A “New Control Theory to Face the Challenges of Modern

CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE

A “New" Control Theory to Face the Challengesof Modern Technology

Romeo OrtegaLSS-Supelec, France

“A control theorist’s first instinct in the face of a new problem is to find a way to use the tools he

knows, rather that a commitment to understand the underlying phenomenon. This is not the failure of

individuals but the failure of our profession to foster the development of experimental control science.

In a way, we have become the prisoners of our rich inheritance and past successes”.

Y. C. Ho (1982).

Supelec, January 2012 – p. 1/91


Layout

Introduction and Motivation

Control by Interconnection

Examples

Passivity–based Control

Mathematical Formulation

Applications

Mechanical systems

Power electronic systems

Electromechanical systems

Energy management

Transient stability of power systems

Wind speed estimation



Introduction and Motivation



Control Challenges in the Modern World

New engineering applications (including biomedical and others) exhibit:

Strong coupling between subsystems.

Mutually interacting, instead of cause–effect, relations.

Need for accurate, non–isolated, models.

Existing control theory, which adopts a signal–processing viewpoint, is inadequate toface those challenges.

Objective: Provide a new control paradigm, based on considerations of

energy,

dissipation and

interconnection.

Paradigmatic examples

Modern electrical (smart) grid

Interacting mechanical systems, e.g. teleoperators, cooperative robots...

Transportation systems

Bio–medical applications...



Classical Control Theory

A strict causality relation, motivated by the presence of sensors and actuators, isadopted. Consequences:

Overall system is “closed and isolated".

Difficult, if not impossible, to couple with other systems.

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

Focus on the details of the system, neglecting the interactions. Rationalized via:

Time–scale separation arguments, and

“high impedance" considerations.

Mathematical models

x = f(x, u)

y = h(x, u),

and analysis/design tools, e.g., Lyapunov theory, not suitable to incorporate

interconnection,

model uncertainty



Prevailing Signal–processing Viewpoint of Control

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

1

2

+

+

-

+u y

u

y

1

2

1e

e 2

G

G2

1

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

Even robustness to model uncertainty, captured via the “Σ−∆ paradigm", isrepresented with a signal processing block.

Control by Interconnection (CbI): View systems and controllers as open, energyprocessing multiports.






Controllers by Interconnection are as Old as Control Itself



They’re Pervasive and Efficient



Even in your Privy



Interconnection Works Even if We Don’t Know Why!

(Spong’06): Synchronization by interconnection, an openproblem since the 17th century.♥



Passivity: The Key Articulating Concept

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 Passivity–based Control (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 900 times and

is the 13th most highly cited paper out of 4520 published in Automatica since1989.

PBC has more than 100,000 hits in Google scholar.



Passivity–Based Control: An Energy–Processing Viewpoint

View plant as energy–transformation multiport devices

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.



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).



Advantages of PBC

Energy and dissipation are additive.

Applicable to NL systems.

Suitable to handle interconnections of open systems.

Model uncertainty, e.g., friction, naturally captured.

Shaping energy and dissipation there’s 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 conservation is a universal property, hence CbI is applicable to multi–domainphysical systems.

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)



Applications

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, windmill generators.

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

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

Hybrid systems: switched systems, hybrid passivity.

...



Mathematical Formulation of CbI



Controllers as Multiport Cyclo–passive Systems

cΣ-

+

-

+ Σu c

yc y

uΣ I

Plant (Σ) and controller (Σc), with states x ∈ Rn, ζ ∈ R

m, are cyclo–passive, that is,∃H : Rn → R, Hc : Rm → R, such that

H = u>y − d

Hc = u>c yc − dc,

where d, dc ≥ 0 are dissipations.



Adding the Energies

Interconnection subsystem (ΣI ) is power–preserving (lossless)

y>u+ y>c uc = y>v (⇐ u = −yc + v, uc = y).

Interconnected system satisfies

H + Hc = v>y − d− dc

⇒ H(x) +Hc(ζ) is the new energy and d+ dc new dissipation.

Problem:Although Hc(ζ) is free, not clear how to affect x?

Energy functions “coupled" via the generation of invariant spaces.

