Gain Scheduling for LPV Systems

1
Gain Scheduling for LPV Systems

• Brian D O Anderson
• Research School of Information Sciences and
  Engineering, Australian National University
• and
• National ICT Australia

2
OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches and tools for seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

Aim of presentation
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AIM OF PRESENTATION

• To present a design method for constructing
  gain-scheduled controllers for LPV systems
• What is different
• Controller is not constant but parameter
  dependent
• Arbitrarily fast variation of parameters is not
  assumed
• Stability of closed loop is guaranteed, even with
  time variation of parameters
• Parameters enter underlying system in a special
  way

4
OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches and tools for seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

5
Systems considered

• Underlying system is SISO (SIMO can be done),
  linear and time-invariant except for dependence
  on a parameter
• The parameter may be time-varying (environment
  change, operating point change of a nonlinear
  system, etc). Bound on change rate is known.
• We shall restrict attention to SISO plants of the
  form N(s) etc are stable rational
• where ? is a vector multilinear in some scalar
  ?i,

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Systems considered

• We shall restrict attention to SISO plants of the
  form
• where ? is a vector multilinear in scalar ?i
• For example,
• The multiaffine dependence of the numerator and
  denominator of P(s) on entries of ? is typical
  when those entries are physical parameter values
  (mass, friction coefficient, capacitance etc) or
  their inverses.

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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches and tools for seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

8
Possible Approaches to seeking a controller

• Find a controller which is parameter independent.
  (Robust controller problem may be highly
  non-optimal, may not exist). Usually parameters
  vary arbitrarily fast.
• Find a controller which tracks the parameter
  variations and yields stability for arbitrarily
  fast parameter variations (May not exist
  requirement for stability with arbitrarily fast
  variations is too strong!)
• Find a controller which tracks variations and
  yields stability for bounded-rate-of-variation of
  parameters. (Our approach)

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Key tools to develop controller

• Robust Hurwitz stability, using affine or
  multiaffine parameter sets (Anderson, Dasgupta,
  Khargonekar, Kraus and Mansour Rantzner and
  Megretski)
• Positive realness and LMIs
• Retention of stability when parameters become
  time-varying in time-invariant stable affine
  (extendable to multiaffine) systems (Freedman and
  Zames Anderson and Moore Dasgupta, Anderson,
  Chockalingam and Fu)

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Combination of key tools
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
Controller design with parameter dependence with
closed loop char. poly multiaffine dependence
Closed loop design using LMIsTolerance of
parameter Variation, and char poly multi- affine
parameter dependence
11
OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

12
Robust Hurwitz Stability

• Let p0(s) be a Hurwitz polynomial and let qi(s),
  qij(s), etc be polynomials of lower degree than
  that of p0. Consider the polynomial set

• Then all such polynomials are stable if there
  exists    polynomials m(s) and n(s) such that
  either of the    following 2 equivalents holds

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Robust Hurwitz Stability 2

• Consider a set of stable transfer functions

• Here p0, entries of Q and r are all polynomials,
  and     entries of Q have lower degree than p0

• Then all such transfer functions have stable
   numerator    if there exist polynomials m(s),
  n(s) such that

• If p0 r, the condition is if there exists SPR
  ?(s) with

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Robust Hurwitz Stability 2

• Consider a set of stable transfer functions

Whole set

• Here p0, entries of Q and r are all polynomials,
  and     entries of Q have lower degree than p0

• Then all such transfer functions have stable
   numerator    if there exist polynomials m(s),
  n(s) such that

Corners

• If p0 r, the condition is if there exists SPR
  ?(s) with

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Robust Hurwitz Stability 3

• Consider a set of stable transfer functions

Whole set

• Here p0, entries of Q and r are all polynomials,
  and     entries of Q have lower degree than p0.
•   Is asking for an SPR condition asking for a lot?

• If the numerator p(?,s) is affine in ?, rather
  than being    multiaffine, the     SPR conditions
  is not just an if condition    but also an only
  if condition for stability. For many multiaffine
     dependencies, the SPR condition is easy to
  establish.

