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Fig. 1. Level curves of Lyapunov functions corresponding to the conditions (1.2), (1.3) and (1.4) in Proposition 1: If Aξ < 0, then V (x) = maxi(xi/ξi) is a Lyapunov function with rectangular level curves. If T , then T is a linear z A < 0 V (x) = z x x3 Lyapunov function. Finally if AT P + PA ≺ 0 and P ≻ 0, then V (x)= xT Px is a quadratic Lyapunov function for the system x˙ = Ax. Fig. 2. A graph of an interconnected system. In Example 1 the interpretation is a transportation network and each arrow indicates a transportation link. In Example 2 the interpretation is instead a vehicle formation and each arrow Hurwitz if all eigenvalues have negative real part. It is Schur indicates the use of a distance measurement. if all eigenvalues are strictly inside the unit circle. Finally, the matrix is said to be Metzler if all off-diagonal elements are T T nonnegative. The notation RH represents the set of rational gives (A P + P A)ξ = A z + PAξ < 0 so the symmetric ∞ T ✷ functions with real coefficients and without poles in the closed matrix A P + P A is Hurwitz and (1.4) follows. right half plane. The set of n m matrices with elements in Example 1. Linear transportation network. Consider a RH is denoted RHn×m. × dynamical system interconnected according to the graph il- ∞ ∞ lustrated in Figure 2:
x˙ 1 −1 − ℓ31 ℓ12 0 0 x1 III. DISTRIBUTED STABILITY VERIFICATION x˙ 0 −ℓ − ℓ ℓ 0 x 2 = 12 32 23 2 The following well known characterizations of stability will x˙ 3 ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ34 x3 x˙ 4 0 0 ℓ43 −4 − ℓ34 x4 be used extensively: (1) Proposition 1: Given a Metzler matrix A Rn×n, the The model could for example be used to describe a transporta- following statements are equivalent: ∈ tion network connecting four buffers. The states x1, x2, x3, x4 (1.1) The matrix A is Hurwitz. represent the contents of the buffers and the parameter ℓ Rn ij (1.2) There exists a ξ such that ξ > 0 and Aξ < 0. determines the rate of transfer from buffer j to buffer i. Such ∈Rn T (1.3) There exists a z such that z > 0 and z A< 0. transfer is necessary to stabilize the content of the second and (1.4) There exists a diagonal∈ matrix P 0 such that ≻ third buffer. AT P + P A 0. Notice that the dynamics has the form x˙ = Ax where A ≺ (1.5) The matrix A−1 exists and has nonnegative entries. is a Metzler matrix provided that every ℓij is nonnegative. − Hence, by Proposition 1, stability is equivalent to existence of Moreover, if ξ = (ξ1,...,ξn) and z = (z1,...,zn) satisfy numbers ξ1,...,ξ4 > 0 such that the conditions of (1.2) and (1.3) respectively, then P = −1 − ℓ31 ℓ12 0 0 ξ1 0 diag(z1/ξ1,...,zn/ξn) satisfies the conditions of (1.4). 0 −ℓ − ℓ ℓ 0 ξ 0 12 32 23 2 < Remark 1. Each of the conditions (1.2), (1.3) and (1.4) ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ34 ξ3 0 0 0 ℓ −4 − ℓ ξ 0 corresponds to a Lyapunov function of a specific form. See 43 34 4 Figure 1. Given these numbers, stability can be verified by a distributed test where the first buffer verifies the first inequality, the second Remark 2. One of the main observations of this paper is that buffer verifies the second and so on. In particular, the relevant verification and synthesis of positive control systems can be test for each buffer only involves parameter values at the local done with methods that scale linearly with the number of node and the neighboring nodes, so a global model is not interconnections. For stability, this claim follows directly from needed anywhere. ✷ Proposition 1: Given ξ, verification of the inequality Aξ < 0 requires a number of scalar additions and multiplications that Example 2. Vehicle formation (or distributed Kalman is directly proportional to the number of nonzero elements in filter). Another system structure, which can be viewed as a the matrix A. In fact, the search for a feasible ξ also scales dual of the previous one, is the following: linearly, since integration of the differential equation ξ˙ = Aξ x˙ 1 = x1 + ℓ13(x3 x1) with ξ(0) = ξ0 for an arbitrary ξ0 > 0 generates a feasible − − x˙ = ℓ (x x )+ ℓ (x x ) ξ(t) in finite time provided that A is Metzler and Hurwitz. 2 21 1 − 2 23 3 − 2 (2) x˙ 3 = ℓ32(x2 x3)+ ℓ34(x4 x3) Proof of Proposition 1. The equivalence between (1.1), (1.2), − − x˙ 4 = 4x4 + ℓ43(x3 x4) (1.4) and (1.5) is the equivalence between the statements G20, − − I27, H24 and N38 in [2, Theorem 6.2.3]. The equivalence be- This model could for example be used to describe a formation tween (1.1) and (1.3) is obtained by applying the equivalence of four vehicles. The parameters ℓij represent position adjust- between (1.1) and (1.2) to the transpose of A. Moreover, if ments based on distance measurements between the vehicles. ξ = (ξ ,...,ξ ) and z = (z ,...,z ) satisfy the conditions of The terms x and 4x reflect that the first and fourth 1 n 1 n − 1 − 4 (1.2) and (1.3) respectively, then P = diag(z1/ξ1,...,zn/ξn) vehicle can maintain stable positions on their own, but the 3
second and third vehicle rely on the distance measurements for Proof. It is well known that g 2−ind = maxω G(iω) 2−ind stabilization. Again, stability can be verified by a distributed for general linear time-invariantk k systems. Whenkg(t) k0, the test where the first vehicle verifies the first inequality, the maximum must be attained at ω =0 since ≥ second vehicle verifies the second inequality and so on. ✷ ∞ G(iω)w g(t)e−iωt dt w A discrete time counterpart to Proposition 1 is given next: | |≤ 0 · | | Proposition 2: For B Rn×n, the following statements are Z ∞ + G equivalent: ∈ = g (t)dt w = (0) w 0 · | | | | (2.1) The matrix B is Schur stable. Z for every w Cm. This completes the proof for p = 2. For (2.2) There is a ξ Rn such that ξ > 0 and Bξ <ξ. p =1, the fact∈ follows from the calculations (2.3) There exists a∈z Rn such that z > 0 and BT z