Port-controlled Hamiltonian Systems: modelling and control · 2018. 2. 8. · Vilanova i la Geltru Arnau Do`ria-Cerezo PCHS: modelling and control 5/33. Introduction Port-controlled

IntroductionPort-controlled Hamiltonian systems. Modelling

Port-controlled Hamiltonian systems. Control

Port-controlled Hamiltonian Systems:

modelling and control

Arnau [email protected]

Arnau Doria-Cerezo PCHS: modelling and control 1/33



Outline

1 Introduction

2 Port-controlled Hamiltonian systems. Modelling

Port-controlled Hamiltonian systems

PCHS examples

3 Port-controlled Hamiltonian systems. Control

Stability properties of the PCHS

Interconnection and Damping Assignment-PBC

Advances on IDA-PBC




Outline

1 Introduction






Affiliation

Universitat Politecnica de Catalunya

http://www.upc.edu

Teaching tasks

Department of Electrical Engineering

Research tasks

Institute of Control and Industrial Engineering (Research institute)

Advanced Control on Energy Systems (Research group)




UPC - Universitat Politecnica de Catalunya

The UPC is present in 9 cities near to Barcelona:

Barcelona

Canet de Mar

Castelldefels

Igualada

Manresa

Mataro

Terrassa

Sant Cugat del Valles

Vilanova i la Geltru




ACES Research group

Advanced Control on Energy Systems

Keyboards: Complex systems, non-linear systems, power systems, power electronics, control theory.

Strategy Goals

Research: modelling and control of complex systems, and its applicationto problems related to the generation, conditioning, management andstorage of electrical energy.

Technology transfer: Diffusion of technological advances to the Industry

Training of specialized personnel: Participation in

three PhD programs: Advanced Automation and Robotics, AppliedMathematics, and Applied Physics.four Master programs: Robotics and automation, Appliedmathematics, Engineering mathematics and Electronics.




ACES Staff

Universitat Politecnica de Catalunya

Carles Batlle: M.Sc. and Ph.D. degrees in Physics

Domingo Biel: M.Sc. and Ph.D. degrees in Telecommunication Engineering

Ramon Costa-Castello: M.Sc. and Ph.D. degrees in Computer Science

Arnau Doria-Cerezo: M.Sc. degree in Electrical Eng. and Ph.D. in Automatic Control

Enric Fossas: M.Sc. and Ph.D. degrees in Mathematics (person in charge)

Jaume Franch: M.Sc. and Ph.D. degrees in Mathematics

Robert Grino: M.Sc. degree in Electrical Eng. and the Ph.D. degree in Automatic Control

Josep Maria Olm: M.Sc. degree in Physics and Ph.D. in Automatic Control

Ester Simo: M.Sc. and Ph.D. degrees in Mathematics

Consejo Superior de Investigaciones Cientıficas

Jordi Riera: M.Sc. and Ph.D. degrees in Mechanical Engineering (person in charge)

Maria Serra: M.Sc. degree in Physics and Ph.D. in Chemical Engineering

Multidisciplinary group




ACES resources

Human resources

PhD 11

Lab Technicians 4

PhD students ∼10

ACES Labs

Laboratory of Industrial Power Converters

Laboratory of Electrical Machines

Laboratory of Fuel Cells




Research topics

Main research

Modelling: Bond graph, Hamiltonian systems, linear identification,complementary systems, order reduction.

Analysis: Nonlinear systems, variable structure systems, bifurcations andchaos.

Design: Linear control, repetitive control, passivity-based control,flatness, sliding mode control, digital control.

Implementation: DSP, FPGA, LabView, RT-Linux.

Mainly applied to electromechanical systems and fuel cells.




Port-controlled Hamiltonian systemsPCHS examples

Outline

1 Introduction



PCHS examples







Port-controlled Hamiltonian systems...

are a class of system description based on the network modelling of physicalsystems.

Attractive aspects...

Linear or non-linear systems.

Multidomain description (electrical, mechanical, thermodynamics,...).

This formalism underscores the physics of the system: it contains theenergy storage, dissipation, and interconnection structure of the system,

The interconnection between Hamiltonian systems results in aHamiltonian system.

Hamiltonian systems are close to the classical Lagrangian methods, bothtechniques use the state dependent energy or co-energy functions tocharacterize the dynamics of the different elements.






Why an PCH model?

Allows to model complex systems,

with clear interconnection rules, and

preserving the physical structure of the system.

Energy-based description is an appropriate modeling framework to designpassivity-based controllers.





PCHS form

An explicit PCHS have the from1

x = (J(x) − R(x))∂xH(x) + g(x)uy = gT (x)∂xH(x)

where

x ∈ Rn is the vector state, or Hamiltonian variables.

u, y ∈ Rm are the port variables.

