CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 20 Years of Passivity–Based Control (PBC): Theory and Applications David Hill/Jun Zhao (ANU), Robert Gregg (UTDallas) and Romeo Ortega (LSS) Contents: Preliminaries on passivity (DH). PBC: History, main principles and recent developments (RO). Interconnection and Damping Assignment PBC (RO). PBC in bipedal locomotion (RG). Passivity of switched systems (JZ). CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 1/85
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
20 Years of Passivity–Based Control (PBC):Theory and Applications
David Hill/Jun Zhao (ANU), Robert Gregg (UTDallas)and Romeo Ortega (LSS)
Contents:
Preliminaries on passivity (DH).
PBC: History, main principles and recent developments (RO).
Interconnection and Damping Assignment PBC (RO).
PBC in bipedal locomotion (RG).
Passivity of switched systems (JZ).
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 1/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
What is PBC?
Why is passivity important?
For physical systems it is a restatement of energy conservation.
Is a natural generalization (to NL dynamical systems) of positivity of matrices andphase-shift of LTI systems—sign preserving property.
Term PBC introduced in
R. Ortega and M. Spong, Adaptive Motion Control Of Rigid Robots: A Tutorial,Automatica, Vol. 25, No. 6, 1989, pp. 877-888,
to define a controller methodology whose aim is to render the closed–loop passive.
It was done in the context of adaptive control of robot manipulators.
Natural, because mechanical systems and parameter estimators define passive maps.
The paper
has been cited more than 600 times and
is the 13th most highly cited paper out of 4250 published in Automatica since1989.
PBC has 2500 hits in Google scholar.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 2/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Passivity as a Design Tool: Foundational Results
(Moylan and Anderson, TAC’73): Optimal systems define passive maps. Nonlinearextension of Kalman’s inverse optimal control result.
(Fradkov, Aut and Rem Control’76): Necessary and sufficient conditions for passivationof LTI systems via state feedback.
(Takegaki and Arimoto, ASME JDSM&C’81), (Jonckheere, European Conf Circ. Th.and Design’81): Potential energy shaping and damping injection as design tools formechanical and electromechanical systems—new energy function as Lyapunovfunction.
(Kokotovic and Sussmann, S&CL’89): Stabilization of a NL system in cascade with anintegrator using positive realness.
(Ortega, Automatica’91): Extension to cascade of two NL systems using Hill/Moylantheorem.
(Byrnes, Isidori and Willems, TAC’91): Complete geometric characterization ofpassifiable systems—via minimum phase and relative degree conditions.
Backstepping and forwarding are passivation recursive designs that overcome theobstacles of relative degree and minimum phase for systems with special structures.See e.g., (Astolfi, Ortega and Sepulchre, EJC’02).
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
PBC Provides a Paradigm Shift for Controller Design
Definition We say that the m–port system with state x ∈ Rn, and power port variables
u, y ∈ Rm
Σ :
x = f(x) + g(x)u
y = h(x)
is cyclo–passive if there exists storage (energy) function H : Rn → R such that
H[x(t)]−H[x(0)]︸ ︷︷ ︸
stored energy
≤
∫ t
0u>(s)h(x(s))ds
︸ ︷︷ ︸
supplied energy
If H(x) = 0 then the system is passive with port variables (u, y) and storage function H(x).
Remark For passive systems we have
−
∫ t
0u>(s)y(s)ds ≤ H[x(0)] <∞ ⇒
amount of energy that can be extracted from a passive system is bounded.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Stabilization via Energy Shaping and Damping Injection
With u(t) ≡ 0, we have
H[x(t)] ≤ H[x(0)] ⇒
Trajectories tend to converge towards points of minimum energy
If the minima are strict H(x) qualifies as a Lyapunov function for them
To operate the system around some desired equilibrium point, say x∗, PBC shapes theenergy to assign a strict minimum at this point.
Furthermore, if we terminate the port with
u = −Kdiy, Kdi = K>di > 0
we get
H ≤ −y>Kdiy ≤ 0.
