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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

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.

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)

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)

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

Stabilization of LPV Systems Packard and Becker, ASME Winter Meeting, 1992. Discover S R nn and ER mn 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.

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.

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 nn , E i R mn 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

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?

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.

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 nn 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 .

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

LDV Controller for the Henon Map

H Control for LDV Systems Bohacek and Jonckheere SIAM J. Cntrl & Opt. Objective :

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 .

X May Become Discontinuous as is Reduced

LPV with Rate Limited Parameter Variation Suppose that | f( )- | < and where S i R nn , E i R mn 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.

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.

LSVDV frameworks sort 1 disappointment ostensible sort 2 disappointment

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

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

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

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

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

Worst Case Cost piece 2 piece 1 non-quadratic cost piece-wise quadratic

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

Piecewise Quadratic Approximation of the Cost not a LMI

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

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

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.

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

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.

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 iN 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

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:

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

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

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