Another alternative is to make Hc(x, ζ).



Invariant Function Method

Principle: Restrict the motion to a subspace of (x, ζ)

1

x (t)

x (o) x

x

2

ξ

Define the set

Ωκ , (x, ζ)|ζ = F (x) + κ,

(κ determined by the controllers ICs).Then, in Ω0,

Hd(x) , H(x) +Hc[F (x)]

It can be shaped selecting Hc(ζ).Mathematical problem Let

C(x, ζ) , F (x)− ζ

Finding F (·) that renders Ω invariant ⇔

d

dtC|C=0 ≡ 0, (PDE)



Suitable Model: 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 formulation using Dirac structures.

Most nonlinear cyclo–passive systems can be written as PH systems. Actually, in(network) modeling is the other way around!



Modularity of CbI

PSfrag replacements

+

+

+

–

–

–

v

Σ(u,y) ΣC

ΣI

y

ycu

uc



Mechanical Systems



Application to Underactuated Mechanical Systems

Flexible Joint Robots

q c

q *

δ( )

q q

p1

p2

K 2

Rc

K 1m=1

D

Plant energy:

H(qp, pp) =1

2p>p M

−1(qp)pp+V (qp)

Controller energy:

Hc(qc, pc, qp2) =1

2|pc|

2 +

+1

2(qc − qp2)

>K2(qc − qp2) +

+1

2(qc − δ)>K1(qc − δ).

Controller Rayleigh dissipationfunction: F(q) = 1

2q>c Rcqc

Rigid case solved in (Kelly’93), withflexibility in (Ailon/Ortega, SCL’93)



Pendular Systems

Acrobot

g

Actuator

l

l

c1

c2

link 2

link 1

q2

m m1, 2l , l1 2I I, 21

lc1 , lc2

= link masses

= link lengths

= link moments of inertia

= centers of masses

1q∼

Furuta’s Pendulum

q1

m

l

2q R

M

u

g



Performance Improvement via Energy Shaping

Transient Performance of the Ball and Beam

Steep energy function ⇒ quick response but bad transient ♥

“Wider" energy function ⇒ slower response without overshoot transient ♥

No regulation of transient excursions ⇒ ball gets off the bar ♥

Limiting level sets ⇒ ball remains in the bar ♥

Smooth Swing–up of Underactuated Mechanical Systems

Acrobot ♥ In (Viola, et al.’08) first proof of stability including the lower half plane.

Furuta’s Pendulum ♥

Orbital Stabilization of Passive Walking Robot

Problem cannot be posed in terms of tracking, a natural approach is to regulatethe kinetic–potential energy exchange ♥

‘Achieved assigning an energy function like a "Mexican sombrero" ♥



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 PBC law ensures almost global asymptotic stability of the desired equilibrium(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



Bilateral Teleoperators

Two mechanical systems, a human–controlled master and a teleoperated slave.

Slave should follow the master’s position and the master “feel" the slave’s forces.

No causality relation for the human nor the environment interaction!



cont’d

Instability arises due to the transmission delays.

Passivity–based solution (Anderson and Spong, ’89)

Assume operator and environment are passive, i.e.,

−

∫ t

0F (τ)q(τ)dτ ≤M.

Transform the transmission delays into a transmission line, which is also passive!(Anderson/ Spong, ’89)

TRANSMISSION DELAY

CODIFICATION SCHEME

ENVIRONMENT

CONTACT

RC filtermaster side

RC filterslope side

i(t,0)

v(t,0)

Bs

Km

Bm

Mm Bm Ms

Bs

Ks

i(t,l)

+

−

+ +

− −

+

−

h e

s

OPERATOR

HUMAN+

1 1mi (t)i (t)

v(t,l)v (t) v (t)

Interconnection of passive systems is stable.