Hence, x(t) → 0 if h(x) is detectable (for the closed–loop system). That is, if
h(x(t)) ≡ 0 ⇒ x(t) → 0.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Ex. 1 State–feedback PBC: PI Control of Power Converters
A large class of power converters are modeled by
x =
(
J0 +
m∑
i=1
Jiui − R
)
∇H(x) +
(
G0 +
m∑
i=1
Giui
)
E (SW )
where x ∈ Rn is the converter state (typically containing inductor fluxes and capacitor
charges), u ∈ Rm denotes the duty ratio of the switches, the total energy stored in
inductors and capacitors is
H(x) = 12x>Qx , Q = Q> > 0
∇ = ∂∂x
, Ji = −J>i i ∈ m := 0, . . . ,m are the interconnection matrices,
R = R> ≥ 0 represents the dissipation matrix, and the vector Gi ∈ Rn contains the
(possibly switched) external voltage and current sources.
The control objective is to stabilize and equilibrium x? ∈ Rn.
It is desirable to propose simple, robust controllers.
(Hernandez, et al., IEEE TCST’09)
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
A Three-phase Rectifier
PSfrag replacements
E
vs
rL L
IC rc
is
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
An Incremental Passivity Property
Let x∗ ∈ Rn be an admissible equilibrium point, that is, x∗ satisfies
0 =
(
J0 +m∑
i=1
Jiu∗i − R
)
∇H(x∗) +
(
G0 +m∑
i=1
Giu∗i
)
E,
for some u∗ ∈ Rm. The incremental model of the system for the output y = Cx, where
C :=
E>G>1 − (x∗)>QJ1
...
E>G>m − (x∗)>QJm
Q ∈ R
m×n,
is passive. More precisely, the system verifies the dissipation inequality V ≤ y>u, wherey∗ = Cx∗ and the (positive definite) storage function is given by
V (x) :=1
2(x− x∗)>Q(x− x∗).
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Corollary: Global Asymptotic Stabilization with a PI
Consider a switched power converter described by (SW) in closed loop with the PI controller
z = −y
u = −Kpy +Kiz,
where Kp,Ki ∈ Rm×m are symmetric positive definite matrices, y = Cx. For all initial
conditions (x(0), z(0)) ∈ Rn+m the trajectories of the closed–loop system are bounded and
such that
limt→∞
Cx(t) = 0.
Moreover,
limt→∞
x(t) = x∗,
if y is detectable, that is, if for any solution x(t) of the system the following implication is true:
Cx(t) ≡ Cx∗ ⇒ limt→∞
x(t) = x∗.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Ex. 2 of Control by Interconnection: A Flexible Pendulum
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Ex. 3 PBC via Passive Subsystems Decomposition
Electromechanical Systems
u
i
y
g
m
λ
• λ is flux, θ position, u voltage
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Dynamic Behavior
Model (assuming linear magnetics, i.e., λ = L(θ)i)
λ+ Ri = u
mθ = F −mg
F =1
2
∂L
∂θ(θ)i2
Total energy: H =1
2
λ2
L(θ)︸ ︷︷ ︸
electrical, He(λ,θ)
+m
2θ2 +mgθ
︸ ︷︷ ︸
mechanical, Hm(θ,θ)
Rate of change of energies
He =λ
L(θ)(−Ri+ u)︸ ︷︷ ︸
λ
−1
2
λ2
L2(θ)︸ ︷︷ ︸
i2
∂L
∂θθ = −Ri2 + iu− F θ, Hm = θF
Adding up: H = −Ri2 + iu⇒ is cyclo–passive. However, H is not bounded frombelow, hence not passive!
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Passive Sub–Systems Feedback Decomposition
The maps Σe : (u, θ) → (i, F ) and Σm : (F −mg) → θ are passive (with storagefunction He ≥ 0, m
2θ2 ≥ 0, resp.)
u
F-mgy
λ
Σ
Σ
e
m
Designing a PBC that “views" Σm as a passive disturbance, suggests a nested-loopcontrol configuration
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Classical Nested–Loop Controller
Often employed in applications, where the inner–loop is designed “neglecting" themechanical part. This is rationalized via time–scale separation arguments.
yu
F
-mg
λ
Col
Fd
Cil Σe Σm
y*
Passivity provides a rigorous formalization of this approach, without this assumption,see (Ortega et al’s Book, ’98).
