evaluating regions of attraction of lti systems with saturation...
TRANSCRIPT
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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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