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

Introduction

n Saturated control of a linear system

x = Ax+Bu , u = sat(Ky) , y = Cx

l Assume K designed for the linear system (no saturation)

l System with saturation: Stability is (in general) only local

l Problem: find (largest possible) set of x(0) such that x(∞) = 0

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

l Can ”system augmentation” relaxations provide less conservative results ?

1 SKIMAC / Jan 22-24, 2013

Topological separation - [Safonov 80]

n Well-posedness of a feedback loopF(w, z) = z

z

z

w

w

G(z, w) = w

l 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

n iff exists a topological separator θ

l Negative on the inverse graph of one component

l Positive definite on the graph of the other 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||)}

s Issues: How to choose θ ? How to test the separation inequalities ?

2 SKIMAC / Jan 22-24, 2013

Example : the small gain theorem

n Well-posedness of a feedback loopF(w, z) = z

z

z

w

w

G(z, w) = w

l In case of causal G(z, w) : w = ∆z , ∆ ∈ RHm×l∞and stable proper LTI F (w, z) : z = H(s)w

l Necessary and sufficient (lossless) choice of separator

θ(w, z) = ‖w‖2 − γ2‖z‖2

l Separation inequalities:

θ(w, z) = ‖w‖2 − γ2‖z‖2 ≤ 0 , ∀w = ∆z ⇔ ‖∆‖2∞ ≤ γ2

θ(w, z) = ‖w‖2 − γ2‖z‖2 > 0 , ∀z = H(s)w ⇔ ‖H‖2∞ <1

γ2

3 SKIMAC / Jan 22-24, 2013

Integral Quadratic Separation (IQS)

n Choice of an Integral Quadratic Separator

θ(w, z) =

⟨(z

w

) ∣∣∣∣∣Θ(z

w

)⟩=

∫ ∞0

(zT (t) wT (t)

)Θ(t)

(z(t)

w(t)

)dt

l Identical choice to IQC framework [Megretski, Rantzer, Jonsson]

θ(w, z) =

∫ +∞

−∞

(zT (jω) wT (jω)

)Π(jω)

(z(jω)

w(jω)

)dω

s Π is called a multiplier. θ(w, z) ≤ 0 is called an IQC.

s Conservatism reduction in IQC framework : ω-dependent multipliers:

Π(jω) =[1 Ψ1(jω)∗ · · · Ψr(jω)∗

]Π

1

Ψ1(jω)...

Ψr(jω)

4 SKIMAC / Jan 22-24, 2013

Integral Quadratic Separation (IQS)

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

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

l One component is a linear application, can be descriptor form F (w, z) = Aw−Ezs can be time-varyingA(t)w(t)−E(t)z(t) or frequency dep. A(ω)w(ω)−E(ω)z(ω)

sA(t), E(t) are bounded and E(t) = E1(t)E2 where E1(t) is full column rank

l The other component can be defined in a set

G(z, w) = ∇(z)− w , ∇ ∈ ∇∇

s∇∇ must have a linear-like property

∀(z1, z2) , ∀∇ ∈ ∇∇ , ∃∇ ∈ ∇∇ : ∇(z1)−∇(z2) = ∇(z1 − z2)

s∇∇ need not to be causal

n The matrix Θ must satisfy an IQC over∇∇ + an LMI involving (E ,A)

5 SKIMAC / Jan 22-24, 2013

Examples - Topological Separation and Lyapunov

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

F(w, z) = z

z

z

w

w

G(z, w) = wG(z = x, w = x) =

∫ t0 z(τ)dτ − w(t),

F (w, z, t) = f(w, t)− z(t)

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

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

the state remains bounded around the equilibrium≡ global stability

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

l IQS: θ(w, z) =∫∞

0

(zT (t) wT (t)

) 0 −P (t)

−P (t) −P (t)

z(t)

w(t)

dt

s θ(w, z) ≤ 0 for all G(z, w) = 0 iff P (t) ≥ 0

s θ(w, z) > 0 for all F (w, z) = 0 iff AT (t)P (t) + P (t)A(t) + P (t) < 0

6 SKIMAC / Jan 22-24, 2013

Examples - Topological Separation and Lyapunov

n 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

0x(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

−1z

n IQS applies for linear f1, f2

l Dead-zone embedded in a sector uncertainty∇∇∞ = {∇∞ : 0 ≤ ∇∞(g) ≤ g}

GI2 = {(v, g) : G2(g, v) = 0} ⊂ {(v, g) : v = ∇∞(g) , ∇∞ ∈ ∇∇∞}

s This is the only source of conservatism

l LMI conditions obtained for the IQS defined by

Θ =

0 0 −P 0

0 0 0 −p1−P 0 0 0

0 −p1 0 2p1

, P > 0,

p1 > 0.