Adopting this perspective allows to prove that the industry standard field orientedcontrol of induction motors is a particular case of PBC. Hence, rigourously prove itsglobal stability.
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State–feedback PBC
Consider the system
Σ :
x = f(x) + g(x)u
y = h(x),
Definition [The set PBC] The state-feedback uSF : Rn → Rm is said to be a PBC
(uSF ∈ PBC) if and only if there exist functions Hd : Rn → R and hd : Rn → Rm such
that u = uSF + v renders the closed–loop system
Σd :
x = fd(x) + g(x)v, fd(x) := f(x) + g(x)uSF(x)
yd = hd(x)
cyclo-passive with storage function Hd(x). That is, if it verifies
Hd ≤ y>d v (DPE)
From Hill-Moylan’s Theorem, the new power balance becomes Hd = y>d v − dd, where
Remark Besides the energy and the dissipation, the output has also beenmodified—is a natural way to satisfy the vector relative degree requirement and toovercome the minimal phase restriction on the plant—corresponds to the addition of acurrent source h− hd.
(Castanos and Ortega, SCL’09)
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Electrical Circuit Analog of PBC
PSfrag replacements
Σ
Σd
+
+
–
–
vh− hd
y yd
u
uSF
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Energy-Balancing PBC is Output-Dissipation Preserving
The most natural desired storage function candidate is the difference between thestored and the supplied energies:
Hd(x(t)) = H(x(t))−
∫ t
0h>(x(s))uSF(s)ds.
Definition [Energy-Balancing] A PBC for the cyclo-passive system Σ is said to be EB(i.e., uSF ∈ PBC ∩ EB) if and only if
−y>uSF = Ha .
Proposition uSF ∈ PBC ∩ EB if and only if, the output and the dissipation remaininvariant. That is, if and only if (DPE) holds with
yd = y, dd = d.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
The Dissipation Obstacle
The storage function is typically used as a Lyapunov function, so it is required that
x? = argminHd .
Since ∇H?d = 0 is a necessary condition it is clear that y?d = 0 and d?d = 0.
EB PBCs, which preserve output and dissipation, impose to the open-loop system that
d? = −(∇H?)>f? = 0, y? = (g?)>∇H? = 0.
This is the so-called dissipation obstacle.
Extracted power should be zero at equilibrium ⇒ EB–PBC applicable only for systemswithout pervasive damping.
OK in regulation of mechanical syst. where power = F>q, but very restrictive forelectrical or electromechanical syst.: power = v>i.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Role of Dissipation
• Non–pervasive
1
1C
1L
1C
L
u
2R
q C
ϕL
Equilibria: (iL∗, vC∗) = (0, ∗) ⇒
zero extracted power!
• Pervasive1L
1Cu 2R 1
L
1CqC
ϕL
Only the dissipation has changed.
iL∗ 6= 0 ⇒ nonzero power
limt→∞
|
∫ t
0vC(s)iL(s)ds| = ∞
for any stabilizing controller (rundown the battery!)
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Application of EB–PBC to Mechanical Systems
EB–PBC are widely popular for potential energy shaping of mechanical systems. Inthis case
x =
q
p
, H(q, p) =1
2p>M−1(q)p+V (q), F =
0 I
−I −R
, g(q) =
0
G(q)
,
where (q, p) are the generalized coordinates and momenta, M =M> > 0 is theinertia matrix, R = R> ≥ 0 is the dissipation due to friction, G is the input matrix andV is the open–loop potential energy.
The added energy is Ha(q) = Vd(q)− V (q), Vd is the desired potential energy.
(EB-PDE)
g⊥F>
g>
∇Ha = 0 becomes
G⊥(∇Vd −∇V ) = 0,
known as the potential energy matching equation.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
EB–PBC: An Alternative Viewpoint
Given
x = f(x) + g(x)u, y = h(x).
If we can find a vector function u : Rn → Rm such that the PDE
(∂Ha
∂x(x)
)>
[f(x) + g(x)u(x)] = −h>(x)u(x),
can be solved for Ha : Rn → R, and Hd(x)4= H(x) +Ha(x), is such that
x? = argminHd(x), then, the state–feedback u = u(x) is an EBC, i.e., x? is stable withLyapunov function
Hd(x) = H(x)−
∫ t
0u>(x(s))h(x(s))ds.
