20 Years of Passivity–Based Control (PBC): Theory and ...webpages.lss.supelec.fr › perso ›...

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


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.



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).



PBC Provides a Paradigm Shift for Controller Design

Classical formulation: Signal–processing viewpoint

1

2

+

+

-

+u y

u

y

1

2

1e

e 2

G

G2

1

System model and controller are signal processors: G1 : e1 → y1, G2 : e2 → y2.

Control specifications in terms of signals: tracking, disturbance attenuation, etc.

Presumes the presence of actuators and sensors—not always (physically) true or(conceptually) convenient

Fine for “isolated" systems but cannot easily accommodate “interactions"—e.g., theΣ−∆ paradigm to model uncertainty.

At a more philosophical level: there’re no inputs and outputs in nature!



Passivity–Based Control: An Energy–Processing Viewpoint

View plant as energy–transformation multiport device

Physical systems satisfy (generalized) energy–conservation:

Stored energy = Supplied energy + Dissipation

Control objective in PBC: preserve the energy–conservation property but with desiredenergy and dissipation functions

Desired stored energy = New supplied energy + Desired dissipation

In other words

PBC = Energy Shaping + Damping Assignment

For general systems achieve a passivation objective

Three possible formulations.

State feedback either to passivize or to change the energy function (anddissipation).

Control by Interconnection (CbI)—plant and controller are energy–transformationdevices, whose energy is added up.

Decompose the system into passive (or passifiable) sub-blocks and design PBCsfor each one of them.



Advantages of PBC

Advantages of energy–shaping (over nonlinearity cancellation and high gain)a

Handle on performance, not just stability

Respect, and effectively exploit, the structure of the system to

incorporate physical knowledge,

provide physical interpretations to the control action

Energy serves as a lingua franca to communicate with practitioners

There’s an elegant geometrical characterization of

power–conserving interconnections (via Dirac structures) and

passifiable NL systems (in terms of stable invertibility and relative degree)

aEuphemistically called “nonlinearity domination”.



Applications of PBC

Mechanical systems: walking robots, bilateral teleoperators, pendular systems.

Chemical processes: mass–balance systems, inventory control, reactors.

Electrical systems: power systems, power converters.

Electromechanical systems: motors, magnetic levitation systems.

Transportation systems: underwater vehicles, surface vessels, (air)spacecrafts.

Control over networks: formation control, synchronization, consensus problems.

Hybrid systems: switched systems, hybrid passivity.

...



Key Property: Cyclo–Passivity

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.



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.



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)



A Three-phase Rectifier

PSfrag replacements

E

vs

rL L

IC rc

is



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∗).



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∗.



Ex. 2 of Control by Interconnection: A Flexible Pendulum

q c

q *

δ( )

q q

p1

p2

K 2

Rc

K 1m=1

D

Plant energy: H(qp, pp) =12p>p D

−1(qp)pp + V (qp)

Controller energy:Hc(qc, pc, qp2) =

12|pc|2 + 1

2(qc − qp2)>K2(qc − qp2) +

12(qc − δ)>K1(qc − δ)

Controller Rayleigh dissipation function: 12q>c Rcqc

(Ailon/Ortega, SCL’93)



Ex. 3 PBC via Passive Subsystems Decomposition

Electromechanical Systems

u

i

y

g

m

λ

• λ is flux, θ position, u voltage



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!



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



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.



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

yd = hd = g>∇Hd, dd(x) = −∇H>d (x)(f(x) + g(x)uSF(x))



Algebraic Characterization of PBCs

Assumption Σ is cyclo-passive. That is, there exists H : Rn → R, d : Rn → R+

such that

H = y>u− d(x).

Proposition uSF ∈ PBC if and only if there exist functions Ha : Rn → R andda : Rn → R, with

da(x) ≥ −d(x),

such that

h>(x)uSF(x) = −∇H>a (x)(f(x) + g(x)uSF(x))− da(x) .

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)



Electrical Circuit Analog of PBC

PSfrag replacements

Σ

Σd

+

+

–

–

vh− hd

y yd

u

uSF



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.



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.



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!)



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.



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.



