w w 0 G ( z 0 , w 0 ) = 0 w : ( w, - - PDF document

Evaluating regions of attraction of LTI systems with saturation in IQS framework

Dimitri Peaucelle Sophie Tarbouriech Martine Ganet-Schoeller Samir Bennani 7th IFAC Symposium on Robust Control Design / Aalborg / June 20, 2011

Introduction

■ Integral Quadratic Separation framework ■ Launcher attitude control: local stability of linear system with saturation ■ IQS methodology for the given problem and results

1 IFAC ROCOND / Aalborg / June 20-22, 2012

Topological separation - [Safonov 80]

■ Well-posedness of a feedback loop

F(w, z) = z z z w w G(z, w) = w

• Uniqueness and boundedness of internal signals for all bounded disturbances

∃γ : ∀( ¯ w, ¯ z) ∈ L2 × L2 ,

• w − w0

z − z0

• ≤ γ
• ¯

w ¯ z

• ,

G(z0, w0) = 0 F(w0, z0) = 0

■ iff exists a topological separator θ

• Negative on the inverse graph of the other component
• Positive definite on the graph of one component of the loop

GI( ¯ w) = {(w, z) : G(z, w) = ¯ w} ⊂{ (w, z) : θ(w, z) ≤ φ2(|| ¯ w||)} F(¯ z) = {(w, z) : F(w, z) = ¯ z} ⊂{ (w, z) : θ(w, z) > −φ1(||¯ z||)} L Issues: How to choose θ ? How to test the separation inequalities ?

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Integral Quadratic Separation (IQS)

■ Choice of an Integral Quadratic Separator θ(w, z) =

• z

w

• Θ
• z

w

• =

∞

• zT (t)

wT (t)

• Θ(t)
• z(t)

w(t)

• dt
• Identical choice to IQC framework [Megretski, Rantzer, J¨
• nsson]

■ IQS is necessary and sufficient under assumptions (proof based on [Iwasaki 2001])

• One component is a linear application, can be descriptor form F(w, z) = Aw − Ez

L can be time-varying A(t)w(t)−E(t)z(t) or frequency dep. ˆ A(ω) ˆ w(ω)− ˆ E(ω)ˆ z(ω) L A(t), E(t) are bounded and E(t) = E1(t)E2 where E1(t) is full column rank

• The other component can be defined in a set

G(z, w) = ∇(z) − w , ∇ ∈ ∇ ∇ L ∇ ∇ must have a linear-like property ∀(z1, z2) , ∀∇ ∈ ∇ ∇ , ∃ ˜ ∇ ∈ ∇ ∇ : ∇(z1) − ∇(z2) = ˜ ∇(z1 − z2) ■ The matrix Θ must satisfy an IQC over ∇ ∇ + an LMI involving (E, A)

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Examples - Topological Separation and Lyapunov

■ Global stability of a non-linear system ˙ x = f(x, t)

F(w, z) = z z z w w G(z, w) = w

G(z = ˙ x, w = x) = t

0 z(τ)dτ − w(t),

F(w, z, t) = f(w, t) − z(t)

• ¯

w plays the rle of the initial conditions, ¯ z are external disturbances

• Well-posedness: for all bounded initial conditions and all bounded disturbances,

the state remains bounded around the equilibrium ≡ global stability

■ For linear systems ˙ x(t) = A(t)x(t), ∇ = s−11, s−1 ∈ C+

• IQS: θ(w, z) =

∞

• zT (t)

wT (t)

• ⎡

⎣ −P(t) −P(t) − ˙ P(t) ⎤ ⎦ ⎛ ⎝ z(t) w(t) ⎞ ⎠ dt L θ(w, z) ≤ 0 for all G(z, w) = 0 iff P(t) ≥ 0 L θ(w, z) > 0 for all F(w, z) = 0 iff AT (t)P(t) + P(t)A(t) + ˙ P(t) < 0

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Examples - Topological Separation and Lyapunov

■ Global stability of a system with a dead-zone

F(w, z) = z z z w w G(z, w) = w

G1( ˙ x, x) = t

0 ˙

x(τ)dτ − x(t), G2(g, v) = dz(g(t)) − v(t), F1(x, v, ˙ x, t) = f1(x, v, t) − ˙ x(t), F2(x, v, g, t) = f2(x, v, t) − g(t)

1 w 1 z

■ IQS applies for linear f1, f2

• Dead-zone embedded in a sector uncertainty ∇

∇∞ = {∇∞ : 0 ≤ ∇∞(g) ≤ g} GI

2 = {(v, g) : G2(g, v) = 0} ⊂{ (v, g) : v = ∇∞(g) , ∇∞ ∈ ∇

∇∞} L This is the only source of conservatism

• LMI conditions obtained for the IQS defined by

Θ = ⎡ ⎢ ⎢ ⎣

−P −p1 −P −p1 2p1

⎤ ⎥ ⎥ ⎦,

P > 0, p1 > 0.