Remark: A necessary condition for the solvability of the PDE is
f(x) + g(x)u(x) = 0 ⇒ h>(x)u(x)︸ ︷︷ ︸
power
= 0,
i.e. extracted power at the equilibrium x should be zero.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Mechanical Systems: Physical Interpretation
Passivity H ≤(
∂H∂p
)>G(q)u
Full actuation m = n,G(q) = I wecan assign any function of q withu(q) = − ∂Ha
∂q(q).
Asymptotic stability withv = −Kdi q, Kdi > 0
Underactuated case cannot solve
G(q)u(q) = −∂Ha
∂q(q)
equivalent to the matching equation.Need total energy shaping.
In (Ailon/Ortega, ’93) done withdynamic extension to inject dampingwithout measuring speeds.
q *
q *
k
k
d
p
q= mg
δ( )
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Flexible Pendulum
q c
q *
δ( )
q q
p1
p2
K 2
Rc
K 1m=1
D
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Overcoming the Dissipation Obstacle
Proposition [Interconnection and Damping Assignment (IDA PBC)] Fix
dd(x) = ∇H>d (x)Rd(x)∇Hd(x)
with Rd : Rn → Rn×n, Rd = R>
d ≥ 0.
(i) uSF ∈ PBC if and only if
g(x)uSF(x) = −f(x)− Rd(x)∇Hd(x) + α(x)
for some function α : Rn → Rn such that
α>∇Hd is identically zero,
α? = 0
(ii) For any Jd : Rn → Rn×n, Jd = −J>
d , the function α(x) = Jd(x)∇Hd(x), satisfiesboth restrictions: α? = 0 and α>∇Hd = 0. Furthermore, the closed-loop system, Σd,takes the port-Hamiltonian (PH) form
Σd :
x = Fd(x)∇Hd(x) + g(x)v , Fd(x) := Jd(x)−Rd(x)
yd = g>(x)∇Hd(x).
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Control by Interconnection
Control by interconnection viewpoint,
cΣ-
+
-
+ Σu c
yc y
uΣ I
Subsystems:
Σc (control) and Σ (plant) are PH systems
ΣI (interconnection).
Principle:Select ΣI such that we can “add" the energies of Σ and Σc.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Controllers by Interconnection are as Old as Control Itself
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They’re Pervasive and Efficient
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Adding Energies
Definition: The interconnection is power preserving if
Proposition:ΣI power preserving, Σ, Σc cyclo–passive with states x ∈ R
n, ζ ∈ Rnc , and
energy–functions H(x), Hc(ζ), resp. Then, interconnection cyclo–passive with newenergy–function
H(x) +Hc(ζ).
Problem:Although Hc(ζ) is free, not clear how to affect x?
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Invariant Function Method
Principle: Restrict the motion to a subspace of (x, ζ)
1
x (t)
x (o) x
x
2
ξ
Say Ωκ , (x, ζ)|ζ = F (x) + κ,(κ determined by the controllersICs).
Then, in Ω0,
Hd(x) , H(x) +Hc[F (x)]
It can be shaped selecting Hc(ζ).
Problem Let C(x, ζ) , F (x)− ζ.Finding F (·) that renders Ω
invariant ⇔
d
dtC|C=0 ≡ 0, (PDE)
Stymied by pervasive damping forsolution of (PDE) but can be over-come selecting new port variables.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Port–Hamiltonian Systems
PH model of a physical system
Σ(u,y) :
x = [J (x)−R(x)]∇H + g(x)u
y = g>(x)∇H
u>y has units of power (voltage–current, speed–force, angle–torque, etc.)
J = −J> is the interconnection matrix, specifies the internal power–conservingstructure (oscillation between potential and kinetic energies, Kirchhoff’s laws,transformers, etc.)
R = R> ≥ 0 damping matrix (friction, resistors, etc.)
g is input matrix.
PH systems are cyclo–passive
H = −∇H>R∇H + u>y.
Invariance of PH structure Power preserving interconnection of PH systems is PH.
Nice geometric structure formalized with notion of Dirac structures.
