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

A Deeper Look at LPV

Stephan Bohacek

USC

General Form of Linear ParametricallyVarying (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))

linear parts

nonlinear part

xRn

u Rm

- compact

A, B, C, D, and f are continuous functions.

How do LPV Systems Arise?

• Nonlinear tracking(k+1)=f((k),0) – desired trajectory

(k+1)=f((k),u(k)) – trajectory of the system under control

Objective: find u such that

| (k)-(k) | 0 as k .

(k+1)= f((k),0) + f((k),0) ((k)- (k)) + fu((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 ?

• Gain Scheduling

x(k+1) = g(x(k), (k), u(k)) gx(0,(k),0) x(k) + gu(0,(k),0) u(k)

(k+1) = 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)

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()- |< - 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 ,)}

nominal type 1failure

type nfailure

nominal type 1failure

type nfailure

ball of radius centered at n

S (A S + B E )T (C S)T (D S)T

AS + B E S

C S I

D S I

Stabilization of LPV SystemsPackard and Becker, ASME Winter Meeting, 1992.

Find SRnn and ERmn such that

x(k+1) = (A+B (ES-1)) x(k)(k+1) = f((k))

In this case, is stable.

If is a polytope, then solving the LMI for all is easy.

> 0

for all

xT X 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

Cost

For LPV systems, you only get an upper bound on the cost.

x(0) X x(0) = k[0,] |Cj[0,k](A+BF)x(0)|2

+ |DF(j[0,k](A+BF))x(0)|2

where X = ATXA - ATXB(DTD + BTXB)-1BTXA + CC

}depends on

For LTI systems, you get the exact cost.

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

S + i{1,n} iSi (A S + B E )T (C S)T (D S)T

AS + B E S

C S I

D S I

LPV with Rate Limited Parameter VariationWu, Yang, Packard, Berker, Int. J. Robust and NL Cntrl, 1996

Gahinet, Apkarian, Chilali, CDC 1994

Suppose that | f()- | < and

S = i{1,N} bi() Si

E = i{1,N} bi() Ei

> 0 for all and |i|<

x(k+1) = (A(k) + B(k)E(k) X(k))x(k)

where X = (S)-1

then is stable.

where Si Rnn, Ei Rmn and {bi} is a set of orthogonal functions such that |bi () - bi (+)| < .

We have assumed solutions to the LMI have a particular structure.

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)

You still only get an upper bound on the cost

Might the solution to the LMI be discontinuous?

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.

|x(k+j)| (0)(0)|x(k)|

Linear Dynamically Varying (LDV) Systems Bohacek and Jonckheere, IEEE Trans. AC

Assume that f is known.

Def: The LDV system defined by (f,A,B) is stabilizable if there exists

F : Z Rmn

such that, if x(k+1) = (A(k) + B(k)F((0),k)) x(k)

(k+1) = f((k))

then j

for some (0) < and (0) < 1.

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 continuous functions.

Continuity of LDV Controllers

X = AXA + CC - AXB(DD + BXB)-1BXA T T T T T T

Theorem: LDV system (f,A,B) is stabilizable if and only if there exists a bounded solution X : Rnn to the functional algebraic Riccati equation

In this case, the optimal control is

Since X is continuous, X can be estimated by determining X on a grid of .

and X is continuous.

u(k) = - (D (k) D (k) + B (k) X (k) B (k))-1B (k) X (k) A(k) x(k)T T T

Continuity of X implies that if |1- 2| is small, then

k

jjfkf

Tkf

k

j

Tjf

clcl ACCA0

1110 1

...0

2220 2

o

k

jjfkf

Tkf

k

j

Tjf

xACCA clcl

021

k

Too

To xxXXx

which only happened when f is stable, 0lim21

jfjfjclcl AA

Which is true if

or k

k

jjf

clA

0

where and are independent of , which is more than stabilizability provides.

Continuity of LDV Controllers

is small.

kk

kkkuk

12

22

11

3.01

)4.1(11

LDV Controller for the Henon Map

kx

kxFFku kk

2

121

1F 2F

kfk

kuDkxCkz

kwBkuBkxAkx

kk

kkk

1

1 21

Objective:

0for 0such that Find 2 wkxlu

2

2

2 ,z

and lww l

l

H Control for LDV SystemsBohacek and Jonckheere SIAM J. Cntrl & Opt.

