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Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design

Franco Blanchini 1, Daniele Casagrande 2 Stefano Miani 2 and Umberto Viaro 2

1Dipartimento di Matematica e Informatica

Universit´ a degli Studi di Udine

2Dipartimento di Ingegneria Elettrica Gestionale e Meccanica

Universit´ a degli Studi di Udine

August 30, 2011

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Transfer function versus state–space

Plant: ˙ x(t) = Ax(t)+Bu(t) y(t) = Cx(t) Control: ˙ z(t) = Fz(t)+Gy(t) u(t) = Hz(t)+Ky(t)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Transfer function versus state–space

Plant: y(s) = P(s)u(s) Control: u(s) = K(s)y(s) Under suitable “no zero–pole cancellation” assumptions det(I −K(s)P(s)) = 0, ∀s ∈ C + ⇔ A+BKC BH GC F

• is such that Re(λi) < 0

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Transfer function versus state–space

Closed–loop stability does not depend on the chosen realization; Optimality does not depend on the chosen realization; (if you do not consider numerical problems)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Parametric design

Plant: ˙ x(t) = A(w)x(t)+B(w)u(t) y(t) = C(w)x(t) w ∈ W is a constant parameter Control: ˙ z(t) = F(w)z(t)+G(w)y(t) u(t) = H(w)z(t)+K(w)y(t) A(w)+B(w)K(w)C(w) B(w)H(w) G(w)C(w) F(w)

• must be Hurwitz for all w ∈ W .

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV design

u y A(w) B(w) C(w) K(w) w G(w) F(w) H(w)

Figure: Gain–scheduling control

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV design

Plant: ˙ x(t) = A(w(t))x(t)+B(w(t))u(t) y(t) = C(w(t))x(t) w(t) is an arbitrary time–varying parameter w(t) ∈ W Control: ˙ z(t) = F(w(t))z(t)+G(w(t))y(t) u(t) = H(w(t))z(t)+K(w(t))y(t)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

w(t)

OPTIMALITY STABILITY

Pointwise optimality LPV stability1

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

STABILITY OPTIMALITY

w(t)

Pointwise optimality LPV stability

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

p h1 y u

˙ y(t) = −α y(t)+w(t)u(t), α > 0, w− ≤ w ≤ w+

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

p h1 y u

˙ y(t) = −α y(t)+w(t)u(t), α > 0, w− ≤ w ≤ w+

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

p h1 y u

˙ y(t) = −α y(t)+w(t)u(t), α > 0, w− ≤ w ≤ w+

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

Compensator: κ(w)

s+β , β > 0,

κ(w) = −κ0

w , κ0 > 0,

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

Compensator: κ(w)

s+β , β > 0,

κ(w) = −κ0

w , κ0 > 0,

Closed loop polynomial: d(s) = s2 +(β +α)s +αβ +κ0.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

Compensator: κ(w)

s+β , β > 0,

κ(w) = −κ0

w , κ0 > 0,

Closed loop polynomial: d(s) = s2 +(β +α)s +αβ +κ0. Realizations Σ1(w) =

• β

1 −κ0/w

• , Σ2(w) =
• β
• κ0/w

1

• Blanchini Casagrande Miani Viaro

Stable LPV realization for LPV design

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Motivation: parametric synthesis

Compensator: κ(w)

s+β , β > 0,

κ(w) = −κ0

w , κ0 > 0,

Closed loop polynomial: d(s) = s2 +(β +α)s +αβ +κ0. Realizations Σ1(w) =

• β

1 −κ0/w

• , Σ2(w) =
• β
• κ0/w

1

• Closed–loop systems

A1(w) = −α −κ0 1 −β

• LPV

stable

, A2(w) =

• −α

w −κ0/w −β

• LPV

unstable

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Motivation: parametric synthesis

Compensator: κ(w)

s+β , β > 0,

κ(w) = −κ0

w , κ0 > 0,

Closed loop polynomial: d(s) = s2 +(β +α)s +αβ +κ0. Realizations Σ1(w) =

• β

1 −κ0/w

• , Σ2(w) =
• β
• κ0/w

1

• Closed–loop systems

A1(w) = −α −κ0 1 −β

• LPV

stable

, A2(w) =

• −α

w −κ0/w −β

• LPV

unstable

Stability depends on the compensator realization: Rugh and Shamma (2000).