Most nonlinear cyclo–passive systems can be written as PH systems. Actually, in(network) modeling is the other way around!
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Basic CbI for PH Systems
Given a PH system,
Σ(u,y)
x = F (x)∇H(x) + g(x)u
y = g>(x)∇H(x),⇒ H ≤ u>y
where we defined F (x) := J (x)−R(x), J = −J>, R = R> ≥ 0.
PH controller (nonlinear integrators), ζ ∈ Rm
Σc :
ζ = uc
yc = ∇ζHc(ζ),⇒ Hc = u>c yc
Standard negative feedback interconnection
ΣI :
u
uc
=
0 −1
1 0
y
yc
+
v
0
⇒ H + Hc ≤ v>y
For ease of presentation, and with loss of generality, we have taken ζ ∈ Rm .
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Conditions for CbI
Proposition Assume there exists a vector function C : Rn → Rm such that
F>
g>
∇C =
g
0
(CbI − PDE)
Then, for all functions Φ : Rm → R, the following cyclo–passivity inequality is satisfied
W ≤ v>y,
where the shaped storage function W : Rn × Rm → R is defined as
W (x, ζ)4= H(x) +Hc(ζ) + Φ(C(x)− ζ).
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Comparison of CbI and State Feedback PBC: Applicability
(CbI)
F
g>
∇C =
−g
0
(CbISM)
g⊥F
g>
∇C = 0
(Basic CbIPS) F∇C = −g
(CbIPS) Fd∇C = −g plus (PO–PDE)(F∇H = Fd∇HPS)
(Basic CbISMPS
)
g⊥F∇C = 0
(CbISMPS
)
g⊥Fd∇C = 0
plus (PO-PDE).
(EBC)
g⊥F
g>
∇Ha = 0
(Basic IDA)
g⊥F∇Ha = 0
(PS)
g⊥Fd∇Ha = 0
plus (PO–PDE)
(IDA)
g⊥Fd∇Ha = g⊥(F − Fd)∇H
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Final Implication DiagramPSfrag replacements
CbI
CbIsm
Basic CbIps
Basic CbIsmps CbI
smps
CbIps
EB Basic IDA PS IDA(Ortega, et al., TAC’08)
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
State Feedback PBC and CbI: Connections
CbIDynamic feedback control u = −yc + v = −∇ζHc(ζ) + v,
ζ controllers state with energy Hc(ζ) free,
Generate Casimir functions, C, that make Ω = (x, ζ)|ζ = C(x) invariant
⇒ For arbitrary Φ
H(x) + Hc(ζ) + Φ(C(x)− ζ) ≤ v>y
State–feedback PBCSolve some PDE on Ha and define a static state feedback, u(x), that ensures
H + Ha ≤ v>y
FactsState–feedback PBC is the projection of CbI into the invariant manifold.
There is no advantage of dynamic extension from minimum assignment viewpoint.(Astolfi and Ortega, SCL’09)
Simpler controllers with CbI.
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Open Problems
Is there a CbI version of IDA? What is the modification that is needed to add thisdegree of freedom?
We have fixed the order of the dynamic extension to be m. There are someadvantages for increasing their number. Also, we have taken simple nonlinearintegrators.
CbI does not help for minimum assignment, but certainly has an impact onperformance and simplicity.
Can CbI be used—as an alternative to the current “perturbation" framework—toformulate the problem of control over networks?
Key question: Impact of the network topology on the ability to shape the energy, i.e., togenerate the Casimir functions.
