Ryozo Nagamune - KTHryozo/psfiles/newmlsemi.pdf · Small jSj implies good disturbance attenuation Performance limitations (area formula, water-bed e ect etc.)) impossible to reduce

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Robust control design based on Nevanlinna-Pick

interpolation with degree constraint

Ryozo Nagamune

Post-doc, Institut Mittag-Leffler

joint work with

A. Blomqvist, G. Fanizza, A. Lindquist

Optimization and Systems Theory, KTH

Ryozo Nagamune 1 April 24th, 2003

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Outline

1. Nevanlinna-Pick interpolation

• Classical theory

(Caratheodory, Schur, Pick, Nevanlinna etc.)

• Degree constraint (Byrnes, Georgiou, Lindquist etc.)

• Computation

2. Robust control applications


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Classical Nevanlinna-Pick interpolation

Given {(zj, wj)}n

j=0 ⊂ C2 with zj ∈ D (open unit disc), distinct

find all f s.t.

1. positive real (i.e., f is analytic and Ref(z) ≥ 0 in z ∈ D)

2. interpolation conditions f(zj) = wj , j = 0, 1, . . . , n

1

w2

w0

z 0

z 2

z 1

w

Re

Im

Re

Im

0

f

1

i

Note: There are many essentially equivalent problem formulations.


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Essentially equivalent problems

• derivative conditions f (k)(zj) = wjk

• different domains and ranges

Discrete−time control problems

Classical math problem

Continuous−time control problems

Circuit theory

f

Im

0 Re 1

i

Re

Im

f

1

i

Re

Im

1

i

Re

Im

f

1

i

Re

Im

1

i

Re

Im

f

Im

0 Re

Im

0 Re


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Solvability and parameterization of f

Existence of f ⇐⇒ Pick matrix P :=[

wi+wj

1−zizj

]n

i,j=0≥ 0

• P is singular: unique rational f

• P > 0: infinitely many f{

f(z) =t11(z)g(z) + t12(z)

t21(z)g(z) + t22(z), g ∈ H∞, ‖g‖

∞≤ 1

}

tij : determined by given data, rational, deg tij ≤ n

– We prefer simple (rational, low degree) f in applications.

– Which g gives a simple f? (not trivial...)


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Nevanlinna-Pick with degree constraint

Given {(zj, wj)}n

j=0 ⊂ C2 with zj ∈ D, distinct and P > 0

find all f s.t.

1. positive real

2. interpolation conditions

3. degree constraint: real rational, deg f ≤ n

Note: (degree bound n) = (# data pairs)−1

P > 0 =⇒ SNPDC:= {f : f satisfies 1,2,3}6= ∅

Question: What characterizes the set SNPDC?


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What characterizes SNPDC?

Suppose f ∈ SNPDC.

Cond. 3 (f : real rational) ⇒ f(z) = β(z)/α(z)

(deg f ≤ n) deg α ≤ n, deg β ≤ n

Cond. 2 (f(zj) = wj) ⇒ β = κ(α) (κ : linear map)

Cond. 1 (f is positive real) ⇒

α [κ(α)]∗ + [κ(α)] α∗ ≥ 0

∀ |z| = 1 (α∗(z) := α(z−1))

α + κ(α) 6= 0, ∀ |z| < 1

For any f ∈ SNPDC, there corresponds

a nonnegative trigonometric polynomial of degree at most n:

α [κ(α)]∗+ [κ(α)] α∗ =:

∑n

j=0 dj(zj + z−j) ≥ 0, ∀ |z| = 1


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Characterization of SNPDC by Georgiou

The map G : A → D is bijective.

G(α) := α [κ(α)]∗+ [κ(α)] α∗

A := {α : deg α ≤ n, κ(α)/α is in SNPDC, α(0) > 0}

D :=

d(z, z−1) :=

n∑

j=0

dj(zj + z−j) ≥ 0, ∀ |z| = 1

d0 > 0, dj ∈ R, j = 0, 1, . . . , n

• homeomorphism (Blomqvist, Fanizza & Nagamune)

• diffeomorphism between interior sets (Byrnes & Lindquist)

Question: For each d ∈ D, how to find the unique α ∈ A?


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A continuation method

• For a specified d ∈ D, solve G(α) = d (nonlinear).

• For a special d0 ∈ D, G(α) = d0 becomes linear.

GPSfrag replacements

α(0)

α(1)

α(ν)

d

d0

(1 − ν)d0 + νd

A D: convex set

Follow the trajectory {α(ν)}1ν=0 from ν = 0 to ν = 1 !

Question: What are the properties of the trajectory?


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Properties of the trajectory {α(ν)}1

ν=0

• continuously differentiable in the interval [0, 1)

and continuous at ν = 1

• no turning point in interval [0, 1)

• no bifurcation

turning pointbifurcation

^α(ν)

0 1ν

^

0 1ν

α(ν)

Q: How to follow the trajectory? −→ predictor and corrector steps


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Predictor-corrector steps

1. Predictor step: Given α(ν)

α(ν + δν) = α(ν) + v(α(ν)) · δν(v(α(ν)) := dα

dν(ν)

)

We need v(α(ν)) (computable) and δν (efficient way to determine).