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stabilizability: separation principle

Theorem P = (A(w),B(w),C(w)) LPV is stabilizable via linear LPV control iff it is possible to build an LPV state observer and an LPV stabilizing state feedback (dual problems).

x

PLANT A(w) B(w) C(w)

u y

OBSERVER (w) ESTIMATED STATE FEEDBACK (w)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stabilizability: duality

Theorem P = (A(w),B(w),C(w)) LPV is stabilizable iff A(w)X +B(w)U(w) = XP(w) (state feedback eq.), RA(w)+L(w)C(w) = Q(w)R (state observer eq.). X full row rank, R full column rank, µ1(P(w)) < 0, µ∞(Q(w)) < 0.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stabilizability: duality

Theorem P = (A(w),B(w),C(w)) LPV is stabilizable iff A(w)X +B(w)U(w) = XP(w) (state feedback eq.), RA(w)+L(w)C(w) = Q(w)R (state observer eq.). X full row rank, R full column rank, µ1(P(w)) < 0, µ∞(Q(w)) < 0. Theorem P = (A(w),B(w),C(w)) is quadratically stabilizable iff PA(w)T +A(w)P +B(w)U(w)+U(w)TB(w)T < 0, A(w)TQ +QA(w)+Y (w)C(w)+C T(w)Y (w)T < 0, with P > 0 and Q > 0.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stable synthesis

Assumption (A(w),B(w),C(w)) is (quadratically) LPV stabilizable.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stable synthesis

Assumption (A(w),B(w),C(w)) is (quadratically) LPV stabilizable. Question: Given a plant (A(w),B(w),C(w)) and a parametric compensator transfer function R(s,w) such that the closed–loop system is stable for constant values w ∈ W , is it possible to realize this compensator in such a way that the closed loop is LPV stable?

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stable synthesis

Assumption (A(w),B(w),C(w)) is (quadratically) LPV stabilizable. Question: Given a plant (A(w),B(w),C(w)) and a parametric compensator transfer function R(s,w) such that the closed–loop system is stable for constant values w ∈ W , is it possible to realize this compensator in such a way that the closed loop is LPV stable? Theorem Yes, it is.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV stable realization

Definition Given T(s,w) = N(s,w) d(s,w) , w ∈ W , the systems ˙ z(t) = F(w)z(t)+G(w)ω(t), ξ(t) = H(w)z(t)+K(w)ω(t), is a parametric realization of T(s,w) if T(s,w) = H(w)(sI −F(w))−1G(w)+K(w), ∀w ∈ W .

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Basic result

Definition LPV stable realization. Assuming that d(s,w) is a Hurwitz polynomial for all w ∈ W , the realization Σ(w) = F(w) G(w) H(w) K(w)

• ,

is LPV stable if ˙ z(t) = F(w(t))z(t) (1) is asymptotically stable for any function w(t) ∈ W .

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Basic result

Definition LPV stable realization. Assuming that d(s,w) is a Hurwitz polynomial for all w ∈ W , the realization Σ(w) = F(w) G(w) H(w) K(w)

• ,

is LPV stable if ˙ z(t) = F(w(t))z(t) (1) is asymptotically stable for any function w(t) ∈ W . Theorem Each parametric transfer function T(s,w), with d(s,w) Hurwitz, admits an LPV stable realizations.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Procedure

Procedure

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Procedure

Procedure

1 Take any realization

˜ F(w) ˜ G(w) ˜ H(w) ˜ K(w)

• Blanchini Casagrande Miani Viaro

Stable LPV realization for LPV design

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Procedure

Procedure

1 Take any realization

˜ F(w) ˜ G(w) ˜ H(w) ˜ K(w)