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( )



Flexible Pendulum

q c

q *

δ( )

q q

p1

p2

K 2

Rc

K 1m=1

D



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).



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.



Controllers by Interconnection are as Old as Control Itself



They’re Pervasive and Efficient



Adding Energies

Definition: The interconnection is power preserving if

y>u+ y>c uc = 0.

Simplest example: Classical feedback interconnection

u

uc

=

0 −1

1 0

y

yc

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?



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.



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!



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 .



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).



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



Final Implication DiagramPSfrag replacements

CbI

CbIsm

Basic CbIps

Basic CbIsmps CbI

smps

CbIps

EB Basic IDA PS IDA(Ortega, et al., TAC’08)



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.



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.



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.



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.



IDA PBC as a State–Modulated Interconnection

Controller as an (infinite energy) source

Σc :

ζ = uc

yc = ∂Hc

∂ζ(ζ)

with energy function Hc(ζ) = −ζ.

State–modulated (power–preserving) interconnection ΣI

u(s)

uc(s)

=

0 −u(x)

u(x) 0

y(s)

yc(s)

Overall interconnected PCH system (with total energy H(x) +Hc(ζ))

x

ζ

=

J(x)−R(x) −g(x)u(x)

u>(x)g>(x) 0

∂H∂x

(x)∂Hc

∂ζ(ζ)



When is IDA an Energy–Balancing–PBC?

Plant is a PCH system

x = [J(x)−R(x)]∂H

∂x+ g(x)u, y = g>(x)

∂H

∂x

The PDE becomes

g⊥(x)[Jd(x)−Rd(x)]∂Ha

∂x= −g⊥(x) [Jd(x)− J(x)−Rd(x) +R(x)]

∂H

∂x

to be solved for Ha(x) and form Hd(x) = H(x) +Ha(x).

If Rd(x) = R(x), and satisfies

R(x)∂Ha

∂x(x) = 0,

that is, no shaping of coordinates with damping, then

Ha = −u>y ⇔ IDA–PBC is energy balancing.



Some Practical Considerations: Integral Action

PropositionIDA–PBC with integral action u = ues + udi + v where

v = −KIg>∂Hd

∂x

with KI = K>I > 0, preserves stability.

Proof Let

W (x, v)4= Hd +

1

2v>K−1

I v

The closed–loop

x

v

=

Jd −Rd gKI

−KIg> 0

∂W∂x

∂W∂v

is clearly PCH.



Damping Injection with "Dirty Derivatives"

To inject additional damping, instead of the passive output, we can feed back itsintegral, preserving stability.

Of particular interest to obviate velocity measurements in mechanical systems wherethe passive output is q.

PropositionIf

u = ues +Kdig>∂Hd

∂x

is asymptotically stable, then u = ues + udi, where

udi = −1

τudi −

Kdi

τg>

∂Hd

∂x

with τ > 0, also ensures convergence of x∗.



Proof

With the energy

W (x, udi)4= Hd +

τ

2Kdi

u2di

we have

x

udi

=

Jd

Kdi

τg

−Kdi

τg> −Kdi

τ2

∂W∂x

∂W∂udi

yielding

W = −uTdiK−1diudi



Existing Approaches to Solve the Matching Equation

Non–Parameterized IDA One extreme case:

fix Jd(x), Rd(x) and g⊥(x),

(ME) becomes a PDE for Hd(x),

among the family of solutions select one that assigns minimum.

Algebraic IDA At the other extreme:

fix Hd(x),

(ME) becomes an algebraic equation in g⊥(x), Jd(x) and Rd(x).

Parameterized IDA Restrict the desired energy function to a certain class,

for instance, for mechanical systems

Hd(q, p) =1

2p>M−1

d(q)p+ Vd(q),

(ME) becomes a PDE in Md(q), Vd(q),

imposes some constraints on Jd(x), Rd(x).

Application of Poincare’s Lemma (applicable for systems NL in u):

∇Hd(x) = F−1d

(x)f(x, u).



Application to a MEMS Actuator

b k

u

R

q

q

m+-

Energy function

H(q, p,Q) =1

2k(q−q?)

2+1

2mp2+

q

2AεQ2.

