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arXiv:math/0504275v1 [math.OC] 13 Apr 2005

Diagonal Stability for a Class of Interconnected Passive Systems

Murat Arcak Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute Troy, NY 12180 Email: arcakm@rpi.edu February 1, 2008

Abstract We consider a class of matrices with a speci?c structure that arises, among other examples, in dynamic models for biological regulation of enzyme synthesis [6]. We ?rst show that a stability condition given in [6] is in fact a necessary and su?cient condition for diagonal stability of this class of matrices. We then revisit a recent generalization of [6] to nonlinear systems given in [5], and recover the same stability condition using our diagonal stability result. Unlike the input-output based arguments employed in [5], our proof gives a procedure to construct a Lyapunov function. Finally we study static nonlinearities that appear in the feedback path, and give a stability condition that mimics the Popov criterion.

Main Result

The results of this note were triggered by the recent paper [5] and by several discussions with its author. We give our main diagonal stability result in Theorem 1 below, and present its implications for stability of a class of interconnected systems in the form of corollaries to this theorem.

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Theorem 1 A matrix of the form

?

?1 γ2 0 . . . 0

A=

? ? ? ? ? ? ? ? ?

··· . ?1 . .

0

0

?γ1 0 . . . 0 ?1

? ? ? ? ? ? ? ? ? ?

. γ3 ?1 . . .. .. .. . . . · · · 0 γn

γi > 0, i = 1, · · · , n,

(1)

is diagonally stable; that is, it satis?es DA + AT D < 0 for some diagonal matrix D > 0, if and only if γ1 . . . γn < sec(π/n)n . (2) 2 The proof is given in an appendix. The results surveyed in [3, 2] for diagonal stability of various classes of matrices do not encompass the speci?c structure exhibited by (1). In particular, the sign reversal for γ1 in (1) rules out the “M-matrix” condition, which is applicable when all o?-diagonal terms are nonnegative. We now apply Theorem 1 to characterize the stability of the feedback interconnection in Figure 1. When each block Hi is a ?rst-order linear system with transfer function Hi (s) = γi /(τi s + 1), γi > 0, τi > 0, then a statespace representation of the interconnection would be obtained from the Amatrix in (1) by multiplying its ith row by 1/τi for i = 1, · · · , n. Since row multiplications by positive constants do not change diagonal stability, Theorem 1 recovers the result in [6] which states that if (2) holds then the feedback interconnection in Figure 1 is asymptotically stable. Theorem 1 further shows that stability can be proven with a Lyapunov function V = xT P x in which P is diagonal. The linear result in [6] has been extended in [5] to the situation where Hi ’s are not restricted to be linear, and instead, characterized by the output feedback passivity (OFP) [4] (a.k.a. output strict passivity [7]) property: ? β ≤ ? yi

2

+ γi < ui , yi > 2

(3)

where · and < ·, · > denote, respectively, the norm and inner product in the extended L2 space, and β ≥ 0 represents the bias due to initial conditions. Using this property, [5] proves that the secant condition (2) insures stability of the feedback interconnection in Figure 1. Unlike the input-output proof given in [5], we now assume that a storage function Vi is available for each block in Figure 1, and show that a weighted sum of these Vi ’s,

n

V =

i=1

d i Vi ,

(4)

where di > 0 are chosen following the procedure below, is a Lyapunov function for the closed-loop system. Indeed, a storage function verifying the OFP property (3) satis?es 2 ˙ i ≤ ?yi V + γi u i yi (5) which, when substituted in (4) along with the interconnection conditions u1 = ?yn , results in ui = yi?1 , i = 2, · · · n, ˙ ≤ ?y T DAy V (6)

where A is as in (1), and D is a diagonal matrix comprising of the coe?cients di in (4). It then follows from Theorem 1 that positive di ’s that render the right-hand side of (6) negative de?nite indeed exist if (2) holds: Corollary 1 Consider the feedback interconnection in Figure 1 and let ui , xi and yi denote the input, state vector, and output of each block Hi . Suppose, further, there exist C 1 storage functions Vi (xi ), satisfying (5) with γi > 0 along the state trajectories of each block. Under these conditions, if (2) holds then there exist di > 0, i = 1, · · · , n, such that the Lyapunov function (4) satis?es n ˙i ≤ ??|y |2 ˙ di V V =

i=1

for some ? > 0.

2

Corollary 1 still holds when some of the blocks are static nonlinearities satisfying the sector condition

2 0 ≤ ?yi + γi u i yi ,

γ i > 0,

(7)

3

H1 ?

