A Deeper Look at LPV

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A Deeper Look at LPV

• Stephan Bohacek
• USC

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General Form of Linear ParametricallyVarying
(LPV) Systems
x(k1) A?(k)x(k) B?(k)u(k) z(k)
C?(k)x(k) D?(k) u(k) ?(k1) f(?(k))
linear parts
nonlinear part
x?Rn u ?Rm ??? - compact
A, B, C, D, and f are continuous functions.
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How do LPV Systems Arise?

• Nonlinear tracking
• ?(k1)f(?(k),0) desired trajectory
• ?(k1)f(?(k),u(k)) trajectory of the system
  under control
• Objective find u such that
• ?(k)-?(k) ? 0 as k ? ?.
• ?(k1) f(?(k),0) f?(?(k),0) (?(k)- ?(k))
  fu(?(k),0) u(k)
• Define x(k) ?(k) - ?(k)
• x(k1) A?(k)x(k) B?(k)u(k)

?
?
A?(k)
B?(k)
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How do LPV Systems Arise ?

• Gain Scheduling
• x(k1) g(x(k), ?(k), u(k)) ? gx(0,?(k),0) x(k)
  gu(0,?(k),0) u(k)
• ?(k1) f(x(k), ?(k), u(k)) models variation
  in the parameters
• Objective find u such that x(k)? 0 as k? ?

?
?
A?(k)
B?(k)
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Types of LPV Systems Different amounts of
knowledge about f lead to a different types of
LPV systems.

• f(?) ?? - know almost nothing about f (LPV)
• f(?)- ?lt? - know a bound on rate at which ?
  varies (LPV with rate limited parameter
  variation)
• f(?) - know f exactly (LDV)
• ? is a Markov Chain with known transition
  probabilities (Jump Linear)
• f(?)?? where f(?) is some known subset of ?
  (LSVDV)
• f(?)?0, ?1, ?2,, ?n
• f(?)B(?0,?), B(?1,?), B(?2,?),, B(?n ,?)

type 1 failure
ball of radius ? centered at ?n
type n failure
nominal
type n failure
type 1 failure
nominal
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Stabilization of LPV SystemsPackard and Becker,
ASME Winter Meeting, 1992.

• Find S?Rn?n and E?Rm?n such that

for all ???
gt 0
x(k1) (A?B? (ES-1)) x(k) ?(k1) f(?(k))
In this case,
is stable.
If ? is a polytope, then solving the LMI for all
??? is easy.
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Cost
For LTI systems, you get the exact cost.
x(0) X x(0) ?k?0,? C?j?0,k(ABF)x(0)2
DF(?j?0,k(ABF))x(0)2 where X
ATXA - ATXB(DTD BTXB)-1BTXA CC
For LPV systems, you only get an upper bound on
the cost.

xT X x ? ?k?0,? C?(k)?j?0,k(A?(j)B?(j)F)x2
D?(k)F(?j?0,k(A?(j)B?(j)F))x2 wh
ere XS-1
depends on ?

• If the LMI is not solvable, then
• the inequality is too conservative,
• or the system is unstabilizable.

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LPV with Rate Limited Parameter VariationWu,
Yang, Packard, Berker, Int. J. Robust and NL
Cntrl, 1996Gahinet, Apkarian, Chilali, CDC 1994
Suppose that f(?)- ? ? lt ? and
where Si? Rn?n, Ei? Rm?n and bi is a set of
orthogonal functions such that bi (?) - bi
(??) lt ??.
S? ?i?1,N bi(?) Si E? ?i?1,N bi(?) Ei
We have assumed solutions to the LMI have a
particular structure.
for all ??? and ?ilt ??
gt 0
x(k1) (A?(k) B?(k)E?(k) X?(k))x(k)
then
is stable.
where X? (S?)-1
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Note that X(t)(?i?1,n bi(t)Si)-1 So
X(td)X(t)X(z)d by the mean value thm. Hence
X(td)X(t) d/dx (?i?1,n bi(t)Si)-1 X(t)
(?i?1,n bi(t)Si)-1 ?i?1,n (dbi/dx)di
(?i?1,n bi(t)Si)-1 Therefore, post and
premultiplying the 1-1 element of the above
matrix by X(t) Yeilds X(td)
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Cost
You still only get an upper bound on the cost
x(0) X?(0) x(0) ? ?k?0,? C?(k)?j?0,k(A?(j)B
?(j)F?(k))x(0)2 D?(k)F?(k)(?j?0,k(A?(j)
B?(j)F?(k)))x(0)2 where X (?i?1,N bi(?)
Si)-1 and F?(k) E?(k) X?(k)

• If the LMI is not solvable, then
• the assumptions made on S are too strong,
• the inequality is too conservative,
• or the system is unstabilizable.

