter is needed, slnce the choice s ~ v-I allo\vs the parameter to reach any posjtion in its interval of definition in one

discrete-time step. This last exanlple shows that the proposed approach can be used in the discrete-time domain to reduce the conservatiSlTI of the quadratic stability approach, also when no bound is available on the pa:ranleter variation rate. _

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