7 SKIMAC / Jan 22-24, 2013

Launcher model

n Launcher in ballistic phase : attitude control

l neglected atmospheric friction, sloshing modes, ext. perturbation, axes coupling: Iθ = T

l Saturated actuator: T = satT (u) = u− Tdz( 1Tu)

l PD control u = −KP θ −KDθ

n Global stability LMI test fails

s Sector uncertainty includes∇∞ = 1 for which the system is Iθ = 0 (unstable)

l LMI test succeeds (whatever g <∞) if dead-zone is restricted to belong to

∇∇g = {∇g : 0 ≤ ∇g(g) ≤ 1−gg g}

zz

!1

1!zw

1

s Useful if one can prove for constrained x(0) that |g(θ)| ≤ g holds ∀θ ≥ 0.

n How can one prove local properties in IQS framework ?

8 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS

n Well-posedness of a feedback loopF(w, z) = z

z

z

w

w

G(z, w) = w

l 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

s How to introduce initial conditions x(0) and “final” conditions g(θ) in IQS framework?

n Square-root of the Dirac operator: linear operator such that

x 7→ ϕθx :< ϕθx|Mϕθx >=

∫∞0 ϕθx

T (t)Mϕθx(t)dt = xT (θ)Mx(θ)

< ϕθ1x|Mϕθ2x >= 0 if θ1 6= θ2

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

9 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS

n System with initial and final conditions writes asϕ0x

Tθx

Tθg

ϕθg

=

0 0 0 1

A 0 B 0

C 0 0 0

0 C 0 0

Tθx

ϕθx

Tθv

ϕ0x

s Tθx is the truncated signal such that Tθx(t) = x(t) for t ≤ θ and = 0 for t > θ.

l The integration operator is redefined as a mapping

Tθxϕθx

= I

ϕ0x

Tθx

10 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS

ϕ0x

Tθx

Tθg

ϕθg

=

0 0 0 1

A 0 B 0

C 0 0 0

0 C 0 0

Tθx

ϕθx

Tθv

ϕ0x

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

Tθv = ∇gTθgz

z!1

1!zw

1

11 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS

ϕ0x

Tθx

Tθg

ϕθg

=

0 0 0 1

A 0 B 0

C 0 0 0

0 C 0 0

Tθx

ϕθx

Tθv

ϕ0x

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

s i.e. find sets 1 ≥ xT (0)Qx(0) =< ϕ0x|Qϕ0x > s.t. |g(θ)| = ‖ϕθg‖ < g

s reformulated as well posedness problem where ϕ0x = ∇ciϕθg defined by

wci = ∇cizzi : g2 < wci|Qwci >≤ ‖zci‖2

12 SKIMAC / Jan 22-24, 2013

Initial conditions dependent IQS

ϕ0x

Tθx

Tθg

ϕθg

=

0 0 0 1

A 0 B 0

C 0 0 0

0 C 0 0

Tθx

ϕθx

Tθv

ϕ0x

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

∇ =

I

∇g∇ci

l IQS framework applies and gives conservative LMI conditions

l Equivalent to LaSalle invariance principle with V (x) = xTQx (ellipsoidal sets of IC)

13 SKIMAC / Jan 22-24, 2013

System augmentation with derivatives

n How to reduce conservatism ?

l Needed a description of the dead-zone better than sector uncertainty

l Needed to have dead-zone dependent sets of initial conditions

n 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

l For IQS, link between v and g is embedded in v = ∇{0,1}g, with∇{0,1} ∈ {0, 1}.l Also needed to specify that v is the integral of v (thus descriptor form)

14 SKIMAC / Jan 22-24, 2013

System augmentation with derivatives

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 −C 0 0 0 1 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

ϕ0x

ϕ0v

Tθ xTθ v

Tθg

ϕθg

Tθ g

ϕθg

=

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

A B 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0

C 0 0 0 0 0 0 0 0

0 0 C 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 C 0 0 0 0 0 0

0 1 0 0 −1 0 0 0 0

0 0 0 1 0 −1 0 0 0

TθxTθvϕθx

ϕθv

Tθv

ϕθv

Tθ v

ϕ0x

ϕ0v

n Problem defined in this way is a well-posedness problem with∇ composed of 5 blocs

l IQS framework applies and gives less conservative LMI conditions

l Equivalent to LaSalle invariance principle with

V (x) =

x

v

T

Qa

x

v

=

x

dz(Cx)

T

Qa

x

dz(Cx)

15 SKIMAC / Jan 22-24, 2013

Application to the launcher model

n LMIs tested on the launcher example

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!"#!$

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l Sets of initial conditions for which |g(θ)| ≤ 8 is guaranteed

l Improvement thanks to piecewise quadratic sets of initial conditions

16 SKIMAC / Jan 22-24, 2013

Conclusions

n IQS framework can handle local stability issues

l Provides LMI tests - conservative

l System augmentation + descriptor modeling = reduction of conservatism

n Prospectives

l Improved construction of the IQS≡ “generalized sector conditions”

l Further system augmentation with higher derivatives (?)

l Simultaneous handling of saturation, uncertainties, delays...

l Hybrid systems ?

17 SKIMAC / Jan 22-24, 2013