# A More profound Take a gander at LPV.

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Category: General / Misc
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(k 1) = f(x(k), (k), u(k)) models variety in the parameters ... x(k 1) = (A B (ES-1)) x(k) (k 1) = f( (k)) For this situation, is steady. ...
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﻿A Deeper Look at LPV Stephan Bohacek USC

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General Form of Linear Parametrically Varying (LPV) Systems x(k+1) = A (k) x(k) + B (k) u(k) z(k) = C (k) x(k) + D (k) u(k) (k+1) = f((k)) straight parts nonlinear part x R n u R m  - minimized A, B, C, D , and f are ceaseless capacities.

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How do LPV Systems Arise? Nonlinear following (k+1)=f((k),0) – craved direction (k+1)=f((k),u(k)) – direction of the framework under control Objective: discover u such that | (k)- (k) | 0 as k  . (k+1)= f((k),0) + f  ((k),0) ((k)- (k)) + f u ((k),0) u(k) Define x(k) = (k) - (k) x(k+1) = A (k) x(k) + B (k) u(k)  A (k) B (k)

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How do LPV Systems Arise ? Pick up Scheduling x(k+1) = g(x(k), (k), u(k))  g x (0,(k),0) x(k) + g u (0,(k),0) u(k) (k+1) = f(x(k), (k), u(k)) – models variety in the parameters Objective: discover u such that |x(k)| 0 as k   A (k) B (k)

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Types of LPV Systems Different measures of learning about f lead to an alternate sorts of LPV frameworks. f( )  - know nothing about f (LPV) |f( )- |< - know a bound on rate at which  differs (LPV with rate constrained parameter variety) f( ) - know f precisely (LDV)  is a Markov Chain with known move 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 ,)} sort 1 disappointment chunk of span  focused at  n sort n disappointment ostensible sort n disappointment sort 1 disappointment ostensible

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Stabilization of LPV Systems Packard and Becker, ASME Winter Meeting, 1992. Discover S R nn and ER mn such that for all  > 0 x(k+1) = (A  +B  (ES - 1 )) x(k) (k+1) = f((k)) For this situation, is steady. On the off chance that  is a polytope, then understanding the LMI for all  is simple.

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Cost For LTI frameworks, you get the accurate expense. x(0) X x(0) =  k[0,] |C  j[0,k] (A+BF)x(0)| 2 + |DF ( j[0,k] ( A+BF )) x(0)| 2 where X = A T XA - A T XB(D T D + B T XB) - 1 B T XA + CC For LPV frameworks, you just get an upper bound on the expense. } x T x   k[0,] |C (k)  j[0,k] (A (j) +B (j) F)x| 2 + |D (k) F ( j[0,k] ( A (j) +B (j) F )) x| 2 where X=S - 1 relies on upon  If the LMI is not resolvable, then the disparity is excessively moderate, or the framework is unstabilizable.

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LPV with Rate Limited Parameter Variation Wu, Yang, Packard, Berker, Int. J. Hearty and NL Cntrl, 1996 Gahinet, Apkarian, Chilali, CDC 1994 Suppose that | f( )-  |  <  and where S i  R nn , E i  R mn and { b i } is an arrangement of orthogonal capacities such that | b i () - b i (+)| <   . S  =  i{1,N} b i () S i E  =  i{1,N} b i () E i We have accepted answers for the LMI have a specific structure. for all  and |  i |<   > 0 x(k+1) = ( A (k) + B (k) E (k) X (k) ) x(k) then is steady. where X  = (S  ) - 1

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Cost You still just 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] b i () S i ) - 1 and F (k) = E (k) X (k) If the LMI is not feasible, then the suppositions made on S are excessively solid , the disparity is excessively traditionalist, or the framework is unstabilizable. Might the answer for the LMI be intermittent?

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Linear Dynamically Varying (LDV) Systems Bohacek and Jonckheere, IEEE Trans. Air conditioning Assume that f is known. x(k+1) = A (k) x(k) + B (k) u(k) z(k) = C (k) x(k) + D (k) u(k) (k+1) = f((k)) A, B, C, D and f are persistent capacities. Def: The LDV framework characterized by ( f,A,B ) is stabilizable if there exists F :   Z  R m  n x ( k+1 ) = ( A  ( k ) + B  ( k ) F ( (0),k ) x ( k )  ( k+1 ) = f (  ( k )) such that, if | x ( k+j )|   (0)  (0) | x ( k )| then j for some   (0) <  and   (0) < 1.