H(x) : Rn → R is the Hamiltonian function (or energy function).

J(x) ∈ Rn×n is the interconnection matrix (J(x) = −J(x)T ).

R(x) ∈ Rn×n is the dissipation matrix (R(x) = RT ≥ 0).

g(x) ∈ Rn×m is the external port connection matrix.

1The ∂x operator defines the gradient of a function of x, and in what follows wewill take it as a column vector.





Example: DC-motor

-

+

-

+if ia

vf va

rf ra

Lf

La

ω

Hamiltonian variables:

xT = [λf , λa, p], where p = Jmω

Hamiltonian function:

H = 1

2Lfλ2

f + 1

2Laλ2

a + 1

2Jmp2

Interconnection matrix:

J(x) =

2

4

0 0 00 0 −Laf if0 Laf if 0

3

5

PCHS form

x = (J(x) − R(x))∂xH(x) + g(x)u

Dissipation matrix:

R =

2

4

rf 0 00 ra 00 0 Br

3

5

Port connection matrix:

g = I3

Port input:

uT = [vf , va, τL]

Passive outputs:

yT = [if , ia, ω]





Example: levitation ball

+

-

v i r

L(y)

y

m

g


xT = [λ, p, y], where p = my


H = 1

2k(a + y)λ2 + 1

2mp2 − mgy

PCHS form

x = (J(x) − R(x))∂xH(x) + g(x)u

Interconnection and dissipationmatrices:

J − R =

2

4

−r 0 00 0 −10 1 0

3

5


gT =ˆ

1 0 0˜

Port input:

u = v

Passive output:

y = i





Example: DC-DC boost converter

+

u

r L

C

i

Rl

s


xT = [λ, q]


H = 1

2Lλ2 + 1

2Cq2

Interconnection matrix:

J =

»

0 −s

s 0

–

s ∈ {0, 1}

PCHS form

x = (J(u) − R(x))∂xH(x) + g(x)u

Power converters are variable structuresystems

Dissipation matrix:

R =

»

r 00 1

Rl

–


gT =ˆ

1 0˜

Port input:

u = v

Passive output:

y = i




Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC

Outline

1 Introduction





Advances on IDA-PBC






Consider a PCHS

x = (J(x) − R(x))∂xH(x) + g(x)u

with the energy function bounded from below (H(x) > c), then its derivative

H(x) = (∂H)T x = (∂H)T (J(x) − R(x))∂H + (∂H)T g(x)u

with J(x) = −J(x)T and y = gT (x)∂H the power-balance equation is recovered

H(x)| {z }

Stored power

= − (∂H)T R(x)∂H| {z }

Dissipated power

+ yT u|{z}

Supplied power

with R(x) = RT (x) ≥ 0 and considering u = 0, the system is asymptotically stable

H(x) ≤ 0.

The Hamiltonian function H(x) is a Lyapunov function.






Consider a PCHS

x = (J(x) − R(x))∂xH(x) + g(x)u




H(x)| {z }

Stored power

= − (∂H)T R(x)∂H| {z }

Dissipated power

+ yT u|{z}

Supplied power


H(x) ≤ 0.







Consider a PCHS

x = (J(x) − R(x))∂xH(x) + g(x)u




H(x)| {z }

Stored power

= − (∂H)T R(x)∂H| {z }

Dissipated power

+ yT u|{z}

Supplied power


H(x) ≤ 0.







Consider a PCHS

x = (J(x) − R(x))∂xH(x) + g(x)u




H(x)| {z }

Stored power

= − (∂H)T R(x)∂H| {z }

Dissipated power

+ yT u|{z}

Supplied power


H(x) ≤ 0.







Consider a PCHS

x = (J(x) − R(x))∂xH(x) + g(x)u




H(x)| {z }

Stored power

= − (∂H)T R(x)∂H| {z }

Dissipated power

+ yT u|{z}

Supplied power


H(x) ≤ 0.







Consider an affine dynamical system: x = f(x) + g(x)u.

+ +Σc ΣI Σ

yc

uc

y

u

Control by interconnection:ΣC + ΣI + Σ = PCHS

The whole system behaves as x = (Jd(x) − Rd(x))∂Hd where

Hd(x) > c has a local minimum at x∗,

Jd = −JTd and Rd = RT

d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd

The control law, u = β(x)

β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)







+ +Σc ΣI Σ

yc

uc

y

u





d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)







+ +Σc ΣI Σ

yc

uc

y

u





d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)







+ +Σc ΣI Σ

yc

uc

y

u





d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)







+ +Σc ΣI Σ

yc

uc

y

u





d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)







+ +Σc ΣI Σ

yc

uc

y

u





d ≥ 0.

