On Moment Problems in Robust Control, Spectral Estimation, Image - - PowerPoint PPT Presentation

On Moment Problems in Robust Control, Spectral Estimation, Image Processing and System Identification

Anders Lindquist Shanghai Jiao Tong University, China, and the Royal Institute of Technology, Stockholm, Sweden

University of Maryland May, 2017

Special recognition

Christopher Byrnes Tryphon Georgiou + Sergei Gusev, Alexei Matveev, Alexandre Megretski, Giorgio Picci, Per Enqvist, Johan Karlsson, Ryozo Nagamune, Anders Blomqvist, Vanna Fanizza, Enrico Avventi, Axel Ringh

What is the talk about

• A classical problem – the moment problem – with

a decidedly non-classical twist motivated by engineering applications.

• What is new are certain rationality constraints

imposed by applications that alter the mathematical problem and make it nonlinear.

• A global-analysis approach that studies the class of

solutions as a whole.

• A powerful paradigm for smoothly parameterizing,

comparing, and shaping solutions to specifications.

The moment problem

Chebyshev Markov Lyapunov

space of positive measures dµ ∈ M+

t1 t2 t3 t4 x1 x2 x3 x4 a t b x µ

• Ex. 1

1 µ step function

N

• j=1

αk(tj)xj = ck, k = 0, 1, . . . , n N > n      α0(t1) α0(t2) · · · α0(tN) α1(t1) α1(t2) · · · α1(tN) . . . . . . ... . . . αn(t1) αn(t2) · · · αn(tN)           x1 x2 . . . xN      =      c0 c1 . . . cn      infinitely many solutions

• Ex. 2

spectral density dµ = Φ(eiθ)dθ 2π Z π

−π

eikθΦ(eiθ)dθ 2π = ck, k = 0, 1, . . . , n Re Im

z = eiθ

θ

!"

!"

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Where do we find moment problems in applications?

• spectral estimation
• speech synthesis
• system identification
• image processing
• optimal control
• robust control
• model reduction
• model matching problems
• simultaneous stabilization
• optimal power transfer

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2 2

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generator coupling load

… and why do these problems require a nonclassical approach?

• Solution must be of bounded complexity (such as

rational of a bounded degree) so that one can realize it by a finite-dimensional device

w(z) u y

• Classical theory does not provide natural para-

meterizations of rational solutions of bounded degree

Prototype problem: Covariance extension

ck = E{y(t + k)y(t)}, k = 0, 1, 2, . . . , where y stationary stochastic process

Carathéodory Schur Kalman

· · · (ii) is rational of degree at most n r f ∈ C+

f

2 · · (i) is a Carath´ eodory function f(z) = 1

2c0 + c1z + · · · + cnzn + cn+1zn+1 + · · ·

Given c0, c1, . . . , cn, find an infinite extension cn+1, cn+2, . . . such that

Trigonometric moment problem

f MOMENT PROBLEM: Find Φ of degree at most 2n such that

Φ(z) = P(z) Q(z), P, Q trigonometric polynomials of degree n

Spectral estimation by covariance extension

w(z) u y white noise stationary process with spectral density where

• bserved data

Since , we use ergodic estimate Hence, we can only estimate Remains to determine

Modeling speech

w(z) u y Excitation signal Speech w(z) varies with time w(z) constant on each (30 ms) subinterval

A 30 ms frame of speech for the voiced nasal phoneme [ng] Periodogram (FFT) of voiced nasal phoneme [ng] filter determined by few parameters 6 4 Construct a rational filter with a rational w(z)

• f low degree, modeling the window of speech

w(z)

y u y u

Linear Predictive (LPC) Filtering

yields Szegö polynomial and modeling filter where

Linear Predictive (LPC) Filtering

w(z)

y u y u

n = 10

Choose an excitation signal u from a code book with 1024 = 210 entries (10 bits) Send coefficients in ϕ10(z) and number

• f “best” signal

w(z) = σ(z) a(z)

However, can we construct the general solution?

w(z) = σ(z) a(z)

