A Deeper Look at LPV

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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)) 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.

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

centered at n

type n

failure

nominal

type n

failure

type 1

failure

nominal

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.

Cost

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

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  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.

Continuity of LDV Controllers

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 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

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.

LPV with Rate Limited Parameter Variation

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.

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.

LSVDV systems

type 1 failure

nominal

type 2 failure

1 - Step Cost

For example, let f()={1, 2}

alternative 1

alternative 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X1A + CC

T

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X2A + 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

conservative

Worst Case Cost

piece 2

piece 1

Piecewise Quadratic Approximation of the Cost

Define X(x,) := maxiN xTXi()x

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)  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

Optimal Control of LSVDV Systems

only the direction is important

the optimal control is homogeneous

Summary

LPV

increasing computational complexity

increasing conservativeness

LPV with rate limited parameter variation

optimal in the limit

LSVDV