Marco Liserre [e-mail]

1
Passivity based control applied to power
converters
Marco Liserre liserre_at_poliba.it
2
Passivity based control history

• 70s definition of dissipative systems (Willems)
• 1981 application to rigid robots (Arimoto e
  Takagi)
• in power electronics . . .
• 1991 first theoretical paper (Ortega, Espinoza
  others)
• 1996 first experimental paper (Cecati, others,
  IAS Annual Meeting)
• 1998 first book Passivity-Based Control of
  Euler-Lagrange Systems
• (Ortega and Sira-Ramirez, Springer, ISBN
  1852330163)
• 1999 application to active filters (Mattavelli
  and Stankovic, ISCAS 99)
• 2002 Brayton-Moser formulation (Jeltsema and
  Scherpen, Am. Control Conf.)
• 2002 application to multilevel converters
  (Cecati, DellAquila, Liserre, Monopoli, IECON
  2002)

3
Contribution of my research group on the topic

• collaborations with
• University of LAquila (Prof. Cecati)
• University of Delft (Prof. Scherpen)

• main papers
• C. Cecati, A. DellAquila, M. Liserre, V. G.
  Monopoli A passivity-based multilevel active
  rectifier with adaptive compensation for traction
  applications IEEE Transactions on Industry
  Applications, Sep./Oct. 2003, vol. 39, no. 5.
• A. DellAquila, M. Liserre, V. G. Monopoli, P.
  Rotondo An Energy-Based Control for an
  n-H-Bridges Multilevel Active Rectifier IEEE
  Transactions on Industrial Electronics, June
  2005, vol. 52, no. 3.

4
Basic idea of the Passivity-based approach

• The basic idea of the PBC is to use the energy to
  describe the state of the system
• Since the main goal of any controller is to feed
  a dynamic system through a desired evolution as
  well as to guarantee its steady state behavior,
  an energy-based controller shapes the energy of
  the system and its variations according to the
  desired state trajectory
• If the controller is designed aiming at obtaining
  the minimum energy transformation, optimum
  control is achieved
• The PBC offers a method to design controllers
  that make the system Lyapunov-stable
• The energy approach is particularly suitable
  when dealing with
• electromechanical systems as electrical machines
• grid connected converters (non-linear model)

5
The introduction of damping

• The control objective is usually achieved through
  an energy reshaping process and by injecting
  damping to modify the dissipation structure of
  the system
• From a circuit theoretic perspective, a PBC
  forces the closed-loop dynamics to behave as if
  there are artificial resistors the control
  parameters connected in series or in parallel
  to the real circuit elements
• When the PBC is applied to grid connected
  converters, harmonic rejection is one of the main
  task, hence the passive damping can be
  substituted by a dynamic damping (i.e. virtual
  inductive and capacitive elements should be
  added)
• The point of view is always the energy reshaping
  (i.e. the energy associated to the harmonics)

6
The Eulero-Lagrange formulation

• Passivity-based control has been firstly
  developed on the basis of Eulero-Lagrange
  formulation
• One of the major advantages of using the EL
  approach is that the physical structure (e.g.,
  energy, dissipation, and interconnection),
  including the nonlinear phenomena and features,
  is explicitly incorporated in the model, and thus
  in the corresponding PBC
• This in contrast to conventional techniques that
  are mainly based on linearized dynamics and
  corresponding proportional-integralderivative
  (PID) or leadlag control

7
The Passivity Based Controller design

• In the context of EL-based PBC designs for power
  converters, two fundamental questions arise
• which variables have to be stabilized to a
  certain value in order to regulate the output(s)
  of interest toward a desired equilibrium value?
  In other words, are the zero-dynamics of the
  output(s) to be controlled stable with respect to
  the available control input(s), and if not, for
  which state variables are they stable?
• where to inject the damping and how to tune the
  various parameters associated to the energy
  modification and to the damping assignment stage?

