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# A Deeper Look at LPV - PowerPoint PPT Presentation

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)) linear parts nonlinear part x R n u R m  - compact

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### 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

xRn

u Rm

 - compact

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

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)

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 ,)}

type 1

failure

centered at n

type n

failure

nominal

type n

failure

type 1

failure

nominal

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

Find SRnn and ERmn such that

for all 

> 0

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.

For LTI systems, you get the exact cost.

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 = 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)x|2 + |D(k)F(j[0,k](A(j)+B(j)F))x|2

where X=S-1

depends on 

• If the LMI is not solvable, then

• the inequality is too conservative,

• or the system is unstabilizable.

LPV with Rate Limited Parameter VariationWu, Yang, Packard, Berker, Int. J. Robust and NL Cntrl, 1996Gahinet, Apkarian, Chilali, CDC 1994

Suppose that | f()-  | <  and

where Si Rnn, Ei Rmnand {bi} is a set of orthogonal functions such that |bi() - bi(+)| < .

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 |i|< 

> 0

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

then

is stable.

where X = (S)-1

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?

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

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 continuous functions.

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

F :   Z  Rm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.

X = AXA + CC - AXB(DD + BXB)-1BXA

T

T

T

T

T

T

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

T

T

T

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 .

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

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 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.

X May Become Discontinuous as  is Reduced

Suppose that | f()-  | <  and

where Si Rnn, Ei Rmnand {bi} is a set of orthogonal functions such that |bi() - bi(+)| < .

S = i{1,N}bi() Si

E = i{1,N}bi() Ei

for all  and |i|< 

> 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.

(LSVDV) Systems

Bohacek and Jonckheere, ACC 2000

set valued dynamical system

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

 is compact.

type 1 failure

nominal

type 2 failure

For example, let f()={1, 2}

alternative 1

alternative 2

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X1A + CC

T

T

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X2A + CC

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2

-1.5

-1

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2

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

piece 2

piece 1

Define X(x,) := maxiN xTXi()x

3

2

1

0

-1

-2

-3

-3

-2

-1

0

1

2

3

3

2

1

0

-1

-2

-3

-3

-2

-1

0

1

2

3

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

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

2

1.5

Theorem: If

1. the system is uniformly exponentially stable,

2. X : Rn    R solves

3. X(x, )  0,

thenX is uniformly continuous.

• Hence, X can be approximated:

• partition Rn into N cones, and

• grid  with M points.

X(x,,T,N,M)  maxf()X(Ax,,T-1,N,M) + xTCCx

T

Define X(x,,T,N,M) := maxiNxTXi(,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 

X can be Found via Convex Optimization

The cone centered around first coordinate axis

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

depends N, the number is cones

convex optimization:

X can be Found via Convex Optimization

The cone centered around first coordinate axis

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

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

only the direction is important

the optimal control is homogeneous

LPV

increasing computational complexity

increasing conservativeness

LPV with rate limited parameter variation

optimal in the limit

LSVDV