A Dual Problem

Overvoltage Problem The presence of long cables between a fast–sampling actuatorand plant induces oscillations: The cables behave like a transmission line.

v(t,l)v(t,o)

i(t,0) i(t,l)

+ +

− −

ACTUATOR

TRANSMISSION LINEFast sampled − data

PLANT

(Ortega/Spong, US Patent 07): Transform transmission line into delays

ACTUATOR PLANTCOMPENSATORTRANSMISSION

LINE

+

−

i(t,l)

v(t,l)

−

+

i(t,0)

v(t,0)

−

+

i(t)

v(t)~

~



Synchronization of Uncertain Euler–Lagrange Systems

N Euler–Lagrange systems described by

Mi(qi)qi + Ci(qi, qi)qi + gi(qi) = τi,

where all parameters are unknown.

Interconnected through a communication channel

With unknown transmission delays.

Only weakly connected.

Objective: Find a dynamics controller that achieves either

synchronization, that is, qi(t) → qd(t), or

consensus, qi(t)− qj(t) → 0.

Reported in (Nuño, et al.’10)

Experiments between the UdG (Guadalajara) and UPCatalonya (Barcelona)interconnected via Internet

Three coordinated Phantoms ♥

Teleoperation between Guadalajara and Barcelona ♥



Electrical Systems



PI Control of Power Converters



Problem Formulation

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

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’10; ISIE’10)



Boost Converter

+

L+

+E

u=0

u=1

C+

R

PH Model: x = col(φL, qC)

x =

0 −u

u −1R

︸︷︷︸J (u)−R

∇H +

E

0



Cuk Converter

PH Model

x =

0 −(1− u) 0 0

1− u 0 u 0

0 −u 0 −1

0 0 1 − 1R

∇H +

E

0

0

0



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∗.



Application to a Quadratic Converter

+

_

+

_

+

_

PSfrag replacements

E

L1 L2

C1 C2

D1

D2 D3

iL1 iL2

vC1 vC2rLu′

r1

r2

r3

Element Value

Mosfet IRFS38N20D

Diodes Schottky MBRB20200CT

Current Sensor HSX-50

Capacitances CKG57NX7R2A475M



Port–Hamiltonian Model

x = (J0 + J1u−R)∂H

∂x+ B,

x =(iL1 iL2 vC1 vC2

), B =

(EL1

0 0 0)>

R =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1rLC2

2

, Q =

L1 0 0 0

0 L2 0 0

0 0 C1 0

0 0 0 C2

J0 =1

C1L2

0 0 0 0

0 0 1 0

0 −1 0 0

0 0 0 0

J1 =

0 0 − 1L1C1

0

0 0 0 − 1L2C2

1L1C1

0 0 0

0 1L2C2

0 0

.



Incrementally Passive Output

The goal is to regulate the capacitor voltage x4 to the constant value x?4 = vd.

The admissible equilibria can be parameterized by the reference x?4 as follows

x? :=[

1rL(u?)2

1rLu? u? 1

]>x?4

where u? =√

Ex?4

is the corresponding constant control.

The output

y = −√Evdx1 − vdx2 +

v2dErL

x3 +vd

rL

√vd

Ex4,

is incrementally passive and detectable

The equilibrium x? can be rendered globally asymptotically stable with a PI controller:

The only parameters that are required are rL and E, and that the tuning gains cantake arbitrary positive values.

Observers and adaptation added to estimate the state x and rL, preserving thestability properties.



Experimental Setup

Q1

+

_

+

_

+

_

PWM

PSfrag replacements

E

L1 L2

C1

C2

D1

D2 D3

iL1

iL2vC1

vC2

rL

u′

r1

r2

r3

DSPIC33F128GP802



Experimental Results

A voltage reference step change from vd = 80V , to vd = 120V

A change in the load resistance, from rL = 330Ω to rL = 198Ω

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

Amperes

seconds

iL1

0 0.5 1 1.5 2 2.5 3 3.5 4

20

40

60

80

100

120

140

Volts

seconds

ReferenceOutput Voltage

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7x 10

−3

Mho

seconds

Theta estimated



A Three-phase Rectifier

PSfrag replacements

E

vs

rL L

IC rc

is

The technique yields Akagi’s PQ instantaneous power controller (Perez, et al.,IEEE-TCST’04).



Some Additional Results

An adaptive observer–based PBC for the SEPIC, (Jaafar, et al., ACC’12).

Control of converters in discontinuous conduction mode, (Allawieh, et al., IFAC’11).