Continuity of the H Controller

Theorem: There exists a controller such that

22

2 ,z

lww l

l

if and only if there exists a bounded solution to

X = CC + AXf()A - L(R)-1L

In this case, X is continuous.

TT T

X May Become Discontinuous as is Reduced

LPV with Rate Limited Parameter Variation

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.

S + i{1,n} iSi (A S + B E )T (C S)T (D S)T

AS + B E S

C S I

D S I

Suppose that | f()- | < and

S = i{1,N} bi() Si

E = i{1,N} bi() Ei

> 0 for all and |i|<

where Si Rnn, Ei Rmn and {bi} is a set of orthogonal functions such that |bi () - bi (+)| < .

1 kfk

1

kuD

kxCkz

kuBkxAkx

k

k

kk

Linear Set Valued Dynamically Varying (LSVDV) SystemsBohacek and Jonckheere, ACC 2000

A, B, C, D and f are continuous functions.

is compact.

set valued dynamical system

1

2

0

nominal

type 1 failure

type 2 failure

ff

f

LSVDV systems

xCCAXAxxJ TTT

f

max:,

For example, let f()={1, 2}

xCCAXAxxCCAXAxxJ TTTTTT 21 ,max,

alternative 1 alternative 2

1 - Step 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

Cost if Alternative 1 Occurs

1: xxxQxTwhere Q = AX1A + CC TT

-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

Cost if Alternative 2 Occurs

1: xxxQxTwhere Q = AX2A + CC

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

xCCAXAxxCCAXAxxJ TTTTTT 21 ,max,

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 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

•non-quadratic cost•piece-wise quadratic

Worst Case Cost

piece 1

piece 2

nN

iii CNiC R

0such that ,,0: cones Define

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

0C

1C

NC

quadratic

Piecewise Quadratic Approximation of the Cost

Define X(x,) := maxiN xTXi()x


xXxxIAQAx jT

iTT

such that : find ,:Given NjXPiQ ji

1,0:for 1

xcxCxn

lllj

cone positive - jC

not an LMI

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

xQxT1

xXxxIAQAx Ti

TT

i0max 0Cx

0C-cone

xQx iT

imax

xXxT0

xQxT2


definite-semi positivenot is :Note 1X

1C - cone


-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

xXxxIAQAx Ti

TT

i1max 1Cx

-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 definite Xi permits good approximation.

-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

1.5

2

2.5 XX ii

max

,max xXxIAQAx iTT

i


,,max, xCCxxAXxX TT

f

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.

The Cost is Continuous

such that

numberof cones

number ofgrid points

in

timehorizon


Define X(x,,T,N,M) := maxiN xTXi(,T,N,M)x

X(x,,T,N,M) maxf() X(Ax,,T-1,N,M) + xTCCxT

X(x,,0,N,M) = xTx.

X(x,,0,N,M) X(x,) as N,M,T Would like

The cone centered around first coordinate axis

fx

QCCKMNTxA

QQKKMNTX

N

T

jjj

nnRTQQ

,for

,,,1,,X:subject to

1logminarg,,,,

1

1,1,11

C

convex optimization:

X can be Found via Convex Optimization

C1 := {x : > 0, x = e1 + y, y1=0, |y|=1}

depends N, the number is cones

The cone centered around first coordinate axis

fx

QCCKMNTxA

QQKKMNTX

N

T

jjj

nnRTQQ

,for

,,,1,,X:subject to

1logminarg,,,,

1

1,1,11

C

convex optimization:

X can be Found via Convex Optimization

C1 := {x : > 0, x = e1 + y, y1=0, |y|=1}

depends N, the number is cones

21

,,,,,,,0 K

TxXKMNTxX

Theorem: X(x,,0,N,M,K) X(x,) as N,M,T,K

related to the continuity of X

In fact,

,, 2 xXxX

,, ** xuxu the optimal control is homogeneous

,,, *** yuxuyxu but not additive

only the direction is important

Optimal Control of LSVDV Systems

Summary

LPV

LPV with rate limited parameter variation

LSVDV

LDV

increasing knowledge

about f

increasing computational

complexity

increasing conservativeness

optimal in the limit

optimalmight not be that bad

Documents

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