2. Corrector step: Given α(ν + δν), get α(ν + δν).

α(0)^

α(1)^

α(ν)^

0 1ν+δνν

α(ν+δν) α(1)^

α(0)^

α(ν)^

10 ν ν+δν

α(ν+δν)^α(ν+δν)

Predictor step Corrector step


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Remark on multivariable extension

• Matrix-valued Nevanlinna-Pick interpolation

F (zj) = Wj, j = 0, 1, . . . , n

Conjecture: The map G : A → D has “nice” properties:

G(A) := [κ(A)]∗

A + A∗ [κ(A)]

A :=

A :

deg A ≤ n, κ(A)A−1 is in SNPDC

A(0) : upper-triangular with positive diagonals

D :=

D(z, z−1) :=

n∑

j=0

(Djz

j + DTj z−j

)≥ 0, ∀ |z| = 1

D0 + DT0 > 0, Dj ∈ R

`×`, j = 0, 1, . . . , n


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Outline



• Degree constraint

• Computation


• Basic ideas

• Examples

– Sensitivity shaping in H∞ control

– Robust regulation with robust stability

– Robust stabilization for gain uncertainty


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Basic ideas

1. Robust control problems reducible to N-P interpolation

• Gain-margin maximization (Khargonekar & Tannenbaum)

• Robust stabilization (Kimura)

• Model matching (Francis & Zames, Chang & Pearson)

2. Theory and algorithm for N-P with degree constraint

We apply 2 to 1 → a controller set with some degree bound

• How small is the degree bound?

• How to choose our design parameters (d ∈ D)?


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Outline




• Computation


• Basic ideas

• Examples





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Sensitivity function

- d-C(z) -P (z)- d -?d:disturbance+

y:output6−r

• P (z) : SISO discrete-time real rational plant

• Sensitivity function S

d → y : S(z) =1

1 + P (z)C(z)

• Small |S| implies good disturbance attenuation

• Performance limitations (area formula, water-bed effect etc.)

⇒ impossible to reduce |S| over all frequencies

⇒ trade-off between frequencies


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Sensitivity shaping problem

- d-C(z) -P (z)- d -?d+

y S =1

1 + PC6−r

• Given

– a plant P (z) with distinct unstable zeros zi, i = 1, . . . , nz and

unstable poles pj , j = 1, . . . , np

– γ: global upper bound, γ: upper bound in Θ ⊂ [0, π]

• design an S(z) s.t. the closed-loop system satisfies

– internal stability ⇔ S: stable, S(zi) = 1, S(pj) = 0

– specification in the frequency domain

-

6|S|

θ0 Θ

1

πγ

γ∣∣S(eiθ)

∣∣ < γ, ∀θ ∈ [0, π]

∣∣S(eiθ)

∣∣ ≤ γ, ∀θ ∈ Θ


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Weighted H∞ minimization approach

1. Select W with high gain on Θ.

2. Solve

-

6|W |

θ0 Θ

1

πγ

γ@

@@inf

S

∥∥∥S

∥∥∥

∞s.t.

S : stable

S(zi) = W (zi), S(pj) = 0

and set S = W −1S.

3. Check if S meets the spec. If not, change W and go to Step 2.

4. C(z) = (1 − S(z))/P (z)S(z)

• deg C ≈ deg P + deg W


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Proposed approach

• Problem: find a stable S satisfying

(1) S(zi) = 1, i = 1, . . . , nz, S(pj) = 0, j = 1, . . . , np

(2) |S(eiθ)| < γ, ∀θ ∈ [0, π]

(3) |S(eiθ)| ≤ γ, ∀θ ∈ Θ

(1)+(2)→ Nevanlinna-Pick (NP) interpolation problem

• Our approach

1. focus on the set of NP interpolants with degree constraint:

SNPDC := {S : (1), (2), rational, deg S ≤ nz + np − 1}

2. pick up an S from the set SNPDC which also meets (3).

Question: How to characterize SNPDC?


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Characterization of SNPDC (bounded-real version)

The map G : A → D is bijective.

G(α) := γ2αα∗ − [κ(α)] [κ(α)]∗

A := {α : deg α ≤ n, S := κ(α)/α is in SNPDC, α(0) > 0}

D :=

d(z, z−1) := d0 +

n∑

j=1

dj(zj + z−j) > 0, ∀ |z| = 1

d0 > 0, dj ∈ R, j = 0, 1, . . . , n

Question: How to choose d ∈ D to get S with a desired shape?

γ2αα∗ − [κ(α)] [κ(α)]∗

= d ⇐⇒ γ2 − SS∗ =d

αα∗


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Shaping strategy I

γ2 − S(z)S(z−1) =d(z, z−1)

α(z)α(z−1)

d(λ, λ−1) = 0, λ ≈ eiθ1

⇒ γ2 −∣∣∣S(eiθ1)

∣∣∣

2

≈ γ2 − S(λ)S(λ−1) =d(λ, λ−1)

a(λ)a(λ−1)= 0

⇒∣∣S(eiθ1)

∣∣ ≈ γ

A root of d near the unit circle with angle θ1 pins up |S| around θ1.