• 2 Solve ˜

F T(w)P(w)+P(w) ˜ F(w) = −I

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Procedure

Procedure

1 Take any realization

˜ F(w) ˜ G(w) ˜ H(w) ˜ K(w)

• 2 Solve ˜

F T(w)P(w)+P(w) ˜ F(w) = −I

3 Factorize P(w) = RT(w)R(w) (Cholesky). Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Procedure

Procedure

1 Take any realization

˜ F(w) ˜ G(w) ˜ H(w) ˜ K(w)

• 2 Solve ˜

F T(w)P(w)+P(w) ˜ F(w) = −I

3 Factorize P(w) = RT(w)R(w) (Cholesky). 4 The LPV stable realization is

F(w) G(w) H(w) K(w)

• =

R(w) ˜ F(w)R−1(w) R(w) ˜ G(w) ˜ H(w)R−1(w) ˜ K(w)

• Blanchini Casagrande Miani Viaro

Stable LPV realization for LPV design

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Procedure

Procedure

1 Take any realization

˜ F(w) ˜ G(w) ˜ H(w) ˜ K(w)

• 2 Solve ˜

F T(w)P(w)+P(w) ˜ F(w) = −I

3 Factorize P(w) = RT(w)R(w) (Cholesky). 4 The LPV stable realization is

F(w) G(w) H(w) K(w)

• =

R(w) ˜ F(w)R−1(w) R(w) ˜ G(w) ˜ H(w)R−1(w) ˜ K(w)

• 5 Indeed F(w)T +F(w) = −R(w)−TR(w)−1

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV Stabilizing compensator structure

u y

PLANT A(w)

W(s,w)

B(w) C(w)

Figure: Observer-based controller parametrization.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV Stabilizing compensator structure

u y

PLANT A(w)

We can decide

• nly here!

the realization

B(w) C(w)

Figure: Observer-based controller parametrization.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Compensator structure, quadratic stabilizability

u y x

• +

+

+ + − +

ν

OBSERVER PLANT

• ESTIM. STATE FEEDBACK

C(w)

ustab

A(w) B(w) C(w) x’ =[A(w)+L(w)C(w)] x − L(w) y + B(w) u J(w)

T(s,w)

Figure: Quadratic stability (Zhou and al., Sznaier and Sanchez Pena)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Compensator structure, quadratic stabilizability

u y x

• +

+

+ + − +

ν

OBSERVER PLANT

• ESTIM. STATE FEEDBACK

C(w)

ustab

A(w) x’ =[A(w)+L(w)C(w)] x − L(w) y + B(w) u J(w) G (w) F (w) H (w) H (w)

T T T T

LPV STABLE B(w) C(w)

Figure: Quadratic stability (Zhou-Doyle-Glover and Sznaier-Sanchez Pena)

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV Stabilizing compensator structure

u y x

• +

+

+ + − +

ν

OBSERVER PLANT

• ESTIM. STATE FEEDBACK

C(w)

ustab

G (w)

SF

F (w)

SF

H (w)

SF

K (w)

SF

Q(w) RB(w) A(w) −L(w) B(w) C(w) M

T(s,w)

Figure: Observer-based controller parametrization.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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LPV Stabilizing compensator structure

u y x

• +

+

+ + − +

ν

OBSERVER PLANT

• ESTIM. STATE FEEDBACK

C(w)

ustab

G (w)

SF

F (w)

SF

H (w)

SF

K (w)

SF

Q(w) RB(w) A(w) −L(w) B(w) C(w) M K (w)

T

F (w)

T

G (w)

T

H (w)

T

K (w)

LPV stable

Figure: Observer-based controller parametrization.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Compensator structure, quadratic stabilizability

The Youla-Kucera Parameter is given by F (T)(w) G (T)(w) H(T)(w) K (T)(w)