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Continuity of LDV Controllers X  = A  X  A  + C  C  - A  X  B  ( D  D  + B  X  B  ) - 1 B  X  A  T u ( k ) = - ( D  (k) D  (k) + B  (k) X  (k) B  (k) ) - 1 B  (k) X  (k) A (k) x ( k ) T Theorem: LDV framework ( f,A,B ) is stabilizable if and just if there exists a limited arrangement X :   R nn to the practical mathematical Riccati comparison For this situation, the ideal control is and X is consistent . Since X is constant, X can be assessed by deciding X on a matrix of .

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Continuity of LDV Controllers Continuity of X infers that if | 1-2| is little, then is little. Which is valid if which just happened when f is steady, where  and  are free of , which is more than stabilizability gives. then again

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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 just if there exists a limited answer for X  = C  C  + A  X f() A  - L  (R  ) - 1 L  T For this situation, X is constant .

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X May Become Discontinuous as  is Reduced

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LPV with Rate Limited Parameter Variation Suppose that | f( )-  |  <  and where S i  R nn , E i  R mn and { b i } is an arrangement of orthogonal capacities such that | b i () - b i (+)| <   . S  =  i{1,N} b i () S i E  =  i{1,N} b i () E i for all  and |  i |<   > 0 If the LMI is not reasonable, then the set {b i } is too little (or  is too little), the imbalance is excessively moderate, or the framework is unstabilizable.

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Linear Set Valued Dynamically Varying (LSVDV) Systems Bohacek and Jonckheere, ACC 2000 set esteemed dynamical framework A, B, C, D and f are constant capacities.  is minimal.

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LSVDV frameworks sort 1 disappointment ostensible sort 2 disappointment

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1 - Step Cost For instance, let f ( )={ 1 ,  2 } elective 1 elective 2

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Cost if Alternative 1 Occurs 2 1.5 1 0.5 0 - 0.5 - 1 - 1.5 - 2 - 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2 where Q = A  X 1 A  + C  C  T

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Cost if Alternative 2 Occurs 2 1.5 1 0.5 0 - 0.5 - 1 - 1.5 - 2 - 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2 where Q = A  X 2 A  + C  C 

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Worst Case Cost 2 1.5 1 0.5 0 - 0.5 - 1 - 1.5 - 2 - 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2

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The LMI Approach is Conservative 2 1.5 1 0.5 0 - 0.5 - 1 - 1.5 - 2 - 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2 traditionalist

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Piecewise Quadratic Approximation of the Cost Define X ( x , ) := max iN x T X i () x quadratic

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Piecewise Quadratic Approximation of the Cost not a LMI

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Piecewise Quadratic Approximation of the Cost 3 2 1 0 - 1 - 2 - 3 - 3 - 2 - 1 0 1 2 3

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Piecewise Quadratic Approximation of the Cost 3 2 1 0 - 1 - 2 - 3 - 3 - 2 - 1 0 1 2 3

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Piecewise Quadratic Approximation of the Cost 1 0.8 0.6 0.4 0.2 0 - 0.2 - 0.4 - 0.6 - 0.8 - 1 - 1 - 0.8 - 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 0.8 1 Allowing non-positive distinct X i allows great guess.

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Piecewise Quadratic Approximation of the Cost 2.5 2 1.5 1 0.5 0 - 0.5 - 1 - 1.5 - 2 - 2.5 - 2.5 - 2 - 1.5 - 1 - 0.5 0 0.5 1 2 1.5

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The Cost is Continuous Theorem: If 1. the framework is consistently exponentially steady, 2. X : R n    R understands 3. X ( x , )  0, then X is consistently constant . Henceforth, X can be approximated: allotment R n into N cones, and lattice  with M focuses.

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Piecewise Quadratic Approximation of the Cost X ( x , , T,N,M )  max  f () X ( A  x ,, T-1,N,M ) + x T C  C  x T Define X ( x , , T,N,M ) := max iN x T X i (, T,N,M ) x such that X ( x , , 0,N,M ) = x T x. X ( x , , 0,N,M )  X ( x , ) as N,M,T   Would like time skyline number of cones number of matrix focuses in 

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X can be Found by means of Convex Optimization The cone based on first facilitate pivot C 1 := {  x :  > 0, x = e 1 +  y , y 1 =0, | y |=1} depends N , the number is cones curved enhancement:

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X can be Found through Convex Optimization The cone revolved around first organize hub C 1 := {  x :  > 0, x = e 1 +  y , y 1 =0, | y |=1} depends N , the number is cones arched streamlining: X ( x , , 0,N,M,K )  X ( x , ) as N,M,T,K   Theorem: truth be told, identified with the coherence of X

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Optimal Control of LSVDV Systems just the course is imperative the ideal control is homogeneous yet not added substance

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Summary LPV expanding information about f expanding computational intricacy expanding conservativeness LPV with rate restricted parameter variety ideal in the cutoff LSVDV won\'t not be that awful ideal LDV

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