H2

···

Hn

Figure 1: Feedback interconnection for Corollary 1.

rather than the dynamic property (5). To see this we let I denote the subset of indices i which correspond to dynamic blocks Hi satisfying (5), and employ the Lyapunov function V = d i Vi . (8)

i∈I

For the static blocks, that is Hi , i ∈ / I , we note from (7) that the sum

n 2 di (?yi + γ i u i y i ) di > 0

(9)

i=1 i∈ /I is nonnegative and, hence, ˙ ≤ V

i∈I

˙i + di V

n 2 di(?yi + γi u i yi ) ≤

n 2 di (?yi + γi ui yi) = ?y T DAy i=1

i=1 i∈ /I

(10) as in (6). Then, as in Corollary 1, condition (2) insures existence of a D > 0 ˙ ≤ ??|y |2 for some ? > 0. such that V

A Popov Criterion

A special case of interest is the feedback interconnection in Figure 2, where Hi , i = 1, · · · , n, are dynamic blocks as in (5), and the feedback nonlinearity ψ (t, ·) satis?es the sector property:

2 0 ≤ yn ψ (t, yn ) ≤ κyn ,

(11)

4

rewritten here as 0 ≤ ?ψ (t, yn )2 + κψ (t, yn )yn . (12) If we treat the feedback nonlinearity as a new block yn+1 = ψ (t, yn ), and note from (12) that it satis?es (7) with γn+1 = κ, we obtain from Corollary 1 and the ensuing discussion the stability condition: κγ1 · · · γn < sec(π/(n + 1))(n+1) . (13)

u ?

H1

H2

···

Hn

yn

ψ (t, ·) Figure 2: The feedback interconnection for Corollary 2. This condition, however, may be conservative because it does not exploit the static nature of the feedback nonlinearity. Indeed, using the Popov Criterion, the authors of [6] obtained a relaxed condition in which n + 1 in the right-hand side of (13) is reduced to n when Hi ’s are ?rst-order linear blocks of the form Hi (s) = γi /(τi s + 1) and the feedback nonlinearity is time-invariant. To extend this result to the case where Hi ’s are OFP as in (5), and not necessarily linear, we recall that the main premise of the Popov Criterion is that a time-invariant sector nonlinearity, when cascaded with a ?rst-order, stable, linear block preserves its passivity properties. This means that, by only restricting Hn to be linear, and combining it with the feedback nonlinearity as in Figure 3, the relaxed sector condition of [6] holds even if H1 , · · · , Hn?1 are nonlinear: Corollary 2 Consider the feedback interconnection in Figure 2 where Hi , i = 1, · · · , n ? 1, satisfy (5) with C 1 storage functions Vi and γi > 0, Hn is a 5

linear block with transfer function H n (s ) = γn , τn s + 1 τn > 0, γn > 0, (14)

the feedback nonlinearity ψ (·) is time-invariant and satis?es the sector property (11). Under these assumptions, if κγ1 · · · γn < sec(π/n)n , then there exists a Lyapunov function of the form

n?1

(15)

V =

i=1

d i Vi + d n

yn 0

ψ (σ )dσ,

di > 0, i = 1, · · · , n,

(16)

satisfying for some ? > 0.

˙ ≤ ??|(y1 , · · · , yn?1 , ψ (yn ))|2 V 2

Proof: Rather than treat Hn and ψ (·) as separate blocks, we combine them as in Figure 3: τn y ˙n = ?yn + γn yn?1 ?n : H (17) y ?n = ψ (y n ), and de?ne Vn = κτn which, from (17), satis?es ˙ n = ?κyn ψ (yn ) + κγn ψ (yn )yn?1. V Because ?κyn ψ (yn ) ≤ ?ψ (yn )2 from (12), we conclude

2 ˙ n ≤ ?ψ (yn )2 + κγn ψ (yn )yn?1 = ?y + κγn y ?n yn?1 , V ?n 0 yn

ψ (σ )dσ

(18)

(19)

(20)

? n is OFP as in (5), with γ which shows that H ?n = γn κ. The result then follows from Corollary 1. 2 Corollary 2 can be further generalized to the situation where other nonlinearities exist in between the blocks Hi , i = 1, · · · , n, in Figure 2. If such a nonlinearity is in the sector [0, κi+1 ], and is preceded by a linear block γi Hi (s) = τi s , then the two can be treated as a single block with γ ?i = κi+1 γi , +1 thus reducing n in the right-hand side of (2). 6

u ?

H1

H2

···

H n?1

y n?1

y ?n ψ (· )

yn ?n H

Hn

Figure 3: An equivalent representation of the feedback system in Figure 2. n , its series interconnection with the When Hn is a linear block Hn (s) = τnγ s+1 ? n which satis?es [0, κ] sector nonlinearity ψ (·) constitutes a dynamic block H (5) with γ ?n = κγn .

