Chap. 4. Singular Perturbations, Hierarchical Control
1. Singularly Perturbed Systems
1.1. Standard Form
ẋ1 = εf1(x1, x2, u, ε)
ẋ2 = f2(x1, x2, u, ε)
where dim x1 = n1, dim x2 = n2, dim u = m,
ε: “small” parameter, real > 0.
f1 and f2: C ∞ with respect to (x1, x2, u, ε).
The vector field f2 is, around an equilibrium point, of an order
of magnitude (in ε) larger than εf1.
Changing the time-scale: Set τ = εt (τ slower than t).
dx1
= f1(x1, x2, u, ε)
dτ
dx2 1
= f2(x1, x2, u, ε)
dτ
ε
• In the original time-scale t:
– the 1st component of the vector field, εf1, is 0(ε)
– the 2nd component, f2, is 0(ε0).
• In the slow time-scale τ = εt:
– the 1st component is of order 0 (0(ε0)),
– the 2nd component of order -1 (0(ε−1)).
Remark The standard form depends on the choice of coordinates.
Set:
z1 = x1 + x2, z2 = x1 − x2,
then
z1 + z2 z1 − z2
z1 + z2 z1 − z2
ż1 = εf1(
,
, u, ε) + f2(
,
, u, ε)
2
2
2
2
z1 + z2 z1 − z2
z1 + z2 z1 − z2
ż2 = εf1(
,
, u, ε) − f2(
,
, u, ε)
2
2
2
2
Theorem Consider the system
ẋ = F (x, u, ε)
with rank ∂F
∂x (x, u, 0) = n2 < dim x. In a neighborhood of the
equilibrium manifold Σ0 = {(x, u)|F (x, u, 0) = 0}, one can
find a diffeomorphism x = ϕ(x1, x2, ε) such that, in the new
coordinates, the system is expressed in standard form.
Proof: Consequence of the center manifold Theorem.
When ε = 0, Equilibrium Manifold:
Σ0 = {(x1, x2, u)|f2(x1, x2, u, 0) = 0}
For ε 6= 0:
Σε = {(x1, x2, u)|f1(x1, x2, u, ε) = 0, f2(x1, x2, u, ε) = 0}
thus dim Σε < dim Σ0 : singular perturbation.
By the Implicit Function Theorem, if, in a neighborhood of
∂f2
= n2, we have:
Σ0, rank ∂x
2
Σ0 = {(x1, x2, u)|x2 = X2(x1, u)} .
Moreover, Σ0 is an invariant manifold for ε = 0.
What can be said for ε small?
1.2. Persistance of the Invariant Manifold
For ε 6= 0, sufficiently small, and u slowly varying, i.e. u̇ = εv
with v such that supt∈R kv(t)k < +∞, set
Σ0,ε = {(x1, x2)|f2(x1, x2, u, ε) = 0}.
∂f2
= n2, there exists a smooth function X 2 such
Since rank ∂x
2
that
(x1, x2) ∈ Σ0,ε ⇐⇒ x2 = X 2(x1, u, ε).
We can verify that Σ0,ε is invariant by the slow dynamics:
ẋ1 = εf1(x1, X 2(x1, u, ε), u, ε)
u̇ = εv.
and remains close to Σ0 for small ε.
This property is called persistance.
Approximation:
∂f2
has all its eigenvalues with strictly
Theorem If the matrix ∂x
2
negative real part at every point (x1, x2, u, 0) of a neighborhood V (Σ0) of Σ0, for all sufficiently small ε, and if u is slowly
varying, the manifold Σ0,ε is approximated at the first order in
ε, in V (Σ0), by:
x2 = X2(x1, u)
−1 ∂f2
∂X2
∂X2
∂f2
+ε
f1 +
v−
(x1, X2(x1, u), u, 0)
∂x2
∂x1
∂u
∂ε
+0(ε2).
Moreover, the slow dynamics is given, at the first order in ε, by
ẋ1 = εf1(x1, X2(x1, u), u, 0) + 0(ε2).
f2 (x1 , x2 , u, ε)
( x1 , x2 )
f2 (x1 , x2 , u, 0) ( x1 , x2)
εf1 (x1 , X2 (x1 , u), u, 0)
Σ0,ε
ε small
Σ0
ε=0
“The Shadow Lemma” : For every initial condition in a
neighborhood of Σ0,ε, the integral curves fastly converge to
Σ0,ε, without necessarily entering in Σ0,ε, but are approximated
by trajectories remaining in Σ0,ε (the shadow) to which they
converge.
1.3. Robustness of Stability
For ε = 0: non hyperbolic system. However:
∂f2
Theorem Assume that ∂x
has all its eigenvalues with strictly
2
negative real part for all (x1, x2, u, 0) in V (Σ0).
Denote by F1(x1, u) = f1(x1, X2(x1, u), u, 0).