• =

R(w) I

• ×

  A(w)+B(w)K(w)C(w) B(w)H(w) G(w)C(w) F(w) B(w)K(w)−L(w) G(w) −J(w)+K(w)C(w) H(w) K(w)  × R−1(w) I

• Blanchini Casagrande Miani Viaro

Stable LPV realization for LPV design

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Comments

Comments

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Comments

Comments

1 No upper bounds for the compensator complexity; Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Comments

Comments

1 No upper bounds for the compensator complexity; 2 For quadratic stability, the compensator order is 2n+nc where

nc is the “original compensator” order.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Comments

Comments

1 No upper bounds for the compensator complexity; 2 For quadratic stability, the compensator order is 2n+nc where

nc is the “original compensator” order.

3 For compensators based on Youla–Kucera parameter

• ptimization the algorithm is direct.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Comments

Comments

1 No upper bounds for the compensator complexity; 2 For quadratic stability, the compensator order is 2n+nc where

nc is the “original compensator” order.

3 For compensators based on Youla–Kucera parameter

• ptimization the algorithm is direct.

4 Suitable for interpolation-based schemes. Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 1: on–line parameter tuning

+ + − r(t) d(t) P(s) ω

Figure: LTI plant with notch filter

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 1: on–line parameter tuning

+ + − r(t) d(t) P(s) ω

Figure: LTI plant with notch filter

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 1: on–line parameter tuning

5 10 15 −10 10

ylpv

5 10 15 −50 50

ymin

5 10 15 10 20 30

ω

Figure: ylpv: output with the proposed realization; ymin: output corresponding minimal realization of the compensator; ω: tuning parameter subject to a piecewise sinusoidal variation.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 2: LPV stability within the Hurwitz region

A(w) B(w) C(w)

• =

  1 −(1+ρw) −ξ 1 1   PI controller Wcomp(s) = k + h

s

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 2: LPV stability within the Hurwitz region

A(w) B(w) C(w)

• =

  1 −(1+ρw) −ξ 1 1   PI controller Wcomp(s) = k + h

s

for time varying 0 ≤ w(t) ≤ 1 stability is assured;

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 2: LPV stability within the Hurwitz region

A(w) B(w) C(w)

• =

  1 −(1+ρw) −ξ 1 1   PI controller Wcomp(s) = k + h

s

for time varying 0 ≤ w(t) ≤ 1 stability is assured; the proportional and integral gain can be changed online, within the region for which stability is assured for all fixed w R = {h ≥ ε and h ≤ ξ(1+k)−ε}

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 2: LPV stability within the Hurwitz region

A(w) B(w) C(w)

• =

  1 −(1+ρw) −ξ 1 1   PI controller Wcomp(s) = k + h

s

for time varying 0 ≤ w(t) ≤ 1 stability is assured; the proportional and integral gain can be changed online, within the region for which stability is assured for all fixed w R = {h ≥ ε and h ≤ ξ(1+k)−ε} Extension to LPV stability!

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Application 3: Pointwise optimality and LPV stability

˙ x(t) =     1 1 1 w(t)    x(t)+     1    u(t), y(t) =

• 1
• x(t),

w(t) ∈ [20,980]. Cost function: J =

∞

0 x(t)2 +0.01|u(t)|2dt

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Pointwise optimality and LPV stability

Figure: System output with the LQG controller applied directly (solid line) and with the LQG controller realized according to the proposed technique (dashed line).

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Robust design

u y A(w) B(w) C(w) K(w) w G(w) H(w) F(w) ~ ~ ~ ~

Figure: Robust control

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Robust design

u y A(w) B(w) C(w) K(w) w G(w) H(w) F(w) ~ ~ ~ ~

Figure: Robust control

No equivalence (unless for state feedback) between robust and LPV stabilizability

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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Robust design

u y A(w) B(w) C(w) K(w) w G(w) H(w) F(w) ~ ~ ~ ~

Figure: Robust control

No equivalence (unless for state feedback) between robust and LPV stabilizability No separation principles, no duality.

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

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The end

THANKS !

Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design