Might the solution to the LMI be discontinuous?
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Linear Dynamically Varying (LDV) Systems Bohacek
and Jonckheere, IEEE Trans. AC
Assume that f is known.
x(k1) A?(k)x(k) B?(k)u(k) z(k)
C?(k)x(k) D?(k) u(k) ?(k1) f(?(k))
A, B, C, D and f are continuous functions.
Def The LDV system defined by (f,A,B) is
stabilizable if there exists
F ? ? Z ? Rm?n
x(k1) (A?(k) B?(k)F(?(0),k)) x(k) ?(k1)
f(?(k))
such that, if
x(kj) ? ??(0)??(0)x(k)
then
j
for some ??(0) lt ? and ??(0) lt 1.
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Continuity of LDV Controllers
Theorem LDV system (f,A,B) is stabilizable if
and only if there exists a bounded solution X ?
? Rn?n to the functional algebraic Riccati
equation
In this case, the optimal control is
and X is continuous.
Since X is continuous, X can be estimated by
determining X on a grid of ?.
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Continuity of LDV Controllers
Continuity of X implies that if ?1- ?2 is
small, then
is small.
Which is true if
which only happened when f is stable,
where ? and ? are independent of ?, which is
more than stabilizability provides.
or
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LDV Controller for the Henon Map
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H? Control for LDV Systems Bohacek and Jonckheere
SIAM J. Cntrl Opt.
Objective
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Continuity of the H? Controller
Theorem There exists a controller such that
if and only if there exists a bounded solution to
X? C?C? A?Xf(?)A? - L?(R?)-1L?
T
T
T
In this case, X is continuous.
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X May Become Discontinuous as ? is Reduced
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LPV with Rate Limited Parameter Variation
Suppose that f(?)- ? ? lt ? and
where Si? Rn?n, Ei? Rm?n and bi is a set of
orthogonal functions such that bi (?) - bi
(??) lt ??.
S? ?i?1,N bi(?) Si E? ?i?1,N bi(?) Ei
for all ??? and ?ilt ??
gt 0

• If the LMI is not solvable, then
• the set bi is too small (or ? is too small),
• the inequality is too conservative,
• or the system is unstabilizable.

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Linear Set Valued Dynamically Varying (LSVDV)
Systems Bohacek and Jonckheere, ACC 2000
set valued dynamical system
A, B, C, D and f are continuous functions.
? is compact.
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LSVDV systems
type 1 failure
nominal
type 2 failure
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1 - Step Cost
For example, let f(?)?1, ?2
alternative 1
alternative 2
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Cost if Alternative 1 Occurs
where Q A?X1A? C?C?
T
T
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Cost if Alternative 2 Occurs
where Q A?X2A? C?C?
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Worst Case Cost
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The LMI Approach is Conservative
conservative
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Worst Case Cost
piece 2
piece 1

• non-quadratic cost
• piece-wise quadratic

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Piecewise Quadratic Approximation of the Cost
Define X(x,?) maxi?N xTXi(?)x
quadratic
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Piecewise Quadratic Approximation of the Cost
not an LMI
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Piecewise Quadratic Approximation of the Cost
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Piecewise Quadratic Approximation of the Cost
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Piecewise Quadratic Approximation of the Cost
Allowing non-positive definite Xi permits good
approximation.
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Piecewise Quadratic Approximation of the Cost
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The Cost is Continuous
Theorem If
1. the system is uniformly exponentially stable,
2. X Rn ? ? ? R solves
3. X(x, ?) ? 0,
then X is uniformly continuous.

• Hence, X can be approximated
• partition Rn into N cones, and
• grid ? with M points.

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Piecewise Quadratic Approximation of the Cost
Define X(x,?,T,N,M) maxi?N xTXi(?,T,N,M)x
such that
X(x,?,0,N,M) xTx.
X(x,?,0,N,M) ? X(x,?) as N,M,T ? ?
Would like
time horizon
number of cones
number of grid points in ?
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X can be Found via Convex Optimization
The cone centered around first coordinate axis
C1 ?x ? gt 0, x e1 ?y, y10, y1
depends N, the number is cones
convex optimization
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X can be Found via Convex Optimization
The cone centered around first coordinate axis
C1 ?x ? gt 0, x e1 ?y, y10, y1
depends N, the number is cones
convex optimization
X(x,?,0,N,M,K) ? X(x,?) as N,M,T,K ? ?
Theorem
In fact,
related to the continuity of X
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Optimal Control of LSVDV Systems
only the direction is important
the optimal control is homogeneous
but not additive
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Summary
LPV
increasing knowledge about f
increasing computational complexity
increasing conservativeness
LPV with rate limited parameter variation
optimal in the limit
LSVDV
might not be that bad
optimal
LDV