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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 ,)}

type 1

failure

ball of radius

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 SRnn and ERmn 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.

Cost

For LTI systems, you get the exact 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

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 Rnn, Ei Rmnand {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

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?

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 Rmn

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

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 : Rnn 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 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

Continuity of the H Controller

Theorem: There exists a controller such that

if and only if there exists a bounded solution to

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

T

T

T

In this case, X is continuous.

X May Become Discontinuous as is Reduced

LPV with Rate Limited Parameter Variation

Suppose that | f()- | < and

where Si Rnn, Ei Rmnand {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.

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.

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 = AX1A + CC

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 = AX2A + CC

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

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 definite Xi permits good approximation.

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

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

2

1.5

The Cost is Continuous

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.

Piecewise Quadratic Approximation of the Cost

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

T

Define X(x,,T,N,M) := maxiNxTXi(,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

Optimal Control of LSVDV Systems

only the direction is important

the optimal control is homogeneous

but not additive

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

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