8
Dissipativity definition
dissipativity definition
9
Passivity definition
10
Definitions

• Supply Rate speed of the energy flow from a
  source to the system
• Storage function energy accumulated in a system
• Dissipative systems systems verifying
  dissipation inequality
• Along time trajectories of dissipative systems
  the following relationship holds
• energy flow storage function
• (In other words, dissipative systems can
  accumulate less energy than that supplied by
  external sources)
• The basic idea of PBC is to shape the energy of
  the system according to a desired state
  trajectory, leaving uncontrolled those parts of
  the system not involved in energy
  transformations, this result can be obtained only
  working on strictly passive systems

11
Feedback systems decomposition

• dividing the system into simpler subsystems, each
  one identifying those parts of the system
  actively involved in energy transformations
• each subsystem has to be passive introducing
  energy balances, expressed in terms of the
  Eulero-Lagrange equations

passivity invariance
12
Feedback systems decomposition

• The full order model describing the system is
  divided into simpler subsystems identifying those
  parts actively involved in energy transformation
• Hence, energy balances, expressed in terms of the
  Eulero-Lagrange equations (based on the
  variational method and energy functions expressed
  in terms of generalised coordinates), are
  introduced
• The system goes in the direction where the
  integral of the Lagrangian is minimized
  (Hamilton's principle)

13
Feedback systems decomposition

• This formulation highlights active, dissipative
  and workless forces i.e. the active parts of the
  system (those which energy can be modified by
  external forces), those passive (i.e. dissipating
  energy, e.g. thermal energy), and those parts
  which do not contribute in any form to control
  actions and can be neglected during controller
  design
• Because of the energy approach, it is quite
  straightforward to obtain fast response under
  condition that the control "moves" the minimum
  amount of energy inside the system
• Moreover, because global stability is ensured by
  passivity properties, a simple a effective
  controller can be designed

14
Eulero-Lagrange formulation

• The eulero-lagrange formulation is particularly
  suited for the control of electromechanical
  systems as electrical motors
• In fact different subsystems are related by their
  ability to transform energy, therefore it is a
  good thing to define energy functions for each
  one, expressed in terms of generalised
  coordinates qi.
• In electric motor case
• qm mechanical position (for mechanical
  subsystems)
• qe electric charge (for electrical subsystems)
• Using variational approch we can introduce
  Lagrangian equations of the system and apply
  Hamilton's principle. This method highlights
  subsystems interconnections and their various
  energies dissipated, stored and supplied

15
Eulero-Lagrange formulation
induction motor formulation
The mechanical subsystem does not take an active
part in control actions, i.e. it doesn't produce
energy but only transforms and dissipates the
input energy, for design purposes its
contribution can be considered as an external
disturbance for the electrical subsystem and the
controller has to compensate for this
disturbance, in order to maintain electrical
equation balance. In passivity terms, it
defines a passive mapping around the electrical
subsystem, it can be neglected during controller
design and the attention can be focused on the
electrical subsystem.
16
Eulero-Lagrange formulation
The electrical subsystem is simply passive, then
its evolution can be corrupted by any external
disturbance leading to instability. Therefore, in
order to obtain global stability, it is an
important step of the approach to make it
strictly passive by means of the addition of a
suitable dissipative term (damping injection)
17
Passivity-based control of the H-bridge converter

• PBC has been successfully applied to d.c./d.c.
  converters, active rectifiers and multilevel
  topologies
• Particularly the single-phase Voltage Source
  Converter (VSC) also called H-bridge or full
  bridge can be used as universal converter due to
  the possibility to perform dc/dc, dc/ac or ac/dc
  conversion
• Moreover it can be used as basic cell of the
  cascade multilevel converters
• In the following it will be reviewed the
  application of the PBC to H-bridge single phase
  inverters (one-stage and multilevel) using the
  Brayton-Moser formulation which is the most
  suitable for the converter control