The Shortage of Passivity in a Cascade of OFP Systems

When the blocks H1 , ..., Hn each satisfy the OFP property (5), their cascade interconnection in Figure 4 inherits the sum of their phases and loses passivity. The following corollary to Theorem 1 quanti?es the “shortage” of passivity in such a cascade: Corollary 3 Consider the cascade interconnection in Figure 4. If each block Hi satis?es (5) with a C 1 storage function Vi and γi > 0, then for any δ > γ1 · · · γn cos(π/(n + 1))(n+1) , the cascade admits a storage function of the form (4) satisfying ˙ ≤ ??|y |2 + δu2 + uyn . V for some ? > 0. (22) 2 (21)

Inequality (22) is an input feedforward passivity (IFP) property [4] where the number δ represents the gain with which a feedforward path, if added from u to yn in Figure 4, would achieve passivity. Corollary 3 thus shows that the cascade of OFP systems (5) in which γi > 0 represents an “excess” 7

u

H1

H2

···

Hn

yn

Figure 4: The cascade interconnection for Corollary 3.

of passivity, satis?es the IFP property (22) with a “shortage” characterized by (21). Proof of Corollary 3: Using (4), (5), and substituting ui = yi?1 , i = 2, · · · , n, we rewrite (22) as

2 d1 (?y1 + γ1 y1 u ) +

1 2 di (?yi + γiyi yi?1 ) + δ (?u2 ? uyn ) ≤ ??|y |2 . (23) δ i=2

n

To show that di > 0, i = 1, · · · , n, satisfying (23) indeed exist, we de?ne

?

?1 γ1 0 . . . 0

?= A

? ? ? ? ? ? ? ? ?

··· . ?1 . .

0

0

?1 δ 0 . . .

? ? ? ? ? ? ? ? ? ?

. γ2 ?1 . . .. .. .. . . . 0 · · · 0 γn ?1

(24)

and note that the left-hand side of (23) is ?A ? [u y T ]D u y (25)

? := diag {δ, d1 , · · · , dn }. Because A ? is of the form (1) with dimension where D ? rendering (n + 1), an application of Theorem 1 shows that a diagonal D ) < sec(π/(n + 1))(n+1) . (25) negative de?nite exists if and only if (γ1 · · · γn 1 δ Because this condition is satis?ed when δ is as in (21), we conclude that such ? > 0 exists and, thus, (22) holds. aD 2

8

APPENDIX: Proof of Theorem 1

Necessity follows because, as shown in [6], if (2) fails then A is not Hurwitz. To prove that (2) is su?cient for diagonal stability, we de?ne r := (γ1 . . . γn )1/n > 0 (26) γ2 γ2 γ3 γ · · · γ γ · · · γ 2 i 2 n ? := diag 1, ? , , · · · , (?1)i+1 i?1 , · · · , (1)n+1 n?1 r r2 r r and note that

?

1 r

0 1

? ??1 A? =

Thus, with the choice we get

? ? ? ? ? ? ? ? ?

··· .. .

0

(?1)n+1 r 0 . . . 0 1

?

.. . 0 r 1 . . . . . . .. .. .. 0 ··· 0 r D = ??2

? ? ? ? ?. ? ? ? ?

(27)

(28) (29)

DA + AT D = ??1 (??1 A? + ?AT ??1 )??1

which means that DA + AT D < 0 holds if the symmetric part of (27), given by 1 (???1 A? ? ?AT ??1 ), (30) 2 is positive de?nite. To show that this is indeed the case, we note that (27) exhibits a circulant structure [1] when n is odd, and a skew-circulant structure when n is even. In particular, it admits the eigenvalue-eigenvector pairs λk = 1 + rei n k when n is odd; and λk = 1 + rei( n + n k)

π 2π 2π

vk =

2π 2π 2π 1 [1 e?i n k e?i2 n k · · · e?i(n?1) n k ]T n

k = 1, · · · , n

vk =

1 2π π 2π π 2π π [1 e?i( n + n k) e?i2( n + n k) · · · e?i(n?1)( n + n k) ]T n

when n is even. Since, in either case, (27) is diagonalizable with the unitary matrix V = [v1 · · · vn ], the eigenvalues of the symmetric part (30) coincide with the real parts of λk ’s above. Finally, because

k =1,···n

min Re{1 + rei n k } = min Re{1 + rei( n + n k) } = 1 ? r cos(π/n),

k =1,···n

2π

π

2π

9

we conclude that if (2) holds, that is r < sec(π/n), then all eigenvalues of (30) are positive and, hence, (30) is positive de?nite and (29) is negative de?nite. 2

References

[1] P.J. Davis. Circulant Matrices. John Wiley & Sons, 1979. [2] E. Kaszkurewicz and A. Bhaya. Matrix Diagonal Stability in Systems and Computation. Birkh¨ auser, Boston, 2000. [3] R. Redhe?er. Volterra multipliers - Parts I and II. SIAM Journal on Algebraic and Discrete Methods, 6(4):592–623, 1985. [4] R. Sepulchre, M. Jankovi? c, and P. Kokotovi? c. Constructive Nonlinear Control. Springer-Verlag, New York, 1997. [5] E.D. Sontag. A generalization of the secant condition to passive systems. Submitted to the IEEE Conference on Decision and Control, 2005. [6] J.J. Tyson and H.G. Othmer. The dynamics of feedback control circuits in biochemical pathways. In R. Rosen and F.M. Snell, editors, Progress in Theoretical Biology, volume 5, pages 1–62. Academic Press, New York, 1978. [7] A. J. van der Schaft. L2 -gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, New York and Berlin, second edition, 2000.

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