An adaptive PBC for a unity power factor rectifier, (Escobar, et al., IEEE TCST’01).

Experimental comparison of several PWM controllers for a single-phase ac-dcconverter, (Karagiannis, et al., IEEE-TCST’03).

An adaptive controller for the shunt active filter considering a dynamic load and the lineimpedance, (Valdez, et al., IEEE-TCST’09).



Transient Stability of Power Systems



Model of the Power System

We consider a large–scale power system consisting of n generators interconnectedthrough a transmission network which we assume is lossy.

The dynamics of the i-th machine with excitation is represented by the classicalthree–dimensional flux decay model

δi = ωi

ωi = −Diωi + Pi −GiiE2i −Ei

n∑

j=1,j 6=i

EjYij sin(δi − δj + αij)

Ei = −aiEi + bi

n∑

j=1,j 6=i

Ej cos(δi − δj + αij) +Efi + ui

δi, ωMi: rotor angle and speed, E′qi: the quadrature axis internal voltage, ufi: the

field excitation signal, GMij = GMji, BMij = BMji and GMii: conductance,susceptance and self-conductance of the generator i, Efsi: constant component ofthe field voltage, Pmi: constant mechanical power, xdi, x′di, ωi0 and DMi:direct-axis—synchronous and transient—reactances, synchronous speed anddamping coefficient



Problem Formulation and Solution

Assume the model with ui = 0 has a stable equilibrium point at [δi∗, 0, Ei∗], withEi∗ > 0. Find a control law ui such that in closed–loop

an operating equilibrium is preserved,

a Lyapunov function for it is given and,

it is asymptotically stable with a well–defined domain of attraction.

Two additional requirements are that the domain of attraction of the equilibrium isenlarged by the controller and that the Lyapunov function has an energy–likeinterpretation.

For lossless lines we can assign

Hd(δ, ω,E) = ψ(δ) +1

2|ω|2 +

1

2(E − E∗)

>Γ(E −E∗),

If the lines are lossy we have to introduce a cross-term in the energy function.

Hd(δ, ω,E) = ψ(δ) +1

2|ω|2 +

1

2[E − λ(δ)E∗]

>Γ[E − λ(δ)E∗].

(Ortega, et al., TAC’08)



Simulations

Consider the two machines system

3

1

5

6

4

2

j0.5

0.012+j0.10

0.012+j0.20

.005+j0.05

.005+j0.10 j0.5E2

d2

1.0

+j0

1.0

+j0

G2G1

E1

d1

F

X’ =0.5

X =1.8

T =6s

M =5s

d1

d1

d01

1

D =11

X’ =0.5

X =2.3

T =7s

M =7s

D =0.2

d2

d2

d02

2

2

The disturbance is a three-phase fault in the transmission line that connects buses 3

and 5, cleared by isolating the faulted circuit simultaneously at both ends.

This modifies the topology of the network an consequently induces a change in theequilibrium point.

Without control the system is highly sensitive to the fault and the critical clearing timeis almost zero.



Load Angle and Internal Voltage, tcl = 80 m sec

0 0.5 1 1.5 2 2.5 3 3.5 40.2

0.4

0.6

0.8

1δ

(rad

)

0 0.5 1 1.5 2 2.5 3 3.5 40.5

1

1.5

2

time (sec) →

E (

pu)

δ1

δ2

E1

E2



Other Results

Derivation of a Lyapunov function for the full model (Zonetti, et al., ACC’12).

Actuation via flexible AC transmission systems, (Manjarekar, et al., Electrical PowerSystems Research’10, EJC’11).

Structure preserving models

Cyclo-dissipativity properties have been established and a linear controllerproposed – leads to an LMI test, (Guisto, et al., CDC’08).

“Full solution" using Lyapunov–based designs (Dib, et al., TAC’10), (Casagrande,et al., CDC’11).

Alternative formulation as a synchronization, not stabilization, problem, (Dib, et al.,ACC’11).

Immersing a pendular dynamics.

Existence solution for n machines.



Electromechanical Systems



Energy Shaping of 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.