Dynamical model

q =p

m

p = −k(q − q?)−Q2

2Aε−

b

mp

Q = −qQ

RAε+

1

Ru

Desired equilibrium (q?, 0, 0)

p is not measurable.



Algebraic IDA

Fix quadratic energy function

Hd(q, p,Q) =γ1

2(q − q?)

2 +1

2mp2 +

γ2

2Q2.

(ME) solvable with γ1 = km and

Jd(Q) =

0 1m

0

− 1m

0 − Q2Aεγ2

0 Q2Aεγ2

0

, Rd =

0 0 0

0 − bm

0

0 0 −r3

and γ2, r3 > 0 free parameters. Note J23 = − Q2Aεγ2

6= 0.

Control law

u(q, p,Q) = γ1pQ− γ2Q+1

AεqQ

where γi > 0 are arbitrary.



Parameterized IDA–PBC

Fix Hd = 12m

p2 + ϕ(q,Q) and Jd −Rd =

0 1 0

−1 −b 0

0 0 −r3

.

Solution of the PDE:

ϕ(q,Q) =1

2k(q − q?)

2 +1

2AεqQ2 + ψ(Q),

with ψ(Q) free for the equilibrium assignment.

Output–feedback control

u(q,Q) = −(r3R− 1)1

AεqQ− r3Rψ

′

Contains, as a particular case with r3 = 1R

and ψ(Q) quadratic, the well–known linearcharge feedback controller.



Poincare’s Lemma: Boost Converter

Procedure There exists Hd : Rn → R such that ∇Hd(x) = F−1d

(x)f(x, u) if and only if

∇[

F−1d

(x)f(x, u(x))]

= (∇[

F−1d

(x)f(x, u(x))]

)>.

Model (under fast switching), x(0) ∈ R2>0

x =

0 −u

u −1R

︸︷︷︸

J(u)−R

∂H

∂x(x) +

E

0

, H(x) =1

2Lx21 +

1

2Cx22.

Control objective: Regulate 1Cx2 to a desired constant value V∗ > E, verifying

C.1 Only x2 measurable.

C.2 u ∈ (0, 1).

C.3 x ∈ R2>0.

C.4 R is unknown.

Main contribution: Stabilization via IDA–PBC with a simple static nonlinear outputfeedback.



Proposition

For all R > 0 the IDA–PBC

u = u∗

(x2

x2∗

)α

, 0 < α < 1

yields

(i) x∗ = ( LRE

V 2∗ , CV∗) is asymptotically stable with Lyapunov function

Hd(x) =1

2Lx21 +

1

2Cx22 + κ1x

2(1−α)2 − (κ2 + κ3x1)x

1−α2

(ii) Domain of attraction: Ξα4= x|x ∈ R2

>0 and Hd(x) ≤ Hd(0, x2∗) is such thatx(0) ∈ Ξα ⇒ x(t) ∈ Ξα and limt→∞ x(t) = x∗

(iii) Saturation: ∀x∗, ∃α ∈ (0, 1) s.t.

x(0) ∈ Ξα ⇒ 0 < u(t) < 1



Proof

• IDA Select Ra = diagRa,−1R ⇒ Rd = diagRa, 0.

• Integrability Key PDEa

K4=∂Ha

∂x(x) =

1

β(x2)

− 1

RCx2

− 1LRax1 − E + Ra

RCx2

β(x2)

PDE solvable ⇔ ∂K2

∂x1(x) = ∂K1

∂x2(x) ⇔

dβ

dx2(x2) =

α

x2β(x2)

where α4= 1− RaRC

L. Thus,

u = c1xα2

aAssuming J(β(x))−Rd is invertible.



Proof cont’

• Equilibrium assignment: c1 such that

∂Hd

∂x(x∗) =

∂H

∂x(x∗) +

∂Ha

∂x(x∗) = 0

This yields c1 = u∗

xα

2∗

.

• Hessian condition

∂2Hd

∂x2(x∗) =

1L

− Ra

u∗L

− Ra

u∗L1C

+(Rax1∗+EL)α

u∗Lx2∗+

(1−2α)Ra

u∗2RC

Positive definite ⇔ −1 < α < 1.