1 has all its eigenvalues with strictly negative
If the matrix ∂F
∂x1
real part at every point (x1, u) of a suitable open subset, then
the system is L-asymptotically stable for all ε ≥ 0 sufficiently
small, i.e. its stability is robust.
1.4. Application to Modelling
Interest: Reduction of the model dimension.
Example: DC motor
dI
L = U − RI − Kω
dt
dω
J
= KI − Kv ω − Cr .
dt
Inductance L small: L = ε.
Σ0 = {U − RI − Kω = 0}.
Slow Dynamics :
dω
J
=−
dt
K2
+ Kv
R
!
K
ω − Cr + U .
R
Σ0,ε given at the order 1 in L by
U − Kω
I=
R L K
U − Kω
+ 2
K
− Kv ω − Cr − U̇
R
R J
+0(L2)
Then for U = Kω, the current converges to
L
LK
I0 = − 2 (Kv ω + Cr ) − 2 U̇ + 0(L2)
JR
R
To measure the motor dry friction (friction force opposing to
the motor start up): it suffices to tune U̇ to have I0 = 0. We
thus obtain
J
Cr = U̇ + 0(L).
K
2. Hierarchical Control
2.1. Principle
Two possible basic designs depending on the use of “small
gains” (control of the slow dynamics) or “high gains” (control
of the slow dynamics through the fast ones).
2.1.1. Control of the Slow Dynamics
This technique is also called reduced model control.
We go back to the standard form
ẋ1 = εf1(x1, x2, u, ε)
ẋ2 = f2(x1, x2, u, ε)
∂f2
with all the eigenvalues of ∂x
with strictly negative real part at
2
every point (x1, x2, u, 0) of a neighborhood V (Σ0) of Σ0, for all
sufficiently small ε.
We have seen that if u is given by (small gain)
u̇ = εv
the fast state x2 may be replaced by its equilibrium point, approximated at the first order in ε, in V (Σ0), by:
x2 = X2(x1, u)
−1 ∂X2
∂X2
∂f2
∂f2
f1 +
v−
(x1, X2(x1, u), u, 0)
+ε
∂x2
∂x1
∂u
∂ε
+0(ε2).
where X2 is the local solution to f2(x1, x2, u, 0) = 0, and the
slow dynamics is given, at the first order in ε, by
ẋ1 = εf1(x1, X2(x1, u), u, 0) + 0(ε2),
u̇ = εv.
v is the new control to track a reference trajectory of x1 or, in
particular, stabilize x1 around 0.
2.1.2. Indirect Control of the Slow Dynamics by
High-Gain
Consider the 2-dimensional single input system:
ẋ1 = f1(x1, x2)
ẋ2 = f2(x1, x2) + u.
One can assign x2 to follow a given reference trajectory x?2 by
high gain control
k
u = − (x2 − x?2 )
ε
with ε small and k finite real positive. In fast time-scale τ = εt ,
dx1
= εf1(x1, x2)
dτ
dx2
= εf2(x1, x2) − k(x2 − x?2 )
dτ
Thus, since k > 0, x2 = x?2 + kε f2(x1, x?2 ) = x?2 + 0(ε) is an
attractive invariant manifold at the order 0 in ε.
Slow dynamics at the first order in ε:
dx1
= εf1(x1, x?2 )
dτ
The reference x?2 becomes the (fictitious) input of the slow
dynamics.
One can approach x2 = x?2 as close as one wants thanks to the
choice of ε.
u
u
system
low-level
subsystem
x2*
x1
high-level
subsystem
x1
2.3. Applications
Controller design of general purpose actuators. In particular:
• control of hydraulic jacks (e.g. for the orientation of the
wing flaps of an aircraft);
• control of electromagnetic valves (e.g. for the fluid flow rate
in the active dampers of a vehicle suspension, or for the inlet
products of a chemical reactor);
• control of the current in the coils of an electromagnet (e.g.
for the levitation of a vacuum pump rotating shaft);
• control of electric drives (AC or DC, synchronous or asynchronous) (e.g. for mechanical systems positionning: electric windows, windshield wipers, positionning tables in 2 or
3 dimensions, cranes, robots, etc.)
• etc.
Example: DC drive controlling the stirring rod of a chemical
reactor.
Cr
Cm
dI
L = U − RI − Kω
dt
dω
J
= KI − Kf ω 2.
dt
Inductance L small.
We want to follow the reference angular speed ω ∗, assumed to
be constant or slowly varying.
Low-level loop:
k1
U = − (I − vI )
L
Resulting slow dynamics:
dω
J
= KvI − Kf ω 2.
dt
High-level loop:
vI = vI∗ − k2(ω − ω ∗)
avec vI∗ = Kf (ω ∗)2. We get
dω
J
= −Kk2(ω − ω ∗) − Kf (ω + ω ∗)(ω − ω ∗)
dt
∗
= − Kk2 + Kf (ω + ω ) (ω − ω ∗)
which ensures the desired convergence.