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IDA–PBC: A Matching Perspective
Consider x = f(x) + g(x)u. Assume there are matrices
g⊥(x), Jd(x) = −J>
d (x), Rd(x) = R>
d (x) ≥ 0,
where g⊥(x)g(x) = 0, g⊥(x) full rank, and a function Hd(x), with x? = argminHd(x), thatverify the matching equation
g⊥(x)f(x) = g⊥(x)[Jd(x)−Rd(x)]∇Hd (ME)
Then, the closed–loop system with u = u(x), where
u(x) = [g>(x)g(x)]−1g>(x)[Jd(x)−Rd(x)]∇Hd − f(x),
takes the port controlled Hamiltonian (PCH) form
x = [Jd(x)−Rd(x)]∇Hd,
with x? a stable equilibrium.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Universal Stabilizing Property of IDA–PBC
LemmaIf x∗ is asymptotically stable for x = f(x), f(x) ∈ C1 then ∃Hd(x) ∈ C1, positive definite,and C0 functions
Jd(x) = −J>d (x),Rd(x) = R>
d (x) ≥ 0
such that
f(x) = [Jd(x)−Rd(x)]∂Hd
∂x
CorollaryIf ∃u(x) ∈ C1 that asymptotically stabilizes the PCH system, then
∃Jd(x),Rd(x) ∈ C0 and Hd(x) ∈ C1 which satisfy the conditions of theIDA–PBC theorem.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 43/85
Consider the p–dimensional dynamic state–feedback controller
ζ = F3(x, ζ)∇xH(x, ζ) + F4(x, ζ)∇ζH(x, ζ)
u = [g>(x)g(x)]−1g>(x)F1(x, ζ)∇xH(x, ζ) + F2(x, ζ)∇ζH(x, ζ)− f(x),
with arbitrary matrices F3, F4. The closed–loop system is PH
x
ζ
= F (x, ζ)
∇xH(x, ζ)
∇ζH(x, ζ)
Furthermore, if:
(B2) F (x, ζ) + F>(x, ζ) ≤ 0;
(B3) (∇H)? = 0 and (∇2H)? > 0, for some ζ? ∈ Rp;
then (x?, ζ?) is a (locally) stable equilibrium with Lyapunov function H(x, ζ).
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 62/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Dynamic Extension is Unnecessary for Stabilization
Observation It is clear that we have to compare the sets of solutions of the PDEs (A1),(B1)—subject to the constraints, (A2), (A3) and (B2), (B3), respectively.
Proposition The following statements are equivalent:
(S1) There exists a matrix F : Rn → Rn×n and a function H : Rn → R such that (A1)–(A3)
hold.
(S2) There exists a positive integer p, matrices F1, F2 and a function H such that (B1)–(B3)hold.
Consequently, the equilibrium x? is stabilizable via static state–feedback IDA–PBC if andonly if the equilibrium (x?, ζ?) is stabilizable via dynamic (state–feedback) IDA–PBC.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 63/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Proof
((S1) ⇒ (S2))Trivial.
((S2) ⇒ (S1)) Assume conditions (B1)–(B3) hold. Now, since both PDEs have the same lefthand side
We now construct functions F (x) and H(x) that satisfy (KE)—hence (A1)—and conditions(A2), (A3). First, notice that, in view of (B3), one has
[∇ζH(x, ζ)]? = 0, det[∇ζζH(x, ζ)]? > 0.
Therefore, application of the Implicit Function Theorem to the function ∇ζH(x, ζ) proves theexistence of a function γ : Rn → R
p such that
[∇ζH(x, ζ)]|ζ=γ(x) = 0
in some open neighborhood of (x?, ζ?). Notice, also, that ζ? = γ?.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 64/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
cont’d
Replacing in (KE) yields
g⊥(x)[F (x)∇H(x)− F1(x, γ(x))W (x)] = 0,
where, W (x) := ∇xH(x, γ(x)),. Now, select F (x) = F1(x, γ(x)) that, in view of (B2),necessarily satisfies (A2). Replacing F (x) above yields
g⊥(x)F (x)[∇H(x)−W (x)] = 0.
A function H(x) that satisfies the equation above is given as
H(x) := H(x, γ(x)).
To complete the proof it only remains to show that the function H(x) verifies condition (A3).For, note that
(∇H)? =W? = 0, (∇2H)? = (∇W )? > 0
where (B3), the definition of W (x) and the fact that ∇W (x) = ∇xx[H(x, γ(x))], have beenused.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 65/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Remarks
An extension of the (static state–feedback) IDA–PBC technique that incorporates an(arbitrary) dynamic extension has been presented.
It has been shown that, for the purposes of (local) Lyapunov stabilization, noadvantage is gained with this extension.
The proof of the equivalence relies on the Implicit Function Theorem, from which thelocal nature of our result is inherited. Hence, the domain of stability of the equilibriumfor static feedback may be smaller that the one obtained using dynamic feedback.