On the Role of Jd(x)

Propagates the damping via interconnection of strictly passive and passive subsystems

x =

0 −α(x)

α(x) −r

x ⇔

Σ1 is passive and Σ2 strictly passive (if r > 0)

Property is preserved for “concatenated structures", for instance

x =

0 β(x) 0

−β(x) 0 −α(x)

0 α(x) −r

x



Choosing Jd(x) From Physical Considerations

Isotropic (smooth rotor) synchronous motors are more efficient that indented rotor motors

• Open–loop

J(x) =

0 0 x2

0 0 −(x1 + Φ)

−x2 x1 +Φ 0

Φ is the dq back emf constant.

• Virtual behavior in closed–loop

Jd(x) =

0 L0x3 0

−L0x3 0 −Φ

0 Φ 0

L0 free parameter, represents stator in-ductance (Ld = Lq).



cont’d

Enforce a coupling between electrical and mechanical coordinates in magnetic levitationsystems, coordinates x = [λ, y,my]>

u

i

y

g

m

λ

Open-loop interconnection

J =

0 0 0

0 0 1

0 −1 0

Desired interconnection

Jd =

0 0 −α

0 0 1

α −1 0

with α chosen by the designer.



Other Degrees of Freedom: Change of Coordinates

Define x = φ(z), where φ is a diffeomorphism, and apply the construction.

Modified (ME)

g⊥(z)(∇φ)−1f(φ(z)) = g⊥(z)[

Jd(z)− Rd(z)]

∇Hd

where

Jd(z) = (∇φ)−1Jd(φ(z))(∇φ)−> = −J>

d (z),

Rd(z) = (∇φ)−1Rd(φ(z))(∇φ)−> = R>

d (z) ≥ 0,

g(z) = (∇φ)−1g(φ(z))

The choice of φ is a new degree of freedom.

New control laws for power systems.



Adding Dynamics to IDA–PBC

Proposition Consider the system x = f(x) + g(x)u. Define g⊥ : Rn → R(n−m)×n to be

a full–rank left annihilator of g(x), i.e., g⊥(x)g(x) = 0, and rank g⊥(x) = n−m for all x.Let x? ∈ R

n be an assignable equilibrium. Assume there exists a matrix F : Rn → Rn×n

and a function H : Rn → R such that the following holds.

(A1) g⊥(x)f(x) = g⊥(x)F (x)∇H(x)

The system in closed–loop with the static state–feedback

u = [g>(x)g(x)]−1g>(x)F (x)∇H(x)− f(x),

can be written in port–Hamiltonian (PH) form

x = F (x)∇H(x).

Furthermore, if

(A2) F (x) + F>(x) ≤ 0;

(A3) (∇H)? = 0 and (∇2H)? > 0;

then x? is a (locally) stable equilibrium with Lyapunov function H(x).



IDA–PBC with Dynamic Extension

Proposition Assume there exists F1 : Rn+p → Rn×n, F2 : Rn+p → R

n×p andH : Rn+p → R such that:

(B1) g⊥(x)f(x) = g⊥(x)[F1(x, ζ)∇xH(x, ζ) + F2(x, ζ)∇ζH(x, ζ)]

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).



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.



Proof

((S1) ⇒ (S2))Trivial.

((S2) ⇒ (S1)) Assume conditions (B1)–(B3) hold. Now, since both PDEs have the same lefthand side

g⊥(x)[F (x)∇H(x)− F1(x, ζ)∇xH(x, ζ)− F2(x, ζ)∇ζH(x, ζ)] = 0. (KE)

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?.



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.



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.



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



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.



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.



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)



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.



Parametrization of the Kinetic Energy PDE

Assumption A.1 Underactuation degree one: m = n− 1.Assumption A.2 G⊥∇q(p>M−1p) = 0

Then, kinetic energy PDE becomes

n∑

i=1

γi(q)∂Md

∂qi= −[J (q)A>(q) +A(q)J>(q)],

where J (q) is free,

γ4=M−1Md(G

⊥)>, A4=

[

W1

(

G⊥)>

,W2

(

G⊥)>

, . . . ,Wno

(

G⊥)>]

with n04= n

2(n− 1) and Wi = −W>

i , e.g., for n = 3

W14=

0 1 0

−1 0 0

0 0 0

, W2

4=

0 0 1

0 0 0

−1 0 0

, W3

4=

0 0 0

0 0 1

0 −1 0

.



