Can Systems be Certified Distributively? Scalable Analysis Methods - - PDF document

1 Scalable Analysis Methods for Sparse Large-scale Systems

Anders Rantzer

LCCC — Lund Center for Control of Complex engineering systems Automatic Control LTH, Lund University, Sweden

Can Systems be Certified Distributively?

Can a global performance certificate be split into component specifications without introducing conservatism?

Can Systems be Certified Distributively?

What freedom do we have to modify a component without redesigning the whole system?

Outline

○ Introduction

• Distributed Positive Test for Matrices

○ Distributed Nonconservative System Verification ○ A Scalable Robustness Test

A Matrix Decomposition Theorem

The sparse matrix on the left is positive semi-definite if and only if it can be written as a sum of positive semi-definite matrices with the structure on the right.

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= +

x x x x x x x x x x x x x x x x x x

+ ... +

x x x x x x x x x

Proof idea

The decomposition follows immediately from the band structure

• f the Cholesky factors:

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

=

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

[Martin and Wilkinson, 1965]

Example

The simplest decomposition is to just split each coefficient equally between the squares where it belong. This could work if the matrix is diagonally dominant:

= + ... +

t11 t11 t12 t12 t13 t13 t21 t21 t22 t23 t24 t31 t31 t32 t33 t34 t35 t42 t43 t44 t45 t46 t53 t54 t55 t56 t57 t57 t64 t65 t66 t67 t67 t75 t75 t76 t76 t77 t77

t22 2 t23 2 t32 2 t33 3 t66 2 t65 2 t56 2 t55 3

Generalization

Cholesky factors inherit the sparsity structure of the symmetric matrix if and only if the sparsity pattern corresponds to a “chordal” graph.

= + + +

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

7 1 2 3 4 5 6

[Blair & Peyton, An introduction to chordal graphs and clique trees, 1992]

2 Example: Non-chordal graph Example: Chordal graphs

If T is a tree, then T k is chordal for every k ≥ 1. T T2

A Theorem on Positive Extensions

A matrix with entries specified according to a chordal graph has a positive definite completion if and only if all fully specified principal minors are positive definite. [Grone, et.al, 1984]

1 3 2 1 2 1 4 2 2 4 * * * * * * * * * * * * * * * * * * * * 3 5 4 3 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1

Outline

○ Introduction ○ Distributed Positive Test for Matrices

• Distributed Nonconservative System Verification

○ A Scalable Robustness Test

A System with Tridiagonal Structure

w1 w2 wn x1 x2 x2 x3 xn−1 xn      ˙ x1(t) ˙ x2(t) . . . ˙ xn(t)      =       a11 a12 a21 a22 ... ... a(n−1)n an(n−1) ann            x1(t) x2(t) . . . xn(t)      +      w1(t) w2(t) . . . wn(t)     

A Sparse Stability Test

For the sparse matrix A, let the left hand side illustrate the structure of (sI − A)∗(sI − A). Then the matrix is stable if and

• nly if the right hand side split can be done with all squares

positive definite for s in the right half plane.

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= +

x x x x x x x x x x x x x x x x x x

+ ... +

x x x x x x x x x

• (sI−A)∗(sI−A)

Hence global stability can always be verified by local tests!

A Sparse Stability Test

Find stable minimum-phase rational Φ1(s),... , Φm(s) with

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= +

x x x x x x x x x x x x x x x x x x

+ ... +

x x x x x x x x x

• (sI−A)∗(sI−A)
• Φ1(s)∗Φ1(s)
• Φ2(s)∗Φ2(s)
• Φm(s)∗Φm(s)

for s = iω. Then A is Hurwitz iff equality holds also for Re s > 0. Proof idea: If A Hurwitz, the maximum modulus theorem gives

(sI − A)∗(sI − A) Φ1(s)∗Φ1(s) + ⋅ ⋅ ⋅ + Φm(s)∗Φm(s) Re s > 0

A Sparse Gain Bound

Solutions to ˙ x(t) = Ax(t) + w(t), x(0) = 0 satisfy T x(t)2dt ≤ γ 2 T w(t)2dt if and only if

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= +

x x x x x x x x x x x x x x x x x x

+ ... +

x x x x x x x x x

• γ 2(sI−A)∗(sI−A)−I

where the terms on the right hand side are positive definite for s in the right half plane.

3 A Sparse Passivity Test

Suppose ˙ x = Ax + Bx + w x(0) = 0 y = Cx Then T

• γ 2u(t)y(t) + w(t)2

dt ≥ 0 for all u,w, T if and only if the matrix (sI − A)∗(sI − A) γ 2CT − (sI − A)∗B γ 2C − B∗(sI − A) BT B

• is positive semi-definite for Re s ≥ 0.

Passivity can be tested componentwise without conservatism!

Outline

○ Introduction ○ Distributed Positive Test for Matrices ○ Distributed Nonconservative System Verification

• A Scalable Robustness Test

Robustness Analysis for Chained System

✛ q q q q q ✲δ m

− 1

✛

δ m

✲ ✛ ✛ ✲ ✛ ✲ ✛ ✲ ✛

δ1

✲ ✛ ✲ δ2 ✛ ✲ ✛ ✲ ✲ q Many robustness analysis problems can be reduced to proving that (I − ∆(s)G(s))−1 is stable for ∆ = diag{δ 1,... ,δ m} with δ i(iω) ≤ 1. This can be done by finding X (ω) = diag{x1(ω),..., xm(ω)} ≻ 0 with X (ω) ≻ G(iω)X (ω)G(iω)∗ where G(iω) =         1 h1 f1 2 h2 f2 ... ... ... m−1 hm−1 fm−1 m         Note that each xi influences at most nine elements of X − G X G∗.

Scalable Distributed Computations

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= + + ... +

W1 W2 Wm

− 2

The matrix G X G∗ − X is negative definite if and only if there exist yi, zi,wi such that the following are negative definite:

W1 =   1 h1 f1 2 f2  

• x1

x2   1 h1 f1 2 f2  

∗

−   x1 x2 + w1 y1 y∗

1

z1   W2 =   h2 3 f3   x3   h2 3 f3  

∗

+   w1 y1 y∗

1

z1 − w2 − x3 −y2 −y∗

2

−z2   . . . Wm−2 = ...

Can Systems be Certified Distributively?

Yes, distributed tests can be constructed without conservatism! Local trade-offs between accuracy and complexity! Chordal graphs are common and natural! Nonlinear versions?!