Cellular telephone: envelope in purple

All the solutions

• Existence proved by Georgiou 1983; conjectured uniqueness
• The rest proved by Byrnes, Lindquist, Gusev, Matveev 1993
• These first proofs were nonconstructive, but there are now

constructive proofs based on optimization

THEOREM. The solutions of the rational covariance extension problem are completely parameterized by the zeros of the corres- ponding shaping filter, i.e., given an arbitrary monic stable polynomial σ(z) there is one and only one stable polynomial a(z) such that w(z) = σ(z) a(z) is a shaping filter for c0, c1, . . . , cn. The correspondence is a diffeomorphism.

zeros/poles envelope zeros/poles envelope

10th degree LPC 20th degree LPC 6th degree filter with appropriate zeros

Optimization approach

• THEOREM. There is a unique minimum.

Then where Given σ and c, minimize

• ver all q such that

The space of all rational Carath´ eodory functions f

• f degree at most n is a 2n-dimensional manifold.

unique solution Φ = |σ|2

Q

min

q

Jσ(q)

Nonlinear coordinates

Normalize so that c0 = 1 A foliation with one leaf for each choice of σ (fast Kalman filtering)

! " ! "

A foliation with one leaf for each choice of c = (c1, c2, . . . , cn)

! "

A global analysis approach

• Find complete parameterization

Complete class of solutions satisfying complexity constraint Sets of tuning parameters

• For any choice of tuning parameters, determine the

corresponding solution by convex optimization

• Choose a solution that best satisfies additional design

specifications (without increasing the complexity) smooth bijection

Other applications leading to moment problem with rationality constraints

Nevanlinna-Pick interpolation

f

Tuning by moving interpolation points

Covariance extension corresponds to N-P interpolation with all interpolation points at the origin (z = 0) maximum entropy (LPC) We moved the spectral zeros from the origin closer to the unit circle Next we tune by moving the interpolation points closer to the unit circle

A tunable high resolution spectral estimator (THREE)

G (z) G (z) G (z)

1

2 n

y v v v

(1) (2) (n)

x1 x2 xn y

Estimation of spectral lines in colored noise

Robust control

f

• Carathéodory function
• f analytic for |z| ≤ 1
• Re{f (z)} > 0 for |z| ≤ 1

f

• f analytic for |z| ≤ 1
• |f(z)| < 1 for |z| ≤ 1
• Schur function
• f analytic for |z| ≥ 1
• Re{f (z)} > 0 for |z| ≥ 1

f

• positive real function

Loop shaping in robust control

G(z) K(z) d u y S(z) d y f

f

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!6

!6

!6

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The interpolants of degree at most n are parmeterized by the spectral zeros (σ) in a 1 − 1 fashion

Example: Sensitivity shaping

Design a strictly proper K so that the closed-loop system is

• internally stable and
• satisfies the specifications:
• settling time at most 8 seconds
• overshoot at most 10%
• Doyle, Francis

Tannenbaum G(s) u K(s) r y step reference

• A. Blomqvist and R. Nagamune

Try to achieve (DFT) Internal stability requires (unstable plant poles) (nonminimum-phase plant zeros) (strictly proper controller) DFT: deg K = 8 NPDC: deg K = 4

• 3
• 3
• 2
• 2
• 1
• 1

1 2 3 20 20 40 40 60 60 80 80 100 100 120 120

Choose:

|σ|2

• settling time at most 8 seconds
• overshoot at most 10%
• DFT: deg C = 8

NPDC: deg C = 4

Multidimensional moment problems

Z

K

αkdµ = ck, k = 1, 2, . . . , n Z

• dµ nonnegative measure on a compact subset K of Rd
• α1, α2, . . . αn linearly independent basis functions defined on K

Image compression

Model reduction

Original system: Antoulas-Sorensen: Global-analysis approach:

A large-scale problem: A CD player

Model reduction: Antoulas-Sorensen solutions:

Global-analysis solution Antoulas-Sorensen solutions

Conclusions

• A global analysis approach that studies the class of

solutions as a whole.

• Complete parameterizations of solutions with

smooth tuning strategies. An enhanced theory for generalized moment problems that incorporates rationality constraints prescribed by applications.

• Convex optimization for determining solutions.