Solving the Kinetic Energy PDE

The expression above characterizes all solutions to the KE-PDE

Assumption A.3 G is function of a single element of q, say qr , r ∈ 1, . . . , n

A.3 satisfied (for partially–linearized systems) if the column of M corresponding to theunactuated coordinate depends only on qr .

A subset, for which KE–PDE becomes algebraic and we can find explicit solutions, is

γrdMd

dqr= −2AJ> ⇒

dMd

dqrei ∈ Im A

Furthermore, G⊥A = 0 ⇔ A ∈ Im G, suggesting dMd

dqrei ∈ Im G

Proposition For all desired (locally) positive definite inertia matrices

Md(qr) =

∫ qr

q?r

G(µ)Ψ(µ)G>(µ)dµ+M0d

where Ψ = Ψ> and M0d = (M0

d )> > 0, may be arbitrarily chosen, there exists J2

such that the kinetic energy PDE holds.



Solving the Potential Energy PDE

Recalling PE–PDE:

G⊥∇V −MdM−1∇Vd = 0

Can be written as

γ>(q)∇Vd = s(q), s4= G⊥∇V.

Remarks concerning s:

For all admissible equilibria q, we have s(q) = 0.

G⊥∇V are forces that cannot be (directly) affected by the control.

Since G,Md depends on qr it is reasonable:

Assumption A.4 γ, s are functions of qr only. (⇐M =M(qr)

A generic condition is needed to ensure that the PDE admits a well–defined solution:

Assumption A.5 γr(q?r ) 6= 0



Proposition

Under Assumptions A.1–A.5 and Md(qr) =∫ qrq?r

G(µ)Ψ(µ)G>(µ)dµ+M0d all solutions of

the PE–PDE are given by

Vd(q) =

∫ qr

0

s(µ)

γr(µ)dµ+Φ(z(q)),

with z(q) , q −∫ qr0

γ(µ)γr(µ)

dµ the characteristic of the PE–PDE, and Φ an arbitrary

differentiable function.

Remarks

Identify a set of assignable energy functions parameterized by Ψ,M 0d ,Φ.

γr is the element of the “coupling term", G⊥M−1Md, through which we can modifythe (unactuated coordinates of the) open–loop potential energy.

For stability, since Φ(z) is arbitrary, restrictions will only be imposed on∫

sγr

. Namely,that its second derivative, evaluated at q?r , is positive.



Main Stabilization Result

Assumption A.6 γr(q?r )dsdqr

(q?r ) > 0

ensures (q?, 0) is a locally stable equilibrium with Lyapunov function Hd(q, p).

Assumption A.7 |G>M−1er(q?r )| 6= 0,

makes it asymptotically stable. Furthermore, if we select

Φ(z(q)) =1

2[z(q)− z(q?)]> P [z(q)− z(q?)]

with P = P> > 0, the control law is of the form

u = A1(q)PS(q − q?) +

p>A2(qr)p

...

p>An(qr)p

+An+1(qr)−KvAn+2(qr)p

where Kv = K>v > 0 is free, S ∈ R

(n−1)×n is obtained removing the r–th row from then–dimensional identity matrix.



Pendulum on a Cart

Model

ml2q1 +ml cos q1q2 −mgl sin q1 = 0

(M +m)q2 +ml cos q1q1 −ml sin q1q12 = v.

Can be transformed into

q = p

p = a sin q1e1 +

−b cos q1

1

u

Notice that G⊥(q1) = [1, b cos q1].

It can be shown that ψ(q1) cannot be a constant. Propose ψ = −k sin q1.

Independently of Ψ,M0d ,Φ assumptions cannot be satisfied outside (− π

2, π2).



