qThis paper was not presented at any IFAC meeting. This paperwas recommended for publication in revised form by Associate EditorF. Blanchini under the direction of Editor Paul Van den Hof.

*Corresponding author. Tel.: #349-86812677; fax: #349-86812116.

E-mail address: mosquera@gts.tsc.uvigo.es (C. Mosquera).

Automatica 37 (2001) 625}628

Technical Communique

On the strengthened robust SPR problem for discrete-time systemsq

Carlos Mosquera*, Fernando PeH rezDept. Tecnologn&as de las Comunicaciones, ETSI Telecomunicacio& n, Universidad de Vigo, 36200-Vigo, Spain

Received 25 May 2000; received in "nal form 9 October 2000

Abstract

The well-known necessary and su$cient condition to "nd a solution to the robust strictly positive real (SPR) problem, namely, thatthe phase di!erence of the polynomials involved must be less than p, was found to be also necessary and su$cient in the strengthenedrobust SPR problem by Anderson and Landau in a recent work, only for the continuous-time case. In this brief communication, wederive a necessary condition in the discrete-time case, which cannot be formulated in phase terms, and which is more stringent thanthe condition of the robust SPR problem, thereby establishing a fundamental di!erence between both continuous-time anddiscrete-time domains. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Adaptive algorithms; Convergence; Discrete-time systems; Robustness

1. Introduction and problem formulation

Anderson and Landau (1994) presented the so-called strengthened robust strictly positive real (SPR)problem. This problem, here also referred to as the robustc-SPR problem, is de"ned as that of seeking a transferfunction C(s) such that C(s)/A(s)!c is SPR for allA(s)"sN#a

1sN~1#2#a

0inside a given set, i.e.

C(s)/A(s) must be stable with strictly positive real part onthe ju-axis. This problem is motivated by a number ofadaptive problems, which require the veri"cation of anSPR condition for guaranteeing convergence, and forwhich some a priori knowledge may be possible. Ourmain interest is the discrete-time case, for which someadaptive recursive algorithms which require c'0 in theSPR condition above are the pseudolinear regression(c"1

2) (Shynk, 1989) and least-squares with a forgetting

factor o (c"12o) (Landau, 1979), among others. The

fundamental result that Anderson and Landau were ableto prove was that the robust SPR problem (i.e. withc"0) and the robust c-SPR problem are equivalent (if0(c(1) for the continuous-time case. In other words,

the necessary and su$cient conditions for the existenceof a solution in both problems are the same. The follow-ing result states such a condition for a "nite number ofpolynomials (Anderson & Landau, 1994).

Theorem 1. Let us consider a set of polynomials A(s), suchthat C(s)/A(s)!c is SPR provided that C(s)/A

i(s)!c,

i"1,2, r is SPR, where A1(s),2,A

r(s) are known as

extreme polynomials. Then there exists a rational functionC(s) such that C(s)/A(s)!c is SPR for all A(s)3A(s) if andonly if

maxi/1,2,r

[argMAi( ju)N]! min

i/1,2, r

[argMAi( ju)N](p. (1)

This necessary and su$cient condition was "rst de-rived in Anderson, Dasgupta, Khargonekar, Kraus, andMansour (1990) for the robust SPR problem (i.e. c"0) inboth continuous- and discrete-time cases, although nopractical constructive procedures were provided. In anaccompanying paper (Mosquera & PeH rez, 2000), we de-rive an algebraic method to solve the robust SPR prob-lem for some sets of polynomials A(s) valid for bothcontinuous- and discrete-time cases. Anderson andLandau (1994) pointed out that the discrete-timecounterpart to the continuous-time robust c-SPR prob-lem was unresolved. Here we show that the followingdiscrete-time counterpart of condition (1)

maxi/1,2, r

[argMAi(e+u)N]! min

i/1,2, r

[argMAi(e+u)N](p, (2)

0005-1098/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 9 7 - 7

necessary and su$cient condition for the existence ofa polynomial C(z) solving the robust SPR problem(Anderson et al., 1990), is not su$cient for the existenceof a solution to the robust c-SPR problem, as suggestedby Anderson and Landau (1994) in their concludingremarks. We present a necessary condition that, asopposed to the continuous-time case, depends on c, thusrestricting the number of possible sets A(z) for whichthere exists C(z)"1#c

1z~1#2#c

Pz~P such that

C(z)/A(z)!c is SPR for all A(z) in A(z) and for someinteger P. Note that the sought C(z) is monic; otherwise,the solution would be trivial after the solution of theproblem for the case c"0.

In the next section, we will derive a necessary condi-tion for the existence of C(z) solving the robust c-SPRproblem, for the case of the convex combination of twopolynomials.

2. Derivation of a necessary condition

Condition (2) will be proved to be not su$cient for theexistence of a solution to the robust c-SPR problem, with0(c(1. For this purpose, we will consider the case ofthe convex combination of two polynomials:

A(z)"MA(z)"jA1(z)#(1!j)A

2(z), 04j41N (3)

with

A1(z)"(1!az~1)N,

(4)A

2(z)"(1!bz~1)N, !1(b(a(1.