18
Passivity-based control of the H-bridge converter

• Control of one H-bridge-based active rectifier
• G. Escobar, D. Chevreau, R. Ortega, E. Mendes,
  An adaptive passivity-based controller for a
  unity power factor rectifier, IEEE Trans. on
  Cont. Syst. Techn., vol. 9, no. 4, July 2001, pp.
  637 644
• Control of two (multilevel) H-bridge-based active
  rectifier
• C. Cecati, A. Dell'Aquila, M. Liserre and V. G.
  Monopoli, "A passivity-based multilevel active
  rectifier with adaptive compensation for traction
  applications", IEEE Trans. on Ind. Applicat.,
  vol. 39, Sept./Oct. 2003 pp. 1404-1413
• the two dc-links are not independent !
• Control of n (multilevel) H-bridge-based active
  rectifier
• A. DellAquila, M. Liserre, V. G. Monopoli, P.
  Rotondo An Energy-Based Control for an
  n-H-Bridges Multilevel Active Rectifier IEEE
  Transactions on Industrial Electronics, June
  2005, vol. 52, no. 3.
• the n dc-links are independent !

19
Brayton-Moser Equations

• Brayton and Moser, introduced in 1964 a scalar
  function of the voltages across capacitors and
  the currents through inductors in order to
  characterize a given network
• This function was called the Mixed-Potential
  Function P(iL, vC) and it allows to analyze the
  dynamics and the stability of a broad class of
  RLC networks
• These equations can be considered an effective
  alternative to Euler-Lagrange formulation
• This formulation has a main advantage over the
  counterpart in case of power converter control
  it allows the controllers to be implemented using
  measurable quantities such as voltages and
  currents.

20
Brayton-Moser Equations

• Topologically Complete Networks networks which
  state variables form a complete set of variables
• Complete Set of Variables set of variables
  that can be chosen independently without
  violating Kirchhoffs laws and determining either
  the current or voltage (or both) in every branch
  of the network
• Additionally for Topologically Complete Networks
  it is possible to define two subnetworks
• One subnetwork has to contain all inductors and
  current-controlled resistors
• The other has to contain all capacitors and
  voltage controlled resistors

21
Brayton-Moser Equations
For the class of topologically complete networks
it is possible to construct the mixed-potential
function directly. For this class it is known
that the mixed potential is of the form

• R(iL) is the Current Potential (Content) and is
  related with the current-controlled resistors and
  voltage sources
• G(vC) is the Voltage Potential (Co-content) and
  is related with the voltage-controlled resistors
  and current sources
• N(iL,vC) is related to the internal power
  circulating across the dynamic elements

22
Brayton-Moser Equations
The components of the Mixed-Potential Function
can be analysed in more detail as follows

• PR is the Dissipative Current Potential
• PG is the Dissipative Voltage Potential

23
Brayton-Moser Equations
The dissipative current and voltage potentials
can be calculated as follows
In case of linear resistor PR is half the
dissipated power expressed in terms of inductor
current, and PG is half the dissipated power
expressed in terms of capacitor voltages.

• PE is the total supplied power to the voltage
  sources E
• PJ is the total supplied power to the current
  sources J

24
Brayton-Moser Equations

• PT is the internal power circulating across the
  dynamic elements and is represented by

In this representation ? denotes the
interconnection matrix and it is determined by
KVL and KCL
25
Brayton-Moser Equations

• Finally the expression of the mixed-potential
  function

can be rewritten as follows
PD(x) PR(x)- PG(x) is the Dissipative
Potential PF(x) PJ(x)- PE(x) is the Total
Supplied Power
26
Brayton-Moser Equations
The dynamic behaviour of topologically complete
networks is governed by the following
differential equations

• iL (iL1 , . . . , iL? )T are the currents
  through the ? inductors
• vC (vC1 , . . . , vC? )T are the voltages
  across the ? capacitors.