Desired equilibrium (q?, 0, 0)

p is not measurable.

Problem solved assigning theenergy Hd = 1

2mp2 + ϕ(q,Q),

ϕ(q,Q) =1

2k(q−q?)

2+1

2AεqQ2+ψ(Q).

ψ(Q) is free allowing to improveperformance.



PBC via Passive Subsystems Decomposition

λ is flux, θ position, u voltage, i current.

u

i

y

g

m

λ



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.



Induction Motor

Model

Σe

λ = L(θ)i ∈ R4

λ+Ri =

vs

0

Σm

jω = τ − τL

τ = 12i>

∂L(θ)∂θ

i

where L(θ) =

LsI2 Lsre

Jnpθ

Lsre−Jnpθ LrI2

is inductance matrix, λ =

λs

λr

flux,

i =

is

ir

current, θ rotor angle, vs stator voltage, τ torque, τL load torque.

Proposition The IM defines a passive operator with port variables

(V, I)4= (

vs

−ω

,

is

τ

),

and storage function electric energy: 12λ>L−1(θ)λ.



Proposed PBC

Controller in implicit form

Σc

λd +Rid =

vs

0

λd = L(θ)id

τd = 12i>d

∂L(θ)∂θ

id

Proposition Σc defines a passive operator with port variables

(V, Id)4= (

vs

−ω

,

isd

τd

).

Fact Given τd, βd, Σc is realizable via

λdr = eJρd

βd

0

, ρd =

Rr

npβ2d

τd, vs = vs(λd, τd, βd, θ).




∑E

∑C

∑

I dI

v -

+

I~

Yields error dynamics Σ such that I → 0 (exp).

Extended to the generalized machine (Nicklasson, et al., TAC’97).




Sensorless control of PMSM with guaranteed stability properties, (Shah, et al.,IFAC’11).

PBC of doubly–fed induction machines, for energy management applications (Battle, etal., EJC’07).

CbI of doubly–fed induction machines, (Battle, et al., IJC’11).



Dynamic Energy Router



Formulation of the Energy Transfer Problem

Energy management between storing, generating and load units interconnected throughpower electronic devices



Current Practice, Limitations and Objective

Assume that system operates in steady state

Translate power demand into current or voltage references

Track references with PI controllers in the power converters

Discriminate between fast and slow changing power demand via linear filtering

⇒ Behavior below par during transients and for fast changing demands

Our objective is to propose a ΣI that

Does not rely on steady–state considerations

Allows to incorporate dynamic restrictions of the units

Handles dissipation



Mathematical Formulation of the Problem

Units modeled as multiports Σj with port variables vj(t), ij(t) ∈ Rm

They verify the energy conservation law

Hj(t)−Hj(0) =

∫ t

0v>j (s)ij(s)ds− dj(t),

Hj(t) is the stored energy,

The supplied energy is, ∫ t

0v>j (s)ij(s)ds.

dj(t) ≥ 0 is the dissipation.



Duindam–Stramigioli Dynamic Energy Router

DS–DER is a power–preserving interconnection ΣI that transfer instantaneously theenergy from one unit to the other. (Sanchez, et al., IEEE–CSM’10)

Assume, for simplicity, two ports

ΣI is power preserving selecting

ΣI :

i1(t)

i2(t)

=

0 Γ(t)

−Γ>(t) 0

v1(t)

v2(t)

Indeed,

i>1 v1 + i>2 v2 = 0,

for any Γ ∈ Rn×n.



cont’d

Now, neglecting dissipation,

H1 = v>1 i1 = v>1 Γv2

H2 = v>2 i2 = −v>2 Γ>v1.

How to select Γ? Take, for instance

Γ(t) = α(t)v1(t)v>2 (t), α(t) ∈ R

then

H1 = α|v1|2|v2|

2

H2 = −α|v1|2|v2|

2.

α > 0 transfers all energy from Σ2 to Σ1, (α < 0, viceversa).

Selecting the “shape" of α(t) we can regulate the energy transfer rate.



Power Electronics Implementation of DER

Design a control law for the DER switches, which ensures that the currents track theirdesired references

i?1(t)

i?2(t)

=

α(t)v1(t)v22(t)

−α(t)v2(t)v21(t)

.