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Launcher model

■ Launcher in ballistic phase : attitude control

• neglected atmospheric friction, sloshing modes, ext. perturbation, axes coupling: I ¨

θ = T

• Saturated actuator: T = sat ¯

T (u) = u − ¯

Tdz( 1

¯ T u)

• PD control u = −KP θ − KD ˙

θ ■ Global stability LMI test fails L Sector uncertainty includes ∇∞ = 1 for which the system is I ¨ θ = 0 (unstable)

• LMIs succeeds (whatever ¯

g < ∞) if dead-zone is restricted to belong to ∇ ∇¯

g = {∇¯ g : 0 ≤ ∇¯ g(g) ≤ 1−¯ g ¯ g g}

• L Useful if one can prove for constrained x(0) that |g(θ)| ≤ ¯

g holds ∀θ ≥ 0. ■ How can one prove local properties in IQS framework ?

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Initial conditions dependent IQS

■ Well-posedness of a feedback loop

F(w, z) = z z z w w G(z, w) = w

• Uniqueness and boundedness of internal signals for all bounded disturbances

∃γ : ∀( ¯ w, ¯ z) ∈ L2 × L2 ,

• w − w0

z − z0

• ≤ γ
• ¯

w ¯ z

• ,

G(z0, w0) = 0 F(w0, z0) = 0

L How to introduce initial conditions x(0) and “final” conditions g(θ) in IQS framework? ■ Square-root of the Dirac operator: linear operator such that x → ϕθx : < ϕθx|Mϕθx >= ∞

0 ϕθxT (t)Mϕθx(t)dt = xT (θ)Mx(θ)

< ϕθ1x|Mϕθ2x >= 0 if θ1 = θ2

• Such operator is also used for PDE to describe states on the boundary

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Initial conditions dependent IQS

■ System with initial and final conditions writes as ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ϕ0x Tθ ˙ x Tθg ϕθg ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 A B C C ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Tθx ϕθx Tθv ϕ0x ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ L Tθx is the truncated signal such that Tθx(t) = x(t) for t ≤ θ and = 0 for t > θ.

• The integration operator is redefined as a mapping of (ϕ0x, Tθ ˙

x) to (Tθx, ϕθx).

• Restricted sector constraint assumed to hold up to t = θ (i.e. Tθv = ∇¯

gTθg)

• Goal is to find sets 1 ≥ xT (0)Qx(0) =< ϕ0x|Qϕ0x > s.t. g(θ) = ϕθg < ¯

g. ■ Problem defined in this way is a well-posedness problem with ∇ composed of 3 blocs

• IQS framework applies and gives conservative LMI conditions
• Equivalent to LaSalle invariance principle with V (x) = xT Qx (ellipsoidal sets of IC)

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System augmentation with derivatives

■ How to reduce conservatism ?

• Needed a description of the dead-zone better than sector uncertainty
• Needed to have dead-zone dependent sets of initial conditions

■ Both features derived via descriptor modeling of system augmented with ˙ v and ˙ g v = dz(g) : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ 

if g > 1

v = g − 1 ˙ v = ˙ g

if |g| ≥ 1

v = 0 ˙ v = 0

if g < −1

v = g + 1 ˙ v = ˙ g

• For IQS, link between ˙

v and ˙ g is embedded in ˙ v = ∇{0,1} ˙ g, with ∇{0,1} ∈ {0, 1}.

• Also needed to specify that v is the integral of ˙

v (thus descriptor form)

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System augmentation with derivatives

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 1 1 1 1 1 −C 1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

ϕ0x ϕ0v Tθ ˙ x Tθ ˙ v Tθg ϕθg Tθ ˙ g ϕθg

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 1 A B 1 C C C 1 −1 1 −1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Tθx Tθv ϕθx ϕθv Tθv ϕθv Tθ ˙ v ϕ0x ϕ0v

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ■ Problem defined in this way is a well-posedness problem with ∇ composed of 5 blocs

• IQS framework applies and gives less conservativeLMI conditions
• Equivalent to LaSalle invariance principle with

V (x) = ⎛ ⎝ x v ⎞ ⎠

T

Qa ⎛ ⎝ x v ⎞ ⎠ = ⎛ ⎝ x dz(Cx) ⎞ ⎠

T

Qa ⎛ ⎝ x dz(Cx) ⎞ ⎠

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Application to the launcher model

■ LMIs tested on the launcher example

• Sets of initial conditions for which |g(θ)| ≤ 8 is guaranteed
• Improvement thanks to piecewise quadratic sets of initial conditions

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Conclusions

■ IQS framework can handle local stability issues

• Provides LMI tests - conservative
• System augmentation + descriptor modeling = reduction of conservatism

■ Prospectives

• Improved construction of the IQS ≡ “generalized sector conditions”
• Further system augmentation with higher derivatives (?)
• Simultaneous handling of saturation, uncertainties, delays...

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