These differential equations correspond with
Kirchhoffs voltage and current laws
27
Brayton-Moser Equations
The previous equations can be expressed in a more
compact way as follows
with the state vector x?Rn R?? defined as
and with the diagonal square matrix Q(x) ?
R(??)x(??) defined as
28
Brayton-Moser Equations
When a circuit contains only linear passive
inductors and capacitors, then the diagonal
matrices L(iL) ? R?x? and C(vC) ? R?x? are of the
form
The Brayton-Moser equations are closely related
to the co-Hamiltonian H(iL, vC) (that represents
the total co-energy stored in the network). If
the co-Hamiltonian is known, then the matrices
L(iL) and C(vC) can be easily found as follows
29
Switched Brayton-Moser Equations
For a circuit with one or more switches it is
possible to obtain a single Switched
Mixed-Potential Function by properly combining
the individual mixed-potential functions
associated to each operating mode.
u0
P0(x)
u1
P1(x)
Then it is possible to obtain one Switched
Mixed-Potential Function parameterized by u as
The Switched Mixed-Potential Function is
consistent with the individual Mixed-Potential
Functions
30
Switched Brayton-Moser Equations
It is worth to notice that the only difference
between each individual Mixed-Potential Function
and the Switched Mixed-Potential Function will be
in the term
and in particular in the interconnection matrix
? which becomes a function of u, ? (u)
31
Average State Model
When the switching frequency is sufficiently
high, it is possible to prove that the average
state model of a circuit with a single switch can
be derived from the discrete model by only
replacing the discrete variable u?0,1 with the
continuously varying duty-cycle variable µ?0,1
. Additionally, to show that the model is a state
average model, the state vector x is replaced by
the state average vector z
Discrete Model
Average State Model
32
Average State Model
The former result can be extended to circuits
with multiple switches. In this case the matrix ?
(u) assumes as many configurations as the
possible combinations of the status of the
switches are (e. g. for an H-bridge converter ?
is a mono-dimensional matrix and may assume
three distinct values -1,0,1)
Discrete Model
Average State Model
33
Passivity Based Control Procedure
To design a Passivity Based controller the
average co-energy function H(z) and the
dissipative potential PD(z) have to be modified.
To this purpose the Brayton-Moser equations can
be rewritten as
The first two derivative terms are still function
of z, in the sense that the partial derivatives
of PT(z) and PD(z) are still dependent on z The
third term is constant meaning that the partial
derivative of PF(z) is not dependent on z anymore
f(z)
constant
The following step is to rewrite the previous
equations by replacing the state variables z with
an auxiliary system of variables ? which
represent the desired state trajectories for
inductor currents and capacitor voltages
The first two derivative terms are still function
of ? The third term is constant and is obviously
equal to the partial derivative of PF(z)
f(?)
constant
34
Passivity Based Control Procedure
being z? z - ? the average state errors, it is
possible to write
Assuming that the first two derivatives are
linear functions of z and the last two
derivatives are linear functions of ?, yields
The previous expression represents the error
dynamics and it could be obtained from
by simply replacing the variable z with the error
variable z? and eliminating the derivative of PF
35
Passivity Based Control Procedure
The next step is to add a damping term to the
error dynamics to ensure asymptotic stability
This injection can be seen as an expansion of the
dissipative potential
Considering z? (i?L, v?C)T where i?L (z?1 . .
. z??)T are the error-currents through the
inductors v?C (z??1 . . . z???)T are the
error-voltages across the capacitors
The injected dissipation can be decomposed as
follows
The injected dissipation together with the
dissipative potential of the system, gives the
Total Modified Dissipation Potential PM
36
Passivity Based Control Procedure
Subtracting
from
the controller dynamics are obtained
37
Passivity Based Control Procedure
Two theorems ensure the stability of the closed
loop system.
THEOREM 1 (R-Stability) If RS is a
positive-definite and constant matrix, and
with 0 lt?lt 1, then for all (i?L, v?C) the
solutions of
tend to zero as t ? 8
where closed-loop resistance matrix RS is
38
Passivity Based Control Procedure
THEOREM 2 (G-Stability) If GP is a
positive-definite and constant matrix, and
with 0 lt?lt 1, then for all (i?L, v?C) the
solutions of
tend to zero as t ? 8.
where closed-loop conductance matrix GP is
39
Passivity Based Control Procedure

