port-controlled hamiltonian systems: modelling and control · 2018. 2. 8. · vilanova i la geltru...
TRANSCRIPT
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
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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
Arnau Doria-Cerezo PCHS: modelling and control 2/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Outline
1 Introduction
2 Port-controlled Hamiltonian systems. Modelling
3 Port-controlled Hamiltonian systems. Control
Arnau Doria-Cerezo PCHS: modelling and control 3/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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)
Arnau Doria-Cerezo PCHS: modelling and control 4/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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
Arnau Doria-Cerezo PCHS: modelling and control 5/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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.
Arnau Doria-Cerezo PCHS: modelling and control 6/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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
Arnau Doria-Cerezo PCHS: modelling and control 7/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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
Arnau Doria-Cerezo PCHS: modelling and control 8/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
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.
Arnau Doria-Cerezo PCHS: modelling and control 9/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
Outline
1 Introduction
2 Port-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems
PCHS examples
3 Port-controlled Hamiltonian systems. Control
Arnau Doria-Cerezo PCHS: modelling and control 10/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
Port-controlled Hamiltonian systems
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.
Arnau Doria-Cerezo PCHS: modelling and control 11/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
Port-controlled Hamiltonian systems
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.
Arnau Doria-Cerezo PCHS: modelling and control 12/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
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.
Arnau Doria-Cerezo PCHS: modelling and control 13/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
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]
Arnau Doria-Cerezo PCHS: modelling and control 14/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
Example: levitation ball
+
-
v i r
L(y)
y
m
g
Hamiltonian variables:
xT = [λ, p, y], where p = my
Hamiltonian function:
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
Port connection matrix:
gT =ˆ
1 0 0˜
Port input:
u = v
Passive output:
y = i
Arnau Doria-Cerezo PCHS: modelling and control 15/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Port-controlled Hamiltonian systemsPCHS examples
Example: DC-DC boost converter
+
u
r L
C
i
Rl
s
Hamiltonian variables:
xT = [λ, q]
Hamiltonian function:
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
–
Port connection matrix:
gT =ˆ
1 0˜
Port input:
u = v
Passive output:
y = i
Arnau Doria-Cerezo PCHS: modelling and control 16/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Outline
1 Introduction
2 Port-controlled Hamiltonian systems. Modelling
3 Port-controlled Hamiltonian systems. Control
Stability properties of the PCHS
Interconnection and Damping Assignment-PBC
Advances on IDA-PBC
Arnau Doria-Cerezo PCHS: modelling and control 17/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Stability properties of the PCHS
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.
Arnau Doria-Cerezo PCHS: modelling and control 18/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Stability properties of the PCHS
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.
Arnau Doria-Cerezo PCHS: modelling and control 18/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Stability properties of the PCHS
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.
Arnau Doria-Cerezo PCHS: modelling and control 18/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Stability properties of the PCHS
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.
Arnau Doria-Cerezo PCHS: modelling and control 18/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Stability properties of the PCHS
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.
Arnau Doria-Cerezo PCHS: modelling and control 18/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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)
Arnau Doria-Cerezo PCHS: modelling and control 19/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
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.
Arnau Doria-Cerezo PCHS: modelling and control 20/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd
Non-Parameterized IDA
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
Arnau Doria-Cerezo PCHS: modelling and control 21/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd
Non-Parameterized IDA
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
Arnau Doria-Cerezo PCHS: modelling and control 21/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
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)
Arnau Doria-Cerezo PCHS: modelling and control 22/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
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)
Arnau Doria-Cerezo PCHS: modelling and control 22/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd
Interlaced Algebraic-Parameterized IDA
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
Arnau Doria-Cerezo PCHS: modelling and control 23/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Interconnection and Damping Assignment-PBC
Solving the matching equation...
f(x) + g(x)u = (Jd(x) − Rd(x))∂Hd
Interlaced Algebraic-Parameterized IDA
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
Arnau Doria-Cerezo PCHS: modelling and control 23/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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.
Arnau Doria-Cerezo PCHS: modelling and control 24/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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.
Arnau Doria-Cerezo PCHS: modelling and control 24/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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.
Arnau Doria-Cerezo PCHS: modelling and control 24/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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).
Arnau Doria-Cerezo PCHS: modelling and control 25/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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).
Arnau Doria-Cerezo PCHS: modelling and control 25/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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).
Arnau Doria-Cerezo PCHS: modelling and control 25/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Example: The control design
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.
Arnau Doria-Cerezo PCHS: modelling and control 26/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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]
gamma=0.1gamma=1gamma=3
Arnau Doria-Cerezo PCHS: modelling and control 27/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Example: Simulations
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
gamma=0.1gamma=1gamma=3
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).
Arnau Doria-Cerezo PCHS: modelling and control 28/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Example: Simulations
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
Arnau Doria-Cerezo PCHS: modelling and control 29/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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.
Arnau Doria-Cerezo PCHS: modelling and control 30/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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]
Arnau Doria-Cerezo PCHS: modelling and control 31/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
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!!!
Arnau Doria-Cerezo PCHS: modelling and control 32/33
IntroductionPort-controlled Hamiltonian systems. Modelling
Port-controlled Hamiltonian systems. Control
Stability properties of the PCHSInterconnection and Damping Assignment-PBCExampleAdvances on IDA-PBC
Port-controlled Hamiltonian Systems:
modelling and control
Arnau [email protected]
Arnau Doria-Cerezo PCHS: modelling and control 33/33