(e )iθ

0Re

γ

1θ

S

θ

xλi

1

θ1

Im

-1

-i

| |


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Shaping strategy II

Redefine SNPDC with an extra condition:

SNPDC := {S ∈ H∞ : S(zi) = 1, i = 1, . . . , nz, S(pj) = 0, j = 1, . . . , np

‖S‖∞

< γ}∩ {S : S(λ) = η} ∩ {S : deg S ≤ nz + np}

• (degree bound) = #(interpolation conditions) minus one

• λ and η are design parameters. λ ≈ eiθ2 ⇒∣∣S(eiθ2)

∣∣ ≈ |S(λ)| = |η|

An extra condition near the unit circle with angle θ2 fixes |S| to |η| at θ2.

2θ

2θ

(e )iθ

0Re

γ

S

i

1

Im

-1

-i

|η|xλ

| |


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Controller degree bound

Suppose S ∈ SNPDC.

SNPDC =

S ∈ H∞ :

S(zi) = 1, i = 1, . . . , nz, S(pj) = 0, j = 1, . . . , np

S(λk) = ηk, k = 1, . . . , nex

‖S‖∞

< γ, deg S ≤ nz + np − 1 + nex

The controller C(z) = (1 − S(z))/P (z)S(z) satisfies a degree bound:

deg C ≤ deg P − nz − np + deg S

≤ deg P − 1 + nex

NPDC deg C ≤ deg P − 1 + nex

weighted H∞ deg C ≈ deg P + deg W


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An example

• plant P (z) = 1/(z − 1.05)

• specifications: |S| < −20 dB (θ ∈ [0, 0.3]), ‖S‖∞

< 6.02 dB

|T | < −6.02 dB (θ ∈ [2.5, π])

• extra conditions: S(1.01e±0.3i) = 0, S(−1.01) = 1

−1 0 1

−1

0

1

Re

Im

0 0.3 1 2 2.5 3−40

−30

−20

−10−6.02

0

6.0210

Frequency (rad/sec)G

ain

(dB

)

|S||T|

Roots of d(z, z−1) Frequency responses

• deg C = 3 (deg C ≤ deg P − 1 + nex)


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A numerical example (cont’d)

• mixed weighted H∞ minimization:

minC(z)

∥∥∥∥∥∥

W1S

W2T

∥∥∥∥∥∥

∞

sub. to internal stability

deg W1 = 5, deg W2 = 1

0 0.3 1 2 2.5 3−40

−30

−20

−10−6.02

0

6.0210

Frequency (rad/sec)

Gai

n (d

B)

|S| |T| 1/|w1|1/|w2|

• deg C = 6


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Outline




• Computation


• Basic ideas

• Examples





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Robust regulation with robust stability (RRRS)

• Example

Exo.1 - ee-C(s)

u-

Exo.2

?P (s) - e -?

Exo.3

+ y

6−

– Plant: P (s) = (−s − 2)/(s2 + s − 2)

– Exo.1: step reference

– Exo.2 & 3: sinusoidal disturbances (known frequencies)

Problem: Design C to achieve RRRS.


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Example of RRRS (cont’d)

• The design problem → Nevanlinna-Pick interpolation problem

• A controller set for RRRS with a degree bound

deg C ≤ deg (Exosystems)︸︷︷︸

sufficient for RR

+ deg P − 1︸︷︷︸

sufficient for RS

• Errors e(t) for the nominal system

0 20 40 60 80 100−10

−5

0

5

10

0 20 40 60 80 100−10

−5

0

5

10

controller 1 controller 2

• Potential to improve system performance without increasing deg C

• How to use d ∈ D for pole-placement?


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Outline




• Computation


• Basic ideas

• Examples





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Robust stabilization for gain uncertainty

d-C(s)- k -P (s) -6−

[by Khargonekar & Tannenbaum]

• k: uncertain gain in [a, b] of a given nominal plant P

• Design C that robustly (i.e. for any k) stabilizes the

c.l. system

−→ Nevanlinna-Pick interpolation

We can obtain a controller set with a degree bound:

deg C ≤ deg P + nz + np − 2

nz : #(unstable zeros of P )

np : #(unstable poles of P )


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Conclusions

• Robust control based on N-P interpolation with degree

constraint

– Computational procedure (continuation method)

– Examples

∗ Sensitivity shaping in H∞ control

∗ Robust regulation with robust stability

∗ Robust stabilization for gain uncertainty

Future subjects

• Automatic tuning of design parameters

• User-friendly software

• Applications to real-world engineering problems


Documents

Ryozo Nagamune - KTHryozo/psfiles/newmlsemi.pdf · Small jSj implies good disturbance attenuation Performance limitations (area formula, water-bed e ect etc.)) impossible to reduce