If we can satisfy the strengthened SPR condition at thetwo extremes, then it will hold for the whole set A(z).This follows from the following proposition (Basso, Tesi,Vicino, & Zappa, 1995).

Proposition 2. C(z)/A(z)!c is SPR if and only if

KA(e+u)C(e+u)

!

1

2c K(1

2c(5)

for all u.

The same argument was used in Anderson and Landau(1994) for reducing the robust c-SPR problem to thecorners of a convex polytope in the continuous-time case.

Let us state "rst the phase condition (2) for A1(z) and

A2(z) in (4). This condition will be strengthened later for

c'0.

Lemma 3. If N'1, there exists C(z) such that C(z)/A1(z)

and C(z)/A2(z) are SPR if and only if

a!b

J(1!a2)(1!b2)(tan

pN

. (6)

Proof. Condition (2) can be rewritten in this case as

maxu|*0,2p)

CNAarctana sinu

1!a cosu!arctan

b sinu1!b cosuBD(p.

Setting the derivative to zero in the equation above to"nd the frequency u

.for which the maximum takes

place, we obtain that

a cosu.!a2

1#a2!2a cosu.

"

b cosu.!b2

1#b2!2b cosu.

,

giving

cosu."

a#b

1#ab.

Now the maximum value can be evaluated, consideringthat

sinu."S

(1!a2)(1!b2)

(1#ab)2,

maxu|*0,2p)

[argMA1(e+u)N!argMA

2(e+u)N]

"NAarctan aS1!b2

1!a2!arctan bS

1!a2

1!b2B. (7)

Then, by using (7) in (2), we get

a!b

J(1!a2)(1!b2)(tan

p

N,

which completes the proof. h

Note that, for N"1, A1(z) and A

2(z) are already SPR.

For b"!a condition (6) reduces to

a(tanp

2N. (8)

Next, we will show how condition (6), and consequently,condition (2), is not su$cient to "nd C(z) such thatC(z)/A

1(z)!c and C(z)/A

2(z)!c are simultaneously

SPR. We will consider a and b such that ab(0, since wehave in this case that ImMA

1(e+u)N ImMA

2(e+u)N(0 for all

u3(0,p)X(p, 2p), provided that condition (6) is satis"ed.Then, for u3(0,p), A

1(e+u) stays in the "rst and second

quadrants, whereas A2(e+u) does not abandon the third

and fourth quadrants, following the opposition of thesigns of their imaginary parts. This property will prove tobe useful for the derivation of a new necessary condition.The c-SPR condition for both C(z)/A

1(z) and C(z)/A

2(z)

can be expressed, for all u3[0, 2p), as

ReMC(e+u)NReMA1(e+u)N#ImMC(e+u)N ImMA

1(e+u)N

'cDA1(e+u)D2 (9)

626 C. Mosquera, F. Pe& rez / Automatica 37 (2001) 625}628

f (u)OGReMA

1(e+u)N ImMA

2(e+u)N!ReMA

2(e+u)N ImMA

1(e+u)N

DA1(e+u)D2 ImMA

2(e+u)N!DA

2(e+u)D2 ImMA

1(e+u)N

for u3(0,p)X(p, 2p),

1

(1!b)Nfor u"0,

1

(1#a)Nfor u"p,

(13)

and

ReMC(e+u)NReMA2(e+u)N#ImMC(e+u)N ImMA

2(e+u)N

'cDA2(e+u)D2. (10)

The simultaneous veri"cation of both inequalities yieldsthe following:

ReMC(e+u)NReMA1(e+u)N

ImMA1(e+u)N

!

ReMC(e+u)NReMA2(e+u)N

ImMA2(e+u)N

'cCDA

1(e+u)D2

ImMA1(e+u)N

!

DA2(e+u)D2

ImMA2(e+u)ND (11)

for u3(0,p)X(p, 2p). If c"0, condition (11) reduces tocondition (2). The cases u"0 and u"p require addi-tional considerations. At these frequencies ImMA

1(e+u)N"

ImMA2(e+u)N"0, and the c-SPR condition reads as

C(e+u)A

i(e+u)

'c, i"1,2, u"0, p. (12)

Now, if we de"ne the function f (u) as

then we can rewrite (11) and (12) in a more compact formas

c(ReMC(e+u)N f (u), ∀u. (14)

The function f (u) possesses two discontinuities at u"0and u"p. This can be shown by computing its limits,which after some algebraic manipulations give

limu?0

f (u)Of0"

a!b

a(1!b)N`1!b(1!a)N`1, (15)

limu?p

f (u)Ofp"

a!b

a(1#b)N`1!b(1#a)N`1. (16)

A continuous function g(u) with the same integral as f (u)will be used later. It is de"ned as

g(u)OGf (u) for u3(0,p)X(p, 2p),

f0

for u"0,

fp

for u"p.

(17)

Note that f (u) and g(u) are positive for all u providedthat ab(0 and (6) is veri"ed.