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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

17
Constraints on the controller

• Recall the plant transfer function has numerator
  and denominator which are multiaffine in ?.
• The controller will also be taken in this form.
• In general this results in a closed-loop
  characteristic polynomial which can be quadratic
  in individual ?i.
• We will impose some constraints which make it
  multiaffine in ?.
• This will allow application of the machinery we
  have recalled robust Hurwitz stability

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Controller transfer function

• Recall that the plant is

• We assume a controller of the same
  form--numerator and    denominator stable,
  rational and multiaffine in ?

• The nominal controller

is designed for the nominal plant

• The closed loop is stable for a fixed ? if and
    only if the  following expression has a stable
  numerator

• To make this multiaffine in ? we require that

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Controller transfer function

• To make this affine in ? we require that

• Define

and suppose F has normal rank r. Choose
with full row rank and
Then for arbitrary
and
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Controller Summary
Closed Loop stability given by numerator of
We ensure that
by constraining the parameter dependent part of
the controller via
with arbitrary
and
Closed Loop stability now given by numerator
(affine in ?) of
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Controller Summary
Closed Loop stability now given by numerator
(affine in ?) of
by constraining the parameter dependent part of
the controller via
with arbitrary
and
Simplify by assuming the Bezout identity
Then with ? and ? shorthand for quantities
definable from the plant and the nominal
controller, closed loop stability is given by
numerator of
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Local Summary

• Nominal plant is used to design nominal
  controller
• Plant with multiaffine parameter dependence is
  used to parametrise set of controllers using
  stable transfer functions which
• Include the nominal controller
• Have multiaffine parameter dependence
• Give closed-loop characteristic polynomial with
  multiaffine dependence on parameter vector
• Next task is to choose acceptable controller from
  the set.

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Review of key tools use
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
Controller design with Parameter dependence with
closed loop char. poly multiaffine dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
24
Review of key tools use
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
DONE
Controller design with Parameter dependence
with closed loop char. poly multiaffine
dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
25
Review of key tools use
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
DONE
Controller design with Parameter dependence with
closed loop char. poly multiaffine dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
DONE
26
Review of key tools
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
DONE
Controller design with Parameter dependence
with closed loop char. poly multiaffine
dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
DONE
NOW FOR LMIs!
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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

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Choosing the controller from the allowed set

• With the following quantities determined from the
  plant and the nominal controller

together with the constraints
and
select ? and ? so that
has stable numerator. Equivalently, select ? and
? and an SPR transfer function ?(s) so that the
following finite set is strictly positive real
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Choosing the controller from the allowed set
Select ? and ? and an SPR transfer function ?(s)
so that the following finite set is SPR
Equivalently, select ?(s), ?(s) ?(s)?(s), ?(s)
?(s)?(s), all stable and with ?(s) skew so
that
If this holds, ?(s) is automatically SPR. Now
introduce a Laguerre basis matrix
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Choosing the controller from the allowed set
Equivalently, select ?(s), ?(s) ?(s)?(s), ?(s)
?(s)?(s), all stable and with ?(s) skew so
that
Introduce Laguerre basis
where m is unspecified. Approximate ?(s), ?(s)
and ?(s) by
with
We require
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Final controller parametrisation
parametrise the controller and must satisfy

•  This is a finite set of inequalities
•  One specifies an ?-grid and attempts to solve,
  starting with small    values for the integers
  N1, N2 and N3
• One increases the integers until the LMIs can be
  solved for the     parameters.
•  There is an alternative way of treating the LMIs
  without gridding

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Review of key tools
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
NEXT
DONE
Controller design with Parameter dependence with
closed loop char. poly multiaffine dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
DONE
LMIs DONE
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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

34
Robust stability for parameter-varying systems

• Consider
•  Assume the characteristic polynomial of A(?)
  is affine in the elements of ? and for all ? ? ?
  box
• the equation has exponential degree of
  stability ?. Then the LTV system obtained by
  letting ? vary (in the box) has exponential
  degree of stability ? if the average logarithmic
  rate of variation of ? is bounded

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Robust stability for parameter-varying systems

• Consider
•  Assume the characteristic polynomial of A(?)
  is multiaffine in the elements of ? and for all
  ? ? ? box
• the equation has exponential degree of
  stability ?. If the characteristic polynomial is
  p(?, s) and there exists an SPR condition for
  p(?, s- ?)
•     the same tolerance of time variation applies
  as for the affine case.