• With these theorems lower bounds are found for
  the control parameters (RS and/or GP )
• These lower bounds ensure a reasonably nice
  response in terms of overshoot, settling-time,
  etc
• If just one of these theorems is satisfied, the
  system is stable. This means there are two
  damping injection strategies that can be
  selected

Series Damping (damping on inductor currents)
Parallel Damping (damping on capacitor voltages)
Although it is sufficient to use only one of
these strategies, the equations could contain
both the series damping injection term and the
parallel damping injection term
40
Passivity Based Control Procedure
Finally, if n is the number of minimum phase
states it is possible to modify n equations of
the ?? differential equations in
To this purpose n minimum phase states have to be
found. Consequently the remaining ??-n state
variables will be indirectly controlled through
the control of the n selected states
For the n selected variables it is possible to
set the derivative of reference value to zero
obtaining n algebraic equations
Controller Equations
and ??-n differential equations
41
Passivity Based Control Procedure
Controller Implementation
At the beginning initial values of the n control
inputs have to be set
Using these values the differential
equations can be solved to obtain the time
evolution of the auxiliary variables for the
indirectly controlled variables.
The former references are needed to solve the
algebraic equations
which solutions are the set of values for the
control inputs to be applied in the next
switching period.
42
PBC of an H-bridge
The passivity-based controller will be designed
by inspection, identifying the potential functions
43
PBC of an H-bridge
44
PBC of an H-bridge
45
PBC of an H-bridge
LKT
LKC
controller
46
PBC of an H-bridge damping injection
47
PBC of an H-bridge damping injection
48
PBC of an H-bridge control variables

• which variables have to be stabilized to a
  certain value in order to regulate the output(s)
  of interest toward a desired equilibrium value?
• in other words, are the zero-dynamics of the
  output(s) to be controlled stable with respect to
  the available control input(s), and if not, for
  which state variables are they stable?

49
PBC of an H-bridge zero-dynamics

• The steady-state solution in case of direct
  control of the dc-voltage or in case of indirect
  control of the dc-voltage (through the grid
  current) should be found

Switching function in case of direct control
Switching function in case of indirect control

• A stable system can be obtained only by
  indirectly controlling the dc voltages through
  the ac current i

50
PBC of an H-bridge

• A stable system can be obtained only by
  indirectly controlling the dc voltages through
  the ac current i
• This means that

• From the power balance it results that

dc voltage reference
load conductance
grid voltage amplitude
controller
and
reference voltage Vd and load conductance ?
reference current i
switching function µ
internally generated ?2
algebraic
power balance
ODE
51
PBC of an H-bridge

• Since it is possible to control directly the
  grid current,

average KVL on the a.c. side

• The d.c. voltage is controllable only
  indirectly, through an internal variable ?2

average KCL on the d.c. side

• Then it is necessary to estimate the d.c. load

52
PBC of an H-bridge damping tuning
53
Passivity Control of the Multilevel Converter

• The use of the passivity-based control (PBC)
  properly fits stability problems related to this
  type of converter
• Two approaches for the PBC design have been
  considered
• the first is developed considering the overall
  multilevel converter
• the second is developed by splitting the system
  into n subsystems and controlling them
  independently
• As regards the second, the partition of the
  multilevel converter is done on the basis of
  energy considerations
• The main advantage of the second approach is the
  separate control of the different DC-links and a
  flexible loading capability

54
Mathematical model of the system

• The converter is controlled with a discrete
  switching function ui (i1,2,...,n) for each
  H-bridge

55
PBC of an H-bridge

• Since it is possible to control directly the
  grid current,

average KVL on the a.c. side

• The d.c. voltage is controllable only
  indirectly, through an internal variable ?2

average KCL on the d.c. side

• Then it is necessary to estimate the d.c. load

56
PBC of an n-H-bridge converter

• It is not possible a simple extension of the PBC
  of one H-bridge active rectifier to n-bridge
  active rectifier
• In fact the PBC of an H-bridge active rectifier
  needs
• one algebraic equation and one differential
  equation