We have to maximize minuMReMC(e+u)N f (u)N for allpossible C(z) so as to get a necessary condition on c. Themonicity constraint on C(z) can be translated into

1

2p P2p

0

ReMC(e+u)Ndu"1. (18)

This constraint will be used in the maximization process:

c( maxC(z),(1@2p): 2p

0 R%(C(%+u)) $u/1Gmin

uMReMC(e+u)N f (u)NH. (19)

We consider the term ReMC(e+u)N f (u) in (14) boundedfrom below by its minimum

mC"min

uMReMC(e+u)N f (u)N. (20)

Then

ReMC(e+u)N f (u)5mC

∀u,

ReMC(e+u)N5m

Cf (u)

∀u.

Now, if we consider the fact that C(z) is monic, then

1

2p P2p

0

ReMC(e+u)Ndu"151

2pm

CP2p

0

1

f (u)du,

mC4

2p:2p0

(1/f (u)) du"

2p:2p0

(1/g(u)) du. (21)

The maximum value of mC

for all C(z) such that(1/2p):2p

0ReMC(e+u)Ndu"1 provides an upper bound for

c. Note that the maximum of the right-hand side of (21) isachieved for

ReMC(e+u)N"2p

g(u):2p0

(1/g(u)) du, (22)

where g(u) is used by the continuity of C(e+u).With all these considerations in mind, the following

result can be stated:

Theorem 4. Let A1(z) and A

2(z) be two polynomials with

all their roots inside the unit circle, verifying (2), and suchthat ∀u3(0,p)X(p, 2p), Im(A

1(e+u)) ) Im(A

2(e+u))(0.

Then, a monic compensator C(z) such that C(z)/A1(z)!c

and C(z)/A2(z)!c are simultaneously SPR does not exist

C. Mosquera, F. Pe& rez / Automatica 37 (2001) 625}628 627

Fig. 1. Upper bounds for N"2 and 4, with b"!a.

if

c52p

:2p0

(1/g(u)) du, (23)

where g(u) is expressed in (17).

In Fig. 1 the upper bound of c is plotted versus a, forA

1(z)"(1!az~1)N, A

2(z)"(1#az~1)N, N"2 and

N"4, respectively. As intuitively expected, the bound onc gets to zero as condition (8) is violated.

There still remains an important question: doesthere exist C(z) such that the minimum value ofReMC(e+u)/A

1(e+u)N and ReMC(e+u)/A

2(e+u)N is as close as

desired to the bound speci"ed by Theorem 4? The answeris negative, as we will show next using the frequenciesu"0 and u"p. At u"0, the value of the real part ofC(e+u) speci"ed by expression (22) is

C(e+0)"2p

g(0):2p0

(1/g(u)) du, (24)

while the bound on c imposed in (14) for u"0 is

c(ReMC(e+0)N f (0)"C(e+0)

(1!b)N. (25)

It turns out that the substitution of (24) into (25) givesa more restrictive condition than that expressed in

Theorem 4. This follows from some basic algebraicmanipulations:

C(e+0)

(1!b)N"

2p

(1!b)Ng(0):2p0

(1/g(u)) du(

2p

:2p0

(1/g(u)) du(26)

based on the fact that

a!b

a(1!b)N`1!b(1!a)N`1(1!b)N'1

for !1(b(0, 0(a(1. (27)

Similar considerations apply for u"p. This shows thatthe bound expressed in Theorem 4 is a necessary condi-tion on c, but not su$cient. At this point, a synthesisprocedure for "nding a monic compensator C(z) solvingthe robust c-SPR problem is unknown in the discrete-time case.

3. Concluding remarks

The result presented in Theorem 4 complements thework in Anderson and Landau (1994), explicitly showinghow the interesting results obtained in Anderson andLandau (1994) for the continuous-time case cannot beextended to the discrete-time case. This illustrates a fun-damental di!erence in the strengthened robust SPRproblem between both continuous-time and discrete-time cases, showing how the well-known phase conditionis no longer su$cient for solving the discrete-time problem.

References

Anderson, B. D. O., Dasgupta, S., Khargonekar, P., Kraus, F. J., &Mansour, M. (1990). Robust strict positive realness: Characteriza-tion and construction. IEEE Transactions on Circuits and Systems,37(7), 869}876.

Anderson, B. D. O., & Landau, I. D. (1994). Least squares identi"cationand the robust strict positive real property. IEEE Transactions onCircuits and Systems I, 41(9), 601}607.

Basso, M., Tesi, A., Vicino, A., & Zappa, G. (1995). Some analysis toolsfor the design of robust strict positive real systems. Proceedings ofthe 34th IEEE conference decision and control, New Orleans, LA,pp. 176}181.

Landau, I. D. (1979). Adaptive control. New York: Marcel Dekker.Mosquera, C., & PeH rez, F. (2001). Algebraic solution to the robust SPR

problem for two polynomials. Automatica, 37(5), in press.Shynk, J. J. (1989). Adaptive IIR "ltering. IEEE ASSP Magazine, 6(2),

4}21.

628 C. Mosquera, F. Pe& rez / Automatica 37 (2001) 625}628