Σ1,Σ2 are supercapacitors.



Experimental Results

α(t) controls the direction and rate of change of energy flow.

2 3 4 5 6 7 8 9 10 11

−0.01

−0.005

0

0.005

0.01R

efer

enceα

Time (s)

Fundamental problem: The power balance of the DER becomes

HI(t) = v1(t)i1(t) + v2(t)i2(t)︸︷︷︸=0

− dI(t) ≤ 0,

energy decreases and it becomes non-operational.



Compensating the Dissipation

Adding outer–loop PI’s to a feedback linearizing control to regulate the voltage vC(t):

wj(t) = −kp ij(t)− ki

∫ t

0ij(s)ds− kpv vC(t)− kiv

∫ t

0vC(s)ds, j = 1, 2.

0 2 4 6 8 10−150

−100

−50

0

50

100

150

Pow

er(W

)

Time (s) (a)

0 2 4 6 8 102000

2200

2400

2600

2800

3000

En

ergy

(J)

Time (s)(b)



Currents of ΣI and their References

0 2 4 6 8 10−15

−10

−5

0

5

10

15

Cu

rren

t(A

)

Time (s)(a)

0 2 4 6 8 10−15

−10

−5

0

5

10

15

Cu

rren

t(A

)

Time(s)(b)

2.2 A

2.2 A

2 A

2 A0.8 A

0.8 A



Voltage of DC Link

0 2 4 6 8 100

5

10

15

20

Vol

tage

(V)

Time (s)



Proposed Solution: Abandon Power Preservation

Define mappings Fj(v) for the current references:

i?j (t) = Fj(v(t)), j ∈ N ,

Two different objectives:

Ensure the desired power dispatch, P ?j (t) = v>j (t)Fj(v(t)).

Compensate dissipation,∑N

j=1 v>j (t)Fj(v(t)) = dI (t).

Possible choice

Fj(v) = δjΠNk=1,k 6=j |vk|

2vj ,

N∑

j=1

δj(t) = dI(t).

If |vj(t)| ≥ ε > 0, fix

Fj(vj(t)) =P ?j (t)

|vj(t)|2vj(t),

with∑N

j=1 P?j (t) = dI(t).



Geometric Interpretation of the New DER and the DS–DER

Given v and dI , the set F defines the admissible vectors F (v), that satisfy

N∑

j=1

v>j (t)Fj(v(t)) = dI (t).



Simulation Results of the New DER

A battery is added as a third port, to compensate the losses.

Same energy pattern for the supercapacitors as before, i.e. P ?1 (t) = −P ?

2 (t), but

P ?3 (t) = dI (t) = R1i

21(t) + R2i

22(t) + R3i

23(t).



Power of Multiports

0 1 2 3 4 5 6 7 8 9 10 11−100

−50

0

50

100

Pow

er1

(W)

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11−100

−50

0

50

100

Pow

er2

(W)

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11

0

20

40

60

Pow

er3

(W)

Time (s)(a)



Voltage of DC Link

0 2 4 6 8 10 1218.8

19

19.2

19.4

19.6

19.8

20

20.2

Volt

age

(V)

Time (s)




Power factor compensation is equivalent to cyclo–dissipasivation: The nonlinearnon–sinusoidal case, (Garcia, et al., IEEE-CSM’07).

Proof of passivity of a PEM fuel cell model, (Talj and Ortega, Automatica’11).

Passivity and robust PI control of the air supply system of a PEM fuel cell model, (Talj,et al., IEEE TIE’10).



Wind Speed Estimation in WindmillSystems



Mathematical Model of the Windmill System

System is a wind turbine and a generator.

The mechanical dynamics,

Jωm = Tm − Te, (Σ)

with J the rotor inertia, P is the number of pole pairs, ωm the mechanical speed, Tethe electrical torque, and the mechanical torque

Tm =Pw

ωm.

The mechanical power at the windmill shaft

Pw =1

2ρACp(

rωm

vw)v3w,

with Cp(λ), λ := rωm

vw, the power coefficient and vw the unknown wind speed, which

enters nonlinearly.



