[PPT] - Evaluating regions of attraction of LTI systems with saturation in PowerPoint Presentation

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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 Seminar SKIMAC 2013 - jan 22-24

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Introduction

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

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

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Topological separation - [Safonov 80]

■ Well-posedness of a feedback loop

∃γ : ∀( ¯ w, ¯ z) ∈ L2 × L2 ,

z − z0

w ¯ z

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

■ iff exists a topological separator θ

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||)} ▲ Issues: How to choose θ ? How to test the separation inequalities ?

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Example : the small gain theorem

■ Well-posedness of a feedback loop

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

θ(w, z) = w2 − γ2z2

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

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

γ2

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Integral Quadratic Separation (IQS)

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

w

w

∞

wT (t)

w(t)

θ(w, z) = +∞

wT (jω)

w(jω)

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

Ψ1(jω)∗ · · · Ψr(jω)∗

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])

▲ 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

G(z, w) = ∇(z) − w , ∇ ∈ ∇ ∇ ▲ ∇ ∇ must have a linear-like property ∀(z1, z2) , ∀∇ ∈ ∇ ∇ , ∃ ˜ ∇ ∈ ∇ ∇ : ∇(z1) − ∇(z2) = ˜ ∇(z1 − z2) ▲ ∇ ∇ need not 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)

G(z = ˙ x, w = x) = t

F(w, z, t) = f(w, t) − z(t)

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

the state remains bounded around the equilibrium ≡ global stability

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

∞

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 ▲ 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

G1( ˙ x, x) = t

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

∇∞ = {∇∞ : 0 ≤ ∇∞(g) ≤ g} GI

∇∞} ▲ This is the only source of conservatism

Θ =    

   ,

P > 0, p1 > 0.

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Launcher model

■ Launcher in ballistic phase : attitude control

θ = T

Tdz( 1

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

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

▲ 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

∃γ : ∀( ¯ w, ¯ z) ∈ L2 × L2 ,

z − z0

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 >= ∞

< ϕθ1x|Mϕθ2x >= 0 if θ1 = θ2

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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.

  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        

Tθv = ∇¯

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Initial conditions dependent IQS

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

▲ 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

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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 ∇¯

∇ci     

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System augmentation with derivatives

■ How to reduce conservatism ?

■ 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

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

v (thus descriptor form)

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System augmentation with derivatives

          

                  

        =           

                    

          ■ Problem defined in this way is a well-posedness problem with ∇ composed of 5 blocs

V (x) =   x v  

Qa   x v   =   x dz(Cx)  

Qa   x dz(Cx)  

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Application to the launcher model

■ LMIs tested on the launcher example

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Conclusions

■ IQS framework can handle local stability issues

■ Prospectives