57
PBC of an n-H-bridge converter

• It is not possible a simple extension of the PBC
  of one H-bridge active rectifier to n-bridge
  active rectifier
• In fact the PBC of an H-bridge active rectifier
  needs
• one algebraic equation and one differential
  equation
• Thus a simple extension of this control needs n
  algebraic equations and n differential equations.
  However this is not possible since the n
  H-bridges are connected in series on the grid
  side and the ac voltage equation results in only
  one algebraic equation

58
PBC of an n-H-bridge converter

• It is not possible a simple extension of the PBC
  of one H-bridge active rectifier to n-bridge
  active rectifier
• In fact the PBC of an H-bridge active rectifier
  needs
• one algebraic equation and one differential
  equation
• Thus a simple extension of this control needs n
  algebraic equations and n differential equations.
  However this is not possible since the n
  H-bridges are connected in series on the grid
  side and the ac voltage equation results in only
  one algebraic equation

• We have proposed two PBC approaches
• one algebraic eq. plus n differential eq. (with
  ?2,1 .. ?2,n)
• n algebraic eq. (based on n virtual KVLs) plus
  n differential eq. (with ?2,1? .. ? ?2,n)

59
Passivity-based control approach 1

Indirect control of output voltages
reference voltage Vd and load conductance ? equal
for all the bridges
reference current i
switching function µ
algebraic
internally generated ?2, equal for all the bridges
power balance
ODE
60
PBC 2 Model formulation in subsystems
61
PBC 2 Model formulation in subsystems
62
Passivity-based control approach 2

Indirect control of each output voltage achieved
via the separate control of each bridge leading
to n passivity-based controllers related through
bi
supervisor
n controllers for n H-bridges
63
Passivity-based control approach 2

only changing the parameters of the controllers
64
Harmonic compensation

• In case of harmonics, the design results in the
  use of an RLC active damping branch very
  effective in harmonic rejection
• The damping is made by a resistance and a
  band-pass filter

energy function includes energy related to
harmonics
65
Harmonic compensation
bandpass filters

66
Set-up for the multilevel active rectifier
67
PBC tuning voltage error damping GDP
GDP 0.1 GDP 1 GDP 10
dc-link voltage due to a laod step change
68
PBC tuning estimate parameter ?
? 0.01
? 0.01
estimate dc-link load resistance due to a load
step change it has a strong influence on the
dc-link dynamic
69
Steady-state (PBC 1 2)
grid voltage
grid current
dc-link voltage
dc-link voltage
70
Dynamical test
PBC 1 PBC 2
Measured DC voltages 10 V/div consequent to a
load step change from half to full load on both
the DC-links (330 mF)
71
Dynamical test (PBC 2)
dc voltage reference step on one bus
dc load steps on the two buses leading to
different loads
Measured DC voltages 50 V/div and grid current
4 A/div (2330 mF)
72
Dynamical test for active load (PBC 2)
a dc motor has been used as active load
73
Unbalance loads on the two dc-links (PBC 2)
Steady-state behavior of PBC2 with full load on
DC bus 1 and half load on DC bus 2
multilevel ac voltage 200 V/div
grid voltage 100V/div grid current 10 A/div
74
Computational efforts comparison

• PBC 2 needs p-3 equations more than PBC 1
• However PBC 1 employs a division by the reference
  current that leads to computational problems
• with p number of desired levels

75
Harmonic compensation
R-damping
RLC-damping
76
Conclusions

• Passivity-based theory offers a straightforward
  approach to design controllers without
  linearazing the system
• physical and intuitive representation of the
  control problem
• design method to make the system Lyapunov-stable
• feedback decomposition useful for
  electromechanical systems

• Eulero-Lagrange formulation more suitable for
  electrical motors
• Brayton-Moser formulation more suitable for power
  converters (tuning procedure)
• Optimal results can be obtained with RLC damping
  of harmonics (similar to those obtained with
  generalized integrators resonant controllers
  linear approach)