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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

A Deeper Look at LPV

Stephan Bohacek

USC

general form of linear parametrically varying lpv systems
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
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 arise4
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)

slide5
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 systems packard and becker asme winter meeting 1992
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.

slide7
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.
slide8
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

slide9
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 systems bohacek and jonckheere ieee trans ac
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
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 controllers12
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

slide14

HControl for LDV Systems

Bohacek and Jonckheere SIAM J. Cntrl & Opt.

Objective:

continuity of the h controller
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.

slide17

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.
slide18

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
LSVDV systems

type 1 failure

nominal

type 2 failure

1 step cost
1 - Step Cost

For example, let f()={1, 2}

alternative 1

alternative 2

slide21

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

slide22

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

slide23

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

slide24

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

slide25

Worst Case Cost

piece 2

piece 1

  • non-quadratic cost
  • piece-wise quadratic
piecewise quadratic approximation of the cost
Piecewise Quadratic Approximation of the Cost

Define X(x,) := maxiN xTXi()x

quadratic

piecewise quadratic approximation of the cost30
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 cost31
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
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 cost33
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
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 optimization35
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
Optimal Control of LSVDV Systems

only the direction is important

the optimal control is homogeneous

but not additive

summary
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