Stability Result

The IDA–PBC

u = A1(q1)P (q2 − q2∗) + p>A2(q1)p−KvA3(q1)p+A4(q1)

ensures asymptotic stability of the desired equilibrium (0, q2∗, 0, 0) with a domain ofattraction containing the set (− π

2, π2)× R

3 and Lyapunov function Hd(q, p) where

Md(q1) =

kb2

3cos3 q1 − kb

2cos2 q1

− kb2

cos2 q1 k(cos q1 − 1) +m022

,

Vd(q) =3a

kb2 cos2 q1+P

2

[

q2 − q2∗ +3

bln(sec q1 + tan q1) +

6m022

kbtan q1

]2

,

(which is radially unbounded on the set (− π2, π2).)



Simulations

Trajectories with [q(0), p(0)] = [π/2− 0.2,−0.1, 0.1, 0]—pendulum starting near thehorizontal ♥

0 20 40 60 80

0

20

40

60

Time (sec)

Pos

ition

s (q

)

0 20 40 60 80−5

0

5

10

15

Time (sec)

Vel

ociti

es (

q)

0 20 40 60 80−20

−10

0

10

20

Time (sec)

Con

trol

0 20 40 60 800

2000

4000

6000

Time (sec)

Ene

rgy

−1 0 10

20

40

60

q1 (rad)

q 2 (m

)

−1 0 1−1.5

−1

−0.5

0

0.5

q1 (rad)

q 1 (ra

d/se

c)

°

°

q2

q1



Strongly Coupled VTOL Aircraft

θ

x

y ε v

2

v1

g

COG

Model (ε 6= 0, possibly large)

x = − sin θv1 + ε cos θv2

y = cos θv1 + ε sin θv2 − g

θ = v2

Can be transformed intoq = p

p =

1 0

0 1

1εcos θ 1

εsin θ

u+ g

εsin θe3

Objective: Characterize assignable energy functions with (x∗, y∗, 0, 0, 0, 0)

asymptotically stable.



Proposition

A set of assignable energy functions is characterized by

Md(q3) =

k1ε cos2 q3 + k3 k1ε cos q3 sin q3 k1 cos q3

k1ε cos q3 sin q3 −k1ε cos2 q3 + k3 k1 sin q3

k1 cos q3 k1 sin q3 k2

with k1 > 0 and

k3 > 5k1ε,k1

ε> k2 >

k1

2ε

and the potential energy function

Vd(q) = −g

k1 − k2εcos q3 +

1

2‖

q1 − q1∗ − k3

k1−k2εsin q3

q2 − q2∗ + k3−k1εk1−k2ε

(cos q3 − 1)

‖P .

Moreover, the IDA–PBC law ensures almost global asymptotic stability of the desiredequilibrium (q1∗, q2∗, 0, 0, 0, 0).



Simulations

Effect of tuning (matrix P )

−8 −6 −4 −2 0 2 4 6 8−7

−6

−5

−4

−3

−2

−1

0

1

2

3

x(m)

y(m

)

g

−8 −6 −4 −2 0 2 4 6 8−7

−6

−5

−4

−3

−2

−1

0

1

2

3

x(m)

y(m

)

g



cont’d

Upside down simulation. ♥

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

x(m)

y(m

)g



Some Results at the Frontier

General input–output theory(Rantzer, SCL’06) Connection between S–procedure and KYP Lemma.

(Sontag, SCL’08) Secant condition.

(Janson and Petersen, TAC’08) Passivity F → q.

(Iwasaki and Hara, TAC’08) Finite frequency response positive realness.

(Willems at al., EJC’08) Behavioral framework.

(Ito and Jiang, TAC’09) IiSS and small gain theorem.

Various applications(Arcak and Kokotovic, Automatica’01) Observer design.

(Pavlov and Marconi, SCL’08) Convergent systems and output regulation.

(Alonso et al., SCL’09) Reaction systems.

Ships, UAV’s, reaction systems, fuel–cells,...

Port Hamiltonian systems: Chen, van der Schaft, Stramigioli, Fujimoto, Ortega...



cont’d

Synchronization and networks(Arcak and Wen, INFOCOM’03) Network flow control.

(Pogromsky and Nijmeijer, TCS’06) Notion of almost passivity.

(Chopra and Spong, CDC’06) Output synchronization of passive systems.

(Arcak/Sontag, CDC’09) Incrementally passive systems.

(Nunio, et al., ACC’10) Euler–Lagrange systems.


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