Evaluating regions of attraction of LTI systems with saturation...
Transcript of Evaluating regions of attraction of LTI systems with saturation...
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