Integralnoe kvadratiqnoe razdelenie: opisanie podhoda, svz s funkcimi - - PowerPoint PPT Presentation

Integralьnoe kvadratiqnoe razdelenie:

• pisanie podhoda, svzь s funkcimi Lpunova i S-proceduro

i kuboviqa, priloжenie k analizu usto iqivosti sputnika s ograniqeniem po vhodu

Dimitri PEAUCELLE / Dmitri

i Жanoviq Poselь -Konovalov

LAAS-CNRS - Université de Toulouse - FRANCE

Sankt-Peterburg Ma i 2013

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

Dimitri Peaucelle Sophie Tarbouriech Martine Ganet-Schoeller Samir Bennani Presented first at 7th IFAC Symposium on Robust Control Design / Aalborg

Introduction

■ Saturated control of a linear system ˙ x = Ax + Bu , u = sat(Ky) , y = Cx

• Assume K designed for the linear system (no saturation)
• System with saturation: Stability is (in general) only local
• Problem: find (largest possible) set of x(0) such that x(∞) = 0

■ Goal of this presentation : formalize the problem in the IQS framework

• Can "system augmentation" relaxations provide less conservative results ?
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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

• ,
• with

   G(z0, w0) = 0 F(w0, z0) = 0

solution to the system without perturbations

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Topological separation - [Safonov 80]

■ Well-posedness of a feedback loop

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

■ Theorem: Well-posed iff exists a topological separator θ

• ‘Negative’ on the inverse graph of one component

GI( ¯ w) = {(w, z) : G(z, w) = ¯ w} ⊂ {(w, z) : θ(w, z) ≤ φ2(|| ¯ w||)}

• ‘Positive definite’ on the graph of the other component of the loop

F(¯ z) = {(w, z) : F(w, z) = ¯ z} ⊂ {(w, z) : θ(w, z) > −φ1(||¯ z||)} ▲ Issue 1: How to choose θ ? Answer: S-procedure. ▲ Issue 2: How to test the separation inequalities ? Answer: LMIs.

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Example : the small gain theorem

■ Well-posedness of a feedback loop

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

• In case of causal G(z, w) : w = ∆z , ∆ ∈ RHm×l

∞

and stable proper LTI F(w, z) : z = H(s)w

• Necessary and sufficient (lossless) choice of separator

θ(w, z) = w2 − γ2z2

• Separation inequalities:

θ(w, z) = w2 − γ2z2 ≤ 0 , ∀w = ∆z ⇔ ∆2

∞ ≤ γ2

θ(w, z) = w2 − γ2z2 > 0 , ∀z = H(s)w ⇔ H2

∞ < 1

γ2

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Example : stability of passive interconnected systems

■ Well-posedness of a feedback loop

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

• In case of passive G(z, w) : w = ∆z

and stable LTI F(w, z) : z = H(s)w

• Necessary and sufficient (lossless) choice of separator

θ(w, z) = − < w|z >

• Separation inequalities:

θ(w, z) = − < w|z >≤ 0 , ∀w = ∆z ⇔ ∞ wT (t)z(t)dt ≥ 0 θ(w, z) = − < w|z > 0 , ∀z = H(s)w ⇔ H∗(jω) + H(jω) < 0 , ∀ω

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Integral Quadratic Separation (IQS)

■ From topological separation to 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, Jönsson]

θ(w, z) = +∞

−∞

• zT (jω)

wT (jω)

• Π(jω)
• z(jω)

w(jω)

• dω

▲ Π is called a multiplier. θ(w, z) ≤ 0 is called an IQC. ▲ Conservatism reduction in IQC framework : ω-dependent multipliers: Π(jω) =

• 1

Ψ1(jω)∗ · · · Ψr(jω)∗

• ˆ

Π         1 Ψ1(jω)

. . .