The version of IDA–PBC considered here does not presume any particular structure forthe desired energy function. It is known that for some classes of systems, for instance,mechanical, it is convenient to “parameterize" the solutions. Studying the effect of adynamic extension in that case leads to a problem different from the one studied here.
Considering dynamic extensions the possibility to improve the transient performanceand to remove the need to measure the full state is opened up.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 66/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
IDA–PBC of Mechanical Systems
Model
q
p
=
0 In
−In 0
∂H∂q
∂H∂p
+
0
G(q)
u
where H(q, p) = 12p>M−1(q)p+ V (q), rank(G) = m < n.
Desired energy is parameterized
Hd(q, p) =12p>M−1
d(q)p+ Vd(q), Md(q) =M>
d (q) > 0
q? = argminVd(q).
Desired interconnection and damping matrices
Jd(q, p) =
0 M−1(q)Md(q)
−Md(q)M−1(q) J2(q, p)
= −J>d (q, p)
Rd(q) =
0 0
0 G(q)KvG>(q)
≥ 0, Kv > 0
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 67/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Proposition
Assume there is Md(q) =M>d (q) ∈ R
n×n and a function Vd(q) that satisfy the PDEs
G⊥
∇q(p>M−1p)−MdM
−1∇q(p>M−1
dp) + 2J2M
−1dp
= 0
G⊥∇V −MdM−1∇Vd = 0,
for some J2(q, p) = −J>2 (q, p) ∈ R
n×n and a full rank left annihilator G⊥(q) ∈ R(n−m)×n
of G, i.e., G⊥G = 0 and rank(G⊥) = n−m. Then, the system in closed–loop with
u = (G>G)−1G>(∇qH −MdM−1∇qHd + J2M
−1dp)−KvG
>∇pHd,
takes the desired Hamiltonian form. Further, if Md > 0 (in a neighborhood of q?) and
q? = argminVd(q),
then (q?, 0) is a stable equilibrium point with Lyapunov function Hd.
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 68/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Proof
G⊥
G>
p =
G⊥
G>
(−∇qH +Gu)
=
G⊥
G>
(−1
2∇q(p
>M−1p)−∇V +Gu)
≡
G⊥
G>
(−MdM−1∇qHd + (J2 −GKvG
>)∇pHd)
=
G⊥
G>
(−MdM−1[
1
2∇q(p
>M−1dp) +∇Vd] + (J2 −GKvG
>)M−1dp.
To prove stability: Hd is positive definite and
Hd ≤ −λminKv|G>M−1
dp|2 ≤ 0.
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CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Connection with Controlled Lagrangians
PDE’s (with J2(q, p) = 12
∑nk=1 Uk(q)pk, Uk = −U>
k )
G⊥∂>
∂q(M−1
(·,k))−MdM
−1 ∂>
∂q(M−1
d)(·,k) + UkM
−1d
= 0
G⊥∂V
∂q−MdM
−1 ∂Vd
∂q = 0
If J2(q, p) =MdM−1
[∇q(MM−1dp)]> −∇q(MM−1
dp)
M−1Md, we recover the
controlled–Lagrangian method
All matrices that “preserve mechanical structure" (arbitrary Q(q))
J2(q, p) = “J2 above” +MdM−1[
[∇qQ]> −∇qQ]
M−1Md
Gyroscopic (intrinsic) terms are added to the Lagrangian
Lc(q, q) =1
2q>M(q)M−1
d(q)M(q)q + q>Q(q)− Vd(q)
CDC Workshop, Shanghai, PRC, 15/12/2009 – p. 70/85
CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE
Constructive Solution For Underactuation Degree One Systems
Identification of a class of underactuation degree one mechanical systems for whichthe PDEs are explicitly solved.
The KE–PDE becomes an algebraic equation and we give a set of solutions.
Assume that the inertia matrix and the force induced by the potential energy (on theunactuated coordinate) are independent of the unactuated coordinate.
One condition for stability—an algebraic inequality—that measures our ability toinfluence, through the modification of the inertia matrix, the unactuated component ofthe force induced by potential energy.
Suitable parametrization of assignable energy functions—via two free functions and again matrix—to address transient performance and robustness issues.
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