Assumptions for Wind Speed Estimation

Assumption 1 The power coefficient is aknown, smooth, function, e.g.,

Cp(λ) = e−cp1λ

( cp2λ

− cp3

)+ cp4λ, ,

which verifies

C′p(λ)

> 0 for λ ∈ [0, λ?)

= 0 for λ = λ?

< 0 for λ ∈ (λ?, λM ],

where λ? := argmaxCp(λ).

λ

Cp

λ∗

Cp∗

Assumption 2 The wind speed vw is an unknown positive constant.

Assumption 3 The electrical torque Te and the motor speed wm are measurable.

Assumption 4 For all λ ∈ (0, λ?), the power coefficient verifies

κ(λ) :=3

λCp(λ)− C′

p(λ) > 0.



Some Remarks

Cp(λ) can be easily obtained from experimental data, and the algorithm implementedfrom a table look–up.

Constant wind speed assumption only needed for the theory. An on–line estimator isable to track slowly–varying parameters, assumption justified by the time scaleseparation between the wind dynamics and the mechanical and electrical signals.

On–line estimators average the noise—in contrast with differentiator–based orextended Kalman filter schemes currently used.

Measuring wm and Te is standard practice in windmill systems.

Theory applicable also if blade pitch β is included, i.e., Cp(λ, β), or for more completedescriptions of the mechanical dynamics.

Assumption 4 is satisfied in normal operating range (for Region 2), where the torquecoefficient has negative slope. Indeed,

CT (λ) :=1

λCp(λ),

satisfies

C′T (λ) ≤ 0 ⇒ Assumption 4.



Immersion and Invariance Parameter Estimators

Proposition (Liu, et al., TAC’10, SCL’11) Consider the system

x = F (t) + Φ(x, θ),

where x ∈ R, the function F (t) and the mapping Φ : R× R → R are known, and θ ∈ R is aconstant unknown parameter. Assume there exists a smooth mapping β : R → R such thatthe parameterized mapping

Qx(θ) := β′(x)Φ(x, θ)

is strictly monotone increasing. The I&I estimator

˙θI = −β′(x)

[F (t) + Φ(x, θI + β(x))

]

θ = θI + β(x),

is asymptotically consistent. That is, limt→∞ θ(t) = θ. for all (x(0), θI(0)) ∈ R× R, andF (t) such that (x(t), θ(t)) exist for all t ≥ 0.



Verifying the Monotonicity Condition

Apply Proposition with x = ωm, θ = ωm and F (t) = − 1JTe(t).

Key step is to construct a function β(·), such that the parameterized function

Qωm(vw) = β′(ωm)Φ(ωm, vw)

is strictly monotonically increasing.

The latter is true if and only if its derivative is positive,

Q′ωm

(vw) = β′(ωm)∂Φ(ωm, vw)

∂vw

=ρAr

2Jvwβ

′(ωm)

[3vw

rωm

Cp

(rωm

vw

)− C′

p

(rωm

vw

)].

The term in brackets, is precisely κ(λ), hence the need for Assumption 4.



Main Estimation Result

Proposition Consider the system (Σ), verifying Assumptions 1–4. The I&I estimator

˙vIw = γ

[Te −

ρA

2

(vIw + γωm)3

ωm

Cp

(rωm

vIw + γωm

)]

vw = vIw + γωm,

where γ > 0, is an adaptation gain, is asymptotically consistent, that is,

limt→∞

vw(t) = vw.

Question Can Assumption 4 be relaxed?

It is not (globally) satisfied for several known turbines.

Analysis when it does not hold?



Simulation Results: Periodic Wind

Done in Vestas professional software, with a full model including torsional modes, look–uptable for Cp(λ), and real wind data.

−100 0 100 200 300 400 500 6008

10

12

14

16

18

20

22

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS



Simulation Results: Turbulent Wind

−100 0 100 200 300 400 500 60010

12

14

16

18

20

22

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS



Simulation Results: Gust

140 160 180 200 220 240 260 280

12

13

14

15

16

17

18

19

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS


Documents

A “New Control Theory to Face the Challenges of Modern