Matching equation

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


β(x) = (gT (x)g(x))−1g

T (x)((Jd(x) − Rd(x))∂Hd − f(x)






Solving the matching equation...

f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd

Three degrees of freedom:

Jd(x): Interconnection assignment

Rd(x): Damping assignment

Hd(x): Energy shaping

Proposed methods to solve the ME

Non-Parameterized IDA

Algebraic IDA

Interlaced Algebraic-Parameterized IDA

There is not a best method to solve the matching equation. Each controlproblem requires an individual study to find out which of the above strategiesprovides an acceptable solution of the matching equation.







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


1 the Jd(x) and Rd(x) are fixed,

2 the matching equation is pre-multiplied by a left annihilator of g(x)(g(x)⊥g(x) = 0)

3 and the resulting PDE in Hd is then solved.

g(x)⊥f(x) = g(x)⊥(Jd(x) − Rd(x))∂Hd

First proposed method

+ Nice and original controllers

- Difficulty on to solve some PDE







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


1 the Jd(x) and Rd(x) are fixed,

2 the matching equation is pre-multiplied by a left annihilator of g(x)(g(x)⊥g(x) = 0)

3 and the resulting PDE in Hd is then solved.

g(x)⊥f(x) = g(x)⊥(Jd(x) − Rd(x))∂Hd

First proposed method

+ Nice and original controllers

- Difficulty on to solve some PDE







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd

Algebraic IDA

1 the Hd function is first selected (for example a quadratic function in theerror terms)

2 and then the resulting algebraic equations are solved for Jd and Rd.

+ Easy solutions

+ Control law visible along the desing process

- Feedback linearization term (Robustness)







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd

Algebraic IDA

1 the Hd function is first selected (for example a quadratic function in theerror terms)

2 and then the resulting algebraic equations are solved for Jd and Rd.

+ Easy solutions

+ Control law visible along the desing process

- Feedback linearization term (Robustness)







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


1 the PDE is evaluated in some subspace (where the solution can be easilycomputed)

2 and then matrices Jd , Rd are found which ensure a valid solution of thematching equation.

+ Tailored solutions

- Extra degree of freedom







f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd


1 the PDE is evaluated in some subspace (where the solution can be easilycomputed)

2 and then matrices Jd , Rd are found which ensure a valid solution of thematching equation.

+ Tailored solutions

- Extra degree of freedom





Example: The model

Consider the following 2-dimensional nonlinear system

x1 = −x1 + ξx2

2

x2 = −x1x2 + u,(1)

where ξ > 0. The control objective is to regulate x2 to a desired value xd2. The

equilibrium of (1) corresponding to this is given by

x∗1 = ξ(xd

2)2, u∗ = ξ(xd2)3.

This system can be cast into PCHS form

x = (J − R)∂H + gu (2)

with

J =

»0 x2

−x2 0

–

, R =

»1 00 0

–

, g =

»01

–

H(x) =1

2x2

1 +1

2ξx2

2.





Example: The model


x1 = −x1 + ξx2

2

x2 = −x1x2 + u,(1)



x∗1 = ξ(xd

2)2, u∗ = ξ(xd2)3.


x = (J − R)∂H + gu (2)

with

J =

»0 x2

−x2 0

–

, R =

»1 00 0

–

, g =

»01

–

H(x) =1

2x2

1 +1

2ξx2

2.





Example: The model


x1 = −x1 + ξx2

2

x2 = −x1x2 + u,(1)



x∗1 = ξ(xd

2)2, u∗ = ξ(xd2)3.


x = (J − R)∂H + gu (2)

with

J =

»0 x2

−x2 0

–

, R =

»1 00 0

–

, g =

»01

–

H(x) =1

2x2

1 +1

2ξx2

2.





Example: The control design

Using the IDA-PBC technique, within the algebraic approach, we match (2) to

x = (Jd − Rd)∂Hd

and the Hamitonian function is selected as

Hd(x) =1

2(x1 − x∗

1)2 +1

2γ(x2 − xd

2)2,

where γ > 0 is an adjustable parameter. Interconnection and damping matrices havethe form

Jd =

»0 α(x)

−α(x) 0

–

, Rd =

»1 00 r

–

,

where α(x) is a function to be determined by the matching procedure and r > 0. Fromthe first row of the matching equation (J − R)∂H + gu = (Jd − Rd)∂Hd one gets

−x1 + ξx2

2 = −(x1 − x∗1) +

α

γ(x2 − xd

2),

from whichα(x) =

γ

x2 − xd2

(ξx2

2 − x∗1) = . . . = γξ(x2 + xd

2).