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Robust stability for parameter-varying systems

• There is also a result which says if a system is
  stable for all fixed values of a parameter in a
  compact set, it will be stable for sufficiently
  slow variations of the parameter
• The result using the rate of variation bound
  yields a far less conservative bound on the
  allowed rate.

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Tolerating parameter variation
We have arranged for the closed-loop
characteristic polynomial to be the numerator of
the following, and to be stable for all ? ? ?box
It is easy to write down a zero-input linear
state variable equation with A(?) matrix having
this characteristic polynomial
Hence the results on tolerance of time-variation
in the parameter for the state variable equation
apply to our closed loop.
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Last adjustment to design

• Assume the original plant we wish to control is
  and define a shifted plant

• Design a controller for the shifted plant
  according to the    procedure.    Call this
  controller

• Define the actual controller to use with the
  original plant by

•  The closed-loop characteristic polynomial for
  the actual     plant and controller
  interconnection is still multiaffine in ?,
       but has degree of stability ? due to the
  shifting trick above.     This ensures tolerance
  of parameter time-variation.

39
Review of key tools
Robust stability results based on using
Strict Postive Real ideas for Time invariant
systems
Tolerance of time-varying parameters with rate
bound, Robust Stability and char poly
multiaffine dependence
REVIEWED
DONE
Controller design with Parameter dependence with
closed loop char. poly multiaffine dependence
Closed loop design using LMIsTolerance of
parameter variation, and char poly multi- affine
parameter dependence
DONE
LMIs, PARAMETER VARIATION TOLERANCE and CHAR POLY
MULTI AFFINE DEPENDENCE--ALL DONE!
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Another Summary Design and Theory tools

• Robust Hurwitz with multiaffine parameter
  dependence treatable with SPR ideas.
• SPR conditions hold in parameter box if and only
  if they hold at corners
• Multiaffine dependence on parameter in CLCP means
  state-variable equation can capture this CLCP
• State variable view allows use of good
  time-variation tolerance result

• Do pole shift of plant
• Design nominal controller for new nominal plant
• Parametrise set of controllers giving multiaffine
  CLCP, using certain transfer functions
• Same, except use constant matrices (Laguerre
  basis)
• Convert to finite set of PR conditions
• Convert to LMIs using ?-griddingSolve LMI, get
  controller, do pole shift back to get real
  controller

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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

42
Example-affine dependence only
Do pole shift by ? .38. Design observer-based
state space compensator to position poles at s
-1, -2,-8. This gives nominal controller C(0,s).
When poles are shifted back, resulting nominal
controller gives minimum degree of stability of
-1.38 at the nominal parameter value. Stability
also is secured at extreme parameter values and
random points in ?box. The following parameters
destabilise the closed loop
Note that the associated frequency of variation
is ? 5 gtgt 1.38!
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Example continued
Hence
The nominal controller is given by
The Strict Positive Real conditions are set up,
and the parametrising transfer functions
approximated by Laguerre expansions. One has a
constant term only, one has a first order factor,
with pole at -40.
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Example continued
The parameter controller with parameter
dependence is defined by
before reversing the pole shift by ? .38. The
relevant definitions are
As expected, simulations show a stable closed
loop with time varying parameters.
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OUTLINE

• Aim of Presentation
• Background on LPV systems
• Approaches to seeking a controller
• Robust stability review
• First description of all acceptable controllers
• Defining acceptable controllers via LMIs
• Achieving parameter variation tolerance
• Example
• Open questions

46
Open questions

• Introducing further design constraints beyond
  stability in selecting the parameter -dependent
  part of the controller
• Other ways to acquire controller structure, based
  e.g. on estimator/state feedback law structure.