Ψr(jω)        

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Integral Quadratic Separation (IQS)

■ Main IQS result (both for ω or t or k dependent signals) ■ 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

▲ 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) ▲ 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 , ∇ ∈ ∇ ∇ ▲ ∇ ∇ must have a linear-like property ∀(z1, z2) , ∀∇ ∈ ∇ ∇ , ∃ ˜ ∇ ∈ ∇ ∇ : ∇(z1) − ∇(z2) = ˜ ∇(z1 − z2) ▲ ∇ ∇ does not need to be causal ■ 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 role 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

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Examples - Topological Separation and Lyapunov

■ Global stability of a linear TV system ˙ x = A(t)x

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

G(z = ˙ x, w = x) = t

0 z(τ)dτ − w(t) = s−1z − w,

F(w, z, t) = A(t)w(t) − z(t)

• IQS: θ(w, z) =

∞

• zT (t)

wT (t)

• 

 −P(t) −P(t) − ˙ P(t)     z(t) w(t)   dt ▲ θ(w, z) ≤ 0 for all G(z, w) = 0 iff P(t) ≥ 0

• x(0) = 0 ,

t ( ˙ xT Px + xT ˙ Px + xT P ˙ x)dτ = xT (t)P(t)x(t)

• ▲ θ(w, z) > 0 for all F(w, z) = 0 iff AT (t)P(t) + P(t)A(t) + ˙

P(t) < 0

• zT Pw + wT ˙

Pw + wT Pw = wT (AT P + PA + ˙ P)w

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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

• Dead-zone embedded in a sector uncertainty ∇

∇∞ = {∇∞ : 0 ≤ ∇∞(g) ≤ g} GI

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

∇∞} ▲ Choosing θ IQS w.r.t. ∇ ∇∞ rather than w.r.t GI

2, is a source of conservatism

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Examples - Topological Separation and Lyapunov

■ IQS applies for linear f1, f2 ■ 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) = Ax(t) + Bv(t) − ˙ x(t), F2(x, v, g, t) = Cx(t) + Dv(t) − g(t)

• LMI conditions obtained for the IQS defined by

Θ =    

−P −p1 −P −p1 2p1

   ,

P > 0, p1 > 0.

• Result is exactly identical to circle theorem result
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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

G1( ˙ x, x) = t

0 ˙

x(τ)dτ − x(t), G2(g, v) = dz(g(t)) − v(t), F1(x, v, ˙ x, t) =   1 −KP −KD   x(t) +   − ¯ T   v(t) − ˙ x(t), F2(x, v, g, t) =

• − KP

¯ T

− KD

¯ T

• x(t) − g(t)
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Launcher model

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

• LMI test succeeds (whatever ¯

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

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

▲ 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

▲ 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         ▲ 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

  Tθx ϕθx   = I   ϕ0x Tθ ˙ x  

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Initial conditions dependent IQS

        ϕ0x Tθ ˙ x Tθg ϕθg         =         1 A B C C                 Tθx ϕθx Tθv ϕ0x        

• Restricted sector constraint assumed to hold up to t = θ:

Tθv = ∇¯

gTθg z z !1 1!z w 1

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Initial conditions dependent IQS

        ϕ0x Tθ ˙ x Tθg ϕθg         =         1 A B C C                 Tθx ϕθx Tθv ϕ0x        

• Goal is to prove the restricted sector condition holds strictly at time θ (whatever θ)

▲ i.e. find sets 1 ≥ xT (0)Qx(0) =< ϕ0x|Qϕ0x > s.t. |g(θ)| = ϕθg < ¯ g ▲ reformulated as well posedness problem where ϕ0x = ∇ciϕθg defined by wci = ∇cizzi : ¯ g2 < wci|Qwci >≤ zci2 ∇ci is a non-causal, virtual, operator, used to define the problem in IQS framework

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Initial conditions dependent IQS

        ϕ0x Tθ ˙ x Tθg ϕθg         =         1 A B C C                 Tθx ϕθx Tθv ϕ0x         ■ Problem defined in this way is a well-posedness problem with ∇ composed of 3 blocs ∇ =      I ∇¯

g

∇ci     

• 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

• All system equations:

      

Tθ ˙ x = ATθx + BTθv Tθg = CTθx Tθ ˙ g = CTθ ˙ x ϕθg = Cϕθx

,    

Tθx Tθv ϕθx ϕθv

    = I    

ϕ0x ϕ0v Tθ ˙ x Tθ ˙ v

    ,

Tθv = ∇¯

gTθg

ϕθv = ∇¯

gϕθg

Tθ ˙ v = ∇{0,1}Tθ ˙ g   ϕ0x ϕ0v   = ∇ciϕθg

• Gives a descriptor matrix linear transformation

          

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

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System augmentation with derivatives

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

V (x) =   x v  

T

Qa   x v   =   x dz(Cx)  

T

Qa   x dz(Cx)   ▲ Such result cannot be obtained when applying classical IQC results

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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...
• Hybrid systems ?
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