Hd(x) =1

2(x1 − x∗

1)2 +1

2γ(x2 − xd

2)2,


Jd =

»0 α(x)

−α(x) 0

–

, Rd =

»1 00 r

–

,


−x1 + ξx2

2 = −(x1 − x∗1) +

α

γ(x2 − xd

2),

from whichα(x) =

γ

x2 − xd2

(ξx2

2 − x∗1) = . . . = γξ(x2 + xd

2).









Hd(x) =1

2(x1 − x∗

1)2 +1

2γ(x2 − xd

2)2,


Jd =

»0 α(x)

−α(x) 0

–

, Rd =

»1 00 r

–

,


−x1 + ξx2

2 = −(x1 − x∗1) +

α

γ(x2 − xd

2),

from whichα(x) =

γ

x2 − xd2

(ξx2

2 − x∗1) = . . . = γξ(x2 + xd

2).






Substituting α(x) into the second row of the matching equation

−x1x2 + u = −α(x1 − x∗1) −

r

γ(x2 − xd

2),

yields the feedback control law

u = x1x2 − γξ(x1 − x∗1)(x2 + xd

2) −r

γ(x2 − xd

2). (3)

This control law yields a closed-loop system which is Hamiltonian with (Jd, Rd, Hd),

and which has (x∗1, xd

2) as a globally asymptotically stable equilibrium point.





Example: Simulations

x1 and x2 behaviour

0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

gamma=1

x 2

time [s]

r=0.5r=1r=5

0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

r=1

x 2

time [s]

gamma=0.1gamma=1gamma=3

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5gamma=1

x 1

time [s]

r=0.5r=1r=5

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5r=1

x 1

time [s]







Phase portrait plots

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Phase Portrait, gamma=1

x 2

x1

r=0.5r=1r=5

0 0.5 1 1.5 2 2.5 3 3.5−3

−2

−1

0

1

2

3Phase Portrait, r=1

x 2

x1


Notice that the γ parameter has more influence on the trajectories. This is due to the fact that γ modifies the Hamiltonian in the x2

direction and tuning this parameter makes trajectories of x2 restricted (or semi-bounded).






The energy function Hd = 1

2(x1 − x∗

1)2 + 1

2γ(x2 − xd

2)2

–2

–1

0

1

2

3

4

x1

–2

–1

0

1

2

3

4

x2

0

2

4

6

8γ = 0.1

γ = 1

γ = 3





Advances on the IDA-PBC technique

Simultaneous IDA-PBC technique

The SIDA-PBC approach suggests

f(x) + g(x)u = Fd(x)∂Hd(x)

where Fd(x) + Fd(x)T ≤ 0 instead of f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd(x).

Application of SIDA-PBC also yields a closed-loop PCH system of the formx = (Jd(x) − Rd(x))∂Hd(x) with

Jd(x) =1

2(Fd(x) − F T

d (x)), Rd(x) =1

2(Fd(x) + F T

d (x)).

The interconnection and damping matrices are fixed simultaneously.

One restriction Fd(x) + Fd(x)T ≤ 0

instead of two Jd(x) = −Jd(x)T , Rd(x) + Rd(x)T ≥ 0.

SIDA-PBC enlarge the set of systems that can be stabilized via IDA-PBC.

SIDA-PBC allows to design output feedback, as opposed to state feedback,controllers.





SIDA-PBC: Example

Consider the previous dynamical system,

x1 = −x1 + ξx2

2

x2 = −x1x2 + u

where x2 is the desired output variable and x1 not-measurable.

The standard IDA-PBC the control law is...

u = x1x2 − γξ(x1 − x∗1)(x2 + xd

2) − r

γ(x2 − xd

2)

which depends on x1.

Applying the SIDA-PBC approach

u = x∗1x2 − k(x2 − xd

2) with k ≥ 1

4ξxd

2

we obtain a simpler and output feedback controller(only x2 measures are required).

Simulation parameters:ξ = 2, xd

2= 1 and k = 1.

0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

2

2.5

3

x 1

0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

2

x 2

time [s]





Advances on the IDA-PBC technique

More advances on the IDA-PBC technique...

Robustness is the main problem of the controllers obtained via IDA-PBC.

Mainly static controllers are obtained. Some works deals on dynamic extensionsto improve the performance and robustness.

The basic IDA-PBC is restricted to regulation problems, tracking remains anopen issue.

To increase the range of applications where the IDA-PBC is used.

...

... are coming soon!!!





Port-controlled Hamiltonian Systems:

modelling and control

Arnau [email protected]


Documents

Port-controlled Hamiltonian Systems: modelling and control · 2018. 2. 8. · Vilanova i la Geltru Arnau Do`ria-Cerezo PCHS: modelling and control 5/33. Introduction Port-controlled